Computer-Aided Tuning and Design of Microwave Circuits Using Fuzzy Logic by Vahid Miraftab A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Electrical and Computer Engineering Waterloo, Ontario, Canada, 2005 ©Vahid Miraftab, 2005 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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Conformement a la loi canadienne sur la protection de la vie privee, quelques formulaires secondaires ont ete enleves de cette these. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. i*i Canada Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract This dissertation introduces, for the first time, the application o f Fuzzy Logic Systems (FLS) to the tuning and design o f microwave filters. Fuzzy logic deals with artificial intelligence (AI), and can bring human intelligence into problems in a systematic way. Several different algorithms are proposed based on Fuzzy Logic for tuning and design of microwave circuits. The methods take advantage o f the ability o f Fuzzy Logic systems to incorporate numerical information and linguistic (expert) information, as well as the ability o f Fuzzy Logic systems to act as function approximators. One of the most interesting applications o f FLS is extracting human expert knowledge in the process o f post-production tuning o f microwave filters, and thus making an automated filter tuning system based on expert rules. Tuning o f complex microwave structures have been always a challenge in the microwave engineering area. Fuzzy Logic Systems (FLS) and controllers are found as great means to circumvent this challenge. Promising results for tuning 4-pole chebyshev and 8-pole elliptic filters are obtained for both slightly detuned and highly detuned cases. More robust fuzzy systems based on numerical data are developed using Sugeno type fuzzy logic systems and subtractive clustering to work for both cases o f slightly detuned and highly detuned filters at the same time. The FLS techniques are also successfully applied to the design o f microwave circuits including couplers, 3-pole and 6pole filters. The approach could be considered a fast and efficient synthesis method. A method to show the ability o f capturing human expertise in the form o f fuzzy logic controllers is proposed and applied in tuning o f 4-pole Chebyshev filters. Moreover, a unique method for microwave filter tuning based on linguistic rules extracted from an expert is developed. The method employs two types o f fuzzy controllers, one based on phase response for coarse tuning and the other based on the magnitude response for fine-tuning. This method is shown experimentally for tuning 7-pole and 3-pole Chebyshev iris-coupled waveguide filters by designing a totally automated hardware/software interface. The interface includes high resolution smart motors, custom designed flexible couplings to transfer the motion from iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the motor shafts to the tuning screws, flexible mounting plates to mount and position the motors, and programming for communication between VNA, PC and motors using GPIB card and serial port. The theoretical and experimental results given in this thesis verify the validity o f the proposed concepts. The thesis has been a pioneer in applying Fuzzy Logic techniques to the tuning and design o f microwave circuits. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I would like to take this opportunity to thank all the individuals who directly or indirectly contributed for this accomplishment. On academic level, I would like to thank Prof. R. R. Mansour for his great supervision and support during my Ph.D. program. His support was not only in technical level but also in personal aspects. I would also like to extend my gratitude to my defence committee members, professor Q. J. Zhang, Dr. Ming Yu, professor Safieddin Safavi-Naeini, professor Fakhri Karray, and professor Sujeet Chaudhuri for accepting to be on the committee and for reading my thesis. I am also thankful for their valuable suggestions and comments for the future work. I am specially thankful to professor Q. J. Zhang for traveling from Ottawa to Waterloo to attend my defence. I am also very grateful to Bill Jo lle y , CIRFE lab manager, Brian Keats my colleague, Michael Haber from Heli-cal products, Ed Spike, engineering lab assistant, and Minh Tran from Aniamtics company for their technical support and consultation for the experimental tuning setup realization. I am deeply thankful to Wendy Boles and her staff in the graduate office o f the Electrical and Computer Engineering department for their great patience and support throughout the course my study. This research was supported in part by NSERC and COM DEV, Cambridge, Ontario, Canada. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my beloved wife: Narges and My Parents For their continuous support Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents Abstract..................................................................................................................................................iii Acknowledgements................................................................................................................................. v Table of Contents..................................................................................................................................vii List of Figures......................................................................................................................................... x List of Tables........................................................................................................................................ xv Chapter 1 Introduction............................................................................................................................ 1 1.1 Motivation............................................................................................................................... 1 1.2 Thesis Organization................................................................................................................4 Chapter 2 Literature Review................................................................................................................... 5 2.1 Existing Techniques for Tuning..............................................................................................5 2.1.1 Time Domain Techniques................................................................................................6 2.1.2 Frequency Domain Techniques.......................................................................................8 2.2 Existing Techniques for Design.......................................................................................... 11 2.3 Fuzzy Logic Systems............................................................................................................ 18 2.3.1 Fuzzy Sets...................................................................................................................... 19 2.3.2 Building the Fuzzy Logic System..................................................................................22 2.3.3 Rules..............................................................................................................................24 2.3.4 Fuzzy Inference Engine.................................................................................................25 2.3.5 Fuzzification..................................................................................................................26 2.3.6 Defuzzification...............................................................................................................28 2.3.7 Fuzzy Basis Functions and FLS Formulation............................................................... 29 Chapter 3 Computer-Aided Tuning of Microwave Filters Using Fuzzy Logic....................................33 3.1 Filter Tuning Using the Wang & Mendel Fuzzy Method.....................................................33 3.1.1 Introduction....................................................................................................................33 3.1.2 The Filter Tuning Problem.............................................................................................34 3.1.3 Generating Fuzzy Rules from Numerical Data..............................................................36 3.1.4 Setting up the Tuning Problems.....................................................................................38 3.1.5 Assigning the Membership Functions........................................................................... 41 3.1.6 Calculation of Output Parameters..................................................................................43 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.7 Tuning Results for the Slightly De-tuned 4-Pole Chebyshev Filter.............................. 43 3.1.8 Tuning Results for the Highly De-tuned 4-Pole Chebyshev Filter............................... 45 3.1.9 Tuning Results for the Slightly De-tuned 8-Pole Elliptic Filter.................................... 47 3.1.10 Tuning Results for the Highly De-tuned 8-Pole Elliptic Filter......................................50 3.2 Filter Tuning Using the Sugeno Method................................................................................54 3.2.1 Introduction.................................................................................................................... 54 3.2.2 Definition of the Problem..............................................................................................55 3.2.3 The Fuzzy Logic System................................................................................................56 3.2.4 Rule Identification Based on Fuzzy Subtractive Clustering..........................................57 3.2.5 Identification of the Fuzzy Logic System for the 8-pole Elliptic Filter Problem 3.2.6 Tuning Results for Highly De-tuned and Slightly De-tuned 8-pole Elliptic Filters .... 63 3.2.7 A 4-pole Chebyshev Filter Example with Detuned Resonators.....................................67 60 Chapter 4 Computer-aided Design of Microwave CircuitsUsing FuzzyLogic....................................70 4.1 Introduction............................................................................................................................ 70 4.2 Microstrip Coupler Design Problem......................................................................................72 4.2.1 4.3 Design Results for the Coupled-line Coupler................................................................73 Microstrip 3-pole filter design...............................................................................................76 4.3.1 Identification of the Fuzzy Logic System for the Filter Design Problem......................77 4.3.2 Design Results for the 3-pole Microstrip Filters............................................................79 4.4 Microstrip 6-pole filter design............................................................................................... 81 4.4.1 The Design Procedure Using Synthesis for the 6-pole Microstrip Filter.......................83 4.4.2 Design Results for the 6-pole Microstrip Filter Using the Fuzzy Logic System 4.5 85 Design of a 3-pole HTS Filter................................................................................................ 88 Chapter 5 Tuning of Microwave Filters by Extracting HumanExperience Using Fuzzy Logic 90 5.1 Introduction............................................................................................................................ 90 5.2 The Tuning Procedure............................................................................................................91 5.3 Implementing the Tuning Procedure......................................................................................93 5.4 Results.................................................................................................................................... 95 5.5 Results by considering Tuning both self couplings and mutualcouplings using the fuzzy logic approach................................................................................................................................... 98 Chapter 6 A Novel Filter Tuning Method Using Multi-Level Fuzzy Controllers Based on Linguistic Rules................................................................................................................................................... 100 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1 Introduction.......................................................................................................................... 100 6.2 A General Resonator Tuning Procedure Based on Phase/Group Delay Response 102 6.2.1 Proof of Concept Using Coupling Matrix Formulation...............................................103 6.2.2 Error Calculation..........................................................................................................109 6.2.3 Mathematical Formulation for tuning the second resonator........................................110 6.2.4 Error Calculation..........................................................................................................112 6.2.5 Higher order verification by an eight pole elliptic filter example..............................113 6.2.6 The Experimental Setup for an Experimental 7-pole Chebyshev Waveguide Filter .119 6.3 Transmission Line Effect..................................................................................................... 123 6.4 Two-step Tuning Method Using Fuzzy Controllers....................................................... 124 6.4.1 First Step Based on Linguistic Rules......................................................................... 124 6.4.2 Designing the fuzzy controller.....................................................................................125 6.4.3 The performance of the fuzzy controllers....................................................................128 6.4.4 Second Step Based on Linguistic Rules....................................................................... 128 6.4.5 Designing the fuzzy controller................................................................................... 130 6.4.6 Results for 7-pole filter.............................................................................................. 134 6.4.7 Results for 3-pole filter................................................................................................ 139 Chapter 7 Concluding Remarks....................................................................................................... 142 7.1 Summary and Contributions................................................................................................ 142 7.2 Future Work......................................................................................................................... 144 Bibliography......................................................................................................................................145 ix Reproduced with permission of the copyright owner. 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List of Figures Figure 2-1: Tuning Steps........................................................................................................................ 6 Figure 2-2: Design Steps....................................................................................................................... 13 Figure 2-3: Fuzzy Logic System (FLS)................................................................................................ 18 Figure 2-4 Functions for X(Age)={Very young, Young, Middle age, Old, Very old}.The shape of the membership functions as well as their degree of overlap is quite arbitrary..................................20 Figure 2-5: Fuzzy inference engine as a system...................................................................................25 Figure 3-1: A generalized model for coupled resonator filters.............................................................35 Figure 3-2: A sample practical filter response to show the concept.....................................................37 Figure 3-3: Two examples of slightly de-tuned and highly de-tuned 4-pole Chebyshev filter characteristics................................................................................................................................ 39 Figure 3-4: The ideal 8-pole elliptic filter characteristic......................................................................39 Figure 3-5: Two examples of slightly de-tuned and highly de-tuned 8-pole elliptic filtercharacteristics ...................................................................................................................................................... 40 Figure 3-6: General shape of input membership functions for 4-pole Chebyshev filter example....... 41 Figure 3-7: Output membership functions for yi corresponding to the 4-pole Chebyshev filter......... 42 Figure 3-8: General shape of input membership functions for 8-pole elliptic filter example.............. 42 Figure 3-9: Output membership functions for yi corresponding to the 8-pole elliptic filter................ 42 Figure 3-10: A Comparison between Experimental and extracted performance using fuzzy logic for the slightly de-tuned filter............................................................................................................. 44 Figure 3-11: A Comparison between Experimental and extracted performance using fuzzy logic for the highly de-tuned filter a) S2i, b) Sn.......................................................................................... 46 Figure 3-12: Comparison between Experimental and extracted performance of the 8-pole filter using fuzzy logic with 9 inputs for the slightly de-tuned filter a) S2j, b) Sn..........................................51 Figure 3-13: A Comparison between Experimental and extracted performance of the 8-pole filter using fuzzy logic with 17 inputs for the slightly de-tuned filter a) S21 , b) Sn.............................. 52 Figure 3-14: A Comparison between Experimental and extracted performance of the 8-pole filter using fuzzy logic with 17 inputs for the highly de-tuned filter.....................................................53 Figure 3-15: Typical Gaussian membership functions for inputs.........................................................56 Figure 3-16: Error variation for training and checking data.................................................................61 Figure 3-17: Comparison between the desired and extracted m/2 for a set of training data pairs........ 62 x Reproduced with permission of the copyright owner. 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Figure 3-18: Comparison between the desired and extracted m/2 for a set of checking data pairs...... 63 Figure 3-19: Comparison between the desired and extracted S21 response for the slightly de-tuned filter example................................................................................................................................ 65 Figure 3-20: Comparison between the desired and extracted Sn response for the slightly de-tuned filter example................................................................................................................................ 65 Figure 3-21: Comparison between the desired and extracted S2i response for the highly detuned filter example......................................................................................................................................... 66 Figure 3-22: Comparison between the desired and extracted Sn response for the highly de-tuned filter example......................................................................................................................................... 67 Figure 3-23: The ideal 4-pole filter response........................................................................................68 Figure 3-24: The mistuned experimental response...............................................................................68 Figure 3-25: S2i Comparison between the desired response and extracted response employing the fuzzy logic approach.....................................................................................................................69 Figure 3-26: Sn Comparison between the desired response and extracted response employing fuzzy logic approach............................................................................................................................... 69 Figure 4-1: The coupled microstrip lines structure...............................................................................72 Figure 4-2: Error variation for training and checkingdata for the coupler design problem................74 Figure 4-3: Comparison between fuzzy logic and synthesis results for 10-dB coupling.............. 75 Figure 4-4: Comparison between fuzzy logic and synthesis results for 15-dB coupling..............76 Figure 4-5: The 3-pole Chebyshev filter structure................................................................................76 Figure 4-6: Filter design responses with different bandwidths.............................................................77 Figure 4-7: Error variation for training and checkingdata.................................................................. 78 Figure 4-8: Comparison between the performances obtained from synthesis and the fuzzy logic system for BW=0.6%.................................................................................................................... 80 Figure 4-9: Comparison between the performances obtained from synthesis and the fuzzy logic system for BW=0.8%.................................................................................................................... 80 Figure 4-10: Comparison between the performances obtained from synthesis and the fuzzy logic system for BW= 1.13%.................................................................................................................. 81 Figure 4-11: The 6-pole Chebyshev filter structure..............................................................................81 Figure 4-12: Filter design response with the bandwidth of 2% ............................................................82 Figure 4-13: The capacitive coupling section of the filter....................................................................83 Figure 4-14: The Y-parameter representation of the capacitive coupling section................................ 84 xi Reproduced with permission of the copyright owner. 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Figure 4-15: The resultant resonator structure......................................................................................84 Figure 4-16: Error variation for training and checking data for the 6-pole filter problem....................86 Figure 4-17: Comparison between the Sn responses obtained from synthesis and the fuzzy logic system for the 6-pole filter (BW=2%).......................................................................................... 87 Figure 4-18: Comparison between the S2i responses obtained from synthesis and the fuzzy logic system for the 6-pole filter (BW=2%).......................................................................................... 87 Figure 4-19: The layout of the 3-pole HTS filter....................................... 88 Figure 4-20: Picture of the experimental 3-pole HTS filter..................................................................89 Figure 4-21: Measured results of the HTS filter...................................................................................89 Figure 5-1: Human expert (technologist) who tunes a microwave filter...............................................92 Figure 5-2: Replacing the human expert with a Fuzzy Logic Controller..............................................92 Figure 5-3: The control system composed of several simple FL controllers........................................ 93 Figure 5-4: The general decision making Fuzzy Sets with 5 membership functions............................94 Figure 5-5: Gaussian membership functions for FLS inputs................................................................ 94 Figure 5-6: The flow diagram for detailed steps of the tuning procedure using FL controller approach. ...................................................................................................................................................... 95 Figure 5-7: The simple case of decision-making Fuzzy Sets with 2 membership functions................96 Figure 5-8: The initial detuned response vs. ideal response: a) Sn b) S2i ............................................97 Figure 5-9: The final response after using FLS tuning algorithm vs. ideal response: a) mag(Su) b) phase(Sn) c) mag(S2i) d) phase(S2!) .............................................................................................98 Figure 5-10: The graphical user interface with a) Detuned response, b) Resonators approximately tuned, and c) After all parameters are tuned.................................................................................99 Figure 6-1: Frequency response of the ideal 3-pole filter example..................................................... 104 Figure 6-2: Frequency response of the 3-pole filter when all resonators are highly detuned 106 Figure 6-3: Frequency response of the 3-pole filter when only the first resonator is tuned...............108 Figure 6-4: Frequency response of the 3-pole filter with tuned first and second resonators 112 Figure 6-5: Frequency response of the ideal 8-pole filter example.....................................................114 Figure 6-6: The 8-pole filter example response with all resonators highly detuned...........................115 Figure 6-7: The 8-pole filter example with the first resonator tuned.................................................. 116 Figure 6-8: The 8-pole filter example with the first two resonators tuned......................................... 116 Figure 6-9: The 8-pole filter example with the first three resonators tuned.......................................117 Figure 6-10: The 8-pole filter example with the first four resonators tuned....................................... 117 xii Reproduced with permission of the copyright owner. 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Figure 6-11: The 8-pole filter example with the first five resonators tuned................................118 Figure 6-12: The 8-pole filter example with the first six resonators tuned..................................118 Figure 6-13: The 8-pole filter example with the first seven resonators tuned.............................119 Figure 6-14: The automated filter tuning setup.................................................................................. 120 Figure 6-15: A 7-pole Chebyshev waveguide filter with tuning screws............................................. 121 Figure 6-16: The graphical user interface for computer-aided tuning................................................ 122 Figure 6-17: Tuning steps using multi-level FL controllers based on linguistic rules........................122 Figure 6-18: Transmission lines at input and output ports.................................................................. 123 Figure 6-19: The control block diagram for tuning based on the first type FLS................................124 Figure 6-20: Phase at center frequency for the res. 4 (even) and 5 (odd) vs. the screw positions 125 Figure 6-21: Input fuzzy sets: (a) Odd number resonators, (b) Even number resonators...................126 Figure 6-22: Fuzzy sets for the second input variable........................................................................ 126 Figure 6-23: Output fuzzy sets............................................................................................................ 126 Figure 6-24: An illustration to show the regions where each rule is dominant..................................127 Figure 6-25: Time response for tuning the resonators........................................................................ 129 Figure 6-26: The control block diagram for tuning based on the second type FLS............................129 Figure 6-27: Return loss objective function with respect to screw adjustments.................................130 Figure 6-28: Fuzzy sets for the first input........................................................................................... 132 Figure 6-29: Fuzzy sets for the second input...................................................................................... 132 Figure 6-30: Fuzzy sets for the third input.......................................................................................... 132 Figure 6-31: Fuzzy sets for the output................................................................................................ 132 Figure 6-32: An illustration to show the regions where each rule is dominant..................................133 Figure 6-33: Original case when all the resonators are highly detuned.............................................. 134 Figure 6-34: Coarse tuning results when the first 5 resonators are tuned........................................... 135 Figure 6-35: Coarse tuning results when all resonators are tuned except the last one........................135 Figure 6-36: Filter response when the coarse tuning is complete for all resonators...........................135 Figure 6-37: Group delay response when: (a) all the resonators are mistuned; (b) resonator 1 is tuned; (c) 1 and 2 are tuned; (d) 1-3 are tuned; (e) 1-4 are tuned; (f) 1-5 are tuned; (g) 1-6 are tuned; (h) 1-7 are tuned............................................................................................................................... 136 Figure 6-38: Filter response when the fine tuning is complete with a 25 dB return loss target 137 Figure 6-39: Filter response after 1 iteration of fine-tuning............................................................... 137 Figure 6-40: Filter response after 2 iterations of fine-tuning (meets the spec, for RL of 20 dB ) xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 Figure 6-41: Filter response after 5 iterations of fine-tuning (meets thespec, for RL of 25 dB ) 138 Figure 6-42: The 3-pole waveguide filter setup.................................................................................. 139 Figure 6-43: The response of filter without tuning............................................................................. 140 Figure 6-44: Response after coarse tuning using first type fuzzy controllers.....................................140 Figure 6-45: 3-pole Filter response after 1 fine-tuning iteration......................................................... 141 Figure 6-46: 3-pole Filter response after 2 fine-tuning iterations....................................................... 141 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 3-1: M,deai of the 8-pole elliptic filterexample.............................................................................49 Table 3-2: Mexampiei of the 8-pole elliptic filter example (slightly de-tuned)....................................... 49 Table 3-3: Mexlracled of the 8-pole filter example (slightly de-tuned, 9 inputs).................................... 49 Table 3-4: Mextracted of the 8-pole elliptic filter example (slightly de-tuned,17 inputs)........................ 50 Table 3-5: Mexampiei of the 8-pole elliptic filter example (highly detuned)...........................................53 Table 3-6: Mextracted of the 8-pole elliptic filter example (highly de-tuned)..........................................53 Table 3-7: Desired coupling elements for the slightly de-tuned filter example....................................64 Table 3-8: Extracted coupling elements for the slightly de-tuned filter example.................................64 Table 3-9: Desired coupling elements for the highly de-tuned filter example......................................66 Table 3-10: Extracted coupling elements for the highly de-tuned filter example.................................66 Table 3-11: The coupling elements for an ideal 4-pole Chebyshev filter.............................................67 Table 4-1: A comparison between the dimensions extracted by synthesis and our fuzzy model for the coupled-line coupler...................................................................................................................... 75 Table 4-2: Physical dimensions extracted using filter synthesis...........................................................79 Table 4-3: Physical dimensions extracted using the optimized fuzzy logic system............................ 79 Table 4-4: A comparison between the dimensions extracted by synthesis and our fuzzy model for the 6-pole filter.................................................................................................................................... 86 Table 6-1: The coupling elements for the ideal 8-pole elliptic filter..................................................115 Table 6-2: Detuned diagonal elements................. 115 Table 6-3: The linguistic fuzzy rules to find the proper phase. (Example: Rule 3- /FPhase(fO) is NEG and Delta is PB, THEN Am is PS ).............................................................................................. 127 Table 6-4: The linguistic fuzzy rules to find the current best return loss in pass-band. (Example: Rule 6- 7F(ARL(t-l) is POS) and (ARL(t) isNEG)and (Am(t) is POS), THEN Am(t+1) is PS) xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 133 Chapter 1 Introduction 1.1 Motivation Computer-aided tuning and design are very essential in the fabrication o f complex microwave filters. Tuning is almost necessary for any manufactured microwave circuit due to lack o f highly accurate design models, manufacturing tolerances and design uncertainties. Computer-aided tuning helps to speed up the tuning process and can be incorporated to improve the design model. Currently, most o f the post-production tuning o f complex structures is done by human experts who have in-depth experience with tuning o f those specific circuits and yet still sometimes takes a few days to tune one unit. Therefore, an automated tuning procedure, which includes expert heuristics, would make the process much faster and cost-efficient. The importance o f the design o f microwave circuits is also obvious in engineering, where computer-aided design also helps us avoid the unnecessary experimental work. For many o f the real-world engineering problems two different forms o f knowledge about the problem exist: 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1) Objective knowledge, which could be in the form o f mathematical model, based on either a theoretical or experimental model. 2) Subjective knowledge, which represents linguistic information that is not usually possible to be quantified using traditional mathematics, such as rules, expert information and design requirements. Examples o f objective knowledge are: equations o f motion o f a mass; Maxwell equations to predict the electromagnetic behavior o f a system; and convolutional model that describes a communication channel. Examples o f subjective knowledge are: the following rule for driving a car— If the speed is too low, then push the gas pedal moderately; and the following rule that might be used in tuning o f circuit— If the reflection coefficient o f the filter at center frequency is low, then turn the coupling screws slightly to reduce coupling. Both types o f knowledge can be utilized to solve engineering problems. The two forms o f knowledge can be coordinated in a logical way using fuzzy logic. Similarly, for most real-world microwave tuning/design problems, the information regarding design, evaluation, realization, etc., can be classified into two types: numerical information obtained from mathematical models or measurements, and linguistic information obtained from human experts. Most current intelligent control approaches combine the standard processing methods using the numerical data with expert systems. Fuzzy logic theory allows us to incorporate the expert information into the tuning/design problem. Most o f the tuning/design techniques in microwave circuits are based on implementing a mathematical model that is capable o f interpreting the measured data. The Fuzzy Logic approaches also allow a mathematical model to be used in generating the fuzzy rules, which in turn are used to interpret the measured data. Fuzzy Logic systems however have the additional flexibility o f allowing the integration o f mathematical models with information obtained from human experts in terms o f linguistic rules. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fuzzy set theory (FST) was first introduced by Zadeh [37]. In classical logic, sets are defined in a crisp manner, i.e., an element either belongs to a set or does not belong to it. In fuzzy logic (FL), a membership value between “0” and “ 1” is assigned to each element o f the set. “0” means the element does not belong to the set at all, whereas “ 1” means the element totally belongs to that set. Fuzzy logic interprets the numerical data as linguistic rules. The extracted rules will then be used as a kind o f system specification to calculate the output values o f the system. The procedure o f creating fuzzy sets from numerical data is called “fuzzification,” and the process o f calculating the output values from the output fuzzy sets based on some linguistic rules is called “defuzzification.” More details about these procedures are described in Chapter 2. Different from conventional approaches, fuzzy logic systems (FLS) [45], use if-then rules that could be generated from a mathematical model, measured data, expert information or any combination o f the three. A number o f rules will be fired with various strengths corresponding to the match between the inputs and the antecedents o f the fuzzy rules. The invoked fuzzy rule actions are combined by a defuzzification mechanism to generate a final output. The antecedents o f the fuzzy rules decompose the range o f the input into a number o f fuzzy regions; then the consequences approximate the system in each region via a simple model. This makes the fuzzy models capable o f aggregating the local actions fired from rules to globally describe the circuit performance. The term “Fuzzy” may look very misleading in the sense that it implies inaccuracy. But, in fact, it indicates that even though human does not calculate everything numerically to drive a car or tune a microwave filter, he/she manages to meet the targets (for example: stopping the car at an intersection, or tuning a microwave filter to meet the design specs) accurately. In this research, we show how the intelligent techniques in fuzzy logic can be applied to microwave tuning/design problems. 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 Thesis Organization Chapter 1 outlines the research motivation and organization o f the thesis. Chapter 2 includes a literature survey on different design and tuning techniques for microwave circuits as well as an introduction to fuzzy logic systems. Chapter 3 applies two different types of fuzzy logic systems based on objective information for diagnosis and tuning o f 4-pole and 8pole microwave filters using theoretical filter models. Chapter 4 shows the feasibility o f fuzzy logic techniques as robust tools in the design o f microwave circuits. The method is applied to couplers, 3-pole filters and 6-pole filters. Chapter 5 shows a fuzzy logic control algorithm based on extracting human expert data. The technique is demonstrated by considering the tuning o f 4-pole filters. Chapter 6 introduces a unique method for microwave filter tuning based on group delay/phase and magnitude response o f the filters. It also applies unique automation methods to 3-pole and 7-pole Chebyshev waveguide filters based on linguistic rules extracted from an expert implemented in fuzzy controllers. Finally, chapter 7 outlines thesis conclusions and future research. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Literature Review 2.1 Existing Techniques for Tuning There have been many different methods proposed for tuning. They can be categorized into two main categories, time domain techniques and frequency domain techniques. In both methods, the effort is to evaluate the circuit parameters. The tuning procedure for any circuit can be generalized in the following steps: 1) Measure the performance o f the circuit under tuning. 2) Extract the circuit parameters [ 1]. 3) Compare the extracted parameters [3] with the required ones to determine discrepancy. 4) Change the parameters considering the discrepancy obtained from the previous step. 5) Repeat the procedure from step 1 to step 4, for fine-tuning Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2-1 shows the flow diagram for these tuning steps. The main issue in tuning is the second step; extraction o f the circuit parameters. Therefore, the difference among different tuning techniques mostly arises from this step. Start M easure the perform ance of the circuit under tuning ye» End Measured Required Response = Response Change the parameters according to the discrepancy obtained from the previous step no Extract the circuit parameters and subtract from the required parameters Figure 2-1: Tuning Steps 2.1.1 Time Domain Techniques Filter tuning using time domain is described by Dunsmore [10]-[12], The method shows an approach for tuning filters based on the time domain response o f the return loss, where the time domain response is obtained by a special type o f discrete inverse Fourier transform o f 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the frequency response. The method demonstrates that the time domain response can distinguish the individual responses o f each resonator as well as the sequential couplings. Dunsmore has demonstrated a time domain tuning approach by considering 5-pole and 8pole chebyshev filters. The time response is obtained by a special type o f discrete inverse Fourier transform o f the frequency response. The idea o f the time domain technique is that the return loss response in time domain will contain some dips exactly corresponding to each resonator in the filter. Moreover, the peaks in the same response correspond to the coupling between resonators. Using these information, one can find the element causing the de-tuning by looking at the time domain response. For tuning process, a real filter target response is first established by a “golden” filter standard that is tuned by an experienced technologist. The reported tuning time for the 8-pole filter with 7 adjustable couplings was about 6 minutes. In [12], Dunsmore has considered more complex filters with cross-coupled resonators in which he uses both time domain response and frequency response in order to do the tuning. The tuning procedure o f course becomes more complicated since the time domain response no longer has the simple relationship to filter tuning. This is because in this case each node frequency is no longer at the same frequency as the center frequency is. Moreover, changing the cross-coupling element will also change the resonant frequencies o f the resonators associated the cross-coupling. The tuning procedure could be summarized as follows: 1) Set up a template filter called “Golden Filter” using either an already tuned filter or simulation response and display the time domain response in the VNA display. 2) Approximately tune all the resonators to the bandwidth using the frequency response. 3) Using the null positions o f the template time domain response, tune all the resonators. 4) Starting from input and output ports o f the filter tune the couplings to the center o f the filter. Note that the peak values account for the coupling values. 5) At each coupling adjustment, the adjacent tuning screws may require little adjustment. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. One of the features o f time domain method is that we can create a new filter shape without knowing the coupling elements o f the filter. The method, however, does not show any feasibility to be generalized to any microwave circuit, since it only describes the elements o f coupled-resonator filters. The tuning method still needs a human operator. Furthermore, the method works well if there exists a properly tuned filter, and there is a difficult challenge to create the filter template in the first place. 2.1.2 Frequency Domain Techniques The traditional approach to the alignment o f a microwave device in frequency domain is the empirical adjustment o f a set o f variable elements in order to bring the network response within some predefined specification. Specific tuning strategies have been developed to aid in the alignment o f microwave filters [l]-[4]. These procedures, however, are prone to be very time consuming, and usually require experienced and skilled operators. Computer aided tuning (CAT), therefore, plays an important role in the time and cost-effective production of microwave circuits, and is the only way to automate circuit tuning. Some computer aided tuning techniques were developed to extend the previous methods to an automatic technique, or to make the tuning procedure easy and fast for an inexpert operator [5]-[9]. Atia and Yao [13], have presented a method for the determination o f the individual resonant frequencies and coupling coefficients o f a system consisting o f cascaded coupled resonators. The method applies a recursive algorithm to polynomials to extract the resonant frequencies and coupling factors. In their approach, they consider the cases where the design involves different resonant frequencies. However, they neglect the effect o f cross-couplings. Coupling elements and resonant frequencies are derived from the frequencies o f zeros and poles o f the cascaded resonator input impedance with short circuit at one port. As an application, they apply their method to an 8-pole quasi-elliptic function filter. The accuracy o f the method highly depends on the accuracy o f the extracted zeros and poles as well as the 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. number o f resonators involved. Since there is not a unique model for microwave circuits in general and this method also uses the coupled resonators model, the method is limited to filter tuning problems. M. Kahrizi et al. [14], have suggested a similar approach based on polynomial parameter extraction and coupling element extraction. For both steps, they use an optimization procedure. They have considered an 8-pole Chebyshev filter with two transmission zeros as an example. They also add a small perturbation in the form o f noise to the coupling elements. The major disadvantage o f the approach is that they have considered a small perturbation from the ideal design, whereas in practical cases we may encounter a highly detuned filter characteristic. Therefore, considering the ideal design parameters, as the initial points in the optimization process may not lead to convergence. Harscher and Vahldieck [15] have presented an automatic approach for filter tuning. They have considered the effect o f tuning screws in their analysis. Then according to a sensitivity analysis o f the effect o f each screw on Sn, S22 and S2 1 , a tuning algorithm is built. For the parameter extraction part o f the tuning, they use gradient optimization with a mean-square error function. They consider a 3-pole Chebyshev filter to show their method. The tuning time was reported 5 minutes. The method however still needs a little user interaction, and may be much more time-consuming for higher order filters considering more data storage time and longer optimization process. Another limitation to this method is that the method needs a basic position for the filter, which must be within the tuning range. In other words, the initial filter response must resemble a band pass filter. This means that perhaps an initial manual tuning is needed at the first time. This is also necessary for such optimization procedures to converge. Another tuning method using a machine learning approach was suggested by A. R. Mirzai et al. [9]. The method is based on machine learning system (MLS) theory in which an adaptive algorithm is used in such a way that it can assist an unskilled operator to perform accurate and fast tuning o f the filters. The overall system can be thought o f as an equivalent 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to an expert system, where there is a “training mode” and a “use mode”. In the training mode, the expertise o f a skilled operator is represented and stored in such a way that it can be accessed later in the use mode. In MLS approach, pattern recognition and adaptive signal processing is employed to obtain and represent the data. They use two methods called “Distance Classifiers” and “Adaptive Combiners” to tune the filter. The tuning methods for this approach could be summarized as: 1) All the screws are removed. 2) The tuning screws are inserted on at a time to bring the resonant frequency within the bandwidth. 3) At this step the tuning screws are adjusted to minimize the return loss response o f the filter as much as possible. 4) Finally, the coupling screws are inserted. At this stage the operator needs to adjust all the screws in such a way that Sn meets the customer specifications. The author indicated at the conclusion o f the paper that they reduced the number o f adjustable parameters to the tuning screws, assuming that the coupling screws are correctly adjusted and are left untouched. They also mention that work is in progress to investigate the effects o f the coupling screws and how to adapt the MLS for the first stage o f tuning process i.e. step 3. So basically, the paper only uses Adaptive Combiners for the step 4, and that assuming the coupling screws are already adjusted. H. Hsu et al. [16] propose a method based on measuring or computing the refection coefficient phase for tuning microwave filters. The method is effective for tuning and diagnosis o f coupled resonator filters. They consider the coupled resonator model with two ports consisting o f n cascaded coupled resonators. The procedure looks at the phase o f input impedance when we have shorted the output port. Shorting the output port is not an accurate approach since the reference plane position is unknown. So, more manual tuning is needed. G. Pepe et al. [17] proposes a diagnosis approach for filters that can be described with coupled resonator model. The authors use optimization technique but yet in a systematic way 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to avoid divergence o f the optimization. The approach is based on a sequential tuning procedure. They consider different coupling matrices and define sub-filters. Several other tuning techniques for different types o f electrical and electronic circuits could be found in [ ] - [], with their own merits and shortcomings. 2.2 Existing Techniques for Design The design procedure for any circuit can be generalized in the following steps: 1) Specify the desired circuit performance (Design Goal). 2) Choose a hardware configuration that can give the desired performance. 3) Choose an appropriate model for your circuit (Modeling). 4) Choose an optimization procedure to find the desired parameters in the model. 5) Find the physical dimensions by having the model parameters. Figure 2-2 shows the flow diagram for these design steps. Most o f the effort in different suggested design problems deals with the step 3 and 4. So any type o f modeling or optimization technique can be considered as a design technique. In microwave circuit design, we usually deal with two types o f models: 1) Fine models, which are very accurate models based on Maxwell equations. These kinds o f models, however, are very computationally expensive; 2) Coarse models, which are less accurate, but very fast. Examples o f these kinds o f models are: circuit level models and coarse numerical electromagnetic (EM) models. We can assume three types o f design methods based on the above-mentioned models: 1) Circuit level (coarse) simulation + Optimization 2) EM (fine) simulation + Optimization 3) Circuit simulation + EM simulation + Advanced techniques + Optimization 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first and second approaches are not usually practical ones, because o f inaccuracy and intensive computation respectively. The third approach tries to combine the advantages o f the previous two methods to obtain the desired accuracy along with less computation. Many o f the recent design techniques are based on the third approach. Here we review some o f the more important techniques in microwave design appeared in literature: Roan and Zaki [24] have described an iterative method for the design o f high order microwave low-pass elliptic filters. The essence o f their method is to build an iterative algorithm to adjust the filter element values to match the desired filter characteristics o f the filter such as zeros, poles and scale factor. They start with finding the desired insertion loss characteristic o f the filter in their method. Then, they replace all the microstrip sections with closed form models. They also take into account the discontinuities in their model. Cascading all sections, and using chain matrix representation, they come up with the modeled transfer function o f the filter. Using the function, the algorithm then performs the Jacobian matrix with respect to variable network parameters, and applies the Newton method to give the necessary correction to the variable vector. In this approach, the design optimization goal is set to match the transmission zeros and poles to the desired ones. To verify their design procedure, the authors have considered a seven-element low-pass elliptic function filter with a 0.17 dB passband ripple, 40 dB cutoff band attenuation, with cutoff frequency o f 4 GHz. The calculated and measured responses are in good agreement. The method works well for low-pass filters. However, since it uses the cascaded model to develop the filter response, it does not take into account the effects o f stray couplings. Stray coupling may not have much effect for low pass structures, while it is very crucial for band pass filters. Another disadvantage for this method appears when there is not an accurate closed form equation model for distributed elements o f the filter. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Start Specify the design goal, hardware configuration and an appropriate (fast and accurate) model for your circuit Choose an optimization procedure to find the desired parameters in the model Find the physical dimensions by having the model parameters End Figure 2-2: Design Steps Peik et al. [25] have extended the one-dimensional Cauchy approach to multidimensional approach with respect to both frequency and physical dimensions. The Cauchy interpolations model shows very close agreement with a complete EM simulation. Multi-dimensional Cauchy method enables us to extrapolate the performance o f the circuit with respect to frequency and dimensions using far fewer sampling points. The method also allows application o f adaptive sampling, which reduces the required number o f samples even further. The authors have proposed two different approaches to reach to their formulation. The first approach is a recursive algorithm, which solves the multidimensional interpolation using 1-D Cauchy interpolation. The second approach is to implement the interpolation by a single multidimensional rational polynomial. The authors then apply their method to three 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. different examples. The method is very useful to be used for design optimization, since it is much less computationally expensive. As an example, if the optimization were done using full EM simulation, it would require several days, while the proposed method just takes a few minutes. The disadvantage o f the method is that it is only useful for microwave circuits where the system function is close to a rational function polynomial. If the system function has a more nonlinear type o f performance e.g. exponential function, then the interpolation may not work properly. Moreover, Cauchy method fails when the system is very noisy. A very general and useful optimization technique for microwave circuits called Space Mapping (SM) technique was introduced by Bandler et al. [26]. This technique has been further advanced since then, and is also used for response interpolation [27]. The method tries to reduce the effort o f optimization using electromagnetic (EM) models, which are very accurate but very computationally intensive. The method suggests considering two different model types defined as coarse model and fine model. The coarse model is a circuit level model or an inaccurate EM model. The simulation and optimization in coarse model is very fast but is not accurate enough. The fine model is an accurate EM-based model, which is very slow. The method tries to decrease the optimization effort dramatically by finding an approximate mapping between coarse and fine model parameters, and finding the design parameters by an iterative method. The method can be summarized by the following steps: 1) Do the optimization using the coarse model, and find the resulting input vector. 2) Use the input vector from the previous step as fine model input vector, and perform fine model simulation. 3) Do the coarse model optimization using the fine response obtained from step 2, and find a coarse model input vector. 4) Find a mapping function from input vector at fine model input space obtained from step 1 to input vector at coarse m odel input space obtained from step 3. 5) Find the inverse o f the function from the previous step, and calculate another fine model input vector from the optimized coarse model input obtained from step 1. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6) Perform fine model simulation for the resulting input vector, and check if the response is within the specifications. 7) If the response is within the specifications, stop. Otherwise, go back to step 3. The method, however, does not converge in some conditions where the coarse model does not have the elements to describe a specific behavior o f the circuit such as stray couplings. In other words, there is no guarantee to find an optimization solution in steps 1 and 3. Shen and Mansour [28] later proposed an alternative technique, which is more efficient and easier to implement. In this technique, which is a parameter extraction method, the circuit is split into individual components. Then a full wave electromagnetic (EM) simulation for each part is done. The individual components then will be cascaded. The model up to this step is missing the electromagnetic interactions between nonadjacent elements. Therefore, this method introduces some additional circuit blocks to provide paths between the nonadjacent blocks. The next effort in this method would be the proper modeling o f the additional parts. Using this idea, the authors build a design algorithm using EM simulation and circuit optimization. During the iterative algorithm, the adjusted model will be changed since we have a local model at each iteration. The authors apply their method to two examples o f three-pole and four-pole Chebyshev microstrip filters. The results show perfect agreement between the modified cascaded circuit and EM simulations. Finally, the design EM response is very close to that o f measurement for the four-pole filter. One o f the key points in this technique is having a good understanding o f the interactions among the design parts. The method is limited, since for more complicated microwave circuits, it is very difficult to find a good physical interpretation to apply the modification. Peik and Mansour [29] proposed another design algorithm based on coupling matrix alteration for microwave planar filters to overcome the problem associated with space mapping method. In their method, they adjust the coarse model during the design process. To include certain electromagnetic phenomena such as stray couplings, they suggest a hybrid optimization based on coupling matrix concept. In their coarse model, they also apply multidimensional Cauchy method for each section. Then they apply the algorithm to a 6-pole 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C-band filter design problem to demonstrate the technique. The design steps can be summarized as follows: 1) Do the optimization using the coarse model, and find the resulting input vector (neglect the cross couplings), and extract the coarse coupling elements ( M coarse ). 2) Use the input vector from the previous step as fine model input vector, and perform fine model (EM) simulation. 3) Using the EM response, extract the coupling matrix ( M fme). 4) Calculate: AM = M fme - M coa„e. 5) Adjust the coarse model as: M adjmed = M COfl„e + A M . 6) Find the desired coupling matrix M jdeal . 7) Perform the optimization for the adjusted model such that M adjmled - M ideal , and find the input design vector. 8) Using the input obtained from the previous step do the fme model simulation. 9) If the response in within the desired specifications, stop. Otherwise, go back to step 3. The approach works well for passive planar filter designs where coupling matrix models can be efficiently used. However, for other types o f microwave circuits where there is not such an accurate model, this method cannot be used anymore. Ureel et al. [30] have introduced adaptive frequency sampling o f scattering parameters obtained by electromagnetic simulation. The method tries to employ Cauchy method with optimized selection o f the frequency points where there is more information about the system, and thus minimize the computation effort. The procedure is that we define a specific error function, and find the frequency at which there is maximum error. Then, the next sampling point will be selected that frequency point. The authors apply the method to a ring resonator. From the examples they show, the computation time can be reduced by a factor o f 3 to 30 using this method. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Another method for modeling which is used in design o f microwave circuits is neural network models. In recent years several CAD approaches based on neural network models have been introduced for microwave impedance matching, modeling, simulation and optimization [31]-[33]. Neural models can be much faster than original detailed EM/physics models, more accurate than polynomial and empirical models, allow more dimensions than table lookup models and are easier to develop when a new device/technology is introduced. The above-mentioned neural network models are extracted from training data. The main disadvantage o f these methods is that usually a large amount o f training data is needed to ensure model accuracy. Generating large amount o f training data could be very expensive for microwave problems because the original detailed simulation/measurement has to be performed for many combinations o f different values o f geometrical/material/process parameters in the EM or device physics problems. Wang and Zhang [34], proposed a new microwave-oriented knowledge based neural network (KBNN) in which microwave knowledge in the form o f empirical functions or analytical approximations are incorporated into neural networks. They take advantage o f the fact that in microwave modeling areas many o f the important knowledge are in the form o f functions. By adding prior knowledge into neural networks, the model can be improved and simplified. The advantage o f the method is that it enhances the neural model accuracy specially for unseen data, and reduces the need for a large set o f training data. However, when the circuit becomes more complicated, even this method will need a large amount o f training data. In the space mapping method, there is a parameter extraction step, which can be done using different methods (optimization methods). Another recently proposed approach for the parameter extraction part is using neural network modeling methods. This new method named Neural Space Mapping (NSM) was proposed by Bakr [35]. The proposed NSM optimization algorithm tries to efficiently approximate mapping from the fme to the coarse input space. Also, an enhanced model for non-linear microwave devices based on the NeuroSM method is presented by Zhang et al. [36]. Their method aims to increase the accuracy o f the non-linear device models while maintaining the computation speed. These methods, 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. however need a large amount o f training data and cannot include linguistic rules in their models. 2.3 Fuzzy Logic Systems A fuzzy logic system (FLS), in general is a nonlinear mapping o f an input data vector into a scalar output. If we have a vector output, we can decompose it into a collection o f independent multi-input/single-output systems. The richness o f FL is that there are lots of possibilities that lead to many different mappings. A fuzzy logic system can also be described as a function approximator. Figure 2-3 shows a fuzzy logic system. RULES Crisp Outputs Crisp Inputs DEFUZZIFIER FUZZIFIER Fuzzy input sets FUZZY INFERENCE SYSTEM Fuzzy output sets Figure 2-3: Fuzzy Logic System (FLS) As is seen in the figure, FLS maps crisp input into crisp outputs. The system has four components: fuzzifier, rules, inference engine, and defuzzifier. Once the rules are established, the FLS can be viewed as a function y=f(x). Rules can be expert rules, or rules extracted from numerical data. In either case, engineering rules are expressed as a collection o f IF-THEN statements. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The fuzzifier maps crisp input numbers into fuzzy sets. This lets the rules to act on the input data, since they are in terms o f linguistic variables that have fuzzy sets associated with them. The inference engine maps fuzzy sets into fuzzy sets. It is responsible to combine the rules in a specific way. There are many different types o f fuzzy logic inferential procedures. However, small number o f them are usually being used in engineering applications. This is similar to human way o f making decisions, where different types o f inferential procedures are used to understand things. The defuzzifier maps fuzzy output sets into crisp output numbers. This step is necessary since we need to obtain a crisp number in most o f the engineering applications. 2.3.1 Fuzzy Sets The first paper on FL was written by Lotfi A. Zadeh [37], who is considered to be the founding father o f the entire field o f FL. He introduces fuzzy set theory as a formal way to represent uncertainty mathematically. Recall that a crisp set A in a universe o f discourse U can be defined as A={x | x meets some condition}. This tells us that if x meets the specific condition, then it belongs to set A; otherwise it does not belong to set A. Alternatively, we can introduce a zero-one membership function fiA(x) to describe the membership o f x to A. If x e A , then juA(x)=l\ and if x g A , then M a (x )—0. A fuzzy set F in a universe o f discourse U is characterized by a membership function jU f( x ) which can take values in the interval [0,1]. A fuzzy set is a generalization o f a crisp set whose membership function only can accept zero or one. A membership function in this case measures the degree o f similarity o f an element in U to a fuzzy subset. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A fuzzy set F in U can be represented as: F={(x, p F(x)) | xeU }. When U is continuous, F is usually written as F = ^ / u F( x ) l x . In this equation, the integral sign does not mean integration; it denotes a collection o f all points x associated with their related membership functions ju f(x ). When x is discrete, F is written as F = ^ u juF( x ) / x . In this equation, the summation sign does not mean summation; it denotes the collection o f all points x associated with their membership functions jurfx). As an example, consider Figure 2-4, which shows different membership functions for the variable “Age”. We can interpret “Age” as a linguistic variable. This linguistic variable can be decomposed into the following set o f terms: X(Age)={Very young, Young, Middle, Old, Very old}, where each term in X(Age) is characterized by a fuzzy set in the universe of discourse U={0, 80}. For example as can be seen from Figure 2-4, we can interpret Young as an age close to 20. Another important point to mention is that a person with the age 25 belongs to both fuzzy sets “Young” and “Middle”, but with different membership functions o f 0.7 and 0.3 respectively. This example demonstrates that in fuzzy logic an element can reside in more than one set to different degrees o f similarity. Very Young Young Middle age Old Very old 1 0.7 0.3 0 20 25 x(Age) Figure 2-4 Functions for X(Age)={Very young, Young, Middle age, Old, Very old}. The shape of the membership functions as well as their degree of overlap is quite arbitrary. The most commonly used shapes for membership functions are triangular, trapezoidal, piecewise linear, and Gaussian. The membership functions can be chosen based on user’s experience. The membership functions can be designed using an optimization procedure, but they are usually chosen based on user’s experience. The number o f membership functions is 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. up to the designer. By choosing more membership functions we can achieve greater resolution at the price o f more computational complexity. After defining fuzzy sets, we can now perform set operations (analogous operations for crisp sets). The main set operators are union, intersection and complement. In order to define these operations for fuzzy sets, we start applying mathematical formulas for crisp operations. Then we can find fuzzy set operation formulas by getting motivated from these crisp definitions. These definitions can be different, as long as it satisfies the crisp set operator conditions. As an example, one o f the widely used definitions is given below: U B => /uA{jB(x) = max[ju A(x ), juB(*)] (2-1) A fl B => jUmB (x) = m in IX (x), jUB(*)] (2-2) ^ ( * ) = 1~ M a ( x ) (2-3) A Although these equations can be used for both fuzzy and crisp sets, we have to remember that for fuzzy sets can only be characterized by their membership functions, whereas crisp sets can be characterized either by membership functions or by a description o f their elements. The fuzzy operations A and B are fuzzy sets,which lead to membership function values in the interval [0,1]. It can be easily seen by an example that the laws o f Contradiction ( A U A = U ) and Excluded Middle ( A f ) A = 0 ) in crisp set theory are broken in fuzzy set theory. This is one o f the main differences between crisp set theory and fuzzy set theory. The “max” and “min” operators, which were used in equations (2-1) and (2-2), are not the only ones that can be used to model fuzzy union and intersection. Zadeh [37], in his pioneering paper, suggests some other operators for union and intersection. The union operator in fuzzy logic is called t-conorm or s-norm, denoted ©, and intersection operator is called t-norm, denoted *. The different t-norms, t-conorms and complements, available in fuzzy set theory, provide us with richness and lots o f choices to choose from. However, in 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. most o f the engineering applications o f fuzzy sets, min or algebraic product is used for tnorm, max operator for t-conorm and 2.3.2 (1 -jU a (x )) for fuzzy complement. Building the Fuzzy Logic System As can be seen from Figure 2-3, one o f the major components of our FLS is the Rules, which are in the form o f logical implications o f IF-THEN statements. As a simple case we can consider the rules in the form of: If w is ^4, THEN v is B, which represents a kind o f relation between sets A and B. This relation can be explained with a two-dimensional membership function juA^ B(x, y ). The question here is how to define this membership function. If we try to follow the crisp set theory to get to a formulation for /uA^ B(*, y ) , we will see that we will not get the proper ones [38]. The core reason for this is that in engineering, cause and effect is the basis o f modeling, whereas in traditional prepositional logic, this is not the case. In fuzzy reasoning as in traditional prepositional logic, an important inference rule is defined, which is called “Modus Ponens”. Modus Ponens in crisp logic has two premises and a consequence: Premise 1: “ u is A ” Premise 2: “ IF u is A THEN v is B ” Consequence: “ v is B ” In fuzzy logic, Modus Ponens is in the following form and is called “Generalized Modus Ponens” : Premise 1: “ u is .4*” Premise 2: “ IF u is A TH EN v is B ” Consequence: “ v is B* ” 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As can be seen by comparison between Modus Ponens and Generalized Modus Ponens, in the latter fuzzy set A* does not have to be the same as the fuzzy set A in the antecedent o f Premise 2. Also, fuzzy set B* does not have to be the same as the rule consequence fuzzy set B. In crisp logic a rule will be fired only if the first premise is exactly the same as the antecedent o f the rule, and the result o f that is the actual rule’s consequence. However, in fuzzy logic, a rule will be fired as long as there is a nonzero degree o f similarity between the first premise and the antecedent o f the rule. The result o f this kind o f rule firing will be a consequent that has a nonzero degree o f similarity to the rule’s consequent. The membership function o f Consequence in Generalized Modus Ponens can be formulated in the following form [38]: f i B. (y) = s u p ^ . (*) * / j A^ B(x, y)] (2_4) xeA where “sup” operator is called supremum and is usually a t-conorm, and is a t-norm. As was mentioned above, the definition o f juA^ B(x ,y ) plays an important role. The problem o f defining this membership function according to crisp logic theory in brief is that the resulting membership function for the output usually does not make engineering sense, since it will be nonzero over the whole universe o f discourse. Mamdani [48] is the first person to suggest a solution to this problem. He introduces the following definition for n A_+B( x , y ) : Later, Larsen [39], proposed the following product implication: A = (2_6) Their reason to choose these choices was mostly due to achieve more simplicity rather than cause and effect. However, by building a truth table it can be shown that these two inferences preserve cause and effect, i.e. juA_>B(x, y) will be fired when the antecedent and consequence 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. are both true. Minimum and Product inferences have been the most widely used inferences in the engineering applications o f fuzzy logic. 2.3.3 Rules The IF-THEN rules in a fuzzy rule base can be expressed as: R (l) : IF x x is F ' and x 2 is F[ a n d ... x p is F pl , THEN y is G l (2-7) There are two ways to extract rules from numerical data: Let the data specify the fuzzy sets for the variables in the antecedents and consequences o f the rules. Specify the fuzzy sets for the antecedents and consequences regardless o f the data positions, rather establishing domain intervals for all input and output variables. Suppose that we have n input-output data pairs as in equation (2-8): where p is the number o f input variables, and n is the number o f data pairs available. Using the following steps [45], we can generate fuzzy rules from a given data: Determine the membership values o f the elements For this purpose, we must find the membership values o f x& with respect to all o f the fuzzy sets defined for the corresponding x t variable. We have to do the same thing for the output y. Assign each variable to the fuzzy set with maximum degree, i.e. largest membership function value. Obtain one rule from each pair o f input-output data. Up to now, we have obtained n rules. However, it is highly probable that we will have som e conflicting rules. Conflicting rules are rules with the same antecedents but different consequents. To resolve this problem, we assign each rule a degree. This degree determines the firing strength o f the rule. Then we accept only the rule from a conflicting group that has maximum degree. We define the degree associated with each rule as: 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D ( R (J)) = ju{x\j) )//(x (2j) )...ju(x(pj) )ju(yU)) (2' 9) Note that for multi-input multi-output case, which leads to multiple-antecedent multiple consequent rules, we can always decompose them into a set o f multi-input single-output rules. Therefore, it is enough to study the multi-input single-output case. I Some fuzzy logic systems use rules where the consequent is given in terms o f a functional relation of inputs. This is known as Sugeno fuzzy rules: F (/) : IF x, is F / and x 2 is F2 a n d ... x p is Fp, THEN y = f ( x ],...,x p) (2-10) Sugeno models are suited for modeling nonlinear systems by interpolating multiple linear models, and are not well suited for intuitive rules [65]. 2.3.4 Fuzzy Inference Engine In fuzzy inference engine section o f a fuzzy logic system, fuzzy logic principles are used to combine fuzzy IF-THEN rules in order to map input fuzzy sets into an output fuzzy set. In other words, fuzzy inference engine is a system that maps fuzzy sets into fuzzy sets by means o f juA^ B(x, y ) . Figure 2-5 shows such a system. A* fist* ) W Inference Engine B* W M B- O') F ig u r e 2-5: F u zzy in feren ce en g in e as a system Note that in this case x is a vector containing all input variables, i.e. x = ('x\,X2 , ...,xp). So, for rule 1 we have the following membership function: 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. M R ( , ) ( x , y ) = M A ^ B ( x >y) = m f; O i ) * ••• * mfi (x p ) * juG>(y) (2 1 1) where * is assumed to be a product or min t-norm. The p-dimensional input to Raj belongs to the fuzzy set Ax whose membership function is: M a, ( * ) = ( * i ) * • • • * Mx„ ( x p ) (2 -1 2 ) where X k c U k (k = 1,.,.,p) are fuzzy sets describing the inputs. By applying each rule to these inputs and using the equation (2-4), we can find an output fuzzy set Bl with the following membership function: ( y ) = s u p ^ [/uAx (x) * Ma^ b ( x , y)] (2-13) This fuzzy set is due to exciting one rule. In order to get the final fuzzy set B, we need to combine these individual results. The rules are usually combined using a t-conorm, with good results for engineering applications. 2.3.5 Fuzzification The fuzzifier part in a FLS maps a crisp point x = (xi,X2 , ...,xp) into a fuzzy set A* in U. The most widely used fuzzifier is a fuzzy singleton. A fuzzy singleton is a fuzzy set with membership function that is 1 at a single point x and zero at other points. The reason for popularity o f singleton fuzzifier is that in this case equation (2-13) will be tremendously simplified. Since we have just one non-zero membership function for input, equation (2-13) will be simplified to: (2-14) MB' { y ) = BA^B( x ' , y ) 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Singleton fuzzification, however, may not always be enough, specially when the observed data is noisy, and therefore the observed data has some errors. Non-singleton fuzzification handles such uncertainties. A non-singleton fuzzifier is one for which ju , (V) = 1 and fi A. (x) decreases from unity when x gets away from x \ In fact in non-singleton fuzzification we have a “fuzzy number” . The examples o f such membership functions are Gaussian and triangular. Once we have a broader membership function, it means thatthere ismore uncertainty about the data. In the case o f non-singleton fuzzification, equation (2-14) is no longer valid. Substituting (2-11) and (2-12) into (2-13), we can find the membership function corresponding to the 1th rule: juB, (y ) = sup juXi (*, ) * . . .j u Xpi ( x p ) (2-15) xsU (y ) Since the supremum operator is only over x e U , then we can take out the /uB, {y) term and rewrite the equation (2-15) as: t*Bi i y ) = MgI 0 0 * sup f i Xi ( x ,) *... n Xpx (x p ) (2-16) xeU *ju Fi( x ]) * ...ju F, ( x p ) Since a t-norm is a two-place function from [0 ,l]x [0 ,l], we can consider each t-norm in equation (2-16) to act on a pair o f membership functions: Mb' ( y ) = jUgi (y ) *siip[^ (*,) *n (x,)] xeU x*u 1 (2-17) ( xp) * l “ Fi (*„)] By properties o f supremum, the supremum will be attained when each term in braces in (2-17) attains its supremum. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.3.6 Defuzzification Defuzzifier block in Figure 2-3 produces a crisp output from the fuzzy sets we have got from the output o f the inference block. In other words, it is system that transforms fuzzy sets into crisp values. The most commonly used defuzzifiers for engineering applications are the following five defuzzifiers: Maximum Defuzzifier: This defuzzifier chooses the output variable y in the output fuzzy set B where it has maximum membership function value. This defuzzifier fails to give an output when the membership function has more than one maximum. Mean o f Maxima Defuzzifier: This defuzzifier first finds the values o f y at which giB(y) is maximum. Then it calculates the mean o f these values as its output. Centroid Defuzzifier: This defuzzifier determines the center o f gravity (centroid) o f output fuzzy set B, and this value will be considered as the output o f the FLS. The output will be calculated as: y = ]yf*B(y )dy / (2-18) j/ 'a O O # where S is the support (the values at which the membership function is non-zero) o f jUB(y). Height Defuzzifier: This defuzzifier is one o f the most widely used forms o f defuzzification, and is formulated as follows: M y>, = /=i M I > * ' O'') (2-19) where y l is the center o f gravity o f the fuzzy set Bl associated with the activation o f rule R(l). M is the number o f rules. The advantage o f this defuzzifier is that if we have a specific form o f membership function for the output, such as symmetric triangular, Gaussian and symmetric trapezoidal, we know its center o f gravity without any need for recalculating that. 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Equation (2-19) makes use o f the entire shape o f each antecedent’s membership function, since this information is embodied in juB, ( y l ). But from the consequence it only uses the center o f the gravity y l . This value does not change whether the consequent membership function is narrow or broad. To fix this problem a modified height defuzzifier is defined. Modified Height Defuzzifier: This defuzzifier is defined as: (2-20) yh 1=1 where S 1 is a measure o f the spread o f the consequence membership function R(l). For triangular and trapezoidal membership functions this value could be the support o f the triangle or trapezoid, and for a Gaussian membership function it could be the standard deviation o f the Gaussian function. 2.3.7 Fuzzy Basis Functions and FLS Formulation In this section we try to find a mathematical formula in the form o f y=f(x) that represents our fuzzy logic system (FLS). In order to write such a formula, we have to make specific choices for fuzzifier, membership functions, composition, inference and defuzzifier. As an example [38] if we choose singleton fuzzification, max-product composition, product inference, and height defuzzification, leaving the choice o f membership functions open, we can show that y =f.M ) = M p M p 1=1 (=1 l= \ 1=1 (2-21) This can be obtained by starting from the equation (2-19) and substituting for ju , ( y ‘), where 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p 1=1 ' (2-22) p 1=1 ' , and in the above equation it is assumed that the membership functions are normalized, so that juG, ( y l ) = 1. If we change the product inference to minimum inference in the previous example, following the same procedure we obtain the function as: y = fs(* ) (2-23) mm Suppose that we use Gaussian membership functions, i.e. p F, (x,-) = exp{-[(x(. - x j ) / cr! ]2}, where i = 1,2,...,p and / = 1,2, ...,M (p is the dimension o f input vector, and M is the number o f rules). As another choice, if we choose non-singleton fuzzification, max-product composition, product inference, Gaussian membership functions for both input variables and fuzzy sets, and height defuzzification, we can obtain a system function o f the form [53]: j (2-24) y = fns(x)= Z ^ ' n ^ ^ m a x ) / where (2-25) and A MQ'k(xk) = Mxk(xk) p F,(xk) ( 2 - 26 ) 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparing (2-21) and (2-24), we can observe the strong similarity between the two function structures. The FLS functions in (2-21), (2-23) and (2-24) can also be written in the following form: M (2-27) where (j)l (x) is called a fuzzy basis function (FBF). For the functions o f equations (2-21) and (2-24) the FBF’s are: (2-28) (for singleton fuzzification) ^(X) = n i=l k Kmax) / i n / /=1 k=1 M qI k (2-29) (^A.max) (for non-singleton fuzzification). The fuzzy logic system in this form can be referred as a fuzzy basis function expansion. This kind o f representing a fuzzy logic system is very useful, because it can be placed in a more global perspective o f a function approximation. These representations are valid for the specific choices we have made in our FLS. For many other choices, formulas similar to equations (2-28) and (2-29) can be derived. A comparison o f fuzzy basis functions with some other basis functions have been extensively studied in [51]. They are more general than radial basis functions, generalized radial basis functions, and hyper-basis functions. For very special choices o f parameters o f fuzzy basis functions with singleton fuzzification, they have structural resemblance to generalized regression neural networks [38]. As can be seen from the equation (2-27), the summation is done over the number of rules. It shows the direct dependence o f a fuzzy logic system and its basis function on each 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rule. Rules can come from numerical data or from expert linguistic knowledge. There are two ways to combine the rules obtained from numerical data and rules obtained from linguistic information [38]: We can separate these two types o f rules and perform two different FLS functions. Then we can add these two outputs, i.e. y = f(x) = f^(x)+fi(x). If we have a higher degree o f belief in one set o f rules over the other, we can combine the two functions in the following way: y = f(x) = ccfN(x)+(l-a)fL(x) where 0 < a < 1. We can put all the rules together in one set o f rules and create the fuzzy basis functions. In this case, the denominator o f the FBF’s will be affected by both kinds o f rules at the same time, which provides strong coupling between the numerical information and linguistic information, and might be o f more interest in some applications. Fuzzy logic systems can be viewed as universal approximators, i.e. can approximate any real continuous nonlinear function to arbitrary degree o f accuracy. For many combinations o f the parameters o f a FLS, this has been proved. For example,for singleton fuzzification, product inference, product implication, Gaussian membership functions and height defuzzification, this has been proved by Wang and Mendel [52]. The design degrees o f freedom that control the accuracy o f a FLS are, number o f inputs, number o f rules, number o f fuzzy sets for each input variable, and membership function shapes o f fuzzy sets. If we consider p input variables, and divide each o f them into r overlapping regions, i.e. r fuzzy sets, then a complete fuzzy rule bank must contain r p rules. As the resolution parameter r increases, the size o f the fuzzy rule bank increases enormously, making the system very complex. This will lead to a tradeoff between resolution and complexity. However, in actual practice, we almost never need a complete fuzzy rule bank o f r p rules. This is because, in practical applications there all usually large regions o f the input space that are never seen during the actual operation o f a system; therefore, rules are not needed for such regions. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Computer-Aided Tuning of Microwave Filters Using Fuzzy Logic 3.1 Filter Tuning Using the Wang & Mendel Fuzzy Method 3.1.1 Introduction Over the past years, several papers[10]-[15] have been published on computer-aided tuning o f microwave filters employing different techniques. These techniques can be basically divided into two main categories: time domain techniques and frequency domain techniques. Filter tuning using time domain is described by Dunsmore [10]-[12], while different theoretical and computational frequency domain techniques were proposed in [13 ]-[l5]. All the above techniques are based on implementing a mathematical model that is capable o f interpreting the measured data. The Fuzzy Logic approach also allows a mathematical model 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to be used in generating the fuzzy rules, which in turn are used to interpret the measured data. The approach however has the additional flexibility o f allowing the integration o f the mathematical model with information obtained from human experts. In this section we present an application o f the algorithm based on Fuzzy Logic for tuning microwave filters [42]-[43]. The approach is demonstrated by considering two filters: a 4-pole Chebyshev filter and an 8-pole elliptic filter. Each filter is then de-tuned to perform two examples: one is slightly de-tuned and the other is highly de-tuned. 3.1.2 The Filter Tuning Problem Consider the generalized filter network shown in Figure 3-1. The filter performance can be described by a coupling matrix M whose elements are identified in Figure 3-1. The generalized matrix is shown in equation (3-1). To minimize the tuning effort, accurate determination o f individual resonant frequencies and coupling coefficients is essential. Tuning the filter by adjusting each parameter individually as proposed in [12]-[13] may not lead to a convergent solution in some filters, particularly in structures, where the resonant frequency o f the resonator is strongly dependent on the coupling values to the adjacent resonators. The fuzzy logic approach deals with the adjustment o f all filter parameters taking into consideration the dependency o f the parameters on each others. mu mn m 1n m 2X m 22 m 2n M = (3-1) m n1 m n2 m. The typical structure shown in Figure 3-1 consists o f n coupled loss-less resonators. M y: Mji denotes the frequency-independent coupling between resonator i and j. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3-1: A generalized model for coupled resonator filters Following the analysis in [44], we can get the scattering parameters in terms o f the coupling elements: *^2i — ^J\JR]R2 l-d (3-2) ]„i: (3-3) where, A = AI - jR + M (3-4) /o BW fo (3-5) f where I is the unity matrix, and R is a matrix with all elements zero, except Rn= Ri and Rnn~R2The importance o f the coupling elem ents in filter tuning is that the coupling elem ents are directly related to the position o f adjustable screws. Once one o f these elements deviate from the desired value, we can easily turn it back to the desired value by turning the corresponding screw. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In case o f a de-tuned filter, we have access to the S-parameters o f the de-tuned filter. The extraction o f the coupling elements with the knowledge o f scattering parameters will help us find the corresponding screws to be adjusted for tuning. In next sections we will show how by using this mathematical model, we can extract the coupling elements using the fuzzy logic approach. 3.1.3 Generating Fuzzy Rules from Numerical Data In this chapter, the fuzzy rules are generated using the method proposed by Wang and Mendel as described in chapter 2, since it allows combining both numerical and linguistic information into a common framework— a fuzzy rule base [45]. We consider the M-Matrix coupling coefficients as outputs, whereas the S-parameters o f the filter at different frequencies considered as inputs. Suppose we have p frequency sampling points i.e. p inputs and q unknown coupling coefficients as outputs (see Figure 3-2). We can either extract the input information from S21 or Sn. The inputs then will be in the form S(fi) ... S(fp), which can be written in the form xi, x 2 ... xp for simplicity. The outputs, which are the coupling coefficients, could also be written in the form yi, y 2 ... y q for simplicity. We can now alter each coupling coefficient around the ideal design depending on the degree o f mistuning, and generate a number o f input-output data pairs: ( x l ' \ x ^ . . . , x ^ - , y \ ]\ y ^ . . . , y ^ ) , (x ? \x ? \..,x f^ \y ? ...,y f\ (3-6) (x \n\ x [ n\ . . , x y ^ \ n\ y ? . . . , / ? ) . For each input and output, we should define membership functions. Using the membership functions, for each data pair we obtain a rule in the format: 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I F (x, is f s x]) a n d (x 2 is f s x2) a n d (x p is f s xp), T H E N (3-7) ( y ] is f s y]) . . . a n d ( y q is f s yq) where f s is a fuzzy set among the fuzzy sets o f each input/output variable. 11 -1 0 -40 -50 \ -60 -70 11.92 11.94 11.98 12,02 Frequency (GHz) 12.04 12.06 12.08 Figure 3-2: A sample practical filter response to show the concept Basically, we get n rules corresponding to n data pairs. However, in practice it is highly probable that there will be some conflicting rules, i.e., rules that have the same IF part but a different TH E N part. To resolve the conflict, we will choose the rule with maximum degree, i.e. most probable one, among the conflicting rules. In this way, not only the conflict problem is resolved, but also the number o f rules is greatly reduced. In order to find the rules, there is another step, which is to assign membership functions to any o f the input/output variables. The input membership functions are selected considering the difference between the ideal and experimental input values to get proper domain intervals for each input. Note that the variables are also allowed to lie outside their domain intervals. If a data pair fits in all o f the input intervals corresponding to inputs, then that data pair will take effect in output calculation. In other words, the corresponding rule to the data pair will be fired. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We should also choose the output domain intervals such that most probably the output values will lie in those intervals. The firing rules correspond to different data pairs that resemble the experimental performance o f the circuit i.e. S-parameters in this case. The defuzzification part o f the fuzzy logic system (FLS) will combine the fired rules information to calculate the output o f the system. In our fuzzy logic system (FLS) we need to choose singleton fuzzification, sum-product composition and product inference. 3.1.4 Setting up the Tuning Problems To illustrate the proposed fuzzy logic approach in this section, first we consider the tuning o f a 4-pole band-pass Chebyshev filter. The coupling matrix (M-matrix) is a symmetrical 4x4 matrix with all elements zero except mi 2, m 23 , m34. Figure 3-3 depicts S 21 versus frequency o f two de-tuned filters, one with a slight deviation and the other with a high deviation from the ideal filter performance. As a more complicated example we consider an 8-pole band-pass elliptic filter, which also has stray coupling elements. The coupling matrix for this example is a symmetrical 8x8 matrix with all elements zero except the ones shown in equation (3-8). Figure 3-4 shows S u and S22 o f the ideal design filter with the center frequency o f 12 GHz. 0 m \2 m \2 0 0 0 0 0 0 0 0 "*23 0 0 0 0 0 "*34 0 0 "*36 0 "*27 0 0 0 0 0 "*23 0 0 0 "*23 0 0 0 0 0 0 m 21 "*36 0 0 0 0 "*34 0 "*45 0 "*45 0 0 "*34 0 0 0 "*34 0 "*23 0 "*12 0 "*12 0 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 - - Example 1 Example 2 ■5 -10 ml 2=0.7....... m23=1.6 -15 m34.=0.a....... -20 -25 -30 -35 1.12 1.14 1.16 1.18 1.2 1.22 Freq(Hz) 1.24 1.26 1.28 1.3 x 10« Figure 3-3: Two examples of slightly de-tuned and highly de-tuned 4-pole Chebyshev filter characteristics S11 S21 -10 -20 -30 -40 -50 -60 -70 •j...... -80 _901---------- 1---------- 1---------- 1---------- 1---1.192 1.194 1.196 1.198 1.2 Freq(Hz) 1.202 1.204 1.206 1.208 x 1 0 1° Figure 3-4: The ideal 8-pole elliptic filter characteristic 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 i r -10 -20 J/ / / / \i ■i i i // .... ........ ..../.J...... -30 -40 / / -50 -60 ...... ,y -70 ' j -80 \ .\ . \\ ........... Example 1 \ ' A \ \ ’ /I ^ . i -90 _J__ _ _ _ _ _ _ _ _ _ 1_ Ti 92 1.194 1.196 _i______________l_ 1.198 1.2 Freq(Hz) 1.202 1.204 1.206 1.208 x 10 Figure 3-5: Two examples of slightly de-tuned and highly de-tuned 8-pole elliptic filter characteristics We also assume that the filter is symmetrical, thus: m (« +!-/),(n+l-y) (3-9) As a result, the nonzero variables in the M matrix are: ml2,m n ,m i4,m 45,m 26 and m21. Figure 3-5 depicts the frequency response o f the 8-pole filter in two cases o f slightly de-tuned and highly de-tuned. These two examples represent the experimental data o f two de-tuned filters each with two different deviations. In order to use the tuning procedure, we need to extract the M-matrix elements associated with the experimental results. Then, with the knowledge o f the ideal coupling matrix one can identify the elements that caused the de-tuning. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.5 Assigning the Membership Functions In assigning the membership functions or fuzzy sets, we use 4-5 input fuzzy sets, 5 output fuzzy sets, and triangular membership functions. For the 4-pole filter example, we choose 5 fuzzy sets for each input with the third fuzzy set centered on the measured value o f the input. The membership functions are in the shape depicted in Figure 3-6. m(x) 0.8 0.6 0.4 0.2 c-d c+d x(dB) Figure 3-6: General shape of input membership functions for 4-pole Chebyshev filter example In Figure 3-6, c is the measured value o f S 21 at a specific frequency. The domain interval, ba, is selected considering the difference between the measured and ideal S 21 . The value d is usually a small fraction o f the domain interval to let the measured input value belong to the three middle fuzzy sets at the same time. The values e and/ are chosen around the middle o f a,c and b,c respectively. The output membership functions are symmetrical triangular functions with centers at coupling element values by which the data pairs are generated. As an example, Figure 3-7 depicts the output membership function o f the coupling element yi i.e. For the 8-pole filter example, we choose 4 fuzzy sets for each input with the measured value o f the input at the middle o f two centers o f adjacent fuzzy sets as shown in Figure 3-8. In Figure 3-8, c is the measured value o f S21 at a specific frequency. The domain interval, ba, is also selected considering the difference between the measured and ideal S2 1 - The value d 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is usually a small fraction o f the domain interval to let the measured input value belong to the two middle fuzzy sets at the same time. m(y1) 0.8 0.6 0.4 0.2 0.75 1.25 y1 Figure 3-7: Output membership functions for yi corresponding to the 4-pole Chebyshev filter The output membership functions in this example are also chosen as symmetrical triangular functions each centered at the coupling element sampling points. As an example, Figure 3-9 shows the output membership functions of the coupling element yi i.e. mn. m(x) 0.8 0.6 0.4 0.2 c-d c+d b x(dB) Figure 3-8: General shape of input membership functions for 8-pole elliptic filter example m(y1) 0.8 0.6 0.4 0.2 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 Figure 3-9: Output membership functions foryv corresponding to the 8-pole elliptic filter 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.1.6 Calculation of Output Parameters To calculate any o f the outputs, we use the centroid defuzzification formula: K T dm Jy ! yt =—K (3-io) 7=1 m j = ntj (x, )m j (x2)... m,j (x9) (3-11) where y j denotes the center value o f the fuzzy set corresponding to rule j, and output y j . The x k values are the input values at which the output is desired. The term m j( x k)is the membership value of x k to the fuzzy set corresponding to the rule j, and input x k . K is the number o f rules. Note that as long as the output membership functions are symmetrical, the shapes of the individual membership functions are arbitrary, and y j is simply the center of each membership function. If we chose a non-symmetrical membership, then we would need to calculate the center o f gravity o f each membership function as y j . By choosing symmetrical membership functions, we will need less computation. 3.1.7 Tuning Results for the Slightly De-tuned 4-Pole Chebyshev Filter The ideal coupling matrix o f the filter is given in equation (3-12), while the coupling matrix o f the slightly de-tuned filter (example 1) is given in equation (3-13). The performance associated with this coupling matrix represents the experimental performance o f a slightly de-tuned filter. 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By defining all membership functions for inputs and outputs, extracting the rules from the generated data, and using the defuzzification formula, we extracted the coupling matrix of the slightly de-tuned filter. The fuzzy logic approach required 70 rules and only 9 frequency sampling points i.e. 9 inputs to perform the extraction. Example 1 Extracted m12=1.28 m2?=1.03 m34=1.118 mil 2=1.3 -10 / " m23=i:0 '/ mi34=1 -25 -30 1.12 1.14 1.16 1.18 1.22 1.24 1.26 1.28 x 10 10 Figure 3-10: A Comparison between Experimental and extracted performance using fuzzy logic for the slightly de-tuned filter. The extracted coupling matrix is given in equation (3-14), while Figure 3-10 shows the extracted performance calculated using equation (3-14). The extracted coupling matrix provides a response that is fairly close to the experimental filter response. 0 1.2 0 0 1.2 0 0.95 0 0 0.95 0 1.2 0 0 1.2 0 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.3 0 0 1.3 0 1.05 0 0 1.05 0 1 0 0 1 0 0 1.28 0 0 1.28 0 1.03 0 0 1.03 0 1.118 0 0 1.118 0 " 0 M M example\ extracted (3-13) (3-14) Note that for this example the inputs are selected to be the magnitude o f S 21 at different frequencies with 7 frequencies inside the pass-band and the other 2 outside the pass-band. 3.1.8 Tuning Results for the Highly De-tuned 4-Pole Chebyshev Filter The coupling matrix o f the highly de-tuned filter (example 2) is given in equation (3-15). We also used only 9 frequency points and 70 rules for this example. Equation (3-16) gives the extracted coupling matrix, while Figure 3-11 illustrates a comparison between the fuzzy logic extracted performance and the experimental performance for both S 21 and Sj/. A very good match between the two filter characteristics is achieved. By comparing the ideal matrix given in equation (3-15) and the extracted matrix given in equation (3-16) one can easily identify the coupling coefficients, which caused the de-tuning. 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — Example 2 Extracted -10 r/ -15 '■j m I -20 hn12=0.7 ' rrrl 2=0.75'' m23=1.645 m34=0.759 -25 -30 -35 1.12 1.14 1.16 1.18 1.22 1.24 1.26 1.28 Freq(Hz) 1.3 ..in x 10 (a) — Example 2 - Extracted -10 -15 TJ-20 -25 -30 -35 1.12 1.14 1.16 1.18 1.22 Freq(Hz) 1.24 1.26 1.28 1.3 x 101D (b) Figure 3-11: A Comparison between Experimental and extracted performance using fuzzy logic for the highly de-tuned filter a) S2i, b) Su. 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For this example we used the same frequency sampling points as the slightly de-tuned example. We also used example 2 M extracted 3.1.9 as inputs. 0.7 0 0 0.7 0 1.6 0 0 1.6 0 0.8 _0 0 0.8 0 0 0.75 0 0 0.75 0 1.645 0 0 1.645 0 0.759 0 0 0.759 0 ' 0 M S21 (3-15) (3-16) Tuning Results for the Slightly De-tuned 8-Pole Elliptic Filter The ideal coupling matrix o f the filter is given in Table 3-1, while the coupling matrix o f the slightly de-tuned filter (example 1) is given in Table 3-2. The performance associated with this coupling matrix represents the experimental performance o f a slightly de-tuned filter. By defining all membership functions for inputs and outputs, extracting the rules from the generated data, and using the defuzzification formula, we extracted the coupling matrix o f the slightly de-tuned filter. For each o f the 6 unknowns i.e. coupling elements in equation (3-8), we chose 5 sampling points within the range o f ±10% of the ideal coupling elements. We assume that the coupling elements to be extracted are within this range. Therefore, the number o f data pairs in this case is 56=15625. We first consider 9 frequency points i.e. 9 inputs. To obtain a better selectivity in the case o f slightly de-tuned filter we choose the magnitude o f S n at frequency points in the pass-band as inputs. After omitting the contradicting rules, the number o f rules reduces to 1016 rules. For the inputs we have chosen, 24 o f these rules are fired to extract the outputs. 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The extracted coupling matrix is given in Table 3-3, while Figure 3-12 shows the extracted performance calculated using Table 3-3. The extracted coupling matrix provides a close response to the experimental filter response o f S2i as shown in Figure 3 -la, but not as much close response to the experimental response o f S u as shown in Figure 3 -12b. To perform a better response, we increase the number o f frequency points to 17. Adding the number o f frequency points or inputs can benefit us in two ways: 1) The possibility o f rule contradiction will decrease, and thus we get more rules out of the basic rules extracted from the data pairs. More rules could give us a more accurate system. Note that by adding the number o f inputs, the number o f rules cannot exceed the number o f sampling points. 2) For a rule to be fired, we need all the conditions o f equation (3-7) at the antecedent to be satisfied. Therefore, more inputs lead to more conditions to be satisfied. This makes the probability for a rule to be fired decrease. In other words, the rules will be more selective. For this case after omitting the contradicting rules, we get 2235 rules, which is about two times more than the number o f rules for 9 inputs. For the same input, in this case 49 rules are fired to calculate the outputs. The extracted coupling matrix for 17 inputs is given in Table 3-4, while Figure 3-1 shows the extracted performance calculated using Table 3-1. The extracted coupling matrix provides very close responses to the experimental responses o f S 21 and S u as shown in Figure 3-13a and Figure 3-13b respectively. A comparison between Figure 3-12 and Figure 3-13 demonstrates the effect o f increasing the number o f inputs as it shows a better match for both S2i and S u characteristics. 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3-1: M ideai of the 8-pole elliptic filter example 0 .8231 0 0 0 0 0 0 .8231 0 .5917 0 0 0 -.0251 0 0 .5917 0 .5516 0 .0781 0 0 0 0 .5516 0 .4925 0 0 0 0 0 0 .4925 0 .5516 0 0 0 0 .0781 0 .5516 0 .5917 0 0 -.0251 0 0 0 .5917 0 .8231 0 0 0 0 0 0 .8231 0 Table 3-2: 3 /^ ,,^ / of the 8-pole elliptic filter example (slightly de-tuned) 0 .8000 0 0 0 0 0 0 .8000 0 .6000 0 0 0 -.0263 0 0 .6000 0 .6000 0 .0720 0 0 0 0 .6000 0 .5000 0 0 0 0 0 0 .5000 0 .6000 0 0 0 0 .0720 0 .6000 0 .6000 0 0 0 -.0263 0 0 .6000 0 .8000 0 0 0 0 0 0 .8000 0 Table 3-3: M extracted of the 8-pole filter example (slightly de-tuned, 9 inputs) 0 .8004 0 0 0 0 0 0 .8004 0 .5762 0 0 0 - .0263 0 0 .5762 0 .5559 0 .0765 0 0 0 0 .5559 0 .4945 0 0 0 0 0 0 .4945 0 .5559 0 0 0 0 .0765 0 .5559 0 .5762 0 0 - .0263 0 0 0 .5762 0 .8004 0 0 0 0 0 0 .8004 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3-4: M extracted of the 8-pole elliptic filter example (slightly de-tuned, 17 inputs) 0 .8026 0 0 0 0 0 0 .8026 0 .6013 0 0 0 -.0254 0 0 .6013 .5781 0 .0744 0 0 0 0 0 .5781 0 .4886 0 0 0 0 0 0 .4886 0 .5781 0 0 0 0 .0744 0 .5781 0 .6013 0 -.0254 0 0 0 .6013 0 0 .8026 0 0 0 0 0 0 .8026 0 3.1.10 Tuning Results for the Highly De-tuned 8'Pole Elliptic Filter The coupling matrix of the highly de-tuned filter (example 2) is given in Table 3-5. We used 17 frequency points as well. The number o f rules after resolving the contradictions becomes 3066. For the inputs we have chosen, 10 rules are fired to extract the outputs. Table 3-6 gives the extracted coupling matrix, while Figure 3-14 illustrates a comparison between the fuzzy logic extracted performance and the experimental performance for S2i- A very good match between the two filter characteristics is achieved. Comparing the number o f firing rules in the case o f highly de-tuned with the case o f slightly de-tuned filter, we can observe that although we have more rules in this example, the number o f firing rules are less. The reason for this is that in the case o f highly detuned filter, we need bigger domain intervals for inputs and outputs, while keeping the number o f membership functions the same. This will result to having less data pairs that resemble the experimental performance o f the filter or less firing rules. By comparing the ideal matrix given in Table 3-1 and the extracted matrix given in Table 3-6 one can easily identify the coupling coefficients, which caused the de-tuning. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Example 1 Extracted (9 inputs) -10 -20 -30 -40 m TJ -50 ?4 W -60 -70 -80 -90 Freq(Hz) x 10 10 (a) — Example 1 Extracted (9 Inputs) -10 -20 .-30 m 36=0.10765 m27=-Q.02626 -50 -60 -7 0 1— 1.192 1.194 1.196 1.202 1.198 1.204 1.206 Freq(Hz) 1.208 x 10 10 (b) Figure 3-12: Comparison between Experimental and extracted performance of the 8-pole filter using fuzzy logic with 9 inputs for the slightly de-tuned filter a) S2i, b) Sn. 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 — Example 1 Extracted (17 inputs) -10 -20 -30 m34i0.6 m45i0.5 ^ -40 -m 36=kt.072 m274-0.026 -50 -60 -70 -80 -90 -100 1.19 1.192 1.194 1.196 1.198 1.2 1.202 Freq(Hz) 1.204 1.206 1.208 1.21 (a) Example 1 Extracted (17 inputs) -10 -20 .-30 CO- 4 0 -50 -60 -70'— 1.192 1.194 1.196 1.198 1.202 Freq(Hz) 1.204 1.206 1.208 10 x 10 (b) Figure 3-13: A Comparison between Experimental and extracted performance of the 8-pole filter using fuzzy logic with 17 inputs for the slightly de-tuned filter a) S21, b) Su. 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3-5: M exampki of the 8-pole elliptic filter example (highly detuned) 0 .5000 0 0 0 0 0 0 5000 0 .7000 0 0 0 .1000 0 0 -.1000 0 0 .1000 0 0 0 0 1.0000 .7000 0 0 1.0000 0 0 0 1.0000 0 0 1.0000 0 .7000 0 0 0 0 0 -.1000 .1000 0 0 .1000 0 0 0 .7000 0 0 0 0 0 0 0 0 .5000 .5000 0 Table 3-6: M extracle(i of the 8-pole elliptic filter example (highly de-tuned) 0 .4722 0 0 0 0 0 0 4722 0 .6324 0 0 0 .0972 0 0 .6324 0 0 0 0 0 -.1222 0 0 0 .9280 0 0 .9280 0 .1222 0 0 0 .1222 0 .9280 0 0 0 0 -.1222 0 .6324 0 0 .9280 0 .6324 .4722 .0972 0 0 0 0 0 0 0 0 0 .4722 0 — Example 2 Extracted -10 -20 -30 -50 -60 -70 1.21 Freq(Hz) x 10 10 Figure 3-14: A Comparison between Experimental and extracted performance of the 8-pole filter using fuzzy logic with 17 inputs for the highly de-tuned filter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 3.2.1 Filter Tuning Using the Sugeno Method Introduction This section introduces an improved algorithm based on Fuzzy Logic for tuning microwave filters [46]. The approach is demonstrated by considering slightly-detuned and highlydetuned 8-pole elliptic function filters with tuned resonators and a 4-pole Chebyshev filter with mistuned resonators. The parameters o f the fuzzy system are methodically adjusted to provide an optimized system. Unlike the method in previous section, only one fuzzy logic system is adequate to deal with both cases o f slightly de-tuned and highly de-tuned filters. The achieved results demonstrate the validity o f the proposed approach in identifying the filter elements that cause the de-tuning. In particular, the approach is very useful in dealing with highly de-tuned filters where the majority o f conventional parameter extraction or optimization techniques fail to converge to the correct solution. Different from conventional approaches, fuzzy logic systems (FLS) [47], use if-then rules that could be generated from a mathematical model, measured data, expert information or any combination o f the three. A number o f rules will be fired with various strengths corresponding to the match between the inputs and the antecedents o f the fuzzy rules. The invoked fuzzy rule actions are combined by a defuzzification mechanism to generate a final output. The antecedents o f the fuzzy rules decompose the range o f the input into a number o f fuzzy regions; then the consequences approximate the system in each region via a simple model. This makes the fuzzy models capable o f aggregating the local actions fired from rules to globally describe the filter performance. In addition to the if-then rules, the main other elements o f the FLS include membership functions and fuzzy inference system (FIS). The choices o f member-ship functions along with the rules are key components in optimizing the performance o f the fuzzy logic system. In the previous section [42]-[43], we used an approach based on a Mamdani-type FIS [48], 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where the if-then rules have linguistic antecedents and linguistic consequences. The antecedent variables were the scattering parameters at the sampling frequencies, whereas the consequent variables were the filter coupling elements. The input/output fuzzy sets were also set to triangular membership functions. In order to reduce the number o f data pairs required to build the fuzzy system, the approach reported in [42]-[43] necessitated the use o f two different fuzzy logic systems to deal with the highly de-tuned and slightly de-tuned problems. In this section, we employ a robust fuzzy logic system based on Sugeno-type FIS [49], with Gaussian member-ship functions. In the new proposed approach, one fuzzy logic system is used to predict the highly de-tuned and slightly de-tuned performances at the same time. The Sugeno rules contain linguistic antecedents but the consequent is a function o f input variables usually characterized by a linear function. Therefore each rule’s consequent has coupling elements as a linear function o f sampled scattering parameters. We also use an efficient and automated method based on subtractive clustering [50] to minimize the number o f rules and to determine the centers o f the membership functions. The standard deviation (a) o f the membership functions is adjusted to obtain an optimized fuzzy logic system. The proposed approach lends itself to adaptive techniques, since its parameters can be easily modified to accommodate the data-pairs available for the filter-tuning problem at hand. 3.2.2 Definition of the Problem We consider the generalized filter network based on coupling element representation explained in section 3.1.2. The scattering value at each frequency is considered as an input among the inputs o f our fuzzy logic system. Once we alter the coupling elements o f the filter, these scattering values at sampled frequencies also change. The outputs o f our fuzzy logic system are set to be the coupling elements o f the filter. Using these facts, we obtain a record o f input-output data pairs for building our fuzzy logic system (see section 3.1.3). In next sections we explain how we build our FLS using the generated data pairs. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2.3 The Fuzzy Logic System We built our new FLS based on Sugeno fuzzy inference system [49]. The consequent o f rules are no longer fuzzy sets but mathematical functions. The most commonly used type o f these systems contain rules with linear functions: IF X, is A, & X 2 is A2. . . & X . is A. &...THEN J Yl isBl &Y2isB2...&Yj isBj ... (3-17) 1 where Xj is the j th input variable and Yj is the j th output variable, and Bj is in the form: B j = a 0 +fl,x, + a 2x 2 +••■ (3-18) The input fuzzy sets, Aj, are characterized by Gaussian membership functions as depicted in Figure 3-15. Each membership function has a function o f the form: / _ \2 1 X-Xj jUA. (x ) = exp (3-19) 2 \ <y:J where 3t. is the center o f the membership function for fuzzy set Aj, and o] is the standard deviation o f the Gaussian function. The center and standard deviation o f the Gaussian functions are two major parameters to be determined to achieve an optimum system. 1 0.5 0 X (input variable) Figure 3-15: Typical Gaussian membership functions for inputs 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The output o f these systems has to be calculated using a weighted average o f each rule’s output similar to centroid defuzzification: t* Y „ = — ------ (3-20) 1=1 where //, is the firing strength o f the ith rule and is calculated by equation (3-19), Yy is the output value corresponding to the ith rule and j th output variable, yj.mod is the output obtained by the fuzzy model for the j th output variable and c is the number o f rules. Models that employ the Sugeno type rules have been shown to be able to accurately represent complex behavior with only a few rules [54], and therefore the complexity o f the system decreases dramatically. In next sections, we will show how the rules along with the parameters o f membership functions are determined and optimized. 3.2.4 Rule Identification Based on Fuzzy Subtractive Clustering One o f the efficient methods to determine the rules in a fuzzy system is subtractive clustering [50]. The purpose o f clustering is to extract natural groupings o f data from a large data set, giving a concise representation o f system’s behavior. Subtractive clustering is a fast clustering method designed for high dimension problems with moderate number o f data points. This method automatically generates fuzzy rules by clustering the data. Each cluster center represents a fuzzy rule. Subtractive clustering is a very useful approach, since it avoids the complexity o f the system to a great extent in comparison to other clustering techniques. The subtractive clustering method assumes each data pair is a potential cluster center and calculates a measure o f the potential for each data pair based on the density o f the surrounding data pairs. The algorithm selects the data pair with the highest potential as the first cluster center and then reduces the potential o f data pairs near the first cluster center. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The data pair with highest remaining potential is then selected as next cluster. This procedure is repeated until the potential o f remaining data fall below a threshold. Assume a data set with n data points containing filter data as in (3-6), where each data pair is denoted by a vector, z\, o f p+q variables, including the output variable values, where the first p dimensions correspond to input variables and the last q dimensions correspond to output variables. The algorithm [50] begins by normalizing the data set: x„ = - Zjj - m in (z. ) 1---------— ------ (3-21) m a X , {Zy ) - mitt,- {Zy ) Let Xj be the vector o f normalized values for the ith data pair. Next, the scaled distance between all pairs o f data is measured: f , x2V/2 ( X ij d ( x „ x ,) M *1] X l i )• (3-22) rj . where V7 is the range influence o f each variable, called the radius o f the j th variable. Note that rj is a term proportional to the standard deviation (a) o f the membership functions [50]. The initial potential for each data point is calculated as: P m ( x i) = Y j e~ad{x"x,) (3-23) /= i where a is a positive constant. The first cluster center is determined by calculating the maximum energy obtained from (3-23): P m* = m ax(/,0)(x;)) / (3-24) The first cluster center is x 0)*, which is the data pair with maximum initial potential value, where P 0)* = P (1,(x(1)* ). To find the next cluster center, we need to eliminate the effect o f the previous cluster center. Normally previous cluster center is surrounded by a number o f data 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. pairs that also have high potential values. To remove this effect, we revise the potential formula for the remaining data pairs: p (*> (x .) = p (*_1)( x (. ) - P (*_1)* (3-25) where /3 is a positive constant. To stop the process o f acquiring new cluster centers, the algorithm in [50] is used, which defines the criteria for accepting/rejecting the cluster centers. Consider the Sugeno-type fuzzy system with the rules o f the form (3-17). As a result o f subtractive clustering, we obtain a collection o f cluster centers (x\k)* ,x(2k)* ,...,x c(k)*), each representing a collection o f points in the data space corresponding to the kth fuzzy rule o f the form: IF x , IS CLOSE TO x\k)' AND x 2 IS CLOSE TO x[k)' AND x p IS CLOSE TO x (k)t, THEN (3-26) y, = a {, + a \x x + a \x 2 + ... + a'px p y q - Cq + a xq X\ + a q2x 2 +... + a qpx p where p is the number o f input variables, q is the number o f output variables and xj(k)* represents a cluster center. We can implement the term “IS CLOSE TO " with a Gaussian membership function. The parameters o', can be determined using a linear least squares estimation method [50], For the kth rule, the membership o f the input xi is: (3-27) Recall from (3-20) that the output is calculated as: 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i* y , yj _m06=— c (3-28) I >, 1=1 where j is the output number, and Yy are linear functions. So far, we have determined the centers o f membership functions as the cluster centers. The design o f our fuzzy system will finish by determination o f rj and a'. parameters. For this purpose, we divide the data pairs into two different parts, one to build the fuzzy system, and the other to check the validity o f the system function. We call the former training data pairs, and the latter checking data pairs. Using the training data pairs and plugging them into the output function in (3-28), we obtain a set o f linear equations, with a'j parameters as unknowns. O f course, the number of equations are more than the number o f unknowns, and this problem can be solved as a leastsquares estimation problem [50]. To find the a ' parameters using the least-squares method, we need to have access to parameter rj. As we will explain in the next section, to find the optimal value for rj, we take advantage o f the checking data pairs by making a comparison between training and checking errors for different 3.2.5 values. Identification of the Fuzzy Logic System for the 8-pole Elliptic Filter Problem To illustrate the proposed approach, we consider an 8-pole band-pass elliptic filter, which has coupling between nonadjacent resonators. The coupling matrix for this example is a symmetrical 8x8 matrix with all elements zero except the ones shown in Table 3-1. Figure 3-4 shows S u and S 22 o f the ideal design filter with the center frequency o f 12 GHz. We also assume that the filter is symmetrical, thus: 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As a result, the nonzero variables in the M matrix are: m'2’mj 3’ ’m36 an<^ . For this problem, we need to build a fuzzy system with 6 outputs, since we have 6 unknown coupling elements. We use 9 frequency samples o f £2 / as inputs to our system. To generate the data, we assume a tolerance o f 50% for each coupling element. For this purpose, we use uniformly distributed random numbers for a set o f input-output data. Then we follow the procedure explained in previous sections to obtain each FLS by subtractive clustering. In our algorithm, we also change the standard deviation (a) o f the Gaussian membership functions to find out the best fuzzy system. In order to get a measure o f which fuzzy system is most appropriate, we separate the data pairs into training and checking pairs. The root square error has been used as a measure for checking and training data error. To show the performance o f our method, here we generate 5000 training data pairs to build the fuzzy system. We change the range influence parameter rj and checked the error for the checking data consisting 500 data pairs. To find the optimal fuzzy system for different r,values, we choose the system with minimal checking error. The error variation for training and checking data is depicted in Figure 3-16. 0.025 T raining error C hecking data error 0.02 0.015 8 LtLI 0.01 0.005 0.5 0.6 0.7 0.8 0.9 Range of influence (r) Figure 3-16: Error variation for training and checking data 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As can be seen from Figure 3-16, when rj decreases or in other words the complexity o f the system increases, the error for the training data set goes down. The checking error however doesn’t have the same trend. It decreases by decreasing rj up to a certain point, and increases afterwards. This phenomenon is referred as over-fitting. Over-fitting is when the system works extremely well for the training data with a very low error, while it has a big error for the checking data. In order to avoid the over-fitting problem, the best system is the fuzzy system with minimal checking error. By considering the two curves in Figure 3-16, we can extract this optimal point, which is corresponding to rj=0.68, where we have the optimal fuzzy system. For this optimized result, the training error is 7.7x10-3 and the checking error is 8.1x10-3. The optimized fuzzy system has 28 rules. Figure 3-17 shows the comparison between the desired and extracted coupling element mi 2 for a set o f training data pairs. Figure 3-18 shows the comparison between the desired and extracted coupling element ml2 for a set o f checking data pairs. - a - D e s ire d O " E x tr a c t e d 0.95 0.9 0.85 0.8 0.7 0.65 0.6 - • « .... 0.55 0.5, Training data pair number Figure 3-17: Comparison between the desired and extracted m t2 for a set of training data pairs 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. «• 0.95 D e s ire d E x tr a c te d 0.9 0.85 0.8 -- 0.75 0.7 0.65 0.6 0.55 0.5. Checking data pair number Figure 3-18: Comparison between the desired and extracted m I2 for a set of checking data pairs The comparison for both training and checking data pairs shows a fairly close match between the desired and extracted elements. 3.2.6 Tuning Results for Highly De-tuned and Slightly De-tuned 8- pole Elliptic Filters To see how accurate the optimized fuzzy system is, we consider two examples o f slightly de tuned and highly de-tuned filters from outside o f the training data set. Note that we don’t have to build different fuzzy systems for each type o f slightly de-tuned and highly de-tuned performances. Table 3-7 shows the desired coupling matrix for the slightly de-tuned example. The extracted coupling matrix using our fuzzy system is shown in Table 3-8, while Figure 3-19 and Figure 3-20 illustrate a comparison between the fuzzy logic extracted performance and the experimental performance for S 21 and S n respectively. The results show a very good match between the actual and desired responses along with a good match between the coupling element values. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3-7: Desired coupling elements for the slightly de-tuned filter example 0 0 0 0 0 0.9417 0 0 0 0 0.1534 0 0.7349 0 9417 0 0 0.1336 0 0 0.7349 0 0.9179 0 0 0.9179 0.4283 0 0 0 0 0 0 0.9179 0 0 0 0 0.4283 0 0.7349 0 0.9179 0 0 0 0.1336 0 0.7349 0 0.94 0.1534 0 0 0 0 0 0 0.9417 0 0 0 0 0 Table 3-8: Extracted coupling elements for the slightly de-tuned filter example 0 0.9548 0 0 0 0 0 0 0.7379 0.9548 0 0 0 0 0.1515 0 0.7379 0 0 0.9473 0 0.1477 0 0 0 0 0.9473 0 0.4841 0 0 0 0 0 0 0.4841 0 0.9473 0 0 0 0 0.1477 0 0.7379 0 0.9473 0 0 0.1515 0 0 0.7379 0 0 0.95' 0 0 0 0 0 0 0.9548 0 As for the highly de-tuned filter example, consider the coupling matrix in Table 3-9. The extracted coupling matrix using our fuzzy system is shown in Table 3-10, while Figure 3-21 and Figure 3-22 illustrate a comparison between the fuzzy logic extracted performance and the experimental performance for S 21 and S u respectively. The results show a very good match between the actual and desired responses along with a good match between the coupling element values. 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Desired Extracted -10 -15 m X3 V" CN CO -20 -25 -30 -35 = 0.1534 i -40 -45 11.94 11.96 12.02 11.98 12.04 12.06 12.08 Freq (GHz) Figure 3-19: Comparison between the desired and extracted S21 response for the slightly de tuned filter example Desired - Extracted m12= 0.9417 m = 0.7349 m-2=-0-.9548 m = 0.7379 m =0,9473..... m.4 5 = 0.4841 m ■o w m .= 0.9173 m45= 0.4283 m-jg—4--1336m = 0.1534 m^Q.1477 m =0.1515 "1^.92 11.94 11.96 11.98 12 12.02 Freq (GHz) 12.04 12.06 12.08 Figure 3-20: Comparison between the desired and extracted Sn response for the slightly de tuned filter example 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3-9: Desired coupling elements for the highly de-tuned filter example 0 0.5959 0 0 0 0 0 0 0.5959 0 0.7151 0 0 0 0.0585 0 0 0.7151 0 1.0260 0 0.0752 0 0 0 0 1.0260 0 0.8982 0 0 0 0 0 0 0.8982 0 1.0260 0 0 0 0 0.0752 0 1.0260 0 0.7151 0 0 0.0585 0 0 0 0.7151 0 0.5959 0 0 0 0 0 0 0.5959 0 Table 3-10: Extracted coupling elements for the highly de-tuned filter example 0 0.5469 0 0 0 0 0 O ' 0.5469 0 0.7267 0 0 0 0.1361 0 0 0.7267 0 1.1114 0 0.0691 0 0 0 0 1.1114 0 0.9036 0 0 0 0 0 0 0.9036 0 1.1114 0 0 0 0 0.0691 0 1.1114 0 0.7267 0 0 0.1361 0 0 0 0.7267 0 0.5469 0 0 0 0 0 0 0.5469 0 Desired Extracted -10 -20 -30 « -40 -50 r r i ' = 0.0752 mj = 0.0585 -60 .92 11.94 11.96 11.98 Freq (GHz) 12.02 12.04 12.06 12.08 Figure 3-21: Comparison between the desired and extracted S2i response for the highly detuned filter example 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — D esired - Extracted -10 -20 nii12= 0.5469 ■ffi23= 07267 -40 ni45= 0.9036 m. = 0.0691 njt =0.136|1 -50 11.94 11.96 11.98 12.02 12.04 12.06 12.08 Freq (GHz) Figure 3-22: Comparison between the desired and extracted Sn response for the highly de tuned filter example 3.2.7 A 4-pole Chebyshev Filter Example with Detuned Resonators The examples discussed in the previous sections are cases where the filter resonators are fully tuned, but the other couplings are de-tuned. In this section, by considering another example, we show the feasibility o f the fuzzy logic method to predict the coupling elements for the cases where all resonators are de-tuned. For this purpose, we consider a 4-pole Chebyshev filter with the coupling matrix shown in Table 3-11, and corresponding response plotted in Figure 3-23. Table 3-11: The coupling elements for an ideal 4-pole Chebyshev filter 0 0.9106 0 0 0.9106 0 0.7000 0 0 0.7000 0 0.9106 0 0 0.9106 0 As for an experimental example, we consider the response in Figure 3-24, which represents an experimental response, which is mistuned in comparison to the ideal response plotted in 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3-23. As can be seen from the figure, the response is also shifted to the left due to the mistuned resonators. We want to see how well the fuzzy logic approach can predict the coupling elements o f the de-tuned response. We followed the same steps as discussed in previous sections. The resulting fuzzy logic system has 15 inputs and 7 outputs, where the outputs are: m u , ni 2 2 , m 3 3 , 11144 , nr 12 , m 23 and ni3 4 . For this purpose, we used 6000 training data pairs and 500 checking data pairs. The resulting fuzzy logic system has 21 rules. Note that we have also optimized the fuzzy logic system according to the procedure explained in previous sections. -10 -20 ft -30 -50 -60 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 4.05 Freq (GHz) Figure 3-23: The ideal 4-pole filter response - - s. -10 -20 -30 S -4 0 -50 -60 -70 i? 9 5 3.96 3.97 3.98 4.01 4.02 4.03 4.04 4.05 Freq (GHz) Figure 3-24: The mistuned experimental response 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 3-25 and Figure 3-26 show the coupling values extracted using the fuzzy logic approach along with the desired coupling values. It also shows the corresponding responses for desired and extracted coupling elements. 0 /- 4 . -4 I | rh ^ = r- 1 — 1.4955 Desired Extracted -10 X -20 \ m .. = “ 7000 \ ni = 2.1000 .... .......... rn‘ 3 - 1 .5000........ rri =2.3P00 / -30 / rti33 = 1.881 4 1)1^ = 1.9363 rin12 = 0.8119 rti23 = 0.8242 n,” “ U.09S4 34 r -40 23 = 0.9000 rr 3. = 0.9000 nr -50 |\x -60 -70 3.96 3.97 3.98 3.99 4 4.01 Freq (GHz) 4.02 4.03 4.04 4.05 Figure 3-25: S2i Comparison between the desired response and extracted response employing the fuzzy logic approach o Desired -5 -10 \\ i; -15 -20 S' s -25 -3 0 -3 5 - .........n j i . . = 1.7000 >• \ ^22 = 2.1000 \ ti33 = i.gooO = 2-8000 rti12= 0.51000 m „ = ).9955 m22 = 1.7815 ndjj = 1,8814 m44.. = i .9363 .m,2 =. 0.6119... m23 = 6.8242 .m.-..=.Q8984--- nil.. = 0.9j000 nji^ = 0,^000 -4 0 -4 5 3,98 3.99 4 4.01 Freq (GHz) 4.02 4.03 4.04 4.05 Figure 3-26: Sn Comparison between the desired response and extracted response employing fuzzy logic approach The results show that the fuzzy logic approach can approximate the coupling values even in the cases where the resonators are mistuned. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Computer-aided Design of Microwave Circuits Using Fuzzy Logic 4.1 Introduction Over the past years, several techniques based on neural networks [31],[35] and Cauchy methods [29]-[30] have been introduced as fast and flexible EM-based tools for microwave modeling. With the use o f training data generated from the EM simulator, these techniques have been successful in building a model that can be used to replace the EM simulator. However, the role o f these models has been limited to simulation allowing the prediction o f the scattering parameters o f the circuit for given physical dimensions, i.e. the forward problem. Although these models are fast and efficient, they still need to be integrated with optimization tools to complete the design process. More recently, the feasibility o f using Fuzzy Logic Systems (FLS) in diagnosis and tuning of microwave filters has been demonstrated in chapter 3 [42],[43] and [46]. FLS techniques can be implemented to deal directly with the reverse problem i.e. for given design 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. specifications, which could also be described in terms o f scattering parameters, the FLS model would predict the physical dimensions. In fact, the circuit design problem and the filter diagnosis problem can be basically viewed as the same problem where the de-tuned filter response is replaced by the required ideal circuit response while the coupling elements are replaced by the circuit physical dimensions. In the diagnosis problem, a theoretical coupling matrix model can be combined with measured data as well as expert information to generate the if-then rules o f the FLS, while in the design problem, the if-then rules are generated using an EM simulator. As explained in chapter 2, the key components in an FLS are fuzzy rules, membership functions and fuzzy inference system (FIS). The choices o f these elements have a key impact on the performance o f the fuzzy logic system. In this chapter, we consider the robust fuzzy logic system based on Sugeno-type FIS [49], with Gaussian membership functions. These types o f systems are also addressed in chapter 3. The Sugeno rules contain linguistic antecedents but the consequent is a linear function o f input variables. Therefore, each rule’s consequent has design dimensions as a linear function o f our design specifications. Models that employ the Sugeno-type rules have been shown to be able to accurately represent complex behavior with only a few rules [54], and therefore the complexity o f the system decreases dramatically. Moreover, the formulation o f the system is such that the parameters o f the fuzzy system can be easily modified to optimize the performance o f the FLS. Subtractive clustering [50], which is an efficient method in data grouping is another improvement in building our FLS from numerical data. The clustering helps the Sugeno-type FLS to be initiated by proper membership functions and minimal number o f rules. In this chapter, we demonstrate how fuzzy logic techniques can be used in the design o f microwave circuits. To show the feasibility o f the proposed approach, the design o f different microwave circuits are considered. These circuits include a microstrip coupled-line coupler, 3-pole microstrip filters, a 6-pole microstrip filter and an HTS 3-pole filter. Data pairs were generated using HP ADS [55] and SONNET [56], which are then grouped using the subtractive clustering technique to minimize the number o f rules. With the use o f Sugenotype fuzzy logic techniques, the fuzzy membership functions were optimized using the initial 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. set o f data pairs as well as a set o f checking data pairs. The fuzzy logic model was then built to synthesize the dimensions o f the circuits under design for different design specifications. The FLS is valid over range o f design specifications such as different couplings for the microstrip coupled-line coupler, or different bandwidths and return loss specifications for the filters. 4.2 Microstrip Coupler Design Problem In this section, we demonstrate how to use the fuzzy logic technique for the synthesis o f the physical dimensions o f the microstrip coupler shown in Figure 4-1, or how to determine w, s for a given even-mode and odd-mode impedances Zoe and Zq0. W Figure 4-1: The coupled microstrip lines structure Since Z0e and Z0o are related to the maximum coupling value and Zo [57], for a 50 Q system, the synthesis problem reduces to the problem o f evaluating w and s for a given maximum coupling value. In order to build an FLS for this problem, we need to generate some data pairs using Sonnet software. We perform our simulations for different 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. combinations o f w and s, and obtain the magnitude o f S 31 to obtain the coupling. Note that change o f w would slightly shift the frequency response, since it also changes the effective permittivity o f the lines. In order to extract the exact maximum coupling associated with the circuit, we need to detect the peak value o f S 3 1 . After performing the simulations, we can obtain a record o f input-output data pairs for building our fuzzy logic system. The data pairs are in the form: (4-1) / v(") v(«) -(«).,.(*) -,(«) ,A,2 "•'i'A'p 9✓1 9^2 ) where we have n data pairs for a system with p sampled scattering parameters as inputs, and q unknown physical dimensions as outputs. In the coupler design p - 1 and q=2. This shows that the fuzzy logic system to be designed will have 1 input, which is the coupling, and 2 outputs, which are the two design dimensions w and s. In next section, we will see how we can build the FLS, and will show the results for the design o f 10-dB and 15-dB couplers, along with a comparison with traditional synthesis techniques. 4.2.1 Design Results for the Coupled-line Coupler Using the algorithm explained in previous sections, we design our FLS. For this purpose, we divide the data pairs into two different parts, one to build the fuzzy logic system, and the other to check the validity o f the system function. We call the former training data pairs, and the latter checking data pairs. Using the training data pairs for different standard deviations, we obtain a set o f fuzzy logic systems. To find the optimal value for standard deviations, we take advantage o f the checking data pairs by making a comparison between training and 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. checking errors for different standard deviations. The root square error has been used as a measure for checking and training data error. As described, we build our FLS with 1 input and 2 outputs. For this design we considered a range o f [0.07-0.1 lcm] for w, and a range o f [0.02-0.15] for s. According to the generated data, our fuzzy system works well for couplings in the range o f 10-20 dB. We generated 82 data pairs, among which 70 data pairs are used for training, and 12 data pairs are considered as checking data. Building the FLS as described for different standard deviations, we can obtain Figure 4-2, which compares the errors for training and checking data. The checking data error is minimal when rj=0.205, which makes our optimal FLS. This FLS has 31 rules. Note that rj is proportional to the standard deviation o f membership functions [50], 0.012 - Training d ata error - Checking d a ta error 0.008 e 0.006 0.004 0.002 0.16 0.17 0 .18 0.19 0.2 0.21 R an g e o f influence (r ) 0 .22 0 .23 0 .24 0.25 Figure 4-2: Error variation for training and checking data for the coupler design problem To check the validity o f our approach, we have considered the design o f 2 couplers, one with 10-dB coupling and the other with 15-dB coupling. Using our optimized FLS, we obtain the design dimensions. Table 4-1 shows the extracted design dimensions obtained using conventional design synthesis procedure [57] and those extracted using our FLS. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4-1: A comparison between the dimensions extracted by synthesis and our fuzzy model for the coupled-line coupler Design Comparison Z0e(Q) Z0o(Q) 69.37 36.04 Synthesis C= lOdB Fuzzy logic Synthesis C= 15dB Fuzzy logic 59.85 41.77 w(cm) s(cm) 0.0800 0.0300 0.1077 0.0185 0.1000 0.0600 0.0835 0.0680 In order to compare the performance o f the two design methods, we use HP-ADS Momentum to perform their coupling responses. Note that here we used HP-ADS instead o f SONNET, since in order for SONNET to simulate the exact circuit dimensions indicated in Table 4-1, a very small cell size is needed, which makes the simulations too long. Figure 4-3 and Figure 4-4 show the S 31 magnitude response for the data provided in Table 4-1. The peak values o f these curves determine the coupling values for our designs. As it is clear from the figures, the fuzzy logic approach gives the exact couplings o f 10-dB and 15-dB for the two cases, whereas the couplings as a result o f synthesis are not accurate. -10 S y n th e s is - S F u z z y L o g ic 12- -1 4 - w -16 -1 8 - -20 1.0 1.5 2.0 2.5 3.0 3.5 Freq (GHz) 4.0 4.5 5.0 F ig u r e 4-3: C o m p a riso n b etw een fu zzy lo g ic an d sy n th esis resu lts fo r 10-d B co u p lin g 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -25■26— i i 1.0 i i | i i m 1.5 | i 2.0 | m 2.5 i i i i i | i i i i | ii 3.0 3.5 i i | i ii i 4.0 [ i i i 4.5 i 5.0 Freq (GHz) Figure 4-4: Comparison between fuzzy logic and synthesis results for 15-dB coupling 4.3 Microstrip 3-pole filter design We consider designing a 3-pole Chebyshev microstrip filter having the layout shown in Figure 4-5. As can be seen from Figure 4-5, there are 4 key parameters in the design o f this filter structure. These parameters are: d\, d 2, h, and h. Figure 4-5: The 3-pole Chebyshev filter structure To start the design process, we need to generate the proper data pairs to build our FLS. For this problem, we take some frequency samples at different frequencies o f our desired performance to capture the most important features o f the filter. These scattering values at the sampled frequencies are considered as inputs to our fuzzy logic system. Once we alter the 4 dimensions in Figure 4-5, the scattering values at sampled frequencies also change. The 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. outputs o f our fuzzy logic system are set to be the design dimensions including the resonator lengths I], h, and gap spacings di, d 2, which represent the sequential couplings o f the filter. With the use o f this information, we obtain a record o f input-output data pairs for building our fuzzy logic system. The data pairs are in the form o f equation (4-1). The data pairs are obtained using HP-ADS. For our design problem, we consider here designing filters with a center frequency o f 2 GHz. We also consider bandwidth variations o f 0.6-1.2 percent. The desired return loss for this problem is 15 dB. Figure 4-6 shows 3 different examples with 0.6, 0.8, and 1.13 percent bandwidths respectively. Our goal is to build a fuzzy logic system to extract the physical dimensions o f the filter for different bandwidths. - 20 - BW = 1 . 13% r\i /) 1 -6 0 - BW 0. 8% EW = 0 . 6% -80 1.90 1.92 1.94 1.96 1.98 2.00 2.02 2.04 2.06 2.08 2.10 2.12 Freq (GHz) Figure 4-6: Filter design responses with different bandwidths In next sections, we will show how we can get a fuzzy logic system to directly extract the design dimensions, and will compare the results with regular filter synthesis results. 4.3.1 Identification of the Fuzzy Logic System for the Filter Design Problem Using the algorithm explained in previous sections, we design our FLS. For this purpose, we divide the data pairs into two different parts, one to build the fuzzy logic system, and the 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. other to check the validity o f the system function. We call the former training data pairs, and the latter checking data pairs. Using the training data pairs for different standard deviations, we obtain a set o f fuzzy logic systems. To find the optimal value for standard deviations, we take advantage o f the checking data pairs by making a comparison between training and checking errors for different standard deviations. For this problem, we need to build a fuzzy system with 4 outputs, since we have 4 unknown dimensions. We use 15 frequency samples o f S 21 as inputs to our system. To generate the data, we first determine the range o f the dimension values where the final design can take. Next, we use uniformly distributed random numbers for a set o f input-output data. Then we obtain the sampled scattering parameters using HP-ADS. To find the optimal fuzzy system, we separate the data pairs into training and checking pairs. The root square error has been used as a measure for checking and training data error. For this problem, we generate 800 training data pairs to build the fuzzy system. We change the standard deviation (a), and check the error for a checking data set consisting 200 data pairs. The error variation for training and checking data is depicted in Figure 4-7. x 104 Training data error Checking data error 0.8 gJ U 0.6 0.4 0.2 ' JL '\ ; i n r, n (ii O O 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95' Range of influence (rp Figure 4-7: Error variation for training and checking data 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As can be seen from Figure 4-7, the FLS is optimum, when we have the smallest error for checking data, which is corresponding to rj=0.94, where rj-2.83a. The optimized fuzzy system has 12 rules. 4.3.2 Design Results for the 3-pole Microstrip Filters To illustrate the performance o f our FLS in the design o f our 3-pole Chebyshev microstrip filter, we consider designing 3 different filters with the response plotted in Figure 4-6. We also use the regular filter synthesis procedure [58] to design the same filters. Table 4-2 shows the physical dimensions extracted using filter synthesis, while Table 4-3 shows the extracted physical dimensions using our optimized fuzzy system for different bandwidths. Table 4-2: Physical dimensions extracted using filter synthesis BW di(mm) d2(mm) li(mm) l2(mm) 0.6% 0.580 2.290 17.6286 17.7534 0.8% 0.500 2.040 17.5971 17.7525 1.13% 0.250 1.670 17.4618 17.7466 Table 4-3: Physical dimensions extracted using the optimized fuzzy logic system BW di(mm) d2(mm) li(mm) l2(mm) 0.6% 0.510000 2.17100 17.6100 17.7600 0.8% 0.445100 2.00000 17.5801 17.7600 1.13% 0.329900 1.80000 17.5199 17.7600 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figures below show a comparison between the scattering parameters corresponding to the synthesized physical dimensions and those extracted using the fuzzy logic approach. - 10 - - 20 Synthesis - Fuzzy Logic -30— -4 0 -50 1.94 1.96 1.98 2.00 2.02 2.04 2.06 freq, GHz Figure 4-8: Comparison between the performances obtained from synthesis and the fuzzy logic system for BW=0.6% - 10- - 20 — Synthesis / -3 0 - Fuzzy Logic -4 0 -50 1.94 1.96 1.98 2.00 2.02 2.04 2.06 freq, G H z Figure 4-9: Comparison between the performances obtained from synthesis and the fuzzy logic system for BW=0.8% 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 10 - - 20 - Synthesis ^ GO -o -3 0 - Fuzzy Logic -4 0 -5 0 -60 1.94 1.96 1.98 2.00 2.02 2.04 2.06 freq, GHz Figure 4-10: Comparison between the performances obtained from synthesis and the fuzzy logic system for BW=1.13% As it is evident from the results, the responses obtained using fuzzy logic are slightly better than the ones obtained using synthesis. One fuzzy logic system is enough to generate all the designs with different bandwidth. 4.4 Microstrip 6-pole filter design We consider designing a 6-pole Chebyshev microstrip filter having the layout shown in Figure 4-11. As can be seen from Figure 4-11, there are 7 key parameters in the design o f this filter structure. These parameters are: d\, d 2 , d 3 , ch, //, I2 and 13 . Figure 4-11: The 6-pole Chebyshev filter structure 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To start the design process, we need to generate the proper data pairs to build our FLS. For this problem, we take some frequency samples at different frequencies o f our desired performance to capture the most important features o f the filter. These scattering parameter (S 2 1 ) values at the sampled frequencies are considered as inputs to our fuzzy logic system. Once we alter the 7 dimensions in Figure 4-11, the scattering parameter values at sampled frequencies also change. -10 -20 -30 g -4 0 -50 -60 -70 1.97 1.98 2.01 1.99 2.02 2.03 2.04 Freq (GHz) Figure 4-12: Filter design response with the bandwidth of 2% The outputs o f our fuzzy logic system are set to be the design dimensions including the resonator lengths //, h, I3 , and gap spacings dj, d 2 , d3 , d 4, which represent the sequential couplings o f the filter. The data pairs are in the form o f equation (4-1), and are obtained using HP-ADS. For our design problem, we consider here designing filters with a center frequency o f 2 GHz. We also consider bandwidth variations o f 1-3 percent. The desired return loss for this problem is 20 dB. Figure 4-12 shows the desired filter response with the bandwidth o f 2%. 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.1 The Design Procedure Using Synthesis for the 6-pole Microstrip Filter The popular synthesis procedure is based on J-inverter concept described in [58]. The first step o f the design is to get lumped element g values for the Chebyshev low-pass prototype filters. By using these low-pass element values, we can compute the J-inverter values. These values are corresponding to the mutual couplings o f the adjacent resonators. For the symmetrical 6-pole filter shown in Figure 4-11, we will have 4 J-inverter values corresponding to the 4 mutual couplings with the filter specifications provided. To continue with the synthesis process, we need to decompose the filter structure into coupling sections as shown in Figure 4-13, connected by regular microstrip lines. Figure 4-13: The capacitive coupling section of the filter With the use o f an EM simulator [55], one can find the S-parameters o f the capacitive coupling section with reference planes as shown in Figure 4-13. By transforming the Sparameters to Y-parameters, the symmetric microstrip circuit shown in Figure 4-13 can be represented by the model given in Figure 4-14. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. J-lnverter I----------------------------------------------------------------------------------- 1 -Y Figure 4-14: The Y-parameter representation of the capacitive coupling section. Since we have already calculated the J values (or Y 12 values) for each capacitive coupling section o f the filter, we can find the corresponding gap size for each coupling section using the EM simulation results. Once all the gap sizes are evaluated, we can also find the corresponding 4>values. These values represent the loading from adjacent capacitive coupling sections and are necessary to find the length o f each resonator. Figure 4-15 shows the resultant resonator structure including the loading from the adjacent capacitive coupling sections. -<&, TL -<P2 Figure 4-15: The resultant resonator structure The circuit in Figure 4-15 must resonate at the filter center frequency. This can be done easily by equating the overall length to A / 2 . Thus all filter dimensions are calculated. In the following section, we will compare the results from the above EM-based synthesis approach with those obtained from our proposed fuzzy logic approach. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.4.2 Design Results for the 6-pole Microstrip Filter Using the Fuzzy Logic System For the 6-pole filter design problem, we need to build a fuzzy system with 7 outputs, since we have 7 unknown dimensions. We used 17 frequency samples o f S 21 as inputs to our system. These samples are uniformly distributed to cover mainly the pass-band and some outside o f the band o f the frequency response. The minimum number o f samples could be determined by (2n + 1 ) , where n is the number o f poles. This is based on the fact that in Cauchy method [25] (Pate polynomials) we can get the whole S-parameter response by only having (2n+l) samples. Therefore, at least 13 samples are needed to represent the whole response for the 6-pole filter. To generate the data, we first determine the range o f the dimension values where the final design can take. Next, we use uniformly distributed random numbers for a set o f input-output data. Then we obtain the sampled scattering parameters using HP-ADS. We generate 5500 training data pairs to build the fuzzy system. We also used 500 checking data pairs. The error variation for training and checking data is depicted in Figure 4-16. It shows that our optimum FLS is when rj=0.74, where the checking data error is minimum. The optimized fuzzy system has 78 rules. To evaluate our FLS in the design o f our 6-pole Chebyshev microstrip filter, we have considered designing a Chebyshev 6-pole filter with center frequency o f 2 GHz, and a bandwidth o f 2% as shown in Figure 4-12. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4-16: Error variation for training and checking data for the 6-pole filter problem We also use the regular filter synthesis procedure [58] to design the same filter. Table 4-4 shows the physical dimensions extracted using filter synthesis along with the results obtained by our fuzzy logic system. Table 4-4: A comparison between the dimensions extracted by synthesis and our fuzzy model for the 6-pole filter BW=2% d, (mm) d2 (mm) d} (mm) Synthesis 0.040 1.390 Fuzzy Logic 0.071 1.441 d4 (mm) I, (mm) h (mm) h (mm) 1.610 1.650 17.19 17.76 17.77 1.653 1.651 17.38 17.78 17.78 Figure 4-17 shows a comparison between the scattering parameters (return loss) corresponding to the synthesized physical dimensions and those extracted using the fuzzy logic approach. Figure 4-18 shows the same comparison by looking at S 2 1 . As it is evident from both figures, the results extracted from our fuzzy logic system is closer to our desired response in Figure 4-12 compared to synthesis, but needs a bit o f fine 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. tuning. In fact our FLS has managed to track the magnitude shape o f S 21 plotted in Figure 4-12 very closely to obtain the design dimensions. - 10- -20 -30-----4 0 - Synthesls -5 0 - Fuzzy Logic -60 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 freq, GHz Figure 4-17: Comparison between the S u responses obtained from synthesis and the fuzzy logic system for the 6-pole filter (BW=2%) - 10- - 20- -3 0 - Synthesis '■ Fuzzy Logic -4 0 -50 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 freq, GHz Figure 4-18: Comparison between the S 2i responses obtained from synthesis and the fuzzy logic system for the 6-pole filter (BW=2%) 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 Design of a 3-pole HTS Filter In this section, we design a 3-pole HTS filter with the layout structure as shown in Figure 4-19. We have 4 parameters for design, including the two gap sizes and the two resonator lengths shown in Figure 4-19. We follow the same steps as section 4.4.1 for synthesis and the same steps as section 4.4.2 for the fuzzy logic approach. The FLS is designed for filters with a center frequency o f 3.95 GHz, bandwidth range o f 0.5%-0.75%, and a return loss o f 20dB. The FLS is made with 15 frequency samples as inputs, and 4 layout dimension outputs and 800 data-pairs [59]. n i______________________________________________________________________ l Figure 4-19: The layout of the 3-pole HTS filter The design is done for a filter bandwidth o f 0.6% once by the regular synthesis method explained in section 4.4.1, and once by the Fuzzy Logic approach. Figure 4-20 shows the fabricated 3-pole HTS filter including the metal box. The measurement results are shown in Figure 4-21, which further proves the feasibility o f our approach. The small discrepancy between the design and measurement is due to manufacturing tolerances and/or input-output port mismatching. 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4-20: Picture of the experimental 3-pole HTS filter — OQ TJ. S11 S21 -10 V) <v £ -30 -40 -50 3.85 3.9 3.95 4 4.05 4.1 4.15 Frequency (GHz) Figure 4-21: Measured results of the HTS filter 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Tuning of Microwave Filters by Extracting Human Experience Using Fuzzy Logic 5.1 Introduction Fuzzy logic systems (FLS) have been proven to be strong tools and reliable models for tuning and design o f microwave circuits. The feasibility o f using Fuzzy Logic Systems (FLS) in diagnosis, tuning and design o f microwave circuits has been demonstrated in chapters 3 and 4 [43],[46] and [59]. Fuzzy logic deals with artificial intelligence (AI), and is unique for the ability o f dealing with objective (mathematical or measurement information) and subjective knowledge (rule-based knowledge) at the same time. The FLS in our previous microwave tuning and design methods were based on a model whose inputs were the circuit responses at different frequency points, whereas the outputs were the tuning/design variables. The FLS’s used in these problems have to be designed deliberately since there is no iteration 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. involved after the FLS is provided. Therefore, there is a need to have enough data-pairs to build the FLS. Using an iterative approach combined with FLS adds more flexibility to our approach, while needing fewer data pairs resulting in a simpler overall FLS. Use o f human expert knowledge adds intelligence to our system, which results to simpler and more efficient fuzzy logic systems at the same time. Since the FLS in this approach goes through iterations or in other words, we have a feedback system, we refer to the FLS’s as FL Controllers. Fuzzy control has been used as one o f the most successful applications o f fuzzy theory. Among the first people who implemented fuzzy control are Chang and Zadeh in 1972 [63], and Mamdani in 1974 [48]. Fuzzy control possesses some remarkable merits and has been successfully applied in many kinds o f systems. A fuzzy controller is a real-time expert system, which models the thinking processes an expert might go through in the course o f manipulating process. The FL Controllers in this chapter are based on fuzzy clustering technique [50] associated with Sugeno type fuzzy logic systems [49], The membership functions are o f Gaussian type with variable standard deviations. 5.2 The Tuning Procedure Consider the filter coupling matrix model described in [60], which relates coupling elements to the filter response. Figure 5-1 shows a block diagram o f the process o f tuning by a human expert using simulated results generated by this coupling matrix model. The expert applies his/her own experience as if he/she is observing the experimental response. The adjustable coupling elements on the interface represent the tuning screws positions. This has been implemented by a user interface program. Usually, the human expert adjusts one coupling element at each step. Meanwhile, a computer program records the necessary samples o f the frequency response o f the filter as well as the amount o f change in the altered coupling element. Input data for the expert here is 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the frequency response at different frequency points as well as a comparison with the ideal response, whereas output data is the coupling element changes or equivalently, the change in the position o f the selected screw. The recorded input/output data will act as the data-pairs to generate rules and thus a fuzzy logic system. The FLS in turn can be used instead o f the human expert to tune the de-tuned circuit response. pc D ISPLA Y HU m o Adjustments O n C o m p u te r / \ Figure 5-1: Human expert (technologist) who tunes a microwave filter. If we take a look at the general feedback control systems structure [65], we can show the control process with the block diagram shown in Figure 5-2. The fuzzy controller is a replacement for an expert (technologist). A human expert in general compares the actual output with the desired output o f the system and based on the difference decides to change the input parameters o f the circuit. Fuzzy Controller also compares the actual output with the desired output o f the system, and accordingly changes the input parameters o f the circuit. Desired Response Fuzzy Logic Controller Microwave Filter Actual Response Figure 5-2: Replacing the human expert with a Fuzzy Logic Controller. 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 Implementing the Tuning Procedure In order to apply the method, we need to ask the expert to tune the filter for different scenarios, i.e. given different filter responses; the expert starts the tuning process. Our program keeps track o f the moves the expert takes on the tuning screws to tune the filter. At each step within a scenario, a set o f input/output data is recorded. As shown in Figure 5-2, the inputs are the differences between the Actual Response and Desired Response (Ideal Response) at different frequency points. The actual response in this paper is the return loss o f the filter. The outputs are the amount o f change to the coupling elements (or tuning screws) at each step. Once the tuning process is finished for a specific scenario, the corresponding input/output data pairs are collected to build the corresponding fuzzy logic system. Then another different scenario is given to the expert, and the same procedure is repeated to build the next FLS. Figure 5-3 shows the FLS controller based on several sub-controllers. There are three advantages to having several FLS controllers rather than making one FLS out o f the collected data: 1- It avoids the possibility of conflicting rules. Therefore, for a given response there is only one action suggested by the fuzzy system. 2- The fuzzy systems are simpler due to fewer numbers o f data pairs associated with them and thus easier to deal with. 3- The overall number o f data pairs does not affect the complexity o f the FLS’s, and therefore the number o f data pairs can increase without any limitation as needed. FLS 1 FLS 2 Microwave Desired _ Response F ilte r FLSn Actual Response Figure 5-3: The control system composed of several simple FL controllers. 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 5-4 shows a general decision making Fuzzy system to decide which FLS in Figure 5-3 is most similar to the current scenario. The fuzzy set with the greatest membership is selected for a specific input value. Once the fuzzy sets are determined, their centers are corresponding to an FLS, which is the selected FLS for the tuning process. The decision making fuzzy sets follow the procedure in [45]. S2 S1 CE B1 B2 0.1 0.45 0.8 1.15 1.5 Figure 5-4: The general decision making Fuzzy Sets with 5 membership functions. The selected FLS’s (FL controllers) are first order Sugeno type Fuzzy Inference Systems (FIS) that benefit from Fuzzy Subtractive clustering, Gaussian membership functions and Centroid defuzzification [50], [49]. The Fuzzy Subtractive Clustering groups the data pairs into a number o f rules; with Gaussian membership functions as in Figure 5-5 and the rules in the form: IF X. is A & X 2is A2&... THEN (5-1) y ’ Yl isBl &Y2isB2... where Xj is the j th input variable and Yj is the j th output variable, and Bj is in the form: B j = a 0 + a xx x + a 2x 2 + " - (5-2) 0 .5 X (Input var i abl e) Figure 5-5: Gaussian membership functions for FLS inputs. 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The flow chart diagram shown in Figure 5-6 shows the detailed steps for the proposed filter tuning method. The procedure has the capability o f learning from a new scenario, which is referred to as online learning. (START) m O ) T .. ________ Step 5: Compare the experimental response with the desired one and give it as an input to the FL Controller Stepl: Prepare a User Interface for the human expert to capture the tuning steps he/she takes I Step 2: Give the expert Afferent tuning scenarios representing different experimental responses. Choose two fuzzy regions for each coupling element input variable to cover the wfiole input space Step 6: Choose the closest FLS based on the fuzzy regions chosen at Step 2, and go through n iterations (this is called one cycle) x Step 7: Find the change in the coupling Step 3: Tune each filter. Track and save the corresponding tuning steps data related to each scenario elements as the output of the selected FLS and thus find the new coupling element values (screw positions) Step 4: Build the FLS’s corresponding to each scenario using Sugeno Fuzzy Model/ Subtractive Clustering X Step 8: Find the closest filter response compared to the ideal one for then iterations and record the local optimal response (S11, S21) as vuell as the corresponding couplings (screw positions) ( 2 ) Ci") x Step 9: Does the < response meet the design requirements? - Yes -■■►{ end ) No Step 10: Is the maximum number of cycles reached? No Yes I Step 11: Give the optimal response to the expert to tune and thus add another FLS to the control system (online learning) Figure 5-6: The flow diagram for detailed steps of the tuning procedure using FL controller approach. 5.4 Results To show the feasibility o f the concept, we consider tuning o f a 4-pole Chebyshev filter. The coupling matrix is in the form: m ll M = mn m\2 m22 m 23 m 23 m 33 m 34 m 34 m 44 (5-3) 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with the input/output couplings o f R} and R 2 . Following the analysis described in [60], the fdter scattering parameters could be determined given the coupling element values: S 2l = - 2 j (5-4) - JR + M )-1]„,, (5-5) S n = \ + 2 j R x[ ( M - j R + M y X , l where I is the unity matrix, and R is a matrix with all elements zero, except R u= R / and We have assumed a synchronously detuned problem, where the diagonal elements o f the coupling matrix are zero. Therefore, the process consists o f tuning the sequential couplings as well as input/output couplings i.e. mu, m 23 , m34, Rj and R 2 . The center frequency is at 2 GHz, with a bandwidth o f 15 MHz and return loss o f 20 dB. For the case o f decision-making, we use 2 Fuzzy Sets centered at 0.45 and 1.15 for an input range o f [0.1 1.5]. The case o f Fuzzy Sets (membership functions) is identical for the 5 variables. Fig. 7 shows the shape of the mentioned Fuzzy Sets. S1 B1 0.45 1.15 Figure 5-7: The simple case of decision-making Fuzzy Sets with 2 membership functions. This results in an initial set of 25=32 FLS’s corresponding to 32 scenarios. We followed the procedure described in the flow diagram o f Figure 5-6 and added 16 m ore F L S’s to make the system obtain a robust overall system (online learning). The human expert took approximately 15 moves for each scenario to tune. To check the robustness o f our method, we tested our system for more than 20 random detuned filter scenarios and our program 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. managed to perfectly tune all o f them. Therefore, by using only 48 tuning scenarios and thus building 48 FLS’s using the Sugeno Type FLS’s and subtractive clustering, we could make a very robust system for any initial value o f the 5 coupling elements representing the position o f the 5 coupling screws. As an example, we consider tuning an experimental response with the coupling values of: ' M = 0 0.35 0.35 0 0.65 0.65 0 0.8 0.8 0 (5-6) 0.7, R2 = 0.9 The ideal design has a coupling matrix of: ' M ■ 0 0.91 0.91 0 0.7 0.7 0 0.91 0.91 0 (5-7) 1.07, R2 =1 07 Figure 5-8 shows a comparison between the ideal response (desired) and the initial detuned response. Our FL program managed to tune the filter through 4 cycles each with 7 iterations. Figure 5-9 shows a comparison between the response obtained by our proposed FLS method and the ideal design response. The response after tuning perfectly meets the design requirements for magnitude and phase. \ : i i i [ !m i.' 7»i [-—Initial] IdeajJ i— Initial! Ideal; / 1 t I / ~-30 CD •o ^■-40 \ \ \ J / \ \ / CM ir|!| H M t W -5 0 -60 X ** -70 £95 Freq (GHz) 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 F req (GHz) (a) (b) Figure 5-8: The initial detuned response vs. ideal response: a) Sn b) S2i 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.04 4.05 FL Controler Ideal -10 •FL Controller] Ideal I C c(0O ■D -20 E c m ^D. w-40 a -50 3.97 4.01 4.01 4.02 4.05 4.02 Freq (GHz) Freq (GHz) (b) (a) o,— •FL Controller! Ideal | -10 Ideal (0 c "O -20 E c CM -40 -50 4.01 Freq (GHz) 4.02 4.05 Freq (GHz) (d) (c) Figure 5-9: The final response after using FLS tuning algorithm vs. ideal response: a) mag(Sn) b) phase(Sn) c) mag(S20 d) phase(S20 5.5 Results by considering Tuning both self couplings and mutual couplings using the fuzzy logic approach In this section, we explain how we can also consider the effect o f the resonators in the process of tuning. We considered the same procedure as in the previous sections except that we do that in two steps. In the first step, we capture the expert moves for rough tuning the resonators (to bring th response within the bandwidth) and make a simple FLS based on that. In the second step, we consider all variables including self couplings and mutual couplings. Then we track human expert moves for different scenarios until the filter is tuned. For this 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. case, we considered 100 random scenarios for the expert to tune. Each scenario was tuned in an average o f 20 steps. The FLS managed to do the second step tuning in 6 cycles each with 10 iterations. The first step tuning is a one step FLS similar to the ones explained in previous chapters. It roughly tunes the resonators. "i*" <...<{»’ Va Figure 5-10: The graphical user interface with a) Detuned response, b) Resonators approximately tuned, and c) After all parameters are tuned 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 A Novel Filter Tuning Method Using Multi-Level Fuzzy Controllers Based on Linguistic Rules 6.1 Introduction Computer-aided tuning o f microwave filters has been the focus o f microwave engineers over the past years [60]-[61]. This is because o f the demand to have a fast and cost-effective production line in the current fast-growing market. These methods are either based on optimization [14],[17] or synthesis [16] o f the coupling matrix mathematical model. These techniques, however, have their own shortcomings. These are mainly initiated from three basic facts. First o f all, the mathematical model is an approximate model and does not directly reflect the effect o f the tuning screws and dispersion. Secondly, there are always manufacturing tolerances involved, and finally the post-production tuning lines are usually not fully automated. These shortcomings are usually magnified when dealing with more 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. complicated structures. As a result, there is still a need for a human operator who has expertise in the tuning o f such structures to further tune the structure to meet the customer needs. Fuzzy logic systems (FLS) have been proven to be strong tools and reliable models for tuning and design o f microwave circuits. The feasibility o f using Fuzzy Logic Systems (FLS) in diagnosis, tuning and design o f microwave circuits has been demonstrated in chapters 3 and 4 [43],[46] and [59]. The FLS’s used in these chapters are used as strong parameter extraction techniques and have to be designed deliberately since there is no iteration involved after the FLS is provided. Therefore, there is a need to have enough data-pairs to build the FLS. Use o f human expert knowledge in tuning microwave filters was first introduced in chapter 5 [62]. The method is based on fuzzy controller concept, which actually models the thinking processes an expert might go through in the course o f manipulating process. Fuzzy control has been used as one o f the most successful applications o f fuzzy theory and was introduced by Chang and Zadeh in 1972 [63], and Mamdani in 1974 [48]. Fuzzy control possesses some remarkable merits and has been successfully applied in many kinds o f systems. The fuzzy controllers described in chapter 5 are based on numerical data extracted by tracking human expert moves and require enough number o f scenarios for the expert to tune to complete the learning process. The fuzzy controllers in this chapter, however, are based on linguistic if/then rules extracted from an expert. These fuzzy controllers are capable o f including expert rules to accurately tune the filter response in two steps o f coarse tuning and fine tuning. Their mathematical formulation is explained in chapter 2 . A unique hardware/software setup is designed to enable full automation o f the process. The hardware includes programmable motors, adjustable mounting plates, power supply to run the motors, flexible couplings to transfer the movement from motor shafts to the screws, GPIB card to read the network analyzer data and a computer. The software includes a graphical user interface (GUI) to control the motors both manually and automatically, 101 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. communication between the computer and motors through serial port, communication between the computer and the network analyzer through serial port and GPIB card. In this chapter, we also propose a novel approach to approximately tune the resonators based on phase/group delay o f the scattering parameters. This method is proved mathematically following the coupling matrix model for 3-pole filters. The proof could be extended to higher order filters. The concept is similar to what Ness proposed in 1998 [61] but different in the procedure. The method is applied by Fuzzy controllers based on linguistic rules extracted from an expert. The first level fuzzy controllers do a coarse tuning o f the filter, while the second level fuzzy controllers perform a fine tuning procedure to minimize the return loss. The approach is tested on experimental 3-pole and 7-pole waveguide filters. 6.2 A General Resonator Tuning Procedure Based on Phase/Group Delay Response The following tuning steps are initiated from observations o f a human expert while tuning a microwave filter. The method is useful for tuning the resonators and would give a good starting point for a fine-tuning algorithm. The procedure is summarized in this section and then in the following sections, the proof using coupling matrix mathematical formulation is illustrated. 1) Calibrate the filter for accurate phase measurements. 2) Highly de-tune all the resonators by turning the screws clockwise all the way inside the filter, while removing all the coupling screws. 3) Turn the 1st resonator screw counterclockwise while looking at the Sn group delay response. When the resonator is in the tuned position, the Sn group delay would be symmetrical around the center frequency with a larger magnitude. 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4) If there is no transmission lines at the input and output ports, the phase would be ±180 degrees when the first resonator is tuned. For the second resonator, the phase is zero degrees when tuned, and toggles from ±180 to 0 degrees until only the last resonator is mistuned. If there are transmission lines included (internally or externally), there is a constant phase shift, which could be measured by looking at the group delay response. Therefore, we could use a method based on either group delay symmetry around the center frequency or phase value at center frequency. It is easier and more accurate to build a method based on the phase value at center frequency since it is less noisy and has a more defined function. To use the phase method, we need to know the phase shift. To do so, we first tune the first resonator using the group delay response. Then, the phase at center frequency would be the constant phase shift (a). So, the objective would be to find the screw position where phase is (±180 ± a) or a. In the next sections, we prove the concept for 3-pole filters and illustrate the method for both 3-pole and 8 -pole filters. 6.2.1 Proof of Concept Using Coupling Matrix Formulation Consider the following coupling matrix for a 3-pole Chebyshev filter with the response shown in Figure 6-1: M = m \\ m \2 0 ™ \2 m 22 m 23 m 2i m 3i 0 As an example, if we synthesized for a 15 dB return loss target, we would get the following design parameters: 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mu - m22 = m 33 = 0, mn = m2J - 1.22, R i = R 2 = 1.22 (6-2) Following the coupling-matrix mathematical model [60]: S n =1 + 2 , A = XI - j R + M (6-3) fo L - L l BW fo f (6-4) where, /? = R] 0 0 0 0 0 0 0 n, ,A -10 -10 -30 -20 •40 -25 11.5 12.5 j 11.5 Freq (GHz) 12.5 is Freq (GHz) 200 S?100 5-100 11.5 12.5 11.5 Freq (GHz) 12.5 Freq (GHz) Figure 6-1: Frequency response of the ideal 3-pole filter example For the above 3-pole filter the A matrix is given by: (mu + X ) - j r A r 0 0 (m22 + X) r (6-5) (m 33 + A ) - j r 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where r - 1 . 2 2 The objective is to calculate the phase at the center frequency when m 22 and m 33 are highly detuned. Therefore, 2=0. Moreover, assuming|m33| » \jr\, we can neglect the smaller term: mu - j r r 0 r m22 r 0 r m 33 (6 - 6 ) Using the above equations, Sn is then calculated as: s n =1 + 2 j r ■m22m33+r - O n ~ j r ) m 22m33 +{mn - j r ) r + r m33 p _ ( r 2 m33 + m u r 2 - mn m22m3l) + j { r l - rm 22w33) 11 ~ / 2 {r m33+ m n r 2 - mu m22ml3)\ - j ( r 3 - r m 22m33)^ (6-7) (6 - 8 ) At the beginning when all the resonators are highly detuned, by eliminating the lower order terms, equation (6-7) could be approximated as: jr m, 2 *S. . = 1 + (6-9) Therefore, the phase is: <p(Su ) = tan- r 2r' \ mw J 2r (6 - 1 0 ) m, Considering the effect o f the frequency, the phase could be written as: ^ (5 „ ) r mn + A 2 (6 - 1 1 ) The group delay could be obtained by taking the derivate o f the phase with respect to frequency. Using the chain rule: 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. According to the above equation, when the frequency is close to the center frequency and mi i is very large, group delay will be very small. Assume the diagonal elements o f the coupling matrix M are all highly detuned with mi i=10 m 2 2 = 9 11133 = 1 1. Figure 6-2 shows the corresponding frequency response. 0.1 -10 0.05 -20 ---30 -40 -0.05 -50 -0.1 11.5 12 12.5 11.5 12.5 Freq (GHz) Freq (GHz) 0.06 0.05 ;o.o4 :0.03 0.01 11.5 12.5 13 11.5 Freq (GHz) 12.5 Freq (GHz) Figure 6-2: Frequency response of the 3-pole filter when all resonators are highly detuned When the first resonator is tuned, based on the fact that m 2 2 and m 33 are large numbers (highly detuned), we can only keep the larger terms and neglect the other terms in (6-7) and thus: Sn = 1 +- 2 jr mw ~ J r mu + j r (6-13) mu ~ J r The phase then could be calculated as: 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. When mi i is tuned, its value is supposed to be zero and therefore the phase would become ±180 degrees. Therefore, the step 1 tuning method would be to highly detune all the resonators, look at the return loss phase at center frequency, and tune it until getting a phase of ±180 degrees. The key assumption here is to have a very large m 2 2 , which is more dominant than m 33 in the above equations. To calculate the group delay around the center frequency when m u is tuned, we can just replace m u by X in equation (6-14) and take the derivate with respect to frequency: (p(Sn ) « ± ^ + 2 r r —A 1 dcp dk 1 2 r ( r 2 +A2) 1 GD, = ---------—x — = ---------^----- — - x 2 k dA 8f 2 k (r - A ) BW (6-15) 1 + fo 2 f (6-16) At fo the group delay becomes: \ GO, ( / . ) = — (6-17) ^ 7 vkB W It is interesting to see the relative change in the group delay at center frequency to notice the magnitude change: GDl f mx, (detuned) ^ GD0 v (6-18) Equation (6-16) also illustrates an interesting observation. It indicates that the group delay shape is approximately symmetrical around the center frequency. A ssum e two frequencies o f fo±Af We could calculate X as: 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. l+ fo BW I I X fo fo ) _ fo f (A/ ) 2 ± 2 / 0 A/ BW [ / o ( / 0 ± A /) J /o ± A /J (6-19) Since A /is small compared to fo, 2Af X = ±BW ( 6 -2 0 ) Therefore, the magnitude o f A is the same for the two frequencies and thus the group delay is approximately symmetrical. Figure 6-3 shows the corresponding frequency response o f the filter. 0.1 -10 0.05 I-20 -0.05 -40 11.5 12 12.5 13 11.5 Freq (GHz) 12.5 Freq (GHz) 200 2 100 ©0.5 1-100 11.5 12.5 Freq (GHz) Freq (GHz) Figure 6-3: Frequency response of the 3-pole filter when only the first resonator is tuned 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.2.2 Error Calculation Suppose there is a small error due to not having ideal assumptions. Then, instead o f getting a phase o f ±%, we get a phase o f (± 7t+a), where a is a small positive phase. Following equation ( 6 - 8 ), we calculate a and m u drift (or frequency drift) that gives the phase o f ±7t. By neglecting the lower order terms and the fact that m u is zero: (r - j m 22) *^n = " I ^ (r + j m 12) (6 - 2 1 ) The phase offset from ±n is then calculated as: - 2 r 2r a = —-------- « — r - m 22 ( 6 -2 2 ) m 22 Now we want to calculate the phase drift. Since in the original formulation, the A matrix contains the term (m n + l), we can look at the problem by either considering m u as zero and find X at the phase o f ± 7t or considering X as zero and find the equivalent mi i drift. We chose the second one. By finding the phase o f equation ( 6 - 8 ) and solving for mi i when the phase is zero: ( r 3 - rm22mii ) ( - m u m 22mii + mu r 2 + r 2 m33) = 0 (6-23) - r 2 m„ r2 mu = Am,, = — — » -----r - m 2 2 m33 m22 (6-24) This is the error due to not having an ideal detuned condition. This error is equivalent to having a small phase shift in frequency. This phase shift could be calculated as follows: fo BW = Am,, fo (6-25) f 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. f 2 - (Amn B W ) f - / 02 (6-26) =0 Solving for/ gives: (Amu ) B W ± ^ ( A m u B W ) 2 + 4 f 02\ (6-27) (6-28) Substituting (6-24) into (6-28) gives: (6-29) As could be seen from equation (6-29), the frequency shift error is very small when ni2 2 is large. The same procedure could be obtained for tuning the last resonator using the S22 response. 6.2.3 Mathematical Formulation for tuning the second resonator Following equation (6-7), assuming that m 33 is a relatively large number, while m u and ni22 are small numbers, we can simplify the equations as: (6-30) The phase at center frequency is then calculated as: ? ( £ „ ) = tan -1 2 r (6-31) 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. So, when 0133 is large enough (highly detuned), the phase is very close to zero. This suggests the second step o f the tuning, which is leaving 11133 detuned and tuning 11122 until the phase is zero. To calculate the group delay, we substitute m 2 2 with X, considering 1U22 is zero and neglect the smaller terms: p ( S „ ) = ta n 1 -1 + 2 j r- f 2r' 2r V W 33 J m 33 (6-32) Xm33 + r (6-33) = 1+ 7 r 2mji VW 33 The phase is then calculated as: p (S n ) = tan 2r 2X 2 ^ _ m 33 ,™33 2 A (6-34) r The group delay is then calculated as: 1 dcp dl GD-, = ----------- x — 2n dX d f 1 2 2 k 1 r BW l + /° 2 \ (6-35) r Using similar explanations, the group delay formula above is symmetrical around the center frequency. The group delay at center frequency is: GD2( f 0) = (6-36) rnBW The change in the group delay at center frequency with respect to the original detuned scenario is: GD2 gK f mu (detuned)'' (6-37) ~ 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. which gives the same relative magnitude compared to the first resonator. -10 -30 -0.5 11 11.5 12 Freq (GHz) 11.5 12.5 Freq (GHz) 12.5 2.5 200 100 «-100 0.5 11.5 Freq (GHz) 11.5 12.5 Freq (GHz) 12.5 Figure 6-4: Frequency response of the 3-pole filter with tuned first and second resonators 6.2.4 Error Calculation Suppose instead o f getting a phase o f zero, we get a phase o f a, where a is a small positive phase. This is normally due to not having a large enough 11133 . Therefore a is calculated using equation (6-31): 2r a = ----- (6-38) m33 Now, similar to the previous section, we want to calculate the frequency drift. That is calculated when the phase in equation (6-7) is zero. Here we solve equation (6-23) for 0122 - Since m u and m 22 are both very small, the first term could only be zero and thus: ( r 3 - r m 12m 33) = 0 (6-39) 112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. m22 = tsm22 r 2 (6-40) Interestingly, this is very similar to equation (6-24) and suggests that the next adjacent resonator plays the major role. So, the more mistuned the adjacent resonator is, the more accurate the current resonator could be adjusted. We can use equation (6-28) to get the frequency drift except that instead o f Ami l we use Am2 2 : (6-41) 6.2.5 Higher order verification by an eight pole elliptic filter example To verify the method for higher order filters, consider tuning the 8 -pole elliptic filter with the response shown in Figure 6-5. The coupling elements are listed in Table 6-1. Now, consider all the diagonal elements to be highly detuned with the diagonal elements detuned as in Table 6-2. Note that for the coupling elements other than the self-couplings, we would leave the corresponding screws out o f the filter and thus the coupling values will be slightly different from the ideal ones. In general, by increasing the coupling values, the tuning concept would still work, but the error would increase. By decreasing the coupling values, the error would decrease. This could be seen as well for the 3-pole filter error analysis in sections 6.2.2 and 6.2.4. 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -20 11.95 12 12.05 12.1 11.95 Freq (GHz) 12.05 12.1 12.05 12.1 Freq (GHz) 200 120 100 £ 100 to n-100 11.95 12.05 11.95 12.1 Freq (GHz) Freq (GHz) Figure 6-5: Frequency response of the ideal 8 -pole filter example The coupling matrix for the filter is as follows: m, 2 0 0 "*22 "*23 0 "*23 0 0 0 0 0 0 "*27 0 mxi mn 0 M = 0 0 0 0 0 0 0 0 0 "*33 "*34 0 "*34 0 "*44 "*45 "*36 0 "*27 0 0 0 "*45 0 "*55 "*56 0 0 "*66 "*67 0 0 "*56 0 "*78 0 "*67 0 "*77 0 "*78 "*88 "*36 0 0 0 (6-42) The initial detuned filter response is shown in Figure 6 -6 , where the response is detuned completely around the target center frequency o f 12 GHz. Figure 6-7 to Figure 6-13 show the filter response when the resonators get tuned one by one from the first to the 7th. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 6-1: The coupling elements for the ideal 8-pole elliptic filter mn 0 m 66 0 mS4 0.5516 m27 -0.0251 m 22 0 m 77 0 m 45 0.4925 mu 0.0781 mn 0 mm 0 mss 0.5516 R, 0.9824 m 44 0 m ,2 0.8231 m 67 0.5917 Ri 0.9824 mss 0 mu 0.5917 mn 0.8231 Table 6-2: Detuned diagonal elements mu 4 mss 5 m22 6 mM 4.7 mu 4.5 m 77 5.2 m44 5.5 mss 6 -20 -40 ■- -60 -80 -100 11.95 12.05 12.1 Freq (GHz) 200 150 100 c100 -100 11.95 12.05 11.95 12.1 Freq (GHz) 12.05 12.1 Freq (GHz) Figure 6-6: The 8-pole filter example response with all resonators highly detuned 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 - -20 0.2 m -40 ~ -60 oj £ - 0 .4 W - -80 0.6 -100 11.95 12 12.05 12.1 11.95 12.05 12.1 F req (GHz) F re q (G H z) 200 20 100 <0 c c CO n3Qikj O ro -100 12.05 11.95 11.95 12.1 12.1 F req (GHz) F req (GHz) Figure 6-7: The 8-pole filter example with the first resonator tuned 0.04 -20 0.02 g m -40 "O 0 c ~ - 0.02 -60 CM V) -0.04 03 -80 -0.06 -100 "OQPi:9 11.95 12 12.05 11.95 12.1 F req (G Hz) 12.05 12.1 12.05 12.1 F req (G Hz) 40r 200r 100 ¥ CD 10 -100 -20,0, 11.95 12.05 11.95 12.1 F req (G Hz) ’F req (G Hz) Figure 6-8: The 8-pole filter example with the first two resonators tuned 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 x10 -20 0.5 S 1 1 in d B -40 0 CD T3 -0.5 -60 ~ CNJ co -1 -80 -1.5 11.95 12 F req (GHz) 12.05 12.1 11.95 12.05 12.1 12.05 12.1 F req (GHz) P h a s e (S 1 1 ) in d eg rees 200 100 c 40 CO 20.0 11.! • 12.05 11.95 12.1 11.9 11.95 F req (GHz) F req (G Hz) Figure 6-9: The 8 -pole filter example with the first three resonators tuned x 10 -40 I 1 -50 S 1 1 in d B 0.5 ■o -60 0r m CQ - -0.5 -70 C\J -1 1/3 -80 -1.5 -90 .9 11 95 12 F req (GHz) 12.05 -1Q0 , 12 11.95 12.05 12.1 12.05 12.1 F req (GHz) P h a s e (S 1 1 ) in d egrees 100 100 CO c c CO >* ro a> Q 12.05 11.95 12.1 20 11.95 Freq (GHz) F req (G Hz) Figure 6-10: The 8 -pole filter example with the first four resonators tuned 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ,x 1 CL -40 -50 0 CD a, -60 ■a TJ ■ S .1 ~ -70 03 W -80 CN -2 -90 11.95 12 12.05 -100 11 12.1 J 12.05 11.95 12.1 F req (GHz) F req (G Hz) 150 200 100 CO c 100 03 ro a> O TO-100 12.05 11.95 11.9 12.1 12.05 11.95 12.1 F req (GHz) F req (GHz) Figure 6-11: The 8-pole filter example with the first five resonators tuned 0.01 -20 0.005 -10 “ e •E -0.005 03 -60 - 0.01 -100 -0.015 12.05 11.95 12. 11.95 12.05 12.1 12.05 12.1 F req (G Hz) F req (G Hz) 200 200 £C3) 100 co 150 tv CO ■C o <D 100 03 <V o ro -100 r- -2 Q .Q, 11.9 12.05 11.95 12 . 50 11.95 F req (G Hz) F req (GHz) Figure 6-12: The 8-pole filter example with the first six resonators tuned 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -20 - 0.2 -40 CD -100 11.95 12.05 11.95 12.1 F r e q (G H z ) 12.05 12.1 12.05 12.1 F r e q (G H z ) 250 200 200 ~ 150 So-100 12.05 11.95 12.1 F r e q (G H z ) F r e q (G H z ) Figure 6-13: The 8-pole filter example with the first seven resonators tuned As it is clear from the above figures, the phase at center frequency toggles from ±180 to zero degrees. Also the group delay has a relatively symmetrical shape around the center frequency when each resonator is tuned. The area under the group delay curve around the center frequency also gets larger once more resonators are tuned. 6.2.6 The Experimental Setup for an Experimental 7-pole Chebyshev Waveguide Filter Figure 6-14 and Figure 6-15 show the experimental setup for the filter under test. It is a 7pole band pass filter with adjustable 4-40 tuning screws. The screws are connected to servo/step motors by custom designed flexible couplings to connect the screws to motor shafts. The motors are controlled by a GUI (Graphical User Interface) programmed by MATLAB as shown in Figure 6-16. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There is also an interface between the network analyzer and a computer through a GPIB card to be able to extract the network analyzer data. The GUI contains sliders that are programmed to rotate motors and thus the screws back and forth. Once the movement is settled, the network analyzer data containing the scattering parameters at the designated channel is read and plotted as graphs in the GUI. This helps the human operator to go through the tuning process by using the GUI only. Meanwhile, the information can be tracked at each step by saving the corresponding data. The program is designed to initialize the setup for two cases o f slightly detuned and highly detuned scenarios. The hardware includes high resolution programmable servo motors called Smart Motors [64] to facilitate very small rotary movements due to the high sensitivity o f the response with respect to screw positions. The motors are daisy chained and both connected to a power supply and the serial port o f a computer for communication. There are also universal mounting brackets that are designed to be arbitrary positioned on a main aluminum plate. This gives adjustability in 3 dimensions to target any position on a particular device. Figure 6-14: The automated filter tuning setup 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6-15: A 7-pole Chebyshev waveguide filter with tuning screws The GUI is designed to implement the tuning method explained in the previous sections. It also utilizes a fine-tuning algorithm. Both steps are implemented using fuzzy controllers based on linguistic rules extracted from an expert. Using the fuzzy approach, also gives the flexibility o f adding more expert heuristics in terms o f rules. There is option for either running each FLS individually or let the program run the whole sequence. The flow chart diagram in Figure 6-17 shows the tuning steps. These steps could also be summarized as follows: 1) Detune all the resonators completely by turning the screws clockwise. 2) Measure the phase reference looking at Sn and S22 group delay symmetry around the center frequency for first and last resonators and detune them back. 3) Run the corresponding FL controller to tune the last resonator by looking at S2 2 phase at center frequency, record the screw position and then detune it back. 4) Run the corresponding FL controllers from resonator 1 to n-1 to tune the first n-1 resonators successively to the desired phase response. 5) Turn the last resonator (n) back to the recorded position. Coarse tuning is complete. 6 ) Set the objective function to minimize the return loss. Start the fine-tuning algorithm using the second type FL controllers by fine-tuning the first and last resonator. 7) Continue fine-tuning the resonators 2 to n-1 by the second type FL controllers. 8 ) If the response meets the specs, stop the algorithm, otherwise go back to step 6 . 121 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ICWUU PMMBl)*fat!!i.WfW1?Sl».>« ICW*m>A'*AHOjW»!! Figure 6-16: The graphical user interface for computer-aided tuning START Step 1: Detune all the resonators completely by turning the screws clockwise Step 2: Measure the phase reference looking at S n and S 2 2 group delay symmetry around the center frequency for first and last resonators and detune them back Step 3: Run the corresponding FL controller to tune the last resonator by looking at S 2 2 phase at center frequency , record the screw position and then detune it back Step 4: Run the corresponding FL controllers from resonator 1 to n-1 to tune the first n -1 resonators Step 5: Turn the last resonator (n) back to the recorded position. Coarse tuning is complete Step 6: Start the fine-tuning algorithm using the second type FL controllers by fine-tuning the first and last resonator . Step 7: Continue fine-tuning the resonators 2 to n-1 by the second type FL controllers Step 8: Does the response meet the design requirem ents? Yes END No Figure 6-17: Tuning steps using multi-level FL controllers based on linguistic rules 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.3 Transmission Line Effect As explained in previous sections, the coarse tuning is based on tuning each resonator to reach a desired phase. However, in practical cases, we often have a transmission line component at the input/output ports, which could be part o f the circuit or an external part imposed by a system. Therefore, it is important to take into account these effects. Figure 6-18 shows a two port microwave device, which has two transmission lines attached to its input/output ports with constant phase shifts. [S]2 TL Figure 6-18: Transmission lines at input and output ports The resultant scattering parameters could be simply calculated as: i O i 1 1 S 22_ r ...... 1 ft O r ..... V B~j(2^) _^2I o o i ~e~M ft 1 [S]2 = (6-43) oc 12ea-J(fa+fa) S 2Xe Equation (6-43) indicates that the resultant Sn and S2 2 responses would have constant phase shifts. This suggests that these two constant phases should be measured at the beginning o f the tuning process. The constant phase shift, however, does not affect the group delay response as its derivate with respect to frequency would be omitted. As a result, the 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. phase offset is measured by tuning the first and last resonator screws to obtain group delay symmetry around the center frequency as explained in step 6.4 1 in section 6 .2 .6 . Two-step Tuning Method Using Fuzzy Controllers In the following sections, details o f the two FLS-based steps are given. The first step utilizes FL controllers to roughly tune the resonators based on the phase response, while the second step contains the FL controllers to fine-tune the resonators. 6.4.1 First Step Based on Linguistic Rules Figure 6-19 shows the block diagram o f the control process to adjust the screw positions based on the phase response. At each iteration the FL controller determines the amount o f the screw turn based on the phase response and the goal. It is helpful to have the phase response plot for different screw positions to clarify the procedure. Figure 6-20 shows some o f these variations for the 7-pole waveguide filter. The phase reference is measured to be 35 degrees. Therefore, for the odd number resonators the phase goal would be at -145 degrees or +215 degrees unwrapped. The even resonators would be tuned at +35 degrees consequently. P h a s e Ref. + > APh(fo)/Am Fuzzy Controller Waveguide Filter [SI L Ph(t-1) r Phase(Sn) at Center Frequency Figure 6-19: The control block diagram for tuning based on the first type FLS 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 300 200 S CL 100 -100 -150 5000 10000 R e s o n a to r 4 s c re w position -100, 15000 5000 10000 R e s o n a to r 5 s c re w position 15000 Figure 6-20: Phase at center frequency for the res. 4 (even) and 5 (odd) vs. the screw positions 6.4.2 Designing the fuzzy controller Consider the patterns in Figure 6-20. When an expert starts tuning a resonator screw, he follows some basic rules. He doesn’t know the response pattern ahead o f time, but still manages to tune accurately. For example, at the beginning when the screw position is zero, he applies a relatively big screw turn and observes the phase change. As he observes that, he concluded that he needs to continue applying a big turn to get closer to the target. Once the phase gets close to the target, bigger phase changes are noticed, and therefore the expert starts slowing down with the screw turns. Similar rules could be obtained when we are far from the target on the opposite side. Given the facts above, we can assign the inputs to the FL controller as ‘phase at center frequency 1 and the ‘relative phase changes’ with respect to the screw movement. The output would be the ‘screw turn’ with respect to the last position. These are basically the parameters that an expert take into account when tuning each resonator. We follow the procedure described in chapter 2 to design the FLS. We use triangular membership functions for input/output fuzzy sets, max-product inferencing, and centeroid defuzzification. In order to start the design process, we need to assign fuzzy sets or membership functions to each input and output parameter. Figure 6-21 shows the fuzzy sets for the phase input. There are two fuzzy sets assigned, one as positive (POS) and the other one as negative (NEG) with respect to the target. Figure 6-22 shows the fuzzy sets for the 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relative phase input. There are three fuzzy sets assigned to this variable as positive small (PS), positive (P) and positive big (PB). Figure 6-23 shows the fuzzy sets for the output variable, which is the relative change given to the designated screw. There are 6 fuzzy sets assigned as positive small (PS), positive (P), positive big (PB), negative small (NS), negative (N) and negative big (NB). CL 3. NEG Q. 3. 1 POS NEG , POS \ / a 2tt+q T t+ a -TT+a Phase(fo) (a) a tt+ q Phase(fo) (b) Figure 6-21: Input fuzzy sets: (a) Odd number resonators, (b) Even number resonators PB PS 10 Delta = abs(APh(fo)Mm) Figure 6-22: Fuzzy sets for the second input variable s < NB N NS PS p PB A A A A A A. -0.5 -0.1 0.1 0.5 Figure 6-23: Output fuzzy sets Given the fuzzy sets for the variables, we can now develop the fuzzy rule base. Table 6-3 shows the 6 possible rules based on the defined fuzzy sets. Figure 6-24 helps to understand 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. how the rules take a role in finding the target phase. As an example, when we are in the area where rule 4 is fired, it means we are close to the target and need to give a small change towards the negative direction. The output is calculated using centriod defuzzification formula. 100 50 Rule 6 ; -50 -100 -150^ 2000 4000 6000 8000 10000 Resonator 6 screw position 12000 14000 16000 Figure 6-24: An illustration to show the regions where each rule is dominant Table 6-3: The linguistic fuzzy rules to find the proper phase. (Example: Rule 3- IF Phase(fO) is NEG and Delta is PB, THEN Am is PS) THEN IF Rule N um ber Phase(fQ) Delta = |APh(fO)/Am| Am 1 NEG PS PB 2 NEG P P 3 NEG PB PS 4 PO S PB NS 5 PO S P N 6 POS PS NB Note that Delta is not the derivative o f the phase function, but it measures the changes from the previous iteration to the current iteration. It will be close to the derivate only if Am 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. happens to be very small, which will be the case during the last iterations. When the system is ready, the expert should test it to see whether it is duplicating his/her actions properly or not, and can consequently adjust the membership function parameters. 6.4.3 The performance of the fuzzy controllers Figure 6-25 shows the time response for tuning each resonator using the designed fuzzy controllers. It shows the number o f iterations took for every screw to tune. The time for each iteration varies depending on the travel o f the screw. The overall settling time depends on the speed o f the motor, and takes about 30 seconds for each to complete with the available setup. As could be seen from the figures, the fuzzy controller is designed to give an under-damped time response. This is found to be fast and stable at the same time. 6.4.4 Second Step Based on Linguistic Rules Figure 6-26 shows the block diagram o f the control process for fine tuning by small screw adjustments. At each iteration the FL controller determines the amount o f the screw turn based on the Si i magnitude response. When the filter is roughly tuned, the purpose o f the expert is to make some local adjustments around the current state. The fine-tuning process is not an easy task and may take a very long time for the expert to finish. One reason is that once one screw is tuned, the other screws are no longer at their optimum position and need to be adjusted too. The other reason is that the response is very sensitive to small screw turns. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 250 35 Phase Value 215 © 150 <0 > 0) P h a s e T a rg e t -50 V) 100 -100 -15C Iteration Number for Res. 2 Iteration Number for Res. 1 250 Phase Value 215 <D ra 150 > 0) P h a s e T a rg e t v> ra 100 P h a se T a rg e t -50 -100 -15C Iteration Number for Res. 3 Iteration Number for Res. 4 250 Phase Value 215 150 < 3D 5 -50 (0 (B s : P h a s e T a rg e t © 100 CL -100 Iteration Number for Res. 5 Iteration Number for Res. 6 Figure 6-25: Time response for tuning the resonators ARL(t) ARL(t-1) Fuzzy Controller Am(t+1) Waveguide Filter ts] Objective Function Figure 6-26: The control block diagram for tuning based on the second type FLS 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The expert looks at the Sii magnitude response and tries to adjust each screw to minimize Si i magnitude in the pass-band. So, we define the objective function as follows: Obj = max(20Log(S] ,20L o g (S x, ( / 2)) (6-44) where fi is the start frequency and f2 is the stop frequency o f the band-pass filter. At the beginning, the first and last resonators need to be adjusted since they found to have the most significant effect on the objective function. Then, the rest o f the resonators are adjusted. Once all the screws are adjusted approximately for a relatively minimum objective function, the process needs to be repeated since the screws are no longer at their previous minimum, although we are getting closer to our return loss objective globally. It is helpful to see how the objective function changes around the current screw position to get a better understanding o f the problem. Figure 6-27 shows these variations for some resonators o f the 7-pole waveguide filter. One can see from these figures that the curves have a minimum but with different shapes. They are also noisy. This makes it a very good candidate for a fuzzy controller. c O ■ s c 3 cO C ,3 0 0 no S'o UJ LU ■8 > > i 1000 0 500 R e s o n a t o r 2 s c r e w p o s itio n -5 0 0 500 R e s o n a t o r 1 s c r e w p o s itio n Figure 6-27: Return loss objective function with respect to screw adjustments 6 .4 .5 D e s i g n i n g t h e f u z z y c o n t r o lle r Consider the sample patterns in Figure 6-27. When an expert starts fine-tuning the resonators, he follows some basic rules. He doesn’t know the response pattern and the target 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. return loss ahead o f time, but still manages to tune. To successfully tune each screw, the expert keeps track o f these parameters: return loss changes from the previous move, return loss changes from the current one and the amount o f the current move all in linguistic terms. O f course, the output is the next screw turn to be applied. Therefore, we need to build a fuzzy logic system with 3 inputs and 1 output. Here again similar to the first step fuzzy controllers, we use triangular membership functions for input/output fuzzy sets, max-product inferencing, and centeroid defuzzification. In order to start the design process, we need to assign fuzzy sets or membership functions to each input and output parameter. Figure 6-28 shows the fuzzy sets for the first input as ‘return loss changes from the previous step’. There are two fuzzy sets assigned, one as positive (POS) and the other one as negative (NEG). Figure 6-29 shows the fuzzy sets for the second input as ‘return loss changes for the current step’. There are again two fuzzy sets assigned to this variable as positive (POS) and negative (NEG). Figure 6-30 shows the fuzzy sets for the third input variable as ‘screw turn at current step’. There are also two fuzzy sets assigned as positive (POS) and negative (NEG) to account for the amount and direction o f the change. Figure 6-31 shows the fuzzy sets for the output variable, which is the relative change given to the designated screw. There are 4 fuzzy sets assigned as positive (P), positive big (PB), negative (N) and negative big (NB). Note that we can always increase the number o f the fuzzy sets and alter the rules to improve the fuzzy controller performance. Given the fuzzy sets for the variables, we can now develop the fuzzy rule base. Table 6-4 shows the 8 possible rules based on the defined fuzzy sets. Figure 6-32 helps to understand how the rules take a role in going towards the minimum return loss. As an example, when we are in the area where rule 3 is fired, it means we have had a negative change in the previous and current steps and thus need to continue going towards the negative current screw changes. Note that rules 7 and 8 are unlikely to be fired. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6-28: Fuzzy sets for the first input -1 .5 0 1 .5 ARL(t) Figure 6-29: Fuzzy sets for the second input N EG 1 POS - A m (t) 350 -3 5 0 Figure 6-30: Fuzzy sets for the third input t v \ !\ ■S N NS . PS P A A 1\ 3 00 400 -3 0 0 . Am(t+1) Figure 6-31: Fuzzy sets for the output 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 5 I -400 -200 200 400 600 800 Resonator 2 screw position 1000 1200 1400 Figure 6-32: An illustration to show the regions where each rule is dominant Table 6-4: The linguistic fuzzy rules to find the current best return loss in pass-band. (Example: Rule 6- IF (ARL(t-l) is POS) and (ARL(t) is NEG) an d (Am(t) is POS), 77ffiiVAm(t+l) is PS) IF Rule N um ber THEN A R L(t-l) ARL(t) Am(t) Am(t+1) 1 NEG NEG PO S P 2 NEG PO S PO S NS 3 NEG NEG NEG N 4 NEG PO S NEG PS 5 POS NEG NEG NS 6 POS NEG PO S PS 7 PO S PO S NEG P 8 POS PO S PO S N 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4.6 Results for 7-pole filter The method are applied to tune a 7-pole Chebyshev waveguide to be tuned at a center frequency o f 11.2 GHz and the pass-band from 10.85 GHz to 11,55GHz, and a return loss o f 20 dB. The filter is iris coupled with a WR75 flange size and 4-40 tuning screws. Figure 6-33 shows the original case when all the resonators are highly detuned by turning all o f them clockwise inside the filter. As it is evident from the figure, there is no filter response shape within the frequency band o f interest. Figure 6-34 shows the case when the first type fuzzy controllers were applied and the coarse tuning is done for the first 5 resonators. A 25 dB insertion loss is observed. Figure 6-35 is the coarse tuning step when all the resonators are tuned except the last one. The insertion loss has decreased to 7 dB in this case. Finally, Figure 6-36 shows the case when the coarse tuning is complete. The return loss however is around 10 dB in the pass-band. Therefore, there is a need to apply the fine tuning steps using the second type fuzzy controllers designed in the previous sections. ■)j»ul /pale itKjieti'ftonidi HEWLETT PACKAR0^722KS.i.i83917ClM,/.W Hewunr Packard,e^es.ys»i75i907M P»a)frM*«dlwnrg...................................- @ © © iKW’Bl IjwJ ® w. . II 11.5 U Ffsquwcy |!yH/| 125 13 Figure 6-33: Original case when all the resonators are highly detuned 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 6-34: Coarse tuning results when the first 5 resonators are tuned Figure 6-35: Coarse tuning results when all resonators are tuned except the last one Figure 6-36: Filter response when the coarse tuning is complete for all resonators 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10.5 10.9 11.2 11.5 F re q u e n c y [G H z] 12 12.5 13 F re q u e n c y [G H z] [d] [C] 0.06 0.05 <2 . 0.08 0.04 S' 0.06 “ 0.03 o. 0.04 g 0.02 0.02 0.01 12.5 13 F re q u e n c y [G H z] F re q u e n c y [G H z] [e] [f] 0.2 0.2 12.5 0.15 5's2* T O < D Q Q. 3 o 0.05 0.05 F re q u e n c y [G H z] 12.5 13 F re q u e n c y [G H z] [g] 12.5 i [h] <D 1 Q 11 >lr 1\ ■J J / F re q u e n c y [G H z] W’v'I\ F re q u e n c y [G H z] Figure 6-37: Group delay response when: (a) all the resonators are mistuned; (b) resonator 1 is tuned; (c) 1 and 2 are tuned; (d) 1-3 are tuned; (e) 1-4 are tuned; (f) 1-5 are tuned; (g) 1-6 are tuned; (h) 1-7 are tuned 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 4 Rm .S Rkw ft HEWLETTPACKARD,87*2ES,UB3yi/619B,/.B4 R«*.7 HEWLETTPACKAS0.W22E8.U838I/51987 64 200 150 100 60 0 •60 •100 ri 150 10 10.5 11 11. 12 126 13 02 3s* o I Figure 6-38: Filter response when the fine tuning is complete with a 25 dB return loss target Figure 6-39 shows the filter response after 1 iteration o f the fine-tuning procedure. One iteration means that the second type fuzzy controller has tuned all the resonators once in sequence. The return loss is very close to the 20 dB return loss target. and S 21 after 1 iteration -10 -20 -30 S3 -40 -50 -60 -70 1 0 .5 1 0 .8 5 1 1 .5 5 1 2 .5 Figure 6-39: Filter response after 1 iteration of fine-tuning 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. S -it and S n. after 2 iterations 11 21 -10 -20 -25 -30 $ -40 -50 -6 0 -7 0 -80 10.5 12.5 1 0 .8 5 F re q u e n c y [GHz] Figure 6-40: Filter response after 2 iterations of fine-tuning (meets the spec, for RL of 20 dB) Figure 6-40 shows the filter response after 2 iterations of the fine-tuning procedure. It means that the second type fuzzy controller has tuned all the resonators once in sequence twice. The return loss is at 23 dB, which meets the design specifications for the 20 dB return loss target. We could continue the fine-tuning process to get a better return loss. After 5 iterations the 25 dB target is achieved as shown in Figure 6-41. -10 -1 7 -25 -3 2 -50 -60 -70 -80 10.5 1 0 .8 5 1.2 1 1 .5 5 F r e q u e n c y [GHz] 1 2 .5 Figure 6-41: Filter response after 5 iterations of fine-tuning (meets the spec, for RL of 25 dB) The coarse tuning process takes 3-4 minutes to complete with the current setup. Each iteration o f the fine tuning procedure takes about 1 minute to complete resulting in a 5-6 minutes total time for the 20 dB return loss target. 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.4.7 Results for 3-pole filter The 7-pole filter tested in the previous sections did not include the mutual couplings. This was due to hardware limitation since we had only 7 motors available. To show that the method works for resonators and couplings at the same time, we consider design o f a 3-pole waveguide filter with center frequency o f 15 GHz, bandwidth o f 500 MHz, and a return loss o f 20 dB. Figure 6-42 shows the tuning setup. Figure 6-42: The 3-pole waveguide filter setup Figure 6-43 shows the response o f the filter after fabrication without any tuning. As could be seen from the figure, w e are not getting the desired response due to manufacturing tolerances and design imperfections. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 ■..... -———1 ............ 1 "^3 ___ _ / / \ _ S 11 \ -20 — S 21 / -40 CD ■o -60 -80 -ioq t mJ ij J j . * *a A Vf 5 13 14 15 16 Frequency [GHz] 17 18 Figure 6-43: The response of filter without tuning The same procedure as used and the same fuzzy controllers were used for tuning the filter. The coarse tuning fuzzy controllers only use the resonator screws to coarse tune the filter. Figure 6-44 shows the corresponding filter response. _ S -20 £-40 -60 Frequency [GHz] Figure 6-44: Response after coarse tuning using first type fuzzy controllers 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The fine tuning fuzzy controllers (second type) consider all resonator and coupling screws at the same time. After one iteration o f fine-tuning the response changes to the response shown in Figure 6-45. Finally, Figure 6-46 shows the filter response after 2 iterations o f fine-tuning. As could be seen from the figure, the results closely meet the design specifications. _S -2 Oh -40 00 TJ -60 -80 Frequency [GHz] Figure 6-45: 3-pole Filter response after 1 fine-tuning iteration -20 g -4 0 -60 A'*' Frequency [GHz] Figure 6-46: 3-pole Filter response after 2 fine-tuning iterations 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 7 Concluding Remarks 7.1 Summary and Contributions Fuzzy Logic has been applied to the design and tuning o f microwave circuits for the first time with a focus particularly on microwave filters. Different types o f fuzzy logic systems have been successfully used for tuning and design o f microwave circuits. The first FL approach based Wang and M endel’s method has been applied to tune fourpole Chebyshev and eight-pole elliptic filters for two different cases o f slightly detuned and highly detuned. In both cases, a very small number o f measured frequency points were required to identify the coupling coefficients that caused the detuning. An FLS can be considered as a universal function approximator, with the extra capability o f incorporating the human expert information, which makes it unique among other methods. The second FL approach takes advantage o f an improved FLS for tuning microwave filters. The method employs a more efficient fuzzy system based on a Sugeno fuzzy system structure and subtractive clustering. The approach has been successfully applied to tune eight-pole elliptic filters for two different cases o f slightly detuned and highly detuned responses. The approach has also been applied for tuning a four-pole Chebyshev filter with detuned 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. resonators. The proposed FLS is very robust since it works well for slightly detuned and highly detuned cases at the same time. The approach required only a few rules to identify the coupling coefficients that caused the detuning. Although we have obtained a very close approximation for the coupling elements o f the filter, fuzzy logic is not intended to give an exact solution. It rather tries to bring the coupling elements closer to their final solution. Once the close solutions are available, an optimization routine could be used for fine adjustments. This FL approach has also been used in the design o f microwave circuits. We have applied the approach to the design o f a microstrip coupler, a 6 -pole filter and a 3-pole HTS microstrip filter. The fuzzy logic system (FLS) is based also on Sugeno-type rules, and subtractive clustering. The data pairs were obtained using an EM simulator. The standard deviations o f the membership functions are adjusted to find the optimal FLS with minimal error. The design dimensions extracted with the use o f our optimized FLS satisfies the design requirements, while a regular filter synthesis gives a response, which is relatively far from the design goal. The fuzzy logic approach can be easily applied to other microwave design problems. The third FL method realizes the use o f human expert knowledge for tuning o f microwave circuits for the first time. The approach captures the human expert intelligence in the form o f a Fuzzy Logic Controller. The feasibility o f the approach has been proven by considering the tuning of a 4-pole Chebyshev microwave filter. The achieved results confirm the validity o f the proposed method. The fourth FL method takes advantage o f human expert knowledge in terms o f linguistic rules and it manages very successfully to tune a filter in two levels o f coarse and fine tuning. Both o f the FL controllers are based on linguistic rules. The method is applied on 7-pole and 3-pole waveguide filters and the measurements prove the validity o f the approach. A fully automated tuning setup, which could be used for tuning any microwave circuit, has been also successfully designed, fabricated and tested. 143 Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 7.2 • Future Work Extend the FL techniques to deal with a system having a large number o f inputs/outputs. Conventional FL techniques become very computationally expensive and a solution needs to be found. • Extract more linguistic rules from experts to enhance the system. • Combine the models based on linguistic rules, theoretical results and measured data into one comprehensive model. • Investigate interaction with other artificial intelligence techniques such as neural networks, reinforcement learning and pattern recognition. • Apply these techniques to other types o f tunable structures such as tunable MEMS devices. • Develop fuzzy logic models for simulation and design o f different types o f microwave circuits. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] M. Dishal, “Alignment and adjustment procedure of synchronously tuned multiple resonant circuit filters,” Proc. IRE, vol.39, pp. 1448-1455, Nov. 1951. [2] H. A. Wheeler, “Tuning of waveguide filters by perturbing of individual sections,” in Symp. Modem Advances in Microwave Tech., 1954, pp.343-353. [3] A. E. Atia and A. E. Williams, “Nonminimum phase optimum amplitude band pass waveguide filters,” IEEE Trans. Microwave Theory Tech., vol. MTT-22, pp. 425-431, Apr. 1974. [4] A. E. Atia and A. E. Williams, “Measurement of intercavity couplings,” IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 519-522, June 1975. [5] A. E. Williams, R.G. Egri, and R.R. Johnson, “Automatic Measurement of Filter Coupling Parameters,” 1983 IEEE MTT-S International Microwave Symposium Digest, pp.418-420. [6 ] L. Accatino, “Computer-aided tuning of microwave filters,” in IEEE MTT-S Int. Microwave Symp. Dig., 1986, pp. 249-252. [7] H.L. Thai, “Computer aided filter alignment and diagnosis,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 958-963, Dec. 1978. [8 ] T. Ishizaki, H. Ikeda, T. Uwano, M. Hatanaka, and H. Miyake “A computer aided accurate adjustment of cellular radio RF filters,” in IEEE MTT-S Int. Microwave Symp. Dig., 1990, pp. 139-142. [9] A. R. Mirzai, C. F. N. Cowan, and T. M. Crawford, “Intelligent alignment of waveguide filters using a machine learning approach,” IEEE Trans. Microwave Theory Tech., vol. 37, pp. 166173, Jan. 1989. [10] Joel Dunsmore, “Simplify Filter Tuning using Time Domain Transformers”, Microwaves & RF, March 1999. [11] Joel D unsm ore, “T uning Band pass Filters in the Tim e D om ain”, D igest o f 1999 IEEE MTTS Int. Microwave Sym., pp. 1351-1354. [12] Joel Dunsmore, “Advanced filter tuning in the Time Domain”, Conference Proceedings of the 29th European Microwave Conference, Vol. 2, pp. 72-75. 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [13] A. E. Atia and Hui-Wen Yao, “Tuning & Measurement of Couplings and Resonant Frequencies for Cascaded Resonators”, Digest of 2000 IEEE MTTS Int. Microwave Sym., pp. 1637-1640. [14] M. Kahrizi, S. Safavi-Naeini, and S. K. 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