# Optimization of RF/microwave filters and diplexers using hybrid evolutionary programming and model calibration technique

код для вставкиСкачатьOptimization of RF/Microwave Filters and Diplexers Using Hybrid Evolutionary Programming and Model Calibration Technique by Nader Damavandi 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, 2004 ©Nader Damavandi, 2004 R eproduced with perm ission of the copyright owner. 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I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. Signature I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. Signature 11 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Borrower’s Page The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. m R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements The author wishes to express his sincere appreciation to Dr. S. Safavi-Naeini for his encouragement, expert guidance and keen supervision throughout the course of this work. Thanks are extended to Dr. S. K. Chaudhuri, Dr. R. R. Mansour and Dr. K. Ponnambalam, Supervisory Committee members, for their continuing interest. It is author’s pleasure to acknowledge fruitful collaboration and stimulating discussions with his colleagues of the RF/Microwave and Photonics Group at University of Waterloo. Last, but not least, thanks are due to my family, specially my wife Azadeh, for encouragement, understanding, patience and continuing support. The author gratefully acknowledges the financial assistance provided by the Ministry of Training for Colleges and Universities in Ontario through two Ontario Graduate Scholarships, the Communication and Information Technology Ontario (CTTO) and Ericsson Canada Inc. IV R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Abstract This thesis presents efficient and robust optimization techniques for computer aided design and diagnosis of complex RF/microwave filters and diplexers used in modem communication systems. We specifically focus on two areas in the design/verification cycle of microwave filters and diplexers, namely the numerical optimization methods and the methodologies for improving the speed of electromagnetic (EM) optimization. It is shown that the available numerical optimization methods used by many filter designers and implemented in RF and microwave software tools are not suitable for several complex multimodal optimization problems. In order to address this issue, an efficient global optimization algorithm is developed based on the hybrid evolutionary programming (BP). The formulation of conventional EP is reviewed and a methodology for enhancing the robustness of the algorithm using the clustering technique is presented. The modified EP is then combined with the quasi-Newton search method in a novel fashion to improve the speed of optimization process. Furthermore, several specific enhancements for RF and microwave filters are presented. The outline of the developed numerical software based on the hybrid EP is also presented. Performance of the new algorithm was tested using several benchmark problems as well as three practical problems including the synthesis of cross-coupled resonator filters and diplexers, and computer aided diagnosis of a complex microwave filter. In all cases, superior performance was observed in terms of robustness and speed of optimization in comparison to other available optimization techniques. In area of EM optimization of microwave filters, we developed a linear model calibration approach in the framework of space mapping technique. Our approach provides solutions to several bottlenecks in the current space mapping formulations including the failure of the method due to the inaccuracy of the approximate model, the nonuniqueness of the parameter extraction procedure and the complicated numerical implementation of the algorithm. It is shown that by using the reflection function alignment in the process of approximate model calibration, the requirement for use of a very accurate approximate model of the filter is eliminated. Since we use a linear formulation of the model calibration, the implementation of the approach in the available EM simulators is very easy as well. Several guidelines are also given for reducing the nonuniqueness of the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. parameter extraction step. The developed method was successfully used for EM optimization of four microwave filters. The method has performed equally well in all cases including the lossless and lossy filters as well as filters with cascade or cross coupled topology. In all tested problems, only a few number of EM simulations were required for achieving the optimal solutions while using the simplest type of approximate models. 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Table of Contents Borrower’s Page.................................................................................................................. iii Acknowledgements............................................................................................................. iv Abstract............................................................................................................................... v Table of Contents................................................................................................................. vii List of Figures................................................................................................................................. ix List of Tables................................................................................... xiii Chapter 1 Introduction....................................................................................................................1 1.1 Motivations and Contributions.............................................................................................. 4 1.2 Thesis Organization............................................................................................................... 6 Chapter 2 Review of Optimization Methods for RF/Microwave Circuits and Basic Concepts in Filter Optimization........................................................................................................................... 8 2.1 Optimization Methods for RF/Microwave Circuits............................................................... 8 2.1.1 Deterministic Methods.................................................................................................... 8 2.1.2 Stochastic Methods........................................................................................................11 2.1.3 Hybrid Methods.............................................................................................................13 2.2 Methods for Increasing the Speed of Optimization............................................................. 13 2.2.1 Space Mapping and Hybrid Optimization.....................................................................14 2.2.2 Artificial Neural Networks (ANN)................................................................................14 2.2.3 Model Based Parameter Estimation and Adaptive Sampling....................................... 15 2.2.4 Adjoint Network Method...............................................................................................15 2.3 Basic Concepts in Filter Optimization..................................................................................15 2.3.1 Filter Design Specifications and Error Functions..........................................................16 2.3.2 Objective Functions.......................................................................................................17 2.3.3 Parameter Extraction......................................................................................................18 Chapter 3 Hybrid Evolutionary Programming for Filter Optimization....................................... 21 3.1 Review of Conventional Evolutionary Programming.......................................................... 23 3.2 Convergence of E P .............................................................................................................. 29 3.3 Enhancement of Evolutionary Programming Using Clustering Algorithm......................... 30 3.3.1 Clustering Algorithm.....................................................................................................34 3.3.2 Clustering Period.......................................................................................................... 41 vii R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 3.4 Hybridization of the E P ....................................................................................................... 42 3.5 Additional Enhancements for Microwave Filters................................................................45 3.6 Software Design................................................................................................................... 46 3.7 Examples.............................................................................................................................. 48 3.7.1 Test Problems............................................................................................................... 48 3.7.2 Cross-Coupled Resonator Filter Synthesis................................................................... 54 3.7.3 Cross-Coupled Resonator Diplexer Optimization........................................................ 64 3.7.4 Computer Aided Diagnosis of Microwave Filters........................................................ 72 3.8 Conclusions...........................................................................................................................80 Chapter 4 Model Calibration Technique forFilter Optimization................................................. 81 4.1 Linear Model Calibration............................................................. 82 4.2 Model Parameter Extraction.................................................. 85 4.2.1 Existence of Solution in P E .......................................................................................... 86 4.2.2 Uniqueness of Solution................................................................................................. 94 4.3 Examples.............................................................................................................................. 96 4.3.1 Parallel Coupled-Line Filters........................................................................................ 96 4.3.2 Modified Capacitive-Gap Coupled Filter................................................................... 100 4.3.3 Microstrip Cross-Coupled Filter..................................................................................104 4.3.4 Lossy Parallel Coupled-Line Filter.......................................................................... 109 Conclusions............................................................................................................................... 113 Chapter 5 Conclusions and Future Works................................................................................. 114 Future Works............................................................................................................................ 116 Appendix A Prediction of Dissipation Loss in MicrowaveFilters............................................118 Bibliography.......................................................................... VI11 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. List of Figures Fig. 1.1. Design/verification flow for modem RF and microwave filters and diplexers... 3 Fig. 2.1. Conventional space mapping m ethod........................................................... 14 Fig. 2.2. Illustration of a typical mask with upper and lower bonds defined for the response R j ..........................................................................................................................17 Fig. 3.1. The flowchart of a general evolutionary algorithm............................................. 22 Fig. 3.2. Definition of individual and population for a sample combline filter.............. 25 Fig. 3.3. The Conceptual algorithm of E P ............................................................................29 Fig. 3.4. An example of (a) Hard function and (b) Easy function........................ 31 Fig. 3.5. Illustration of convergence of evolutionary programming to a single minimum for a sample multimodal 2-D objective function. The individuals are shown at generation (a) zero, i.e. initial population, (b) 10, (c) 20, (d) 30, (e) 40, and (f) 100. The potential clusters of populations are marked by dashed circles. Two global minima are marked as Gi and G 2 , and two local minima are marked as Li and La.. 33 Fig. 3.6. Two equal distance individuals with (a) large difference, and (b) small difference in fitness value. The two points in (a) belong to different regions and should not be clustered together................................................................................. 36 Fig. 3.7. Illustration of clustering algorithm for a sample 2-D population, (a) The principle component analysis finds the volume of smallest hyper-ellipse containing all the points; (b) hyper-spheres are grown from the points with highest fitness until there is no point between two consecutive spheres........................................................39 Fig. 3.8. Illustration of local search after cluster analysis. The local search may converge to (a) separate solutions inside each cluster, (b) single point inside one cluster, or (c) single point outside the identified clusters, (d) separate solutions outside each cluster....................................................................................................................................43 Fig. 3.9. The flowchart of hybrid evolutionary programming algorithm .......................... 44 Fig. 3.10. The 3D plot of test function in (3.21)....................................................... 48 Fig. 3.11. Contours of function defined in (3.21) and the distribution of populations after 20 generations using conventional EP. Individuals are marked by x..........................49 Fig. 3.12. Contours of function defined in (3.21) and the distribution of population after 20 generations using hybrid EP. All identified clusters are shown by large circles. The population at 19'*’ generation is shown by x and new parents are shown by 0. The global and local minima are shown by G and L respectively............................... 50 Fig. 3.13. (a) A multiple-coupled coaxial cavity filter and (b) its equivalent circuit model.................................................................................................................................... 54 IX R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Fig. 3.14. The synthesized goal response and the circuit response of six-pole filter at first initial point with Chebyshev response..................................................................... ......60 Fig. 3.15. The goal response and the circuit response of six-pole filter after 12 generations...........................................................................................................................60 Fig. 3.16. The goal response and the response of six-pole filter at final global solution. The two responses are indistinguishable......................................................................... 61 Fig. 3.17. The goal response and the response of six-pole filter at a local minimum 61 Fig. 3.18. Distribution of population in the search domain for coupling matrix elements M j, and cross-coupling. The clusters formed around global and local minima are shown separately. The individuals are shown by symbol x ................................... 62 Fig. 3.19. The synthesized goal response and the circuit response of six-pole filter at (a) second initial point, (b) after 40 generation, and (c) the global solution.................... 63 Fig. 3.20. (a) Top view of the DCS 1800 diplexer, and (b) its equivalent circuit model. 64 Fig. 3.21. Initial response of the diplexer before optimization...........................................68 Fig. 3.22. Fitness vs. generation for the best individuals in 4 trials of diplexer optimization......................................................................................................................... 69 Fig. 3.23. The response of optimized diplexer at global solution........................................70 Fig. 3.24. The response of optimized diplexer at a local minimum...................................71 Fig. 3.25. The narrow band equivalent circuit model of a lossy cross-coupled resonator. ............................................................................................................................................... 73 Fig. 3.26. Reference plane shifts at the input/output of bandpass filter (BPF)..................75 Fig. 3.27. Geometry of eight-pole filter. The stray couplings are shown by dashed lines. ............................................................................................................................................... 76 Fig. 3.28. The measured and extracted circuit amplitude response of the degree eight filter.......................................................................................................................................77 Fig. 3.29. The measured and extracted circuit Sn response of the degree eight filter.... 77 Fig. 3.30. The real and imaginary parts of measured and extracted circuit S 21 response of the degree eight filter..........................................................................................................78 Fig. 3.31. The measured and extracted circuit response of the detuned degree eight filter. .............................................................................................................................................. 79 Fig. 4.1. The sample error functions for two model spaces..................................................83 Fig. 4.2. Comparison between the response of a 3-pole Chebyshev filter and four general Chebyshev responses with different transmission zeros............................................... 87 Fig. 4.3. The location of reflection zeros w.r.t. the location of normalized transmission zero for the 3-pole filter.............................................................................................. 88 X R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Fig.4.4. EM response of the fifth degree parallel coupled-line filter....................................91 Fig. 4.5. Pole-zero distribution of scattering parameters of the fifth degree parallel coupled-line filter........................ 91 Fig. 4.6. Magnitude of the scattering parameters for fine model and extracted approximate model of the parallel coupled-line filter................................................... 92 Fig. 4.7. Phase of the reflection function for fine modeland extracted approximate model of the parallel coupled-line filter.......................................... 93 Fig. 4.8. Layout of the HTS parallel coupled-line filte r ........................................96 Fig. 4.9. The optimized approximate model response of parallel coupled-line filter at X*a p x 97 ....................................................................................................................................................................................................................................................................................... * ...................................................................................... Fig. 4.10. EM response of parallel coupled-line filter at initial point ............... 98 Fig. 4.11. EM response of parallel coupled-line filter after one iteration at point x ^ .. 99 Fig. 4.12. Final EM response of the optimized parallel coupled-line filter after four .(5) iterations at point x],^ ^ compared with the approximate model response at em = x fem *■ X cq)X . . 99 ......................* ..................* ...............................................................................................................................................................* .............................................................................................................................................................. Fig. 4.13. Layout of three-pole HTS filter. The fixed dimensions are in mil...................101 Fig. 4.14. Approximate model for half of the 3-pole HTS filter using ADS m odels.... 101 Fig. 4.15. The optimized approximate model response of three-pole HTS filter at . 102 Fig. 4.16. The EM response of three-pole HTS filter at initial point x ^ = x ^ ^ ........... 103 Fig. 4.17. Final optimized EM model response of three-pole HTS filter after four iterations at point x*^ = x ^ ^ compared with the approximate model response at .....................................................................................................................................103 Fig. 4.18. Geometry of the 4-pole cross-coupled microstrip filter.....................................105 Fig. 4.19. The optimal approximate model response of the cross-coupled microstrip filter at ................................................ :.............................................................................. 106 Fig. 4.20. EM response of the cross-coupled filter at initial point x^O - x ^ ^ and the extracted model response at P ( x ® ) .............................................................................106 XI R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.21. Final optimized EM response of the cross-coupled microstrip filter after five iterations at point compared with the optimal approximate model response at x ^ ^ ...............................................................................................................107 Fig. 4.22. Comparison between the wideband responses of the optimized approximate and EM models of the cross-coupled microstrip filter.................................................108 Fig. 4.23. Layout of 4-pole lossy microstrip filter. All dimensions are in m m ................109 Fig. 4.24. The optimal approximate model response of the lossy microstrip filter at . 110 Fig. 4.25. Magnitude of scattering parameters for the EM model of the lossy filter at initial point ~ and the extracted approximate model at P (x^l^) .......... 110 Fig. 4.26. Phase of (a) S 2 1 , and (b) Sn of the lossy filter for the EM model at initial point Xgm - Xapx ^nd the extracted approximate model at P (x lH )................................. I l l Fig. 4.27. Final optimized EM model response of 4-pole lossy microstrip filter after four iterations at point x j„ = x ^ ^ compared with the optimal approximate model response at x*p^............................................................................................................... 112 XU R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 3.1. Comparison of evolutionary algorithm s............................................................... 22 Table 3.2. Number of function evaluations by different algorithms in solving test problems...............................................................................................................................53 Table 3.3. Total CPU time in seconds for each algorithm in solving test problem s 53 Table 3.4. Optimization variables including coupling matrix elements and source/load impedances at global and local minima for the six-pole cross-coupled filter............ 62 Table 3.5. Final value of diplexer circuit components after optimization...........................70 Table 3.6. Extracted circuit components for degree eight filter............................................78 Table 3.7. The extracted circuit components of the detuned eight-pole filter.....................80 Table 4.1. Vectors and P ( x ^^) at each iteration for parallel coupled-line filter. For initial solution Table 4.2 The design variables at x ^^ , P (x^^J) , and x j^ = x f 2 for the 3-pole HTS filter.....................................................................................................................................104 Table 4.3. The design variables at x ^ ^ , P ( x ^ ’) , and x^„ = x ] ^ for the cross-coupled microstrip filter.................................................................................................................. 107 Table 4.4. The design variables at x ^ ^ , P (x^j^) , and Xg„, = x ^ ^ for lossy microstrip filter.............................................................................................. Xlll R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1 Introduction Filtering of high frequency signals is a vital part of any radio frequency (RF) and microwave communications. With continuous advancement of communication techniques and emergence of new technologies over the last few decades, design and modeling methodologies for RF/microwave filters and diplexers’ have also evolved accordingly. Historically, the image parameter and insertion loss methods have been widely used for design of microwave filters since the early forties [9]. In the image parameter method, the filter is treated as cascade of several simpler two-port sections and in the insertion loss method the network synthesis approach is used to design a ladder network of lumped and/or distributed elements that approximates the behavior of the microwave filter. In the literature, these methodologies are commonly referred to as form al synthesis. The formal synthesis approach builds a direct analytical relationship between the value of components in the filter model and the filter response specification. Since the formal synthesis approach works with approximate prototype model of filter, its accuracy is severely influenced by the accuracy of the model in presenting the true physical behavior of the actual filter. On the other hand, improving the accuracy of the lumped/distributed element approximate model of the filter usually leads to a complex network that is practically impossible to synthesize using the formal synthesis approach. During the 1960’s, parallel to development of computers, new class of optimizationbased design approaches emerged with the pioneering works of Temes and Calahan [10] and Bandler [11]. These methods are now commonly known as computer aided design (CAD) approach. The fundamental concept in any CAD approach is the use of computer for fast analysis of the filter model and iterative adjustment of the model parameters in a systematic fashion to achieve the desired response. The advantage of optimization-based ‘ In the follow ing, the terra filte r is used for all filtering structures including conventional two-port filters and three-port diplexers unless w e want to address a specific coraponent. 1 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. design method over the formal synthesis is that it can be used for more complex networks with general topology. Until the early 1980’s, optimization approach was mostly used for design of a simplified physics-based lumped/distributed element model of the filter. The physical dimensions of the actual filter were calculated using empirical or semi-empirical formulas and after fabrication of filter, the necessary tuning or even modification of the design was necessary. This approach is followed even today by many filter designers. Starting from mid 80’s, simultaneous increase in the power of computing machines and the emergence of accurate electromagnetic (EM) analysis tools at the same time brought a new dimension to the microwave filters’ design cycle. These rigorous electromagnetic analysis tools have been extensively used for validation of the designs before fabrication stage. However, despite the dramatic increase in the speed of computers and improvement of the numerical electromagnetic techniques, the direct EM optimization of filter structures is still computationally prohibitive. Over the last decade, several hybrid optimization schemes have been developed to reduce the number of time-consuming EM simulations in this process. Most of these techniques work based on either calibration of an approximate model of the filter using the minimal information from a few EM simulations [82]-[88] or by advanced interpolation techniques [89]-[104]. The last stage of the filter design/verification cycle after EM validation and fabrication is the measurement and tuning. Recently, some techniques have been proposed for automating the tuning process of microwave filters [138]-[141]. One of the most widely used computer aided diagnosis technique is the model parameter extraction using an optimization approach. The parameters of a physics-based model of the filter are extracted through an optimization process that tries to match the response of the model with the measurement. These parameters are then compared with the original design values to identify the mistuned elements. The design/verification product-development cycle for RF and microwave filters is shown in Fig. 1.1. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Set Design Goals/ Response Synthesis Initial Design (from formal synthesis or empirical equations) ► Approximate Model/ Equivalent Circuit Analysis Optimisation No Specs met? Yes Optimization/ Parameter Fxtractlon Approximate Model Calibration Fabrication Electromagnetic Analysis Measurement ' Disgnosis/Parameter Extraction Tuning No Specs met? Yes Stop Fig. 1.1. Design/verification flow for modem RF and microwave filters and diplexers. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 1.1 Motivations and Contributions The computer aided design and diagnosis of complex RF and microwave filters for modem communication systems are challenging tasks. Two areas that can greatly influence the success of the whole process are the optimization techniques used in different stages of the design/diagnosis cycle and the methodologies used fo r reducing the number o f time-consuming EM simulations. In Fig. 1.1, the stages related to these areas are highlighted in gray. As shown in this Figure, three stages in the design/verification flow of microwave filters that involve optimization are: • Synthesis of a circuit or an approximate model of the filter using an optimization approach. • Parameter extraction stage in the process of approximate model calibration of the filter. • Parameter extraction stage in the process of diagnosis and automatic tuning of the filter. Currently, the optimization methods used by filter designers in any of these stages, also implemented in many RF and microwave design tools, are not suitable for many multimodal and nonlinear objective functions with large number of designable variables. These kinds of difficult optimization problems are frequently encountered in many of the today’s complex filters design processes. The designers are usually forced to switch back and forth between several available optimization methods to avoid the entrapment in a local minimum. This process can be very time consuming without any guarantee of finding a true optimal solution. In many cases, a strong local minimum may be identified as the solution mistakenly, leading to an ill-centered design and false diagnosis. The lack of a robust and efficient global optimization method in the arsenal of microwave filter designers is a motivation for the first part of the research work in this thesis. We developed a hybrid algorithm based on the stochastic search method, called evolutionary programming, combined with the well-known quasi-Newton search method. This novel hybridization combined with enhancement of the evolutionary programming using the cluster analysis allows us to find the global solution of the most difficult optimization R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. problems in complex filters design/diagnosis processes in a timely manner and with excellent robustness. In the area of EM optimization, the most widely used techniques for reducing the required number of EM simulations are based on space mapping concept and similar calibration approaches. In these techniques, an existing approximate model of the filter is calibrated using only a limited number of EM simulations. Currently, the robustness and popularity of these techniques are severely affected in three critical areas. First, in current formulations of the space mapping approaches, the success of the method depends on the accuracy of the approximate model. If the approximate model is incapable of predicting the exact behavior of the EM model, the calibration process fails. The failure of the parameter extraction step due to multimodality of the associated objective functions in the current formulations is another bottleneck in the successful application of the available space mapping based techniques. Finally, the implementation of these techniques by the filter designers usually needs development of external optimization modules and a proper link to the existing EM simulators. These tasks are usually very complicated, if not impossible, when it comes to providing a link between the external algorithm and the current commercial EM simulators for a general topology filter. In order to address the above-mentioned challenging problems in the process of EM optimization of RF and microwave filters, we have developed a simple yet efficient linear model calibration approach. The method works based on the space mapping concept with some enhancements. The use of the reflection function of the filter as the preferred response in the approximate model calibration process allows us to use the simplest type of approximate model for the filter without the need for any additional correction networks. Moreover, the use of a linear scheme in our approach eliminates the need for development of any program for linking the EM simulator to the external optimization program. In summary, the specific contributions presented in this thesis are • Formulation and development of evolutionary programming algorithm for optimization and parameter extraction of RF and microwave filters. • Enhancement of the robustness of the evolutionary programming using the clustering algorithm. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Enhancement of the evolutionary programming for specific RF and microwave filter optimization problems. Formulation and development of the hybrid evolutionary programming by novel integration of the quasi-Newton search with the modified evolutionary programming. Implementation of the hybrid evolutionary programming (hybrid_FP) in FORTRAN 90 as a modular independent subroutine. Novel modeling and design-optimization formulation for microwave cross coupled coaxial resonator diplexers with application in mobile base stations. Novel formulation of computer aided diagnosis of microwave filters using direct optimization approach. Formulation and development of a linear model calibration technique based on the reflection function alignment for fast FM optimization of microwave filters. Introduction of several practical guidelines for improvement of the robustness of the parameter extraction step in the linear model calibration algorithm. 1.2 Thesis Organization The organization of the thesis in the next Chapters is as follows: Chapter 2 briefly reviews the existing optimization methods for RF and microwave circuits. We review the deterministic methods, the stochastic methods and the hybrid methods reported in the literature and address the advantage and disadvantage of each method. The major methods for increasing the speed of microwave circuit optimization are also reviewed. The Chapter is concluded by outlining some basic concepts in filter optimization including design specification, definition of error functions and objective functions, and formulation of general parameter extraction. In Chapter 3, a hybrid evolutionary programming is proposed. After presenting the conventional evolutionary programming, it is shown how the cluster analysis approach can be used to improve the robustness of the evolutionary programming. We next R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. introduce the hybridization of the modified evolutionary programming with the quasiNewton search to improve the speed of the algorithm. Several enhancements to the final algorithm for specific microwave filter problems are also presented. The structure of the developed software will also be briefly explained. The performance of the developed algorithm is compared with other global optimization methods in solving several benchmark problems as well as three practical problems including the synthesis of cross coupled coaxial resonator filters and diplexers, and computer aided diagnosis of a complex cross-coupled resonator filter. Chapter 4 deals with EM optimization of microwave filters using model calibration technique. First, the formulation of the linear model calibration technique in the context of space mapping is presented. We then provide the foundations for the effective use of reflection function as the preferred response in the process of approximate model calibration. Several guidelines will be presented to improve the uniqueness of the solution in the parameter extraction stage. The results for EM optimization of four microwave filters will be demonstrated as well. We conclude the thesis in Chapter 5, providing a summary of the research and the suggestions for future works. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Review of Optimization Methods for RF/Microwave Circuits and Basic Concepts in Filter Optimization In this chapter we review some of the existing optimization based design and modeling approaches for RF/microwave circuits. It is tried to briefly mention the advantages and disadvantage of each approach. We will also review some basic concepts in filter optimization such as definition of objective functions and parameter extraction formulation. 2.1 Optimization Methods for RF/M icrowave Circuits Most of the optimization methods for solving the RF/microwave related problems, either in the process of computer-aided design or in model parameter extraction, belong to one of the three classes of deterministic, stochastic, or hybrid techniques. This order nearly reflects the chronological appearance of these methods in the literature. The main reason for this has been the gradual increase in the power of computing machines and development of more sophisticated and efficient stochastic optimization methods. In the following Sections, we review the major works done in each of these three categories. 2.1.1 Deterministic Methods Deterministic methods search the parameter space systematically by moving from one point to another. These methods have been in the center of multidimensional strategies for several types of microwave networks and circuits optimization for many years. Various deterministic methods may be classified in two groups. Methods that do not use the derivatives of the objective function are called direct search methods (zeroth order method), and methods that use the gradient of the objective function in each step are 8 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. called gradient-based methods. From historic point of view, direct search methods have been used earlier than gradient-based methods for solving microwave optimization problems. One of the earliest works in the area of microwave network optimization is done by Bandler [14] for design optimization of a multi-section transmission-line impedance transformer via the pattern search strategy of Hook and Jeeves.. The objective function has been in the form of minimax. It has been shown that the pattern search can fail when the search arrives at a point of discontinuous derivatives. Examples of other direct search methods used for RF/microwave circuit optimizations are Razor Search [15]-[16] which uses multi-step pattern search, and Powell’s method [17], which is an extension of the basic pattern search and works with the conjugate directions in the search space. Powell’s method provides quadratic convergence around the optimum solution. These methods need accurate initial values otherwise they can not provide the true optimal solution. The other groups of deterministic methods are the gradient-based optimization methods. The most widely used gradient-based methods are the steepest-descent (first order method), Newton (and Quasi-Newton-second order method), and Conjugate gradient [11], [13] and [18], The steepest descent method uses the first derivative of the objective function to compute the direction of search in each step. This method is first order since it uses the first order approximation of function around the optimization point (the first two terms in the Taylor series expansion). This method will fail to improve when it faces a saddle point in the landscape because it does not use the information of the curvature of landscape. Moreover, it becomes very slow when the landscape is elongated in one direction. The Newton and Quasi-Newton methods use three terms in the Taylor series expansion of objective function and can provide better accuracy and faster convergence (with quadratic rate of convergence) at price of spending more time on calculation of the Hessian matrix of the objective function. In Quasi-Newton method, the Hessian is approximated by a positive definite matrix calculated from gradient vector, and is updated in each iteration. This will reduce the time spent for calculation of second order derivatives of objective function. The Quasi-Newton method provides superlinear rate of convergence [19]. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 10 Bandler, et al. [20] used a combination of gradient and Quasi-Newton with BFGS update for minimax optimization. They have also applied their method to problem of multisection transmission line transformer and the design of microwave multiplexers. Other examples of using Quasi-Newton optimization methods for microwave problems are the approach proposed by Bandler, et al. [21], which uses Broyden’s rank-one formula for approximating the Jacobi an matrix, and also the method presented by Bandler, et al. [22], which have used Huber objective function. It is shown that the Huber solution features combination of the robustness of the and unbiasedness of the Q norms respectively (these norms will be defined later in this Chapter). Special care should be taken when applying the optimization methods to the Huber function because of the discontinuity of Hessian at some points. Because of this fact, dedicated algorithms should be designed for Huber optimization. Conjugate gradient method (COM) has been used for parameter extraction of a MESFET transistor chip and its package at microwave frequencies [23]. The main advantage of CGM is its low storage requirement in comparison to other gradient-based methods. It is usually suitable for large-scale linear optimization but has been modified to solve nonlinear problems as well [18]. Proper scaling or preconditioning and appropriate restart criteria are usually necessary to ensure convergence. One of the main issues in using gradient-based optimization methods is calculation of gradient. Some works have been done to provide these gradients by sensitivity analysis of the circuit. In [24]-[27] some results regarding the sensitivity analysis of some microwave circuits including multiplexers and multi-coupled cavity filters, are presented. In summary, the direct search methods are suitable for discontinuous objective functions but their speed is not very promising. On the other hand, the gradient-based algorithms can converge to the solution very fast, but their main limitation is that they need the objective function and/or its gradient to be continuous, so they can not be applied to the objective functions that are discontinuous. Moreover, the main disadvantage of both of direct search and gradient-based method is their dependency on the initial estimate of the solution and also their extreme vulnerability to the local minima. These methods may fail to provide the global solution because of their greedy search mechanisms. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 11 2.1.2 Stochastic Methods Stochastic optimization methods choose their path through the parameter space using some random variations. An important feature of these methods is that they accept deterioration in objective function during the iteration process in order to achieve a better robustness. This fact enables them to find the region of the global optimum with high chance of success. Another important characteristic of these methods is their independence on the initial values. Examples of these methods, used in RF/microwave and electromagnetic optimization problems, are pure random search, simulated annealing (SA), genetic algorithms (GA), evolutionary programming (EP), and evolution strategies (ES) [28]-[40]. The enormous increase in computer power has helped these methods to become more popular in the recent years. The pure random search has been used in some microwave simulation and optimization packages such as ADS™ [29] and Ansoft Designer™ [30] as the preferred algorithm for optimization of microwave circuits if there does not exist sufficient a priori information about the location of minimum and also when the conventional gradient optimization fails to provide the global solution. Another stochastic optimization method is Simulated Annealing (SA) which was first introduced in 1950’s. SA treats the optimization problem as the annealing of a molten solid [41]. In [31] and [32], simulated annealing has been used for extraction of parameters of microwave transistors. Courat, et al. [33], Vai et al. [42] and Vincente, et al. [43] have also used SA in microwave optimization problems. The main advantage of SA is its ability to provide global solution in difficult problems. However, SA is very slow compared to other stochastic methods. Another class of stochastic optimization methods that has recently been introduced in microwave and electromagnetic community is Evolutionary Algorithms (EA). Among them, genetic algorithms have been used more extensively compared to the others. The GA imitates nature’s problem solving strategies to achieve an optimal adaptation of a given system to some specific requirements. In classical GA, the optimization parameters are coded in a binary system introducing chromosomes as words of a given length. Algorithm starts with a population of chromosomes and goes through reproduction phase by means of cross-over and mutation. The newborn species are then compared with their R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 12 parents and the individuals with higher fitness will have higher chance to survive and go to next generation. This simple mechanism, inspired by nature, has turned out to be successful in solving some difficult optimization problems, where a little knowledge about the problems and causal relation of parameters and objective function has been available. In [34]-[36] comprehensive investigations about application of GA in electromagnetics and a detailed list of published papers in this field are presented. These applications range from parameter extraction to antenna array design. In real coded GA, instead of binary chromosome, the floating point chromosomes are used and cross-over takes place in the floating point level either in the form of real code gene exchange or arithmetic averaging. It is shown that the utility of GA can be improved when using real coded data representation for the problems with floating point parameters [44]. In [38] GA has been used for optimization of planar microwave circuits such as microstrip bandpass filters. The convergence analysis of GA is performed through schema theory as presented in [45]. The evolutionary strategies and evolutionary programming are the other two optimization techniques in this class that use the principles of organic evolution as rules of optimum seeking [46]-[48]. The details of these algorithms will be presented in Chapter 3. In the area of application, these methods have recently been applied to several antenna optimization problems. Hoorfar [49]-[51] has used EP for design of antennas and filter structures. Chellapila, et al. [52] have applied EP to the problem of design of large size linear arrays and have shown that EP provides better performance and accuracy compared to GA. Convergence analysis of EP and ES has been carried out using Markov chain analysis [53]-[56]. Constraint optimization by EP has been proposed in a series of works by Myung, et al. [57]-[62]. They have used the penalty function and Lagrangian multiplier method for handling different kinds of constraints. Comparison between SA, GA and ES for electromagnetic applications is presented in [65]. In summary, the evolutionary algorithms provide better performance among the other stochastic methods. Although, the evolutionary algorithms, in general, are effective in their search for global optimum, some results show that they have difficulty in locating R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 13 the global solutions in some multimodal problems [48], [66]-[67]. Moreover, they are not fast enough to handle computationally expensive problems very well. 2.1.3 Hybrid Methods The stochastic methods are good at exploring the search space and locating the region of global minimum, but they are slow at exploitation of the solution. On contrary, the deterministic methods are fast at finding the solution when they start within the region of minimum, but they can fail to provide the global solution and can easily be trapped into the local minimum. The deterministic and stochastic methods have complementary properties and it is natural to anticipate a better performance from an optimization method that combines the best features of these two approaches. Some of the recent works in the area of hybridization of deterministic and stochastic optimization methods are presented in [68]-[81]. Most of these articles are published in the last six years indicating that the hybrid techniques are still in their infancies. Some of these methods use the idea of first applying the stochastic methods for finding the region of global minimum and then switching to a deterministic method for finding the final solution very fast [68], [74]-[78]. Other methods, such as those proposed in [77], [79]-[80], start with a deterministic method and switch to a stochastic method when they are trapped in a local minimum. Methods that use the deterministic-like reproduction operator in the genetic algorithm have also been proposed [70]-[71]. In a recent paper by Jones, et al. [68], GA is combined with Powell’s method for design optimization of planar arrays. In this hybrid method, each individual is optimized by Powell’s method in the vicinity of global minimum. 2.2 Methods for Increasing the Speed of Optimization Computation of the objective functions for complex electromagnetic structures need rigorous analysis of the structure, which can be very time consuming. In this section, we briefly review the methods that have been proposed for improving the speed of optimization process by reducing the number of expensive objective function calculations in the optimization loops. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 14 2.2.1 Space Mapping and Hybrid Optimization In order to reduce the number of calls to the expensive rigorous analysis, a methodology, called space mapping, was proposed by Bandler, et al. [82]-[83], In this approach, the main optimization is performed on an approximate inodel of the structure (called coarse model), which is less accurate but computationally inexpensive, and the accurate rigorous model (called fine model) is used to calibrate the coarse model in an iterative process of mapping between the two models as shown in Fig. 2.1. Fme model Param eter extraction , Fine model space (validation space) Coarse model space (optimization space) Fig. 2.1. Conventional space mapping method Although promising results have been reported on the performance of this method, the bottleneck in the application of this technique is in the parameter extraction stage which usually leads to a non-unique solution and finally the divergence or numerical oscillation of the routine. Because of this drawback, it is proposed to either increase the number of calibration points in the fine model space [84] or switch to direct optimization method when the space mapping fails to converge [85]. Other variations of space mapping have recently been proposed [86]-[88] to improve the performance. We will investigate the space mapping method in more details in Chapter 4. 2.2.2 Artificial Neural Networks (ANN) Artificial neural network has been used as a powerful interpolation technique to replace the expensive process of objective function calculation [88]-[91]. The inputs of these networks are design optimization variables and the outputs can be any observable of the system such as 5-parameters. After a period of training, ANN will be able to give an accurate relation between input and output. They can also be designed to accept the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 15 existing knowledge of the system in order to make the model more accurate. These are called knowledge-based ANN [92]. The combination of space mapping and ANN is presented in [93]-[95]. The main drawback of these models is that they do not provide any physical insight for the system under analysis and their accuracy is highly dependent on the training process. 2.2.3 Model Based Parameter Estimation and Adaptive Sampling In [100] and [101], a systematic application of Cauchy method for approximation of system observables by rational functions in the frequency domain is presented. The expensive analysis is performed in only a limited number of frequency points in order to find the unknown coefficients of the polynomials in the rational approximating function. Then, the model can be used to interpolate/extrapolate the response at other frequency points. In [102], the idea of adaptive sampling is used to reduce the number of frequency samples. This method is restricted to frequency response interpolation, and exact calculation of the order of polynomials in the rational functions can not be done analytically. Peik, et al. [104] have extended the idea of Cauchy method to multidimensional rational function approximation. A recursive method is used for calculation of the coefficient in the polynomials. 2.2.4 Adjoint Network Method This method is used for reducing the number of function calls in the process of calculation of gradient for the complex microwave circuits [105]-[107], The complete sensitivity analysis of the circuit can be done by single analysis of the original network and another network called adjoint network. However, the method is only applicable when the adjoint network is available and the gradient of the Y-matrix of the sub-circuits can be found easily. 2.3 Basic Concepts in Filter Optimization In this Section, we review some basic concepts in filter optimization that will be used throughout the thesis. The definition of filter responses and error functions, the formulation of objective functions and parameter extraction will be reviewed. Most of the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 16 definitions and formulation presented in this Section follow the same definitions used in the previous works such as [11]-[13]. 2.3.1 FUter Design Specifications and Error Functions The electrical performance of a filter is specified by its response functions. Examples of these measurable responses are scattering parameters and group delays of the filter. Let us represent these responses as* R (x ,co) = ,Q)),R 2 {x ,co),...,Rq(x ,(o )J (2.1) where x e M” represents the vector of designable parameters and co represents the independent parameter. In this work, we deal with frequency as the only independent parameter of the filter. In the design process, the desired performance of the filter is expressed by a set of specifications which are only function of the independent parameter CO. In the design process, the response of filter is required to satisfy the desired specifications at a set of discrete frequency points. This will give rise to the definition of error functions as the difference between the actual filter response and the desired response. The definition of error functions depends on the type of desired specification we use for the filter. The most commonly used specifications in the filter design process are mask responses that consist of sets of upper and lower bonds on the response functions as shown in Fig. 2.2. In this Figure, the upper and lower specifications for the response Rj are denoted by (co) and G;j (co). The frequency dependent error functions can now be defined as e^j ( x , c o ) = w ( x , co).[R j {x , co)-G ^ j (ty)] €ij ( x , CO) = w ij ( x , co).^G;j (co) - R j ( x ,® )] where w^j and Wij represent the appropriate nonnegative weighting factors. Based on this definition, positive error function value shows a violation of corresponding specification. W e use bold type fonts for vector quantities. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 17 R G ut (® ) ^ u i (® ) Violation of specification CO — Fig. 2.2. Illustration of a typical mask with upper and lower bonds defined for the response Rj Another type of specification used in filter optimization is defined in terms of discrete set of individual specifications. In these cases, the measurable filter quantity should ideally match with these specified values. The error function for each individual specification case can be defined as e j {x, Q}) = w j . R j (x, C O ) - G j (ty)| (2.3) where Wj represent the appropriate nonnegative weighting factor. These kinds of error functions are especially useful in the parameter extraction formulation as will be explained in Section 2.3.3. 2.3.2 Objective Functions The optimization problems are usually formulated as the following minimization minimize U (x ) (2.4) X where f/ is a scalar objective function. The scalar objective function is constructed by an appropriate summation of all the error functions defined in the previous Section. One of the most commonly used summation formulation is based on the -norm. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 18 = Z hC xf (2.5) The value of parameter p has direct effect on the optimization. The most widely used values for p are 2 and 1. In case of p=2, the associated minimization problem is called least square approximation. The advantage of the I 2 norm is that the objective function is differentiable and quadratic if the error functions are linear. The larger values of p emphasize the larger errors. In the limits when p the minimization is referred to as minimax approximation. The minimax optimization tries to minimize the worst case errors in the objective function IHL "max|e,(x)l (2 .6 ) i Using the Ip norm definition of the error function formulations, the objective function for the case of lower and upper specification in filter optimization can be written as ^ = s z Ki i&Jf ^ —1 where and N >r (2-7) are the number of frequency points in each of the upper and lower specifications regions applicable to the response in (2.2). The sets h- . The errors and are defined and 7^ are defined as / „ ( x ) ^ { f |e „ , ( x ) > 0 , i G / , | 7 /(x ) = {i |e ;,;(x )> 0 , i G / J (2 .8 ) / „ + / , ={1,2,...,G} In case of single specification (2.3), the objective function reduces to [ /( x ) = f ; |c ,( x ,t y ) f (2.9) i 2.3.3 Parameter Extraction The goal of parameter extraction is to match sets of filterresponses to given reference responses by changing the designable parameters of the filter.Depending on the problem, the reference responses may come from measurements, rigorous electromagnetic solver. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 19 or any other filter model. The formulation of parameter extraction is very similar to the optimization using single specifications described in the previous Section. Let’s assume we have a filter with a set of Q reference responses (2 . 10) where each individual response is given at a set of frequency points as (2.11) In general, these responses are function of some filter model parameters represented by vector e R ”’ . For example, in case of only two responses, i.e. Q=2, R ^ ^ {x ^,(o ) and R m ^(X m ,^) may be chosen to be the measured Tl„ and 21 responses of the filter. The responses of the extracted model of the filter are written as (2.12) R e ^{x,a )) = where jc e R" is the optimization vector and each individual response is calculated at a set of frequency points as ,.(x,£y) = (2.13) It is important to note that the two vectors x and Xm may belong to completely different model spaces and do not necessary have the same size either. For example, the vector x„ may consists of some physical sizes for a given filter and vector x may consists of the values for a lumped element model of the same filter. As we will see later, in case of parameter extraction in space mapping process, both of these vectors refer to the same designable parameters. Using these responses, the error function is simply defined as the difference between the extraction model response and the reference model response at individual frequencies ejk(x,0)i^)= w J. Rmj(^m^^k)-RextAx,a)k) R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (2.14) 20 Where w j represent the appropriate nonnegative weighting factor. Using the Ip norm definition of the error function formulations, the objective function for parameter extraction can be written as (2.15) j=i k R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Hybrid Evolutionary Progranuning for Filter Optimization In this Chapter we propose a method for improving the diversity of a certain type of EA called evolutionary programming in order to make the method more suitable for multimodal global optimization of complex filter circuits, and then the method is hybridized with a gradient-based search method for improving the speed of optimization. In the mid-sixties, L. J. Fogel et al. described EP for the evolution of finite state machines in order to solve prediction tasks [ 1 1 0 ], but it was not practically used in the optimization community until the late 1980s when D. B. Fogel extended EP for applications involving continuous parameter optimization problems [47], [I I I ] , The motivation behind EP, like other evolutionary algorithms, is to simulate the mechanism of the natural evolution. Evolution is an optimization process in which the highly adapted individuals will be found through the natural selection rule, known as survival o f the fittest. Each generation produces another offspring either by exchange of genetic material within its members or by random perturbation in the genetic codes of the parents. Then the offspring and parents compete with each other and the fittest individuals plus some randomly chosen unfits will be passed to next generation. The flowchart of this simplified version of the simulation of evolution is shown in the Fig. 3.1. These algorithms are very efficient for solving complex optimization problems such as RF/microwave problems. Table 3.1 summarizes the main similarities and differences of the evolutionary algorithms. 21 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 22 I n it ia l iz a ti o n o f population Offspring generation through random variations Fitness evaluation No Tournam ent and selection Term inate? Yes Solution Fig. 3.1. The flowchart of a general evolutionary algorithm. Table 3.1. Comparison of evolutionary algorithms EP Real-valued GA Binary-valued Variances None Self-adaptation ES Real-valued Standard deviations and covariance Objective function value Scaled objective Scaled objective Fitness is function value function value Main operator Only operator Background operator Deterministic Probabilistic Probabilistic Representation Mutation Selection R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 23 The main differences between the three common evolutionary algorithms lie on the reproduction and selection methods. For example in GA, which is a bisexual algorithm, two parents produce two children by exchanging the genetic materials through cross-over and mutation, but in EP and ES, which use asexual reproduction scheme, each parent generates a single child by random mutation. Moreover, the data representation in GA is mainly in binary form while in EP and ES, the data are real-valued. It is shown in [113] that while the computational complexity of EP and ES for an n-dimensional problem is 0 ( e ” ), the computational complexity of GA is 0 ( n ” ), because GA tends to resample the already sampled points with higher probability compared to EP. Additionally,there are other benefits associated with the EP and ES over GA as well: • Implementation of EP will be easier than GA because, first of all the only variation parameter in EP is mutation while in GA both cross-over and mutation should be performed, secondly, the process of coding and decoding for data representation in GA does not exist in EP, • Since EP works with real-valued parameters, it does not suffer from the error associated with GA in the process o f binary coding and decoding, • Development of adaptive variation operators in EP is much easier than GA, • Introduction of hybrid mutation operator is possible only in EP. In the following Section, the outline of the conventional EP with emphasis on the RF/Microwave applications is presented. 3.1 Review of Conventional Evolutionary Programming Without loss of generality, let us assume that our optimization problem has been reduced to the following problem: minimize U (x ) (3.1) s.t. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 24 where U is a scalar objective function, which x = [ x ( l) , x(2), X (3),.... , x (n )f may be discontinuous, and is the vector of optimization variable. The objective function is constructed using the 1^ norm definition as explained in Chapter 2. The only constraint that is considered in (3.1) is the box constraint which is usually the most widely used in the RF/Microwave engineering problems and set the lower and upper limits for the value of each parameter. There may also exists some general constraints that may be included in the objective function in the form of penalty functions [57]-[64]. In the context of EP, a pair of n-dimensional vectors, consisting of parameter-vector x and its variation vector a , is called an individual. The variable parameters of each individual make the genetic structure of the individual and determine its location in the space. In a practical problem, these can be physical parameters or the value of the equivalent circuit elements, etc. as shown in Fig. 3.2. Based on this definition, each individual represents a unique model or circuit. In addition, the variation parameter associated with each parameter controls the relative amount of mutation for that parameter. EP works with a group of individuals called population. The size of population, i.e. the number of individuals, is set at the beginning of the algorithm. Each population with the individuals of the same ages are called generation. Each individual can produce another individual called offspring. This reproduction process is done by randomly changing the value of the individual continuously. Algorithm assigns a fitness value to each individual that shows how well the individual satisfies (3.1). This can be the same as objective function or any other random alteration of objective function plus some scaling. Based on these definitions, EP algorithm consists of five steps: I) Initialization, 2) Fitness evaluation, 3) Reproduction, 4) Tournament, and 5) Selection. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 25 Cl C2 ± ± ± Individual X = [L , Ly. ,w ,Cj ,C2 f , , cr^. , =[cr^, f Population Xi = [18.0,6.0,2,1.2X10"’^, 1X10”^^f , = [0.01,0.005,5x 1 0 ~‘\ 5 x 10“'^ f X2 -[1 8 .5 ,6 .1 ,2 ,1 .1 x 1 0 '^ 1 .4 x lO ^ ^ T , <73 = [0.012,0.003,2x 10"’'^,5x 10“^^f = [0 .0 1 , 0 .0 0 1 , 2 xlO “‘\ 4 xlO^‘^ f x^ =[17.95,6.0,1.8,1.0xl0“‘^ 1 .4 x l0 ^ '^ f , Fig. 3.2. Definition of individual and population for a sample combline filter Initialization At the beginning of algorithm, n individuals are generated using a uniform random distribution to seed the initial population. The size of this population will be kept fixed during the a;=(X;,ff;X optimization V ffj- =[(jj (l), cr, ( 2 ), process. Each individual z € {1, 2, ..., fi] whcrc x,. = [x ; (1),X; (2),....,X; ( n ) f (Jj{n)f e is a real-valued 6 object M” and are n-dimensional design parameter and its corresponding strategy parameter, i.e. standard deviation vector. At the initialization stage, the generation count g is set to zero, and the population P will be the collection of all individuals, i.e. P ( g =0)== [ a / ° \ }. The initial population is chosen uniformly in the range of parameter variation, i.e. x^ £ {x XiU)=x ( j ) + [x where Wj ( 0 ,l) is a random number between (J ) - X 0 and 1 ,x }" or U )] MJ (0,1) (3.2) with uniform distribution, and index j means that this random number should be regenerated for every elements of each R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 26 individual. The vectors a^ are usually initialized based on the complexity of the landscape and the effort that is needed for the exploration phase of the process. For example depending on the sensitivity of objective function with respect to each variable, the corresponding variation parameter can be set to a small value between \x —X m i n i1/1000 andlx —x m m j1/10. I ma x { max Fitness Evaluation A fitness function of each individual F(a) - S ( U (jc)) is the scaled version function U (x) through of objective function S. In many problems, the fitness function is the same as objective function, i.e. F = U ( x ) . Reproduction In EP, each parent ( jc ,- ) creates a single offspring ( j c ',<7 ,') based on the mutation, which is the only variation mechanism. The mutation process is defined by a mutation operator m : I I , m(a) - ( jc ', a ' ) . The mutation formula borrowed from ES is x ' U ) = ^ i U ) + ( ^ i U) N J (0,1) ^ ' r - \ ^ { -\ CT,- ( j ) = (T,- ( j ) [ V (0,l)+r V - (0,1)1 J (3.3) j = l , 2 ,...,n where x(j) and a ( j ) are the jih components of the solution vector and the standard deviation vector, respectively. We have found that this mutation gives a better performance compared to the classical mutation in EP [47] where variances are used as the strategy parameters. In (3.3) N(0,1) is a random variable with Gaussian distribution of mean zero and standard deviation one. The index j in N j (0,1) means that the random variable is generated anew for each value of j. The values for scalars r and r ' are defined as [48]: (3.4) , 1 As can be seen from the second equation of (3.3), the self-adaptive mechanism of the strategy parameter causes an evolution for the amount of mutation during the process, so R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 27 moving from exploration of the landscape at the beginning of the search with larger values of mutations, to exploitation phase at the end of the search, by localized search. The global factor exp(r'A^(0,l)) guarantees an overall change of mutation for all individuals, whereas the factor exp(r N j (0 ,1 )) allows for individual changes of the variations. The mutations that violate the box constraint of (3.1) are called lethal mutation and should be reevaluated. At the end of this step, fitness of each new child is assessed by Tournament In the previous stage, each individual (parent) generated a new member (child). In the tournament phase, we put all the parents and children in a common pool with the capacity of (// + // ) , i.e. P( g) u P 'ig ) . Then we start ordering of this population as follows: For each individual in the pool, q opponents are chosen randomly from the total population of the pool, and the fitness of the selected individual is compared with all of the q selected opponents. For each comparison, if the individual’s fitness is better than the opponent’s, it receives a “win”. Therefore, for the best individual in the pool, the maximum “win” score will be q. This can be formulated as follows i = 1, 2 , 2/j, (3.5) if f ;. < F , otherwise where index r - truncate( 2 .//.w( 0 ,l) + l) will be a random integer between l and 2ju, and m(0,1) is random number between 0 and 1 with uniform distribution. In (3.5), wi will be the final score of ith individual. In EP, survival of the best individual is always guaranteed by assigning a maximum score q. This property of selection mechanism is usually called elitism. As the tournament size q is increased, the selection tums into {fj. + /j) -selection which in the limit when q - 2 j i the // best individuals out of the union of parents and children form the next the parents. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 28 Selection Next, the /J. individuals out of the union of the parents and children with highest “win” score, i.e. the largest Wj, are selected to be parents of next generation. The algorithm goes to reproduction phase and the above process from reproduction to selection is repeated. Termination of an evolutionary programming algorithm is usually controlled either by specifying a maximum number of generations gmax for tho running time or by checking for achieving an acceptable level for the objective function, i.e. , ^ \ true, r(p(g)) = { [false, i f g >g max or \ Ui Xj ) \ <e ® \ ^ otherwise (3.6) The following criteria may also be used as a general stopping condition for the algorithm: • Dijf'erence between the best values o f the fitness function at a set o f generations. This condition is less sensitive to the type of the problem and can be formulated as follows: b ^ <£■ (3.7) where g is the actual generation and gQ is an integer number, which is used to indicates how many generations before g will be considered. Moreover, is the best fitness value at generation g and £ is a small positive number. Difference between the different fitness values in the same generation. This criterion is also less sensitive to the type of the problem and can be written as: - h i . — <£ , Ff <F/, (3.8) where F^_j^ is the ^ h best fitness value after F ^ , and gg is the number of individuals with best fitness taken in the same generations, and f is a small positive number. Fig. 3.3 shows the conceptual algorithm of EP based on the above-mentioned formulations [48]. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 29 Algorithm EP g=0; initialize P ( 0 ) = { a / ° > ,- - ,a / > } e / ^ where and evaluate / = R” a,- (x,-,<7 ,-), F ( 0 ) : {F(x/°>), - , ju}; V re {1 , F (x /> )} where F (x /°^) - S(U (x/°^)); while (r(F (g ))? t tme ) do m utate: V ie {1, evaluate : P 'ig ) = •••, , F ); '■ {F(x;(^>), •••, F(x;^^>)} select: F (g +1) = where F(x'^s^) = 5(U(xf«^)); (P{g) u F'(g)); g ^ g +i end while Fig. 3.3. The Conceptual algorithm of EP 3.2 Convergence of EP The analysis of evolutionary programming is given by Fogel [114]-[116] based on the Markov chain analysis and it indicates that the standard evolutionary programming algorithm will converge asymptotically to the best available solution. This is usually written in the form of convergence with probability one, i.e. lg.^co ) - U ] = 0 -1 (3.9) where U* is the global solution of problem (3.1). Although this result is important theoretically, it does not provide an insight to the required number of generations for discovering the global minimum in a finite period. Until now, no theoretical method that can fully analyze the EP in terms of the specific population size and random selection parameters has been developed. Because of this fact, most authors study the convergence behaviors of evolutionary programming (and evolutionary algorithms in general) using empirical methods by applying different functions and different parameters. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 30 An important factor that affects the rate of convergence of the algorithm is the population size. Although it might seem that increasing the size of population will increase the speed of convergence, it should be understood that any increase in the population size will introduce a delay in the evolution process because of extra time spent on the fitness evaluation. This justifies faster convergence of algorithm with medium size population. In fact, when the size of population is very large, the algorithm will not get the chance to spread the genetic properties of the new generation and improve the population quickly. According to no free lunch theorem in optimization [117], the efforts to find the best population size, the best mutation operator, the best number of rivals for tournament, and so on, in the absence of restricting attention to a particular class of problems, are pointless. For an algorithm to perform better than even random search, it must reflect some peculiar aspects of the structure of the problem it tries to solve. As a consequence, it may become less suitable for some other problems. 3.3 Enhancement of E volutionary Progranuning Using Clustering Algorithm Although the conventional EP, and EA in general, are very efficient in solving different nonlinear multidimensional problems, there are two cases, which cannot be handled by them efficiently; 1) When the global minimum lies in a very narrow valley and at the same time there exists a local minimum with wide valley such that the value of objective function is close to value of objective function at global minimum. We call a function with this behavior ''deceptive” or "hard' function since any algorithm that is based on random or deterministic search will have difficulty in locating its global minimum. Opposite of hard function is the easy function in which the global minimum has the widest basin of attraction. Fig. 3.4 shows the general shape of a one-dimensional hard function and easy function. It is obvious that in case of the hard function, the chance of generating a new offspring that falls in the basin of attraction of global minimum is very low. Due to selection pressure (higher chance of survival for the fitter individuals) in the algorithm, more individuals will tend to gather in the wide valley of local minima marked “L” in the Fig. 3.4(a) and this will finally lead to stagnation of the algorithm in the local minima. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 31 Easy function Hard function (a) (b) Fig. 3.4. An example of (a) Hard function and (b) Easy function 2) In many microwave filter optimization problems, we are looking for not only a single global minimum of the objective function but also the other significant local minima, which are within a certain accuracy level in the multimodal functions. These points can also be accepted as the solution and the knowledge of their locations may be required for further analysis. Unfortunately, the current EP algorithm can not keep a stable subpopulation for all the possible minima in the search space and it will converge to a single minimum. In addition to the above problems, the problem of speed of convergence should also be mentioned. The EP algorithm generates subpopulation in the basin of attraction of the minimum point with the widest and deepest valley relatively fast. However decreasing the value of objective function further down this valley requires a large number of generations. An approach called niching technique has been used to improve the diversity and overcome the problem of premature convergence in EA [118]-[121], In natural ecosystems, the finiteness of resources forces the creatures to evolve in such a way that they can fill each ecological niche. In ecology, a niche is viewed as the smallest unit of habitat that is occupied by an organism. The subdivision of environment based on an organism’s role and habitation reduces inter-species competition for environmental resources, and this reduction in competition helps stabilize sub-population formation around different niches in the environment. W e have had successfully applied the niching technique to several filter optimization and parameter extraction problems [4]-[7]. This technique depends on some fixed parameters which should be assigned at the beginning of the algorithm. Although some general rules R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 32 exist for determining the approximate value of these parameters, our investigation shows that they should be set for each unique problem independently and the performance and success of the method may be severely affected if the values of these parameters are not set correctly. Determination of these parameters usually needs knowledge of shape of landscape of optimization and number of local minima, which are not known before optimization and are difficult to calculate accurately. As an alternative approach, we will use clustering algorithm in order to increase the diversity of EP and prevent premature convergence in multimodal optimization problems. The behavior of population and the pattern of its formation in the process of evolutionary programming optimization reveal that as the algorithm proceeds the individuals form clusters of populations at certain stages of evolution [3]. These clusters can easily disappear if there is not enough pressure for forming stable subpopulations. This behavior is illustrated in Fig. 3.5 for a typical 2-D multimodal objective function. In this Figure, the population is superimposed on the contours of objective function. There are two strong local minima with wide basin of attraction and two global minima, Gi and G 2 , with relatively narrow basin of attraction. As can be seen, after 30 generations the peculation has gathered around three major regions, shown in Fig. 3.5-(d), containing the global and local minima. However, the parents and offspring in the subpopulations number 1 and 2 will have smaller chance of survival when competing with the individuals gathered around the local minima with wider basin of attraction. As a result, these subpopulations will eventually become extinct and the local minima Li will attract most of the individuals as shown in Fig. 3.5-(f). Considering the above mentioned behavior of the evolution of population, we can use the cluster analysis approach to identify the subpopulations generated in the process of evolution. This will allow us to perform the necessary investigation in these regions looking for the possible minima. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 33 .. ■W'A ' 'i /A Gi .-•-■A'' ; A A /' A '■. ■, ^ AA \ 'V' ; i'.i \ \ si\A ' A A \\[a A v'4 /4 '—"N. ‘-H i!! A N '"i 'l I / /'-‘A . (b) (a) <. '\& ., ' \ / (c) ^ ^ V, l'\\ \ A 4 -4 . '; 2 (d) % (e) (f) Fig. 3.5. Illustration of convergence of evolutionary programming to a single minimum for a sample multimodal 2-D objective function. The individuals are shown at generation (a) zero, i.e. initial population, (b) 10, (c) 20, (d) 30, (e) 40, and (f) 100. The potential clusters of populations are marked by dashed circles. Two global minima are marked as Gi and G2 , and two local minima are marked as L] and L2 . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 34 Another drawback of the conventional EP is that it may arrive at the same local solution region several times. This means that much of the effort spent on global optimization is the unnecessary reinvestigation of a local solution that has already been found, whereas this effort could have been spent on exploring other areas of the search space and thus increasing the probability of finding a global solution. Cluster analysis can be used to alleviate these two difficulties. Furthermore, as we will show later, the idea of clustering can be helpful when we want to combine EP with local search algorithms. The basic idea here is to let EP form the subpopulations and determine the regions of local minima by cluster analysis and then apply a fast local search from a point within each region and find the local minimum for each region very fast. 3.3.1 Clustering Algorithm Clustering algorithms are used for dividing a finite set of objects or points into subsets so that each object is more similar to objects within its subset than to the objects outside. We can describe clusters as continuous regions appearing in the space having relatively large mass (high density of points), which are separated from other regions by regions having relatively little mass (a low density of points). This definition is usually used for the natural clusters. On the other hand, there is another definition that relies solely on what could be called the “criterion of closeness” . According to this definition, objects in a cluster should be closer to each other than to objects in other clusters. While this definition is usually used for identification of spherical-like clusters, the definition of natural clusters can be used for any shape of clusters. Most evolutionary algorithms produce the spherical-like clusters when applied to the objective functions of the engineering problems, so it is natural to focus on these types of clusters. Clustering can also be considered as a data-reduction technique. As we mentioned, the goal of cluster analysis is to identify a smaller number of groups such that elements belonging to a given group are in some sense more similar to each other than to elements belonging to other groups. Thus, cluster analysis attempts to reduce the information on the whole set of p objects to information about, say, 5 subgroups, where s<p. We will use density clustering [122] for finding the clusters of population during the optimization process. The goal is to construct a clustering procedure that gives a division into subsets X . , i = 1,2,...,5' for sets X with basically hyper-spherical and well-separated R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 35 clusters. This is done by growing hyper-spheres around best individual points (seed points) until the point density in those hyper-spheres starts to become smaller than the average point density of the whole space containing all points. In this method, we use the Euclidean distance as the dissimilarity measure. In optimizing circuits or empirical models, all optimization variables are not in the same range and need to be normalized before performing the cluster analysis. The conventional standardization procedure normalizes the data to zero mean and unit standard deviation, X: { j ) - m , Xi ( ; ) = ------------- - , i =1,2,...,/?; j = 1,2,...,n (3.10) where it is assumed that there are p observations (typically p ~2ju in EP) on n variables. Also nij and (Tj are the mean and standard deviation of the jth variable. However, our investigations show that this normalization can deteriorate the diversity of the optimization algorithm when combined with the evolutionary programming. The main reason is that this normalization does not take the size of search space into account. As a result, each time the clustering is applied, the algorithm tries to find new clusters even if the individual points are very close to each other with respect to the size of search space. We call this zooming ejfect since the algorithm tries to zoom into each region and find new clusters. This property is very deteriorating to the main objective of using the cluster analysis to identify the real clusters. In order to alleviate this problem, we use another standardization scheme that takes into account the size of search space as well. Each individual is normalized before the clustering as follows i^ l,2 ,...,p where x^^^ij) and J =l , 2, . . . , « (3.11) are the upper and lower bonds of the jth variable. Using this normalization, each variable will be transformed to a dimensionless region between zero and one. As a result, when the points are close in the original domain, they will remain close in the transformed domain as well and no false clusters will be identified. It is also necessary to add the normalized value of objective function to the vector of optimization variable x as its last element in order to prevent closely spaced points with R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 36 large difference in fitness values to be clustered together. Fig. 3.6 shows how the cluster analysis may result in wrong classification of data if value of objective function (fitness value) is not included in the data matrix. For filter optimization and parameter extraction problems, the upper and lower limits of objective function value used in (3.11) are determined by the limits of each individual terms of objective function. These terms consists of some known functions such as scattering parameters which their upper and lower limits can be easily estimated. --^ (b) (a) Fig. 3.6. Two equal distance individuals with (a) large difference, and (b) small difference in fitness value. The two points in (a) belong to different regions and should not be clustered together. In density clustering, we start by growing a cluster from a center (seed point), until the stopping condition for the growth is achieved. This condition should be set carefully because if the threshold size of the cluster is very large, all points will be assigned to a single cluster, and if it is very small, each point will form an isolated cluster. W e define the density of the reference region E containing all individuals and determine the stopping rule for the growth of each cluster. Let’s the volume of E be V. Then the point density p in E is p^pIV (3.12) where V is the volume o f E . The point density in a region determined by the points in a cluster is greater than p . The cluster is grown by enlarging the region around the seed point xq as long as the point density remains greater than p . For EP, the initial seed point will be the point with the best fitness. The consecutive seed points will be the points with best fitness among all the remaining unclassified points. The region will be enlarged successively until the point density in the region remains at least p . These natural R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 37 regions are concentric hyper-spheres. We need to determine the radius of the hyper sphere having a point density p containing i points. Let 5 (n , r;. , x ) b e the hyper-sphere with radius r,- centered at x and W(n, r,-) be the volume of the sphere in an n- dimensional space. Then we will have p W{n,r^) = i, i = l, 2, .... (3.13) Using the formula for the volume of the n-dimensional hyper-sphere defined as W{n, r) = ml _m+l/2 7T r (3.14) 2m+l T{m + 3I2) the radius can be calculated as follows r--M {n,p) f'", i = l, 2, .... (3.15) where ; n= 2m M{n,p) = T im + 212) (3.16) 1/n ; n - 2m +1 p n', m+l / 2 Accuracy of the above cluster analysis depends on the choice of reference region E and especially the volume of E. One choice is the smallest coordinate-oriented body containing a given proportion of all the points. In [122], this volume for a hyper-ellipse body, which its main axes coincide with the principle axes of the data and contains at least 90 percent of the points, is calculated as the product of the square roots of the largest eigenvalues of the covariance matrix of the data points, i.e.. (3.17) i=l where the eigenvalues are sorted in a descending order, i.e. A^>Aj >...>A^-. The constant factor 4" is found empirically to satisfy a wide range of data distributions. We use the standard principle component analysis to find these eigenvalues [136]. If some of R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 38 the eigenvalues are zero or very small, the points will actually be contained in an n space {n < n ) and as a result, we only use the first n eigenvalues that are nonzero or very small. The criterion for inclusion or exclusion of small eigenvalues in the reference volume calculation is found by progressively checking the size of the threshold distance r, in (3.15) as the dimension of space is increased. If the points are approximately contained in a n'-space, we will have r,(1) < T;(2) <... < Tj{ n) > Tj{n +1) (3.18) The distribution of population for a sample 2-D case is illustrated in Fig. 3.7-(a). The smallest ellipse that can contain all the points is shown in this Figure with its axes coinciding with the principle axes of the data matrix. The growth of concentric hyper sphere (in this case circles) from the points with highest fitness is pictured in Fig. 3.7(b). In this Figure, the radius r. represent the radius of the circle that should contain at least i points in order to have a point density which is greater or equal to the average point density of the whole ellipse. The circles are successively enlarged until there is no point between two consecutive circles. For example in cluster 1, starting from r,, there are two points between circles with radii q and radius . So the next circle is created with . Since there are still 5 points between circles with radii and , the 4'’*circle is created. However, there is no point between the circles with radii Tj and r^. As a result, all 9 points within the 3^“^ circle will belong to cluster number 1. The same process is repeated for the rest of unclassified points. In Fig. 3.7-(b) only one additional cluster is identified. Note that we only accept the clusters if they contain at least three points. In general, highly populated clusters have better chance of containing a local niche. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 39 ta'i Cluster 2 Cluster 1 ■ (h i Fig. 3,7. Illustration of clustering algorithm for a sample 2-D population, (a) The principle component analysis finds the volume of smallest hyper-ellipse containing all the points; (b) hyper-spheres are grown from the points with highest fitness until there is no point between two consecutive spheres. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 40 Based on the threshold distance r,-, we can now develop the algorithm that classifies the points in a cluster with a seed point Xq . Clustering Algorithm Step 1- Start from x^ and S{n, rj,Xo) . Step 1-Grow the hyper-sphere successive spheres S (n, until there is no point between the surfaces of r, ,Xq) S(n, r,-_^,,XQ) . Suppose this happens at i=k. Step 2- All points inside the sphere S ( n , r^. ,X q) belong to the cluster with x^ as its seed point. Step 3 -Repeat for unclassified points with a new seed point. The above algorithm can also be formulated in a more practical way as follows. Clustering Algorithm Version 2 Step 2- Start from x^ and S ( n , rj,Xo) . Step 3-Compute the distances between seed point x^ and all the other individuals in the population, i.e. X j , x , , . . . , x ^ . Step 4- Classify points with distances to Xq between successive radii r. starting with 0 and , until a pair and distance falling in the region S (n, rj. ,X q) belong to the cluster with Xg and , is found such that there is no . All points inside the sphere its seed point. Step 5- Repeat for unclassified points with a new seed point. where the Euclidean distance d between two individuals x. = [X;(l),x,.(2),....,x.(n)] and Xj = [Xj{l),Xj{2),....,Xj{n)] is calculated as (3-19) i=i The clustering procedure is repeated until all points have been classified. The total number of distance calculations N j is R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 41 N j =(ju-l) s - ' ^ n ^ k=\ (s-k) (3.20) where nf, is the size of population in kth cluster and s is the number of clusters. Proof of equation (3.20): If the distance between each seed point in a cluster and all the points in other clusters had been calculated independently, the total number of calculations would have been (ju - l ) s . However, there will be some distance calculations that should be subtracted from this value. When the first cluster is formed, it will not be necessary for the seed points of clusters 2 through s to calculate the distance between themselves and any member of cluster one. This reduces the number of distance calculations with the amount of ( 5 -l ) n^ where is the number of points in the first cluster. For the other clusters, the same observation will result in subtraction of similar terms such as (s - 2)^^, (s - 3 )n ., etc. Subtracting all these terms from the term {ju -1)5 will result the equation (3.20). o When s = f i , will be the same as the total number of distance computations as when the whole distance matrix is used, i.e. ju{ / 2 - l ) / 2 . In practice, the numbers of clusters are much less than the size of population and the distance calculations will not be a huge computational overhead. 3.3.2 Clustering Period During the process of EP optimization, clustering is performed every certain number of generations, which is called clustering period. The small clustering period, will cause the algorithm to act more like a random search. The EP needs time to explore the space and form clusters, so any attempt to perform clustering before the actual formation of clusters will be against the potential EP benefits. Moreover, as we will see later, when the method is combined with deterministic search, the small clustering period at early stages of the optimization process may lead to loosing the global solution at all. On the other hand, a large clustering period may lead to the extinction of weak but possibly important clusters. This is due to the fact that in EP, the strong clusters that form around the wide basins of attractions have more chance of survival in a long term. However, the algorithm will still be able to explore the search domain for the global regions eventually. As a result, the main draw back of large clustering period is the waste of computational resources. The R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 42 best value for the clustering period in general depends on the specific properties of the landscape of optimization. Our studies have shown that for most filter optimization problems with the number of unknown variables ranging from 10 to 50, the clustering periods of 10 to 100 will be suitable choices. In general, the clustering period should be increased with the number of optimization variables. For large clustering periods, the size of clusters may become very small. One solution in these cases is to perform the clustering on union of parents and offspring. 3.4 Hybridization of the EP It was mentioned in Section 2.1.3 that the natural choice for an efficient global optimization is a hybrid algorithm that combines the complementary properties of stochastic and deterministic search methods. The simplest implementation of this hybridization process is to allow the stochastic search explore the search space and then switch to a local search for finding the optimum solution fast. Nevertheless, one of the major challenges in this hybridization approach is the lack of information on making the decision in selecting the most appropriate time to make the switch between the two optimization methods. The cluster analysis approach that was proposed in last Section for improving the robustness of the EP provides very good information that can be used for combining the algorithm with a local search as efficient as possible. The clusters of populations potentially form around regions of local or global minima. It is therefore reasonable to consider a fast local search in these regions for finding the exact minima. This can be done, for example, by a gradient-based optimization such as steepest-descent or quasiNewton method. The point with best fitness in each cluster is usually a good candidate for the starting point of each local search. Depending on the landscape of optimization, the local search from center of each cluster may end up in either single or multiple solutions. This is graphically depicted in Fig. 3.8. If the local search does not provide an acceptable solution, the cluster is marked as a forbidden zone and no newborn offspring is allowed in these regions. If the clustering period is large enough, the probability that the global solution resides in a forbidden zone will be very small. A similar local search will be performed for the remaining clusters as well. After all clusters are checked with the local search, a new cycle of EP algorithm R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 43 will start. The populations that existed inside the previous clusters will be moved outside all the forbidden zones by randomly generating points. Since no new point will be allowed inside the forbidden zones during the EP process, the reinvestigation of regions of space that did not provide any solution is prevented, hence greatly improving the robustness of the optimization as well as increasing the speed. Cluster 2 z' ( Cluster 2 / \ / *) \ / ■ -5 m_ _ Cluster 1 (b) m ^ Cluster 2 \ s ’ I Cluster 1 (a) Cluster 2 / \ \ \ . -il m / m / \ S. ^ Cluster 1 Cluster 1 (J) Fig. 3.8. Illustration of local search after cluster analysis. The local search may converge to (a) separate solutions inside each cluster, (b) single point inside one cluster, or (c) single point outside the identified clusters, (d) separate solutions outside each cluster. It was mentioned in the previous Section that the small clustering period may cause the global solution to be overlooked. The main reason is that when the clustering period is small, during the early generations when the population distribution is still not settled in different landscape niches, the clustering algorithm may identify a potentially large region as one cluster. This region may be large enough to cover several local minima and R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 44 even the global solution. Because of this, the deterministic search has very small chance of capturing the global solution within this early formed cluster. As a result, the cluster will be marked as forbidden and the region of global solution will be missed. This is why the clustering period should be chosen large enough especially during the early generations. The flowchart of the hybrid evolutionary programming is shown in Fig. 3.9. Evolutionary Programming No Clustering ^ leriod reached? Clustering Quasi-Newton search Yes No Converged? (Terminate?) Add the cluster to the list of forbidden zones Global solution Move the points outside the forbidden zone using uniform distribution No All clusters tested? Yes Fig. 3.9. The flowchart of hybrid evolutionary programming algorithm. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 45 3.5 Additional Enhancements for M icrow ave Filters In this section we address several modifications to the developed algorithm in order to make it more suitable for microwave filters optimization problems. In general, improving the efficiency of the optimization algorithm for a given RF and microwave filter can be achieved by incorporating knowledge from the specific problem into the optimization routine. This problem specific knowledge can be provided to the algorithm in several ways. In the following, we mention a few of these methods. 1) Parameter constraints. One of the simplest ways to increase the efficiency of any optimization method is to reduce the size of search space. This can be done by limiting the range of optimization variables for the specific problem. In microwave filter optimization, either when the goal is to optimize a lump element network model of the filter or optimizing a physics-based model o f the filter, we can usually find reasonable lower and upper bonds for the optimization variables. In addition, we may have more complex constraints in form of linear or nonlinear relationships between the optimization variables derived from the physics of the filter. For example, we know that in a parallel coupled line microstrip filter, the coupling between the first two resonators is always stronger than the coupling between the second and the third resonator. This property is equivalent of having a simple constraint such as s^ > s^, where .Sj and ^ 2 are spacing between coupled lines. Adding these constraints will reduce the size of search space even further. It should be noted that while incorporating these constraints in a gradient based search method may be challenging, the evolutionary programming makes this task very easy. In EP, it is always possible to check the validity of a newborn offspring against all existing constraints before the objective function calculation is performed. The points that cannot satisfy the given constraints are simply destroyed and a new point is generated with no additional major computational cost. 2) Symmetry. Many filter structures possess some sort of symmetry in their models. Taking advantage of this information can help us to reduce the number of optimization variables hence lowering the computational cost of optimization. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 46 3) Initial solution. Another issue in hybrid EP is the choice of initial solution. Generally, random search methods do not rely on the initial estimate of the solution. However, in many practical cases, such as filter optimization, we have access to a rough estimate of the initial solution. In order to take advantage of this priori knowledge, the EP algorithm is modified to accept the initial solution. At the beginning of the program, this initial solution will be added to the pool of initial parents. If the initial solution is not valid, the clustering approach will find the corresponding zone and continue to explore the remaining regions. In order to provide a higher chance of survival, the initial point does not compete in the tournament during the first few generations. 3.6 Software Design Based on the proposed algorithm in this Chapter, a modular subroutine was developed in Fortran 90 programming language. The unique features o f Fortran 90, such as comprehensive intrinsic functions for matrix manipulations and semi-object-oriented programming capabilities allow us to develop the software in a very efficient manner. The developed subroutine is called HYBRID_EP and is defined as follows SUBROUTINF HYBRID_EP(n, xm in ,xm a x,o b jm in , objmax, x s o l , e r r ) where • n is the number of optimization variables • xmin{n) and xm ax{n) are the lower and upper bond vectors • objmin and objmax are the estimated values for lower and upper limits of the objective function • xsol(n) is the initial estimate of the solution on entry and the final solution on exit • err is the value of objective function at final solution The initial parameters of the algorithm are provided in a separate input initialization file, “ep.ini" containing: • population size • tournament size R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 47 • clustering period • maximum number of generations allowed • error threshold for convergence check • number of generations for stopping criteria calculation, i.e. go in (3.7). The objective function is provided by user as a modular subroutine called F U N C T {n,x,obj) where n is the number of optimization variables, x is the ndimensional vector of optimization variables, and obj is the value of objective function at point X . All the linear and nonlinear constraints on the optimization variables are included in a separate subroutine called CONSTRAINT(x, L). If the input optimization vector does not satisfy the constraints, the parameter L will be flagged and the input variable is rejected. Then a new vector will be generated and passed to this subroutine for checking its validity. The following external subroutine from the standard NAG library [136] are also used in the main module of the program, • Subroutine G03AAF, perform the principle component analysis of the population. • Subroutine MOICAF, sorts the eigenvalues of the covariance matrix of the population. • Subroutine E04JAF, performs the minimization of the objective function using a quasi-Newton algorithm. The gradient o f objective function is computed using finite difference approximation. • Subroutine G05DDF, generates pseudo-random real number taken from a Gaussian distribution. • Subroutine G05CAF, generates pseudo-random real number taken from a uniform distribution. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 48 3.7 Examples In this section we examine the utilities of the developed hybrid EP algorithm in solving several numerical optimization problems. The first sets of examples eonsist of some wellknown mathematical test functions. The process of function minimization will be shown graphically for a 2-D function and compared with the EP alone. The performance of the hybrid EP will also be compared with several optimization methods in solving several test problems. Three practical examples including the optimization of complex cross coupled resonator filters and diplexers, and computer aided diagnosis of a multiplecoupled cavity filter will be presented as well. 3.7.1 Test Problems The first example is a 2-D function minimization. This example is ehosen to visually show the process of a hybrid EP optimization. The function is defined as sm[7i{x 1- -2)]sin[.;r(y -2 ) ] 7T^{x - 2 5 "2 + (x - 7 ) ^ + 2 ( y - i f (3.21) )(y - 2 ) The 3D plot of this function is shown in Fig. 3.10. This function has a global minimum at (2,2) with a very narrow basin of attraction and a very strong local minimum at (7,7) with wide basin of attraction. The value of function is zero at global minimum and 2 at local minimum. 120 - 100 80 - 12 Fig. 3.10. The 3D plot of test function in (3.21). R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 49 This kind of function minimization is usually known as needle-in-a-haystack problem and falls under the category of hard functions as explained in Section 3.3 which is a difficult global optimization problem. We first try to optimize this function using the conventional EP. The size of population jU is set to 15 and the tournament size is 4. The distribution of populations after 20 generations for a sample trial is shown in Fig. 3.11. As can be seen, the EP has converged to the local minimum and the population was not able to gather around the global minimum in the future generations either. Next, the hybrid EP is used for minimization of the same function. A fixed clustering period of 3 is used and the population size is the same as before. Fig. 3.12 shows the population distribution after 20 generations for a sample trial. The last two identified clusters are marked by 1 and 2. The global and local minima are marked by G and L. As can be seen in Fig. 3.12, all individuals in the last generation are outside the 19 marked forbidden zones. This has enabled the algorithm to generate points that have higher chances of getting closer to the global solution. The algorithm successfully found the global solution at the last generation very fast using a quasi-Newton local search on the identified cluster around the global minimum. Fig. 3.11. Contours of function defined in (3.21) and the distribution of populations after 20 generations using conventional EP. Individuals are marked by X . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 50 14 12 10 8 6 4 2 0 ■2 0 2 4 6 8 10 12 14 Fig. 3.12. Contours of function defined in (3.21) and the distribution of population after 20 generations using hybrid EP. All identified clusters are shown by large circles. The population at 19* generation is shown by x and new parents are shown by o. The global and local minima are shown by G and L respectively. Second group o f functions that we report are several popular multidimensional test functions that are constructed heuristically and used by many authors for testing the capabilities of the global optimization methods. These functions usually consist o f a number o f combinations o f fiinctions whose local minima are known. A list o f these functions along with their specifications is given below: GP - Goldstein-Price (n—2). f (x ) — 1+ (x ] + X2 + 1)^ (19 —14x j + 3x j —14x 2 + 6x jX2 + 3x 2 ) 30 + (2 x i - 3x 2 ( 1 8 - 32x, + 12x f + 48 x 3 -36xjX2 + 2 1 x \') - 2 (3.22) <Xj < 2 , i = 1 , 2 This function has a global minimum at (0,-1) and the value o f ftmction is 3. There are four local minima in the minimization region. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 51 B R - Branin (n=2). / (jc) = ( 1 - y 1 2 x 2 + — sin(4;?rx2)-Xj ^ + X2 - ^ s i n ( 2 ;rx i) (3.23) V - 1 0 <x,. < 10 , i = 1 ,2 The global minimum of the function is zero and it is reached at five points in the region: ( 1, 0), (0.148696, 0.402086), (0.402537, 0.287408), (1.59746, -0.287408), (1.85130, -0.402086) S H -S h e k e l (n=4). 10 /u )= -Z ;= 1 1 (x - A,. )(X -, 0 < x , <10, i = 1,2,3,4 - A; ) (3.24) +C; where A = '4 4 4 4 ■ o .r 1 1 1 1 0.2 8 8 8 8 0.2 6 6 6 6 0.4 3 7 3 7 function has 10 c = 0.4 0.6 2 9 2 9 5 5 3 3 0.3 8 1 8 1 0.7 6 2 6 2 0.5 3.6 7 3.6^ 0.5 J This , local minima and one global minimum (4.00075,4.00059,3.99966,3.99951). The global minimum of function is -10.5364. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. at 52 GR - Griewank (n=10). 10 f (x )- -^ 2 — ^ 4 0 0 0 0 n Ffcos f ^ '' ’■ +1, - 6 0 0 < X : <600, j =1,2,...,10 (3.25) f i The function has a global minimum f=0 at x=0 and several thousands local minima. There are four suboptimal minima f ~ 0.0074 at x ~ (± 7r, ± n 4 2 ,0,0,0,0,0,0,0, O f . Results The methods we used for minimization of the above functions are random search algorithm (RSA) [123], genetic algorithm (GA), single-linkage method (SLM) [124][126], and hybrid EP, i.e. the new algorithm. The efficiency of these algorithms in general may depend on some parameters that are usually chosen heuristically. The stopping condition for each algorithm is based on (3.7) with £ = 1x10“^. For each algorithm, the reported result is based on the average of 10 trials. The population size for all cases is set to 20. The cross-over and mutation probability in GA are 0.5 and 0.02 respectively. The tournament size for both GA and EP is equal to 6 . The clustering period for the functions GP, BR, and SH is 5. For function GR, the clustering period is 10. No initial point is provided for the hybrid EP to make the comparison between the methods more realistic. The number of function evaluations during the optimization process for each algorithm is shown in Table 3.2. As can be seen, thanks to hybridization of EP with the Quasi-Newton search method and also prohibiting the algorithm from the redundant search, our method is showing huge decrease in the number of function evaluations compared to other algorithms. This advantage is more evident in case of more complex functions such as SH (n=4) and GR (n=10). Moreover, while other algorithms were only able to find either one local or a global minimum, the hybrid EP was able to locate the major local minima in addition to the global minimum as well. Among different methods the random search algorithm has the poorest performance. The conventional genetic algorithm was not able to locate the global minima in any trials in case of functions SH and GR. The optimization strategy used in the single-linkage method is the closest to the clustering and local search technique we use in hybrid EP. However, since it uses uniform distribution for generating sample points in the space, its performance is poorer compared to the evolutionary programming. It also does not use the concept of forbidden zones as we introduced in the previous Section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 Each algorithm has some auxiliary computations in addition to the function evaluations. In order to include the effect of these overhead computations in our comparison, the absolute CPU time for each function minimization is also reported in Table 3.3. The CPU time in this Table is based on a Pentium 4 CPU running at 1.7GHz. Table 3.2. Number of function evaluations by different algorithms in solving test problems Method GP BR SH GR RSA 2500 1800 4400 12800 GA 1350 1110 2877* 10339* SLM 1055 955 3023 9025 Hybrid EP 703 632 2045 3299 * Only local minimum was found iMe 3.3. Total CPU time in seconds for each algorithm in solving test pro Method GP BR SH GR RSA 2.3 1.7 4.3 13.4 GA 1 .6 1.4 3* 10.7* SLM 1 .1 1 3.4 9.5 Hybrid EP 1 0.9 2.9 4.3 Only local minimum was found Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 3.7.2 Cross-Coupled R esonator Filter Synthesis A. Introduction Cross-coupled resonator filters, also known as multiple-coupled resonator filters, are widely used in many RF and microwave communication systems. These filters may be realized using waveguide cavity resonators, coaxial resonators or dielectric resonators. Fig. 3.13 shows a multiple-coupled coaxial resonator filter with its narrowband equivalent circuit model. Top view (door not shown) Side view (A-A’) A’ -■ (6 ; ('5} ---1.....J i (a) '3N '2 3 L/2 L/2 L/2 L/2 Rl M'2, N IN (b) Fig. 3.13. (a) A multiple-coupled coaxial cavity filter and (b) its equivalent circuit model. Since the introduction of these filters in early 70’s [127], several methods have been developed for synthesis of these filters [128]-[134]. While the success of conventional analytical synthesis approaches is limited to certain types of responses and topologies, the optimization-based synthesis approach may be considered as a general methodology that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 can be used for synthesis of any type of response/topology. However, one of the challenges of using optimization approach for synthesis of such filters is the problem of trapping into local minima which may be very severe depending on the type of response and complexity of the problem. The capability of the developed hybrid EP algorithm in finding the global solution of a complex objective function in a timely manner makes it a very good candidate for addressing this problem. In the following, after driving the governing equations for the filter prototype, we introduce the formulation of the problem and define the objective function. B. Deriving the Governing Equations The first step is to drive the equations used for computation of scattering parameters of the two-port prototype circuit of the filter in terms of its components [134]. Denoting the center frequency of the filter by fo and its corresponding angular frequency by tq, the loop equations for the equivalent circuit of the filter shown in Fig. 3.13-b are \[R ] + j Z , { — - ^ ) [ I ] + ja> [ M j l [ J ] ^ [ E ] [ 0 ), (3.26) J CO where [i?] is a matrix with all its elements equal to zero except R]i=Rs and R m - R i, i-e. the source and load impedances. Matrix [M is the symmetric coupling matrix, and i^ ] represents the currents and [^Jivxiv is the identity matrix. Vector [ j f =[/] the excitation vector is shown by [£]' =[E 0 0 ] '. The two remaining parameters are the internal impedance of resonators Zq =^L!C , and (O^ defined by co^ = 1 / - / l c . For narrowband filters, we can use the approximation co[M] = coq[M] . Denoting the bandwidth of the bandpass filter by BW, and multiplying (3.26) by fQ l{BW .Zo), we will have [ /o { Z qB W roi I -^0 BW / I Q)q (O /o Z qB W Recalling the band-pass to lowpass transformation, i.e. rn_ j -^0 Z qBW B W .{colcOq - co^ /( o) equation (3.27) can be written in terms of normalized angular frequency A in the lowpass as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 { [i?] + j i [ / ] + ; [ M ] } [ 7 ] - [ £ ] (3.28) =[A] where Z^B W [R], [ M] = 0 )^ - g - [M] ^Z^BW (3.29) Z qBW are the normalized components and excitation values. In equation (3.28), the lowpass frequencies A - I and A - - I correspond to the upper and lower bandpass frequencies/h a n d fi respectively where ■The scattering parameters of the filter at lowpass can now be written as 21 ~ ~ ]/V1 (3 30) 11 Note that since matrix [A] in (3.28) can be written as [A] = j { A[ I ] - [ Z] ) where [Z ] = j [ R ] - [ M ] the matrix inversion in (3.30) can be easily carried out using the Cayley-Hamilton theorem [133]. C. Formulation o f Optimization Problem Our approach in formulating the optimization problem is similar to the one used by Amari [134] and Atia et al. [130]. The desired response of the filter is synthesized using general Chebyshev approximation [131]. The synthesized reflection and transmission functions are written in terms of their corresponding zeros, Zi and p, as , 5 E^{s) 5 E , is) where £• is a scaling constant defined by e = loss level in dB. In (3.31), 5 (5 ) = _ o d i L = ^ 2 i----------eE^is) eE,{s) (3.31) 1 — and RL is the prescribed return ^ 1 0 «r/ i o _ i , = jco is the complex frequency variable, N is the degree of filter and N ' < N - 2 is the number of finite transmission zeros. For lossless filters, the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 poles of the reflection and transmission functions are related to the zeros through the following identity is) = is) + (s) (3.32) which is derived from relation s f i + s h = 1. Since the responses of the prototype circuit in Fig. 3.13-b computed using (3.30) are rational functions of degree N, they can be uniquely specified by the location of their reflection and transmission zeros and the scaling factor £. Based on this observation, the objective function for the synthesis of the prototype circuit is given by £ '= Z |^ ,.(a ) f + Z h , K ) f .•=1 ;=i + ( | S i , ( r y - - 1 )| - (3.33) ^ ^ i ) | _ 1 0 -^^/ 2 o where S „ and 5 21 are computed from the analysis of the prototype circuit in Fig. 3.13-b using (3.30). In (3.33), co, = - j z , and = - ) p ,.. The advantage of using this objective function is that it reduces the required number of circuit analysis compared to the objective functions like (2.7) that use the upper and lower specifications. The drawback however is that it produces more local minima in the landscape of optimization and consequently increasing the chance of trapping into a local minimum [134]. Due to this fact, it is imperative to use a global optimization method for solving these problems. D. Results Here, we present the results for the synthesis of a sixth-degree folded canonical bandpass filter for GSM900 base station having a general Chebyshev response with center frequency of 902.5 MHz, bandwidth of 25 MHz and retum-loss level of 25 dB. Two prescribed transmission zeros at 885 MHz and 920 MHz are used to provide the required out of band rejection. The lowpass equivalent of these transmission zeros, , are located at ±1.4. The calculated reflection zeros, O), , at lowpass are ±0.287478, ±0.748611, and ±0.973638. The coupling matrix that should be synthesized for this problem consists of five adjacent couplings and one cross-coupling between resonator and 5 as shown below: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 58 M = 0 0 0 0 z, 0 0 Mb 0 0 0 M, 0 M, 0 0 z, 0 M, 0 M 0 0 0 0 Ms 0 0 M, M, 0 0 Mj 0 0 (3.34) Due to the symmetry of the folded structure, we can reduce the number o f optimization variables. However, in order to test the capability of the algorithm at this stage, we do not use any antidiagonal symmetry for the coupling matrix and allow the algorithm to verify this symmetry by its own investigation. As a result, the number of optimization variables will be 8 (five adjacent couplings, one cross-coupling, and two normalized source and load resistances). The size of population for this problem is chosen to be 20 (i.e. ju- 2 0 ) and the total number of opponents in the tournament is set to q=5. The clustering period is set to 12 generations. The lower and upper bonds for the optimization variables are 0 < M ■< 2 , i =1,..., 5 (3.35) - 0 . 5 < X j <0, 0.1< R ^ < 2 , 0.1 <2 The negative cross-coupling will be necessary for producing two transmission zeros at lower and upper frequency bands. The initial point used for the hybrid EP is chosen as the conventional M j = M j =0.8233, M 2 Chebyshev response with = M ^ =0.6038, M ^ =0.5778 . The response of filter at the initial point is shown in Fig. 3.14 along with the synthesized goal response. Starting from this initial solution, after 12 generations, the hybrid EP algorithm produced the population with its best response shown in Fig. 3.15. At this point, the algorithm identified three clusters and started the local search from center of each clusters. The global solution was successfully found after 20 iterations. The response of filter at global solution is shown in Fig. 3.16. As can be seen, the algorithm has successfully captured transmission and reflection zeros and the whole response matches with the goal response perfectly. In average, the optimization process took less than 17 seconds on a Pentium4 1.7 GHz CPU. The algorithm also found at least two major minima. The distribution of population shown in Fig. 3.18 shows the regions of global and local minima. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 normalized coupling matrix elements for the global solution and the major local minima are shown in Table 3.4. The response of filter at the local minimum is also shown in Fig. 3.17. As can be seen in this Figure, locations of transmission zeros and four reflection zeros are very close to the goal response, however, the whole response deviates from the goal response dramatically. This observation confirms that the computed point is indeed a local minimum considering the behavior of objective function defined in (3.33). In order to study the effect of initial point on the convergence of algorithm, a second optimization is performed starting from another initial point. This time, all optimization variables including the normalized coupling matrix elements and input/output resistances are set to 1 at the initial stage. Other parameters of optimization are similar to the previous trial. The response of filter at this initial point is shown in Fig. 3.19-(a). This response is obviously very far from the goal response as seen in this Figure. Starting from the second initial solution, the optimization algorithm successfully managed to find the global solution. The response of filter after 40 generations, when the first cluster of populations was found, is shown in Fig. 3.19-(b). Due to the distance between the second initial point and the global solution, the algorithm has spend more time on forming the populations around the region of global minimum compared to the first initial solution. The response of optimized filter at global solution matches with the goal response perfectly as shown in Fig. 3.19-(c). Compared to the hybrid EP, the conventional EP with the same population size failed to converge into global minimum even after 500 generations and was trapped into the local minimum. The gradient search method with Chebyshev response as starting point also trapped into the local minimum. We also tested the sequential quadratic programming (SQP) [135]-[137], and multi-start optimization approach in solving this problem. The same Chebyshev response was used as the initial solution. It took the SQP algorithm 27 seconds on the same CPU to find the global solution, which is about 60% slower than the hybrid EP. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 -10 |S „!g o ai -20 -30 -40 -50 -60 -70 -80 840 860 880 900 920 340 980 Frequency (MHz) Fig. 3.14. The synthesized goal response and the circuit response of six-pole filter at first initial point with Chebyshev response. |S^ J1generation 12 -10 iS ,,|g o a l -20 -40 -50 -60 -70 840 860 880 900 920 940 960 Frequency (MHz) Fig. 3.15. The goal response and the circuit response of six-pole filter after 12 generations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 -10 -20 -30 •40 -50 -70 -80^ 840 880 900 920 Frequency (MHz) Fig. 3.16. The goal response and the response of six-pole filter at final global solution. The two responses are indistinguishable. — jS ^ il local rran — |S^^| local min 1^211goal tS^^lQoal -10 -20 -30 ■50 -60 -70 -80 840 860 380 900 920 940 960 Frequency (MHz) Fig. 3.17. The goal response and the response of six-pole filter at a local minimum. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 CM I 0 .tjc a l CO * m'tn. , 11 r: m in. iL o c a l im i. '"1 12 tm M, Fig. 3.18. Distribution of population in the search domain for coupling matrix elements M^, M j, and cross-coupling. The clusters formed around global and local minima are shown separately. The individuals are shown by symbol x. Table 3.4. Optimization variables including coupling matrix elements and source/load impedances at global and local minima for Parameter Global minimum Local minimum Ml 0.9200932 0.99624623 0.5988588 0.56864426 M, 0.7542121 1.08819454 M, 0.5988588 0.56822462 M, 0.9200932 0.99648671 X, -0.1939066 -0.45530240 1.19427 1.210398 x 1 0 ^’ 2.53x10’^ Final error (U) 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 — jSj^j generation 40 |S,^t generation 40 lS2iigoai -10 IS^^I goal -50 640 920 940 860 840 940 F req u e n cy (MHz) F re q u e n c y (MHz) (a) (b) -10 jS^,|goa} -20 -30 -60 -70 -80 840 ) 900 920 i 940 Frequency (MHz) (c) Fig. 3.19. The synthesized goal response and the circuit response of six-pole filter at (a) second initial point, (b) after 40 generation, and (c) the global solution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 960 64 3.7.3 Cross-Coupled Resonator Diplexer Optimization A. Introduction Our next example is the synthesis of a diplexer for DCS 1800 mobile base station using an optimization approach. The diplexer consists of two ninth degree cross-coupled coaxial cavity resonator filters and a common resonator for connecting two filters to the antenna port. Physical geometry and the equivalent circuit model of diplexer are shown in Fig. 3.20. These diplexers are widely employed in the mobile base stations. m RX RXTTX sttj J IX IL(a) '1N L/2 L/2 L/2 LJ2 M', L72 L/2 L72 L72 M', (b) Fig. 3.20. (a) Top view of the DCS 1800 diplexer, and (b) its equivalent circuit model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 The design approach for these diplexers usually consists of several steps including the synthesis of each channel filter separately and then optimizing the whole circuit to achieve the desired response for the circuit model [142]. The use of optimization approach for these diplexers (and almost all types of multiplexers) is inevitable due to the complexity of the circuit and large influence of each channel filter on the response of other filter. This is especially true when the receive and transmit frequency band become very close, i.e. contiguous band, which causes the design to become very sensitive to small variation of the coupling matrices and common cavity couplings. Due to the complexity of this circuit and the large number of optimization variables with high degree of correlations, the landscape of optimization is very multimodal and use of most conventional optimization algorithms will result in trapping into a local minimum. However, it is shown that the hybrid EP can successfully optimize the complete circuit and provide the global solution very efficiently. In the following, after driving the governing equations for the diplexer prototype circuit, we introduce the formulation of the problem and define the objective function. B. Deriving the Governing Equations The derivation of governing equations for the diplexer of Fig. 3.20 is very similar to the approach used for the cross-coupled resonator filter. The loop equations for the diplexer equivalent circuit shown in Fig. 3.20-b are + ]■ + r 1 j COM 0 ••• 0 j coM 0 0 E *5 ja C c 0 h jcoM,^ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ••• h 0 0 0 h 0 { Z j 0 (3.36) 0 liv'xw' } n -_ where matrices [Z„] and [Zj] are defined as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. _ 0 _ 66 Mn ^22 M,3 M 23 ^IN ^ 2N M j3 ^\N M 23 M33 ^ 2N jcoC l^T^N'xN' ~ 1 , ,)U] n'xN' jcoC 'M u M'n M'n M'n M '22 M 23 M w' ]C0_ M'n M '3 ■■ • (3.37) M'in' • ^N'N'+ — JO) The diagonal elements of coupling matrices, i.e. M - and M ^ ., are included in the analysis in order to simulate the effect of deviation of resonant frequency of each resonator with respect to the center frequency of receive (RX) and transmit (TX) filters. For synchronously tuned filters, such as the cross-coupled filter of the previous example, these matrix elements are zero. However, in diplexer synthesis, in order to compensate the effect of loading at common resonator we need to make fine adjustment on the resonant frequencies of the resonators in the vicinity of the common resonator. Moreover, in case of mildly asymmetric responses, the asynchronous tuning can provide the required asymmetry. Choosing E=l, i.e. unit excitation voltage, the scattering parameters of the diplexer can now be written as 5jj —l —2 Rgig S 21 = —2yjR^R^ij^ (3.38) *^31 ” Note that contrary to the analysis equations of the cross-coupled filter, the above analysis is performed at passband frequency and none of the components are normalized. C. Objective Function Due to the interaction between the RX and TX filters, it is not possible to use the objective function based on the location of zeros of the transmission and reflection Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 functions similar to (3.33). Instead, we use the upper and lowers specifications for defining the objective function based on (2.7). For the diplexer of this example, the receive frequency band is 1710-1785 MHz and the transmit band is 1805-1880 MHz. The diplexer is required to have an equiripple response satisfying the following constraints on the magnitude of its scattering parameters [Sj j|<-20 dB, for 1710<f<1785 MHz lSiil<-2 0 dB, for 1805<f<1880 MHz |S2i|<-72dB, for 1600<f<1692 MHz (3.39) |S2,1<-71 dB, for 1803<f<1990 MHz |S3i |<-72 dB, for 1600<f<1787 MHz |S3i |<-71 dB, for 1899<f<1990 MHz An 12 norm was used for the error functions in (2.7). D. Results We used a population of 30 individuals and the clustering period of 100 for this circuit optimization. In this diplexer, a cross-coupling between resonator 3 and 6 in the receive filter is used to provide the required out-of-band rejection by generating two transmission zeros on both sides of the passband. Similarly, two transmission zeros for transmit response are generated using a cross-coupling between resonators 4 and 7 in the transmit filter. For synthesis of the diplexer, first the values of L, L ' and L^. are set to 4 nH and the values of C and C ' are calculated for each filter to have the desired center frequencies at 1747.5 MHz and 1842.5 MHz. These inductance values correspond to the height of rods in the resonators. The size of these rods is usually kept the same for all resonators and the resonant frequency is controlled by changing the capacitance at the top cap by tuning screws. The value of common resonator capacitance Q, is initially set for a resonance at the upper frequency edge, i.e. 1880 MHz. There are 42 optimization variables including the elements of the two coupling matrices, the input/output impedances Rj, R 2 , and Rs, two couplings and M^.2 , and the common resonator capacitance C, . The diagonal elements of both coupling matrices are also considered variable to accommodate for the shift in the resonance frequency of each resonator. During the optimization, the lower and upper limits of all adjacent couplings are set to 0.05 nH and 0.2 nH respectively. For the diagonal elements of coupling Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 matrices these limits are -0.1 nH and 0.1 nH. The limits for cross-coupling elements are set to -0.1 nH and 0. nH. The box constraints for the rest of variables are l < M , . i < 3 nH 1< M ^2 < 3 nH 1.0 <C^. < 3 pF 50 < (3.40) < 500 Ohm 1 < i?i < 50 Ohm 1 < i ? 2 < 50 Ohm It should be noted that the common resonator of the diplexer is heavily loaded in order to have minor effect on the response of each filter. This is the reason behind using large limits for the coupling values and M ^.2 in (3.40). The initial estimates of the circuit components for each filter are determined by first synthesizing each filter separately using the same optimization approach explained in the previous example. The initial response of diplexer is shown in Fig. 3.21 along with the response mask based on (3.39). As can be seen, the interaction of filters on each others through the common resonator causes severe deterioration of the responses. -60 CO -80 -100 -120 -140 -160 1600 1650 1700 1750 1800 1850 Frequency (MHz) 1900 1950 Fig. 3.21. Initial response of the diplexer before optimization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 After applying the hybrid EP to this problem, the algorithm successfully located the global solution in 100 to 200 generations (after 10 trials). About 15 clusters were identified containing the global and local minima in average during the 10 trials. The variations of best fitness value in different generations for 4 trials are shown in Fig. 3.22. As can be seen, in trials 1, 3, and 4, the global solution has been found after performing local search inside the clusters at generation 100. The response of optimized diplexer at the global solution is shown in Fig. 3.23 and as can be seen it satisfies the required mask very well. The response also shows an equiripple behavior atthis global solution. The final values of optimization variables are shown in Table 3.5. 10 x10 - Trial ■ Trial - Trial - Trial 9 8 1 2 3 4 7 6 5 4 3 2 1 0 20 40 60 80 100 120 Generation 140 160 180 200 Fig, 3.22. Fitness vs. generation for the best individuals in 4 trials of diplexer optimization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 — IS -20 -40 -60 m -a -80 i -100 -140 -160[;;^; 1600 ^ 1550 ^ 1700 T750 ^ 1800 1850 ^ ^ 1900 1950 Frequency (MHz) Fig. 3.23. The response of optimized diplexer at global solution. Table 3.5. Final value of diplexer circuit components after optimization. M,, C. Mn Mj2 M22 M23 M 33 M34 M36 M44 M45 M ss Ms6 M ^s M(37 M 77 M 7S M ss M sg M gg 2.387857047 nH 2.341245646 nH 1.738125382 pF 6.5204114962E-02nH 0.1325206251 nH 1.2851916560E-02nH 0.1006308872 nH 8.4786383633E-03 nH 9.3258000455E-02 nH -1.498359536E-02nH 6.1246563780E-03 nH 0.1064621866 nH 6.6950278483E-03 nH 9.0137253349E-02nH 6.3744878847E-03 nH 9.3545993715E-02nH 6.629912715E-03 nH 0.1008207495 nH 9.6995800046E-03 nH 0.1404102029 nH 8.3075169583E-03 nH R. R, R2 M 'n M 'i 2 M 22 M 23 M '3 3 M 34 M 44 M 45 M 47 M 55 M's6 M 'e 6 M'si M 77 M 78 M'%% M \g M '99 368.087933929 Q 1.8082976665 Q 1.8135268680 Q -6.7655 88382E-02nH 0.124852235 nH -3.520333783E-03 nH 9.370646794E-02 nH 3.289125645E-03 nH 8.861793724E-02 nH 4.221754915E-03 nH 8.474630592E-02 nH -1.475690732E-02 nH 4.349796512E-03 nH 0.101208298 nH 4.6866452312E-03 nH 8.683817945 lE-02nH 4.4406711679E-03 nH 9.4613057915E-02nH 4.8106277684E-03 nH 0.1317278007 nH 6.4677483739E-03 nH Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 When this problem was tried by gradient search method using ADS [29] and multi-start SQP [137] starting from the same initial estimate that was used for the hybrid EP, they both trapped into a local minimum with unacceptable response. The diplexer response at one of these local minima found by gradient search method is shown in Fig. 3.24. We also tested the pure random search method in ADS and it could not find the global solution after 50,000 iterations in average either. - - IS^^l - iSjil -20 -40 - 60 - CQ -80 T3 •''i -100 -120 -140 1 I -160 1600 1650 1700 1750 1800 1850 1900 1950 Frequency (MHz) Fig. 3.24. The response of optimized diplexer at a local minimum. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 3.7.4 Computer Aided Diagnosis of Microwave Filters A. Introduction After design and fabrication of almost any microwave filter, if feasible, some fine tuning and adjustments are necessary to achieve the desired response. This process should either rely on the expertise of a trained professional filter technologist for the specific type of filter or on a systematic approach based on numerical optimization. Obviously, the availability of numerical diagnosis method can greatly reduce both cost and time to market for microwave filters and is considered as the preferred method by the industry. These diagnosis tools can also provide valuable insights to the accuracy and sensitivity of approximate circuit model of the filters used in the design process. Another application of the computer aided diagnosis is the coupling matrix extraction that has also recently been used in electromagnetic optimization of microwave filters [146]. Despite the benefits of using a numerical optimization, they have only been used for diagnosis of microwave filter with simple configuration. The main obstacle in using optimization method for more complex and large size filter structures is that the optimization problem becomes so complex that most available optimization techniques either fail to converge or become very slow. The hybrid EP algorithm that we developed in this Chapter is an excellent candidate for addressing these kinds of difficult global optimization problems occurring in the complex filter diagnosis processes. B. Formulation o f the problem In this Section, we demonstrate the performance of our method in extraction of the equivalent circuit parameters of an 8 -pole cross-coupled resonator filter from its measured response. The parameter extraction is formulated to solve the following minimization problem min[or^{|Re(s,, ix,0)^ ))-Re(5y^^^ {x,co,,))!' ^ (x,®^ ))| | J (3.41) k=i" + X |dB (5,, (X , }) - dB ( S ( X , CO, ) ) f , i j = 1 ,2 k=i where S^j and S^j are the simulated and measured scattering parameters of the filter respectively. The vector x represents the vector of optimization variables and cOj, is the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 frequency variable. The scattering parameters are matched at N ^ points. The parameter a is used for appropriate sealing of the first two terms of objective function. Matching both real and imaginary part of scattering parameters in addition to matching the amplitudes in logarithmic scale will allow us to reduce the multimodality of objective function, hence lowering the possibility of trapping into a local minimum. The equivalent circuit used for this filter, shown in Fig. 3.25, is similar to the equivalent circuit used in Section 3.7.2 with two differences. First, each resonator will have an additional series resistor r in order to simulate the effect o f loss in the actual filter. Our investigations show that although in reality the quality factor of each resonator may be slightly different, the value of this resistor can be considered the same for all resonators with a good accuracy and with minor effect on the success of parametric modeling of the filter. Secondly, different resonators will have different equivalent capacitances in order to simulate the effect of asynchronous tuning of resonators. '3N '2 3 L/2 L/2 L/2 L/2 A M r AAA- A A /V '2N IN Fig. 3.25. The narrow band equivalent circuit model of a lossy cross-coupled resonator. Kahrizi, et al. [138] used a two-stage approach for parameter identification of cross coupled filters. First a rational function expansion of the measured response is found using the Cauchy method and then a circuit synthesis step is performed using the calculated zeros and poles of the rational functions. The drawback of this approach is that since the Cauchy method is applied at lowpass, a good knowledge of center frequency and bandwidth of original filter is required to transform the measured response from bandpass to lowpass. However, this information may not be available or may have even changed during the fabrication process. We propose to perform the parameter extraction Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 directly at passband frequency and also eliminate the need for the rational function approximation of the measured response. W e start by writing the loop equations for the equivalent circuit of the lossy filter as j Q) M ] 2 ?i + r + j (a)L — J 0)^/1 1 — ) z't + jco M 2z,i3 +•■■• + j (OM 2 n ^N ' "t" Ri ^+ r + i {a>L — ZI "t j (O A/ 2N ^2 ——) ojC (3.42) = 0 - 0 In order to reduce the dimension of the optimization problem the circuit parameters are normalized to the reactance of resonators at center frequency, i.e. cOqL : CO COr ^ 0) V ^0 y r +i CO,02 CO CO, COq CO yCOQ ^ co„ /y CO + j — ^ ' 23^3 H h j — K zn^n ~ b (3.43) co„ yy = . CO . J — co„ . CO ffij, . • R-2N^2^ ^ R i + r +j CO 0 CO, In V, y = 0 zy where R. zy- The terms A _ , i? - i?, zy^L zy^L zy^L E =— co„L = 1, (3.44) LC, M, z =l, 2, ---, i V; 7 = z + l , z + 2 , - - - , A? in (3.43) are called the coupling coefficients. We define Sco. = zzZq,. /zWq as the relative frequency shifts of each resonator from the center frequency zy^. Note that by performing the above normalization, the governing equations become independent of the R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 75 bandwidth of filter. The scattering parameters of the filter is calculated using the similar formula as in (3.30) where the corresponding parameters are substituted from (3.43) and (3.44). Due to the reference plane shifts in the calibration and measurement process of the actual filter and also the effect of input/output probes on the phase response, we need to modify our circuit model to include the effect of these phase shifts. This is achieved by adding two pieces of ideal transmission line sections to the input and output of the filter as shown in Fig. 3.26. W ithout these additional phase shifts, it is impossible to match both amplitude and phase of scattering parameters with measurements using (3.41). Fig. 3.26. Reference plane shifts at the input/output of bandpass filter (BPF). The scattering parameters of filter after including the effect of these reference plane shifts are calculated as ‘S.; .5 ; c' “^12 22. e 0 e~^^- 0 ~Sn 21 0 6 ^12 ^ 22_ 0 (3.45) 12 e~ c ^ 22 C. Results We report the results for a filter originally designed for GSM 900 mobile base station with the center frequency of 947.5 MHz. After fabrication of the filter, its scattering parameters were measured and the hybrid EP was used to extract the parameters of equivalent circuit. The filter is degree 8 with folded canonical topology as shown in Fig. 3.27. Although the filter is originally designed based on only one cross coupling between resonator 3 and 6 , our investigations show that there are indeed some minor stray couplings between nonadjacent resonators as well. Although these parasitic couplings are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 very small, their cumulative effects on the response are large enough to prevent the convergence of parameter extraction process if they are not included in the analysis. The parasitic couplings are shown by dashed lines in Fig. 3.27. Fig. 3.27. Geometry of eight-pole filter. The stray couplings are shown by dashed lines. In using hybrid EP to solve the optimization problem (3.41), we set the population size to 30 and the clustering period to 100 generations. The total number of optimization variables is 25. Four hundred frequency points were used in the parameter extraction process. The initial estimate of the solution is set to the values of circuit components for a conventional Chebyshev design. It should be noted that this initial solution can be used even for highly detuned filters, since the algorithm does not rely merely on the initial point and has a built-in search capability. In order to be able to extract the stray couplings, the nonadjacent couplings that can potentially generate such couplings are also included in the vector of optimization variables. After running the program several times, the algorithm was able to successfully extract the circuit components in all cases in about 400 generations and locating 10 clusters in average. Fig. 3.28 to Fig. 3.30 show different responses of extracted circuit and the measurement at the solution. The responses match very well over the whole frequency band with very good accuracy. The final values of optimization variables are shown in Table 3.6. As can be seen in Table 3.6, the parameter extraction shows existence of stray coupling between nonadjacent resonators, i.e. K 2 6 ’ ^ 3 5 , K ^ , and , due to the specific physical arrangement of the resonators. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 77 -20 — |S j^ | m easured |S^^| m easured |S j , I extracted — |S ,,1 extracted -40 -80 -100 -120 840 860 880 900 920 940 960 980 1000 1020 1040 Frequency (MHz) Fig. 3.28. The measured and extracted circuit amplitude response of the degree eight filter. — <S.. measured 11 <S., extracted XI Frequency (MHz) Fig. 3.29. The measured and extracted circuit Sn response of the degree eight filter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 78 R eJSj^] m easured RefSj^] extracted 0.5 -0.5 860 880 900 920 940 960 980 1000 1020 1040 ImtSj^l m easured lm [S ,,l extracted 0.5 -0.5 840 860 880 900 920 940 960 980 1000 1020 1040 Frequency (MHz) Fig. 3.30. The real and imaginary parts of measured and extracted circuit degree eight filter. 821 response of the Table 3.6. Extracted circuit components for degree eight filter. 0.999005946 Ki 2 3.2625730929E-02 0.999502586 K 23 2.3102041668E-02 0.999176071 K 26 -2.2514784292E-04 ^0 1.001690075 K 34 2.0945597084E-02 0)^,10)^ 1.001412256 K 35 2.8150020105E-03 ® 06 ^ 0.999612278 K 36 -4.1464489042E-03 ® 07 ^ 0.999128580 K 37 -2.1413275737E-04 ® 08 ^ 0.999162171 K 45 2.4914334830E-02 4.0407873140E-02 K 46 3.1783214432E-03 R, 4.4208970129E-02 Ks6 2.0940054362E-02 r 2.7097151398E-04 -0.148194649 K67 2.3216806506E-02 3.4010307149E-02 ®01 ^ (0^2 ! ® 03 ^ ® 04 ^ Rs <j\ (rad.) <h. (rad.) K 78 -0.509251876 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 79 Next, the diagnosis of a detuned version of the same 8 -pole filter is performed. This is done to check the capability of the extraction algorithm in diagnosis of the source of response misalignment. After measuring the scattering parameters of the mistuned filter, the hybrid EP was used to extract the parameters of equivalent circuit model of the filter using the same objective function definition as in (3.41). The number of optimization variables is again 25 and the number of frequency points in parameter extraction is 400. Other parameters of the hybrid EP are kept the same as the original tuned filter diagnosis. In several trials, the algorithm successfully identified the global solution in less than 400 generations. The response of filter after circuit model parameter extraction is compared with the measured response of the detuned filter in Fig. 3.31. The final extracted circuit components of the detuned filter are shown in Table 3.7. In comparison to the extracted model of the tuned filter in Table 3.6, it is concluded that the coupling coefficients K 12 , K45, and Kj 8 , the normalized input/output impedances and the resonant frequency of the eighth resonator are mistuned. — — -20 {82., I m easured |S.j.^j m easured ISj^l extracted IS.,.,I extracted -40 -60 -80 -100 -120 ’840 860 900 920 940 960 980 1000 1020 1040 Frequency (MHz) Fig. 3.31. The measured and extracted circuit response of the detuned degree eight filter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 80 Table 3.7. The extracted circuit components of the detuned eight-pole filter. 0.999I92I6830 Ki 2 3.32498H4757?l-:-02 0.99983107985 K23 2.31547358174E-02 0.99948288456 K26 -1.443119512957E-04 1.00220231938 K 34 2.071431885515E-02 1.00181283779 K 35 2.361139798701E-03 0.99965115247 K 36 -4.381578320394E-03 0.99972088114 K 37 -2.399256991109E-04 1.0()05:9(i3498 K 45 :’.4.T.=6.1'.2S72n3i:-0?. 4.3Ui8860.18h-02 K 46 3.109331063098E-03 3.87M 8‘)652h-0: Ks6 2.105279342714E-02 (rad.) 2.71100200E-04 -0.1106175 K67 K 78 2.296348783213E-02 3.l6N32f>045477h-02 (rad.) -0.5177074 C0 q2 / 0 )(j J / (Dq 0 )^, 1 0 ), Ri f </>2 3.8 Conclusions In this Chapter, a robust and efficient hybrid evolutionary programming algorithm was proposed for global optimization and parameter extraction of complex microwave filters and diplexers. The conventional EP algorithm was modified using a novel integration with clustering algorithm to improve the diversity and to reduce the possibility of premature convergence in multimodal problems. It was shown that a local search from the center of each identified cluster can be performed to find the minimum of objective function in that region very fast. The clustering approach was also used to reduce the size of search domain during the EP process, which helped to improve the speed and robustness of the algorithm. We tested the developed algorithm with several benchmark function minimization problems. In all cases, our algorithm has shown superior performance in terms of robustness and speed of convergence compared to several global optimization methods such as GA, random search, and single-linkage method. The algorithm also showed very good performance in optimization of complex microwave filters and diplexers as well as computer-aided diagnosis of microwave filters. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Model Calibration Technique for Filter Optimization During the recent years, with availability of many commercial electromagnetic simulators, more attempts are being made towards the development of design strategies which take advantage of accurate analysis methods provided by these simulators. Although the processing speed of computers has been increasing continuously, performing direct optimization using such EM simulators is still computationally very expensive. Most optimization strategies that address the above problem use some kind of hybrid scheme. In these methodologies, the full EM simulation is used to systematically calibrate an approximate physics-based model, which can be analyzed very fast. Bandler, et al. [82] proposed the concept of space mapping for EM optimization for the first time, and in the subsequent works their approach was further improved [83]-[89] and successfully applied to optimization of microwave circuits. Space mapping defines two models: a full-wave EM solver that is referred to as fine model, and an approximate circuit-based model that is called coarse model. In general, space mapping tries to find the optimum solution in the fine model space through series of alignments on the solution in the coarse model. Two of the most important factors that can severely affect the success of this process are the accuracy of coarse model and parameter extraction [143]. In [144], an optimization approach has been proposed, which in addition to aligning coarse model and fine model, improves the approximate model for microstrip filters by extracting the parasitic interactions and incorporating them into the coarse model in form of correction circuits between nonadjacent sections. The success of this method depends on the expertise of the user in choosing the right signal path between different sections, and topology of correction circuit. Furthermore, sometimes the cross-coupling between different parts of structure cannot be modeled by simple node to node connection of two 81 R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 82 ports circuits. Recently, an optimization strategy based on coupling matrix correction has been proposed for direct optimization of microwave filters [145]-[146]. In this approach, . the coarse model is iteratively improved by comparing the extracted coupling matrix of fine and coarse models. This is an effective method only when the difference between the coarse and fine model is limited to the additional cross-coupling elements existing in the fine model. In this Chapter, we develop a simple yet efficient technique for direct optimization of microwave filters. We will use linear space mapping between approximate and EM model spaces to calibrate the approximate models. In order to avoid the need for modeling of all unknown stray couplings in the EM model, an appropriate objective function based on reflection function of the filter will be used. The judicious choice of approximate model for the filter can also greatly improve the robustness of the method while dealing with the problem of nonuniqueness during the model calibration process. In the following Sections, we first introduce the main model calibration step. Then the existence of solution and uniqueness of solution in the parameter extraction step are addressed. The verification of proposed method will be presented in Section 4.3 through four microstrip filter CAD examples. 4.1 Linear Model Calibration Consider a filter with certain physical design parameters denoted by n-dimensional real vector X . The approximate model response is shown by ix,co) where co represents the frequency variable. The reference model, which is also known as fine model, is determined by full-wave EM simulation. The fine model response is shown hyR ^^{x,a)) . The two models, determine two spaces that are represented by their corresponding design parameter vectors x ^ ^ and x ^ . The initial optimization is performed on the approximate model space. The goal is to find the design parameters that minimize the difference between R ^ ^ and a given mask response {co) , = arg min [ {x ^ ^ , co) - ^apx We can now define two objective functions i (fy)||) j and as follows. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (4.1) 83 OJ Uem(x)^'Z W^apx - Rapx (4.2) 1k m Our goal is to find the optimum solution ® ) “ ^«px < ^ ^ x ’^ ) in the fine model space, such that ^em = arg |m in(f/^ Fig. 4.1 shows the behavior of two functions (4.3) (X)) and U for two sample one dimensional models. em 'em Fig. 4.1. The sample error functions for two model spaces. When the two error functions have exactly similar shapes but with minima at two different points, the solution can be found in one step. It will only be necessary to R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 84 start from an initial point, lets say U ~^apx-< find the corresponding point on the through parameter extraction. P ) = a r g j i r a n )- U (x)[| (4.4) and since the two functions have similar shapes, we can write P r-r d apx '^em '^em ) (4 St^ ^ which simplifies to Y* em = Y* apx _ p d) \ 4. Vd) —7 '^ em V* apx —P t yd ) 't ^ em ' t4 6f Note that the solution of PE step in (4.4) may be non-unique. For example in Fig. 4.1, the first PF step will have two solutions. However, only the selected solution provides the local negative slope. The following formulation may be used reduce the nonuniqueness problem in the PF step, F (x^,„ ) - arg j m in ( X “ ^apx ( x , ty)||)| (4.7) More comments about the nonuniqueness problem will be provided in the next Section. In practice, not only the two error functions have different minima, they also have different shapes or contours. However, we always assume that the approximate model is a good representation of the general behavior of the fine model in the vicinity of its minimum. In other words, we can approximate the directional derivative of the FM based error function at a point in the fine model space with the directional derivative of approximate model based error function at its corresponding point in the coarse model space when both points are inside a region of trust [147]. In the iterative minimization process of (4.3), we intend to find the direction h and length of search a that reduces the value of U ^ ^ i X g l + a h ^ ‘^). In general, for steepest descent method the vector h is equal to - V I / {x^^) = -^Uapx (P (Xem)) ^ud Step length is chosen equal to one. However, if the nonlinearity of mapping between the two model spaces is not very large, the linear approximation of the steepest descent method with full step length will be equivalent to R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 85 the direction used in (4.6). In other words, at iteration i, the vector h can be approximated as '■'” = 4 , (4.8) Based on this linear approximation, the next point in fine model space can be found using "^em = x"^em ^ ‘^ ^ ^ (4 9) apx The above iteration is equivalent to the aggressive space mapping [83] when the Jacobian matrix is set to one. This approximation will become more accurate when the search gets close to the solution region. The size of this region depends on the accuracy of approximate model. The examples that follow in Section 4.3 verify that in many practical cases, this region is large enough to allow the use of (4.9) confidently. Note that, by using the linear approximation, and relying on the closeness of the shapes of the error functions in the two model spaces, the calculation of the Jacobian matrix of the coarse model has been avoided. Nonetheless, it is possible to improve the approximation in (4.8) by adding the information of the Jacobian matrix similar to the method used in [83]. The iteration process is terminated either when (^em)| < ^ >where t] is acceptable tolerance, or when the iteration number has passed a certain limit. In the next Section, we address the potential problems in the process of parameter extraction and the proper choice of response for alignment of the two parameter spaces. 4.2 M odel Parameter Extraction The success of linear space mapping approach, as formulated in (4.9), depends on two factors: the validity of the approximate model, and the accuracy of parameter extraction. An invalid approximate model will lead to failure of the parameter extraction step. In other words, if the approximate model is invalid/inaccurate and/or the PE is not formulated correctly, the parameter extraction step (4.7) may not have a solution. This is referred to as the “existence of solution in PE” . On the other hand we will show that both of these factors can also contribute to another unwanted feature called “nonuniqueness of the solution in PE” where existence of several solutions may hamper the success of PE . R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 86 4.2.1 Existence of Solution in PE If the EM model response includes certain features that can not be modeled explicitly by the approximate model, the PE may fail. This is a serious limitation in many practical filters and complex structures optimization, where the actual structure includes many unknown effects such as stray couplings, discontinuity effects, etc. However, from the design point of view, most of these unwanted effects that appear in the filter response are usually in regions that do not affect the main functionality of the device as a filter, otherwise the design is not considered feasible. A good example of this phenomenon is the appearance of unwanted transmission zeros due to the unwanted parasitic couplings. If these transmission zeros are interfering with the design goals without the possibility of adjustment with optimization, the design will not be feasible. However, these parasitic couplings are usually very weak and as a result, the corresponding unwanted transmission zeros are not very close to the passband of the filter. This is an important observation that will be used to choose a suitable formulation for PE. In the following, we examine the mathematical models of scattering parameters for both EM and approximate models and show how the reflection function can be effectively used to align two corresponding parameter spaces. We start by examining the effect of transmission zeros on the location of reflection zeros in the framework of response synthesis. As mentioned in Section 3.7.2, the transfer and reflection function of any filter composed of N coupled resonators can be expressed as the ratio of M h degree polynomials N (®) ^ N (®) where the normalized real frequency O) is used instead of the complex frequency s in (3.31). Without loss of generality, we focus our discussion to a 3-pole lossless filter with general Chebyshev (quasi-elliptic) response. When all the transmission zeros are located at infinity, the response reduces to the conventional Chebyshev response with reflection zeros at 0 . and ± 0 .8 6 6 , F ,{(O) = r , {CO) = W -3 q ) = Ao){(o- 0.866)(<y + 0.866), P ^ {a ))^ l R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (4.11) 87 Now assume we add a finite transmission zero to the response at normalized frequency Q},. This plays the role of an unwanted transmission zero generated by unwanted parasitic couplings. Using the synthesis approach in [131], it can be shown that the corresponding polynomials for the new general Chebyshev response are (4.12) P3 { a } ) - 1 - (ol 0), As can be seen, when 1<», the polynomials in (4.12) reduce to those in (4.11). Fig. 4.2 plots the synthesized response of a 3-pole Chebyshev filter having a return loss level of 20 dB along with the response of four general Chebyshev filters having different transmission zeros ranging from = 2 to ty, = 5 , in normalized frequency scale. As can be seen, the effect of transmission zero on the return loss response of the filter is very minimal. -10 -20 -30 “ -40 -50 -60 — -70 ISg.,! Chebyshev |S^^| Chebyshev General Chebyshev |S^^| General Chebyshev -80 -10 Normalized frequency (co) 4.2. Comparison between the response of a 3 -pole Chebyshev filter and four general Chebyshev responses with different transmission zeros. Fig. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 88 The variations in the location of reflection zeros with respect to the normalized transmission zero is also plotted in Fig. 4.3. This plot clearly shows that F* and 3’^^’ reflection zeros are barely affected and the middle reflection zero is only slightly moved when transmission zero gets very close to the band edge, i.e. a ? - I . The first implication of this observation on the model calibration technique is that it is usually not necessary to develop an approximate model that incorporates all the unwanted transmission zeros as long as the reflection function is used in parameter extraction step. 1.5 First root Second root Third root 0.5 ■ "o1 ° 0 0 Cd -0.5 Transmission zero (op Fig. 4.3. The location of reflection zeros w.r.t. the location of normalized transmission zero for the 3-pole filter. The above observation can also be verified using a different analysis/diagnosis approach. In this approach we start by writing the general pole-zero expansion of the scattering parameters of the approximate model of a filter with N cross-coupled resonators, after transforming to the lowpass region, as M ' 11 apx , {s) = 1=1 N ' 21, (s) = != 1 N eY [{s-p ^) i=i i=l R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (4.13) 89 where s = jco is the complex frequency variable, N is the degree of filter and M < N - 2 . For lossless filters, the poles and zeros are related through the r e l a t i o n + | 5 2 i f = 1. When the response of the same filter is measured or computed using a rigorous EM analysis technique, the mathematical representation of the response over the same band of frequency will need more number of poles and zeros. In other words, over the finite frequency range, the number of poles/zeros required for mathematical modeling of the EM response will be higher than N. This is mainly due to the additional poles/zeros created by unwanted effects such as parasitic couplings, discontinuities, higher order modes, etc. Our investigation, confirmed by several examples presented in the following Section, show that in most cases, the major impact of the above-mentioned unwanted effects is mostly on Sji response and occurring outside the selection band of filter. This observation can be used to write the pole-zero expansion of the EM model response for the region of interest as, ----------- ^ (4-14) H ( ^ - ipi + ^Pi (=1 k=l M Q +A y,))n(^ - n ) 52. (^) - 4 r ------------------------¥ ---------r i ( ^ - (Pi + A/?, ))]][(5 k=i The additional reflection poles and zeros, (4-15) ) andcr^, are very close to each other, producing minor deviation from the reflection function of approximate model. However, the additional transmission zeros, , will have major impact on the response of filter in the band-reject region. The perturbations of the original poles and zeros o f approximate model (A^, A/7 , A v ) are minimal especially when the unwanted transmission zeros are not very close to the band-edge of filter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 90 In order to demonstrate the validity of above expansion, a brief verification through an example is presented. Fig. 4.4 shows the insertion loss and return loss of a fifth degree microstrip parallel coupled filter achieved by full-wave EM simulation. The complete information about this filter will be provided in the Section 4.3.1. Although the filter was originally designed to have a Chebyshev response, the plot clearly shows the existence of an unwanted transmission zero at 3.975 GHz. The Model Based Parameter Estimation (MBPE) [138] approach is used to find the rational function expansion of scattering parameters of the corresponding lowpass prototype given in (4.14) and (4.15). The minimum order of polynomials (N+L) for modeling the EM response was 9. Fig. 4.5 shows the graphical distributions of poles and zeros for this filter. Poles and zeros in Fig. 4.5 can be classified into two groups. The first group consists of five poles and five zeros located in the region - £ - j < s < - £ + j where 0 . < £ ’< 0.5. These account for the poles and zeros of the original Chebyshev design. The second group includes the rest of poles and zeros. These poles and zeros are due to the stray couplings and other non-circuital electromagnetic effects in the filters, including dispersion, discontinuity effects, etc. The extracted transmission zero on the imaginary axis located at - j l .S is responsible for the finite frequency transmission zero at 3.975 GHz. There are two complex transmission zeros and four pairs of reflection poles/zeros in this group as well. As can be seen the poles and zeros in each pair are close to each other, as explained before, making it possible to cancel out when approximating the reflection function in (4.14) with the one in (4.13). The above discussions conclude that an approximate model with less complexity can be effectively used in the model calibration process as long as the two models are aligned using the information of the reflection function over the whole band of frequency. This will eliminate the need for improving the approximate model to include the stray couplings as discussed in [144]. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 91 0 T 1^ 21 -10 em^ -20 -30 -40 ^ -50 -60 -70 -80 -90 -101 _1__________L- .8 3.85 3.9 3.95 4 4.05 4.1 Frequency (GHz) 4.15 4.2 Fig. 4.4. EM response of the fifth degree parallel coupled-line filter. A A X() O X X -2 O -4 O X A O O -1 0 -10 -5 S.J ^ zero S.|.| pole zero 0 10 Re[s] Fig. 4.5. Pole-zero distribution of scattering parameters of the fifth degree parallel coupled-line filter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 92 In order to verify the above conclusion, we try to extract the physical parameters of the approximate model of the same 5-pole parallel coupled-line filter to match its reflection function with the reflection function in the EM response of Fig. 4.4. The approximate model for the filter is constructed from cascading microstrip parallel coupled-line sections [149]. Fig. 4.6 compares the amplitude response of fine model and extracted approximate model, and Fig. 4.7 compares the phase of reflection function for two models after parameter extraction. The magnitude and phase of reflection function match very well over the whole range of frequency. This clearly shows that although the accuracy of the approximate model is not high enough to model the 5 , 1 , it is still capable of modeling the reflection function of fine model with good accuracy. This important property can be effectively used in the process of PE allowing us to calibrate the approximate model successfully. -10 -20 -30 -40 “ -50 -60 -70 -80 -90 Frequency (GHz) Fig. 4.6. Magnitude of the scattering parameters for fine model and extracted approximate model of the parallel coupled-line filter. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 93 200 150 100 50 ) E D (U) 0 Q <s,1 1 e m <s.1 1 a p x -50 -100 -150 -200 Frequency (GHz) Fig. 4.7. Phase of the reflection function for fine model and extracted approximate model of the parallel coupled-line filter. It should be noted that the situation for filters with intentional prescribed transmission zeros is different. For these filters, the transmission zeros are indeed part of the design to provide higher rejection and the physical mechanism that produces these zeros are known in advance. As a result, the approximate model will be developed to model the transmission zero as well, while for the filters with unwanted transmission zeros this modeling is not straight forward as explained before. Consequently, for filters with intentional transmission zeros, the reflection function plus the information of the S 21 response only at the location of these transmission zeros is used in the PE process without any limitation. An example of this type of filters is presented in Section 4.3.3. It should also be mentioned that although most of the discussions in this Section were based on the lossless assumption, the results are easily applicable to the lossy filters as well. The reason behind this is the fact that as shown in Appendix A, the effect of losses in the filter can be modeled by shifting of all poles and zeros horizontally to the left side of the complex plane by an amount inversely proportional to the quality factor of the resonators. This shifting will not have any major impact on the relative arrangement of pole/zeros as was used in the above discussion to drive the conclusion about using the reflection R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 94 function in the PE step. An example of EM optimization of a lossy parallel coupled-line filter using this approach will be presented in Section 4.3.4. 4.2.2 Uniqueness of Solution The parameter extraction step (4.7) potentially provides more than one solution (nonuniqueness problem) if appropriate measures are not taken. One of the methods to reduce this problem is Multipoint Parameter Extraction (MPE) as proposed in [143] and [148]. However, this method requires additional EM simulation to reduce the number of acceptable solutions iteratively and is potentially time consuming. Before resorting to a MPE approach, the following guidelines can be used to alleviate the nonuniqueness problem; a) The choice o f objective function can have direct influence on the success of PE. The more comprehensive the information used for matching the responses during the PE in (4.7) is, the fewer the number of possible solutions will be, hence reducing the nonuniqueness. Considering this, the following objective function is proposed to maximize the probability of convergence while reducing the nonuniqueness 2 + P *+ = arg{m in[a^ ' [1 ^ 1 , ft)-)) - Im (s, (X, ft).)) Zjk >) 1=1 I 1 1 =1 1 Acu-j |k (sa.„. )) - ' ' s (^ 21 , (4.16) + )) ]} The first part of objective function in (4.16) matches the real and imaginary part of the reflection function over the whole frequency band at points. This ensures that both magnitude and phase are matched at the same time. Inside the passband of the filter, both real and imaginary part of become very small making it very difficult to effectively match the responses compared to the rest of frequencies. To solve this problem the magnitude of responses are also matched R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 95 over the whole frequency band in logarithmic scale. Note that a weighting factor a is used for scaling the first part of objective function appropriately to the maximum value of the second part. Finally, in case of optimizing a filter with intentional transmission zeros, the third part of objective function is added to match the magnitude of Sj, response in logarithmic scale only at a small frequency region Ao)^. around each transmission zero, with N , representing the total number of transmission zeros. b) By using a priori knowledge of the problem, the size of search domain in (4.16) can be reduced, leading to reduction of the nonuniqueness in the PE step. The information can be added to the optimization either in the form of additional constraints derived from the physics of the problem or a suitable initial point. For example, if we know that a certain formula used in predicting the coupling between two microstrip lines in the approximate model always underestimate the value of coupling versus the width, then we will expect that the valid solution for the width during the parameter extraction step should always be larger than the original width used in the EM simulation. For each problem, a similar type of information can be found and embedded into the optimization problem to facilitate the elimination of strong local minima from objective function. c) The approximate model should be developed in such a way that the simultaneous effect of several design parameters on a certain property of response is minimized as much as possible. For example, in developing the coarse model for a microstrip half wavelength resonator, if the open-end effects are included in the model, the resonance frequency of the resonator will be a function of both length and width of the resonator. For instance, for a given length of resonator, if the width of resonator increases, the open-end capacitance will increase as well, leading to a lower resonant frequency. This is an unwanted numerical effect that will produce nonuniqueness in PE. To avoid this, no end effect corrections are included in the approximate model. Instead, the shift in the center frequency of the filter will be corrected after one or two iterations very easily. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 96 4.3 Examples The proposed method was successfully applied to many design examples. In order to demonstrate the capabilities of our method, the results for four filters are reported. The first two examples are EM optimization of parallel coupled-line and modified capacitivegap coupled microstrip bandpass filters. These are classic examples of cascaded resonator filters which their EM responses exhibit unwanted effects such as unwanted transmission zeros. The third example will be EM optimization of a cross-coupled microstrip filter with intentional transmission zeros. This example is chosen to demonstrate the validity of the proposed approach in optimization of filters with intentional transmission zeros. Finally, an example of lossy parallel coupled-line microstrip filter is given to verify the applicability of the developed approach to lossy filters. 4.3.1 Parallel Coupled-Line Filters The first example is a high-temperature superconductive (HTS) microstrip parallel coupled-line bandpass filter designed for the center frequency of 4.032 GHz, bandwidth of 64 MHz and minimum passband return loss of 18 dB. The insertion loss of filter should satisfy the following requirements: 15 ^1 1< -4 0 dB, for 3.8 GHz < / < 3.95 GHz IS2 1 1< -4 2 dB, 4.12 GHz < / < 4.25 GHz for (4.17) Based on these specifications, a fifth degree Chebyshev filter will be designed. Eig. 4.8 shows the layout of filter with its surrounding metallic box. 200 mil Line of symmetry is , I i i „ :s3 150 mi L, I Parasitic coupling \ i L3 Fig. 4.8. Layout of the HTS parallel coupled-line filter R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 97 All resonators and the input/output feed lines have the same width, W. The filter is designed on 20 mil thick YAIO 3 substrate with dielectric constant of 15.5. The filter is inside a box with the height of 300 mil. The approximate model used in this example consists of the cascade of microstrip line segment models for input/output lines and six coupled-lines section approximate models [149]. No open-end model is included for the same reason mentioned in the Section 4.2.2-c. The initial solution is obtained by starting from the conventional design approach for parallel coupled-lines with Chebyshev response followed by optimization using hybrid EP for achieving the required response. In hybrid EP, the size of population is 20, the tournament size is 6 and clustering period is 20. The vector of optimization variables is x =[W,Si,S 2 , , 1^,1^]. Fig. 4.9 shows the response of approximate model at the solution point x . -1 0 h -20 h -30h ...... “ -50 -m -80 -90 -100 Frequency (GHz) Fig. 4.9. The optimized approximate model response of parallel coupled-line filter at x apx The fine model is constructed by performing full-wave EM simulation using Momentum [150]. Fig. 4.10 shows the EM response of filter at the initial point x^H - x '^^^. The transmission zero at the lower part of frequency response is due to the parasitic couplings among the resonators as shown in Fig. 4.8. This transmission zero cannot be eliminated by fine model optimization. However, using the proposed optimization approach we try to get as close as possible to the goal response of Fig. 4.9. The next optimization steps are performed according to the proposed procedure and the error function defined in R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 98 (4.16). We use hybrid EP for parameter extraction steps. The size of population is 20, the tournament size is and clustering period is 20. The initial EM response of the filter is 6 shown in Fig. 4. 1 0 . After four iterations, the closest EM response to the goal response was achieved and further iterations did not improve the response. The EM responses of filter after the first and forth iterations are shown in Fig. 4.11 and Fig. 4.12 respectively. The Table 4.1 lists the optimization vector at each iteration along with the corresponding vector of extracted approximate model P { x ^ ^ ) . As can be seen, EM response of the optimized filter satisfies the requirements very well. Or _10( 21 em ‘ 5., 11 em'I . . 2 qL . . ; -3 0 r .40 3 ! 3 i -50b-'" -6 0 1...................... -70^ ... -80 r 8 ............... 3.85 3.9 3.95 4 4.05 4.1 Frequency (GHz) 4.15 4.2 Fig. 4,10. EM response of parallel coupled-line filter at initial point x ^^ = x apx ' R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 99 ern^ em' 21 11 -20 h ->30j’......... ^ 0 CQ -Q -50 -70 h ■ -80 Frequency (GHz) (2 ) Fig. 4.11. EM response of parallel coupled-line filter after one iteration at point x em ■ -20 ■GO -40 v///////w//////////m v//:v///////yy/////^. -60 -80f Frequency (GHz) Fig. 4.12. Final EM response of the optimized parallel coupled-line filter after four iterations at - ^( 5) compared with the approximate model response at x point apx R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 10 0 Table 4.1. Vectors and P { x ^^) at each iteration for parallel coupled-line filter. For initial solution X® i= l 10.8021 s, em (mil) (mil) 18.4927 i= 2 i= 3 i=4 i= 5 9.5208 18.1154 9.4875 9.4791 9.4819 18.0601 18.0689 18.0576 71.9549 85.716 S2 80.6724 69.5125 72.9113 71.7865 Ss 98.7988 79.8786 87.4861 84.4129 L, 233.186 230.209 230.39 230.405 Li 232.86 229.792 229.963 230.001 230.408 229.992 Ls 232.815 229.77 229.981 229.994 229.996 W 12.0834 10.8354 10.8105 10.7993 Si 18.87 18.548 18.504 S2 91.8323 112136 18.4839 81.7972 80.504 Ss 117.719 91.1913 101.872 97.4957 Li 236.163 233.005 233.171 233.183 L2 235.928 232.869 Ls 232.689 232.604 232.822 235.86 232.802 232.813 - 4.3.2 Modified Capacitive-Gap Coupled Filter The next example is a three-pole Chebyshev filter with 1% bandwidth [144]. Fig. 4.13 shows the layout of filter. The substrate of this HTS microstrip filter is LaAlOs with thickness of 20 mil and dielectric constant of 23.5. In [144], the approximate model was developed using cascade of subsections analyzed with EM simulator. Here we use the conventional transmission line models available in AD S as shown in Fig. 4.14. Since the width of lines for gaps are outside the acceptable range of the ADS model, we approximate the gap model with three parallel gap models instead to bring the width size to an acceptable range. Other errors in the model due to the closeness of values to the acceptable limits of models, such as ratio of widths for the input/output steps or small gap size for Si are automatically corrected during the optimization process. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 101 Line of Symmetry 200 150 1 -1 5 0 - L3 - i_ t 100 - 180 100 i i u L_ > Parasitic coupling Fig. 4.13. Layout of three-pole HTS filter. The fixed dimensions are in mil. MLIN TLO M STEP Step1 MUN TL1 -O Q -i • K ID - MGAP Gap1 W=60mti S=s1 MGAP G ap6 W=60mi] S = s2 h id H MGAP G ap2 W=60mii mj MLIN TL2 S te p 2 h j o -j MGAP G ap3 W=60mil M STEP S tep3 MUN TL3 MGAP G ap5 W=60mil MUN TL4 S te p 4 H J D-^ MGAP Gap4 W=60mil Fig. 4.14. Approximate model for half of the 3-pole HTS filter using ADS models. At the initial step, the approximate model is optimized to achieve the required response as given in [144], The vector of optimization variables is x =[Sy,S2 ,Ly,L 2 ,L j,L ^ ] . Fig. 4.15 shows the final optimized response at = [2.505, 23.592, 9.882, 49.855, 55.256, 52.502] mil R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. (4.18) 10 2 3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2 Frequency (GHz) Fig. 4.15. The optimized approximate model response of three-pole HTS filter at x The EM response of filter at the initial point x® = computed using Momentum is shown in Fig. 4.16. The unwanted transmission zero due to stray coupling between the first and third resonator is seen in this Figure. In [144], this parasitic coupling was modeled by adding a two-port circuit between resonator 1 and 3. However, our approach eliminates the need for modeling of this parasitic coupling and eliminates additional computational burden of correction circuit parameter extraction. Our proposed method was applied to the problem and after four iterations, the FM response satisfied the design goals. Fig. 4.17 compares the final FM response with the goal response Rapxi^lpx^ ■The values of optimization variables at , P ( x ® ) , and x^„, = x®^ are shown in Table 4.2. The difference between this value and the final solution in [144] is due to the difference between the accuracies of two FM simulators and the size of box. In comparison to the method in [144], three more FM simulations are performed, but since no FM simulation is required for approximate model calculations the overall optimization will be faster. Moreover, no correction circuits were required in our approach, making it more suitable for optimization of filters when modeling of electromagnetic effects in the form of correction circuits is very difficult or not possible at all. R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission. 103 21 e m ' -10 lle m ' -20 -30 -40 ■o -50 -60 -70 -80 -90 3.8 3.85 3.9 4.05 3.95 Frequency (GHz) 4.15 Fig. 4.16. The EM response of three-pole HTS filter at initial point x 4.2 (I) apx -1 0 -20 -30 -40 -50 -60 -70 IS,2 1 e m -80 I^21apxl -90 - 3.85 3.9 3.95 4.05 - 4.1 |S lla p x i 4.15 4.2 Frequency (GHz) Fig. 4.17. Final optimized EM model response of three-pole HTS filter after four iterations at point compared with the approximate model response at x apx * Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 4 , P(x^^^ ) , and Table 4.2 The design variables at Parameter Si S2 Li L2 Ls L4 X (1) = em X * X apx 2.505 23.592 9.882 49.855 55.256 52.502 for the 3-pole HTS filter. 1.362 24.097 8.829 49.340 54.384 50.951 * em = X (X) em 3.831 23.016 9.162 47.32 60.252 54.11 All values are in mil 4.3.3 Microstrip Cross-Coupled Filter The next example is EM optimization of a four-pole cross-coupled microstrip filter. The geometry of filter is shown in the Fig. 4.18. The filter has a vertical symmetry and the widths of all resonators are the same. The substrate used for this filter is 1.27 mm thick and has a dielectric constant of 10.8. The folded resonators in this type of filter provide the possibility of having controlled cross-coupling [151]. In Fig. 4.18, cross-coupling between resonator 1 and resonator 4 produces two transmission zeros below and above passband. This coupling is controlled by the value of spacing i'g. The approximate model of the filter consists of transmission lines, coupled lines, gap, bend and T-junction models from ADS library. The filter is designed for the extended GSM 900 frequency (880-915 MHz) with the return loss level of at least 20 dB. The approximate model is optimized to satisfy these specifications and also provide two prescribed transmission zeros at 832 MHz and 950 MHz for higher rejections. The width of the feed line and the width of the resonators are kept fixed at =1.1 mm and w - 1 .4 mm during optimization. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 Fig. 4.18. Geometry of the 4-pole cross-coupled microstrip filter. The final optimization vector for the approximate model is ^apx — ^2’ = [1.37,4.62,3.6,0.8,0.92,20.3,9.32,14.92] (4.19) mm Fig. 4.19 shows the optimal approximate model response of the filter at and Fig. 4.20 shows the EM response of the filter, calculated using Momentum, at the initial point x^ll = . The initial EM response has two transmission zeros at 8 6 8 MHz and 945 MHz. As explained before, based on objective function (4.16), 5n will be matched over the whole band and the | Sji | response will also be matched at the location of these transmission zeros during parameter extraction. The response of approximate model after PE is also shown in Fig. 4.20 and as can be seen, the reflection functions are perfectly matched and the two transmission zeros are also extracted very accurately. The process of linear model calibration for this example converges after five iterations and the EM response matches with the optimal response with a very good accuracy as shown in Fig. 4.21. The values of optimization variables at x ® = x^^^ , P (x^'*’) , and X*em =x^^) em are shown in Table 4.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 0 -10 -20 -30 -40 S -50 -60 -70 -80 -90 -lOg^ 7 0.75 0.8 0.85 0.9 0.95 Frequency (GHz) 1.05 1.1 Fig. 4.19. The optimal approximate model response of the cross-coupled microstrip filter at “apx ■ 0 21e m 11e m 21ap x -10 -20 -30 -40 S -50 -60 -70 -80 -90 -101 '8 0.75 0.8 0.85 0.9 0.95 Frequency (GHz) Fig. 4.20. EM response of the cross-coupled filter at initial point model response at P 1.05 1.1 and the extracted ). R eproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 0 21 em -10 -20 21 apx* m -30 -40 “ -50 -60 -70 -80 -90 -101 0.75 0.8 0.85 0.9 0.95 Frequency (GHz) 1.05 1.1 Fig. 4.21. Final optimized EM response of the cross-coupled microstrip filter after five iterations at point compared with the optimal approximate model response at x apx Table 4.3. The design variables at x ^ ^ , P (x^^) , and x^^ —^em coupled microstrip filter. Parameter Si S2 S3 81 82 Li L2 Lj (1) * = X (6 ) X em em * ^ em ~ ^ apx 1.37 4.62 3.6 0.80 0.92 20.3 9.32 14.92 the cross 1.345 6.33 3.464 1.311 1.768 21.53 8.194 15.396 1.009 3.683 3.665 0.701 0.58 21.272 8.515 15.529 All values are in mm It should be mentioned that this filter also generates an additional transmission zero in form of an extracted pole. Tapping the input line into the first ring resonator creates two pieces of open end microstrip transmission lines. At certain resonant frequencies, each Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 8 line produces a short circuit at the input tap junction which can potentially generate transmission zeros at the corresponding resonant frequency. However, the effect of coupling to the second and forth resonators dramatically reduces the loaded Q of the top open end line. Due to this reason, only the lower open end line will produce a significant transmission zero. The wideband frequency sweep of both approximate and EM model of this filter after optimization is shown in Fig. 4.22. 0 :n ;:i If ^^ •- i/r ^\V 1 Iji ! -10 I 'i -20 ( -30 -40 OQ XJ -50 -60 - -70 - (X* )| I S l t e m O 'e m ) ! I^21apx^^*apx^l -80 - I^11apx^^ apx^l -90 1.5 0 Frequency (GHz) Fig. 4.22. Comparison between the wideband responses of the optimized approximate and EM models of the cross-coupled microstrip filter. Both responses clearly show evidence of additional transmission zeros at upper frequency band between 1.1 GHz and 1.4 GHz. The smaller attenuation at transmission zero predicted in EM response is due to lower Q of the open end line when modeled by EM simulator. Since we only used the information around the two main transmission zeros to calibrate the approximate model with the EM response, the additional transmission zero that fell outside the optimization frequency band was not matched on the two responses. On the other hand, this additional transmission zero plays the same role that the unwanted Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 transmission zero in the parallel-coupled line example did play. However, in this case, since approximate model is also capable of producing the same transmission zero, it is possible to include that in the optimization process as well. From practical point of view, the current configuration of filter does not give a high degree of freedom in positioning this extracted pole transmission zero, as a result it is rarely included as part of design process. 4.3.4 Lossy Parallel Coupled-Line Filter In this example, we present the EM optimization of a 4-pole lossy parallel-coupled line filter. The filter will be designed on a microstrip substrate with dielectric constant of 10.2 and thickness of 1.27 mm. The thickness of metal on microstrip substrate is 35 micron and the dielectric loss tangent is 0.0027. The filter is designed to have Chebyshev response with center frequency of 3.95 GHz and bandwidth of 500 MHz. The passband return loss level is 20 dB. The layout of filter is shown in Fig. 4.23. The resonators are tilted in order to reduce the width of metallic enclosure. The approximate model of this filter is constructed by cascading parallel coupled-line sections similar to the example in Section 4.3.1. Fig. 4.23. Layout of 4-pole lossy microstrip filter. All dimensions are in mm. Since we use the lossy model for the microstrip substrate, the approximate model will be able to predict the effect of finite Q of each resonator. The response of approximate model after conventional design and optimization process is shown in Fig. 4.24. The insertion loss of filter in the passband is around 1 dB. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 We use the model calibration procedure for EM optimization of this lossy filter. The magnitude and phase of the scattering parameters for the EM model of the filter at apx , computed using Momentum, along with the response of approximate model at the extracted point are shown in Fig. 4.25 and Fig. 4.26. -10 -15 “ -20 -25 -30 -35 -40 4.5 3.5 Frequency (GHz) Fig. 4.24. The optimal approximate model response of the lossy microstrip filter at x apx '21em‘ ’21apx^ 'l1apx^ -10 -15 “ - 20- -25 -30 -35 -40 3.4 3.6 3.8 4.2 Frequency (GHz) 4.4 4.6 Fig. 4.25. Magnitude of scattering parameters for the EM model of the lossy filter at initial point xfi> = x opx and the extracted approximate model at P (x ^ ). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill 200 100 -100 -200 3 .4 3 .6 4 .4 4 ,2 3 ,8 4 .6 F re q u e n c y (GHz) 200 ! ^ E \ 100 I <s i! 0 -100 -200 3 .4 3 .6 3 .8 4 .4 4 4 .2 F req u en cy (GHz) 4 .6 (b) Fig. 4.26. Phase of (a) S21 , and (b) Sn of the lossy filter for the EM model at initial point ( 1) * X em = X c^x and the extracted approximate model at . The model calibration process for this filter converged after 4 iterations. The final EM response of filter at is compared with the approximate model response at in Fig. 4.27. The final EM response matches the goal response with very good accuracy. This proves that developed EM optimization approach can be effectively used for lossy filters as well. Similar to the example in Section 4.3.1, the final EM response shows the existence of an unwanted transmission zero in the response. The values of optimization variables at , P (xl^^), and x’em - ^em ^^e shown in Table 4.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 2 -10 -20 -30 m -40 -50 -60 -70 3 .2 Fig. 3.4 ).8 4 4.2 F re q u e n c y (GHz) 3.6 4 .4 4 .6 4.27. Final optimized EM model response of 4-pole lossy microstrip filter after four iterations at point compared with the optimal approximate model response at x apx , P { x f^ ) , and x^^ = x^^ for lossy microstrip filter, Table 4.4. The design variables at Parameter Si S2 S3 L, L2 Ls Jt 0 )= em X * * = X (5) X em em apx 0.131 1.097 1.371 7.354 7.182 7.253 0.193 1.181 1.495 7.907 7.501 7.709 0.089 0.999 1.282 6.970 6.722 6.939 All values are in mm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 3 Conclusions In this Chapter, a robust and efficient CAD technique in the frame work of space mapping employing the EM optimization and linear model calibration for microwave filters was presented. The approximate model was iteratively calibrated with series of linear approximations of search direction in the EM model space. It was shown that in this process, the reflection functions of the EM and approximate models can be matched with very good accuracy even when the approximate model does not include all the unwanted electromagnetic effects of the actual filter including the unwanted transmission zeros. The nonuniqueness of parameter extraction step was addressed by proposing series of guidelines that could be used for improving the robustness of the process. Through several numerical examples, we have shown that the proposed approach can effectively minimize the number of full-wave EM simulations in CAD of microwave filters without any need for developing a complex approximate model for the filter. The method is applicable to both lossless and lossy filters. The method has also proven to be very successful in the EM optimization of microwave filters with intentional transmission zeros. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Conclusions and Future Works This thesis has presented efficient and robust optimization techniques for computer aided design and diagnosis of RF/microwave filters and diplexers. The main focus of our work has been on developing optimization methodologies that can address several challenges in the RF/microwave filter design cycle. One of these areas was development of a global optimization method for complex problems either in the process of approximate model design optimization or model parameter extractions. We also introduced an efficient model calibration approach in the framework of linear space mapping that can efficiently reduce the computational cost of full EM optimization for microwave filters. A brief review of the optimization techniques used by the majority of RF/microwave filter designers, which are also implemented in most commercial software suites, was presented in Chapter 2. The advantages and disadvantages of each method were briefly discussed. We also reviewed some basic concepts in filter optimization such as design specifications, definition of error functions, and formulation of parameter extraction. Among several existing optimization techniques, we showed that the unique properties of evolutionary programming make it more suitable for construction of an efficient global optimization method. After outlining the fundamentals of conventional EP in Chapter 3, we showed that the conventional EP cannot provide the true global minimum in many complex multimodal optimization problems especially in case of complex filters and diplexers. In order to address this issue, we proposed a novel integration of the EP with density clustering algorithm. This integration allowed us to not only identify the potentially important regions of the search space by analysis of the population but also reduce the redundancy in the search process. One of the drawbacks of the EP and any algorithm that merely rely on stochastic approach is the slow speed of convergence. Our proposed novel hybridization of the EP with the gradient-based search technique proved to be very efficient in solving many complex optimization problems. W e also proposed 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 several guidelines to enhance the efficiency of the hybrid EP specifically for microwave filters and diplexers optimization problems. The utilities of the hybrid EP was compared with several existing global optimization methods in solving several benchmark problems and in all cases, the new algorithm outperformed the other algorithms. The in-depth investigation of three practical examples was also presented in Chapter 3. The developed global optimization approach was successfully used for circuit synthesis of cross-coupled coaxial resonator filters. It was shown that the algorithm is capable of detecting the global solution as well as the strong local minima in these problems. The results for design optimization of a mobile base station diplexer with complex topology and large number of designable variables was presented. While our hybrid EP algorithm managed to find the global solution in a timely manner, the other optimization algorithms tested in this work were either trapped in the strong local minima or did not improve the solution after a large number of iterations. We also reported successful application of the developed hybrid EP in the computer aided diagnosis of an 8 -pole cross-coupled resonator filter. A proper formulation of the problem was presented to reduce the number of unknown. The algorithm successfully extracted the circuit model values for both tuned and mistuned filters. The performance of our method in terms of speed and robustness was favorably higher than the other algorithms that we tested. One of the advantages of the hybrid EP, which we presented in this dissertation, is that in addition to the RF/microwave filters and diplexers, it can be easily extended for solving a wide variety of global optimization problems in science and engineering as well. We have successfully used the hybrid EP for parametric modeling of flip-chip interconnects, planar antennas [5]-[6]. Chapter 4 of this thesis was concerned with development of a linear calibration method for EM optimization of microwave filters. It was shown how the proper formulation of the parameter extraction in the process of approximate model calibration can eliminate the need for inclusion of the unknown electromagnetic effects in the model of filter. The model calibration through reflection function response alignment showed to be a very powerful technique in EM optimization of microwave filter. The linear formulation of the proposed method makes it very suitable for implementation in the existing commercial EM simulators without the need for any additional link to an external program. Several Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 guidelines were proposed for reducing the nonuniqueness problem in the parameter extraction process. We tested the performance of our linear model calibration technique with several microwave filter design problems. In all cases, the developed method was able to successfully find the global EM solution with only a few EM simulations of the whole structure while using only simple approximate models of the filters. It was shown that the method can be equally applied to both lossless and lossy filters. We also showed that the method can be easily applied to EM optimization of cross-coupled filters which have intentional transmission zeros in their response. This was achieved by including minimal information from the transmission response around each transmission zero in the model calibration process. In summary, the hybrid EP can be used as an efficient and robust global optimization method for most difficult microwave filters and diplexers design and diagnosis problems. Nevertheless, the real advantage of the hybrid EP will be evident even more in cases that either a very good estimate of the initial solution does not exist or direct use of local search techniques such as gradient or quasi-Newton may not provide any optimal solution due to the sever multimodality of the associated objective function. After the design optimization of the approximate of the filters, the linear model calibration technique can be used to finalize the design cycle by EM optimization of the filter structure very fast. Future Works Based on the knowledge and experiences achieved in the course of this research we anticipate the following developments: • Currently, the forbidden zone scheme in the hybrid EP is applied only to the main EP algorithm. Use of these constraints can be extended to the local search stage through nonlinear programming approach. This will improve the speed of convergence by preventing the local searches inside the forbidden zones. • Incorporating the knowledge of the derivatives of scattering parameters into the local search in hybrid EP. When possible, the analytic sensitivity of the scattering Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 parameters with respect to the designable variables can be provided to the quasiNewton method to increase the speed of optimization by eliminating the extra effort for numerical approximation of the gradients. Use of the hybrid EP for design optimization of phase-equalized cross-coupled microwave filters. The optimization of these filters is potentially very challenging due to the sever multimodality of the objective functions. Investigating the influence of other unwanted effects such as radiation loss, dispersion, surface wave, etc. on the performance of model calibration technique when only the reflection function is used for calibration of the low order approximate model. Extending the linear calibration method to EM optimization of other electromagnetic structures through proper formulation of the parameter extraction step. Optimization of challenging multi-modal multi-layered printed antenna problems using Hybrid EP. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 Appendix A Prediction of Dissipation Loss in Microwave Filters The effect of loss on response of microwave filters can be simulated in many ways. The easiest method is by adding series resistors to the inductances or parallel resistors to the capacitances in the equivalent circuit of the filter. These additional resistances are function of the quality factor of the resonators in the filter. From mathematical point of view, the effect of these additional resistances is equivalent to shifting of imaginary axis in the complex plane. In order to show this, consider an inductor L with the reactance sL = jcoL , where s is the complex frequency variable and a> is the real angular frequency of operation. After adding a series resistor, r, to this inductance, the total impedance will be r + j coL . This new impedance can be achieved from the original lossless inductance by a simple shift of imaginary axis in complex plane to the right by cr ~ r IL . The new complex frequency variable will he s = a + jo ) and the reactance of the inductance after this shift will be sL ~(<j + jco)L ~ { r I L + jco)L = r + j coL , which is exactly the same as the impedance of lossy inductance. This process is graphically shown in Fig. A. 1. Lossv Lossless jcoL jo)L Im[s] i Im[s] 5 = j(0 t s =cr + j ' h- ■Re[s] ol <7 = r /L Re[5 ] L sL = j 0)L 1 sL —r + jO)L Fig. A, 1. Illustration of simulating the effect of loss by shifting the imaginary axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 The effect of loss on the capacitance can be modeled in the same way. However, it should be noted that the dominant loss in most microwave filters is due to finite conductivity of the metallic structures, which is eventually modeled by the resistors in series with the inductive elements of the equivalent circuit model of filter. For a filter with given zeros and poles, the shifting of imaginary axis to the right in complex plane will be equivalent to shifting of the zeros and poles to the left of complex plane. For passive structures, all the poles and zeros are on the left half plane. The smaller negative real value for poles (and zero) is equivalent of the higher loss. Based on the above discussions, determining the amount of shift in complex plane requires a priori knowledge of circuit elements which may not be available before the circuit synthesis stage. However, we will show that the amount of this shift can be found without the knowledge of circuit elements as well. We start by rewriting the loop equations for the lossy filter in Fig. 3.25 as |[i? ] + M + y Z ,(— - ^ ) [ / ] + 7 ® [ M ] \ [ J ] ^ [ E ] I (A.1) J O) where it is assumed that all resonators have the same capacitance, C = C , , and Zq = sI l / C is the internal impedance of the resonators. The frequency shift of each resonator is simulated by inclusion of the diagonal values in the coupling matrix [M ]. Assuming a similar quality factor, Q, for all the resonators, the matrix [r] is equal to r[I] where the value of series resistance, r, is equal to Zg IQ . Other parameters in (A .l) are the same as equation (3.26). After multiplying by /g !{BW .Zg), we will have [Z g B W ] + 7T— Q .B W 1+ i BW 0)g CO 1 + Z^BW ]}[/] = J Z^BW 1 (A .2) Recalling the band-pass to low-pass transformation, equation (A.2) can be written in terms of normalized angular frequency X in the low-pass as { [ R ] + i(X + j X ) U ] + j [ M ] ] [ i] = [ i ] where Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (A.3) 120 C7 = — ^ ----Q. B W (A.4) and other parameters are defined in (3.29). For the lossless filter, 2 = (or cr = 0 ), and the equation (A.3) reduces to {[R]+jA[I] + j [ M] } [J] = [E] (A.5) which is exactly the same as equation (3.28). 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