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

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Optimization 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
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Borrower’s Page
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this thesis. Please sign below, and give address and date.
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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
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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
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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.
VI
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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.
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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
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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.
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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
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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].
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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.
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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
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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
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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.
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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
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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
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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.
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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.
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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.
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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)
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(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
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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
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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
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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.
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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.
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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
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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
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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.
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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].
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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.
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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.
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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
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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.
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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 .
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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.
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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.
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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
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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.
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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).
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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 .
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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.
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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.
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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.
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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
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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
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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
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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
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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:
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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
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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.
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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
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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.
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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.
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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
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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.
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(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
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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
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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 .
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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
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(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.
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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
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(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.
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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].
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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.
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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.
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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
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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
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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.
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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
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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
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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 '
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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
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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.
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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
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(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.
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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
*
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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.
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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.
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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
).
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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
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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
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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.
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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 ^ ).
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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.
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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
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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.
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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
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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
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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
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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.
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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). Comparison between (A.3) and (A.5) shows
that inclusion of loss in the filter can be modeled by changing the normalized complex
frequency variable j A to a + j A. From the point of view of locations of poles and
zeros, if the imaginary axis of complex plane is shifted to the right by a , poles and zeros
in the new plane will reproduce the lossy filter response. This is equivalent to shifting all
zeros and poles to the left by the amount of c r.
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