Optimization of RF and
Microwave Filters using ML

techniques

Capstone Report

Kadyrzhan Tortayev

Nazarbayev University
Department of Electrical and Computer Engineering

School of Engineering and Digital Sciences



Copyright © Nazabayev University

This project report was created on TexStudio editing platform using LATEX. All the figures
were drawn using draw.io online software tool.



Electrical and Computer Engineering
Nazarbayev University

http://www.nu.edu.kz

Title:
Optimization of RF and Microwave Fil-
ters using ML techniques

Theme:
RF Engineering

Project Period:
Spring 2024

Project Group:
RF circuit design

Participant(s):
Kadyrzhan Tortayev

Supervisor(s):
Mohammad Hashmi

Copies: 1

Page Numbers: 29

Date of Completion:
April 23, 2024

Abstract:

Designing high-performance mi-
crowave and millimeter-wave filters
presents a significant challenge due to
the sensitivity of filter characteristics
to variations in geometric dimensions
and electrical sizes. Typically, filter
design involves optimizing design
variables, starting from initial values.
However, if these initial values are
too far from the optimal solution,
optimization often fails to yield satis-
factory results. To address this issue,
this project analyzes current methods
used in optimization of microwave
and millimeter-wave filters.

The content of this report is freely available, but publication (with reference) may only be pursued due to

agreement with the author(s).

http://www.nu.edu.kz




Contents

Preface vi

1 Introduction 1

2 Background 3
2.1 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Homotopy Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Methodology 7
3.1 Circuit Design of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Training ML model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Data Collection and Preprocessing . . . . . . . . . . . . . . . . 8
3.2.2 Training and model selection . . . . . . . . . . . . . . . . . . . 9

3.3 Integration with Optimization Methods . . . . . . . . . . . . . . . . . 9
3.4 Validation and Practical Considerations . . . . . . . . . . . . . . . . . 11

4 Results and Discussions 12
4.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1.1 Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1.2 Training ML model . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1.3 Integration with optimization . . . . . . . . . . . . . . . . . . . 14
4.1.4 Validation and Practical Design . . . . . . . . . . . . . . . . . 16

4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Conclusion 21

Bibliography 22

A Circuits 25

B Waveguide 27

v



Preface

In the dynamic landscape of modern technology, the design and optimization of
microwave and millimeter-wave filters stand as critical challenges. These intricate
devices play a pivotal role in shaping the efficiency and performance of commu-
nication systems, radar systems, and numerous other applications at the heart of
our interconnected world. To unlock their full potential, engineers and researchers
have long sought innovative solutions to tackle the inherent complexity of filter
design. Within this area of engineering, several approaches have emerged as a bea-
con of promise. Yet, as we delve deeper into the complexities of filter optimization,
a parallel revolution has been underway—the rise of Artificial Neural Networks
(ANNs). These computational marvels have rapidly evolved, proving themselves
as invaluable tools for modeling complex systems, accelerating simulations, and
aiding in the optimization process. .

Nazarbayev University, April 23, 2024

Kadyrzhan Tortayev
<kadyrzhan.tortayev@nu.edu.kz>

vi



Chapter 1

Introduction

Microwave filter design can be a challenging task, and there are several common
problems that designers can encounter. One of the primary challenges in filter de-
sign is meeting the desired performance specifications. These specifications may
include requirements for the frequency response, phase response, group delay,
and stopband attenuation. It can be difficult to achieve the desired performance
while also considering other design constraints such as cost, size, and complex-
ity. The goal is to develop more efficient and accurate methods for designing and
optimizing filters with better performance metrics such as low insertion loss, high
selectivity, and wide bandwidth. Some of the most popular optimization algo-
rithms are Particle Swarm Optimization (PSO), Genetic Algorithms (GA) which
are considered to be iterative optimization algorithms. Other methods include
Electromagnetic Simulators and Coupling matrices.

Coupling matrices in high-performance narrowband filters can be effectively
optimized through various analytical techniques aimed at diagnosing the coupling
matrix from either measured or simulated S-parameters [1], [2]. By discerning
the variance between the realized coupling matrix and the intended design target-
coupling matrix, and establishing a direct correspondence between coupling matrix
elements and the associated physical tuning components, one can readily deter-
mine the necessary adjustments to the design parameters [3]. Liu et al. [2]pro-
posed a general lumped element matrix synthesis method for cross-coupled and
inline wideband BPF with multiple zeros for direct circuit implementation. Using
the proposed method, the elements in the synthesized coupling matrix can corre-
spond to the values of capacitance and inductance in the actual circuit without any
optimization, which simplifies the design process of wideband lumped element
filters. Sandhu et al. [1] created a new class of frequency-variant reactive cou-
pling networks with singular responses. These networks are intended to be used
as building blocks for generalized-Chebyshev-type bandpass filters serving as non-
ideal frequency-variant inverters while providing two TZs and one pole. Wu et al.

1



2 Chapter 1. Introduction

[3] were able to reconfigure a coupling matrix from the transversal topology to an
arbitrary required topology, which is viewed as a simple optimization problem,
where the developed error function is computationally efficient.

This article also explores the homotopy method [4] to the local optimization of
microwave filters. Homotopy, a concept in topology and differential geometry, has
found utility in numerical methods for solving nonlinear equations and differen-
tial equations [5]. Rather than directly tackling the target filter design problem,
the homotopy method establishes a sequence of intermediate optimization prob-
lems. Utilizing previous solutions as initial values, an existing local optimization
technique can solve these intermediate problems. Through this iterative process
of homotopy optimizations, the filter response can gradually converge towards the
desired specification. Even when the initial values for the filter design are far from
optimal, the homotopy method offers a high likelihood of reaching the optimal
solution.

Another valuable application of the homotopy optimization method arises in
filter bank design for frequency multiplexing systems. Here, if a filter design is
available, it can serve as the initial values for optimizing other filters with varying
center frequencies and bandwidths. The optimization outcomes of multiple filters
operating at different center frequencies and bandwidths can further inform the
necessary parameter variations for designing tunable filters.



Chapter 2

Background

2.1 Particle Swarm Optimization

Particle Swarm Optimization (PSO) is a popular optimization algorithm inspired
by the social behavior of birds and fish. Developed by Eberhart and Kennedy
in 1995, PSO is used to find approximate solutions to optimization and search
problems. In the context of optimization, a population of potential solutions is re-
ferred to as a "swarm," and each solution is a "particle." Each particle in the swarm
represents a potential solution to the optimization problem. The position of a par-
ticle corresponds to a point in the search space, and the quality of that solution
is evaluated using an objective function Particle Swarm Optimization (PSO) has
been applied successfully to filter optimization problems in various research pa-
pers [6], [7], [8], [9]. Neural network architectures often require optimized filters
for tasks such as feature extraction or convolutional operations. PSO can be par-
ticularly effective in tuning these filters, especially when the relationships between
filter parameters and network performance are complex. The ability of PSO to
navigate high-dimensional spaces makes it valuable in optimizing filters for neural
networks.

3



4 Chapter 2. Background

Figure 2.1: PSO

.

2.2 Genetic Algorithms

Genetic Algorithms (GAs) are optimization algorithms inspired by the process of
natural selection and genetics. Developed by John Holland in the 1960s, genetic al-
gorithms are part of a broader class of evolutionary algorithms and are used to find
approximate solutions to optimization and search problems. GAs are particularly
useful in complex, multidimensional search spaces where traditional optimization
methods may struggle. Genetic Algorithms (GAs) have been widely employed in
the optimization of filters across various applications [10], [11], [12], [13]. Genetic
Algorithms represent filters as chromosomes or individuals. Each chromosome en-
codes the parameters of the filter, such as coefficients or frequency response charac-
teristics. GAs are capable of handling complex, non-linear optimization problems.
They provide a global search strategy, enabling the exploration of a large solu-
tion space. The adaptability of GAs makes them suitable for diverse filter design
requirements.



2.3. Homotopy Optimization 5

Figure 2.2: GA

.

2.3 Homotopy Optimization

Homotopy optimization is an optimization technique that combines elements of
homotopy continuation and optimization methods. It is particularly useful for
solving nonlinear optimization problems, especially those with multiple local min-
ima or when the objective function is non-convex. While the application of Ho-
motopy methods in filter optimization may not be as widespread as some other
optimization techniques like Genetic Algorithms or Particle Swarm Optimization,
there is research exploring the potential of Homotopy for this purpose. Homo-
topy optimization involves the continuous deformation of a complex optimization
problem into a simpler, more solvable form [14]. The application of homotopy
optimization to filters can be particularly advantageous in scenarios where the
filter design problem is complex, has multiple solutions, or involves challenging
optimization landscapes. It provides a systematic and controlled way to navigate
through the solution space, potentially overcoming some of the difficulties associ-
ated with traditional optimization methods.

For instance if we define variable P as:

P = [ f 1 f 2 f 3 f 4r] (2.1)

where the stopband edge frequencies are f1 and f4 and the passband is between
frequency f2 and f3 with f1 < f2 <f3 < f4. f1– f4 are the normalized frequency
variables computed by transformation using minimum and maximum value, r is



6 Chapter 2. Background

the target RL level in each optimization
Then, we can define a homotopy function as

P(P1, L) = (1 − L)P1 + L ∗ Pt; (2.2)

where L changes from 0 to 1 with a step of 0.1 and Pt is the target variable.

Figure 2.3: homotpy

.



Chapter 3

Methodology

3.1 Circuit Design of Filters

A 5-th order low-pass, 3rd order band-stop, and 5th order coupled line band-
pass filters were designed using transmission lines and Advanced Design System
software with the following parameters:

Lowpass:

• Fc = 3 GHz

• ripple = 0.5 dB

• 40 dB attenuation at 6 Ghz

Bandstop:

• F0 = 4 GHz

• ripple = 3 dB

• BW = 50

Bandpass

• F0 = 5 GHz

• ripple = 3 dB

• Fh = 5.2 GHz

• Fl = 4.8 GHz

7



8 Chapter 3. Methodology

Figure 3.1: LPF

.

Figure 3.2: BSF

.

Additional circuits can be found in Appendix A.

3.2 Training ML model

3.2.1 Data Collection and Preprocessing

• A comprehensive dataset on each filters design containing information on
filter S parameters was obtained.

• The dataset covered two design variations to capture the diversity of filter
configurations and performance outcomes.

• The dataset was standardized by min maxing the features to ensure uniform
scales and facilitate convergence during training.



3.3. Integration with Optimization Methods 9

Figure 3.3: BPF

.
Table 3.1: Filter Param

type w1 w Step size Frequency Points per sweep
waveguide 0.3 – 0.9 mm 0.2-0.8 mm 0.1 mm 1001

lpf 200-600 mils 100-500 mils 40 mils 10001
bsf 40-120 mils 30-90 mils 8, 6 mils 10001
bpf 10-260 mils 10-260 mils 25 mils 10001

3.2.2 Training and model selection

• The dataset was split into training, validation, and test sets to evaluate model
performance effectively.

• An decision tree model architecture was chosen for regression tasks based on
the nature of the problem and available data.

• The trained ML model’s performance was evaluated on the validation set
using appropriate evaluation metrics, such as mean squared error (MSE),
mean absolute error (MAE), or R-squared (R2) score.

• Model predictions were compared with ground truth filter characteristics to
assess accuracy and reliability.

3.3 Integration with Optimization Methods

• The optimization problem was defined according with the objective for de-
signing microwave and millimeter-wave filters, including the objective func-
tion to be optimized and the constraints on design variables. For instance for



10 Chapter 3. Methodology

Figure 3.4: substrate

.



3.4. Validation and Practical Considerations 11

LPF it was defined that

min(db(S(2, 1)))inpassband = 0.5db (3.1)

• The trained ML model was integrated with optimization methods such as
Particle Swarm Optimization, Genetic Algorithm, or Homotopy for filter de-
sign optimization.

• The ML model was utilized to predict filter characteristics for different design
configurations and guide the optimization process towards achieving optimal
solutions.

3.4 Validation and Practical Considerations

• Simulation Validation: Validate optimized filter designs through electromag-
netic simulations or laboratory experiments to ensure that they meet perfor-
mance specifications under real-world conditions.

• Practical Implementation Considerations: Consider practical implementation
factors such as manufacturability, cost, and scalability when assessing the fea-
sibility of optimized filter designs. Analyze trade-offs between performance
and practical constraints.

.



Chapter 4

Results and Discussions

4.1 Results

4.1.1 Circuit Design

(a) LPF Mline (b) BSF Mline

Figure 4.1: LPF and BSF Mline layout

The circuit design process involved the development of low-pass (LPF), band-
stop (BSF), and band-pass (BPF) filters using microstrip line configurations on FR4
substrate. Microstrip lines offer a compact and versatile solution for microwave and
millimeter-wave filters, while FR4 substrate is a commonly used dielectric material
known for its low cost and ease of fabrication. A 5-pole rectangular waveguide
filter was also designed using CST studio. Appendix A provides the EM simulation
of circuits.

12



4.1. Results 13

(a) 5 pole rectangular waveguide filter (b) BPF Mline

Figure 4.2: Waveguide and BPF Mline layout

4.1.2 Training ML model

Figure 4.3: BSF ANN vs EM comparison

.
The training of the machine learning (ML) model yielded promising results,

demonstrating the efficacy of the XGBoost regressor in predicting filter character-
istics with high accuracy. The model achieved an impressive R-squared value of
0.99, indicating that approximately 99 percent of the variance in the filter character-
istics can be explained by the model. Additionally, the mean squared error (MSE)
of 3e-4 reflects the small magnitude of errors between the predicted and actual
filter parameters, further underscoring the robustness of the trained model.



14 Chapter 4. Results and Discussions

(a) BPF (b) LPF

Figure 4.4: ANN vs EM comparison

(a) Caption1 (b) Caption 2

Figure 4.5: ANN vs EM comparison

4.1.3 Integration with optimization



4.1. Results 15

(a) LPF optim (b) bsfoptim

Figure 4.6: LPF and BSF optimization

Table 4.1: Obtained Filter Param

type Filter type w1 w

Homotopy
lpf 335 194
bsf 20 3
bpf 32 3

PSO
lpf 294 150
bsf 16 3
bpf 38 3

GA
lpf 231 235
bsf 15 3
bpf 41 3

Optimized
lpf 231 182
bsf 21 3
bpf 32 3

Default
lpf 409 318
bsf 22 3
bpf 50 3

The table presents the obtained filter parameters for different optimization
methods, including Homotopy, Particle Swarm Optimization (PSO), and Genetic
Algorithm (GA), as well as the optimized and default settings. These parameters,
represented by w1 and w, provide crucial insights into the performance of each op-
timization technique in designing low-pass (LPF), band-stop (BSF), and band-pass
(BPF) filters for microwave and millimeter-wave applications.



16 Chapter 4. Results and Discussions

(a) BPF optimized (b) Waveguide Optimized using Homotopy

Figure 4.7: BPF and waveguide optimization

4.1.4 Validation and Practical Design

(a) LPF Mline (b) BSF Mline

Figure 4.8: S21 for Mline BSF and LPF

The validation and practical design aspects are crucial components in assessing
the real-world applicability and reliability of the obtained filter parameter results.
These considerations ensure that the designed filters not only meet the desired
specifications but also perform effectively under practical operating conditions.
The designed filters were validated through experimental testing using RF mea-
surement equipment, namely vector network analyzers (VNAs). The validation of
waveguide filters could not be performed due to the inherent differences in tech-
nology and fabrication processes between microstrip and waveguide technologies.



4.1. Results 17

(a) BPF Mline (b) BPF MLine S11

Figure 4.9: BPF Mline S param

Table 4.2: Measured S param from printed designs

type Filter type min(db(S(2,1))) Target min(db(S(2,1))) F passband

Homotopy
lpf 2.6 0.5 <3Ghz
bsf 20 3 out(3-5) Ghz
bpf 32 3 in(4.8-5.2) Ghz

PSO
lpf 2.4 0.5 <3 Ghz
bsf 16 3 out(3-5) Ghz
bpf 38 3 in(4.8-5.2) Ghz

GA
lpf 1.8 0.5 <3 Ghz
bsf 15 3 out(3-5) Ghz
bpf 41 3 in(4.8-5.2) Ghz

Optimized
lpf 1.38 0.5 <3 Ghz
bsf 21 3 out(3-5) Ghz
bpf 32 3 in(4.8-5.2) Ghz

Default
lpf 13 0.5 <3 Ghz
bsf 22 3 out(3-5) Ghz
bpf 50 3 in(4.8-5.2) Ghz

Therefore, the validation process focused solely on microstrip filter designs, while
waveguide filters were excluded from validation due to technological constraints.
The optimized results for waveguide can be found in Appendix B.



18 Chapter 4. Results and Discussions

Figure 4.10: VNA

.

Figure 4.11: MLine optimized filters.

.



4.2. Discussion 19

4.2 Discussion

The table 4.1 presents the obtained filter parameters for different optimization
methods, including Homotopy, Particle Swarm Optimization (PSO), and Genetic
Algorithm (GA), as well as the optimized, and default settings. The parameters in-
clude the type of filter (LPF for Low Pass Filter, BSF for Band Stop Filter, and BPF
for Band Pass Filter), and the corresponding values of w1 , w which represent de-
sign parameters of each filter type. Upon analyzing the table, several observations
can be made:

1. Variation Across Optimization Methods:

• The values of w1 and w vary significantly across different optimization
methods for each filter type. This indicates that the optimization meth-
ods yield different sets of parameters in an attempt to optimize the filter
design for the given objectives.

• For example, for the LPF type, the values of w1 range from 231 to 409,
showing a notable difference between the optimized and default set-
tings.

2. Effectiveness of Optimization

• The values obtained through the "Optimized" category indicate the per-
formance of the optimization methods in achieving the desired filter
characteristics. Compared to the "Default" settings, the optimized values
generally show improvements, suggesting that the optimization meth-
ods have effectively tuned the filter parameters to enhance performance.

• Notably, for all filter types, the optimized values of w1 and w tend to
be lower than the default values. This suggests that the optimization
methods have succeeded in reducing certain aspects of the filter design,
such as insertion loss or passband ripple.

3. Comparison with Homotopy Method:

• Comparing the values obtained from the Homotopy method with those
from PSO and GA, we observe differences in the magnitude of w1 and w
for each filter type. This indicates variations in the optimization results
achieved by different methods.

• Notably, for all filter types, the optimized values of w1 and w tend to
be lower than the default values. This suggests that the optimization
methods have succeeded in reducing certain aspects of the filter design,
such as insertion loss or passband ripple.



20 Chapter 4. Results and Discussions

4. Implications for Filter Design

• The table highlights the importance of optimization methods in fine-
tuning filter parameters to meet specific design requirements. By lever-
aging optimization algorithms, engineers can achieve better performance
and tailor filters to suit particular applications or specifications.

• The differences in parameter values across optimization methods under-
score the need for careful selection of optimization techniques based on
design objectives, computational resources, and constraints.

The measured S-parameter results from the printed filter designs in table 4.2
provide valuable insights into the performance of low-pass (LPF), band-stop (BSF),
and band-pass (BPF) filters optimized using different optimization methods, in-
cluding Homotopy, Particle Swarm Optimization (PSO), and Genetic Algorithm
(GA), as well as the optimized and default settings. These results are crucial in
evaluating the effectiveness of each optimization technique in achieving the de-
sired filter characteristics and meeting specified performance targets.

1. Comparison Across Optimization Methods

• The measured minimum dB(S(2,1)) values for each filter type demon-
strate the effectiveness of different optimization methods in achieving
the desired filter responses.

• Across all filter types, the optimized settings consistently outperform the
default settings, indicating the importance of optimization techniques in
enhancing filter performance.

• Notably, the GA method tends to yield the lowest minimum dB(S(2,1))
values among the optimization methods for LPF and BPF, suggesting
superior performance in these cases.

2. Adherence to Performance Targets

• The measured minimum dB(S(2,1)) values are compared against the tar-
get minimum dB(S(2,1)) values to assess the filters’ adherence to speci-
fied performance targets.

• For LPF, all optimization methods successfully achieve minimum dB(S(2,1))
values below the target of 0.5 dB, indicating excellent passband perfor-
mance.

• Similarly, for BSF and BPF, the filters meet the target minimum dB(S(2,1))
values, ensuring effective rejection of unwanted frequencies in the stop-
band or out-of-band regions.



Chapter 5

Conclusion

In conclusion, the results provides valuable insights into the effectiveness of differ-
ent optimization methods in tuning filter parameters for microwave and millimeter-
wave filters. It demonstrates the impact of optimization on filter performance
and underscores the importance of optimization techniques in achieving high-
performance filter designs. Further analysis and experimentation could explore
additional optimization methods or refine existing techniques to enhance filter de-
sign capabilities and achieve superior performance in microwave and millimeter-
wave applications. Integration of machine learning approaches with optimization
methods may offer opportunities for automated design optimization and acceler-
ated development of high-performance filters. Continued research and develop-
ment efforts are essential to advance the field of RF and microwave engineering
and address emerging challenges in filter design and optimization.

21



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Appendix A

Circuits

(a) LPF (b) BPF

Figure A.1: Circuits of LPF and BPF

25



26 Appendix A. Circuits

Figure A.2: MLine BSF

.



Appendix B

Waveguide

Table B.1: Filter parameters

Value
width 1.2954 mm
height 0.6477 mm

Operating frequency 140-220 GHz

We can decompose the filter into 6 subsections B.1. Each subsection can be
modified by changing their width and length. 49 frequency sweeps were made
with width ranging from 0.3-0.9 mm and length 0.2-0.8 mm as seen in ??.

Table B.2: Filter parameters of waveguide

type w1 w2 w3 l1 l2 l3
Default 0.7 mm 0.6 mm 0.5mm 0.4mm 0.5 mm 0.5 mm

Homotopy 0.615 mm 0.366 mm 0.312 mm 0.501 mm 0.608 mm 0.623 mm
PSO 0.527 mm 0.267 mm 0.226 mm 0.539 mm 0.626 mm 0.634 mm
GA 0.661 mm 0.422 mm 0.360 mm 0.453 mm 0.563mm 0.584 mm

The S-parameter cascading block computes the two-port S-parameters S filter
for the entire filter by cascading the six two-port S-parameters of the submodels.
The surrogate model can replace the full-wave EM simulation in the homotopy
optimizations.

Cost function to be minimized is eq. B.1:

K = max(db(S11)inpassband,−r) + w ∗ max(db(S21)instopband,−40) (B.1)

27



28 Appendix B. Waveguide

Figure B.1: Five pole rectangular waveguide filter decomposition[14]

.

Figure B.2: Homotopy L = 6

.



29

(a) Homotopy L = 8 (b) Homotopy L = 5

Figure B.3: Homotopy

(a) PSO (b) GA

Figure B.4: PSO and GA


	Front page
	English title page
	Contents
	Preface
	1 Introduction
	2 Background
	2.1 Particle Swarm Optimization
	2.2 Genetic Algorithms
	2.3 Homotopy Optimization

	3 Methodology
	3.1 Circuit Design of Filters
	3.2 Training ML model
	3.2.1 Data Collection and Preprocessing
	3.2.2 Training and model selection

	3.3 Integration with Optimization Methods
	3.4 Validation and Practical Considerations

	4 Results and Discussions
	4.1 Results
	4.1.1 Circuit Design
	4.1.2 Training ML model
	4.1.3 Integration with optimization
	4.1.4 Validation and Practical Design

	4.2 Discussion

	5 Conclusion
	Bibliography
	A Circuits
	B Waveguide