DESIGN AND IMPLEMENTATION OF

FRACTIONAL-ORDER BUTTERWORTH FILTER

by

Nikhil Avneet Singh

A thesis submitted in fulfillment of the

requirements for the degree of

Master of Science

Copyright c©2018 by Nikhil Avneet Singh

School of Engineering and Physics

Faculty of Science, Technology and Environment

The University of the South Pacific

October 2018

If You Are Not Failing, You Are Not Doing it Right

- - - Nikhil Singh

Abstract

This thesis investigates the importance and novelty in the design and implementation

of fractional order Butterworth filter of order (1 + α). The filter coefficients with de-

sired specification are provided after constraint optimization. The major constraints

set to develop the fractional step filter and assuring better result with least square er-

ror in magnitude responses, less sensitivity to parameter variation, least passband and

stopband errors and -3dB frequency approximately 1 rad/s. Commonly used approx-

imation, namely continued fractional expansion is adopted for fractional Laplacian

operator sα. The transformation technique is verified to extend same for the high-

pass fractional order filter. Finally, experimental results are validated with simulation

for various fractional step values. More importantly, it is realized and implemented

on hardware environment with the help of analog reconfigurable Field Programmable

Analog Array (FPAA). Experimental results have proven the possibility to implement

the actual fractional filter behavior with closest approximation to the theoretical design.

Keywords: Fractional order Butterworth filter, optimization, FPAA, realization, least

square error, passband and stopband errors, continued fractional expansion, fractional

Laplacian operator.

iv

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction 1

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Background and Literature Work 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Approximation for the General Laplacian Operator . . . . . . . . . . 9

2.4 Fractional Order Filters . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 FPAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Fractional Order Low Pass Filter 17

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Fractional-Order Low Pass Filter Transfer Function of (1 + α) Order . 18

3.3 Previously Selected Coefficients in Optimization Framework . . . . . 19

3.4 Modified PSO for Bilevel Optimization . . . . . . . . . . . . . . . . 20

3.5 Numerical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 23

v

3.5.1 Magnitude Response Errors . . . . . . . . . . . . . . . . . . 24

3.5.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.3 -3dB Frequency Analysis . . . . . . . . . . . . . . . . . . . . 27

3.5.4 Stop Band Attenuation . . . . . . . . . . . . . . . . . . . . . 28

3.6 Sensitivity to Parameter Variation . . . . . . . . . . . . . . . . . . . 29

3.6.1 -3dB Frequency Response to Parameter Variation . . . . . . . 30

3.6.2 Stop Band Attenuation Error to Parameter Variation . . . . . 33

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Fractional Order High Pass Filter 34

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Fractional Order Low Pass to High Pass Transformation . . . . . . . . 34

4.3 Evaluation of the High Pass Filter Transfer Functions . . . . . . . . . 36

4.4 Least Square Error Analysis . . . . . . . . . . . . . . . . . . . . . . 37

4.5 Pass Band and Stop Band Error Analysis . . . . . . . . . . . . . . . . 38

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Hardware Implementation and Realization of Fraction Order Filters 41

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Approximation of Fractional Laplacian Operator to Fractional Order

Low Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.3 Input and Output Interface with FPAA . . . . . . . . . . . . . . . . . 42

5.4 Implementation of Fractional Order Filters on FPAA . . . . . . . . . 44

5.4.1 FPAA Implementation of (1 + α) Order Low Pass Filter . . . 45

5.4.2 Experimental Results for (1 + α) Order Low Pass Filter . . . 50

5.4.3 FPAA Implementation of (1 + α) Order High Pass Filter . . . 53

5.4.4 Experimental Results for (1 + α) Order High Pass Filter . . . 55

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

vi

6 Conclusions and Future Work 59

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Bibliography 61

Appendix 66

Fractional Order Butterworth Filter Design Algorithms . . . . . . . . . . . 66

vii

List of Figures

1.1 Magnitude responses of lowpass Butterworth filter for orders n = 1 to

n = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Fractional order element classification diagram . . . . . . . . . . . . 9

2.2 Magnitude and phase response of second order approximation (dashed

line) and third order approximation (dotted line), of sα for the case

α = 0.5 compared with ideal (solid line) . . . . . . . . . . . . . . . . 10

2.3 Magnitude and phase response of Carlson’s, Matsuda’s, Oustaloop’s

and second order CFE approximation methods to approximate sα for

α = 0.5 compared with ideal case . . . . . . . . . . . . . . . . . . . 12

2.4 Functional block diagram of Anadigm FPAA [5] . . . . . . . . . . . 16

3.1 Flow diagram of bi-level PSO algorithm . . . . . . . . . . . . . . . . 22

3.2 k2,3 coefficients to approximate fractional step filters, compared to co-

efficients presented in [17] and [15] respectively . . . . . . . . . . . . 23

3.3 PE index values, 1: by [17], 2: by [15] and 3: by proposed . . . . . . 25

3.4 SE index values, 1: by [17], 2: by [15] and 3: by proposed . . . . . . 26

3.5 Minimum root angle in W-plane for, 1: by [17], 2: [15] and 3: by

proposed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 -3dB frequency using 1: by [17], 2: [15] and 3: by proposed . . . . . 28

3.7 Stopband attenuation for (1 + α) order lowpass Butterworth filter im-

plementation using, 1: [17], 2: [15] and 3: the proposed; for ω =

[1,10](blue) lines and for ω = [10,100](red) lines compared to the

ideal attenuation (green) lines. . . . . . . . . . . . . . . . . . . . . . 29

3.8 Percentage error in -3db frequency: (1) in [17], (2) in [15] and (3) in

Proposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 Percentage error in stopband attenuation: (1) in [17], (2) in [15] and

(3) in Proposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Magnitude characteristics of 1. (4.1), 2. (4.2) and 3. (4.3) . . . . . . . 36

4.2 LSEs from 1.(4.1), 2.(4.2) and 3.(4.3) for α ∈ (0.01, 0.99) . . . . . . 37

4.3 Stopband error index values of 1.(4.1), 2.(4.2), 3.(4.3) for (1+α) from

1.1 to 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

viii

4.4 Passband error index values of 1.(4.1), 2.(4.2), 3.(4.3) for (1+α) from

1.1 to 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 Coefficients k2,3 w.r.t. α values . . . . . . . . . . . . . . . . . . . . . 42

5.2 Circuits to interface FPAA with external signals . . . . . . . . . . . . 44

5.3 Block diagram of test setup . . . . . . . . . . . . . . . . . . . . . . . 45

5.4 (a).Bilinear and biquadratic filter CAMs in Anadigm Designer envi-

ronment, cascaded to implement a fractional order lowpass filter. (b).

FPAA development board . . . . . . . . . . . . . . . . . . . . . . . . 46

5.5 Internal switched capacitor circuit to realize (a) lowpass filter bilinear

cam (b) bi-quadratic filter cam . . . . . . . . . . . . . . . . . . . . . 47

5.6 Setup of parameter (a) bilinear filter CAM for (1 + α) = 1.2 (b) bi-

quadratic filter CAM using PZ parameters for (1 + α) = 1.2 . . . . . . 49

5.7 Experimental setup for hardware implementation . . . . . . . . . . . 50

5.8 Simulation (solid line) and Experimental (dashed line) results for (1 +α) order fractional order lowpass filter . . . . . . . . . . . . . . . . . 51

5.9 Oscilloscope output for input and output response of the designed (1+α) = 1.2 fractional order lowpass filter . . . . . . . . . . . . . . . . . 52

5.10 Implementation of fractional order highpass filter using bilinear and

biquadratic filter CAMs. . . . . . . . . . . . . . . . . . . . . . . . . 55

5.11 Simulated (solid line) and Experimental (dashed line) results for (1 +α) order fractional order highpass filter . . . . . . . . . . . . . . . . 56

5.12 Waveforms for (1 + α) = 1.2 fractional order highpass filter . . . . . . 57

ix

List of Tables

2.1 Fractional integral definition (α > 0) . . . . . . . . . . . . . . . . . . 7

2.2 Fractional derivative definition (α > 0) . . . . . . . . . . . . . . . . 7

2.3 Rational approximation for (sα) . . . . . . . . . . . . . . . . . . . . 11

3.1 Comparison of PE and SE matrices for (1 + α) order filters . . . . . . 24

4.1 Comparison of PE and SE matrices for (1 + α) order highpass filters . 39

5.1 Theoretical and realised biquad and bilinear CAM parameter values

for physical implementation of (1 + α) order fractional lowpass filter 48

5.2 Theoretical and realised biquad and bilinear CAM parameter values

for physical implementation of (1 + α) order fractional highpass filter 54

x

Acknowledgments

Firstly, I would like to thank God for giving me knowledge and strength to complete

this thesis. Secondly, this thesis would not have been possible without the help of

many people. I would like to thank my supervisor, Dr. Utkal Mehta. He has provided

me with endless support and guidance throughout the entire process. I could not have

imagined a better mentor than him, sharing his knowledge and expertise on signal

processing and control systems. His words of motivation has driven me to successfully

compile this thesis to the best of my ability. I also extend my sincere gratitude and

appreciation to my co-supervisor and the Head of School of Engineering and Physics

Professor Maurizio Cirrincione for his continuously support in my research.

I thank the former Dean of the Faculty of Science, Technology and Environment

(FSTE) Associate Professor Dr. Anjeela Jokhan, for keeping track of my project

progress and supporting me when needed. Moreover, my sincere gratitude goes to

my friends Swastika Devi, Arishnil Bali, Shavneel Muttu, Jashvir Bir, Kunal Singh,

Ritesh Naidu and Darrel Lal for encouraging me to push myself further every inch

in this mile long journey. I would specially like to thank Mrs. Kajal Kothari Par-

mar, who has helped me intensively in software simulations and has guided me along

this research. Also, I thank the lab technicians Abdul Shaiyaz Khan and Binal Raj

and chief technician Radesh Lal for providing technical support to complete hardware

experiments.

Finally, I sincerely thank everyone who has supported me in any way towards the

completion of this thesis.

xi

Dedication

I dedicate this thesis to my mother.

Chapter 1

Introduction

1.1 General Introduction

Electronic filters are circuits consisting of resistors, capacitors, inductors and special-

ized elements such as operational amplifiers that perform signal processing actions.

These signal processing actions can be either direct, channel, integrate, separate, de-

lay, differentiate, attenuate and transform signals [49]. The most common and the

basic filter type that would be studied in this research is analog filters. Analog filter is

the basic building block in signal processing. Some of the basic applications of analog

filters in our everyday life are; selecting of a particular radio station from a wide range

of radio stations by choosing a particular frequency and rejecting all the other frequen-

cies, mobile and telephone communication use analog filters to channel signals from

transmitter to receiver and vise versa by rejecting unwanted signals upon frequency

selection.

Most commonly any filter can be classified into active and passive and depending on

shape of the response can be further categorized into five classes that include:

• Low Pass filter: an ideal filter allows signals with frequency less than a specified

cut-off frequency to pass through and prevents those with frequencies above the

cut-off frequency.

• High Pass filter: an ideal filter cuts off or attenuates all signals with frequency

below a specified cut-off frequency and allows signals with frequency higher

than cut-off frequency to penetrate through.

• Band Pass filter: an ideal filter allows signals within a certain bandwidth between

two specified frequencies to pass through and rejects signals with frequencies

outside this specified range.

1

• Band Reject filter: an ideal filter allows signals with any frequency to penetrate

through except those signal that falls under a precisely selected bandwidth be-

tween two critical frequency boundaries.

• All Pass Filter: an ideal filter allows all signals to pass through, therefore its

magnitude response is uninteresting, however the phase response of the signals

passing through the filter can be altered without having any impact on the signal

amplitude.

The four main filter design approximations are Butterworth, Chebyshev, Inverse Cheby-

shev and Elliptical functions [20]. These are mathematically approximated with the

best fit transfer function for the filter design.

In our work we will focus mainly on design and optimization of coefficients of fractional-

order lowpass and highpass Butterworth filters in the frequency domain. This is to note

that Butterworth filter is most commonly utilized for filter applications. It was firstly

presented in 1930 by Stephen Butterworth and since then it is most commonly used

analog filter model [7]. Butterworth filter has magnitude response with passband as

flat as possible in the frequency domain. This filter has monotonic frequency response

and the steepness or roll-off rate from passband to stopband is determined by the order

of filter.

Frequency(rad/s)10-2 10-1 100 101 102

Magnitude(dB)

-300

-250

-200

-150

-100

-50

0

50

n = 2

n = 4

n = 5

n = 6

n = 7

n = 1

n = 3

Figure 1.1: Magnitude responses of lowpass Butterworth filter for orders n = 1 to

n = 7

2

The Butterworth approximation becomes closer approximation of ideal case as the

order, n, increases, where n is an integer. The magnitude response of lowpass Butter-

worth filter is shown in Fig.1.1 for n = 1 to n = 7. The passband becomes ideally flat

as order n increases and becomes closer approximation of ideal lowpass filter response.

1.2 Motivations

Most recently, fractional calculus (FC) has been imported to electronics making it pos-

sible to design any order filter circuits [1, 6, 15]. It has been shown that fractional

order filters provide a precise control of attenuation, -3dB frequency and stopband at-

tenuation. Classical integer order filters yield −20n dB/decade stopband attenuations

(where n is the integer order), however fractional order provides a greater control with

−20(n + α) dB/decade stopband attenuation (where α < 1 and includes positive real

numbers). This precise control of attenuation gradient is a very useful feature in many

engineering areas including biomedical sciences. Non-integer order or fractional order

is one of the most arduously evaluated field nowadays. The theory relating FC to frac-

tional order filter is well established in recent literature. However, there are many lim-

itations that still exists in implementation of this type of non-integer filter. The issues

were also been highlighted in the literature while taking theoretical designed model

into practical hardware realizations. Not just implementation but efficient implemen-

tation of fractional order filters has great potential in many areas such as biomedical

engineering, telecommunication, control and many others. Thus, this provides motiva-

tion for our research. This research work is focussed on implementation of fractional

order Butterworth filters with optimized coefficients on reconfigurable analog proces-

sor namely, Field Programmable Analog Array (FPAA) development kit.

1.3 Proposed Methodology

In order to fulfil above motivations, the research has been conducted with a key objec-

tive to design fractional order Butterworth filter and to verify the implementation on

FPAA platform. The following steps were executed as,

1. Second order approximation for fractional Laplacian operator was used to design

3

lowpass Butterworth filter of order (1 + α).

2. The filter coefficients were obtained by calculating the passband error between

first order Butterworth response and the response of order (1 + α).

3. Metaheuristic search method was used to further minimize both, the passband

error and the stopband error in order to approximate best fractional order Butter-

worth response with following requirements,

(a) minimum least square error along with minimum passband and stopband

errors.

(b) enhanced stability.

(c) enhanced -3dB frequency which is approximately or nearly 1 rad/s.

(d) less sensitivity to parameter variation.

4. The filter transfer function with most optimal coefficients was realized using

FPAA development environment.

After the verification of design in simulation, the filters were tested by implementing

the circuit virtually using the FPAA internal switched capacitors and operational am-

plifiers. Test signals were passed through the virtual filter on FPAA development board

and waveforms were analyzed in frequency domain.

Similarly, the analysis was conducted for highpass Butterworth response.

1.4 Research Objectives

The principal objective of this research thesis is to design a fractional order Butterworth

filter in frequency domain with optimal filter coefficients. It is necessary to verify

its implementation on hardware components. The objective can be divided into the

following specific aims:

• Use second order Continued Fractional Expansion (CFE) method to approximate

fractional order differentiator, sα, for practical realizable transfer functions.

• Searching for coefficients of the approximated transfer function using minimum

least square error approach between first order and order (1 + α) transfer func-

tion.

4

• Develop a bi-level optimization routine to optimize the coefficients to for mini-

mum passband and stopband error, minimum least square error, enhanced -3dB

frequency, better stability and less sensitivity to parameter variation.

• Realize the transfer function using FPAA Anadigm designer2 development en-

vironment.

• Implement the filter using internally switched capacitors and operational ampli-

fiers of FPAA board.

• Observe the designed filter behavior in real time signal environment.

1.5 Thesis Outline

The thesis is organized as follows,

• Chapter 2 provides the background and literature work. The background in-

cludes the work previously carried out on the design of fractional order filters

that include fractional order Butterworth filter with passband peaking and other

type of filters designed with various specifications. Furthermore, it presents frac-

tional calculus background and its relationship to second order transfer function

and analog circuit design.

• Chapter 3 gives fractional order lowpass Butterworth filter design and optimiza-

tion of coefficients using bi-level particle swarm optimization algorithm. Chap-

ter also focuses on the contribution of optimized coefficients on stability, -3dB

frequency and sensitivity to parameter variation.

• Chapter 4 gives fractional order highpass Butterworth filter design and trans-

formation from fractional order low pass filter transfer function, along with op-

timization of coefficients.

• Chapter 5 presents the realization and implementation of fractional order low-

pass and highpass Butterworth filters on FPAA hardware environment.

• Chapter 6 concludes a general summary on the presented techniques with po-

tential future works.

5

Chapter 2

Background and Literature Work

2.1 Introduction

This chapter introduces fractional calculus and its application to design fractional order

filters. An introduction on design specification of the desired filter is also presented

in this chapter. Fractional Laplacian operator and its definition is elaborated further

which is one of the significant tool to approximate fractional differentiator, sα, where

(α ∈ R). Moreover, previously presented works on design and realization of fractional

order lowpass Butterworth filters have been compared along with its advantages and

drawbacks. Section 2.5 explores the basics on Field Programmable Analog Arrays

(FPAAs) that includes its functionality and purpose in this research.

2.2 Fractional Calculus

Fractional calculus (FC) is three centuries old and is a generalization of decimal or

integer order calculus, giving the potential to accomplish what integer-order calculus

cannot [46]. For the past three centuries, this field was only to the interest of mathe-

maticians, however recently it has been applied in many fields of science and technol-

ogy such as in engineering, biomedicine, economics, control theory, diffusion theory,

material science, robotics and signal processing [34, 38, 19, 45, 13]. Fractional order

derivatives and integrals are widely being used currently in many field of engineering

and sciences to model both simple and complex systems. There are many definitions

of fractional order derivatives and integrals in literature. A summary of all the various

definitions are given in Tables 2.1 and 2.2 respectively [35].

6

Table 2.1: Fractional integral definition (α > 0)

Designation Definition

Lioville integral D−αF (t) = 1(−1)αΓ(α)

∫ +∞0

f(t+ τ)τα−1dτ

Riemann integral D−αF (t) = 1Γ(α)

∫ t

0f(τ) · (t− τ)α−1dτ, t > 0

Hadamard integral D−αF (t) = tα

Γ(α)

∫ 1

0f(tτ) · (1− τ)α−1dτ, t > 0

Left side RL integral D−αF (t) = 1Γ(α)

∫ t

af(τ) · (t− τ)α−1dτ, t > a

Righ side RL integral D−αF (t) = 1Γ(α)

∫ b

tf(τ) · (t− τ)α−1dτ, t < b

Left side Weyl integral D−αF (t) = 1Γ(α)

∫ t

−∞ f(τ) · (t− τ)α−1dτ

Right side Weyl integral D−αF (t) = 1Γ(α)

∫ +∞t

f(τ) · (t− τ)α−1dτ

Table 2.2: Fractional derivative definition (α > 0)

Designation Definition

Left side RL derivative DαF (t) = 1Γ(n−α)

dn

dtn

∫ t

af(τ) · (t− τ)α−n−1dτ

Right side RL derivative DαF (t) = (−1)n

Γ(n−α)dn

dtn

∫ b

tf(τ) · (τ − t)α−n−1dτ

Left side Caputo derivative DαF (t) = 1Γ(−v)

[∫ t

0f (n)(τ) · (t− τ)v−1dτ

]Right side Caputo derivative DαF (t) = 1

Γ(−v)

[∫ +∞t

f (n)(τ) · (τ − t)v−1dτ]

Marchaud derivative Dαf(t) = c · ∫∞0

Δkτf(t)τ1+α dτ, α > 0

Generalised function DαF (t) = 1Γ(−α)

∫ t

−∞ f(τ) · (t− τ)−α−1dτ

Left side Grunwald- Letnikov Dα−f(t) = limh→0+

∑∞k=0 (−1)k(α

k )f(t−kh)

hα

Right side Grunwald- Letnikov Dα+f(t) = limh→0+

∑∞k=0 (−1)k(α

k )f(t+kh)

hα

However only a few of them are being commonly used, for example Riemann-Liouville

(RL) [23], Caputo [37], Weyl [48] and Grunwald-Letnikov definitions are most com-

monly used in engineering applications [18]. Caputo definition is most commonly

used to deal with real world problems, where the boundary condition is necessary and

derivative of a constant is zero. The Caputo definition can be defined as,

0Dαt f(t) =

1

Γ(m− α)

t∫0

f (m)(τ)

(t− τ)α+1−ndτ (2.1)

where m − 1 ≤ α ≤ m and m ∈ N , while Γ(•) is the gamma function. In order to

analyze and study the behavior of an electronic circuit, it is very convenient to use the

Laplace transform. Thus applying Laplace transform to (2.1) yields

L{0Dαt f(t)} = sαF (s)−

m−1∑k=0

sα−k−1f (k)(0) (2.2)

7

where f(0) is the initial condition, sα is the fractional Laplacian operator and F (s) is

the Laplace transform of f(t) [33].

Some concepts from FC can be used in signal processing and circuit theory. For ex-

ample, FC has been imported to electronics making it possible to design and realize

fractional order filter circuits [1]. Well known analog filters include inductor and ca-

pacitor whose numbers determine the filter order. However, inductor or capacitor with

fractional impedance can be generalized and these elements in the fractional domain

are called fractance devices. Fractance devices are not available commercially, how-

ever it is possible to emulate them using resistor-capacitor (RC) or resistor-inductor-

capacitor (RLC) trees and using platform like FPAA. In order to realize fractance de-

vices physically as in fractional order circuits and systems, integer-order approxima-

tions of fractional Laplacian operator, sα, have to be used. Fractional devices are said

to have fractional-order characteristics in terms of impedance and conductance. The

impedance function of a fractance device is expressed as:

Z(s) = κsα = (κω)αejαπ2 (2.3)

where κ is a constant and α refers to the order of the fractance element [46]. The value

of α determines whether the element is inductor or capacitor. For the range of α given,

0 < α < 2, the element may represent a fractional-order inductor and for the range of

α, −2 < α < 0, this element represents fractional order capacitor. However, there are

some special cases given as:

• For α = 1, represents a inductor.

• For α = -1, represents a capacitor.

• For α = -2, represents a frequency dependent negative resistor (FDNR).

A FDNR is a circuit element, and used to implement a lowpass filters. It exhibits real

negative impedance in the frequency domain. A summarizing diagram representing

each of these classifications is given in Fig.2.1.

8

Figure 2.1: Fractional order element classification diagram

2.3 Approximation for the General Laplacian Opera-

tor

An important tool in fractional-order filter design is the fractional Laplacian operator,

sα. Realizations of any fractional-order filter can be divided into two categories: first,

approximation to integer order to realize fractional step filters and second, using frac-

tional capacitor with impedance Z = 1/sαC. Here, C refers to pseudo capacitance

[15]. Commonly used approximation of sα in literatures is the continued fractional

expansion (CFE). These approximations were the Carlson’s method [8], Matsuda’s

method [32] and rational approximation methods (curve fitting technique) [2, 12]. All

these methods give different results in terms of accuracy since the approximation in

the frequency domain varies accordingly to the approximation order. According to

[27] CFE is an attractive choice in terms of phase and gain error and so the aforemen-

tioned procedure of the second-order approximation of the CFE will be adopted in this

9

research framework. In theory, the CFE of (1 + x)α can be written as [26],

(1 + x)α = 1− αx

1+

(1 + α)x

2+

(1− α)x

3+

(2 + α)x

4+

(2− α)x

5+ ...... (2.4)

Now, substituting x = (s− 1) in (2.4) and taking up to 10 terms gives rational approx-

imation for sα as per Table 2.3.

Higher order rational approximation can be obtained by taking more terms in (2.4).

The general form of rational approximation of sα is given in (2.5).

(s)α =α0(s)

n + α1(s)n−1 + ...+ αn−1(s) + αn

αn(s)n + αn−1(s)

n−1 + ...+ α1(s) + α0

(2.5)

where n is the order of the approximation.

From Table 2.3, sα can be approximated with 4 terms as

sα ≈−(α2 + 3α + 2) s2 + (8− 2α2) s+ (α2 − 3α + 2)

(α2 − 3α + 2) s2 + (8− 2α2) s+ (α2 + 3α + 2)(2.6)

Frequency(rad/s)10-2 10-1 100 101 102

Magnitude(dB)

-20

0

20Ideal2nd Order Approximation3rd Order Approximation

Frequency(rad/s)10-2 10-1 100 101 102

Phase(degrees)

0

20

40

60

Figure 2.2: Magnitude and phase response of second order approximation (dashed

line) and third order approximation (dotted line), of sα for the case α = 0.5 compared

with ideal (solid line)

Figure 2.2 shows phase and magnitude responses of sα for the case when α = 0.5. It

compares the ideal response with the second order and third order approximations. It

can be observed from plots of second order approximation that for ω = [0.032, 31.53]

rad/s the maximum error in the magnitude response is approximately 1.382 dB; while

10

Table 2.3: Rational approximation for (sα)

No. of terms Rational Approximation for α Design equation of coefficients

2α0(s)+α1

α1(s)+α0

α0 = (1− α)

α1 = (1 + α)

4α0(s)

2+α1(s)+α2

α2(s)2+α1(s)+α0

α0 = (α2 + 3α + 2)

α1 = (8− 2α2)

α2 = (α2 − 3α + 2)

6α0(s)

3+α1(s)2+α2(s)+α3

α3(s)3+α3(s)

2+α1(s)+α0

α0 = (α3 + 6α2 + 11α + 6)

α1 = (−3α3 − 6α2 + 27α + 54)

α2 = (3α3 − 6α2 − 27α + 54)

α3 = (−α3 + 6α2 − 11α + 6)

8α0(s)

4+α1(s)3+α2(s)

2+α3(s)+α4

α4(s)4+α3(s)

3+α2(s)2+α1(s)+α0

α0 = (α4 + 10α3 + 35α2

+50α + 24)

α1 = (−4α4 − 20α3 + 40α2

+320α + 384)

α2 = (6α4 − 150α2 + 864)

α3 = (−4α4 + 20α3 + 40α2

−320α + 384)

α4 = (α4 − 10α3 + 35α2

−50α + 24)

10α0(s)

5+α1(s)4+α2(s)

3+α3(s)2+α2(s)+α5

α5(s)5+α4(s)

4+α3(s)3+α2(s)

2+α1(s)+α0

α0 = (−α5 − 15α4 − 85α3

−225α2 − 274α− 120)

α1 = (5α5 + 45α4 + 5α3

−1005α2 − 3250α− 3000)

α2 = (−10α5 − 30α4 + 410α3

+1230α2 − 4000α− 12000)

α3 = (10α5 − 30α4 − 410α3

+1230α2 + 4000α− 12000)

α4 = (−5α5 + 45α4 − 5α3

−1005α2 + 3250α− 3000)

α5 = (α5 − 15α4 + 85α3

−225α2 + 274α− 120)

for ω = [0.142, 7.00] rad/s the error in the phase response does not exceed 3.194◦.

11

However, for third order approximation, it can be observed from plots that for ω =

[0.0256, 39.07] rad/s the error in the magnitude response does not exceed 0.8936 dB;

while for ω = [0.09541, 10.48] rad/s, the error in the phase response does not exceed

0.0356◦. Although, third order approximation, approximates sα with minimum phase

and magnitude error, it is difficult to implement and realize these higher order approx-

imations due to hardware limitations. Thus, second order approximation of fractional

Laplacian operator is most suitable to implement compared with other approximation

method. In general, CFE can be a good method to approximate (n+α) order fractional

step filters. In addition to this, the second order approximation is also economically

viable to implement in hardware compared with higher approximations using CFEs

[9].

Frequency(rad/s)10-2 10-1 100 101 102

Magnitude(dB)

-20

-10

0

10

20IdealCarlsons methodOustaloop methodMatsuda methodSecond order CFE

Frequency(rad/s)10-2 10-1 100 101 102

Phase(degrees)

0

20

40

60

Figure 2.3: Magnitude and phase response of Carlson’s, Matsuda’s, Oustaloop’s and

second order CFE approximation methods to approximate sα for α = 0.5 compared

with ideal case

The variable sα can also be approximated by other rational approximation methods.

These methods are Carlson’s method [44], Matsuda’s method [25], and Oustaloop’s

method [22] as shown in Fig.2.3. From the magnitude and phase plot presented in

Fig.2.3, it is clear that Oustaloop’s approximation better approximates sα compared

with other approximation methods but its frequency domain expression is complicated

to implement. Therefore, it can be said that these approximations methods comes with

complexity while implementing on hardware, therefore second order approximation

using CFE is an attractive choice. The best approximation which is the Oustaloop’s

approximation, cannot not be implemented easily in hardware as the quality of ap-

12

proximation may not be satisfactory in high and low frequency bands near the fitting

frequency bounds. Therefore, the entire framework of this work would focus on second

order approximation of fractional Laplacian operation sα using CFE.

From Table 2.3, the general expression to approximate variable (τs)α can also be writ-

ten using second order approximation as

(τs)α =α0(τs)

2 + α1(τs) + α2

α2(τs)2 + α1(τs) + α0

. (2.7)

There are two conditions possible which determine if the approximation is used as

differentiator or integrator, as follows.

• Any positive real number of α represents fractional order differentiator (here

consider the range of α as (0 < α < 1)). The magnitude response of fractional

order differentiator is given as H(ω) = (ω/ω0)α. The unity gain frequency is

given as ω0 = 1/τ , where τ is the corresponding time constant. The transfer

function of fractional order differentiator can be generalised as,

H(s) = (τs)α (2.8)

• Any negative real number of α represents fractional order integrators (here con-

sider the range of α as (−1 < α < 0)). The magnitude response of fractional

order integrators is given as H(ω) = (ω0/ω)α. The unity gain frequency is

given as ω0 = 1/τ . The transfer function of fractional order integrator can be

generalised as,

H(s) =1

(τs)α(2.9)

Authors [34, 38, 19, 45] have presented implementation of fractional order oscillators,

impedance emulators and controllers. Signal conditioning in bio-medical engineering

is one of the most common application for fractional order integrators and differentia-

tors.

2.4 Fractional Order Filters

Firstly, fractional-order filters were critically studied in [40, 39] and shown that such

fractional filters (also called fractional-step filters) are realizable with reasonable over-

shoot in the passband region. Most cases α is considered from 0.1 to 0.9. It has

13

been noticed that any second-order filter transfer function leads to the problem of

passband peaking in the magnitude response. In this regard, the magnitude response

of fractional-order Butterworth filter has been explored recently to address passband

peaking problem [1, 4]. Same concept has also being expanded to elliptical and Cheby-

shev filters [16, 17]. More recently, Kubanek and Freeborn [28] have proposed a new

fractional-order low-pass filter (FLPF) design based on second order function with ar-

bitrary quality factors. The work focused on the search of coefficients to approximate

a second order lowpass filter transfer function with arbitrary quality factor Q. In [17]

and [15], coefficients of FLPF transfer function were selected to approximate a flat

bassband response of a first order Butterworth filter. Another method was proposed

in [42] to approximate coefficients for different cases of a normalized FLPF transfer

function, but this method was based on limited search objective functions that focussed

on only few parameters such as transition bandwidth and maximum allowable peak.

In [15] it is shown that fractional-order filters provide a precise control of attenuation,

-3dB frequency and stop- band attenuation. Integer-order filters yield −20n dB/decade

stopband attenuations, where n is the integer order, however fractional-order provides

a greater control with −20(n+ α) dB/decade stopband attenuation where α (0 < α <

1) [15]. Another very important advantage of fractional-order filter circuits is that

they provide possibility to design band pass and band reject filters with asymmetric

stopband characteristics [3]. The advantage of these filters is that they can also been

used as phase discriminators.

2.5 FPAA

Field programmable analog array (FPAA) is basically a type of analog integrated cir-

cuit. Even though digital circuits still rule the electronic market, the role of analog

integrated circuits remains equally important [10]. An FPAA, in its most general form

can be defined as a monolithic collection of analog building blocks, a user controllable

routing network used for passing signals between the building blocks and a collection

of memory elements used to define both the function and structure. For this research

work, Anadigm AN231E04 FPAA kit would be useful to realize and to test designed

fractional order lowpass and highpass Butterworth filters. Fig.2.4 shows a functional

block diagram of a Anadigm FPAA module. Anadigm FPAA is ‘analogue signal pro-

cessors’ consisting of fully configurable analog modules (CAMs) surrounded by pro-

14

grammable interconnect and analogue input and output cells [5].

Many areas of electrical, electronics and computer science engineering have adopted

the use of FPAA, some of which includes [14]:

• Signal processing, particularly signal filtering and signal conditioning.

• Industrial application, in control and automation of processes and precision con-

trol.

• Medical application for signal monitoring and conditioning.

• Analog signal processing.

• Audio signal filtering.

Before performing any application using the FPAAs, it is necessary to implement re-

quired interfacing circuits. Since the Anadigm FPAA has differential input and differ-

ential output, for most application it would be desirable to implement single to differ-

ential and differential to single converters. There are many applications, where FPAAs

can be used. In a recent work presented in [11], FPAAs were used to realize arti-

ficial neural network. A feedforward neural network architecture was implemented

on the FPAA. It was concluded from the realization experiment that FPAAs realizes

data 1400 times faster than software implementation and more complex architecture

can be implemented by incorporating more FPAA chips. In another work authors pro-

posed the accuracy between the simulation and hardware implementation of matched

and adaptive filters using the FPAAs [47]. One of the key importance, of implementa-

tion of filters using FPAA technology proposed by authors was that, the design could

be reconfigured using the reconfigurable blocks in FPAA to meet the specific design

requirements. Authors of [31] have proposed implementation of self-tuning propor-

tional, integral and derivative (PID) controller using FPAAs. In a research work carried

out by authors of [30], FPAA was used in a real time application to control position

of a DC servo motor. The dynamics of the servo motor system was obtained by an

automated tuning technique based on relay feedback and the system was tuned on-line

with PI configuration using a FPAA. In another research work carried out in [14], au-

thors have implemented PID controller using Anadigm PID tool of Anadigm designer

development environment. It can be said that FPAAs has many applications and many

fields can be explored by hardware implementations using FPAAs.

15

Figure 2.4: Functional block diagram of Anadigm FPAA [5]

2.6 Summary

This chapter firstly gives a brief background on fractional calculus and its applicability

in this thesis. Some of the basic definitions from fractional theory are introduced in

this section. A concept on fractional Laplacian operator using continued fractional

expansion (CFE) is discussed with respect to realization prospects. At the end, some

previously reported works on fractional filters have been taken in this section. FPAA

is introduced briefly with its key features. Applications of such analog processors are

briefly discussed in this chapter.

16

Chapter 3

Fractional Order Low Pass Filter

3.1 Introduction

Fractional order lowpass Butterworth filter is studied in this chapter. Butterworth filter

is an interesting choice to explore because it exhibits some useful properties such as

maximally flat magnitude response in passband region and at DC characteristic it pro-

duces a gain of approximately equal to 0 dB [7]. Traditional integer order Butterworth

filter of order n has magnitude response given by [36],

|HnB(ω)| =

√1

1 + (ω/ωc)2n(3.1)

where ωc is the cut-off frequency and both ω and ωc are in rad/s. Likewise (3.1), a

fractional order counterpart representing magnitude response of order (n + α), where

n > 0 can be derived and written by,

∣∣Hn+αB (ω)

∣∣ =√

1

1 + ω2(n+α)(3.2)

From (3.2), the slope of magnitude response of fractional order Butterworth response

is computed by 20 x (n+ α) dB/decade.

Similarly, the magnitude response of fractional order lowpass Butterworth filter in

terms of low frequency gain, κ and pole frequency ω0 can be given as

|H(jω)| = κ√(ωω0

)2α

+ 2(

ωω0

)α

cos(απ2

)+ 1

(3.3)

The aim of this research is to design a fractional order lowpass filter transfer function

of (1 + α) order. The coefficients of this transfer function are required to be optimize

for maximally flat passband. Together, it is necessary to check most important design

17

constraints. The proposed method is developed to obtain minimum least square errors,

minimum passband and stopband errors along with -3dB frequency approximately or

equal to 1 rad/s. The obtained values of coefficients are used to verify further stability

and sensitivity to parameter variations.

3.2 Fractional-Order Low Pass Filter Transfer Func-

tion of (1 + α) Order

Fractional-order lowpass filter transfer function (FOLTF) of (1 + α) order has previ-

ously been studied in [17, 15, 28]. Mostly the work was focused on the design and

implementation of fractional-order transfer function of the form

HLP1+α (s) =

k1s1+α + sαk2 + k3

. (3.4)

The coefficients k2 and k3 are selected to yield a flat passband response while k1 has

been kept to constant value of 1 which lead to DC gain of 1/k3. The transfer function

(3.4) describes low-pass filters with fractional orders between one and two. Any filter

realization is evaluated based on a flat passband with minimum error in magnitudes

during passband and stopband frequencies and also −3dB frequency almost closer to

1 rad/s. This is achieved by minimizing the error objective function. In literature, the

error function is considered as the difference between the magnitude response of the

(1+α) fractional-order transfer function calculated using k2,3 and the ideal normalized

first order Butterworth response, which can be given by

BLP1 (s) =

1

s+ 1(3.5)

It is important while designing filter using (3.4) that the coefficients should be selected

particularly to obtain the desired characteristics like flat passband, stability, parameter

variation robustness and so on.

In this research, novelty lies in the fact that the designed filter satisfies more than one

characteristics together. Therefore, an objective function is required to formulate that

gives the minimum of a magnitude error with flat passband response and also response

reaches −3dB below its DC value at a frequency 1 rad/s. The advantage of global

search and optimal robust result findings feature of Particle Swarm Optimization (PSO)

is investigated in our analysis. In this the presented algorithm plays a two-step problem

18

and so called a bilevel optimization routine. MATLAB simulations of (1 + α) order

lowpass filters with fractional steps from α = 0.01 to α = 0.99 have developed first as

simulation examples. Various simulation results and statistical analysis are carried out

for verifying fractional step filters. Further, the designed results are compared with the

existing fractional step filters reported in recent literature.

3.3 Previously Selected Coefficients in Optimization Frame-

work

Freebon et al. [17] have optimized the coefficients in (3.4) through numerical search

approach. The values are selected based on least squares error (LSE) that compared

with first order Butterworth response over the frequency range ω = 0.01−1 rad/s. The

obtained coefficients yield the minimum cumulative passband error. The numerical

search is limited to 0 < k2 < 2 and 0 < k3 < 1, while k1 is kept to constant value of

1. The linear functions from the collected raw data for k2 and k3 are given by [17]

k2 = 1.0683α2 + 0.161α + 0.3324 (3.6)

k3 = 0.2937α + 0.7122 (3.7)

Another attempt was presented by authors using MATLAB optimization tool based on

a nonlinear least squares fitting [15]. The coefficients k2 and k3 from this optimized

LSE approach are described as,

k2 = 1.008α2 + 0.2867α + 0.2366 (3.8)

k3 = 0.2171α + 0.7914 (3.9)

Unfortunately, the trade-offs between the improved LSE and stability margin are not

systematically analyzed in previous work. In [15], it is noted that the improved LSE

comes at the cost of stability margin. To further evaluate, it is also shown the best

approximation is that the least variation of -3dB frequencies over the full range of

orders.

Thus the trade-offs are hard to be guaranteed in a uniform way against different design

objectives. Because of the above difficulties, an optimization technique needs to tackle

more than one objectives. In our work, a modified Particle swarm optimization (PSO)

19

is developed to work with more than two objectives at a time. Main focus is on the

design of optimal fractional-order transfer function (FOTF) so that the uniformness of

the trade-offs can be guaranteed. Although the computational complexity of the prob-

lem is further increased by the bilevel structure, the desired solution can be achieved

in a finite time. In following section, the the modified PSO with selected coefficients

is presented with a flow chart.

3.4 Modified PSO for Bilevel Optimization

PSO technique for optimization was firstly introduced by Kennedy and Eberhart [24]

in 1995. The key merit obtained by PSO algorithm is inspired by the social behavior

of flock of birds (called particles). Each particle in the swarm is a potential solution

of the problem under consideration in a search space. Each particle is pictured with

its position and velocity vector. The position vector is the desired solution and the ve-

locity vector gives the speed of a particle with which it can travel the optimal solution.

Each particle is evaluated by its fitness value at every iteration. In this process both

the velocity and the position for each particle are updated for next iteration. Many

researchers have developed so far a simple theoretical framework and so it is relatively

easy to program and implement. It is also shown [43] that the PSO is a computation-

ally less expensive and has low memory requirements. In addition to this, the PSO has

relatively small number (3-5) of user-defined parameters and they are not very sensi-

tive to the convergence and final accuracy of the algorithm. PSO algorithm is robust

in solving many continuous nonlinear optimization problems. The basic PSO version

with inertia weight is described in formula below.

ai ← ωai + R(0, ϕ1) ⊗ (pi − xi) + R(0, ϕ2) ⊗ (pg − xi),

xi ← xi + ai(3.10)

where i ∈ N , ω = is an inertia weight factor and N is a number of particles (usu-

ally N <= 40). The other parameters are as follows: xi gives the particle present

location and ai defines the step velocity of the particle. The expression (3.10) has two

parameters ϕ1 and ϕ2 determines the magnitude of the random forces in the direc-

tion of personal best pi and neighborhood best pg, mostly called acceleration coeffi-

cients. R(0, ϕj), j = 1, 2; delivers a vector of random numbers uniformly distributed

in [0, ϕj]. It is generated randomly after each iteration and for each particle. The inertia

20

weight factor ω is updated by

ω = ωmax − (Ik − 1)

(ωmax − ωmin

Im − 1

)(3.11)

in which ωmax and ωmin are maximum and minimum inertia weights, respectively. Ik

is a current iteration and Im is a maximum iteration number.

Suppose at each iteration when a given boundary is violated by any of the particles,

the particle i is returned to its previous position xi. The step ai is reversed with the

same magnitude, but in the opposite direction, i.e. ai = −ai. This simple heuristics

has been tested on many simulated examples and proven to work a robust result.

The algorithm has been developed in MATLAB 8.5 on Windows 10 pro core i7 Intel 8

GB RAM. The stopping criterion can be imposed by, either use a fixed number of iter-

ation or a given tolerance. Generally a fixed number of iteration is easy to implement

and in this optimization the fixed iteration number is set to be 50, which is adequate for

stated optimization task. Population size, N is set to be 35 and (ωmax, ωmin)=(0.9, 0.1).

In this work, k2,3 are fed in as the variables to be optimized in the PSO algorithm. The

main reason for choosing PSO over other optimization routine is that PSO is more

computationally efficient and more adaptable for bi-level optimization of objective

functions. The optimization is carried out with following bilevel objective in order

to balance the tradeoffs between LSE and -3dB variation closed to 1.

Level 1: Minimum least square error, calculated as

|Ec(jω)| =N∑i=1

| |B1(jωi)| −∣∣HLP

1+α (jωi) |∣∣2 (3.12)

Level 2: -3dB frequency closest to 1 rad/s.

The modified bi-level PSO flow diagram is given in Figure 3.1. Here, pbest1 and pbest2

are the individual best position of particles 1 and 2 respectively which in this case are

the two objective functions described above. Likewise gbest1 and gbest2 is the global

best position of particles 1 and 2 respectively. The magnitude responses |B1(jωi)| and∣∣HLP1+α (jωi)

∣∣ are first order Butterworth filter and the fractional order lowpass filter

of order (1 + α) at frequency ωi, respectively and N is the number of samples taken

between frequency 0.01− 1.5 rad/s.

21

Figure 3.1: Flow diagram of bi-level PSO algorithm

22

Using the modified PSO and constraints in (3.12), the optimal set of coefficients are

obtained for all (1 + α) order transfer functions. The search routine was implemented

in simulation MATLAB environment. Aiming to finalize k2,3 coefficients that yield

the lowest LSE for almost all orders and -3dB frequency close to 1 rad/s. The linear

curve-fitted expressions (3.13-3.14) were obtained in terms of parabolic function of

order 3 and as a function of α using the proposed technique.

k2proposed = 0.5293α3 − 0.3156α2 + 0.9672α + 0.2653 (3.13)

k3proposed = −0.1981α3 + 0.2471α2 + 0.2359α + 0.7233 (3.14)

α

0 0.2 0.4 0.6 0.8 1

k2,k3

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6k2 (LSE)

k3 (LSE)

k2 (Optimized LSE)

k3 (Optimized LSE)

k2 (Proposed)

k3 (Proposed)

Figure 3.2: k2,3 coefficients to approximate fractional step filters, compared to coeffi-

cients presented in [17] and [15] respectively

The coefficients k2,3 that yielded the best performance for (3.4) with k1 = 1 when the

order is increased from 1.01 to 1.99 in steps of 0.01 are given in Figure 3.2 as solid

line. The coefficients from LSE [17] and optimized LSE [15] are also given as dashed

and dotted lines, respectively for further comparison.

3.5 Numerical Comparison

The proposed fractional step filter is necessary to examine for basic characteristics.

The detailed numerical comparison is presented in this section and compared the per-

23

formances with previously designed in [15] and [17]. The presented filter offers stop-

band attenuations of −20(1 + α)dB/decade. With the optimized filter coefficients the

analysis proves the superiority of the proposed lowpass Butterworth filter in terms of

passband error, stopband error, stability, -3dB frequency and sensitivity to parameter

variation.

3.5.1 Magnitude Response Errors

The superior magnitude response can match closely the magnitude response character-

istics of ideal fractional order Butterworth filter. We have used the mean square error

(MSE) to compare responses as

MSE =M∑i=1

∣∣|HB1(ωi)| −∣∣HLP

1+α(ωi)∣∣∣∣2 (3.15)

where HB1(ωi) is the magnitude response of first order butterworth filter at frequency,

ωi for 1000 samples taken within the frequency range from 0.01 to 10 rad/s. HLP1+α(ωi)

is the magnitude response of (1 + α) order lowpass butterworth filter. The reason

for choosing the design frequency to be within the range [0.01, 10] rad/s is due to the

fact that more number of points (M) are needed for the fitness function (3.12) in the

optimization algorithm if the frequency range is wider than [0.01, 10] rad/s. It is noted

that by increasing more data points the algorithm takes more time to return the optimal

values.

Table 3.1: Comparison of PE and SE matrices for (1 + α) order filters

ErrorMethods 1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

PE (dB)

[17] -73.3296 -60.1360 -56.1785 -55.2175 -55.8929 -58.8728 -68.2701 -64.6108 -55.2647

[15] -64.2780 -56.5060 -53.5157 -52.8712 -52.8712 -54.8458 -59.5850 -84.4684 -59.1627

proposed -86.6501 -76.4039 -68.6863 -64.9336 -62.7158 -61.8031 -61.6794 -62.1609 -62.9479

SE (dB)

[17] -61.1042 -55.9379 -53.2501 -51.5651 -50.5155 -49.6003 -49.0040 -48.5517 -48.2114

[15] -61.4480 -55.9622 -53.2648 -51.5745 -50.4233 -49.6034 -49.0022 -48.5528 -48.2121

proposed -61.1045 -55.9321 -53.2450 -51.5616 -50.4159 -49.5992 -49.5992 -48.4404 -48.1549

The magnitude response performance of the designed filters are also compared based

on two error matrices namely passband error (PE) and stopband error (SE) as defined

24

following.

PE = 20× log10

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

√√√√√ K∑i=1

∣∣|HB1(ωi)| −∣∣HLP

1+α(ωi)∣∣∣∣2

K

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

dB (3.16)

where, K = 500 and 0.01 ≤ ω ≤ 1.

SE = 20× log10

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

√√√√√ L∑i=1

∣∣|HB1(ωi)| −∣∣HLP

1+α(ωi)∣∣∣∣2

L

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

dB (3.17)

where, L = 500 and 1 ≤ ω ≤ 10.

Both, PE and SE errors of fractional (1 + α) order filters are listed with the coefficient

used in [15] and [17] in Table 3.1. It is also clear from Figs. 3.3 and 3.4 that the

proposed values result the lower errors mostly for all order of filters. Although the

SE value is almost consistent but less than recently reported values in the literature for

each α value from 0.1 to 0.9.

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Passban

dError

(dB)

-110

-100

-90

-80

-70

-60

-50

123

Figure 3.3: PE index values, 1: by [17], 2: by [15] and 3: by proposed

25

1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Stopban

dError

(dB)

-90

-80

-70

-60

-50

-40

123

1.8954 1.8956 1.8958 1.896

-48.226

-48.225

-48.224

-48.223

Figure 3.4: SE index values, 1: by [17], 2: by [15] and 3: by proposed

3.5.2 Stability Analysis

An important criteria that would be examined with new optimized coefficients is sta-

bility margin, which would be described here in terms of pole angle and the region

of instability. It is very important to analyze the stability of (3.4) with new optimized

coefficients as instability could lead to variations in passband response for higher frac-

tional - order filters. To analyze the stability of FLPF with new coefficients we need

to transform the transfer function (3.4) from s-plane to complex W-plane. As defined

in [41] the transformation steps can be used to convert the fractional transfer function

to the W-plane by taking s = Wm and α = k/m, where k and m are selected for the

desired α value.

This transformation changes (3.4) into

H (W ) =k1

Wm+k + k2W k + k3(3.18)

The characteristic equation from (3.18) in W-plane should be ensured that all the poles

obtained with optimized coefficients are in the stable region. It is necessary to observe

further how far the absolute pole angles, |θW |, are from the value π2m

. If any |θW | < π2m

then the system is unstable. The minimum root angles for α=0.01-0.99 calculated with

k=10 to 990 in steps of 10 when m = 1000. First, by equating the denominator

of (3.18) to 0 for all values of α, minimum root angles (|θW |min) were calculated

and plotted in Figure 3.5. The criterion for stability is, |θW | > π2m

and according to

26

α

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Minim

um

phasean

gle(degree)

0.06

0.08

0.1

0.12

0.14

0.16

0.18123y=0.09

STABLE

UNSTABLE

Figure 3.5: Minimum root angle in W-plane for, 1: by [17], 2: [15] and 3: by proposed

chosen k and m, for stability |θW | > π2m

= 0.09◦. The minimum root angle using

the proposed coefficients show a visibly higher margin than others. Interestingly, the

optimized coefficients by LSE in [15] yield a lower stability than the proposed method

even though their method gives a lager LSE. It can be concluded from results that the

bilevel optimization from PSO has obtained better stability with less value of LSE.

For (1 + α) between 1.2 to 1.8, the minimum root angles are furthest away from the

unstable boundary compared with coefficients from [15] and [17].

3.5.3 -3dB Frequency Analysis

It is desired the angular frequency from the transfer function coefficient gives approx-

imately equal to 1 rad/s as -3dB frequency. Thus for each order from 1.01 to 1.99, the

frequency at which the magnitude response reached -3dB is compared. According to

the criterion, the fractional Butterworth filter that reaches -3dB below its DC value at

frequency 1 rad/s is the best choice for implementation.

27

1 + α

1 1.2 1.4 1.6 1.8 2

-3dB

Frequency

(rad/s)

0.85

0.9

0.95

1

1.05

1.1

1.15123

Figure 3.6: -3dB frequency using 1: by [17], 2: [15] and 3: by proposed

The -3dB frequencies for each order are given in Fig. 3.6 and numerically calculated

with different sets of coefficients from [15], [17] and proposed technique for orders

from 1.01 to 1.99 in steps of 0.01. Both coefficients from [17] and [15] show similar

deviations in -3dB frequencies, at order 1.1 < (1 + α) < 1.5 frequency increases and

reaches peak of 1.13 rad/s and 1.09 rad/s, respectively. After that, frequency drops

gradually and at 1.8 order it crosses 1 rad/sec margin in [17] and at 1.7 rad/s in [15].

However, the proposed filter coefficients show the closer agreement to 1 rad/s through

out all orders. It can be seen that the filter observes very less variations in -3dB fre-

quency, between always 1.005 to 0.998 rad/s. Thus, using the bilevel optimization

the desirable filter characteristics can be improved universally that with the lower LSE

value and high stability margin.

3.5.4 Stop Band Attenuation

The transfer function (3.4) has different roll-off characteristics with different sets of co-

efficients. The stopband attenuation determines how the magnitude response changes

from flat passband response to the ideal stopband attenuation of −20(1+α) dB/decade.

Stopband attenuation is another characteristics of the Butterworth response, the slope

of the roll-off characteristics determines the superiority of the design, that is sharper

28

the slope, better the designed filter.

1 + α

1 1.2 1.4 1.6 1.8 2

Attenuation(dB/d

ecad

e)

-40

-35

-30

-25

-20

-15

1.46 1.48 1.5

-27

-26.5

-26

-25.5

1 [ω = 1-10]2 [ω = 1-10]3 [ω = 1-10]Ideal [ω = 1-10]1 [ω = 10-100]2 [ω = 10-100]3 [ω = 10-100]Ideal[ω = 10-100]

Figure 3.7: Stopband attenuation for (1 + α) order lowpass Butterworth filter imple-

mentation using, 1: [17], 2: [15] and 3: the proposed; for ω = [1,10](blue) lines and

for ω = [10,100](red) lines compared to the ideal attenuation (green) lines.

In order to compare the roll-off characteristics from various methods, the slopes of the

magnitude of transfer functions with coefficients from [17], [15] and proposed values

are given in Fig. 3.7. The solid green line is the ideal characteristic of -20(1 + α),

changing from a value of -20 dB/decade when (1 + α) = 1 to -40 dB/decade when

(1 + α)= 2; corresponding to the traditional integer-order attenuations available for

ω = [10, 100] rad/s. The slops between frequencies ω=1 to ω=10 rad/s are shown with

blue lines and ω = 10 to ω = 100 rad/s with red lines for (1+α) =1.01 to 1.99 in steps

of 0.01.The attenuation for all values of α using the proposed approximation shows the

sharpest roll-off rate that could approximate ideal magnitude response for ω = 1 to 10

rad/s as seen in Fig.3.7.

3.6 Sensitivity to Parameter Variation

The transfer function (3.4) when exposed to variation in coefficients may have alter-

ation in its -3dB frequency response and stopband attenuation. To investigate this phe-

nomena, k2 and k3 were varied slightly by 1% and percentage error for -3dB frequency

29

and stopband attenuation were explored.

3.6.1 -3dB Frequency Response to Parameter Variation

The -3dB frequency for (1 + α) order transfer function (3.4) with variation in coeffi-

cients by a deviation of 1% was explored and results are shown in Fig.3.8. We have

examined the parameter sensitivity for only 1% deviation in coefficients, however the

result revealed that the best balance among the flat passband and -3db frequency ob-

tained from the proposed technique. In Fig.3.8(a), k2 was assumed to be changed by

1% and deviation in -3dB frequency calculated by the percentage error. The proposed

filter less affected from 0.1% to maximum of 0.42% for filter orders 1.1 to 1.9. It is

clearly seen that the effect is minimum when we compare to filters by [17] and [15].

Similarly, Fig. 3.8(b) shows the effect from k3 variation by 1%. The proposed coef-

ficients resulted less error as remain constant for 1.1 < (1 + α) < 1.4 at 0.28% and

decreases to minimum value of 0.03% as order increased. In Fig. 3.8(c), both k2 and

k3 were varied by 1% and for all α from 0.1 to 0.9 the proposed coefficients produced

the minimum percentage error when compared to [17] and [15]. It can be concluded

from the plots that the proposed coefficients are most suitable because the variation in

coefficients (1%) shows the least variation of-3 dB frequencies over the full range of

orders.

30

1 + α

1 1.2 1.4 1.6 1.8 2

-3dB

Fre

quen

cy E

rror (

%)

0

0.2

0.4

0.6

0.8

1

1.2123

(a) k2 varied

1 + α

1 1.2 1.4 1.6 1.8 2

-3dB

Fre

quen

cy E

rror (

%)

0

0.2

0.4

0.6

0.8

1123

(b) k3 varied

1 + α

1 1.2 1.4 1.6 1.8 2

-3dB

Fre

quen

cy E

rror (

%)

0.2

0.4

0.6

0.8

1

1.2

1.4123

(c) k2 and k3 varied

Figure 3.8: Percentage error in -3db frequency: (1) in [17], (2) in [15] and (3) in

Proposed.

31

1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Atte

nuat

ion

Erro

r (%

)

10-4

10-3

10-2

10-1

100

1 2 3 1 2 3

ω = 1 - 10 rad/s

ω = 10 - 100 rad/s

(a) k2 varied

1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Atte

nuat

ion

Erro

r (%

)

10-5

10-4

10-3

10-2

10-1

100

1 2 1 1 2 3

ω = 10 - 100 rad/s

ω = 1 - 10 rad/s

(b) k3 varied

1 + α

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Atte

nuat

ion

Erro

r (%

)

10-4

10-3

10-2

10-1

100

1 2 3 1 2 3

ω = 1 -10 rad/s

ω = 10 -100 rad/s

(c) k2 and k3 varied

Figure 3.9: Percentage error in stopband attenuation: (1) in [17], (2) in [15] and (3) in

Proposed.

32

3.6.2 Stop Band Attenuation Error to Parameter Variation

The error in percentage with respect to the ideal attenuation for (1 + α) order transfer

function (3.4) with variation in coefficients by 1% was explored and results are shown

in Fig.3.9. Again same variations in coefficients are considered as previously but also

the effect on attenuation with respect frequency ranges ω ∈ [1, 10] and ω ∈ [10, 100]

rad/s are necessary to be examined. It is resulted from the Fig.3.9 (a) that the proposed

filter gave the the minimum error in stopband attenuation over both frequency ranges.

Similarly Fig. 3.9(b) shows when k3 was varied by 1%. Again, Fig. 3.9(c) proved that

when k2 and k3 both were varied by 1%, the proposed filter was robust with minimum

attenuation error compared to filters by other methods [17, 15].

3.7 Summary

This work proposed the optimal designed method for the (1+α) fractional-order low-

pass filter. A bilevel optimization technique adopted in order to obtain the best co-

efficient values that approximates the passband of a traditional Butterworth response

with fractional-step stopband attenuation. Firstly, an optimization process has ensured

a flatness of magnitude response and -3 dB frequency near to 1 rad/s. The comparative

study with previously reported methods has shown the superiority in terms of passband

error, stopband error, stability, -3dB frequency and parameter sensitivity.

Next chapter introduces transformation of fractional order highpass filter transfer func-

tion from fractional order lowpass filter transfer function. The proposed coefficients

k2,3 in this section would be used to design fractional order highpass filter in the next

section.

33

Chapter 4

Fractional Order High Pass Filter

4.1 Introduction

The characteristics of fractional order highpass Butterworth filter response is studied in

this chapter. We obtain a highpass filter using various transformation techniques from

its lowpass filter transfer function. Analysis is presented with respect to least square

error, passband and stopband errors using magnitude responses of (1 + α) order high-

pass filter and first order Butterworth filter. The validity of the proposed design method

is described by various analysis important in designing fractional order highpass filter.

4.2 Fractional Order Low Pass to High Pass Transfor-

mation

A fractional order highpass filter can be obtained from fractional order lowpass fil-

ter transfer function by using lowpass to highpass transformation highlighted in [29].

There are three different transformations each of which has its own pros and cons.

In section 3, coefficients of fractional order lowpass filter transfer function (3.4) were

chosen using bi-level PSO algorithm and provided maximally flat passband response,

minimum stopband and passband errors and -3dB frequency approximately or equal

to 1 rad/s. The transformation that would better transform lowpass filter transfer func-

tion to highpass filter transfer function with the proposed coefficients from (3.3) will

be chosen to design fractional order highpass filter. The best transformation provides

minimum passband and stopband errors. In following, three transformations are used

to transform lowpass to highpass filter.

34

1. Transformation 1

A transformation can be obtained by multiplying the transfer function of lowpass

filter as stated in (3.4) by s1+α, resulting in the following equation.

HHP11+α (s) =

s1+αk1s1+α + sαk2 + k3

(4.1)

This highpass transfer function is most commonly used in multi-loop feedback

structure. The expression in (4.1) has k1 constant to take unity, therefore the high

frequency passband gain stays also unity, which is 0 dB. Whereas the lowpass

filter transfer function (3.4) provides passband gain of 1/k1, which is not unity

for all values of α. Therefore the lowpass filter (3.4) and the transformed high-

pass filter (4.1) is anti-symmetrical. Hence this transformation is not the optimal

transformation as it does not maintain minimal deviation in the passband and

stopband regions for first order Butterworth functions.

2. Transformation 2

This transformation is similar to transformation 1 and it is obtained by multiply-

ing the transfer function of lowpass filter (3.4) by s1+α while assuming that k1 is

equal to 1, another step to the assumption that leads to this lowpass to highpass

transformation is applying gain correction of 1/k3 which leads to the following

highpass filter transfer function:

HHP21+α (s) =

s1+α/k3s1+α + sαk2 + k3

(4.2)

It is noted that some properties is not similar to previous technique mainly pass-

band and stopband errors, will be discussed later in following section.

3. Transformation 3

This transformation of highpass filter transfer function is obtained just by re-

placing the Laplacian operator s by 1/s. When the complex variable s in (3.4)

is replaced by 1/s yielding the following highpass filter transfer function.

HHP31+α (s) =

s1+αk1s1+αk3 + sk2 + 1

(4.3)

The unity value of coefficient k1 provides a passband gain of 1/k3 in (4.3). While

comparing (3.4) and (4.3) there is an interchange of denominator coefficients and

also the coefficient of middle term k2. The term sα in (3.4) has been replaced by

only s in (4.3). Taking frequency reference at 1 rad/sec, this lowpass to highpass

transformation does provide a symmetrical magnitude response characteristics.

35

4.3 Evaluation of the High Pass Filter Transfer Func-

tions

As discussed in previous section, three highpass fractional step filters are obtained and

represented by transfer functions (4.1), (4.2) and (4.3). It is necessary to evaluate mag-

nitude responses for all highpass fractional order transfer functions. The magnitude

response from each transformed function for α = 0.5 is plotted in Fig.4.1 against first

order highpass Butterworth filter H1(s) = s/(s + 1). The coefficients k2 and k3 are

Frequency (rad/s)10-1 100 101 102 103

Magnitude(dB)

-70

-60

-50

-40

-30

-20

-10

0

101st order123

Figure 4.1: Magnitude characteristics of 1. (4.1), 2. (4.2) and 3. (4.3)

used same as previously obtained for lowpass filter in Section 3.3. The constant k1 is

kept as 1. The first order Butterworth magnitude response is also plotted with solid

black line for comparison. From Fig.4.1 it is clear that each of the transformation do

not produce the same magnitude characteristics. In order to search for the coefficients

for fractional order lowpass filter, the frequency range used was from 0.01 to 1 rad/s,

therefore to evaluate the fractional order highpass filter response, the frequencies is

reciprocated that is, our interested frequency is now from 1 rad/s to 100 rad/s.

As shown in Fig.4.1 the filter (4.1) has a larger gain than the ideal Butterworth re-

sponse in the whole range of ω = [0.1,1000] rad/s. Similarly the filter (4.2) has much

larger gain deviation and more than (4.1). On the contrary, the filter (4.3) has both

the negative and positive error in the frequency range from 1 rad/s to 100 rad/s, while

compared to the ideal Butterworth response. However it is most accurate compared

36

to previous transformation methods. It concludes the last transformation method by

replacing s with 1/s is the most suited type of lowpass to highpass filter conversion.

With even higher frequencies, the filter (4.3) tends to provide even lower error with

passband magnitude gain and approximately is almost 0dB.

4.4 Least Square Error Analysis

The accuracy of fractional order highpass filters has been evaluated by computing the

least square error (LSE) between the magnitude response of first order Butterworth

filter and order (1 + α). The computed result is shown in Fig.4.2. The proposed

coefficients in Section (3.3) were used to compute magnitude responses of (1 + α)

order filter. The frequency range was taken between 1 rad/s to 100 rad/s.

Order (1 + α)1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

LSE

10-5

10-4

10-3

10-2

10-1

100

123

Figure 4.2: LSEs from 1.(4.1), 2.(4.2) and 3.(4.3) for α ∈ (0.01, 0.99)

The LSE was computed using

LSE =m∑i

[|H1+α(ωi)| − |H1(ωi)| ]2 (4.4)

where, |H1+α(ωi)| is the magnitude response of (1 + α) order highpass transfer func-

tion, |H1(ωi)| is the magnitude response of first order highpass butterworth function

at frequency ωi and m is the frequency points. In our calculation m = 500 frequency

points were taken between range from 1 rad/s to 100 rad/s. As per result plotted in

37

Fig.4.2 for α ∈ (0, 1), the filter transfer function (4.3) provides the lowest LSE. Like-

wise the proposed coefficients have also resulted the lowest LSE ranging from 0.0001

to 0.01 in compared to coefficients given in recent literature [29].

4.5 Pass Band and Stop Band Error Analysis

In order to further analyze the proposed fractional highpass filters, two error matrices

namely, passband error (PE) and stopband error (SE) are computed for all three types

of transfer functions. The analytical expressions for SE and PE for fractional order

highpass Butterworth response can be defined as follows.

SE = 20× log10

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

√√√√√ K∑i=1

||H1+α(ωi)| − |HHP1 (ωi)||

2

K

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

dB (4.5)

where, K = 500 and 1 ≤ ω ≤ 10.

PE = 20× log10

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

√√√√√ L∑i=1

||H1+α(ωi)| − |HHP1 (ωi)||

2

L

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

dB (4.6)

where, L = 500 and 10 ≤ ω ≤ 100.

Order (1 + α)1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

StopbandError(dB)

-28.5

-28

-27.5

-27

-26.5

-26

-25.5

-25

-24.5123

Figure 4.3: Stopband error index values of 1.(4.1), 2.(4.2), 3.(4.3) for (1+α) from 1.1

to 2

38

Order (1 + α)1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

PassbandError(dB)

-27.5

-27

-26.5

-26

-25.5

-25

-24.5

-24

123

Figure 4.4: Passband error index values of 1.(4.1), 2.(4.2), 3.(4.3) for (1+α) from 1.1

to 2

As shown in Fig.4.3, the stopband error is minimum for all values of α ∈ (0.01, 0.99)

from the proposed filter (4.3). High pass transfer function (4.1) has stopband error

close to -28dB, whereas the function (4.2) has returned a maximum -25.1dB at α =

0.01 and decreased as frequency from 1 rad/s to 10 rad/s. Same way, Fig.4.4 shows

the filter (4.1) gives fairly constant passband error for all values of α. This is due to

the passband gain of (4.1) is approximately constant around 0dB. However for (4.3),

the passband error shows a slight increase of 0.5dB as α increases from 0.1 to 0.8.

In general, one can see that all three transfer functions are giving better performances

while computing using the proposed coefficients.

Table 4.1: Comparison of PE and SE matrices for (1 + α) order highpass filters

ErrorTransfer functions 1 + α

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

SE (dB)

(4.1) -25.1093 -25.4162 -25.7662 -26.1683 -26.5144 -26.8790 -27.2191 -27.5231 -27.7909

(4.2) -27.8540 -27.7952 -27.7630 -27.7516 -27.7560 -27.7724 -27.8003 -27.8287 -27.8685

(4.3) -28.1703 -28.3383 -28.3999 -28.3951 -28.3515 -28.2882 -28.2169 -28.1431 -28.0668

PE (dB)

(4.1) -27.0708 -27.0668 -27.0691 -27.0703 -27.0718 -27.0733 -27.0754 -27.0755 -27.0764

(4.2) -24.5404 -24.8778 -25.2326 -25.5889 -25.9337 -26.2559 -26.5464 -26.7969 -27.0001

(4.3) -26.8533 -26.6621 -26.5135 -26.4510 -26.4786 -26.5779 -26.7211 -26.8800 -27.0295

Both SE and PE for (1 +α) order filters are listed in Table.4.1. From the table it is ev-

ident that fractional order transfer function (4.3) provides least passband and stopband

39

errors for highpass response and its the best choice for practical implementation and

realization.

4.6 Summary

This chapter presented a simple method to design a fractional order highpass filter.

The procedures were demonstrated to transform the lowpass to highpass filter. The

resulted filter has shown a better response in magnitude characteristic without peaking

in the passband. It was also confirmed that same coefficients obtained for lowpass

fractional order filter could be useful while designing the highpass. Thus, our proposed

values were successfully applicable to extend for the fractional order highpass filters.

Analysis was provided for all three types of transformed highpass filters and evaluated

with various error matrices.

40

Chapter 5

Hardware Implementation and Realization ofFraction Order Filters

5.1 Introduction

This chapter presents the hardware implementation of fractional order lowpass and

highpass filters. Fractional order filters designed in the earlier chapters are practi-

cally implemented on the field programmable analog array (FPAA) platform. The real

time outputs from the designed filters are used to validate in terms of magnitude re-

sponses. The use of second order approximation for fractional Laplacian operator is

also presented to obtain the design equations for hardware implementation. This chap-

ter presents the comparison of the simulated (1+α) order highpass and lowpass filters

and the practical ones obtained through configurable analog array modules. It is shown

the accuracy of the optimized coefficients obtained to acquire minimum passband and

stopband errors, parameter sensitivity and -3dB frequency to 1 rad/s.

5.2 Approximation of Fractional Laplacian Operator

to Fractional Order Low Pass Filter

Fractional Laplacian operator can be used to realize fractional order filters. The second

order approximation for general fractional Laplacian operator was obtained before in

Section 2.3. With the help of approximation (2.6), one can convert (3.4) into following

form.

HLP1+α(s) =

k1s1+α + k2sα + k3

∼= k1(a2s2 + a1s+ a0)

s3 + c0s2 + c1s+ c2

(5.1)

41

where a0 = α2+3α+2, a1 = 8−2α2, a2 = α2−3α+2, c0 = (a1+a0k2+a2k3)/a0,

c1 = (a1 (k2 + k3)+a2)/a0, and c2 = (a0k3+a2k2)/a0. It is to note that the coefficients

used in this transfer function are obtained before in Section 3.3. The coefficients,

k2 and k3 were selected to maintain a flat passband closest to butterworth response

along with minimum passband, stopband and least square errors. The interpolated

α

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

k2,k

3

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

k2k3

k3 = - 0.1981α 3 + 0.2471α 2 + 0.2359α +

0.7233

k2= 0.5293α 3 - 0.3156α 2 + 0.9672α + 0.2653

Figure 5.1: Coefficients k2,3 w.r.t. α values

equations for coefficients k2 and k3 were drawn from raw data through curve fitting

function of MATLAB. With the proposed coefficients from Fig.5.1, fractional order

low and highpass filters can be implemented and realized using FPAA discussed in the

following section.

5.3 Input and Output Interface with FPAA

There is a need to develop special hardware to interface external devices and in-

put/output pins from FPAAs. This is due to the fact that, there are different kinds

of sensors with variety of their output signals and different types of actuators with spe-

cial needs of input signals. The Anadigm FPAA is a single supply device therefore it

cannot handle a negative signal either on its input or outputs. Anadigm FPAA has in-

ternal signal ground that is put to a constant value of +1.5V and is called Voltage Main

Reference (VMR) also called reference voltage Vref . All the analog inputs/outputs

from FPAA are differential and is centered at Vref and is restricted around +5V which

is the supply voltage.

42

In order to get magnitude response for various (1 + α) order low and highpass fil-

ters, an external input signal is provided through the signal generator and the output

is recorded on an oscilloscope. The filter is designed in Anadigm designer2 software

with all the designed parameters and then the filter with desired design is downloaded

onto the FPAA board. The response of the filter is recorded on the oscilloscope and

magnitude response is plotted using the collected data. The AN231E04 FPAA requires

differential, level shifted input signals for processing. In order to process the single

ended signal from the signal generator, it needs to be converted to fully differential

signal. A circuit as shown in Fig.5.2a was designed to achieve the required level shift-

ing. The single ended input signal Vin from the signal generator is converted to fully

differential output signals I1N and I1P shifted with an offset voltage of Vref (which is

internally generated by FPAA and has a value of +1.5V), with a gain of Rf/Ri given

by the (5.2).

I1N = VinRf

Ri

+ Vref

I1P = −(Vin

Rf

Ri

+ Vref

) (5.2)

The differential output signals I1N and I1P , provides input signal to the FPAA. After

the processing, FPAA produces a fully differential output signal which needs to con-

verted to single ended signal in order to be read by the oscilloscope. A circuit as shown

in Fig.5.2b was designed to convert the differential signal from FPAA to single ended.

The circuit converts the differential output signals O3N and O3P to single ended output

signal V0 with a gain of Rf/Ri described by equation (5.3).

V0 =Rf

Ri

(O3P −O3N) (5.3)

Both single to differential and differential to single converters were implemented us-

ing TLE072A Op-Amps from Texas Instruments [21]. Both the converters were im-

plemented with input 4.7kΩ resistors to maintain a constant gain value of 1 so that

the signal remains undisturbed by Op-Amp gains. Fig.5.3 shows the test setup block

diagram used to interface signal generator and oscilloscope.

43

(a) Level-shifting single to differential converter.

(b) Differential to single converter.

Figure 5.2: Circuits to interface FPAA with external signals

5.4 Implementation of Fractional Order Filters on FPAA

Anadigm FPAA are ‘analogue signal processors’ consisting of fully configurable ana-

log modules (CAMs) surrounded by programmable interconnect and analogue input

and output cells [5]. Fully differential switched capacitor circuitry are built in to the

CAMs that allow signal processing, for different purposes such as filtering, gain con-

44

Figure 5.3: Block diagram of test setup

trol, sampling, summing, rectification and more. Anadign Designer tool provides flex-

ibility to reconfigure these CAMs allowing the user to build virtual circuits using the

design CAMs. After successful design of filters, the AnadigmDesigner tool generates

a configuration data file to program the FPAA. In our implementation of approximated

filters, two CAM modules are used namely the bilinear and biquadratic filter CAM

modules. These CAMs will allow implementation of both fractional order low and

highpass filters.

5.4.1 FPAA Implementation of (1 + α) Order Low Pass Filter

Previously in order to implement any filter, it requires the determination of the set of

component values and their decomposed transfer function to realize. But now using

latest features present in AnadigmDesigner 2 development environment we require

only the transfer function of pole-zero (PZ) frequencies and quality factor. Firstly the

transfer function (3.4) is to be decomposed into first and second order using bilinear

and biquadratic filter CAM modules. This can be written into following form,

H (s) = H1 (s)H2 (s) =1

s+ d0

e0s2 + e1s+ e2

s2 + d1s+ d2. (5.4)

The two CAMs are used to implement approximated fractional step filters as shown

in above equation. The cascaded connection between these two CAM modules in

the Anadigm Designer environment is shown in Fig. 5.4a. The prerequisite to realize

any transfer function using biquadratic and bilinear CAM is to decompose them into

45

Bilinear CAM Biquadratic CAM

(a) AnadigmDesigner2 Development

Environment

(b) FPAA Development kit

Figure 5.4: (a).Bilinear and biquadratic filter CAMs in Anadigm Designer environ-

ment, cascaded to implement a fractional order lowpass filter. (b). FPAA development

board

two parts. Thus, first part is obtained with bilinear characteristic and the other with

biquadratic characteristic as in (5.4). Let us represent H1 (s) and H2 (s) as first and

second order transfer functions, respectively. After decomposition PZ frequencies with

quality factor can be calculated easily using (5.4). The coefficients d0,1,2 and e0,1,2 can

be determined through the following sets of equations:

d0 + d1 =a1 + a0k2 + a2k3

a0

d0d1 + d2 =a1 (k2 + k3) + a2

a0

d0d2 =a0k3 + a2k2

a0

e0 = k1a2a0

e1 = k1a1a0

e1 = k1

(5.5)

CAMs have different form of taking in variables from (5.4) as specified in the AN231E04

FPAA datasheet [5]. Before transforming (5.4), the following frequency transforma-

tion s = ( sω0) = (s/2πf0) has to be performed. Transformation is resulted into CAMs

form as follows.

H (s) = T1 (s)T2 (s)

T1 (s) =2πf1G1

s+ 2πf1

T2 (s) = −s2 + 2πf2z

(Q2z )s+ 4π2f2z

2

s2 + 2πf2z

(Q2p)s+ 4π2f2p

2

(5.6)

46

where, T1 is the transfer function of bilinear CAM, T2 is the transfer function of bi-

quadratic CAM, G1 is a gain of T1, f1 is a pole frequency of T1, f2p,z is a PZ frequency

of T2, Q2p,z is a PZ quality factor of T2 and f0 is a de-normalized frequency. T1 and T2

is realized by the switched capacitor technology as shown in Fig.5.5 [5].

(a)

(b)

Figure 5.5: Internal switched capacitor circuit to realize (a) lowpass filter bilinear cam

(b) bi-quadratic filter cam

To implement (1+α) fractional order lowpass filter, the following design equation can

47

be used from [17].

f1 = d0f0

f2z = f0

√e2e0

Q2z =

√e0e2e1

f2p = f0√

d2

Q2p =

√d2d1

G1 =e0d0

(5.7)

Table 5.1: Theoretical and realised biquad and bilinear CAM parameter values for

physical implementation of (1 + α) order fractional lowpass filter

(a)

Design

Parameters

Order (1 + α)

1.2 1.6 1.9

Theoretical Realized Theoretical Realized Theoretical Realized

f1, kHz 0.3352 0.3550 0.4644 0.4650 0.7300 0.7300

f2p , kHz 1.7537 1.7600 1.4891 1.5000 1.1796 1.1800

f2z , kHz 1.3540 1.3600 2.7255 2.7300 7.0775 7.1100

Q2p , kHz 0.4940 0.4960 0.6480 0.6450 0.6788 0.6700

Q2z , kHz 0.2462 0.2510 0.2097 0.2100 0.1220 0.1250

G1 1.6272 1.600 0.2898 0.2860 0.0273 0.0274

(b)

value(1 + α)k2k3

(1 + 0.2)0.460.78 (1 + 0.6)0.890.91 (1 + 0.9)1.290.99

d0 0.3352 0.4644 0.7300

d1 3.5502 2.2981 1.7377

d2 3.0754 2.2173 1.3915

e0 0.5455 0.1346 0.0200

e1 3.0000 1.7500 1.1579

e2 1.0000 1.0000 1.0000

The filter CAMs are cascaded and wired up as shown in Fig.5.4a. Cell 1 is configured

as input cell and cell 3 is configured as output cell. The bilinear filter CAM is con-

figured as lowpass filter CAM with parameters corner frequency and gain as shown in

Fig.5.6a while biquadratic filter CAM is configured with PZ parameters as shown in

Fig.5.6b.

48

(a)

(b)

Figure 5.6: Setup of parameter (a) bilinear filter CAM for (1+α) = 1.2 (b) biquadratic

filter CAM using PZ parameters for (1 + α) = 1.2

Let us realize the FOLPF of orders (1 + α) = 1.2, 1.6, 1.9. The approximated PZ fre-

quencies of bilinear and biquadratic CAMs to realize using Anadigm FPAA are given

in Table 5.1(a) when f0 = 1 kHz. Table 5.1 (b) shows values of d0,1,2 and e0,1,2 from

49

(5.5) that eventually leads to Table 5.1 (a) for same values of α. The values for k2,3

in Table 5.1 (b) are both optimized values obtained through this implementation. The

realized values differ from the theoretical values as there are limitations on the values

that can be implemented by FPAA. The biquadratic and bilinear CAMs cannot realize

all possible values because there is hardware limitations as all corner frequencies, qual-

ity factors and gains are interrelated to the internal switched capacitor circuits of the

FPAA kit. Since the manufacturers only make finite number of capacitors, the Anadig-

mDesigner tool selects the best ratio of switched capacitors, matching to that entered

as design parameters. Sometimes ratio do not match accurately causing errors between

the theoretical and realized values. With the proposed coefficients in our work, the er-

ror between the theoretical and realized values are minimum and better compared to

those proposed in [17] and [15]. The error between the theoretical and realized values

is minimum which suggests that there is high accuracy in implementation fractional

order lowpass Butterworth filter on FPAA.

5.4.2 Experimental Results for (1 + α) Order Low Pass Filter

Fractional order lowpass filter was implemented on the FPAA development board. The

connections were made as shown in Fig.5.3. Experiment was setup as shown in Fig.5.7.

Figure 5.7: Experimental setup for hardware implementation

50

The cut off frequency for the lowpass filter was set to 1kHz. The (1 + α) order with

α = 0.2, 0.6 and 0.9 values were implemented. Frequency was increased from 100Hz

to 10kHz on a logarithmic scale. The corresponding input and output voltages were

recorded from the oscilloscope. Magnitude response corresponding to each particular

frequency was calculated using equation (5.8).

Magnitude (dB) = 20log10Vout

Vin

(5.8)

After calculating magnitude in decibels for each particular frequency from 1kHz to

10kHz, magnitude response of experimental (1 + α) order (dashed line) was plotted

with respect to simulated (1 + α) order (solid line) in MATLAB as shown in Fig.5.8.

The peak amplitude was set at 700mV equivalent to 1.4V peak-peak.

Frequency(Hz)10-2 10-1 100 101 102 103 104

Magnitude(dB)

-50

-40

-30

-20

-10

0

101st Order Butterworth

2nd Order Butterworth

α = 0.6

α = 0.9

α = 0.2

Figure 5.8: Simulation (solid line) and Experimental (dashed line) results for (1 + α)order fractional order lowpass filter

Input and output amplitude for α = 0.2 is given in Fig.5.9 for (a).frequency = 500Hz

(passband region), (b).frequency = 1kHz (cut-off frequency) and (c).frequency = 15kHz

(stopband region).

51

(a) For frequency = 500Hz, Input = 1.44V, Output =

1.18V, Magnitude response (dB) = -1.7296dB

(b) For frequency = 1kHz, Input = 1.42V, Output =

940mV, Magnitude response (dB) = -3.5832dB

(c) For frequency = 15kHz, Input = 1.44V, Output =

80mV, Magnitude response (dB) = -25.1054dB

Figure 5.9: Oscilloscope output for input and output response of the designed (1 + α)= 1.2 fractional order lowpass filter

52

In Fig.5.9, CH1 shows the input waveform on top of the oscilloscope screen, CH2

shows the output waveform at the bottom of the oscilloscope screen. It is evident that

the experimental results show close relationship with the simulation results. It further

confirms the operation of the proposed fractional order lowpass filter on the FPAA.

5.4.3 FPAA Implementation of (1 + α) Order High Pass Filter

Fractional order highpass filter can be implemented in the similar manner by decompo-

sition of the transfer function (3.4) into the form taken in by bi-linear and biquadratic

filter CAMS given by (5.4). The bilinear CAM had inputs parameters as corner fre-

quency and gain, and the biquadratic filter CAM had input parameters set as PZ con-

figuration. The design equations for d0,1,2 and e0,1,2 is slightly differs from lowpass

filter and can be calculated using the equation set (5.9).

d0 =√[(a2k2 + a0k3) x3 − (a1k2 + a1k3 + a2) x2 + (a0k2 + a2k3 + a1) x+ a0]

d1 =a0k2 + a2k3 + a1

a0− d0

d2 =a1k2 + a1k3 + a2

a0− d0d1

e0 = k1

e1 = k1a1a0

e2 = k1a2a0

e1 = k1

(5.9)

It is to note that x is a dummy variable and d0 is the positive real root in (5.9). The

values of k2,3 that are used to calculate d0,1,2 and e0,1,2, are proposed coefficients. The

values for d0,1,2 and e0,1,2 for filter orders (1 + α) = 1.2, 1.6 and 1.9, were calculated

using (5.9) respectively and is given in Table.5.2b along with bilinear and biquadratic

filter CAM parameters.

There is not much difference in implementing fractional order highpass filter on AN231E04

FPAA when compared to fractional order lowpass. Both the bilinear and biquadratic

filter CAMs are used to implement a highpass. Only the bilinear CAM is set in high-

pass configuration while is earlier set on lowpass implementation. The biquadratic

filter CAM was not changed and remained same the PZ configuration. The cascaded

53

Table 5.2: Theoretical and realised biquad and bilinear CAM parameter values for

physical implementation of (1 + α) order fractional highpass filter

(a)

Design

Parameters

Order (1 + α)

1.2 1.6 1.9

Theoretical Realized Theoretical Realized Theoretical Realized

f1, kHz 29.8315 29.800 21.5312 21.5000 13.6993 13.7000

f2p , kHz 12.5449 12.1000 14.0447 14.0000 10.7524 10.5000

f2z , kHz 7.3855 7.16000 3.6690 3.8300 1.4129 1.5000

Q2p , kHz 1.3903 1.47000 2.3048 2.4000 0.9795 0.9870

Q2z , kHz 0.2462 0.2510 0.2097 0.2180 0.1220 0.1230

G1 1 1 1 1 1 1

(b)

value(1 + α)k2k3

(1 + 0.2)0.460.78 (1 + 0.6)0.890.91 (1 + 0.9)1.290.99

d0 2.9832 2.1531 1.3699

d1 0.9023 0.6094 1.0977

d2 1.5737 1.9725 1.1562

e0 1 1 1

e1 3.0000 1.7500 1.1579

e2 0.5455 0.1346 0.0200

connection to implement a highpass filter using bilinear and biquadratic filter CAMs is

shown in Fig.5.10.

54

Figure 5.10: Implementation of fractional order highpass filter using bilinear and bi-

quadratic filter CAMs.

The theoretical and realized values for PZ frequencies, quality factor for biquadratic

CAM; and frequency, gain for Bilinear CAM are given in Table.5.2a. For highpass

filter f0 = 10kHz was used as a cut-off frequency. The realized and theoretical values

are slightly different due to hardware limitations to implement high decimal values.

5.4.4 Experimental Results for (1 + α) Order High Pass Filter

Another attempt is made to implement a proposed fractional order highpass filter on

FPAA board. The connections are shown same as given before in Fig.5.3 and experi-

ment setup is shown in Fig.5.7. The highpass gain for biquadratic filter CAM was kept

to a constant value of 1. Clock frequency was set to 200kHz. The most optimum trans-

formation of fractional order lowpass to highpass filter is used from (4.3). As discussed

in Section (4), the filter was implemented using obtained best coefficients k2 and k3.

The cut-off frequency for highpass filter was set to 10kHz. We have implemented the

(1 + α) orders with α = 0.2,0.6 and 0.9, respectively. Frequency range was setup from

10kHz to 100kHz on a logarithmic scale. The corresponding input and output voltages

were recorded from the oscilloscope. Magnitude gain corresponding to each frequency

was calculated using (5.8). After calculating magnitude in decibels for each particle

frequency from 10kHz to 100kHz, magnitude response of experimental (1 + α) order

(dashed line) was plotted with respect to simulated (1 + α) order (solid line) in MAT-

55

LAB as shown in Fig.5.8. The peak amplitude was set at 700mV equivalent to 1.4V

peak-peak.

Frequency (rad/s)103 104 105

Magnitude(dB)

-50

-40

-30

-20

-10

0

10

1st Order Butterworth

2nd Order Butterworth

α = 0.2

α = 0.6

α = 0.6

Figure 5.11: Simulated (solid line) and Experimental (dashed line) results for (1 + α)order fractional order highpass filter

Fig.5.12 is shown with input and output signals for α = 0.2 and measured for (a) fre-

quency = 5kHz (stopband region), (b) frequency = 10kHz (cut-off frequency) and (c)

frequency = 30kHz (passband region). In this figure, CH1 shows the input waveform

on top of the oscilloscope screen and CH2 shows the output waveform at the bottom

of the oscilloscope screen.

56

(a) For frequency = 5kHz, Input = 1.44V, Output =

256mV, Magnitude response (dB) = -15.0024dB

(b) For frequency = 10kHz, Input = 1.42V, Output =

1.20V, Magnitude response (dB) = -1.5836dB

(c) For frequency = 30kHz, Input = 1.44V, Output =

1.46V, Magnitude response (dB) = -0.11980dB

Figure 5.12: Waveforms for (1 + α) = 1.2 fractional order highpass filter

57

Results are evident again for the close relationship between simulation and experimen-

tal values. Further the presented filter produces a better result in terms of passband

peaking and a flat passband response from the obtained coefficients values for a filter.

5.5 Summary

This chapter discusses the hardware implementation of fractional order lowpass and

highpass filters. Actual fractional order filter was implemented on FPAA board. Re-

sults seen the approximated (n + α) order in integer terms can be realized for a frac-

tional order filter. In our case, second order approximation for fractional Laplacian

operator was used to implement fractional order filters.

The interface issues between the Anadigm AN231E04 development board and mea-

surement devices were simplified by external converters. Fractional order filters of

order (1 + α) was implemented in FPAA board. Results were shown to compare the

actual and simulated values for both highpass and lowpass filters. It concludes the

proposed fractional order filters can be implemented real-time.

58

Chapter 6

Conclusions and Future Work

6.1 Conclusions

This work has focussed on design and implementation of fractional order lowpass and

highpass filters. Practical realization of fractional order filter was the main objective

using analog processor. In our study we considered most related literature proposed

on various ways to design fractional order transfer function parameters in simulation.

A new bi-level constraint optimization routine was simulated in order to obtain the

best optimal values of filter parameters. The proposed filters have shown better per-

formances in compared to previously proposed fractional filters. An analysis has been

carried out to decide a suitable transformation procedure for highpass filter from the

proposed lowpass filter. Further, the optimal order of approximation sα was suggested

using continued fractional expansion in order to implement fractional differentiator in

hardware with acceptable accuracy.

At the end, fractional order low and highpass filters were implemented using CAM

modules in the Anadigm development environment of FPAA. The waveforms from the

proposed filters (both lowpass and highpass) were measured with various ranges of

signal input frequencies. In this way, actual functionality of the fractional order filter

was validated on the analog array board. The performance of fractional order filters of

order (1 + α) has been studied and compared with corresponding integer order filters

through both experimentation and simulation. The obtained results from MATLAB

and real time have verified the implementation and operation of the fractional step

filters. Also it has been observed that the actual fractional filter’s behavior has closely

followed the theoretical.

59

6.2 Contributions

The contribution of this thesis is mainly to design and implement the fractional (non-

integer) order sα representing the differentiator. The effort has been made to obtain the

optimal set of values for filter parameters to perform not only closely with magnitude

characteristics but also robust with parameter variations. The study is conducted to

know the importance of integer order approximation to design fractional order differ-

entiator until fractance devices becomes available in market. The physical realization

of fractional order filter has been discussed in detail and shown how the fractional or-

der filter behaves with respect to an integer order. Analysis of such research can open

a wide range of possibility in applications for system identification and control.

6.3 Future Directions

Following the design technique described in this thesis, a number of possible directions

for extensions to this work are discussed below:

• The other type of filters such as bandpass, Chebyshev, inverse Chebyshev and

Elliptic filters can similarly be investigated as future scope.

• The phase behavior can also be taken into consideration while implementing the

fractional order component. It will bring more interest if a work can be done

in manipulating the phase response while maintaining the desired magnitude

response.

• Since this work is addressed the issues from implementation the fractional (non-

integer) order sα, it will be interesting to test the fractional order controller like

FOPID on analog processor.

• Digital fractional order filter realization may be possible future work bases on

result obtained in this thesis.

60

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Appendix

Fractional Order Butterworth Filter Design Algorithms

MATLAB Code for Lowpass Filter Magnitude Response

%The following code plots lowpass Butterworth filter magnituderesponse for alpha = 0.2,0.6 and 0.9 for Figure 5.8.

close allclear all

fmin=1000; % minimum frequencyfmax=100000; % maximum frequencyf=linspace(fmin,fmax,10000); % logarithmic scalew=2*pi*f; % define omegaf0=10000; % define cut-off frequencyw0=2*pi*f0;s=1i*w/w0;

% declaration of alpha valuesalpha0=0;alpha1=0.2;alpha2=0.6;alpha3=0.9;alpha4=1;

% calling transfer functions for various alpha values[Hs0]=tfvalue(alpha0,s);[Hs1]=tfvalue(alpha1,s);[Hs2]=tfvalue(alpha2,s);[Hs3]=tfvalue(alpha3,s);[Hs4]=tfvalue(alpha4,s);

% Plottingsemilogx(f,20*log10(abs(Hs0)),’b’);hold on;semilogx(f,20*log10(abs(Hs1)),’y’);hold on;semilogx(f,20*log10(abs(Hs2)),’g’);hold on;semilogx(f,20*log10(abs(Hs3)),’m’);hold on;semilogx(f,20*log10(abs(Hs4)),’c’);hold on;

%legend({’Freeborn et al (2010) ’,’Freeborn (2015)’,’Proposed ’,’y=0.09’},’Location’,’northwest’,’FontSize’,10,’FontName’,’Times New Roman’)

xlabel(’frequency(Hz)’), ylabel(’Magnitude (dB)’)

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MATLAB Code for Highpass Filter Magnitude Response

%The following code plots highpass Butterworth filter magnituderesponse for alpha = 0.2,0.6 and 0.9 for Figure 5.11.

close allclear all

fmin=1000; % minimum frequencyfmax=100000; % maximum frequency

f=linspace(fmin,fmax,100000); % logarithmic scalew=2*pi*f; % define omegaf0=10000; % cut off frequencyw0=2*pi*f0;s=1i*(w/w0);

%declaration of alpha valuesalpha0=0;alpha1=0.4;alpha2=0.5;alpha3=0.9;alpha4=1;

% calling transfer functions for various alpha values[Hs0]=highpass_tfvalue3(alpha1,s);[Hs1]=highpass_tfvalue1(alpha2,s);[Hs2]=highpass_tfvalue2(alpha2,s);[Hs3]=highpass_tfvalue3(alpha2,s);

%plottingsemilogx(f,20*log10(abs(Hs0)),’b’);hold on;semilogx(f,20*log10(abs(Hs1)),’g’);hold on;semilogx(f,20*log10(abs(Hs2)),’m’);hold on;semilogx(f,20*log10(abs(Hs3)),’y’);hold on;

legend({’1st order ’,’2 ’,’3 ’,’4’},’Location’,’northwest’,’FontSize’,10,’FontName’,’Times New Roman’)xlabel(’frequency (rad/s)’), ylabel(’Magnitude (dB)’)

MATLAB Code for FPAA Parameters

% The following code is used to get FPAA parameters for physicalrealization of bilinear and biquad filter CAMsas highlighted in Table 5.1

close all

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clear all

a=0.1; % alpha = 0.1f0= 1; % cut - off frequency 1 khz

k1=1;k2 = (((0.5293*(a)^3))-(0.3156*(a)^3)+(0.9672*(a))+(0.2653))k3 = (((-0.1981*(a)^3))+(0.2471*(a)^2)+(0.2359*(a))+(0.7233))

k2 =round(k2*100)/100;k3 =round(k3*100)/100;

a0=((a^2)+(3*a)+2);a1=(8-(2*(a^2)));a2=((a^2)-(3*a)+2);c0=((a1+(a0*k2)+(a2*k3)));c1=((a1*(k2+k3)+a2));c2=((a0*k3)+(a2*k2));

syms xp = (a0*(x^3))-(c0*(x^2))+(c1*(x)-c2);r = roots(sym2poly(p));d0=min(r)

d1 = ((a1+(a0*k2)+(a2*k3))/(a0))-d0d2 = (((a1*k2)+(a1*k3)+a2)/(a0))-(d0*d1)e0 = ((k1)*(a2/a0))e1 = ((k1)*(a1/a0))e2 = k1f1= (d0*f0)f2_z = f0*sqrt((e2/e0))f2_p = f0*sqrt(d2)Q2_z = ((sqrt(e0*e2))/e1)Q2_p= (sqrt(d2))/d1G1 = (e0/d0)

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