Post on 31-Mar-2020
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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