Chebyshev’s IIR Filter using Bilinear Transformation Method






Bilinear Transformation Method:

  1. Poles are transformed by using equation,

Equation

Where, Ts is Sampling Time

  1. Mapping is one to one.
  2. Aliasing effect is not present.
  3. High pass Filter and band Reject Filter can be designed.
  4. Poles as well as zeros can be mapped.
  5. Frequency warping effect is present.

Chebyshev Filter :

Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, [citation needed] but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials.

Using Chebyshev filter design, there are two subgroups,

  1. Type-1 Chebyshev filter: These filters are all pole filters. In the passband, these filters show equi-ripple behaviour and they have monotonic characteristics in the stopband.
  2. Type-2 Chebyshev filter: This Filter Contains zeros as well as poles.

Design equations and Design steps:

From the given specification of digital Filter, obtain equivalent analog filter as follows:

Omega Equation

Here,

Ω= Frequency of analog Filter.

ω= Frequency of digital Filter.

Ts= Sampling Time.

Step 1: Frequency response:

The magnitude squared frequency response of Chebyshev filter is given by-

value of H j omega

Here step 2 = Chebyshev polynomial of order M.

Step 2: Parameter ϵ

It represents ripple parameter in the passband. It is given by,

Epsilon in decibels

If Ap is not in dB then,

epsilon

Step 3: Order of Filter:

At cut-off Frequency Ωc=1, the magnitude is given by,

use of epsilon

When magnitude is in dB then,

H j omega in decibels

When specifications are not in dB then,

order of filter

Step 4: Position of poles of Chebyshev filter:

First calculate regular position of poles using,

theta

The position of poles of chebyshev filter lie at co-ordinates xk and yk given by,

x and y equations

Here r represents minor axis of ellipse and is given by,

and R represents major axis of ellipse and is given by,

Equation for R

Here the parameter is β given by,

Equation for Beta

Then the pole positions are denoted by,

Poles Equation

Step 5: System Transfer Function:

The system Transfer Function of analog Filter is given by,

Transfer Func

After simplification this equation can be written as,

Simplified Transfer Function

Here b0=constant term in the denominator.

Now the value of k is calculated as follows,

Value of K

MATLAB CODE:

Consider Problem: Design a Chebyshev digital IIR using Bilinear transformation method by taking T=1 sec to satisfy the following specifications:

Chebyshev BLT Example

Solution:

The Given Specifications are

Ap= 0.9                 omega_p= 0.28 π

As= 0.24               omega_s= 0.5 π

clear all
clc
ap=0.8;
as=0.2;
p_d=0.2*pi;
s_d=0.32*pi;
t=1;
pass_attenuation=-20*log10(ap);
stop_attenuation=-20*log10(as);
p_a=(2/t)*tan(p_d/2);
s_a=(2/t)*tan(s_d/2);
[n,cf]=cheb1ord(p_a,s_a,pass_attenuation,stop_attenuation,'s')
[bn,an]=cheby1(n,pass_attenuation,1,'s')
disp('normalised transfer function')
hsn=tf(bn,an)
[b,a]=cheby1(n,pass_attenuation,cf,'s')
disp('unnormalised transfer function')
hs=tf(b,a)
[num,den]=bilinear(b,a,1/t)
disp('digital transfer func')
hz=tf(num,den,t)
w=0:pi/16:pi
disp('freq response')
hw=freqz(num,den,w)
disp('magnitude response')
hw_mag=abs(hw)
plot(w/pi,hw_mag,'k')
grid
title('mag response of chebyshew using bilinear')
xlabel('normalized freq')
ylabel('magnitude')

 

OUTPUT:





Chebyshevs BLT output

n =     3
normalised transfer function
hsn =
0.3333
———————————-
s^3 + 0.7489 s^2 + 1.03 s + 0.3333
[showhide type=”post” more_text=”Show more…” less_text=”Show less…”]
Continuous-time transfer function.
unnormalised transfer function
hs =
0.09147
————————————-
s^3 + 0.4867 s^2 + 0.4351 s + 0.09147
Continuous-time transfer function.
digital transfer func
hz =
0.008386 z^3 + 0.02516 z^2 + 0.02516 z + 0.008386
————————————————-
z^3 – 2.274 z^2 + 1.967 z – 0.6263
Sample time: 1 seconds
Discrete-time transfer function.[/showhide]

What is Frequency Warping?

Because of the non-linear mapping, the amplitude response of digital IIR filter is expanded at lower frequencies and compressed at higher frequencies in comparison to the analog filter. This effect is called as frequency warping.

What is aliasing effect?

Aliasing is an effect that causes different signals to become indistinguishable (or aliases of one another) when sampled. It also refers to the distortion or artifacts that result when the signal reconstructed from samples is different from the original continuous signal.


Pursuing Bachelor's Degree in Electronics and Telecommunication Engineering from K. J. Somaiya Institute of Engineering and Information Technology, Mumbai University.