Chebyshev’s IIR Filter using Impulse Invariance Method

Impulse Invariant Method:

  1. Poles are transferred by using the equation,

Transformation to poles

Where,  Pk = P1, P2, P3….. PN are the poles

Ts is Sampling Time

  1. Mapping is many to one.
  2. Aliasing effect is present.
  3. It is not suitable to design High-pass filter and band reject Filter.
  4. Only Poles of the system can be mapped.
  5. No Frequency warping effect.

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 equiripple 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.

Step 1: Frequency response:

Cheby Filter

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

Here  step 2= Chebyshev polynomial of order M.

Step 2: Parameter ϵ

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

Cheby Filter 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,

Formula for H

When magnitude is in dB then,

Formula for H in dB

When specifications are not in dB then,

Nth Order Formula Chebyshev

Step 4: Position of poles of Chebyshev filter:

First calculate regular position of poles using,

Theta Formula

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

x and y definitions

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

r also

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

R in passband

Here the parameter is β given by,

Value for Beta

Then the pole positions are denoted by,





Formula for Poles

Step 5: System Transfer Function:

The system Transfer Function of analog Filter is given by,

Poles in H Transfer Function

After simplification this equation can be written as,

Shortcut poles

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 impulse invariant transformation by taking T=1 sec to satisfy the following specifications:

Chebyshev 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=0.9;
as=0.24;
p_d=0.28*pi;
s_d=0.5*pi;
t=1;
pass_attenuation=-20*log10(ap);
stop_attenuation=-20*log10(as);
p_a=p_d/t;
s_a=s_d/t;
[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]=impinvar(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 chebyshev using impulse invariant')
xlabel('normalized freq')
ylabel('magnitude')

OUTPUT:

Output Image Cheby Impulse

n =     3

normalised transfer function

hsn =

0.5162

———————————-

s^3 + 1.021 s^2 + 1.272 s + 0.5162

[showhide type=”post” more_text=”Show more…” less_text=”Show less…”]

Continuous-time transfer function.

unnormalised transfer function

hs =

0.3513

————————————

s^3 + 0.8984 s^2 + 0.9839 s + 0.3513

Continuous-time transfer function.

digital transfer func

hz =

0.1224 z^2 + 0.09107 z

———————————-

z^3 – 1.686 z^2 + 1.307 z – 0.4072

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.