1. Infinite Impulse Response (IIR) Filters
Recursive Filters:
with constant coefficients.
Advantages: very selective filters with a few parameters;
Disadvantages: a) in general nonlinear phase,
b) can be unstable.

2. Design Techniques: discretization of analog filters
s-plane
z-plane
analog
digital
Problem: we need to map
the derivative operator “s” into the time shift operator “z”,
and make sure that the resulting system is still stable.

3. approximation of “s” Two major techniques
Euler Approximation (easiest),
Bilinear Transformation (best).
Euler Approximation of the differential operator:
approximation of “s”
take the z-Transform of both sides:

4. Example:
take the analog filter with transfer function and discretize it
with a sampling frequency
By Euler’s approximation
The filter is implemented by the difference equation
s-plane
z-plane
analog
digital

5. if Re[s]<0. Problem with Euler Approximation:
it maps the whole stable region of the s-plane into a subset of the stable region in the z-plane
s-plane
z-plane
since
if Re[s]<0.

6. Bilinear Transformation. It is based on the relationship
nT-T nT A B C D
Take the z-Transform of both sides:
which yields the bilinear transformation:

7. Main Property of the Bilinear Transformation: it preserves the stability regions.
s-plane
z-plane
since:

8. Mapping of Frequency with the Bilinear Transformation.
Magnitude:
Phase:
where

9. See the meaning of this:
it is a frequency mapping between analog frequency and digital freqiency.

10. Example: we want to design a digital low pass filter with a bandwith
and a sampling frequency Use the Bilinear Transformation.
Solution:
Step 1: specs in the digital freq. domain
Step 2: specs of the analog filter to be digitized:
or equivalently
Step 3: design an analog low pass filter (more later) with a bandwith ;
Step 4: apply Bilinear Transformation to obtain desired digital filter.

11. Design of Analog Filters
Specifications:
pass
band
transition
band
stop
band

12. Two Major Techniques: Butterworth, Chebychev
Specify from passband, determine N from stopband:

13. + + + + Poles of Butterworth Filter:
which yields the poles as solutions
N=2
and choose the N poles in the stable region. + +
poles + +
s-plane

14. Example: design a low pass filter, Butterworth, with 3dB bandwith of 500Hz and 40dB attenuation at 1000Hz.
Solution:
solve for N from the expression
poles at

15. Chebychev Filters.
Based on Chebychev Polynomials:
Property of Chenychev Polynomials:
within the interval Chebychev polynomials have least maximum deviation from 0 compared to polynomials of the same degree and same highest order coefficient

16. Why? Suppose there exists with smaller deviation then
root
root
root
But: has degree 2 …
… and it cannot have three roots!!!
So: you cannot find a P(x) which does better (in terms of deviation from 0) then the Chebychev polynomial.

17. Chebychev Filter:
Since (easy to show from the definition), then

18. Design of Chebychev Filters:
Formulas are tedious to derive. Just give the results:
Given: the passband, and
which determines the ripple in the passband,
compute the poles from the formulae
s-plane

19. Example: design a Chebychev low pass filter with the following specs:
passband with a 1dB ripple,
stopband with attenuation of at least 40dB.
Step 1: determine The passband frequency
For 1dB ripple,
Step 2: determine the order N. Use the formula
with , to obtain

20. Frequency Transformations
We can design high pass, bandpass, bandstop filters from transformations of low pass filters.
Low Pass to High Pass:
same value at

21. Low Pass to Band Pass:
The tranformation
maps

22. Low Pass to Band Stop

23. How to make the transformation:
Consider the transfer function
then with we obtain
with zeros and poles solutions of
also n-m extra zeros at s where

24. IIR filter design using Matlab
In Matlab there are three functions for each class of filters (Butterworth, Chebytchev1, Chebytchev2):
BUTTAP CHEB1AP CHEB2AP Poles and Zeros of Analog Prototype Filter
BUTTER CHEBY1 CHEBY2 Numerator and Denominator from N and
BUTTORD CHEBY1ORD CHEBY2ORD N and from specifications
Example. We want to design an IIR Digital Filter with the following specifications:
Pass Band 0 to 4kHz, with 1dB ripple;
Stop Band > 8kHz with at least 40 dB attenuation
Sampling frequency 40kHz
Type of Filter: Butterworth.
Using Matlab:
>> [N, fc]=butterord(fp, fs, Rp, Rs); % fp, fs=passband and stopband freq relative to Fs/2
>> [B, A]=butter(N, fc); % B, A vectors of numerator and denominator coefficients.
In our case:
[N, fc]=butterord(4/20, 8/20, 1, 40), would yield N=7, fc=0.2291;
[B, A]=butter(7, ), would yield the transfer function B(z)/A(z).

25. Let’s verify these numbers:
Step 1: specifications in the digital frequency domain:
Step 2: specifications for analog filter from the transformation

26. Step 3: choose (say) Butterworh Filter
with and from the ripple specification
Step 4: determine order N from attenuation of 40dB
with yields N=7

27. Step 5: finally the cutoff frequency, from the equation
Which yields , corresponding to a digital frequency
Step 6: the desired Filter is obtained by the function
[num, den] = butter( 7 , /)

28. Magnitude and Phase Plots: