1. EE422 Signals and Systems Laboratory
Infinite Impulse Response (IIR) filters
Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

2. Filters Filter are designed based on specifications given by:
spectral magnitude emphasis
delay and phase properties through the group delay and phase spectrum
implementation and computational structures
Matlab functions for filter design
(IIR) besself, butter, cheby1, cheby2, ellip, prony, stmcb
(FIR) fir1, fir2, kaiserord, firls, firpm, firpmord, fircls, fircls1, cremez
(Implementation) filter, filtfilt, dfilt
(Analysis) freqz, fdatool, sptool

3. Filter Specifications
Example:Low-pass filter frequency response

4. Filter Specifications
Example Low-pass filter frequency response (in dB)

5. Filter Specifications
Example Low-pass filter frequency response (in dB) with ripple in both bands

6. Difference Equation and TF Examples
Derive the TFs for the following difference equations (y[n] is output and x[n] is input). (Hint: Use delay property of ZT and assume zero for initial conditions).
𝑦 𝑛 =.5𝑥 𝑛 +𝑥[𝑛−1]−.25𝑦 𝑛− 𝑦[𝑛−2]
Show
𝐻 𝑧 = 𝑌 (𝑧) 𝑋 (𝑧) = .5+ 𝑧 − 𝑧 −1 −0.125 𝑧 −2 = .5 𝑧 2 + 𝑧 1 𝑧 𝑧 1 −0.125
Does this represent an FIR or IIR filter? What are the poles and zeros of this system?

7. Difference Equation and TF Examples
Derive the TFs for the following difference equations (y[n] is output and x[n] is input). (Hint: Use delay property of ZT and assume zero for initial conditions).
𝑦 𝑛 =𝑥 𝑛 −.5𝑥 𝑛−1 +.25𝑥[𝑛−2]−.125𝑥[𝑛−3]
Show
𝐻 𝑧 = 𝑌 (𝑧) 𝑋 (𝑧) = 1−0.5 𝑧 − 𝑧 −2 −0.125 𝑧 −3 1 = 𝑧 3 −0.5 𝑧 𝑧 1 − 𝑧 3
Does this represent an FIR or IIR filter? What are the poles and zeros of this system?

8. Difference Equation and TF Examples
Derive the difference equation from the following TF (y[n] is output and x[n] is input). (Hint: Express in terms of negative z powers, use delay property of ZT and assume zero for initial conditions).
𝐻 𝑧 = 𝑧 3 +2 𝑧 1 −1 𝑧 4 −0.1 𝑧 𝑧 2 −0.5 𝑧 1 +1
Show
𝑦 𝑛 =𝑥 𝑛−1 +2𝑥 𝑛−3 −𝑥 𝑛−4 +.1𝑦 𝑛−1 −.5𝑦 𝑛−2 +.5𝑦 𝑛−3 −𝑦[𝑛−4]
Is it stable? How would this be represented as direct form filter in Matlab?

9. Filter Specification Functions
The transfer function magnitude or magnitude response:
The transfer function phase or phase response:
The group delay (envelope delay) :

10. Filter Analysis Example
Consider a filter with transfer function:
Compute and plot the magnitude response, phase response, and group delay. Note pole-zero diagram. What would be expected for the magnitude response?

11. Filter Analysis Example
% Script to illustrate frequency analysis of filters
% given y[n] = x[n-1] y[n-1] y[n-2]
fs = 8000; % Sampling frequency
% Numerator and denominator polynomials to represent filter
b = [0, 1]; % (numerator) first is element time of current output
a = [1, , 0.81]; % (denominator) first element is time of current output

12. Filter Analysis Example
% Frequency response
[h,f] = freqz(b,a,1024,fs);
figure
% Magnitude
plot(f,abs(h)); xlabel('Hz'); ylabel('TF Magnitude');
% Phase
plot(f,phase(h)); xlabel('Hz'); ylabel('TF Phase in Radians');
% Group delay
[d,f] = grpdelay(b,a,1024,fs);
% delay in seconds
plot(f,d/fs); xlabel('Hz'); ylabel('delay in seconds');
% delay in samples
plot(f,d); xlabel('Hz'); ylabel('delay in samples');

13. Filter Analysis Example

14. Filter Design Examples
The following commands generate filter coefficients for basic low-pass, high-pass, band-pass, band-stop filters:
For linear phase FIR filters: fir1
For non-linear phase IIR filter: besself, butter, cheby1, cheby2, ellip
Example: With function fir1, design an FIR high-pass filter for signal sampled at 8 kHz with cutoff at 500 Hz. Use order 10 and order 50 and compare phase and magnitude spectra with freqz command. Use grpdelay to examine delay properties of the filter. Also use command filter to filter a frequency swept signal from 20 to 2000 Hz over 4 seconds with unit amplitude.

15. Filter Design Examples
Example: With function cheby2 design an IIR Chebyshev Type II high-pass filter for signal sampled at 8 kHz with cutoff at 500 Hz, and stopband ripple of 30 dB down. Use order 5 and order 10 for comparing phase and magnitude spectra with freqz command. Use grpdelay to examine delay characteristics. Also use command filter to filter a frequency swept signal from 20 to 2000 Hz over 2 seconds with unit amplitude.
Example: With function butter, design an IIR Butterworth band-pass filter for signal sampled at 20 MHz with a sharp passband from 3.5 MHz to 9MHz. Use order 5 and verify design of phase and magnitude spectra with freqz command.

16. Computational Diagrams
For Direct form I filter below, derive difference equation and TF
x[n]
w[n]
y[n]
z-1 b2 b1 b0
-a1 G0
-a2
1/a0
Hint: Form difference equations around accumulator/register outputs, convert to TF and try to cancel out auxiliary variables.

17. Computational Diagrams
For Direct form II filter below, derive difference equation and TF
x[n]
w[n]
y[n] b2 b1 b0
z-1
-a2
-a1 G0
1/a0
Hint: Form difference equations around accumulator/register outputs, convert to TF and try to cancel out auxiliary variables.