EE422Signals and Systems Laboratory

Infinite Impulse Response (IIR) filters

Kevin D. DonohueElectrical and Computer Engineering

University of Kentucky

FiltersFilter are designed based on specifications given by:spectral magnitude emphasisdelay and phase properties through the group delay and

phase spectrumimplementation 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

Filter Specifications

Example:Low-pass filter frequency response

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Normalize Frequency (fs/2 = 1)

Am

plit

ud

e S

cale

Passband

Transitionband

Stopband

Cutoff Frequency

Ripple

Filter SpecificationsExample Low-pass filter frequency response (in dB)

0 0.2 0.4 0.6 0.8 1-70

-60

-50

-40

-30

-20

-10

0

10

Normalized Frequency (fs/2 = 1)

Sca

le in

dB Passband

Transitionband

Stopband

Cutoff Frequency

Ripple

Filter SpecificationsExample Low-pass filter frequency response (in dB) with ripple in both bands

0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10

0

Normalize Hz

dB

Filter Magnitude Response

Passband Ripple

Stopband Ripple

Transition

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).

Show

Does this represent an FIR or IIR filter? What are the poles and zeros of this system?

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).

Show

Does this represent an FIR or IIR filter? What are the poles and zeros of this system?

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).

• Show

Is it stable? How would this be represented as direct form filter in Matlab?

Filter Specification Functions

The transfer function magnitude or magnitude response:

The transfer function phase or phase response:

The group delay (envelope delay) :

)2exp()H()(H

fjzM zf

)2exp()H()(H

fjzP zf

df

fdf P

P

)(H)(G

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?

21

1

81.02728.11)(ˆ

zz

zzH

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

RE

Z-PlaneIM

Filter Analysis Example

% Script to illustrate frequency analysis of filters

% given y[n] = x[n-1] + 1.2728y[n-1] - 0.81y[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, -1.2728, 0.81]; % (denominator) first element is time of current output

]1[]2[81.0]1[2728.1][81.02728.11

)(ˆ21

1

nxnynynyzz

zzH

Filter Analysis Example% Frequency response

[h,f] = freqz(b,a,1024,fs);

figure

% Magnitude

plot(f,abs(h)); xlabel('Hz'); ylabel('TF Magnitude');

figure

% Phase

plot(f,phase(h)); xlabel('Hz'); ylabel('TF Phase in Radians');

% Group delay

[d,f] = grpdelay(b,a,1024,fs);

figure

% delay in seconds

plot(f,d/fs); xlabel('Hz'); ylabel('delay in seconds');

figure

% delay in samples

plot(f,d); xlabel('Hz'); ylabel('delay in samples');

Filter Analysis Example

0 1000 2000 3000 4000 5000 6000 7000 80000

2

4

6

8

Hz

TF

Mag

nitu

de

Magnitude Response with FREQZ

0 1000 2000 3000 4000 5000 6000 7000 8000-4

-3

-2

-1

0

Hz

TF

Pha

se in

Rad

ians

Phase Response with FREQZ

0 1000 2000 3000 4000 5000 6000 7000 80000

5

10

Hz

Del

ay in

Sam

ples

Group Delay with GRPDELAY

0 500 1000 1500 2000 2500 3000 3500 40000

0.2

0.4

0.6

0.8

1

1.2x 10

-3

Hz

dela

y in

sec

onds

Group Delay

Filter Design ExamplesThe following commands generate filter coefficients for basic low-pass, high-pass, band-pass, band-stop filters:For linear phase FIR filters: fir1For 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.

Filter Design ExamplesExample: 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.

Computational Diagrams

x[n] w[n] y[n]

z-1

z-1

b2

b1

b0

-a1

z-1

z-1

G0

-a2

1/a0

For Direct form I filter below, derive difference equation and TF

Hint: Form difference equations around accumulator/registeroutputs, convert to TF and try to cancel out auxiliary variables.

Computational DiagramsFor Direct form II filter below, derive difference equation and TF

Hint: Form difference equations around accumulator/registeroutputs, convert to TF and try to cancel out auxiliary variables.

x[n] w[n] y[n]

b2

b1

b0

z-1

z-1

-a2

-a1

G0 1/a0