Lecture 9 FIR and IIR Filter design using

Matlab

2007/11/16

Prof. C.M Kyung

FIR and IIR Filter

GOAL Linear-Time-Invariant (LTI) system and Impulse response

z-Transform

Characteristics of FIR and IIR filters

Design procedure of FIR and IIR filters

FIR and IIR Filter

LTI System Input (x(t) or x[n]) and Output (y(t) or y[n]) is defined first

For a given system, it can be either LTI or non-LTI depending on how we define the input and output.

Linearity For arbitrary and , the output of the system for

is the sum the output for and . (superposition)

Time-invariance Time-shift in input results in time-shift in output by same amount for

time-invariant systems.

][1 nx ][2 nx ][][][ 21 nxnxnx ][1 nx ][2 nx

]}[{][]}[{][ mnxoutputmnynxoutputny ..ei

]}[{]}[{]}[][{ 2121 nxoutputnxoutputnxnxoutput ..ei

FIR and IIR Filter

Impulse Response Definition

Impulse response is the output of the system when impulse signal or is applied as the input of the system.

Importance Impulse response “fully describes” the system if the system is LTI.

– Why and how?– Why the impulse response CANNOT fully describe a non-LTI

system?

Fourier transform of the impulse response shows the characteristic of the system is frequency domain.

)(t ][n

FIR and IIR Filter

z-Transform Definition

The z-transform of a sequence is defined as

Example For ,

,

Note z-transform is reduced to discrete-time Fourier transform (DTFT) if is

substituted by . This means that z-transform on the unit-circle on the complex plain is

same as DTFT. Laplace Transform CTFT ~ z-Transform DTFT

n

nznxnxzX ][]}[{)(

zje

][nx

][][ nuanx n

10

1

1

1)(][][]}[{)(

az

azznuaznxnxzXn

n

n

nn

n

n|||| az

FIR and IIR Filter

Ideal frequency-selective filter A filter whose frequency response is unity over a certain

frequency range and zero for other frequencies.

Frequency response of an ideal low-pass filter

However, an ideal low-pass filter is noncausal.

-3 -2 -1 0 1 2 3-0.2

0

0.2

0.4

0.6

0.8

1

frequency

Fre

quen

cy R

espo

nse

Ideal Low pass filter

FIR and IIR Filter FIR / IIR filter

Definition If the length of the impulse response is finite, the filter is an FIR (finite

impulse response) filter. Otherwise, the filter is an IIR (infinite impulse response) filter.

FIR Inherently BIBO (bounded-input, bounded-output) stable Nonzero pole does not exist in its transfer function Easy to implement Can be designed to have linear phase property

IIR Sometimes unstable Nonzero pole exists in its transfer function Lower filter order than a corresponding FIR filter Usually have nonlinear phase property

FIR and IIR Filter

Filter Design Procedure Design continuous-time IIR filter

Obtain desired using Butterworth, Chebyshev methods

Convert it to discrete-time IIR filter using impulse invariance Impulse invariance : ,

if ,

Obtain discrete-time FIR filter by windowing the IIR filter Windowing : Commonly used windows : rectangular, Bartlett, Hanning, Hamming,

Blackman, Kaiser, … However, windowing does not give the optimum solution and other

approaches can be used.

)(sH

)(][ dcd nThTnh )()(d

cj

TjHeH ||

0)( jH c dT/||

][][][ nwnhnh IIRFIR

FIR and IIR Filter

Frequency response of various filters

FIR and IIR Filter

Problem Statements Design several types of FIR and IIR filters

IIR – butterworth, chebyshev type1, chebyshev type 2, … FIR – using different windows ( Hamming, Hanning, Bartlett, … )

Remove the noise in acoustic signal using the filters What are the differences between the filters ?

Understand the effect of sampling frequency on the sampled signal distortion (aliasing)

FIR and IIR Filter

Experiment Requirements PC Matlab software (with signal processing toolbox)

FIR and IIR Filter

References

Fundamentals of Signal & System using the web and matlab - Edward W. Kamen, Bonnie S. Heck

Discrete-Time Signal Processing- Alan V. Oppenheim, Ronald

W. Schafer

http://www.mathworks.com/access/helpdesk/help/toolbox/signal/filterde.html