IIR Filter Design&

FIR Filter Design

Utkarsh KulshresthaKuldeep Saini

IIR FilterFilters are used to remove the noise from the

baseband signals.

Filters gives the flat and smooth frequency responses.

IIR filters are the Infinite impulse response filters.

Filters are of basically two types:-1.Analog filter2. Digital filters

IIR Filterss.no.

Analog Filters Digital Filter

1. Process analog input and analog output

Process digital input and digital output

2. Constructed from active and passive electronic component

Constructed from adder,multiplier and delay unit

3. Described by a differential equations

Described by difference equation

4. Frequency response is changed by varying components

Frequency response is changed by filter cofficient

IIR FilterIIR design methods are as follows:-

1.Impulse Invariant method

2.Billinear method

3.Approximation of Derivatives

Impulse invariant method

IIR FiltersBilinear Transformation:-

1- Bilinear transformation is transformation from analog s-plane to digital z-plane.

2-It avoid the alising problem occurs in impulse invariant method.

3-This conversion maps analog poles to digital poles and analog zeros to digital

zeros.

IIR Filter

• Prewrapping Procedure:-

• Amplitude response of digital filter is expanded at the lower frequencies and compressed at the higher frequencies in comparision to the analog filter.

• Steps to design digital digital filter using bilinear transformation technique :-

• 1-find prewrapping analog frequency W=2/T tan(w/2)• 2-using analog frequencies find H(s)• 3-select the sampling rate of digital filter substitute S=2/T(1-inv z/1+inv z)Into transfer function H(s)

Butterworth Filter

The butterworth filter has magnitude frequency response is flat in passband and stopband.

As the filter order increases then the butterworth response approximates the ideal filter response.

Chebyshev filterType-1 Chebyshev filter they have equiripple behavior In passband and monotonic characterstics in stop band.

Type-2 Chebyshev filter the magnitude response has maximally in passband and equiripple in stop band.

Chebyshev filter• Type-1

Butterworth Filter

Chebyshev Filter

FIR filterIt is a Digital filter which has Finite Impulse response duration.

They have usually no feedback.

It has finite number of non zero terms.

FIR filters are used where linear phase response in passband is required.

It is used for data transmission, speech processing and always stable.

Design technique for FIR filter

1. Fourier series method2. Windowing method

Rectangular windowRaised cosine windowHamming windowHanning windowBlackmann window

Bartlett windowKaiser window

3.Frequency sampling method

Filter Design by Windowing• Simplest way of designing FIR filters• Method is all discrete-time no continuous-time involved• Start with ideal frequency response

• Choose ideal frequency response as desired response• Most ideal impulse responses are of infinite length• The easiest way to obtain a causal FIR filter from ideal is

• More generally

n

njd

jd enheH

deeH21

nh njjdd

else0

Mn0nhnh d

else0

Mn01nw where nwnhnh d

351M Digital Signal Processing

Properties of Windows• Prefer windows that concentrate around DC in frequency

– Less distortion, closer approximation

• Prefer window that has minimal span in time – Less coefficient in designed filter, computationally efficient

• So we want concentration in time and in frequency– Contradictory requirements

• Example: Rectangular window

2/sin2/1Msin

ee1

e1eeW 2/Mj

j

1MjM

0n

njj

Rectangular Window

else0

Mn01nw

• Narrowest main lob– 4/(M+1)– Sharpest transitions at

discontinuities in frequency

• Large side lobs– -13 dB– Large oscillation

around discontinuities

• Simplest window possible

18

Bartlett (Triangular) Window

else0

Mn2/MM/n22

2/Mn0M/n2

nw

• Medium main lob– 8/M

• Side lobs– -25 dB

• Hamming window performs better

• Simple equation

Hanning Window

else0

Mn0M

n2cos1

21

nw

• Medium main lob– 8/M

• Side lobs– -31 dB

• Hamming window performs better

• Same complexity as Hamming

Hamming Window

else0

Mn0M

n2cos46.054.0nw

• Medium main lob– 8/M

• Good side lobs– -41 dB

• Simpler than Blackman

21

Kaiser Window Filter Design Method• Parameterized equation

forming a set of windows– Parameter to change main-

lob width and side-lob area trade-off

– I0(.) represents zeroth-order modified Bessel function of 1st kind

otherwise

MnalphaIBetaInw

,0

2\1||),(|)({

Determining Kaiser Window Parameters• Given filter specifications Kaiser developed empirical

equations– Given the peak approximation error or in dB as A=-20log10

– and transition band width

• The shape parameter alpha should be

• The filter order M is determined approximately by

210

50212107886.0215842.0

507.81102.04.0

A

AAA

AA

alpha

FpFsF

1

F

FDM

Example: Kaiser Window Design of a Lowpass Filter• Specifications• Window design methods assume• Determine cut-off frequency

– Due to the symmetry we can choose it to be

• Compute

• And Kaiser window parameters

• Then the impulse response is given as

001.0,01.0,6.0,4.0 21pp 001.021

5.0c

2.0ps 60log20A 10

653.5alpha 37M

else0

Mn0653.5I

5.185.18n

1653.5I

5.18n5.18n5.0sinnh

0

2

0

24

Example Cont’d

Comparison of IIR & FIR filters

Copyright (C) 2005 Güner Arslan 351M Digital Signal Processing 25

s.no.

characterstics IIR FIR

1. Unit sample response Infinite duration Finite duration

2. Feedback characterstics

Use feedback from the output

Do not use feedback

3. Phase characterstics Non-linear phase response

Linear phase response

4. Number of computations

Less computations More computations

5. Applications Used where sharp cut-off characterstics with minimum order are required

Used where linear phase characterstics are required

More on FIR & IIR Next time…