Presentation onDesign of Digital Filter using MATLAB

By Imtiyaz Mohiuddin

10361A0434

In chargeDr.M.Narayana

Electronics and Communications Engineering

SECTION OUTLINE:Introduction to Digital filter designIntroduction to FIR FilterDesign of FIR Filter using WINDOW

Techniques Introduction to IIR FilterDesign of IIR Filter using Approximation

MethodMATLAB Code of Designed Filters

Introduction:A digital filter is a system that performs

mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal.

In digital signal processing, there are two important types of systems:Digital filters: perform signal filtering in the

time domainSpectrum analyzers: provide signal

representation in the frequency domain

Digital Filter:

xn yn

Digital Filter

Samplingfrequency

fS

ADC

DAC

x(t) y(t)Analog

anti-aliasing

filter

Analogsmoothing

filter

Preliminaries:The design of a digital filter is carried out in

three steps:Specifications: they are determined by the applicationsApproximations: once the specification are defined, we

use various concepts and mathematics that we studied so far to come up with a filter description that approximates the given set of specifications. (in detail)

Implementation: The product of the above step is a filter description in the form of either a difference equation, or a system function H(z), or an impulse response h(n). From this description we implement the filter in hardware or through software on a computer.

Classification:Digital filters are classified into one of two

basic forms, according to how they respond to a unit impulse:

Finite impulse response Infinite impulse response

Finite Impulse Response:In signal processing, a finite impulse

response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time.

FIR digital filters use only current and past input samples, and none of the filter's previous output samples, to obtain a current output sample value

The transfer function is given by

The length of Impulse Response is NAll poles are at Z=0. . Zeros can be placed anywhere on the

z-plane

1

0).()(

N

n

nznhzH

Filter Design by Windowing

Simplest way of designing FIR filters

Start with ideal frequency response

Choose ideal frequency response as desired response Most ideal impulse responses are of infinite length

n

njd

jd enheH

deeH21nh njj

dd

else0Mn0nhnh d

Rectangular:

Bartlett:

Hamming:

Blackman:

Kaiser:

21 NnN

n21

Nn2cos1

Nn2cos46.054.0

Nn

Nn 4cos08.02cos5.042.0

)(1

21 0

2

0 JNnJ

Commonly used windows

Kaiser windowKaiser window

β Transition width (Hz)

Min. stop attn dB

2.12 1.5/N 30

4.54 2.9/N 50

6.76 4.3/N 70

8.96 5.7/N 90

Rectangular Window

else0Mn01nw

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

discontinuities in frequency

Large side lobs– Large oscillation around

discontinuities– -13 dB

Simplest window possible

Hamming Window

else0

Mn0Mn2cos46.054.0nw

Medium main lob– 8/M

Good side lobs– -41 dB– Simpler than

Blackman

Kaiser Window 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

else0Mn0I

2/M2/Mn1I

nw0

2

0

MATLAB CODE: %Design of LPF&HPF using rectangular,hamming and kaiser windows clc;clear all;close all; rp=input('enter attenuation in pass band'); rs=input('enter attenuation in stop band'); fp=input('enter pass band frequency'); fs=input('enter stop band frequency'); Fs=input('enter sampling frequency'); wp=2*pi*fp/Fs; ws=2*pi*fs/Fs; %formula for FIR filter num=-20*log10(sqrt(rp*rs))-13; den=14.6*(fs-fp)/Fs; n=ceil(num/den); disp('order of filter is \n'); disp(n); disp('press any key to continue'); pause; n1=n+1; %For even order if(rem(n,2)~=0) n1=n; end %LPF

%LPF s1=input('enter the value for window 0-rectangularLPF 1-kaiserLPF 2-hammingLPF 3-

rectangularHPF 4-kaiserHPF 5-hammingHPF'); switch(s1); case 0 y=rectwin(n1); [b,a]=fir1(n,wp,'low',y); freqz(b,a,512); case 1 y=kaiser(n1); [b,a]=fir1(n,wp,'low',y); freqz(b,a,512); case 2 y=hamming(n1); [b,a]=fir1(n,wp,'low',y); freqz(b,a,512); case 3 y=rectwin(n1); [b,a]=fir1(n,wp,'high',y); freqz(b,a,512); case 4 y=kaiser(n1); [b,a]=fir1(n,wp,'high',y); freqz(b,a,512); case 5 y=hamming(n1); [b,a]=fir1(n,wp,'high',y); freqz(b,a,512); end

Pros & Cons:FIR filters have the followingadvantages:

Exactly linear phase is possible

Always stable, even when quantized

Design methods are generally linear

Efficient hardware realizations

Startup transients have finite duration

FIR filters have the following disadvantages: • Higher filter order than IIR

filters• Corresponding greater

delays

Infinite Impulse Response Filter:IIR systems have an impulse

response function that is non-zero over an infinite length of time. This is in contrast to finite impulse response (FIR) filters, which have fixed-duration impulse responses

IIR filters may be implemented as either analog or digital filters

Cont..While designing a digital IIR filter , an analog

filter (e.g. Chebyshev filter, Butterworth filter) is first designed and then is converted to a digital filter by applying discretization techniques such as Bilinear transform or Impulse invariance.

Discretization techniques

Chebyshev Filter:Chebyshev filters are analog or digital

filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II)

Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter, but with ripples in the passband

Cont..Type-1 Chebyshev Filter

Type-2 Chebyshev Filter:

Butterworth filterThe Butterworth filter is a type of signal

processing filter designed to have as flat a frequency response as possible in the pass band. It is also referred to as a maximally flat magnitude filter

MATLAB Prototype Filter Design Commands

[B,A] = BUTTER(N,Wn) [B,A] = CHEBY1(N,R,Wn) [B,A] = CHEBY2(N,R,Wn) [B,A] = ELLIP(N,Rp,Rs,Wn)

– N = filter order– R = pass band ripple (cheby1) or stop-band ripple (cheby2) in

dB. (Rp and Rs respectively for the elliptic filter)– Wn = cut-off frequency (radians/sec for analog filters or

normalized digital frequencies for digital filters)– [B,A] = filter coefficients, s-domain (analog filter) or z-domain

(digital filter)

Design Example Filter Specifications:

Butterworth response Pass-band edges = 400 Hz and 600 Hz Stop-band edges = 300 Hz and 700 Hz Pass-band ripple = 1 dB Stop-band attenuation = -20 dB Sampling Frequency = 2000 Hz

Design Example Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

1.2

1.4

Frequency (kHz)

Mag

nitu

de

Magnitude Response

Band Edges (-1dB and -20 dB)

Design ExampleChebyshev II High-Pass Filter Filter specifications:

Chebyshev II response (stop-band ripple) Pass-band edge = 1000 Hz Stop-band edge = 900 Hz Pass-band ripple = 1 dB Stop-band attenuation = -40 dB Sampling frequency = 8 kHz

MATLAB Code for Design Example>> fs=8000;>> Wp=[2*1000/fs]; % Pass-band edge normalized digital frequency>> Ws=[2*900/fs]; % Stop-band edge normalized digital frequency>> [N,Wn]=cheb2ord(Wp,Ws,1,40); % The “order” command>> [B,A]=cheby2(N,40,Wn,'high'); % cheby2 is the “filter” command. In this command % the syntax requires the stop-band attenuation % as the second parameter>> fvtool(B,A)

Design Example Results

0 0.5 1 1.5 2 2.5 3 3.5-100

-80

-60

-40

-20

0

20

Frequency (kHz)

Mag

nitu

de (d

B)

Magnitude Response (dB)

MATLAB CODE: %Design of IIR filters fp1=input('enter pass band frequency'); fs1=input('enter stop band frequency'); Fs1=input('enter sampling frequency'); wp1=fp1/Fs1; ws1=fs1/Fs1; [n1,wn1]=buttord(wp1,ws1,2,60); [x,y]=butter(n1,wn1,'low'); figure; freqz(x,y,512); [n1,wn1]=buttord(wp1,ws1,2,60); [x,y]=butter(n1,wn1,'high'); figure; freqz(x,y,512); [n1,wn1]=cheb1ord(wp1,ws1,2,60); [x,y]=cheby1(n1,3,wn1,'low'); figure; freqz(x,y,512); [n1,wn1]=cheb1ord(wp1,ws1,2,60); [x,y]=cheby1(n1,3,wn1,'high'); figure; freqz(x,y,512);

Summary of IIR Filter:

IIR filters can be design by pole-zero location– Digital oscillators: poles on the unit circle– Notch filters: zeros on the unit circle with nearby poles to

control notch width Classic analog filters can be designed using the

bilinear transformation IIR filters have the advantage of smaller filter order for

a given frequency response. IIR filters have the disadvantages of possible instability

due to coefficient quantization effects and non-linear phase response.

References:“Design of IIR Filter” by K.S Chandra, M.Tech, IIT-Bombay,

Jan-2006“Digital Filter Design” by Prof. A.G. Constantinides,

University of Auckland, 2006“FIR Filter Design”, Gao Xinbo,School of E.E.,

Xidian Univ. xbgao@ieee.org“FIR Filter by Windowing”- The lab Book Pages.com“Digital Signal Processing”, Prof.Ramesh Babu,

Pondicherry Govt. College, TataMcgraw-Hill publication.Wikipedia.org