Matlab programs

MATLAB Programs
Chapter 16
16.1 INTRODUCTION
MATLAB stands for MATrix LABoratory. It is a technical computing environment
for high performance numeric computation and visualisation. It integrates numerical
analysis, matrix computation, signal processing and graphics in an easy-to-use
environment, where problems and solutions are expressed just as they are written
mathematically, without traditional programming. MATLAB allows us to express
the entire algorithm in a few dozen lines, to compute the solution with great accuracy
in a few minutes on a computer, and to readily manipulate a three-dimensional
display of the result in colour.
MATLAB is an interactive system whose basic data element is a matrix that
does not require dimensioning. It enables us to solve many numerical problems in a
fraction of the time that it would take to write a program and execute in a language
such as FORTRAN, BASIC, or C. It also features a family of application specific
solutions, called toolboxes. Areas in which toolboxes are available include signal
processing, image processing, control systems design, dynamic systems simulation,
systems identification, neural networks, wavelength communication and others.
It can handle linear, non-linear, continuous-time, discrete-time, multivariable and
multirate systems. This chapter gives simple programs to solve specific problems
that are included in the previous chapters. All these MATLAB programs have been
tested under version 7.1 of MATLAB and version 6.12 of the signal processing
toolbox.
16.2 REPRESENTATION OF BASIC SIGNALS
MATLAB programs for the generation of unit impulse, unit step, ramp, exponential,
sinusoidal and cosine sequences are as follows.
% Program for the generation of unit impulse signal
clc;clear all;close all;
t522:1:2;
y5[zeros(1,2),ones(1,1),zeros(1,2)];subplot(2,2,1);stem(t,y);

816 Digital Signal Processing
ylabel(‘Amplitude --.’);
xlabel(‘(a) n --.’);
% Program for the generation of unit step sequence [u(n)2 u(n 2 N]
n5input(‘enter the N value’);
t50:1:n21;
y15ones(1,n);subplot(2,2,2);
stem(t,y1);ylabel(‘Amplitude --.’);
xlabel(‘(b) n --.’);
% Program for the generation of ramp sequence
n15input(‘enter the length of ramp sequence’);
t50:n1;
subplot(2,2,3);stem(t,t);ylabel(‘Amplitude --.’);
xlabel(‘(c) n --.’);
% Program for the generation of exponential sequence
n25input(‘enter the length of exponential sequence’);
t50:n2;
a5input(‘Enter the ‘a’ value’);
y25exp(a*t);subplot(2,2,4);
stem(t,y2);ylabel(‘Amplitude --.’);
xlabel(‘(d) n --.’);
% Program for the generation of sine sequence
t50:.01:pi;
y5sin(2*pi*t);figure(2);
subplot(2,1,1);plot(t,y);ylabel(‘Amplitude --.’);
% Program for the generation of cosine sequence
t50:.01:pi;
y5cos(2*pi*t);
subplot(2,1,2);plot(t,y);ylabel(‘Amplitude --.’);
As an example,
enter the N value 7
enter the length of ramp sequence 7
enter the length of exponential sequence 7
enter the a value 1
Using the above MATLAB programs, we can obtain the waveforms of the unit
impulse signal, unit step signal, ramp signal, exponential signal, sine wave signal and
cosine wave signal as shown in Fig. 16.1.

MATLAB Programs 817
Fig. 16.1 Representation of Basic Signals (a) Unit Impulse Signal (b) Unit-step
Signal (c) Ramp Signal (d) Exponential Signal (e) Sinewave Signal ( f )Cosine Wave Signal
−2 −1 0 1 2 0
0
2
2
4
4
6
6 8
nn
nn
0.2
0.2
1
0.2
0.4
0.4
2
0.4
0.6
0.6
3
4
5
6
7
0.6
0.8
0.8
0.8
1
1
1
0
00
0 2 4 6 8
0
(a)
Amplitude
Amplitude
AmplitudeAmplitude
(b)
(d)(c)
1
1
0.5
0.5
0.5
0.5
0
0
1
1
1.5
(e)
(f)
1.5
2.5
2.5
3
3
3.5
3.5
2
2
0
0
−0.5
−0.5
−1
−1
n
n
AmplitudeAmplitude

16.3 DISCRETE CONVOLUTION
16.3.1 Linear Convolution
Algorithm
1. Get two signals x(m)and h(p)in matrix form
2. The convolved signal is denoted as y(n)
3. y(n)is given by the formula
y(n) 5 [ ( ) ( )]x k h n k
k
−
=−∞
∞
∑ where n50 to m 1 p 2 1
4. Stop
% Program for linear convolution of the sequence x5[1, 2] and h5[1, 2, 4]
clc;
clear all;
close all;
x5input(‘enter the 1st sequence’);
h5input(‘enter the 2nd sequence’);
y5conv(x,h);
figure;subplot(3,1,1);
stem(x);ylabel(‘Amplitude --.’);
subplot(3,1,2);
stem(h);ylabel(‘Amplitude --.’);
subplot(3,1,3);
stem(y);ylabel(‘Amplitude --.’);
disp(‘The resultant signal is’);y
As an example,
enter the 1st sequence [1 2]
enter the 2nd sequence [1 2 4]
The resultant signal is
y51 4 8 8
Figure 16.2 shows the discrete input signals x(n)and h(n)and the convolved output
signal y(n).
Fig. 16.2 (Contd.)
AmplitudeAmplitude
n
n
0
0
0.5
1
1
1
1.1
1.2
1.2
1.4
1.3
1.6
1.4
1.8
1.5
2
(a)
(b)
1.6
2.2
1.7
2.4
1.8
2.6
1.9
2.8
2
3
1
2
1.5
3
2
4

MATLAB Programs 819
16.3.2 Circular Convolution
% Program for Computing Circular Convolution
clc;
clear;
a = input(‘enter the sequence x(n) = ’);
b = input(‘enter the sequence h(n) = ’);
n1=length(a);
n2=length(b);
N=max(n1,n2);
x = [a zeros(1,(N-n1))];
for i = 1:N
k = i;
for j = 1:n2
H(i,j)=x(k)* b(j);
k = k-1;
if (k == 0)
k = N;
end
end
end
y=zeros(1,N);
M=H’;
for j = 1:N
for i = 1:n2
y(j)=M(i,j)+y(j);
end
end
disp(‘The output sequence is y(n)= ‘);
disp(y);
Fig. 16.2 Discrete Linear Convolution
AmplitudeAmplitudeAmplitude
n
n
n
0
0
0
0.5
1
2
1
1
1
1.1
1.2
1.2
1.4
1.5
1.3
1.6
1.4
1.8
1.5
2
2
(a)
(b)
(c)
1.6
2.2
2.5
1.7
2.4
3
1.8
2.6
3.5
1.9
2.8
2
3
4
1
2
4
1.5
3
6
2
4
8
n
n
n
0
0
0
0.5
1
2
1
1
1
1.1
1.2
1.2
1.4
1.5
1.3
1.6
1.4
1.8
1.5
2
2
(a)
(b)
(c)
1.6
2.2
2.5
1.7
2.4
3
1.8
2.6
3.5
1.9
2.8
2
3
4
1
2
4
1.5
3
6
2
4
8

stem(y);
title(‘Circular Convolution’);
xlabel(‘n’);
ylabel(‚y(n)‘);
As an Example,
enter the sequence x(n) = [1 2 4]
enter the sequence h(n) = [1 2]
The output sequence is y(n)= 9 4 8
% Program for Computing Circular Convolution with zero padding
clc;
close all;
clear all;
g5input(‘enter the first sequence’);
N15length(g);
N25length(h);
N5max(N1,N2);
N35N12N2;
%Loop for getting equal length sequence
if(N350)
h5[h,zeros(1,N3)];
else
g5[g,zeros(1,2N3)];
end
%computation of circular convolved sequence
for n51:N,
y(n)50;
for i51:N,
j5n2i11;
if(j550)
j5N1j;
end
y(n)5y(n)1g(i)*h(j);
end
end
disp(‘The resultant signal is’);y
As an example,
enter the first sequence [1 2 4]
enter the 2nd sequence [1 2]
The resultant signal is y51 4 8 8
16.3.3 Overlap Save Method and Overlap Add method
% Program for computing Block Convolution using Overlap Save
Method
Overlap Save Method
x=input(‘Enter the sequence x(n) = ’);

MATLAB Programs 821
h=input(‘Enter the sequence h(n) = ’);
n1=length(x);
n2=length(h);
N=n1+n2-1;
h1=[h zeros(1,N-n1)];
n3=length(h1);
y=zeros(1,N);
x1=[zeros(1,n3-n2) x zeros(1,n3)];
H=fft(h1);
for i=1:n2:N
y1=x1(i:i+(2*(n3-n2)));
y2=fft(y1);
y3=y2.*H;
y4=round(ifft(y3));
y(i:(i+n3-n2))=y4(n2:n3);
end
disp(‘The output sequence y(n)=’);
disp(y(1:N));
stem(y(1:N));
title(‘Overlap Save Method’);
xlabel(‘n’);
ylabel(‘y(n)’);
Enter the sequence x(n) = [1 2 -1 2 3 -2 -3 -1 1 1 2 -1]
Enter the sequence h(n) = [1 2 3 -1]
The output sequence y(n) = 1 4 6 5 2 11 0 -16 -8 3 8 5 3 -5 1
%Program for computing Block Convolution using Overlap Add
Method
x=input(‘Enter the sequence x(n) = ’);
h=input(‘Enter the sequence h(n) = ’);
n1=length(x);
n2=length(h);
N=n1+n2-1;
y=zeros(1,N);
h1=[h zeros(1,n2-1)];
n3=length(h1);
y=zeros(1,N+n3-n2);
H=fft(h1);
for i=1:n2:n1
if i<=(n1+n2-1)
x1=[x(i:i+n3-n2) zeros(1,n3-n2)];
else
x1=[x(i:n1) zeros(1,n3-n2)];
end
x2=fft(x1);
x3=x2.*H;
x4=round(ifft(x3));
if (i==1)

y(1:n3)=x4(1:n3);
else
y(i:i+n3-1)=y(i:i+n3-1)+x4(1:n3);
end
end
disp(‘The output sequence y(n)=’);
disp(y(1:N));
stem((y(1:N));
title(‘Overlap Add Method’);
xlabel(‘n’);
ylabel(‘y(n)’);
As an Example,
Enter the sequence x(n) = [1 2 -1 2 3 -2 -3 -1 1 1 2 -1]
Enter the sequence h(n) = [1 2 3 -1]
The output sequence
y(n) = 1 4 6 5 2 11 0 -16 -8 3 8 5 3 -5 1
16.4 DISCRETE CORRELATION
16.4.1 Crosscorrelation
Algorithm
1. Get two signals x(m)and h(p)in matrix form
2. The correlated signal is denoted as y(n)
y(n) 5 [ ( ) ( )]x k h k n
k
−
=−∞
∞
∑
where n52 [max (m, p)2 1] to [max (m, p)2 1]
4. Stop
% Program for computing cross-correlation of the sequences
x5[1, 2, 3, 4] and h5[4, 3, 2, 1]
clc;
clear all;
close all;
x5input(‘enter the 1st sequence’);
y5xcorr(x,h);
subplot(3,1,2);
stem(h);ylabel(‘Amplitude --.’);
subplot(3,1,3);
stem(fliplr(y));ylabel(‘Amplitude --.’);

MATLAB Programs 823
disp(‘The resultant signal is’);fliplr(y)
As an example,
enter the 1st sequence [1 2 3 4]
enter the 2nd sequence [4 3 2 1]
y51.0000 4.0000 10.0000 20.
↑
0000 25.0000 24.0000 16.0000
Figure 16.3 shows the discrete input signals x(n)and h(n)and the cross-correlated
output signal y(n).
n
n
n
0
0
1
1
1
1
1
1.5
1.5
2
2
2
3
3
3
5
2.5
2.5
4
(a)
(b)
(c)
3.5
3.5
6
4
4
7
2
2
3
3
4
4
30
20
10
0
Fig. 16.3 Discrete Cross-correlation
16.4.2 Autocorrelation
Algorithm
1. Get the signal x(n)of length N in matrix form
2. The correlated signal is denoted as y(n)
y(n) 5 [ ( ) ( )]x k x k n
k
−
=−∞
∞
∑
where n52(N 2 1) to (N 2 1)

% Program for computing autocorrelation function
x5input(‘enter the sequence’);
y5xcorr(x,x);
subplot(2,1,2);
stem(fliplr(y));ylabel(‘Amplitude --.’);
disp(‘The resultant signal is’);fliplr(y)
As an example,
enter the sequence [1 2 3 4]
y54 11 20
↑
30 20 11 4
Figure 16.4 shows the discrete input signal x(n)and its auto-correlated output
signal y(n).
( )
AmplitudeAmplitude
n
n
0
1
1
1
1.5
2
2
3
3
(a)
5
2.5
4
(b) y (n)
3.5
6
4
7
2
3
4
0
5
10
15
20
25
30
Fig. 16.4 Discrete Auto-correlation
16.5 STABILITY TEST
% Program for stability test
b5input(‘enter the denominator coefficients of the
filter’);
k5poly2rc(b);
knew5fliplr(k);
s5all(abs(knew)1);
if(s55 1)
disp(‘“Stable system”’);

MATLAB Programs 825
else
disp(‘“Non-stable system”’);
end
As an example,
enter the denominator coefficients of the filter [1 21 .5]
“Stable system”
16.6 SAMPLING THEOREM
The sampling theorem can be understood well with the following example.
Example 16.1 Frequency analysis of the amplitude modulated discrete-time
signal
x(n)5cos 2 pf1
n 1 cos 2pf2
n
where f1
1
128
= and f2
5
128
= modulates the amplitude-modulated signal is
xc
(n)5cos 2p fc
n
where fc
550/128. The resulting amplitude-modulated signal is
xam
(n)5x(n) cos 2p fc
n
Using MATLAB program,
(a) sketch the signals x(n), xc
(n) and xam
(n), 0 # n # 255
(b) compute and sketch the 128-point DFT of the signal xam
(n), 0 # n # 127
(c) compute and sketch the 128-point DFT of the signal xam
(n), 0 # n # 99
Solution
% Program
Solution for Section (a)
clc;close all;clear all;
f151/128;f255/128;n50:255;fc550/128;
x5cos(2*pi*f1*n)1cos(2*pi*f2*n);
xa5cos(2*pi*fc*n);
xamp5x.*xa;
subplot(2,2,1);plot(n,x);title(‘x(n)’);
xlabel(‘n --.’);ylabel(‘amplitude’);
subplot(2,2,2);plot(n,xc);title(‘xa(n)’);
subplot(2,2,3);plot(n,xamp);
%128 point DFT computation2solution for Section (b)
n50:127;figure;n15128;
f151/128;f255/128;fc550/128;
xc5cos(2*pi*fc*n);
xa5cos(2*pi*fc*n);
(Contd.)

−2
−1
0
1
2
Amplitude
0 100 200 300
(iii) n
Fig. 16.5(a) (iii) Amplitude Modulated Signal
Fig. 16.5(a) (ii) Carrier Signal and
−1
−0.5
0
0.5
1
Amplitude
0 100 200 300
(ii) n
0
−2
−1
0
1
2
100
Amplitude
200
(i)
300
n
Fig. 16.5(a) (i) Modulating Signal x (n)
(Contd.)

MATLAB Programs 827
20
Amplitude
40 60 80 100 120
n
1400
−10
−5
0
5
10
15
20
25
Fig. 16.5(b) 128-point DFT of the Signal xam
(n), 0 # n # 127
20 40 60 80 100 120 140
n
0
−5
0
5
10
15
20
25
30
35
Amplitude
Fig. 16.5(c) 128-point DFT of the Signal xam
(n), 0 # n # 99

xamp5x.*xa;xam5fft(xamp,n1);
stem(n,xam);title(‘xamp(n)’);xlabel(‘n --.’);
ylabel(‘amplitude’);
%128 point DFT computation2solution for Section (c)
n50:99;figure;n250:n121;
f151/128;f255/128;fc550/128;
xc5cos(2*pi*fc*n);
xa5cos(2*pi*fc*n);
xamp5x.*xa;
for i51:100,
xamp1(i)5xamp(i);
end
xam5fft(xamp1,n1);
s t e m ( n 2 , x a m ) ; t i t l e ( ‘ x a m p ( n ) ’ ) ; x l a b e l ( ‘ n
--.’);ylabel(‘amplitude’);
(a)Modulated signal x(n), carrier signal xa
(n) and amplitude modulated signal
xam
(n) are shown in Fig. 16.5(a). Fig. 16.5 (b) shows the 128-point DFT of the
signal xam
(n) for 0 # n # 127 and Fig. 16.5 (c) shows the 128-point DFT of the
signal xam
(n), 0 # n # 99.
16.7 FAST FOURIER TRANSFORM
Algorithm
1. Get the signal x(n)of length N in matrix form
2. Get the N value
3. The transformed signal is denoted as
x k x n e k N
j
N
nk
n
N
( ) ( ) for= ≤ ≤ −
−
=
−
∑
2
0
1
0 1
p

% Program for computing discrete Fourier transform
clc;close all;clear all;
x5input(‘enter the sequence’);
n5input(‘enter the length of fft’);
X(k)5fft(x,n);
stem(y);ylabel(‘Imaginary axis --.’);
xlabel(‘Real axis --.’);
X(k)
As an example,
enter the sequence [0 1 2 3 4 5 6 7]
enter the length of fft 8
X(k)5
Columns 1 through 4
28.0000 24.000019.6569i 24.0000 14.0000i 24.0000
1 1.6569i
Columns 5 through 8
24.0000 24.0000 21.6569i 24.0000 24.0000i 24.0000
29.6569i

MATLAB Programs 829
The eight-point decimation-in-time fast Fourier transform of the sequence x(n)is
computed using MATLAB program and the resultant output is plotted in Fig. 16.6.
0−5
−10
−8
−6
−4
−2
0
2
4
6
8
10
5 10 15 20 25 30
Real axis
Imaginaryaxis
Fig. 16.6 Fast Fourier Transform
16.8 BUTTERWORTH ANALOG FILTERS
16.8.1 Low-pass Filter
Algorithm
1. Get the passband and stopband ripples
2. Get the passband and stopband edge frequencies
3. Get the sampling frequency
4. Calculate the order of the filter using Eq. 8.46
5. Find the filter coefficients
6. Draw the magnitude and phase responses.
% Program for the design of Butterworth analog low pass filter
clc;
close all;clear all;
format long
rp5input(‘enter the passband ripple’);
rs5input(‘enter the stopband ripple’);
wp5input(‘enter the passband freq’);
ws5input(‘enter the stopband freq’);
fs5input(‘enter the sampling freq’);

w152*wp/fs;w252*ws/fs;
[n,wn]5buttord(w1,w2,rp,rs,’s’);
[z,p,k]5butter(n,wn);
[b,a]5zp2tf(z,p,k);
[b,a]5butter(n,wn,’s’);
w50:.01:pi;
[h,om]5freqs(b,a,w);
m520*log10(abs(h));
an5angle(h);
subplot(2,1,1);plot(om/pi,m);
ylabel(‘Gain in dB --.’);xlabel(‘(a) Normalised
frequency --.’);
subplot(2,1,2);plot(om/pi,an);
xlabel(‘(b) Normalised frequency --.’);
ylabel(‘Phase in radians --.’);
As an example,
enter the passband ripple 0.15
enter the stopband ripple 60
enter the passband freq 1500
enter the stopband freq 3000
The amplitude and phase responses of the Butterworth low-pass analog filter are
shown in Fig. 16.7.
Fig. 16.7 Butterworth Low-pass Analog Filter
(a) Amplitude Response and (b) Phase Response
0.1
0.1
−250
−200
−150
−4
−2
2
4
0
−100
−50
50
GainindB
0
0.2
0.2
0.3
0.3
0.4
0.4
(a)
(b)
Normalised frequency
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Phaseinradians
0.1
0.1
−250
−200
−150
−4
−2
2
4
0
−100
−50
50
GainindB
0
0.2
0.2
0.3
0.3
0.4
0.4
(a)
(b)
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Phaseinradians

MATLAB Programs 831
16.8.2 High-pass Filter
Algorithm
% Program for the design of Butterworth analog high—pass filter
clc;
format long
[n,wn]5buttord(w1,w2,rp,rs,’s’);
[b,a]5butter(n,wn,’high’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
enter the sampling freq 8000
The amplitude and phase responses of Butterworth high-pass analog filter are
shown in Fig. 16.8.

16.8.3 Bandpass Filter
Algorithm
% Program for the design of Butterworth analog Bandpass filter
clc;
format long
rp5input(‘enter the passband ripple...’);
rs5input(‘enter the stopband ripple...’);
wp5input(‘enter the passband freq...’);
ws5input(‘enter the stopband freq...’);
fs5input(‘enter the sampling freq...’);
Fig. 16.8 Butterworth High-pass Analog Filter (a) Amplitude Response and
(b) Phase Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−400
−4
4
−2
2
0
−300
−200
−100
0
100
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−400
−4
4
−2
2
0
−300
−200
−100
0
100
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

MATLAB Programs 833
[n]5buttord(w1,w2,rp,rs);
wn5[w1 w2];
[b,a]5butter(n,wn,’bandpass’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
enter the passband ripple... 0.36
enter the stopband ripple... 36
enter the passband freq... 1500
enter the stopband freq... 2000
enter the sampling freq... 6000
The amplitude and phase responses of Butterworth bandpass analog filter are
shown in Fig. 16.9.
Fig. 16.9 Butterworth Bandpass Analog Filter (a) Amplitude Response and
(b) Phase Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−1000
−4
4
−2
2
0
−800
−600
−200
−400
0
200
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

16.8.4 Bandstop Filter
Algorithm
% Program for the design of Butterworth analog Bandstop filter
clc;
format long
[n]5buttord(w1,w2,rp,rs,’s’);
wn5[w1 w2];
[b,a]5butter(n,wn,’stop’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Butterworth bandstop analog filter are
shown in Fig. 16.10.

MATLAB Programs 835
16.9 CHEBYSHEV TYPE-1 ANALOG FILTERS
Algorithm
% Program for the design of Chebyshev Type-1 low-pass filter
clc;
format long
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−150
−50
−100
−200
0
50
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Fig. 16.10 Butterworth Bandstop Analog Filter (a) Amplitude Response and
(b) Phase Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−150
−50
−100
−200
0
50
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

[n,wn]5cheb1ord(w1,w2,rp,rs,’s’);
[b,a]5cheby1(n,rp,wn,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 1 low-pass analog filter
are shown in Fig. 16.11.
GainindBPhaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−80
−40
−20
−60
−100
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Fig. 16.11 Chebyshev Type-I Low-pass Analog Filter (a) Amplitude Response
and (b) Phase Response

MATLAB Programs 837
Algorithm
%Program for the design of Chebyshev Type-1 high-pass filter
clc;
format long
[b,a]5cheby1(n,rp,wn,’high’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
Fig. 16.12 Chebyshev Type - 1 High-pass Analog Filter (a) Amplitude Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−100
−50
−150
−200
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−100
−50
−150
−200
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 1 high-pass analog filter
Algorithm
% Program for the design of Chebyshev Type-1 Bandpass filter
clc;
format long
[n]5cheb1ord(w1,w2,rp,rs,’s’);
wn5[w1 w2];
[b,a]5cheby1(n,rp,wn,’bandpass’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,

MATLAB Programs 839
The amplitude and phase responses of Chebyshev type - 1 bandpass analog filter
Algorithm
2. Get the passband and stopband edge frequency
% Program for the design of Chebyshev Type-1 Bandstop filter
clc;
format long
Fig. 16.13 Chebyshev Type-1 Bandpass Analog Filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−3
3
−2
−1
1
2
0
−200
−100
−300
−400
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−3
3
−2
−1
1
2
0
−200
−100
−300
−400
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

wn5[w1 w2];
[b,a]5cheby1(n,rp,wn,’stop’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 1 bandstop analog filter
Fig. 16.14 Chebyshev Type - 1 Bandstop Analog Filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−150
−50
−100
−200
−250
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−150
−50
−100
−200
−250
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

MATLAB Programs 841
16.10 CHEBYSHEV TYPE-2 ANALOG FILTERS
Algorithm
% Program for the design of Chebyshev Type-2 low pass analog filter
clc;
format long
[b,a]5cheby2(n,rs,wn,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 2 low-pass analog filter

Fig. 16.15 Chebyshev Type - 2 Low-pass Analog Filter (a) Amplitude Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−60
−20
−40
−80
−100
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Algorithm
% Program for the design of Chebyshev Type-2 High pass analog filter
clc;
format long
[b,a]5cheby2(n,rs,wn,’high’,’s’);
w50:.01:pi;

MATLAB Programs 843
m520*log10(abs(h));
an5angle(h);
ylabel(‘Gain in dB --.’);xlabel(‘(a) Normalised frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 2 high-pass analog filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−60
−20
−40
−80
0
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Fig. 16.16 Chebyshev Type - 2 High-pass Analog Filter
Algorithm

% Program for the design of Chebyshev Type-2 Bandpass analog filter
clc;
format long
wn5[w1 w2];
[b,a]5cheby2(n,rs,wn,’bandpass’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 2 bandpass analog filter
Fig. 16.17 (Contd.)
GainindB
aseinradians
(a)
0.1
4
−2
2
0
−80
−60
−20
0
−40
−100
20
0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.9 10

MATLAB Programs 845
Algorithm
% Program for the design of Chebyshev Type-2 Bandstop analog filter
clc;
format long
wn5[w1 w2];
[b,a]5cheby2(n,rs,wn,’stop’,’s’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
Fig. 16.17 Chebyshev Type - 2 Bandstop Analog Filter (a) Amplitude Response
Gainind
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−80
−60
−40
−100
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

As an example,
The amplitude and phase responses of Chebyshev type - 2 bandstop analog filter
Fig. 16.18 Chebyshev Type - 2 Bandstop Analog Filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−60
−40
0
20
−20
−80
40
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
16.11 BUTTERWORTH DIGITAL IIR FILTERS
Algorithm

MATLAB Programs 847
% Program for the design of Butterworth low pass digital filter
clc;
format long
[n,wn]5buttord(w1,w2,rp,rs);
[b,a]5butter(n,wn);
w50:.01:pi;
[h,om]5freqz(b,a,w);
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Butterworth low-pass digital filter are
GainindB
Phaseinradians
(a)
0.1
0.1
−4
4
−2
2
0
−300
−200
0
−100
−400
100
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Fig. 16.19 (Contd.)

Algorithm
% Program for the design of Butterworth highpass digital filter
clc;
format long
[n,wn]5buttord(w1,w2,rp,rs);
[b,a]5butter(n,wn,’high’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
Fig. 16.19 Butterworth Low-pass Digital Filter (a) Amplitude Response and
(b) Phase Response
Gain
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−300
−200
−400
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

MATLAB Programs 849
The amplitude and phase responses of Butterworth high-pass digital filter are
16.11.3 Band-pass Filter
Algorithm
% Program for the design of Butterworth Bandpass digital filter
clc;
format long
Fig. 16.20 Butterworth High-pass Digital Filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−250
−200
−150
0
−100
−50
−300
50
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
−250
−200
−150
0
−100
−50
−300
50
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

wn5[w1 w2];
[b,a]5butter(n,wn,’bandpass’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Butterworth band-pass digital filter are
Fig. 16.21 Butterworth Bandstop Digital Filter (a) Amplitude Response and
(b) Phase Response
GainindB
(a)
(b)
Phaseinradians
0.1
0.1
0
− 200
− 100
− 400
− 500
− 600
− 700
4
2
0
− 2
− 4
− 300
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

MATLAB Programs 851
Algorithm
% Program for the design of Butterworth Band stop digital filter
clc;
format long
wn5[w1 w2];
[b,a]5butter(n,wn,’stop’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of the Butterworth bandstop digital filter are

Fig. 16.22 Butterworth Bandstop Digital Filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
100
0
−200
−100
−400
−300
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
100
0
−200
−100
−400
−300
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
16.12 CHEBYSHEV TYPE-1 DIGITAL FILTERS
Algorithm
% Program for the design of Chebyshev Type-1 lowpass digital filter
clc;
format long

MATLAB Programs 853
[n,wn]5cheb1ord(w1,w2,rp,rs);
[b,a]5cheby1(n,rp,wn);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 1 low-pass digital filter
Fig. 16.23 Chebyshev Type - 1 Low-pass Digital Filter (a) Amplitude Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
0
−200
−100
−500
−400
−300
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

Algorithm
% Program for the design of Chebyshev Type-1 highpass digital
filter
clc;
format long
[b,a]5cheby1(n,rp,wn,’high’);
w50:.01/pi:pi;
m520*log10(abs(h));
an5angle(h);
ylabel(‘Gain in dB --.’);xlabel(‘(a) Normalised frequency
--.’);
As an example,
The amplitude and phase responses of Chebyshev type - 1 high-pass digital filter

MATLAB Programs 855
Algorithm
% Program for the design of Chebyshev Type-1 Bandpass digital filter
clc;
format long
Fig. 16.24 Chebyshev Type - 1 High-pass Digital Filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
0
−250
−200
−150
−100
−50
−350
−300
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
0
−250
−200
−150
−100
−50
−350
−300
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

[n]5cheb1ord(w1,w2,rp,rs);
wn5[w1 w2];
[b,a]5cheby1(n,rp,wn,’bandpass’);
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 1 bandpass digital filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
4
−2
2
0
0
−300
−200
−100
−500
−400
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Fig. 16.25 Chebyshev Type - 1 Bandpass Digital Filter (a) Amplitude
Response and (b) Phase Response

MATLAB Programs 857
Algorithm
% Program for the design of Chebyshev Type-1 Bandstop digital filter
clc;
format long
wn5[w1 w2];
[b,a]5cheby1(n,rp,wn,’stop’);
w50:.1/pi:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 1 bandstop digital filter

Fig. 16.26 Chebyshev Type - 1 Bandstop Digital Filter
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−3
−2
−1
1
2
3
4
0
0
− 200
− 150
− 100
−50
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−3
−2
−1
1
2
3
4
0
0
− 200
− 150
− 100
−50
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
16.13 CHEBYSHEV TYPE-2 DIGITAL FILTERS
Algorithm
% Program for the design of Chebyshev Type-2 lowpass digital filter
clc;
format long
[b,a]5cheby2(n,rs,wn);

MATLAB Programs 859
w50:.01:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 2 low-pass digital filter
Fig. 16.27 Chebyshev Type - 2 Low-pass Digital Filter (a) Amplitude Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
−2
2
4
0
−100
−80
−60
−40
−20
0
20
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
−2
2
4
0
−100
−80
−60
−40
−20
0
20
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Algorithm

% Program for the design of Chebyshev Type-2 high pass digital filter
clc;
format long
[b,a]5cheby2(n,rs,wn,’high’);
w50:.01/pi:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 2 high-pass digital filter
GainindB
aseinradians
(a)
0.1
−2
2
4
0
−120
−100
−80
−60
−40
−20
0
0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.9 10
Fig. 16.28 (Contd.)

MATLAB Programs 861
Algorithm
2. Get the passband and stopband edge frequency
% Program for the design of Chebyshev Type-2 Bandpass digital filter
clc;
format long
wn5[w1 w2];
[b,a]5cheby2(n,rs,wn,’bandpass’);
w50:.01/pi:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
Gainin
Phaseinradians
(a)
(b)
0.1
0.1
−4
−2
2
4
0
−120
−100
−80
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Fig. 16.28 Chebyshev Type - 2 High-pass Digital Filter (a) Amplitude Response

As an example,
The amplitude and phase responses of Chebyshev type - 2 bandpass digital filter
Fig. 16.29 Chebyshev Type - 2 Bandpass Digital Filter (a) Amplitude Response
GainindB
Phaseinradians
(a)
(b)
0.1
0.1
−4
−2
4
0
2
−400
−300
−200
−100
0
100
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0
Algorithm

MATLAB Programs 863
% Program for the design of Chebyshev Type-2 Bandstop digital filter
clc;
format long
wn5[w1 w2];
[b,a]5cheby2(n,rs,wn,’stop’);
w50:.1/pi:pi;
m520*log10(abs(h));
an5angle(h);
frequency --.’);
As an example,
The amplitude and phase responses of Chebyshev type - 2 bandstop digital filter
GainindB
Phaseinradians
(a)
0.1
-4
-2
-1
-3
3
0
1
2
−80
−60
−40
−20
0
20
0.2 0.3 0.4
0.5 0.6 0.7 0.8 0.9 10
Fig. 16.30 (Contd.)

16.14
FIR FILTER DESIGN USING WINDOW
TECHNIQUES
In the design of FIR filters using any window technique, the order can be calculated
using the formula given by
N
p s
f f Fs p s
=
− −
−
20 13
14 6
log( )
. ( ) /
d d
where dp is the passband ripple, ds is the stopband ripple, fp
is the passband frequency,
fs
is the stopband frequency and Fs
is the sampling frequency.
16.14.1 Rectangular Window
Algorithm
4. Calculate the order of the filter
5. Find the window coefficients using Eq. 7.37
% Program for the design of FIR Low pass, High pass, Band pass
and Bandstop filters using rectangular window
fp5input(‘enter the passband freq’);
fs5input(‘enter the stopband freq’);
f5input(‘enter the sampling freq’);
wp52*fp/f;ws52*fs/f;
num5220*log10(sqrt(rp*rs))213;
Fig. 16.30 Chebyshev Type - 2 Bandstop Digital Filter (a) Amplitude Response
Gaini
Phaseinradians
(a)
(b)
0.1
0.1
-4
-2
-1
-3
3
0
1
2
−80
−60
−40
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1
1
0
0

MATLAB Programs 865
dem514.6*(fs2fp)/f;
n5ceil(num/dem);
n15n11;
if (rem(n,2)˜50)
n15n;
n5n21;
end
y5boxcar(n1);
% low-pass filter
b5fir1(n,wp,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
subplot(2,2,1);plot(o/pi,m);ylabel(‘Gain in dB --.’);
xlabel(‘(a) Normalised frequency --.’);
% high-pass filter
b5fir1(n,wp,’high’,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band pass filter
wn5[wp ws];
b5fir1(n,wn,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
subplot(2,2,3);plot(o/pi,m);ylabel(‘Gain in dB --’);
xlabel(‘(c) Normalised frequency --’);
% band stop filter
b5fir1(n,wn,’stop’,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
xlabel(‘(d) Normalised frequency --’);
As an example,
enter the stopband ripple 0.04
The gain responses of low-pass, high-pass, bandpass and bandstop filters using
rectangular window are shown in Fig. 16.31.

16.14.2 Bartlett Window
Algorithm
and Bandstop filters using Bartlett window
Fig. 16.31 Filters Using Rectangular Window (a) Low-pass (b) High-pass
(c) Bandpass and (d) Bandstop
0.2
0.2
0.2
0.2
−80
−80 −20
−80
−60
−60 −15
−60
−40
−40 −10
−40
−20
−20 −5
−20
0
0 0
0
20
20 5
20
0.4
0.4
0.4
0.4
(a)
(c)
(b)
(d)
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
0
0
0
0
GainindBGainindB
GainindBGainindB

MATLAB Programs 867
dem514.6*(fs2fp)/f;
n5ceil(num/dem);
n15n11;
if (rem(n,2)˜50)
n15n;
n5n21;
end
y5bartlett(n1);
% low-pass filter
b5fir1(n,wp,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% high-pass filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band pass filter
wn5[wp ws];
b5fir1(n,wn,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
xlabel(‘(c) Normalised frequency --.’);
% band stop filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
xlabel(‘(d) Normalised frequency --.’);
As an example,
Bartlett window are shown in Fig. 16.32.

16.14.3 Blackman window
Algorithm
and Band stop digital filters using Blackman window
Fig. 16.32 Filters using Bartlett Window (a) Low-pass (b) High-pass
− 35
−40 −8
− 30
−30
−6
− 30
−25
−25
−20
−20
−4
− 20
−15
− 15
−10
−10
−2
−0
− 10
−5
−5
0
20
5
0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
(a)
(c)
(b)
(d)
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
0
0
0
0
GainindB
GainindBGainindB
GainindB

MATLAB Programs 869
dem514.6*(fs2fp)/f;
n5ceil(num/dem);
n15n11;
if (rem(n,2)˜50)
n15n;
n5n21;
end
y5blackman(n1);
% low-pass filter
b5fir1(n,wp,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% high-pass filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band pass filter
wn5[wp ws];
b5fir1(n,wn,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band stop filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
subplot(2,2,4);plot(o/pi,m);;ylabel(‘Gain in dB --.’);
As an example,
Blackman window are shown in Fig. 16.33.

16.14.4 Chebyshev Window
Algorithm
% Program for the design of FIR Lowpass, High pass, Band pass
and Bandstop filters using Chebyshev window
Fig. 16.33 Filters using Blackman Window (a) Low-pass (b) High-pass
−120
−120 −8
−150
−100
−100
−100 −6
−80
−60
− 80
−4
−40
−20
− 60
− 40
− 20
−2
0
−50
0
0
20
20
50
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
(a)
(c)
(b)
(d)
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
0
0
0
0
GainindB
GainindB
GainindB
GainindB

MATLAB Programs 871
r5input(‘enter the ripple value(in dBs)’);
dem514.6*(fs2fp)/f;
n5ceil(num/dem);
if(rem(n,2)˜50)
n5n11;
end
y5chebwin(n,r);
% low-pass filter
b5fir1(n-1,wp,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% high-pass filter
b5fir1(n21,wp,’high’,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band-pass filter
wn5[wp ws];
b5fir1(n21,wn,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band-stop filter
b5fir1(n21,wn,’stop’,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
As an example,
enter the ripple value(in dBs)40
Chebyshev window are shown in Fig. 16.34.

16.14.5 Hamming Window
Algorithm
and Bandstop filters using Hamming window
Fig. 16.34 Filters using Chebyshev Window (a) Low-pass (b) High-pass
−100
−120 −12
−100
−40
−20
−60
−80− 80
−100 −10
−60
− 40
− 80
−8
− 20
− 60
− 40
− 20
−6
−4
−2
0
0
20
0
20
20
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
(a)
(c)
(b)
(d)
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
0
0
0
0
GainindB
GainindBGainindB
GainindB

MATLAB Programs 873
dem514.6*(fs2fp)/f;
n5ceil(num/dem);
n15n11;
if (rem(n,2)˜50)
n15n;
n5n21;
end
y5hamming(n1);
% low-pass filter
b5fir1(n,wp,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% high-pass filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band pass filter
wn5[wp ws];
b5fir1(n,wn,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band stop filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
As an example,
Hamming window are shown in Fig. 16.35.

16.14.6 Hanning Window
Algorithm
and Band stop filters using Hanning window
Fig. 16.35 Filters using Hamming Window (a) Low-pass (b) High-pass
−120
−120 −15
−100
−40
−20
−60
−80−100
−100
−10
−5
−80
− 60
− 80
− 40
− 20
− 60
− 40
− 20
0
0
20
0
20
20
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
(a)
(c)
(b)
(d)
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
0
0
0
0
GainindB
GainindBGainindB
GainindB

MATLAB Programs 875
dem514.6*(fs2fp)/f;
n5ceil(num/dem);
n15n11;
if (rem(n,2)˜50)
n15n;
n5n21;
end
y5hamming(n1);
% low-pass filter
b5fir1(n,wp,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% high-pass filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band pass filter
wn5[wp ws];
b5fir1(n,wn,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
% band stop filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
As an example,
Hanning window are shown in Fig. 16.36.

16.14.7 Kaiser Window
Algorithm
5. Find the window coefficients using Eqs 7.46 and 7.47
and Bandstop filters using Kaiser window
beta5input(‘enter the beta value’);
Fig. 16.36 Filters using Hanning Window (a) Low-pass (b) High-pass
− 120
−120 −12
−100
−40
−20
−60
−80− 100
−100 −10
−8
−6
−4
−2
0
−80
−60
−80
−40
−20
−60
−40
−20
0
20
0
0 2
20
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
(a)
(c)
(b)
(d)
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
0
0
0
0
GainindB
GainindBGainindB
GainindB

MATLAB Programs 877
dem514.6*(fs2fp)/f;
n5ceil(num/dem);
n15n11;
if (rem(n,2)˜50)
n15n;
n5n21;
end
y5kaiser(n1,beta);
% low-pass filter
b5fir1(n,wp,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
xlabel(‘(a) Normalised frequency --’);
% high-pass filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
xlabel(‘(b) Normalised frequency --’);
% band pass filter
wn5[wp ws];
b5fir1(n,wn,y);
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
xlabel(‘(c) Normalised frequency --’);
% band stop filter
[h,o]5freqz(b,1,256);
m520*log10(abs(h));
xlabel(‘(d) Normalised frequency --’);
As an example,
enter the beta value 5.8
Kaiser window are shown in Fig. 16.37.

16.15 UPSAMPLING A SINUSOIDAL SIGNAL
% Program for upsampling a sinusoidal signal by factor L
N5input(‘Input length of the sinusoidal sequence5’);
L5input(‘Up Samping factor5’);
fi5input(‘Input signal frequency5’);
% Generate the sinusoidal sequence for the specified length N
n50:N21;
x5sin(2*pi*fi*n);
% Generate the upsampled signal
y5zeros (1,L*length(x));
y([1:L:length(y)])5x;
%Plot the input sequence
subplot (2,1,1);
stem (n,x);
title(‘Input Sequence’);
xlabel(‘Time n’);
ylabel(‘Amplitude’);
Fig. 16.37 Filters using Kaiser Window (a) Low-pass (b) High-pass
−120
−120 −15
−40
−20
−60
−80
−100
−100
−10
−5
0
−80
− 60
− 80
− 40
− 20
− 60
− 40
− 20
0
20
0
0 5
20
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
(a)
(c)
(b)
(d)
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
1
1
1
1
0
0
0
0
GainindB
GainindB
GainindB
GainindB

MATLAB Programs 879
%Plot the output sequence
subplot (2,1,2);
stem (n,y(1:length(x)));
title(‘[output sequence,upsampling factor5‘,num2str(L)]);
16.16
UPSAMPLING AN EXPONENTIAL
SEQUENCE
% Program for upsampling an exponential sequence by a factor M
n5input(‘enter length of input sequence …’);
l5input(‘enter up sampling factor …’);
% Generate the exponential sequence
m50:n21;
a5input(‘enter the value of a …’);
x5a.^m;
% Generate the upsampled signal
y5zeros(1,l*length(x));
y([1:l:length(y)])5x;
figure(1)
stem(m,x);
xlabel({‘Time n’;’(a)’});
figure(2)
stem(m,y(1:length(x)));
xlabel({‘Time n’;’(b)’});
As an example,
enter length of input sentence … 25
enter upsampling factor … 3
enter the value of a … 0.95
The input and output sequences of upsampling an exponential sequence an
are shown
in Fig. 16.38.
Fig. 16.38 (Contd.)

16.17
DOWN SAMPLING A SINUSOIDAL
SEQUENCE
% Program for down sampling a sinusoidal sequence by a factor M
N5input(‘Input length of the sinusoidal signal5’);
M5input(‘Down samping factor5’);
fi5input(‘Input signal frequency5’);
%Generate the sinusoidal sequence
n50:N21;
m50:N*M21;
x5sin(2*pi*fi*m);
%Generate the down sampled signal
y5x([1:M:length(x)]);
subplot (2,1,1);
stem(n,x(1:N));
%Plot the down sampled signal sequence
subplot(2,1,2);
stem(n,y);
title([‘Outputsequencedownsamplingfactor’,num2str(M)]);
16.18
DOWN SAMPLING AN EXPONENTIAL
SEQUENCE
% Program for downsampling an exponential sequence by a factor M
N5input(‘enter the length of the output sequence …’);
M5input(‘enter the down sampling factor …’);
Fig. 16.38 (a) Input Exponential Sequence
(b) Output Sequence Upsampled by a Factor of 3

MATLAB Programs 881
% Generate the exponential sequence
n50:N21;
m50:N*M21;
a5input(‘enter the value of a …’);
x5a.^m;
% Generate the downsampled signal
y5x([1:M:length(x)]);
figure(1)
stem(n,x(1:N));
xlabel({‘Time n’;’(a)’});
figure(2)
stem(n,y);
xlabel({‘Time n’;’(b)’});
As an example,
enter the length of the output sentence … 25
enter the downsampling factor … 3
enter the value of a … 0.95
The input and output sequences of downsampling an exponential sequence an
are
Fig. 16.39 (a) Input Exponential Sequence
(b) Output Sequence Downsampled by a Factor of 3

16.19 DECIMATOR
% Program for downsampling the sum of two sinusoids using
MATLAB’s inbuilt decimation function by a factor M
N5input(‘Length of the input signal5’);
M5input(‘Down samping factor5’);
f15input(‘Frequency of first sinusoid5’);
f25input(‘Frequency of second sinusoid5’);
n50:N21;
% Generate the input sequence
x52*sin(2*pi*f1*n)13*sin(2*pi*f2*n);
%Generate the decimated signal
% FIR low pass decimation is used
y5decimate(x,M,‘fir’);
subplot (2,1,1);
stem (n,x(1:N));
%Plot the output sequence
subplot (2,1,2);
m50:N/M21;
stem (m,y(1:N/M));
title([‘Outputsequencedownsamplingfactor’,num2str(M)]);
16.20 DECIMATOR AND INTERPOLATOR
% Program for downsampling and upsampling the sum of two
sinusoids using MATLAB’s inbuilt decimation and interpolation
function by a factor of 20.
%Generate the input sequence for Fs
5200Hz, f1
550Hz and
f2
5100 Hz
t50:1/200:10;
y53.*cos(2*pi*50.*t/200)11.*cos(2*pi*100.*t/200);
figure(1)
stem(y);
xlabel({‘Times in Seconds’;’(a)});

MATLAB Programs 883
%Generate the decimated and interpolated signals
figure(2)
stem(decimate(y,20));
xlabel({‘Times in Seconds’;’(b)});
figure(3)
stem(interp(decimate(y,20),2));
xlabel({‘Times in Seconds’;’(c)});
Fig. 16.40 (a) Input Sequence, (b) Decimated Sequence and
(c) Interpolated sequence
5
0
AmplitudeAmplitude
0 500 1000
Time in Seconds
(a)
1500 2000 2500
−5
5
0
0 20 40
Time in Seconds
(b)
60 80 100 120
−5
Amplitude
5
0
0 50 100 150 200 250
Time in Seconds
(c)
−5
16.21
ESTIMATION OF POWER SPECTRAL
DENSITY (PSD)
% Program for estimating PSD of two sinusoids plus noise
% Algorithm;
% 1:Get the frequencies of the two sinusoidal waves
% 2:Get the sampling frequency
% 3:Get the length of the sequence to be considered

% 4:Get the two FFT lengths for comparing the corre-
sponding power spectral densities
clc; close all; clear all;
f15input(‘Enter the frequency of first sinusoid’);
f25input(‘Enter the frequency of second sinusoid’);
fs5input(‘Enter the sampling frequency’);
N5input(“Enter the length of the input sequence’);
N15input(“Enter the input FFT length 1’);
%Generation of input sequence
t50:1/fs:1;
x52*sin(2*pi*f1*1)13*sin(2*pi*f2*t)2randn(size(t));
%Generation of psd for two different FFT lengths
Pxx15abs(fft(x,N1)).^2/(N11);
Pxx25abs(fft(x,N2)).^2/(N11);
%Plot the psd;
subplot(2,1,1);
plot ((0:(N121))/N1*fs,10*log10(Pxx1));
xlabel(‘Frequency in Hz’);
ylabel(‘Power spectrum in dB’);
title(‘[PSD with FFT length,num2str(N1)]’);
subplot (2,1,2);
plot ((0:(N221))/N2*fs,10*log10(Pxx2));
16.22 PSD ESTIMATOR
% Program for estimating PSD of a two sinusoids plus noise using
%(i)non-overlapping sections
%(ii)overlapping sections and averaging the periodograms
N5input(“Enter the length of the input sequence’);
%Generation of input sequence
t50:1/fs:1;
x52*sin(2*pi*f1*1)13*sin(2*pi*f2*t)2randn(size(t));

MATLAB Programs 885
%Generation of psd for two different FFT lengths
Pxx15(abs(fft(x(1:256))).^21abs(fft(x(257:512))).^21
abs(fft(x(513:768))).^2/(256*3); %using nonoverlapping
sections
Pxx25(abs(fft(x(1:256))).^21abs(fft(x(129:384))).^21ab
s(fft(x(257:512))).^21abs(fft(x(385:640))).^21abs(fft(
x(513:768))).^21abs(fft(x(641:896))).^2/(256*6); %using
overlapping sections
% Plot the psd;
subplot (2,1,1);
plot ((0:255)/256*fs,10*log10(Pxx1));
subplot (2,1,2);
plot ((0:255)/256*fs,10*log10(Pxx2));
16.23 PERIODOGRAM ESTIMATION
% Periodogram estimate cold be done by applying a non-
rectangular data windows to the sections prior to com-
puting the periodogram
% This program estimates PSD for the input signal of two
sinusoids plus noise using Hanning window
t50:1/fs:1;
w5hanning(256);
x52*sin(2*pi*f1*t)13*sin(2*pi*f2*t)2randn(size(t));
Pxx5(abs(fft(w.*x(1:256))).^21abs(fft(w.*x(129:384))).^
21abs(fft(w.*x(257:512))).^21abs(fft(w.*x(385:640))).^2
1abs(fft(w.*x(513:768))).^21abs(fft(w.*x(641:896))).^2/
(norm(w)^2*6);
Plot((0:255)/256*fs,10*log10(Pxx));
16.24 WELCH PSD ESTIMATOR
% Program for estimating the PSD of sum of two sinusoids plus
noise using Welch method
n50.01:0.001:.1;
x5sin(.25*pi*n)13*sin(.45*pi*n)1rand(size(n));
pwelch(x)

xlabel(‘Normalised Frequency (x pi rad/sample)’);
ylabel(‘Power Spectrum Density (dB/rad/sample)’)
16.25 WELCH PSD ESTIMATOR USING WINDOWS
% Program for estimating the PSD of sum of two sinusoids using
Welch method with an overlap of 50 percent and with Hanning,
Hamming, Bartlett, Blackman and rectangular windows.
fs51000;
t50:1/fs:3;
x5sin(2*pi*200*t)1sin(2*pi*400*t);
figure(1)
subplot(211)
pwelch(x,[],[],[],fs);
title(‘Overlay plot of 50 Welch estimates’)
subplot(212)
pwelch(x,hanning(512),0,512,fs)
title(‘N5512 Overlap550% Hanning’)
figure(2)
subplot(211)
subplot(212)
pwelch(x,hamming(512),0,512,fs)
title(‘N5512 Overlap550% Hamming’)
figure(3)
subplot(211)
subplot(212)
pwelch(x,bartlett(512),0,512,fs)
title(‘N5512 Overlap550% Bartlett’)
figure(4)
subplot(211)
Fig. 16.41 Welch PSD Estimate
5
0
−5
−10
−15
−20
0 0.1 0.2 0.3 0.4 0.5
Welch PSD Estimate
PowerSpectrumDensity(dB/rad/sample)
Normalised Frequency (x pi rad/sample)
0.6 0.7 0.8 0.9 1

MATLAB Programs 887
Fig. 16.42 (b) Welch Estimate with N5512, 50% Overlap Hamming
0
−20
−40
−60
0 50 100 150 200
Frequency (Hz)
Overlay plot of 50 Welch estimates
PowerSpectralDensity(dB/Hz)
250 300 350 400 450 500
−80
0
−20
−40
−60
0 50 100 150 200
Frequency (Hz)
N=512 Overlap = 50% Hamming
250 300 350 400 450 500
−80
0
−20
−40
−60
0 50 100 150 200
Frequency (Hz)
N=512 Overlap = 50% Hanning
250 300 350 400 450 500
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−80
0
−50
−100
−150
Fig. 16.42 (a) Welch Estimate with N5512, 50% Overlap Hanning
subplot(212)
pwelch(x,blackman(512),0,512,fs)
title(‘N5512 Overlap550% Blackman’)
figure(5)
subplot(211)
subplot(212)
pwelch(x,boxcar(512),0,512,fs)
title(‘N5512 Overlap550% Rectangular’)

Fig. 16.42 (d) Welch Estimate with N5512, 50% Overlap Blackman
0
−20
−40
−60
0 50 100 150 200
Frequency (Hz)
Frequency (Hz)
250 300 350 400 450 500
−80
0
−50
−100
−150
0 50 100 150 200
N=512 Overlap = 50% Blackman
250 300 350 400 450 500
−200
0
−20
−40
−60
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−80
0
−20
−40
−60
0 50 100 150 200
Frequency (Hz)
N=512 Overlap = 50% Bartlett
250 300 350 400 450 500
−80
Fig. 16.42 (c) Welch Estimate with N5512, 50% Overlap Bartlett

MATLAB Programs 889
Fig. 16.42 (e) Welch Estimate with N5512, 50% Overlap Rectangular
0
−20
−40
−60
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−80
0
−10
−20
−30
−40
−50
−60
0 50 100 150 200
Frequency (Hz)
N = 512 Overlap = 50% Rectangular
250 300 350 400 450 500
16.26 WELCH PSD ESTIMATOR USING WINDOWS
% Program for estimating the PSD of sum of two sinusoids plus
noise using Welch method with an overlap of 50 percent and with
Hanning, Hamming, Bartlett, Blackman and rectangular windows
fs51000;
t50:1/fs:3;
x52*sin(2*pi*200*t)15*sin(2*pi*400*t);
y5x1randn(size(t));
figure(1)
subplot(211);
pwelch(y,[],[],[],fs);
title(‘Overlay plot of 50 Welch estimates’);
subplot(212);
pwelch(y,hanning(512),0,512,fs);
title(‘N5512 Overlap550% Hanning’);
figure(2)
subplot(211);
subplot(212);
pwelch(y,hamming(512),0,512,fs);
title(‘N5512 Overlap550% Hamming’);

10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
Fig. 16.43 (a) Welch Estimate with N5512, 50% Overlap Hanning
figure(3)
subplot(211);
subplot(212);
pwelch(y,bartlett(512),0,512,fs);
title(‘N5512 Overlap550% Bartlett’);
figure(4)
subplot(211);
subplot(212);
pwelch(y,blackman(512),0,512,fs);
title(‘N5512 Overlap550% Blackman’);
figure(5)
subplot(211);
subplot(212);
pwelch(y,boxcar(512),0,512,fs);
title(‘N5512 Overlap550% Rectangular’);

MATLAB Programs 891
Fig. 16.43 (b) Welch Estimate with N5512, 50% Overlap Hamming
10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
Fig. 16.43 (c) Welch Estimate with N5512, 50% Overlap Bartlett
10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
10
0
−10
−20
−30
0 50 100 150 200
N=512 Overlap = 50% Bartlett
Frequency (Hz)
250 300 350 400 450 500
−40

10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
Fig. 16.43 (e) Welch Estimate with N5512, 50% Overlap Rectangular
Fig. 16.43 (d) Welch Estimate with N5512, 50% Overlap Blackman
10
0
−10
−20
−30
0 50 100 150 200
Frequency (Hz)
250 300 350 400 450 500
−40
10
0
−10
−20
−30
0 50 100 150 200
N=512 Overlap = 50% Blackman
Frequency (Hz)
250 300 350 400 450 500
−40

MATLAB Programs 893
16.27 STATE-SPACE REPRESENTATION
% Program for computing the state-space matrices from the given
transfer function
function [A,B,C,D]5tf2ss(b,a);
a5input (‘enter the denominator polynomials5’);
b5input (‘enter the numerator polynomials5’);
p5length(a)21;q5length(b)21;N5max(p,q);
if(Np),a5[a,zeros(1,N2p)];end
if(Nq),b5[b,zeros(1,N2q)];end
A5[2a(2:N11);[eye(N21),zeros(N21,1)]];
B5[1;zeros(N21,1)];
C5b(2:N11)2b(1)*(2:N11);
D5b(1);
16.28 PARTIAL FRACTION DECOMPOSITION
% Program for partial fraction decomposition of a rational transfer
function
function[c,A,alpha]5tf2pf(b,a);
p5length(a)21;
q5length(b)21;
a5(1/a(1))*reshape(a,1,p11);
b5(1/a(1))*reshape(b,1,q11);
if(q5p),%case of nonempty c(z)
temp5toeplitz([a,zeros(1,q2p)]’,[a(1),zeros(1,q2p)]);
temp5[temp,[eye(p);zeros(q2p11,p)]);
temp5temp/b’;
c5temp(1:;q2p11);
d5temp(q2p12:q11)’;
else
c5[];
d5[b,zeros(1,p2q21)];
end
alpha5cplxpair (roots(a));’;
A5zeros(1,p);
for k51 :p
temp5prod(alpha(k)2alpha(find(1:p5k)));
if(temp550),error(‘repeated roots in TF2PF’);
else,A(k)5polyval(d,alpha(k))/temp;
end
end

16.29 INVERSE z-TRANSFORM
% Program for computing inverse z-transform of a rational transfer
function
function x5invz(b,a,N);
N5input (‘enter the number of points to be computed5’);
[c,A,alpha]5tf2pf(b,a);
x5zeros (1,N);
x(1:length(c))5c;
for k51:length(A),
x5x1A(k)*(alpha(k)).^(0:N21);
end
x5real(x);
16.30 GROUP DELAY
% Program for computing group delay of a rational transfer
function on a given frequency interval
function D5grpdly(b,a,K,theta);
K5input (‘enter the number of frequency response
points5’);
theta5input (‘enter the theta value5’);
a5reshape(a,1,length(a));
b5reshape(b,1,length(b));
if (length(a)551)%case of FIR
bd52j*(0:length(b)21).*b;
if(nargin553),
B5frqresp(b,1,K);
Bd5frqresp(bd,1,K);
else,
B5frqresp(b,1,K,theta);
Bd5frqresp(bd,1,K,theta);
end
D5(real(Bd).*imag(B)2real(B).*imag(Bd))./abs(B).^2;
else %case of IIR
if(nargin553),
D5grpdly (b,1,K)2grpdly(a,1,K);
else,
D5grpdly(b,1,K,theta)2grpdly(a,1,K,theta);
end
end

MATLAB Programs 895
16.31
IIR FILTER DESIGN-IMPULSE INVARIANT
METHOD
% Program for transforming an analog filter into a digital filter using
impulse invariant technique
function [bout,aout]5impinv(bin,ain,T);
bin5input(‘enter the numerator polynomials5’);
ain5input(‘enter the denominator polynomials5’);
T5input(‘enter the sampling interval5’);
if(length(bin)5length(ain)),
error(‘Anlog filter in IMPINV is not strictly proper’);
end
[r,p,k]5residue(bin,ain);
[bout,aout]5pf2tf([],T*r,exp(T*p));
16.32
IIR FILTER DESIGN-BILINEAR
TRANSFORMATION
% Program for transforming an analog filter into a digial filter using
bilinear transformation
function [b,a,vout,uout,Cout]5bilin(vin,uin,Cin,T);
pin5input(‘enter the poles5’);
zin5input(‘enter the zero5’);
T5input(‘enter the sampling interval5’);
Cin5input(‘enter the gain of the analog filter5’);
p5length(pin);
q5length(zin);
Cout5Cin*(0.5*T)^(p2q)*prod(120.5*T*zin)/
prod(120.5*T*pin);
zout5[(110.5*T*zin)./(120.5*T*pin),2ones(1,p2q)];
pout5(110.5*T*pin)./(120.5*T*pin);
a51;
b51;
for k51 :length(pout),a5conv(a,[1,2pout(k)]); end
for k51 :length(zout),b5conv(b,[1,2zout(k)]); end
a5real(a);
b5real(Cout*b);
Cout5real(Cout);
16.33
DIRECT REALISATION OF IIR DIGITAL
FILTERS
% Program for computing direct realisation values of IIR digital filter
function y5direct(typ,b,a,x);
x5input(‘enter the input sequence5’);

b5input(‘enter the numerator polynomials5’);
a5input(‘enter the denominator polynomials5’);
typ5input(‘type of realisation5’);
p5length(a)21;
q5length(b)21;
pq5max(p,q);
a5a(2:p11);u5zeros(1,pq);%u is the internal state;
if(typ551)
for i51:length(x),
unew5x(i)2sum(u(1:p).*a);
u5[unew,u];
y(i)5sum(u(1:q11).*b);
u5u(1:pq);
end
elseif(typ552)
for i51:length(x)
y(i)5b(1)*x(i)1u(1);
u5u[(2:pq),0];
u(1:q)5u(1:q)1b(2:q11)*x(i);
u(1:p)5u(1:p)2a*y(i);
end
end
16.34
PARALLEL REALISATION OF IIR DIGITAL
FILTERS
% Program for computing parallel realisation values of IIR digital
filter
function y5parallel(c,nsec,dsec,x);
a5input(‘enter the denominator polynomials5’);
c5input(‘enter the gain of the filter5’);
[n,m]5size(a);a5a(:,2:3);
u5zeros(n,2);
for i51:length(x),
y(i)5c*x(i);
for k51:n,
unew5x(i)2sum(u(k,:).*a(k,:));u(k,:)5[unew,u(k,1)];
y(i)5y(i)1sum(u(k,:).*b(k,:));
end
end

MATLAB Programs 897
16.35
CASCADE REALISATION OF IIR DIGITAL
FILTERS
% Program for computing cascade realisation values of digital IIR filter
function y5cascade(c,nsec,dsec,x);
a5input(‘enter the denomiator polynomials5’);
c5input(‘enter the gain of the filter5’);
[n,m]5size(b);
a5a(:,2:3);b5b(:,2,:3);
u5zeros(n,2);
for i51 :length(x),
for k51 :n,
unew5x(i)2sum(u(k,:).*a(k,:));
x(i)52unew1sum(u(k,:).*b(k,:))
u(k,:)5[unew,u(k,1)];
end
y(i)5c*x(i);
end
16.36
DECIMATION BY POLYPHASE
DECOMPOSITION
% Program for computing convolution and m-fold decimation by
polyphase decomposition
function y5ppdec(x,h,M);
h5input(‘enter the FIR filter coefficients5’);
M5input(‘enter the decimation factor5’);
1h5length(h); 1p5floor((1h21)/M)11;
p5reshape([reshape(h,1,1h),zeros(1,1p*M21h)],M,1p);
lx5length(x); ly5floor ((1x11h22)/M)11;
1u5foor((1x1M22)/M)11; %length of decimated sequences
u5[zeros(1,M21),reshape(x,1,1x),zeros(1,M*lu2lx2M11)];
y5zeros(1,1u11p21);
for m51:M,y5y1conv(u(m,: ),p(m,: )); end
y5y(1:1y);
16.37 MULTIBAND FIR FILTER DESIGN
% Program for the design of multiband FIR filters
function h5firdes(N,spec,win);
N5input(‘enter the length of the filter5’);

spec5input(‘enter the low,high cutoff frequencies and
gain5’);
win5input(‘enter the window length5’);
flag5rem(N,2);
[K,m]5size(spec);
n5(0:N)2N/2;
if (˜flag),n(N/211)51;
end,h5zeros(1,N11);
for k51:K
temp5(spec (k,3)/pi)*(sin(spec(k,2)*n)2sin(spec(k,1)
*n))./n;
if (˜flag);temp(N/211)5spec(k,3)*(spec(k,2)2
spec(k,1))/pi;
end
h5h1temp;
end
if (nargin553),
h5h.*reshape(win,1,N11);
end
16.38 ANALYSIS FILTER BANK
% Program for maximally decimated uniform DFT analysis filter
bank
function u5dftanal(x,g,M);
g5input(‘enter the filter coefficient5’);
M5input(‘enter the decimation factor5’);
1g5length(g); 1p5floor((1g21)/M)11;
p5reshape([reshape(g,1,1g),zeros(1,1p*M21g)],M,1p);
lx5length(x); lu5floor ((1x1M22)/M)11;
x5[zeros(1,M21),reshape(x,1,1x),zeros(1,M*lu2lx2M11)];
x5flipud(reshape(x,M,1u)); %the decimated sequences
u5[];
for m51:M,u5[u; cov(x(m,:),p(m,:))]; end
u5ifft(u);
16.39 SYNTHESIS FILTER BANK
% Program for maximally decimated uniform DFT synthesis filter
bank
function y5udftsynth(v,h,M);
1h5length(h); ‘1q5floor((1h21)/M)11;
q5flipud(reshape([reshape(h,1,1h),zeros(1,1q*M21h)],M,1q));

MATLAB Programs 899
v5fft(v);
y5[ ];
for m51:M,y5[conv(v(m,:),q(m,:));y]; end
y5y(:).’;
16.40 LEVINSON-DURBIN ALGORITHM
% Program for the solution of normal equations using Levinson-
Durbin algorithm
function [a,rho,s]5levdur(kappa);
% Input;
% kappa: covariance sequence values from 0 to p
% Output parameters:
% a: AR polynomial,with leading entry 1
% rho set of p reflection coefficients
% s: innovation variance
kappa5input(‘enter the covariance sequence5’);
p5length(kappa)21;
kappa5reshape(kappa,p11,1);
a51; s5kappa(1); rho5[];
for i51:p,
rhoi5(a*kappa(i11:21:2))/s; rho5[rho,rhoi];
s5s*(12rhoi^2);
a5[a,0]; a5a2rhoi*fliplr(a);
end
16.41 WIENER EQUATION’S SOLUTION
% Program
function b5wiener(kappax,kappayx);
kappax5input(‘enter the covariance sequence5’);
kappyx5input(‘enter the joint covariance sequence5’);
q5length(kappax)21;
kappax5reshape(kappax,q11,1);
kappayx5reshape(kappayx,q11,1);
b5(toeplitz(kappax)/(kappayx)’;
16.42 SHORT-TIME SPECTRAL ANALYSIS
% Program
function X5stsa(x,N,K,L,w,opt,M,theta0,dtheta);
x5input(‘enter the input signal5’); L5input(‘enter the
number consecutive DFTs to average5’);
N5input(‘enter the segment length5’); K5input(‘enter
the number of overlapping points5’); w5input(‘enter

the window coefficients5’); opt5input(‘opt5’);
M5input(‘enter the length of DFT5’);
theta05input(‘theta05’); dtheta5input(‘dtheta5’);
1x5length(x); nsec5ceil((1x2N)/(N2K)11;
x5[reshape(x,1,1x),zeros(1,N1(nsec21)*(N2K))2lx)];
nout5N; if (nargin 5),nout5M; else,opt5‘n’; end
X5zeros(nsec,nout);
for n51: nsec,
temp5w.*x((n21)*(N2K)11:(n21)*(N2K)1N);
if (opt(1) 55 ‘z’),temp5[temp,zeros(1,M2N)]; end
if (opt(1)55‘c’),temp5chirpf (temp,theta0,dtheta,M);
else,temp5fftshift(fft(temp)); end
X(n,: )5abs(temp).^2;
end
if(L1);
nsecL5floor(nsec/L);
for n51:nsecL,X(n,:)5mean (X((n21)*L11:n*L,:)); end
if (nsec55nsecL*L11),
X(nsecL11,:)5X(nsecL*L11,:); X5X(1:nsecL11),: );
elseif(nsec nsecL*L),
X(nsecL11,:)5mean(x(nsecL*L11:nsec,:));
X5X(1:nsecL11,:);
else,X5X(1:nsecL,:); end
end
LKh
16.43
CANCELLATION OF ECHO PRODUCED ON
THE TELEPHONE-BASE BAND CHANNEL
Echo
Canceler
Echo
path
Estimated
sequence
Desired
sequence
Base band
transmit
filter
Fig. 16.44 Baseband Channel Echo Canceler
% Simulation program for baseband echo cancellation
shown in Fig. 16.44 using LMS algorithm
format short
T5input(‘Enter the symbol interval’);
br5input(‘Enter the bit rate value’);
rf5input(‘Enter the roll off factor’);
n5[210 10];
y55000*rcosfir(rf,n,br,T); %Transmit filter pulse shape
is assumed as raised cosine

MATLAB Programs 901
ds5[5 2 5 2 5 2 5 2 5 5 5 5 2 2 2 5 5 5 5]; % data sequence
m5length(ds);
nl5length(y);
i51;
z5conv(ds(i),y);
while(i)
z15[z, zeros(1,1.75*br)];
z5conv(ds(i11),y);
z25[zeros(1,i*1.75*br),z];
z5z11z2;
i5i11;
end
%plot(z); %near end signal
h5randn(1,length(ds)); %echo path impulse response
rs15filter(h,1,z);
for i51; length(ds);
rs(i)5rs 1(i)/15;
end
for i51: round(x3/3),
rs(i)5randn(1); % rs2echo produced in the hybrid
end
fs5[5 5 2 2 2 2 2 5 2 2 2 5 5 5 2 5 2 5 2]; % Desired data signal
m5length(ds);
nl5length(y);
i51;
z5conv(fs(i),y);
while(i)
z15[z,zeros(1,1.75*br)];
z5conv(fs(i11),y);
z25[zeros(1,i*1.75*br),z];
z5z11z2;
i5i11;
end
fs15rs1fs; % echo added with desired signal
ar5xcorr(ds,ds);
crd5xcorr(rs,ds);
ll5length(ar); j51;
for i5round(11/2): 11,
ar1(j)5ar(i);
j5j11;
end
r5toeplitz(ar1);
l25length(crd); j51;
for i5round(l2/2):12,
crdl(j)5crd(i);
j5j11;
end
p5crd1’;

lam5max(eig(r)); la5min(eig(r)); l5lam/la;
w5inv(r)*p; % Initial value of filter coefficients
e5rs2filter(w,l,ds);
s51; mu51.5/lam;
ni51;
while (s 1 e210)
w15w22*mu*(e.*ds)’ ; % LMS algorithm adaptation
rs
y45filter(w1,1,ds); % Estimated echo signal using
LMS algorithm
e5y42rs; s50; e15xcorr(e);
for i51:length(e1),
s5s1e1(i);
end
s5s/length(e1);
if (y455rs)
break
end
ni5ni11;
w5w1;
end
figure(1); subplot(2,2,1); plot(z); title(‘near end
signal’);
subplot(2,2,2); plot(rs); title(‘echo produced in the
hybrid’);
subplot(2,2,3); plot(fs); title(‘desired signal’);
subplot(2,2,4); plot(fs1); title(‘echo added with
desired signal’);
figure(2); subplot(2,1,1); plot(y4); title(‘estimated
echo signal using LMS algorithm’);
subplot(2,1,2); plot(fs12y4); title(‘echo cancelled
signal’);
16.44
CANCELLATION OF ECHO PRODUCED ON
THE TELEPHONE—PASS BAND CHANNEL
Fig. 16.45 Pass Band Channel Echo Canceler
Echo
Canceler
Echo
path
Estimated
sequence
Desired
sequence
Pass band
transmit
filter
Passband
receive
filter

MATLAB Programs 903
% Simulation program for passband echo cancellation
shown in Fig. 16.45 using LMS algorithm
format long
fd58000; fs516000; fc58000;
f54000; t50:.01:1; %d5sin(2*pi*f*t/fd);
% Near end signal
ns5[5 2 5 2 5 5 2 2 2 5 5 2 2 2 2 2 2 5 5 2 5 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5];
% Near end input signal is digitally modulated and
plotted
y5dmod(ns,fc,fd,fs,‘psk’);
subplot(2,2,1); plot(y); title(‘input signal’);
xlabel(‘Time ———’); ylabel(‘Amplitude ———’);
% Echo is generated due to mismatch of hybrid impedances
h55*randn(1,length(ns));———
rsl5filter(h,1,y);
% for i51; length(ns);
% rsl(i)5rs6(i);
% end
for i51; length(ns);
rs(i)5rs1(i);
end
subplot(2,2,2); plot(rs); title(‘noise signal’);
% Far end signal
fs15[5 5 2 5 2 5 2 5 2 5 2 5 2 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 2 5];
% rs5sign(rs2);
% Far end signal is digitally modulated and plotted
z15dmod(fs1,fc,fd,fs,‘psk’);
for i51:length(ns),
z(i)5z1(i);
end
subplot(2,2,3); plot(z); title(‘far-end signal’);
% Echo and the far end modulated signal is added in the
hybrid
q15z11rs1;
for i51; length(ns);
q(i)5q1(i);;
end
subplot(2,2,4); plot(q); title(‘received signal’);
q25xcorr(q);
% Auto correlation is taken for the near end signal
ar5xcorr(ns);
% cross correction is taken for the near end and far end
signal
crd5xcorr(rs,ns);

l15length(ar); j51;
for i5round(ll/2): l1,
ar1(j)5ar(i)
j5j11;
end
% Toeplitz matrix is taken for the auto correlated
signal
r5toeplitz(ar1);
l25length(crd); j51;
for i5round(l2/2):l2,
crd1(j)5crd(i);
j5j11;
end
p5crd1’;
% Maximum and minimum eigen values are calculated from
the toeplitz matrix
lam5max(eig(r)); la5min(eig(r)); l5lam/la;
% initial filter taps are found using the below relation
w5inv(r)*p;
% The step size factor is calculated
m5length(ns)22.5;
a5(m2.95367)/.274274;
mu5a/lam;
% The initial error is calculated
s51;
e5rs2filter(w,1,ns); ni51; figure(2); subplot(2,2,1);
% Filter taps are iterated until the mean squared error
becomes E225
while (‘s25’ ! s0’)
w15w22*mu*(e.*ns)’;
if (ni5100)
break;
end
rs
y45filter(w1,1,ns)
e5y42rs; s50; el5e.*e;
for i51: length(e1),
s5s1e1(i);
end
s5s/length(e1);
ni5ni11;
w5w1; plot (ni,e); hold on; title(‘ MSE vs no. of
iterations’);
end
end
subplot(2,2,2); plot(y4); title(‘estimated noise signal’);
subplot(2,2,3); plot(q-y4); title(‘noise cancelled signal’);

MATLAB Programs 905
Review Questions
16.1 (a) Generate unit impulse function
(b) Generate signal x(n)5u(n) 2 u(n 2 N)
(c) Plot the sequence, x(n)5A cos ((2p f n)/fs
), where n50 to 100
fs
5100 Hz, f51 Hz, A50.5
(d) Generate x (n)5exp (25n), where n50 to 10.
16.2 Consider a system with impulse response

h n
nn
( )
( / ) ,
.
=
=



1 2 0 to 4
0, elsewhere
16.3 Consider the figure.
Fig. Q16.3
h (n)1 h (n)2
h (n)3 h (n)4
( )
y(n)
( )−
Fig. Q16.4
( ) H1 H2
H3
y (n)3
y(n)
3 (n)
H4
(a) Express the overall impulse response in terms of h1
(n), h2
(n), h3
(n), h4
(n)
(b) Determine h(n) when h1
(n)5{1/2, 1/4, 1/2}
h2
(n)5h3
(n)5(n 1 1) u(n)
h4
(n)5d(n 2 2)
(c) Determine the response of the system in part (b) if x(n)5 d(n 1 2)
1 3d(n 2 1) 2 4d(n 2 3).
16.4 Compute the overall impulse response of the system
for 0 # n # 99. The system H1
, H2
, H3
, H4
, are specified by
H1
: h1
[n]5{1, 1/2, 1/4, 1/8, 1/16, 1/32}
H2
: h2
[n]5{1, 1, 1, 1, 1}
H3
: y3
[n]5(1/4) x(n) 1 (1/2) x(n 2 1) 1 (1/4) x(n 2 2)
H4
: y[n]50.9 y(n 2 1) 2 0.81 y(n 2 2) 1 V(n) 1 V(n 2 1)
Plot h(V) for 0 # n # 99.
16.5 Consider the system with h(n)5an
u(n), 21 , a , 1. Determine the response.
x(n)5u(n 1 5) 2 u(n 2 10)

Fig. Q16.5
h (n)
z−2 h(n)
y (n)
( )−
16.6 (i) Determine the range of values of the parameter ‘a’ for which the linear
time invariant system with impulse response

h n
a n nn
( )
, ,
,
=
≥



0
0
even
otherwise
is stable.
(ii) Determine the response of the system with impulse response
h(n)5an
u(n) to the input signal x(n)5u(n) 2 u(n 2 10)
16.7 Consider the system described by the difference equation y(n)5a (y 1 1) 1
bx (n). Determine ‘b’ in terms of a so that S h(n)51.
(a) Compute the zero-state step response s(n) of the system and choose ‘b’
so that s(`)51.
(b) Compare the values of ‘b’ obtained in parts (a) and (b). What did you
observe?
16.8 Compute and sketch the convolution y(n) and correlation rxh
(n) sequences
for the following pair of signals and comment on the results obtained.
x n h n
x
1 1
2
1 2 4 1 1 1 1 1
( )
, ,
( )
, , , ,
=
↑








=
↑








(( )
, , , ,
( )
/ , , , , /
n h n=
− −
↑








=
↑




0 1 2 3 4 1 2 1 2 1 1 2
2




=
↑








=
↑





x n h n3 3
1 2 3 4 4 3 2 1
( )
, , ,
( )
, , ,



=
↑








=
↑





x n h n4 4
1 2 3 4 1 2 3 4
( )
, , ,
( )
, , , 

16.9 Consider the recursive discrete-time system described by the difference
equation.
y(n)5a1
y(n 2 1) 2 a2
(n 2 2) 1 b0
x(n)
where a1
520.8, a2
50.64, b0
50.866
(a) Write a program to compute and plot the impulse response h(n) of the
system for 0 # n # 49.
(b) Write a program to compute and plot the zero-state step response s(n)
of the system for 0 # n # 100.
(c) Define an FIR system with impulse response hFIR
(n) given by
h n
h n n
FIR ( )
( ),
,
=
≤ ≤



0 19
0 elsewhere
where h(n) is the impulse response computed in part (a). Write a
program to compute and plot its step response.
(d) Compare the results obtained in part (b) and part (c) and explain their
similarities and differences.

MATLAB Programs 907
16.10 Determine and plot the real and imaginary parts and the magnitude and phase
spectra of the following DTFT for various values of r and u
G(z)51/(1 2 2r(cos u) z21
1 r2
z22
) for 0 , r , 1.
16.11 Using MATLAB program compute the circular convolution of two length-N
sequences via the DFT based approach. Using this problem determine the
circular convolution of the following pairs of sequences:
(a) x(n)5{1, 2 3, 4, 2, 0, 2 2} h(n)5{3, 0, 1, 2 1, 2, 1}
(b) x(n)5{3 1 j2, 22 1 j, j3, h(n)5{ 1 2 j3, 4 1 j2, 2 2 j2,
1 1 j4, 23 1 j3}, 23 1 j5, 2 1 j }
(c) x(n)5cos (pn/2) h(n)53n
0 # n # 5
16.12 Determine the factored form of the following z-transforms
(a) H1
(z)5(2z4
1 16z3
1 44z2
1 56z 1 32)/
(3z3
1 3z3
2 15z2
1 18z 2 12)
(b) H2
(z)5(4z4
2 8.68z3
2 17.98z2
1 26.74z 2 8.04)/
(z4
2 2z3
1 10z2
1 6z 1 65)
and show their pole-zero plots. Determine all regions of convergence of each
of the above z-transforms, and describe the type of their inverse z-transform
(left-sided, right-sided, two-sided sequences) associated with each of the
ROCs.
16.13 Determine the z-transform as a ratio of two polynomials in z21
from each of
the partial-fraction expansions listed below:
(a) H z
z z
z1 1 1
3
12
2
16
4
0 5( )
( ) ( )
, .= +
−
−
−
− −
(b) H z
z
z
z
z2 1
1
2
3
3
1 0 5
4
1 0 25
0 5( )
( . )
( )
( . )
, .= +
+ −
−
−
+
−
−
−
(c) H z
z z z
z3 1 2 1 2
20
5 2
10
5 2
4
1 0 9
0 4( )
( ) ( ) ( . )
, .=
+ −
−
+
+
+
− − −
(d) H z
z
z
z z
z4 1
1
1 2
8
10
5 2 6 5
0 4( )
( ) ( )
, .= +
+
+
+ +
−
−
− −
16.14 Determine the inverse z-transform of each z-transform given in Q16.13.
16.15 Consider the system
H z
z z z
z z z
ROC z( )
( )
( )( . )( . )
, .=
− + −
− − −

− − −
− − −
1 2 2
1 1 0 5 1 0 2
0 5
1 2 3
1 1 1
1
(a) Sketch the pole-zero pattern. Is the system stable?
(b) Determine the impulse response of the system.
16.16 Determine the impulse response and the step response of the following
causal systems. Plot the pole-zero patterns and determine which of the
systems are stable.
(a) y n y n y n x n( ) ( ) ( ) ( )= − − − +
3
4
1
1
8
2

(b) y(n)5y (n 2 1) 2 0.5y(n 2 2) 1 x(n) 1 x (n 2 1)
(c) H z
z z
z
( )
( )
( )
=
+
−
− −1 1
3
1
1
(d) y(n)50.6y(n 2 1) 2 0.08y(n 2 2) 1 x(n)
(e) y(n)50.7y(n 2 1) 2 0.1y(n 2 2) 1 2x(n) 2 x(n 2 2)
Ans: (a), (b), (d) and (e) are stable, (c) is unstable
16.17 The frequency analysis of an amplitude-modulated discrete-time signal
x(n)5sin 2p f1
n 1 sin 2p f2
n
where f1
1
128
= and f2
5
128
= modulates the amplitude-modulated siganl is
xc
(n) 5 sin 2p fc
n
where fc
550/128. The resulting amplitude-modulated signal is
xam
(n)5x(n) sin 2p fc
n
(a) Sketch the signals x(n), xc
(n) and xam
(n), 0 # n # 255
(b) Compute and sketch the 128-point DFT of the signal xam
(n),
0 # n # 127
(c) Compute and sketch the 128-point DFT of the signal xam
(n), 0 # n # 99
(d) Compute and sketch the 256-point DFT of the signal xam
(n),
0 # n # 179
(e) Explain the results obtained in parts (b) through (d) by deriving the
spectrum of the amplitude modulated signal and comparing it with the
experimental results.
16.18 A continuous time signal xa
(t) consists of a linear combination of sinusoi-
dal signals of frequencies 300Hz, 400Hz, 1.3kHz, 3.6KHz and 4.3KHz. The
xa
(t) is sampled at 4kHz rate and the sampled sequence is passed through an
ideal low-pass filter with cut off frequency of 1kHz, generating a continuous
time signal ya
(t). What are the frequency components present in the recon-
structed signal ya
(t)?
16.19 Design an FIR linear phase, digital filter approximating the ideal frequency
response

Hd ( )
, /
, /
␻
␻ ␲
␲ ␻ ␲
=
≤
≤




1 6
0 6
for
for
(a) Determine the coefficient of a 25 tap filter based on the window method
with a rectangular window.
(b) Determine and plot the magnitude and phase response of the filter.
(c) Repeat parts (a) and (b) using the Hamming window
(d) Repeat parts (a) and (b) using the Bartlett window.
16.20 Design an FIR Linear Phase, bandstop filter having the ideal frequency response

Hd ( )
, /
, / /
, /
␻
␻ ␲
␲ ␻ ␲
␲ ␻ ␲
=
≤
≤
≤ ≤





1 6
0 6 3
1 3
for
for
for
(a) Determine the coefficient of a 25 tap filter based on the window
method with a rectangular window.
(b) Determine and plot the magnitude and phase response of the filter.

MATLAB Programs 909
(c) Repeat parts (a) and (b) using the Hamming window
(d) Repeat parts (a) and (b) using the Bartlett window.
16.21 A digital low-pass filter is required to meet the following specfications
Passband ripple  1 dB
Passband edge 4 kHz
Stopband attenuation  40 dB
Stopband edge 6 kHz
Sample rate 24 kHz
The filter is to be designed by performing a bilinear transformation on an
analog system function. Determine what order Butterworth, Chebyshev and
elliptic analog design must be used to meet the specifications in the digital
implementation.
16.22 An IIR digital low-pass filter is required to meet the following specfications
Passband ripple  0.5 dB
Passband edge 1.2 kHz
Stopband attenuation  40 dB
Stopband edge 2 kHz
Sample rate 8 kHz
Use the design formulas to determine the filter order for
(a) Digital Butterworth filter
(b) Digital Chebyshev filter
(c) Digital elliptic filter
16.23 An analog signal of the form xa
(t)5a(t) cos(2000 t) is bandlimited to the range
900  F  1100Hz. It is used as an input to the system shown in Fig. Q16.23.
Fig. Q16.23
( )a(t)
A/ DD / A
ϖ(n) a(n)
H ( )ω
R = 2500 cos (0.8 n)π
(a) Determine and sketch the spectra for the signal x(n) and w(n).
(b) Use Hamming window of length M531 to design a low-pass linear
phase FIR filter H() that passes {a(n)}.
(c) Determine the sampling rate of A/D converter that would allow us to
eliminate the frequency conversion in the above figure.
16.24 Consider the signal x(n)5an
u(m), |a| , 1
(a) Determine the spectrum X()
(b) The signal x(n) is applied to a device (decimator) which reduces the
rate by a factor of two. Determine the output spectrum.
(c) Show that the spectrum is simply the Fourier transform of x(2n).
16.25 Design a digital type-I Chebyshev low-pass filter operating at a sampling
rate of 44.1kHz with a passband frequency at 2kHz, a pass band ripple
of 0.4dB, and a minimum stopband attenuation of 50dB at 12kHz using
the impulse invariance method and the bilinear transformation method.
Determine the order of analog filter prototype and design the analog pro-
totype filter. Plot the gain and phase responses of the both designs using

MATLAB. Compare the performances of the two filters. Show all steps
used in the design.
Hint 1. The order of filter

N
A
s p
=
−−
−
cosh ( ( ) /
cosh ( / )
1 2
1
1 ␧
⍀ ⍀
2. Use the function cheblap.
16.26 Design a linear phase FIR high-pass filter with following specifications:
Stopbandedgeat0.5p,passbandedgeat0.7p,maximumpassbandattenuation
of 0.15dB and a minimum stopband attenuation of 40dB. Use each of the
following windows for the design. Hamming, Hanning, Blackman and
Kaiser. Show the impulse response coefficients and plot the gain response of
the designed filters for each case.
16.27 Design using the windowed Fourier series approach a linear phase FIR low-
pass filter with the following specifications: pass band edge at 1 rad/s, stop
band edge at 2rad/s, maximum passband attenuation of 0.2dB, minimum
stopband attenuation of 50dB and a sampling frequency of 10rad/s. Use each
of the following windows for the design: Hamming, Hanning, Blackman,
Kaiser and Chebyshev. Show the impulse response coefficients and plot the
gain response of designed filters for each case.
16.28 Design a two-channel crossover FIR low-pass and high-pass filter pair for
digital audio applications. The low-pass and high-pass filters are of length
31 and have a crossover frequency of 2kHz operating at a sampling rate of
44.1KHz. Use the function ‘fir1’ with a Hamming window to design the low-
pass filter while the high-pass filter is derived from the low-pass filter using
the delay complementary property. Plot the gain responses of both filters
on the same figure. What is the minimum number of delays and multipliers
needed to implement the crossover network?
16.29 Design a digital network butterworth low-pass filter operating at sampling
rate of 44.1kHz with a 0.5dB cutoff frequency at 2kHz and a minimum stop-
band attenuation of 45dB at 10kHz using the impulse invariance method
and the bilinear transformation method. Assume the sampling interval for
the impulse invariance design to be equal to 1. Determine the order of the
analog filter prototype and then design the analog prototype filter. Plot the
gain and phase responses of both designs. Compare the performances of the
filters. Show all steps used in the design. Does the sampling interval have
any effect on the design of the digital filter design based on the impulse
invariance method?
Hint The order of filter is

N
k
k
=
log ( / )
log ( / )
10 1
10
1
1
and use the function ‘buttap’.

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