digital signal processing-chapter 16: matlab programs

7/30/2019 Digital Signal Processing-chapter 16: MATLAB Programs

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MATLAB Programs

Chapter 1616.1 INTRODUCTION

MATLAB stands or MATrix LABoratory. It is a technical computing environment

or high perormance 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 ew dozen lines, to compute the solution with great accuracy

in a ew minutes on a computer, and to readily manipulate a three-dimensional

display o 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

raction o the time that it would take to write a program and execute in a language

such as FORTRAN, BASIC, or C. It also eatures a amily o application specifc

solutions, called toolboxes. Areas in which toolboxes are available include signal

processing, image processing, control systems design, dynamic systems simulation,

systems identifcation, neural networks, wavelength communication and others.

It can handle linear, non-linear, continuous-time, discrete-time, multivariable andmultirate systems. This chapter gives simple programs to solve specifc problems

that are included in the previous chapters. All these MATLAB programs have been

tested under version 7.1 o MATLAB and version 6.12 o the signal processing

toolbox.

16.2 REPRESENTATION OF BASIC SIGNALS

MATLAB programs or the generation o unit impulse, unit step, ramp, exponential,

sinusoidal and cosine sequences are as ollows.

% Program or the generation o 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);


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816 Digital Signal Processing

ylabel(Amplitude --.);

xlabel((a) n --.);

% Program or the generation o 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 or the generation o ramp sequence

n15input(enter the length o ramp sequence);

t50:n1;

subplot(2,2,3);stem(t,t);ylabel(Amplitude --.);

xlabel((c) n --.);

% Program or the generation o exponential sequence

n25input(enter the length o 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 or the generation o sine sequence

t50:.01:pi;

y5sin(2*pi*t);gure(2);

subplot(2,1,1);plot(t,y);ylabel(Amplitude --.);

xlabel((a) n --.);

% Program or the generation o cosine sequence

t50:.01:pi;

y5cos(2*pi*t);

subplot(2,1,2);plot(t,y);ylabel(Amplitude --.);

xlabel((b) n --.);

As an example,

enter the N value 7

enter the length o ramp sequence 7

enter the length o exponential sequence 7

enter the a value 1

Using the above MATLAB programs, we can obtain the waveorms o the unit

impulse signal, unit step signal, ramp signal, exponential signal, sine wave signal and

cosine wave signal as shown in Fig. 16.1.


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MATLAB Programs 817

Fig. 16.1 Representation of Basic Signals (a) Unit Impulse Signal (b)Unit-stepSignal (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

0.21

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

000 2 4 6 8

0

(a)

Amplitude

Amplitude

Amplitude

Amplitude

(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

Amplitude

Amplitude


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16.3 DISCRETE CONVOLUTION

16.3.1 Linear Convolution

Algorithm

1. Get two signalsx(m)and h(p)in matrix orm

2. The convolved signal is denoted asy(n)

3. y(n)is given by the ormula

y(n) 5 [ ( ) ( )]x k h n kk

=

where n50 to m1p2 1

4. Stop

% Program or linear convolution o 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);

gure;subplot(3,1,1);

stem(x);ylabel(Amplitude --.);

xlabel((a) n --.);subplot(3,1,2);

stem(h);ylabel(Amplitude --.);

xlabel((b) n --.);

subplot(3,1,3);

stem(y);ylabel(Amplitude --.);

xlabel((c) n --.);

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 signalsx(n)and h(n)and the convolved output

signaly(n).

Fig. 16.2 (Contd.)

Am

plitude

n

0

0.5

1 1.1 1.2 1.3 1.4 1.5

(a)

1.6 1.7 1.8 1.9 2

1

1.5

2


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MATLAB Programs 819

16.3.2 Circular Convolution

% Program or 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))];

or i = 1:N

k = i;

or j = 1:n2

H(i,j)=x(k)* b(j);k = k-1;

i (k == 0)

k = N;

end

end

end

y=zeros(1,N);

M=H;

or j = 1:N

or i = 1:n2y(j)=M(i,j)+y(j);

end

end

disp(The output sequence is y(n)= );

disp(y);

Fig. 16.2 Discrete Linear Convolution

Amplitude

n

0

1

1 1.2 1.4 1.6 1.8 2

(b)

2.2 2.4 2.6 2.8 3

2

3

4

Amp

litude

n

0

2

1 1.5 2

(c)

2.5 3 3.5 4

4

6

8


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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 or Computing Circular Convolution with zero padding

clc;

close all;

clear all;g5input(enter the rst sequence);


N15length(g);

N25length(h);

N5max(N1,N2);

N35N12N2;

%Loop or getting equal length sequence

i(N350)

h5[h,zeros(1,N3)];

else

g5[g,zeros(1,2N3)];

end

%computation o circular convolved sequence

or n51:N,

y(n)50;

or i51:N,

j5n2i11;

i(j550)

j5N1j;

end

y(n)5y(n)1g(i)*h(j);

end

end

disp(The resultant signal is);y

As an example,

enter the rst 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 or computing Block Convolution using Overlap SaveMethod

Overlap Save Method

x=input(Enter the sequence x(n) = );


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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=t(h1);

or i=1:n2:N

y1=x1(i:i+(2*(n3-n2)));

y2=t(y1);

y3=y2.*H;y4=round(it(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 or computing Block Convolution using Overlap AddMethod

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=t(h1);

or i=1:n2:n1

i i


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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 signalsx(m)and h(p)in matrix orm

2. The correlated signal is denoted asy(n)


y(n) 5 [ ( ) ( )]x k h k nk

=

where n52 [max (m,p)2 1] to [max (m,p)2 1]

4. Stop

% Program or computing cross-correlation o 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);



xlabel((a) n --.);

subplot(3,1,2);

stem(h);ylabel(Amplitude --.);

xlabel((b) n --.);

subplot(3,1,3);

stem(fiplr(y));ylabel(Amplitude --.);


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MATLAB Programs 823

xlabel((c) n --.);

disp(The resultant signal is);fiplr(y)

As an example,enter the 1st sequence [1 2 3 4]

enter the 2nd sequence [4 3 2 1]

The resultant signal is

y51.0000 4.0000 10.0000 20.0000 25.0000 24.0000 16.0000

Figure 16.3 shows the discrete input signalsx(n)and h(n)and the cross-correlated

output signaly(n).

Amplitude

Amplitu

de

Amplitude

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 signalx(n)o lengthNin matrix orm

2. The correlated signal is denoted asy(n)


y(n) 5 [ ( ) ( )]x k x k nk

=

where n52(N2 1) to (N2 1)


10/96


% Program or computing autocorrelation unction

x5input(enter the sequence);

y5xcorr(x,x);



xlabel((a) n --.);

subplot(2,1,2);

stem(fiplr(y));ylabel(Amplitude --.);

xlabel((a) n --.);

disp(The resultant signal is);fiplr(y)

As an example,

enter the sequence [1 2 3 4]

The resultant signal is

y54 11 2030 20 11 4

Figure 16.4 shows the discrete input signal x(n)and its auto-correlated output

signaly(n).

( )

Amplitude

Amplitude

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

2530

Fig. 16.4 Discrete Auto-correlation

16.5 STABILITY TEST

% Program or stability test


b5input(enter the denominator coecients o the

lter);

k5poly2rc(b);knew5fiplr(k);

s5all(abs(knew)1);

i(s55 1)

disp(Stable system);


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MATLAB Programs 825

else

disp(Non-stable system);

endAs an example,

enter the denominator coecients o the lter [1 21 .5]

Stable system

16.6 SAMPLING THEOREM

The sampling theorem can be understood well with the ollowing example.

Example 16.1 Frequency analysis of the amplitude modulated discrete-time

signal

x(n)5cos 2 pf1n1 cos 2pf

2n

where f1

1

128= andf

2

5

128= modulates the amplitude-modulated signal is

xc(n)5cos 2pf

cn

wherefc550/128. The resulting amplitude-modulated signal is

xam

(n)5x(n) cos 2pfcn

Using MATLAB program,

(a) sketch the signals x(n),xc(n) andx

am(n), 0 #n# 255

(b) compute and sketch the 128-point DFT of the signalxam

(n), 0 #n# 127

(c) compute and sketch the 128-point DFT of the signal xam

(n),0 #n# 99

Solution

% Program

Solution or Section (a)

clc;close all;clear all;

151/128;255/128;n50:255;c550/128;

x5cos(2*pi*1*n)1cos(2*pi*2*n);

xa5cos(2*pi*c*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 or Section (b)

n50:127;gure;n15128;151/128;255/128;c550/128;


xc5cos(2*pi*c*n);

xa5cos(2*pi*c*n);(Contd.)


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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

Am

plitude

0 100 200 300

(ii) n

02

1

0

1

2

100

Amplitude

200

(i)

300n

Fig. 16.5(a) (i) Modulating Signal x (n)

(Contd.)


13/96

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


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xamp5x.*xa;xam5t(xamp,n1);

stem(n,xam);title(xamp(n));xlabel(n --.);

ylabel(amplitude);%128 point DFT computation2solution or Section (c)

n50:99;gure;n250:n121;

151/128;255/128;c550/128;


xc5cos(2*pi*c*n);

xa5cos(2*pi*c*n);

xamp5x.*xa;

or i51:100,

xamp1(i)5xamp(i);

endxam5t(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 o the

signal xam

(n) or 0 #n# 127 and Fig. 16.5 (c) shows the 128-point DFT o the

signalxam

(n), 0 #n# 99.

16.7 FAST FOURIER TRANSFORM

Algorithm

1. Get the signalx(n)o lengthNin matrix orm

2. Get theNvalue

3. The transormed signal is denoted as

x k x n e k Nj

Nnk

n

N

( ) ( ) for =

=

2

0

1

0 1p

\\\% Program or computing discrete Fourier transorm

clc;close all;clear all;x5input(enter the sequence);

n5input(enter the length o t);

X(k)5t(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 o t 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


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MATLAB Programs 829

The eight-point decimation-in-time ast Fourier transorm o the sequencex(n)is

computed usingMATLAB program and the resultant output is plotted in Fig. 16.6.

05

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 ripples2. Get the passband and stopband edge requencies

3. Get the sampling requency

4. Calculate the order o the flter using Eq. 8.46

5. Find the flter coefcients

6. Draw the magnitude and phase responses.

% Program or the design o Butterworth analog low pass flter

clc;

close all;clear all;

ormat longrp5input(enter the passband ripple);

rs5input(enter the stopband ripple);

wp5input(enter the passband req);

ws5input(enter the stopband req);

s5input(enter the sampling req);


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w152*wp/s;w252*ws/s;

[n,wn]5buttord(w1,w2,rp,rs,s);

[z,p,k]5butter(n,wn);[b,a]5zp2t(z,p,k);

[b,a]5butter(n,wn,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);

subplot(2,1,1);plot(om/pi,m);

ylabel(Gain in dB --.);xlabel((a) Normalised

requency --.);

subplot(2,1,2);plot(om/pi,an);xlabel((b) Normalised requency --.);

ylabel(Phase in radians --.);

As an example,

enter the passband ripple 0.15

enter the stopband ripple 60

enter the passband req 1500

enter the stopband req 3000


The amplitude and phase responses o the Butterworth low-pass analog flter are

shown in Fig. 16.7.

Fig. 16.7 Butterworth Low-pass Analog Filter(a) Amplitude Response and (b) Phase Response

0.1

250

200

150

100

50

50

GainindB

0

0.2 0.3 0.4

(a)

Normalised frequency

0.5 0.6 0.7 0.8 0.9 10

0.14

2

2

4

0

0.2 0.3 0.4

(b)


0.5 0.6 0.7 0.8 0.9 10

Ph

aseinradians


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MATLAB Programs 831

16.8.2 High-pass Filter

Algorithm

1. Get the passband and stopband ripples

2. Get the passband and stopband edge requencies





% Program or the design o Butterworth analog highpass flter

clc;


ormat long

rp5input(enter the passband ripple);





w152*wp/s;w252*ws/s;[n,wn]5buttord(w1,w2,rp,rs,s);

[b,a]5butter(n,wn,high,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);subplot(2,1,2);plot(om/pi,an);

xlabel((b) Normalised requency --.);


As an example,





enter the sampling req 8000

The amplitude and phase responses o Butterworth high-pass analog flter are

shown in Fig. 16.8.


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16.8.3 Bandpass Filter

Algorithm




4. Calculate the order o the flter using Eq. 8.465. Find the flter coefcients


% Program or the design o Butterworth analog Bandpass flter

clc;


ormat long

rp5input(enter the passband ripple...);

rs5input(enter the stopband ripple...);wp5input(enter the passband req...);

ws5input(enter the stopband req...);

s5input(enter the sampling req...);


Fig. 16.8 Butterworth High-pass Analog Filter(a) Amplitude Response and(b) Phase Response

Phaseinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

GainindB

(a)

0.1400

300

200

100

0

100

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


19/96

MATLAB Programs 833

[n]5buttord(w1,w2,rp,rs);

wn5[w1 w2];

[b,a]5butter(n,wn,bandpass,s);w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);

subplot(2,1,2);plot(om/pi,an);



As an example,

enter the passband ripple... 0.36

enter the stopband ripple... 36

enter the passband req... 1500

enter the stopband req... 2000

enter the sampling req... 6000

The amplitude and phase responses o Butterworth bandpass analog flter 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


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16.8.4 Bandstop Filter

Algorithm







% Program or the design o Butterworth analog Bandstop flter

clc;


ormat long


rs5input(enter the stopband ripple...);

wp5input(enter the passband req...);




[n]5buttord(w1,w2,rp,rs,s);

wn5[w1 w2];

[b,a]5butter(n,wn,stop,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);






As an example,






The amplitude and phase responses o Butterworth bandstop analog flter are

shown in Fig. 16.10.


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MATLAB Programs 835

16.9 CHEBYSHEV TYPE-1 ANALOG FILTERS


Algorithm


2. Get the passband and stopband edge requencies3. Get the sampling requency




% Program or the design o Chebyshev Type-1 low-pass flter

clc;


ormat long

rp5input(enter the passband ripple...);rs5input(enter the stopband ripple...);




GainindB

(a)

0.1

150

50

100

200

0

50

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Fig. 16.10 Butterworth Bandstop Analog Filter(a) Amplitude Response and(b) Phase Response

Phaseinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


22/96



[n,wn]5cheb1ord(w1,w2,rp,rs,s);

[b,a]5cheby1(n,rp,wn,s);w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Chebyshev type - 1 low-pass analog flter

are shown in Fig. 16.11.

GainindB

Phaseinradians

(a)

(b)

0.1

0.14

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 Responseand (b) Phase Response


23/96

MATLAB Programs 837


Algorithm

1. Get the passband and stopband ripples2. Get the passband and stopband edge requencies3. Get the sampling requency4. Calculate the order o the flter using Eq. 8.575. Find the flter coefcients6. Draw the magnitude and phase responses.

%Program or the design o Chebyshev Type-1 high-pass flter

clc;


ormat long







[b,a]5cheby1(n,rp,wn,high,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);subplot(2,1,1);plot(om/pi,m);

Fig. 16.12 Chebyshev Type - 1 High-pass Analog Filter(a) Amplitude Responseand (b) Phase Response

GainindB

(a)

0.1

100

50

150

200

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Ph

aseinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


24/96



requency --.);



As an example,enter the passband ripple... 0.29





The amplitude and phase responses o Chebyshev type - 1 high-pass analog flter



Algorithm

1. Get the passband and stopband ripples2. Get the passband and stopband edge requencies3. Get the sampling requency4. Calculate the order o the flter using Eq. 8.575. Find the flter coefcients


% Program or the design o Chebyshev Type-1 Bandpass flter

clc;


ormat long







[n]5cheb1ord(w1,w2,rp,rs,s);

wn5[w1 w2];[b,a]5cheby1(n,rp,wn,bandpass,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);


xlabel((b) Normalised requency --.);ylabel(Phase in radians --.);

As an example,





25/96

MATLAB Programs 839



The amplitude and phase responses o Chebyshev type - 1 bandpass analog flter



Algorithm


2. Get the passband and stopband edge requency





% Program or the design o Chebyshev Type-1 Bandstop flter

clc;


ormat long





Fig. 16.13 Chebyshev Type-1 Bandpass Analog Filter(a) Amplitude Response and (b) Phase Response

Ga

in

in

dB

a

0.1

200

100

300

400

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Phase

in

radians

(b)

0.13

3

2

1

1

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


26/96




[n]5cheb1ord(w1,w2,rp,rs,s);wn5[w1 w2];

[b,a]5cheby1(n,rp,wn,stop,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);



As an example,






The amplitude and phase responses o Chebyshev type - 1 bandstop analog flter


Fig. 16.14 Chebyshev Type - 1 Bandstop Analog Filter(a) Amplitude Response and (b) Phase Response

GainindB

(a)

0.1

150

50

100

200

250

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Pha

seinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


27/96

MATLAB Programs 841

16.10 CHEBYSHEV TYPE-2 ANALOG FILTERS


Algorithm







% Program or the design o Chebyshev Type-2 low pass analog flter

clc;


ormat long








[b,a]5cheby2(n,rs,wn,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Chebyshev type - 2 low-pass analog flter



28/96


Fig. 16.15 Chebyshev Type - 2 Low-pass Analog Filter(a) Amplitude Responseand (b) Phase Response

GainindB

Phaseinradians

(a)

(b)

0.1

0.14

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





5. Find the flter coefcients6. Draw the magnitude and phase responses.

% Program or the design o Chebyshev Type-2 High pass analog flter

clc;


ormat long








[b,a]5cheby2(n,rs,wn,high,s);

w50:.01:pi;


29/96

MATLAB Programs 843

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);subplot(2,1,1);plot(om/pi,m);

ylabel(Gain in dB --.);xlabel((a) Normalised requency --.);




As an example,




enter the stopband req... 1600enter the sampling req... 10000

The amplitude and phase responses o Chebyshev type - 2 high-pass analog flter


Gain

indB

Phaseinradians

(a)

(b)

0.1

0.14

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(a) Amplitude Response and (b) Phase Response


Algorithm




30/96




5. Find the flter coefcients6. Draw the magnitude and phase responses.

% Program or the design o Chebyshev Type-2 Bandpass analog flter

clc;


ormat long








wn5[w1 w2];

[b,a]5cheby2(n,rs,wn,bandpass,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Chebyshev type - 2 bandpass analog flter


Fig. 16.17 (Contd.)

GainindB

(a)

0.1

80

60

20

0

40

100

20

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


31/96

MATLAB Programs 845


Algorithm







% Program or the design o Chebyshev Type-2 Bandstop analog flter

clc;


ormat long








wn5[w1 w2];

[b,a]5cheby2(n,rs,wn,stop,s);

w50:.01:pi;

[h,om]5reqs(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




Fig. 16.17 Chebyshev Type - 2 Bandstop Analog Filter(a) Amplitude Responseand (b) Phase Response

Phaseinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


32/96


As an example,


enter the stopband ripple... 30enter the passband req... 1300



The amplitude and phase responses o Chebyshev type - 2 bandstop analog flter


Fig. 16.18 Chebyshev Type - 2 Bandstop Analog Filter(a) Amplitude Response and (b) Phase Response

GainindB

Phaseinradians

(a)

(b)

0.1

0.14

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







33/96

MATLAB Programs 847

% Program or the design o Butterworth low pass digital flter

clc;


ormat long







[n,wn]5buttord(w1,w2,rp,rs);

[b,a]5butter(n,wn);w50:.01:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Butterworth low-pass digital flter are


GainindB

(a)

0.1

300

200

0

100

400

100

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Fig. 16.19 (Contd.)


34/96



Algorithm







% Program or the design o Butterworth highpass digital flter

clc;


ormat long







[n,wn]5buttord(w1,w2,rp,rs);

[b,a]5butter(n,wn,high);

w50:.01:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);


xlabel((b) Normalised requency --.);ylabel(Phase in radians --.);

As an example,




Fig. 16.19 Butterworth Low-pass Digital Filter(a) Amplitude Response and(b) Phase Response

Phaseinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


35/96

MATLAB Programs 849



The amplitude and phase responses o Butterworth high-pass digital flter are


16.11.3 Band-pass Filter

Algorithm







% Program or the design o Butterworth Bandpass digital flter


ormat long




Fig. 16.20 Butterworth High-pass Digital Filter(a) Amplitude Response and (b) Phase Response

GainindB

(a)

0.1

250

200

150

0

100

50

300

50

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Phaseinra

dians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


36/96




w152*wp/s;w252*ws/s;[n]5buttord(w1,w2,rp,rs);

wn5[w1 w2];

[b,a]5butter(n,wn,bandpass);

w50:.01:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Butterworth band-pass digital flter areshown in Fig. 16.21.

Fig. 16.21 Butterworth Bandstop Digital Filter(a) Amplitude Response and(b) Phase Response

GainindB

(a)

(b)

Pha

seinradians

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


37/96

MATLAB Programs 851


Algorithm1. Get the passband and stopband ripples






% Program or the design o Butterworth Band stop digital flter

clc;


ormat long







[n]5buttord(w1,w2,rp,rs);wn5[w1 w2];

[b,a]5butter(n,wn,stop);

w50:.01:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);






As an example,






The amplitude and phase responses o the Butterworth bandstop digital flter are



38/96


Fig. 16.22 Butterworth Bandstop Digital Filter(a) Amplitude Response and (b) Phase Response

Phaseinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

GainindB

(a)

0.1

100

0

200

100

400

300

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

16.12 CHEBYSHEV TYPE-1 DIGITAL FILTERS


Algorithm






% Program or the design o Chebyshev Type-1 lowpass digital flter

clc;


ormat long






39/96

MATLAB Programs 853


[n,wn]5cheb1ord(w1,w2,rp,rs);

[b,a]5cheby1(n,rp,wn);w50:.01:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);



ylabel(Phase in radians --.);As an example,






The amplitude and phase responses o Chebyshev type - 1 low-pass digital flter


Fig. 16.23 Chebyshev Type - 1 Low-pass Digital Filter(a) Amplitude Responseand (b) Phase Response

GainindB

Phaseinradians

(a)

(b)

0.1

0.14

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


40/96



Algorithm1. Get the passband and stopband ripples






% Program or the design o Chebyshev Type-1 highpass digital

ilterclc;


ormat long








[b,a]5cheby1(n,rp,wn,high);

w50:.01/pi:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);


ylabel(Gain in dB --.);xlabel((a) Normalised requency

--.);subplot(2,1,2);plot(om/pi,an);



As an example,






The amplitude and phase responses o Chebyshev type - 1 high-pass digital flter



41/96

MATLAB Programs 855


Algorithm






% Program or the design o Chebyshev Type-1 Bandpass digital flter

clc;


ormat long







Fig. 16.24 Chebyshev Type - 1 High-pass Digital Filter(a) Amplitude Response and (b) Phase Response

Phaseinradians

(b)

0.14

4

2

2

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

GainindB

(a)

0.1

0

250

200

150

100

50

350

300

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


42/96


[n]5cheb1ord(w1,w2,rp,rs);

wn5[w1 w2];

[b,a]5cheby1(n,rp,wn,bandpass);w50:.01:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Chebyshev type - 1 bandpass digital flter


GainindB

Phaseinradians

(a)

(b)

0.1

0.14

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) AmplitudeResponse and (b) Phase Response


43/96

MATLAB Programs 857


Algorithm







% Program or the design o Chebyshev Type-1 Bandstop digital flter


ormat long







[n]5cheb1ord(w1,w2,rp,rs);wn5[w1 w2];

[b,a]5cheby1(n,rp,wn,stop);

w50:.1/pi:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Chebyshev type - 1 bandstop digital flter



44/96


Fig. 16.26 Chebyshev Type - 1 Bandstop Digital Filter(a) Amplitude Response and (b) Phase Response

GainindB

(a)

0.1

0

200

150

100

50

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Phaseinradians

(b)

0.13

2

1

1

2

3

4

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

16.13 CHEBYSHEV TYPE-2 DIGITAL FILTERS


Algorithm



3. Get the sampling requency4. Calculate the order o the flter using Eq. 8.67



% Program or the design o Chebyshev Type-2 lowpass digital flter

clc;


ormat long








[b,a]5cheby2(n,rs,wn);


45/96

MATLAB Programs 859

w50:.01:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));an5angle(h);



requency --.);




As an example,


enter the stopband ripple... 35enter the passband req... 1500



The amplitude and phase responses o Chebyshev type - 2 low-pass digital flter


Fig. 16.27 Chebyshev Type - 2 Low-pass Digital Filter(a) Amplitude Response

and (b) Phase Response

Gainind

B

(a)

0.1100

80

60

40

20

0

20

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Phaseinradians

(b)

0.14

2

2

4

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


Algorithm




46/96






% Program or the design o Chebyshev Type-2 high pass digital flter

clc;


ormat long




ws5input(enter the stopband req...);s5input(enter the sampling req...);



[b,a]5cheby2(n,rs,wn,high);

w50:.01/pi:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);




As an example,






The amplitude and phase responses o Chebyshev type - 2 high-pass digital flter


GainindB

(a)

0.1120

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.)


47/96

MATLAB Programs 861


Algorithm


2. Get the passband and stopband edge requency





% Program or the design o Chebyshev Type-2 Bandpass digital flter

clc;


ormat long





s5input(enter the sampling req...);w152*wp/s;w252*ws/s;


wn5[w1 w2];

[b,a]5cheby2(n,rs,wn,bandpass);

w50:.01/pi:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);


ylabel(Gain in dB --.);xlabel((a) Normalisedrequency --.);




Phaseinradians

(b)

0.14

2

2

4

0

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10

Fig. 16.28 Chebyshev Type - 2 High-pass Digital Filter(a) Amplitude Response



48/96


As an example,






The amplitude and phase responses o Chebyshev type - 2 bandpass digital flter


Fig. 16.29 Chebyshev Type - 2 Bandpass Digital Filter(a) Amplitude Responseand (b) Phase Response

GainindB

Phaseinradians

(a)

(b)

0.1

0.14

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







49/96

MATLAB Programs 863

% Program or the design o Chebyshev Type-2 Bandstop digital flter

clc;


ormat long








wn5[w1 w2];

[b,a]5cheby2(n,rs,wn,stop);w50:.1/pi:pi;

[h,om]5reqz(b,a,w);

m520*log10(abs(h));

an5angle(h);



requency --.);



ylabel(Phase in radians --.);As an example,






The amplitude and phase responses o Chebyshev type - 2 bandstop digital flter


GainindB

(a)

0.1

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.)


50/96


16.14FIR FILTER DESIGN USING WINDOWTECHNIQUES

In the design o FIR flters using any window technique, the order can be calculated

using the ormula given by

Np 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 requency,

fs

is the stopband requency andFs

is the sampling requency.

16.14.1 Rectangular Window

Algorithm




4. Calculate the order o the flter

5. Find the window coefcients using Eq. 7.37


% Program or the design o FIR Low pass, High pass, Band passand Bandstop flters using rectangular window




p5input(enter the passband req);s5input(enter the stopband req);

5input(enter the sampling req);

wp52*p/;ws52*s/;

num5220*log10(sqrt(rp*rs))213;

Fig. 16.30 Chebyshev Type - 2 Bandstop Digital Filter(a) Amplitude Response


Phaseinradians

(b)

0.1-4

-2

-1

-3

3

0

1

2

0.2 0.3 0.4


0.5 0.6 0.7 0.8 0.9 10


51/96

MATLAB Programs 865

dem514.6*(s2p)/;

n5ceil(num/dem);

n15n11;i (rem(n,2)50)

n15n;

n5n21;

end

y5boxcar(n1);

% low-passfilter

b5r1(n,wp,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));subplot(2,2,1);plot(o/pi,m);ylabel(Gain in dB --.);

xlabel((a) Normalised requency --.);

% high-passfilter

b5r1(n,wp,high,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));

subplot(2,2,2);plot(o/pi,m);ylabel(Gain in dB --.);


% bandpassfilter

wn5[wp ws];

b5r1(n,wn,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));

subplot(2,2,3);plot(o/pi,m);ylabel(Gain in dB -->);

xlabel((c) Normalised requency -->);

% bandstopfilter

b5r1(n,wn,stop,y);[h,o]5reqz(b,1,256);

m520*log10(abs(h));


xlabel((d) Normalised requency -->);

As an example,


enter the stopband ripple 0.04




The gain responses o low-pass, high-pass, bandpass and bandstop flters using

rectangular window are shown in Fig. 16.31.


52/96


16.14.2 Bartlett Window

Algorithm







% Program or the design o FIR Low pass, High pass, Band passand Bandstop flters using Bartlett window




p5input(enter the passband req);

s5input(enter the stopband req);


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

8020

80

60

6015

60

40

4010

40

20

205

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

Ga

in

in

dB

Ga

in

in

dB

Ga

in

in

dB

Ga

in

in

dB


53/96

MATLAB Programs 867

wp52*p/;ws52*s/;


dem514.6*(s2p)/;n5ceil(num/dem);

n15n11;

i (rem(n,2)50)

n15n;

n5n21;

end

y5bartlett(n1);

% low-passfilter

b5r1(n,wp,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% high-passfilter

b5r1(n,wp,high,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% bandpassfilter

wn5[wp ws];

b5r1(n,wn,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));


xlabel((c) Normalised requency --.);

% bandstopfilter

b5r1(n,wn,stop,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));


xlabel((d) Normalised requency --.);

As an example,




enter the stopband req 2000enter the sampling req 8000


Bartlett window are shown in Fig. 16.32.


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16.14.3 Blackman window

Algorithm






% Program or the design o FIR Low pass, High pass, Band passand Band stop digital flters using Blackman window


rp5input(enter the passband ripple);rs5input(enter the stopband ripple);




Fig. 16.32 Filters using Bartlett Window (a) Low-pass (b) High-pass(c) Bandpass and (d) Bandstop

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40 8

30

306

30

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20

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(a)

(c)

(b)

(d)





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1

1

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GainindB

GainindB

GainindB

GainindB


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MATLAB Programs 869

wp52*p/;ws52*s/;



n15n11;

i (rem(n,2)50)

n15n;

n5n21;

end

y5blackman(n1);

% low-passfilter

b5r1(n,wp,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% high-passfilter

b5r1(n,wp,high,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% bandpassfilter

wn5[wp ws];

b5r1(n,wn,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% bandstopfilter

b5r1(n,wn,stop,y);

[h,o]5reqz(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.


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16.14.4 Chebyshev Window

Algorithm






% Program or the design o FIR Lowpass, High pass, Band passand Bandstop flters using Chebyshev window







Fig. 16.33 Filters using Blackman Window (a) Low-pass (b) High-pass(c) Bandpass and (d) Bandstop

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120 8

150

100100

100 6

80

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804

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50

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(a)

(c)

(b)

(d)





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0.8

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1

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0

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GainindB

GainindB

GainindB

GainindB


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MATLAB Programs 871

r5input(enter the ripple value(in dBs));

wp52*p/;ws52*s/;

num5220*log10(sqrt(rp*rs))213;dem514.6*(s2p)/;

n5ceil(num/dem);

i(rem(n,2)50)

n5n11;

end

y5chebwin(n,r);

% low-passfilter

b5r1(n-1,wp,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% high-passfilter

b5r1(n21,wp,high,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% band-passfilter

wn5[wp ws];

b5r1(n21,wn,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% band-stopfilter

b5r1(n21,wn,stop,y);[h,o]5reqz(b,1,256);

m520*log10(abs(h));



As an example,





enter the sampling req 10000enter the ripple value(in dBs)40


Chebyshev window are shown in Fig. 16.34.


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16.14.5 Hamming Window

Algorithm






% Program or the design o FIR Low pass, High pass, Band passand Bandstop flters using Hamming window



rs5input(enter the stopband ripple);p5input(enter the passband req);



Fig. 16.34 Filters using Chebyshev Window (a) Low-pass (b) High-pass(c) Bandpass and (d) Bandstop

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120 12

100

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(c)

(b)

(d)





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1

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GainindB

GainindB

GainindB

GainindB


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MATLAB Programs 873

wp52*p/;ws52*s/;



n15n11;

i (rem(n,2)50)

n15n;

n5n21;

end

y5hamming(n1);

% low-passfilter

b5r1(n,wp,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% high-passfilter

b5r1(n,wp,high,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% bandpassfilter

wn5[wp ws];

b5r1(n,wn,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% bandstopfilter

b5r1(n,wn,stop,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



As an example,






Hamming window are shown in Fig. 16.35.


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16.14.6 Hanning Window

Algorithm






% Program or the design o FIR Low pass, High pass, Band passand Band stop flters using Hanning window







Fig. 16.35 Filters using Hamming Window (a) Low-pass (b) High-pass(c) Bandpass and (d) Bandstop

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120 15

100

40

20

60

80100

10010

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(c)

(b)

(d)





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GainindB

GainindB

GainindB

GainindB


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MATLAB Programs 875

wp52*p/;ws52*s/;



n15n11;

i (rem(n,2)50)

n15n;

n5n21;

end

y5hamming(n1);

% low-passfilter

b5r1(n,wp,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% high-passfilter

b5r1(n,wp,high,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% bandpassfilter

wn5[wp ws];

b5r1(n,wn,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



% bandstopfilter

b5r1(n,wn,stop,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));



As an example,






Hanning window are shown in Fig. 16.36.


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16.14.7 Kaiser Window

Algorithm




5. Find the window coefcients using Eqs 7.46 and 7.47


% Program or the design o FIR Low pass, High pass, Band passand Bandstop flters using Kaiser window







beta5input(enter the beta value);

Fig. 16.36 Filters using Hanning Window (a) Low-pass (b) High-pass(c) Bandpass and (d) Bandstop

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120 12

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10010

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(c)

(b)

(d)





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GainindB

GainindB

GainindB

GainindB


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MATLAB Programs 877

wp52*p/;ws52*s/;



n15n11;

i (rem(n,2)50)

n15n;

n5n21;

end

y5kaiser(n1,beta);

% low-passfilter

b5r1(n,wp,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));


xlabel((a) Normalised requency -->);

% high-passfilter

b5r1(n,wp,high,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));


xlabel((b) Normalised requency -->);

% bandpassfilter

wn5[wp ws];

b5r1(n,wn,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));


xlabel((c) Normalised requency -->);

% bandstopfilter

b5r1(n,wn,stop,y);

[h,o]5reqz(b,1,256);

m520*log10(abs(h));


xlabel((d) Normalised requency -->);

As an example,





enter the beta value 5.8


Kaiser window are shown in Fig. 16.37.


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16.15 UPSAMPLING A SINUSOIDAL SIGNAL

% Program or upsampling a sinusoidal signal by actor L

N5input(Input length o the sinusoidal sequence5);

L5input(Up Samping actor5);

5input(Input signal requency5);

% Generate the sinusoidal sequence or the specifed length N

n50:N21;

x5sin(2*pi**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(c) Bandpass and (d) Bandstop

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120 15

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(c)

(b)

(d)





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GainindB

GainindB

GainindB

GainindB


65/96

MATLAB Programs 879

%Plot the output sequence

subplot (2,1,2);

stem (n,y(1:length(x)));title([output sequence,upsampling actor5,num2str(L)]);

xlabel(Time n);

ylabel(Amplitude);

16.16UPSAMPLING AN EXPONENTIALSEQUENCE

digital signal processing-chapter 16: matlab programs

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