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Document Document Document Document name:name:name:name: DSP Lab UDSP Lab UDSP Lab UDSP Lab Ussssing ing ing ing MATLABMATLABMATLABMATLAB



DESIGN OF IIR FILTER BUTTERWORTH

clc

wp=input('enter the pass band edge freq.in r/sec=');

ap=input('enter the pass band riple in db=');

ws=input('enter the stop band edge freq.in r/sec=');

as=input('enter the stop band ripple in db=');

Fs=input('enter the sampling freq=');

[n,Wn]=buttord(wp,ws,ap,as,'s');

disp('the order of filter is n=');

disp(n);

[num,den]=butter(n,Wn,'s');

[b,a]=bilinear(num,den,Fs)

freqz(b,a,512,Fs);

grid on;

xlabel('freq in Hz');

ylabel('Gain in db');

title('Frequency response of the filter');



OUTPUT FOR BUTTERWORTH:

enter the pass band edge freq.in r/sec=0.2*pi

enter the pass band riple in db=0.6*pi

enter the stop band edge freq.in r/sec=1.6

enter the stop band ripple in db=11.8

enter the sampling freq=1

the order of filter is n=

2

b =

0.0971 0.1941 0.0971

a =

1.0000 -0.9463 0.3346



DESIGN OF IIR FILTER USING IMPULSE INVARIANCE

wp=input('enter the pass band frequency in rad');

ws=input('enter the stop band frequency in rad');

rp=input('enter the pass band frequency in ripple');

as=input('enter the stop band frequency in ripple');

T=input('enter the time period');

Fs=1/T;

omegap=(2/T)*tan(wp/2);

omegas=(2/T)*tan(ws/2);

[cs,ds]=afd_butt(omegap,omegas,rp,as)

[b,a]=bilinear(cs,ds,Fs);

[cs,ds]=afd_butt(omegap,omegas,rp,as)

func(b,a)=afd_butt(omegap,omegas,rp,as)

if omegas<=0

error('pass band edge must be larger then pass band edge');

end

if omegas<=omegap

error('stop band edge must be larger then pass band edge');

end

if(rp<=0)|(as<0)

error('pass band ripple and pass band attenuation must be larger then zero');

end

a=(10^(0.1*as)-1)/(10^(0.1*rp)-1);

b=sqart(a);

c=log10(b);

d=omegas/omegap;

e=log10(d);

N=ceil(c/e);

fprintf('\n butterworth filter order=%2f\n':N);

omegac=omegap/(((10^(0.1*rp))-1)^(1/(2*N)));

fprintf('\n omegac=%2f\n',omegac);

[b,a]=u_buttapl(N,omegac);

[z,P,k]=buttap(n);

P=P*omegac;

k=k*omagac^N;

b=real(poly(z));

b0=k;

b=k*b;

a=real(poly(p));



OUT PUT FOR IMPULSE INVARIANCE

enter the pass band edge freq.in r/sec=0.2*pi

enter the pass band riple in db=0.6*pi

enter the stop band edge freq.in r/sec=1.6

enter the stop band ripple in db=11.8

enter the sampling freq=1

the order of filter is n=

2

b =

0.0971 0.1941 0.0971

a =

1.0000 -0.9463 0.3346



STABILITY OF LTI SYSTEM:

clc

b=[1 2];

a=[3 4 5];

w=0:0.01:pi/2;

H=freqz(b,a,w);

subplot(3,1,1);

plot(w,H);

xlabel('w');

ylabel('Frequency');

tf(b,a)

A=abs(H);

subplot(3,1,2);

plot(w,A);

xlabel('W');

ylabel('magnitude');

B=angle(H);

subplot(3,1,3);

plot(w,B);

xlabel('w');

ylabel('phase');

% FREQUENCY RESPONSE

b=[1 2];

a=[3 4 5];

w=0:0.01:pi/2;

freqz(b,a,w);

tf(b,a)



OUTPUT FOR STABILITY OF LTI SYSTEM:

Transfer function for time response:

s + 2

---------------

3 s^2 + 4 s + 5

GRAPH FOR TIME RESPONSE:

Transfer function for frequency response:

s + 2

---------------

3 s^2 + 4 s + 5



GRAPH FOR FREQUENCY RESPONSE:



REPRESENTATION OF BASIC SIGNALS:

% sine wave

t=0:0.01:1;

a=2;

b=a*sin(2*pi*2*t);

subplot(3,3,1);

stem(t,b);

xlabel('time');

ylabel('Amplitude');

title ('sinewave');

% Cosine wave

t=0:0.01:1;

a=2;

b=a*cos(2*pi*2*t);

subplot(3,3,2);

stem(t,b);

xlabel('time');


title ('Cos wave');

% Square wave

t=0:0.01:1;

a=2;

b=a*square(2*pi*2*t);

subplot(3,3,3);

stem(t,b);

xlabel('time');


title ('square wave');

% Exponential waveform

t=0:0.01:1;

a=2;

b=a*exp(2*pi*2*t);

subplot(3,3,4);

stem(t,b);

xlabel('time');


title ('exponential wave');

%sawtooth

t=0:0.01:1;

a=2;

b=a*sawtooth(2*pi*2*t);



subplot(3,3,5);

stem(t,b);

xlabel('time');


title ('sawtooth wave');

% unit step signal

n=-5:5;

a = [zeros(1,5),ones(1,6)];

subplot(3,3,6);

stem(n,a);

Xlabel ('time');

Ylabel ('amplitude');

title('Unit step');

% unit impulse

n=-5:5;

a = [zeros(1,5),ones(1,1),zeros(1,5)];

subplot(3,3,7);

stem(n,a);

Xlabel ('time');

Ylabel ('amplitude');

title('Unit impulse');



OUTPUT FOR REPRESENTATION OF BASIC SIGNALS:



COMPUTATION OF CIRCULAR CONVOLUTION:

clc;

clear all;

close all;

x=input('enter the sequence');

h=input('enter the imp response');

% no of samples in x

n1=length(x);

n2=length(h);

n3=n1+n2-1;

n=max(n1,n2);

if(n3>=0)

h=[h,zeros(1,n3)];

else

x=[n,zeros(1,-n3)];

end;

for a =1:n

y(a)=0;

for i=1:n

j=a-i+1;

if(j<=0)

j=n+j;

end;

y(a)=y(a)+[x(i)*h(j)];

end;

end;

t=1:n

stem(t,y);

disp(y);

title('circular convolution');

xlabel('samples');

ylabel('amplitude');



OUTPUT FOR CIRCULAR CONVOLUTION:

enter the sequence[1 2 2 1]

enter the imp response[1 1 1]

t =

1 2 3 4

4 4 5 5

GRAPH FOR CIRCULAR CONVOULTION:



CALCULATION OF FFT:

function[Xk]=dft(x,N);

x=[1 1 1 1 4 5 6 1];

N=8;

n=[0:1:N-1];

k=[0:1:N-1];

a=(-i*2*pi/N);

WN=exp(a);

nk=n'*k;

WNnk=WN.^nk;

Xk=x*WNnk



OUTPUT FOR CALCULATION OF FFT:

Xk =

Columns 1 through 4

20.0000 -5.8284 + 7.8284i -2.0000 - 4.0000i -0.1716 - 2.1716i

Columns 5 through 8

4.0000 + 0.0000i -0.1716 + 2.1716i -2.0000 + 4.0000i -5.8284 - 7.8284i

ans =

Columns 1 through 4

20.0000 -5.8284 + 7.8284i -2.0000 - 4.0000i -0.1716 - 2.1716i

Columns 5 through 8

4.0000 + 0.0000i -0.1716 + 2.1716i -2.0000 + 4.0000i -5.8284 - 7.8284i



VERIFICATION OF SAMPLING THEOREM:

clear all

t=-100:01:100;

fm=0.02;

x=cos(2*pi*t*fm);

subplot(2,2,1);

plot(t,x);

xlabel('time in sec');

ylabel('x(t)');

title('continuous time signal');

fs1=0.02;

n=-2:2;

x1=cos(2*pi*fm*n/fs1);

subplot(2,2,2);

stem(n,x1);

hold on

subplot(2,2,2);

plot(n,x1,':');

title('discrete time signal x(n) with fs<2fm');

xlabel('n');

ylabel('x(n)');

fs2=0.04;

n1=-4:4;

x2=cos(2*pi*fm*n1/fs2);

subplot(2,2,3);

stem(n1,x2);

hold on

subplot(2,2,3);

plot(n1,x2,':');

title('discrete time signal x(n) with fs>2fm');

xlabel('n');

ylabel('x(n)');

n2=-50:50;

fs3=0.5;

x3=cos(2*pi*fm*n2/fs3);

subplot(2,2,4);

stem(n2,x3);

hold on

subplot(2,2,4);

plot(n2,x3,':');

xlabel('n');

ylabel('x(n)');

title('discrete time signal x(n) with fs=2fm');



OUTPUT OF SAMPLING THEOREM:



DESIGN OF FIR FILTER:

LOW PASS FILTER:

clc;

clear all;

wc=input('enter the value of cut off frequency');

N=input('enter the value of filter');

alpha=(N-1)/2;

eps=0.001;

%Rectangular Window

n=0:1:N-1;

hd=sin(wc*(n-alpha+eps))./(pi*(n-alpha+eps));

hn=hd

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h));

hold on

%Hamming Window

n=0:1:N-1;

wh=0.54-0.46*cos((2*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h),'ms');

hold off;

hold on

%Hanning Window

n=0:1:N-1;

wh=0.5-0.5*cos((2*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h),'blue');

hold off;

hold on

%Blackman Window

n=0:1:N-1;

wh=0.42-0.5*cos((2*pi*n)/(N-1))+0.08*cos((4*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h),'green');

hold off;



OUTPUT FOR LOW PASS FILTER:

enter the value of cut off frequency1.6

enter the value of filter11

hn =

Columns 1 through 8

0.0630 0.0092 -0.1057 -0.0090 0.3185 0.5093 0.3178 -0.0095

Columns 9 through 11

-0.1056 0.0094 0.0630

hn =

Columns 1 through 8

0.0050 0.0015 -0.0421 -0.0062 0.2905 0.5093 0.2899 -0.0065


-0.0420 0.0016 0.0050

hn =

Columns 1 through 8

0 0.0009 -0.0365 -0.0059 0.2881 0.5093 0.2875 -0.0062


-0.0365 0.0009 0

hn =

Columns 1 through 8

-0.0000 0.0004 -0.0212 -0.0046 0.2705 0.5093 0.2699 -0.0049


-0.0212 0.0004 -0.0000



GRAPH FOR LPF:



DESIGN FOR HIGH PASS FILTER:

clc;

clear all;

wc=input('enter the value of cut off frequency');

N=input('enter the value of filter');

alpha=(N-1)/2;

eps=0.001;

%Rectangular Window

n=0:1:N-1;

hd=(sin(pi*(n-alpha+eps))-sin((n-alpha+eps)*wc))./(pi*(n-alpha+eps));

hn=hd

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h));

hold on

%Hamming Window

n=0:1:N-1;

wh=0.54-0.46*cos((2*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h),'ms');

hold off;

hold on

%Hanning Window

n=0:1:N-1;

wh=0.5-0.5*cos((2*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h),'blue');

hold off;

hold on

%Blackman Window

n=0:1:N-1;

wh=0.42-0.5*cos((2*pi*n)/(N-1))-0.08*cos((4*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);

plot(w/pi,abs(h),'green');

hold off;



OUTPUT FOR HIGH PASS FILTER:

enter the value of cut off frequency1.6

enter the value of filter11

hn =

Columns 1 through 8

-0.0628 -0.0094 0.1061 0.0085 -0.3175 0.4907 -0.3188 0.0100


0.1053 -0.0091 -0.0632

hn =

Columns 1 through 8

-0.0050 -0.0016 0.0422 0.0058 -0.2896 0.4907 -0.2908 0.0068


0.0419 -0.0015 -0.0051

hn =

Columns 1 through 8

0 -0.0009 0.0366 0.0056 -0.2872 0.4907 -0.2884 0.0066


0.0364 -0.0009 0

hn =

Columns 1 through 8

0.0100 0.0001 0.0350 0.0055 -0.2539 0.4122 -0.2550 0.0064


0.0348 0.0001 0.0101



GRAPH FOR HPF:



DESIGN FOR BAND PASS FILTER:

clc;

Wc1=input('enter the value of Wc1=');


N=input('enter the value of N=');

alpha=(N-1)/2;

eps=0.001;

%Rectangular Window

n=0:1:N-1;

hd=(sin(Wc1*(n-alpha+eps))-sin(Wc2*(n-alpha+eps)*pi))./((n-alpha+eps)*pi);

hn=hd

W=0:0.01:pi;

h=freqz(hn,1,W);

plot(W/pi,abs(h));

hold on;

%Hamming Window

n=0:1:N-1;

Wn=0.54-0.46*cos((2*pi*n)/(N-1));

hn=hd.*Wn

W=0:0.01:pi;

h=freqz(hn,1,W);

plot(W/pi,abs(h),'green');

hold on;

%Hanning Window

n=0:1:N-1;

Wn=0.5-0.5*cos((2*pi*n)/(N-1));

hn=hd.*Wn

W=0:0.01:pi;

h=freqz(hn,1,W);

plot(W/pi,abs(h),'red');

hold off;

%Blackman Window

n=0:1:N-1;

wh=042-0.5*cos((2*pi*n)/(N-1))-0.08*cos((4*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);


hold off;



OUTPUT FOR BANG PASS FILTER:

enter the value of Wc1=0.95


enter the value of N=11

hn =

Columns 1 through 8

-0.0639 0.0272 -0.0314 0.0566 0.5616 -1.0976 0.5617 0.0575


-0.0323 0.0268 -0.0633

hn =

Columns 1 through 8

-0.0051 0.0046 -0.0125 0.0386 0.5123 -1.0976 0.5123 0.0393


-0.0129 0.0045 -0.0051



hn =

Columns 1 through 8

0 0.0026 -0.0108 0.0370 0.5080 -1.0976 0.5081 0.0377


-0.0112 0.0026 0

hn =

Columns 1 through 8

-2.6472 1.1293 -1.3160 2.3884 23.8004 -46.5602 23.8044 2.4297


-1.3548 1.1148 -2.6228



GRAPH FOR BAND PASS FILTER:



DESIGN FOR BAND STOP FILTER:

clc;



N=input('enter the value of N=');

alpha=(N-1)/2;

eps=0.001;

%Rectangular Window

n=0:1:N-1;

hd=(sin(Wc1*(n-alpha+eps))-sin(Wc2*(n-alpha+eps)*pi))./((n-alpha+eps)*pi);

hn=hd

W=0:0.01:pi;

h=freqz(hn,1,W);

plot(W/pi,abs(h));

hold on;

%Hamming Window

n=0:1:N-1;

Wn=0.54-0.46*cos((2*pi*n)/(N-1));

hn=hd.*Wn

W=0:0.01:pi;

h=freqz(hn,1,W);


hold on;

%Hanning Window

n=0:1:N-1;

Wn=0.5-0.5*cos((2*pi*n)/(N-1));

hn=hd.*Wn

W=0:0.01:pi;

h=freqz(hn,1,W);

plot(W/pi,abs(h),'red');

hold off;

%Blackman Window

n=0:1:N-1;

wh=0.42-0.5*cos((2*pi*n)/(N-1))-0.08*cos((4*pi*n)/(N-1));

hn=hd.*wh

w=0:0.01:pi;

h=freqz(hn,1,w);


hold off;



OUTPUT FOR BSF:



enter the value of N=11

hn =

Columns 1 through 8

0.1054 -0.0500 -0.1986 0.0528 0.6325 -1.0544 0.6314 0.0538


-0.1986 -0.0505 0.1055

hn =

Columns 1 through 8

0.0084 -0.0084 -0.0790 0.0360 0.5770 -1.0544 0.5760 0.0367


-0.0790 -0.0085 0.0084



hn =

Columns 1 through 8

0 -0.0048 -0.0686 0.0346 0.5721 -1.0544 0.5711 0.0352


-0.0686 -0.0048 0

hn =

Columns 1 through 8

-0.0169 0.0005 -0.0656 0.0338 0.5059 -0.8857 0.5050 0.0344


-0.0656 0.0005 -0.0169



GRAPH FOR BSF:

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