Notes

Bsf
N=7;
w=0:0.01:pi;
n=0:1:N-1;
eps=0.001;
alpha=(N-1)/2;
wc1=pi/3;
wc2=(2*pi)/3;
A=sin(wc1*(n-alpha+eps));
B=sin(wc2*(n-alpha+eps));
C=sin(pi*(n-alpha+eps));
D=pi*(n-alpha+eps);
hl=A./D;
hh=(C-B)./D;
hbs=(A-B+C)./D;
hbp=(B-A)./D;
wb=blackman(N);
whm=hamming(N);
whn=hanning(N);
wr=boxcar(N);%boxcar fn is which is zero over entire real line except an interval at which it equals A
hbsb=hbs.*transpose(wb);
hbshm=hbs.*transpose(whm);
hbshn=hbs.*transpose(whn);
hbshr=hbs.*transpose(wr);
hbsbf=freqz(hbsb,1,w);
hbshmf=freqz(hbshm,1,w);
hbshnf=freqz(hbshn,1,w);
hbshrf=freqz(hbshr,1,w);
subplot(2,4,1)
plot(w,abs(hbsbf));
title('bsf using blackman window');
xlabel('w');
ylabel('magnitude of bsf');
grid on
subplot(2,4,2)
plot(w,abs(hbshmf),'linewidth',2);
title('bsf using hamming window');
xlabel('w');
ylabel('magnitude of bsf');
subplot(2,4,3)
plot(w,abs(hbshnf),'linewidth',4);
title('bsf using hanning window');
xlabel('w');
ylabel('magnitude of bsf');
subplot(2,4,4)
plot(w,abs(hbshrf),'linewidth',6);
title('bsf using boxcar');
xlabel('w');
ylabel('magnitude of bsf');
subplot(2,4,5)
plot(w,angle(hbsbf));
xlabel('w');
ylabel('phase of bsf');
grid on
subplot(2,4,6)
plot(w,angle(hbshmf),'linewidth',2);
xlabel('w');
ylabel('phase of bsf');
grid on
subplot(2,4,7)
plot(w,angle(hbshnf),'linewidth',4);
xlabel('w');
ylabel('phase of bsf');
grid on
subplot(2,4,8)
plot(w,angle(hbshrf),'linewidth',6);
xlabel('w');
ylabel('phase of bsf');
grid on




Bpf
N=7;
w=0:0.01:pi;
n=0:1:N-1;
eps=0.001;
alpha=(N-1)/2;
wc1=pi/3;
wc2=(2*pi)/3;
A=sin(wc1*(n-alpha+eps));
B=sin(wc2*(n-alpha+eps));
C=sin(pi*(n-alpha+eps));
D=pi*(n-alpha+eps);
hl=A./D;
hh=(C-B)./D;
hbs=(A-B+C)./D;
hbp=(B-A)./D;
wb=blackman(N);
whm=hamming(N);
whn=hanning(N);
wr=boxcar(N);%boxcar fn is which is zero over entire real line except an interval at which it equals A
hbpb=hbp.*transpose(wb);
hbphm=hbp.*transpose(whm);
hbphn=hbp.*transpose(whn);
hbphr=hbp.*transpose(wr);
hbpbf=freqz(hbpb,1,w);
hbphmf=freqz(hbphm,1,w);
hbphnf=freqz(hbphn,1,w);
hbphrf=freqz(hbphr,1,w);
subplot(2,4,1)
plot(w,abs(hbpbf));
title('bpf using blackman window');
xlabel('w');
ylabel('magnitude of bpf');
grid on
subplot(2,4,2)
plot(w,abs(hbphmf),'linewidth',2);
title('bpf using hamming window');
xlabel('w');
ylabel('magnitude of bpf');
subplot(2,4,3)
plot(w,abs(hbphnf),'linewidth',4);
title('bpf using hanning window');
xlabel('w');
ylabel('magnitude of bpf');
subplot(2,4,4)
plot(w,abs(hbphrf),'linewidth',6);
title('bpf using boxcar');
xlabel('w');
ylabel('magnitude of bpf');
subplot(2,4,5)
plot(w,angle(hbpbf));
xlabel('w');
ylabel('phase of bpf');
grid on
subplot(2,4,6)
plot(w,angle(hbphmf),'linewidth',2);
xlabel('w');
ylabel('phase of bpf');
grid on
subplot(2,4,7)
plot(w,angle(hbphnf),'linewidth',4);
xlabel('w');
ylabel('phase of bpf');
grid on
subplot(2,4,8)
plot(w,angle(hbphrf),'linewidth',6);
xlabel('w');
ylabel('phase of bpf');
grid on

circular conv
a=[1 2 1 1];
b=[1 1 1 2];
c=ifft((fft(a,4).*fft(b,4)),4);
d=cconv(a,b,4);
subplot(2,2,1)
stem(a);
ylabel('Amplitude');
title('Signal 1');
grid;
subplot(2,2,2)
stem(b);
ylabel('Amplitude');
title('Signal 2');
grid;
subplot(2,2,3)
stem(d);
ylabel('Amplitude');
title('Convolution using CCONV');
grid;
subplot(2,2,4)
stem(c);
ylabel('Amplitude');
title('Convolution using FFT');
grid;



1) Linearity:
clf;
n = 0:40;
a = 2;
b = -3;
x1 = cos(2*pi*0.1*n);
x2 = cos(2*pi*0.4*n);
x = a*x1+b*x2;
num = [1 -2];
den = [1 0.5];
ic = [0]; %set zero initial conditions
y1 = filter(num,den,x1,ic); %compute the output y1[n]
y2 = filter(num,den,x2,ic); %compute the output y2[n]
y = filter(num,den,x,ic); %compute the output y[n]
yt = a*y1+b*y2;
d = y-yt; %compute the difference output d[n]
%plot the output and the difference signal
subplot(3,1,1)
stem(n,y);
ylabel('Amplitude');
title('Output due to weighted input:a.x_{1}[n]+b.x_{2}[n]');
subplot(3,1,2);
stem(n,yt);
ylabel('Amplitude');
title('Weighted Output: a.y_{1}[n]+b.y_{2}[n]');
subplot(3,1,3)
stem(n,d);
xlabel('Time index n');
ylabel('Amplitude');
title('Difference Signal');


2) Time Variant Invariant:
% Generate the input sequences
clf;
n=0:40;
D=10;
a=3;
b=-2;
x=a*cos(2*pi*0.1*n) + b*cos(2*pi*0.4*n);
xd=[zeros(1,D) x];
num=[1 -2];
den = [1 0.5];
ic = [0]; % Set initial conditions
% Compute the output y[n]
y = filter(num,den,x,ic);
% Compute the output yd[n]
yd = filter(num,den,xd,ic);
% Compute the difference output d[n]
d = y - yd(1+D : 41+D);
% Plot the outputs
subplot(3,1,1);
stem(n,y);
ylabel('Amplitude');
title('Output y[n]');
grid;
subplot(3,1,2)
stem(n,yd(1:41));
ylabel('Amplitude');
title(['Output due to Delayed Input x[n-D',num2str(D),']']);
grid;
subplot(3,1,3)
stem(n,d);
xlabel('Time Index n');
ylabel('Amplitude');
title('Difference Signal');
grid;

3) Time Scaling:
n = 0:1:10;
xn1 = (1/2).^n;
subplot(2,1,1)
stem(n,xn1);
xlabel('time');
ylabel('x1[n]');
title('discrete signal ');
yn1=(1/2).^(2*n);
subplot(2,1,2)
stem(n,yn1);
xlabel('time');
ylabel('y1[n]');
title('scaled signal');

sampling
clf;
n = 0:50;
x = sin(2*pi*0.12*n);
y = zeros(1, 3*length(x));
y([1: 3: length(y)]) = x;
subplot(2,2,1)
stem(n,x);
title('Input Sequence');
xlabel('Time index n');
ylabel('Amplitude');
subplot(2,2,2)
stem(n,y(1:length(x)));
title('Output Sequence');
xlabel('Time index n');
ylabel('Amplitude');
n = 0: 49;
m = 0: 50*3 - 1;
x = sin(2*pi*0.042*m);
y = x([1: 3: length(x)]);
subplot(2,2,3)
stem(n, x(1:50));
axis([0 50 -1.2 1.2]);
title('Input Sequence');
xlabel('Time index n');
ylabel('Amplitude');
subplot(2,2,4)
stem(n, y);
axis([0 50 -1.2 1.2]);
title('Output Sequence');
xlabel('Time index n');
ylabel('Amplitude');


sammpling
t = 0:1/2000:.02;
x = cos(2*pi*60*t); % approx. to continuous-time
t240 = 0:1/240:.02;
n240 = 0:length(t240)-1;
x240 = cos(2*pi*60/240*n240); % fs = 240 Hz
axis([0 4.8 -1 1]) % axis scale since .02*240 = 4.8
t1000 = 0:1/1000:.02;
n1000 = 0:length(t1000)-1;
x1000 = cos(2*pi*60/1000*n1000); % fs = 1000 Hz
subplot(3,1,1)
plot(t,x);
xlabel('time');
ylabel('x[t]');
title('cosine signal');
subplot(3,1,2)
stem(n240,x240);
xlabel('time');
ylabel('x1[n]');
title('low sampling rate');
subplot(3,1,3)
stem(n1000,x1000);
xlabel('time');
ylabel('x2[n]');
title('high sampling rate');




// No Idea What This Is But Since It Was In The File It Has been Included Here
t = 0:1/2000:2;
x1 = cos(pi*t);
x2 = cos(3*pi*t);
tn = 0:1:2;
xn1 = cos(pi*tn);
xn2 = cos(3*pi*tn);
subplot(2,2,1)
plot(t,x1);
xlabel('time');
ylabel('x1[t]');
title('signal 1');
subplot(2,2,2)
plot(t,x2);
xlabel('time');
ylabel('x2[t]');
title('signal 2');
subplot(2,2,3)
stem(tn,xn1);
xlabel('time');
ylabel('x1[n]');
title('discrete signal 1');
subplot(2,2,4)
stem(tn,xn2);
xlabel('time');
ylabel('x2[n]');
title('discrete signal 2');

dft idft
x=input('Enter the input sequence : ');
N=input('Enter the value of N : ');
if length(x)<N
x=[x zeros(1,(N-length(x)))];
end
w=exp(-2*pi*i/N);
for n=1:N
for k=1:N
W(n,k)=w.^((n-1)*(k-1));
end
end
X=W*x';
n=0:N-1;
subplot(2,2,1),stem(n,x);
title('Zero Padded Discrete Time Signal x[n]');
xlabel('n');
ylabel('Amplitude');
subplot(2,2,2);
stem(n,abs(X'));
title('Magnitude Plot of DFT of x[n] : X(k)');
xlabel('Frequency');
ylabel('Amplitude');
subplot(2,2,3);
stem(n,angle(X'));
title('Phase Plot of DFT of x[n] : X(k)');
xlabel('Frequency');
ylabel('Amplitude');
w=exp(2*pi*i/N);
for n=1:N
for k=1:N
W(n,k)=w.^((n-1)*(k-1));
end
end
x=(W*X)/N;
n=0:N-1;
subplot(2,2,4);
stem(n,x');
title('iDFT Plot of DFT of x[n]');
xlabel('n');
ylabel('Amplitude');

quantization
t=0:0.0005:1/60;
y=sin(2*pi*60*t);
subplot(4,1,1)
plot(t,y);
xlabel('t');
ylabel('Magnitude');
t1000=0:1/1000:1/60;
n1000=0:length(t1000)-1;
x1000=sin(2*pi*60/1000*n1000);
subplot(4,1,2)
stem(n1000,x1000);
xlabel('n');
ylabel('Magnitude');
l=-1.125:0.125:1.125;
for n=1:18
lmid(n)=(l(n)+l(n+1))/2;
end
for i=1:17
for j=1:17
if (lmid(j)<x1000(i) & x1000(i)<=lmid(j+1))
xq1000(i)=(lmid(j)+lmid(j+1))/2;
end
end
end
for h=1:17
d(h)=abs(abs(x1000(h))-abs(xq1000(h)));
end
subplot(4,1,3)
stem(n1000,xq1000);
xlabel('n');
ylabel('Magnitude');
subplot(4,1,4)
stem(n1000,d);
xlabel('n');
ylabel('Magnitude');

HPF
N=7;
w=0:0.01:pi;
n=0:1:N-1;
eps=0.001;
alpha=(N-1)/2;
wc1=pi/3;
wc2=(2*pi)/3;
A=sin(wc1*(n-alpha+eps));
B=sin(wc2*(n-alpha+eps));
C=sin(pi*(n-alpha+eps));
D=pi*(n-alpha+eps);
hl=A./D;
hh=(C-B)./D;
hbs=(A-B+C)./D;
hbp=(B-A)./D;
wb=blackman(N);
whm=hamming(N);
whn=hanning(N);
wr=boxcar(N);%boxcar fn is which is zero over entire real line except an interval at which it equals A
hhb=hh.*transpose(wb);
hhhm=hh.*transpose(whm);
hhhn=hh.*transpose(whn);
hhhr=hh.*transpose(wr);
hhbf=freqz(hhb,1,w);
hhhmf=freqz(hhhm,1,w);
hhhnf=freqz(hhhn,1,w);
hhhrf=freqz(hhhr,1,w);
subplot(2,4,1)
plot(w,abs(hhbf));
title('hpf using blackman window');
xlabel('w');
ylabel('magnitude of hpf');
grid on
subplot(2,4,2)
plot(w,abs(hhhmf),'linewidth',2);
title('hpf using hamming window');
xlabel('w');
ylabel('magnitude of hpf');
subplot(2,4,3)
plot(w,abs(hhhnf),'linewidth',4);
title('hpf using hanning window');
xlabel('w');
ylabel('magnitude of hpf');
subplot(2,4,4)
plot(w,abs(hhhrf),'linewidth',6);
title('hpf using boxcar');
xlabel('w');
ylabel('magnitude of hpf');
subplot(2,4,5)
plot(w,angle(hhbf));
xlabel('w');
ylabel('phase of hpf');
grid on
subplot(2,4,6)
plot(w,angle(hhhmf),'linewidth',2);
xlabel('w');
ylabel('phase of hpf');
grid on
subplot(2,4,7)
plot(w,angle(hhhnf),'linewidth',4);
xlabel('w');
ylabel('phase of hpf');
grid on
subplot(2,4,8)
plot(w,angle(hhhrf),'linewidth',6);
xlabel('w');
ylabel('phase of hpf');
grid on

Iir butterworth bandpass
Rs=input('Accepted ripples in passband in dB : ');
Rp=input('Accepted ripples in stopband in dB : ');
Fs=input('Stopband frequncy in Hz : ');
Fp=input('Passband frequncy in Hz : ');
F=input('Enter sampling frequency : ');
Ws=2*Fs/F; %Normalized edge frequencies
Wp=2*Fp/F;
[n,Wn]=buttord(Wp,Ws,Rp,Rs);
[b,a] = butter(n,Wn); %[b,a] = butter(n,Wn,'ftype')
w=0:0.001:pi;
[h,w] = freqz(b,a,w);
Magh=abs(h);
MagdB=20*log10(Magh);
Phang=angle(h);
plot(w/pi,MagdB);
title('Magnitude Plot of Butterworth Filter (PassBand) in dB');
xlabel('Normalized Frequency (Hz)');
ylabel('Magnitude (dB)');

Iir band stop
Rs=input('Accepted ripples in passband in dB : ');
Rp=input('Accepted ripples in stopband in dB : ');
Fs=input('Stopband frequncy in Hz : ');
Fp=input('Passband frequncy in Hz : ');
F=input('Enter sampling frequency : ');
Ws=2*Fs/F; %Normalized edge frequencies
Wp=2*Fp/F;
[n,Wn]=buttord(Wp,Ws,Rp,Rs);
[b,a] = butter(n,Wn,'stop'); %[b,a] = butter(n,Wn,'ftype')
w=0:0.001:pi;
[h,w] = freqz(b,a,w);
Magh=abs(h);
MagdB=20*log10(Magh);
Phang=angle(h);
plot(w/pi,MagdB);
title('Magnitude Plot of Butterworth Filter (StopBand) in dB');
xlabel('Normailzed Frequency (Hz)');
ylabel('Magnitude (dB)');

Iir butterworth high pass
Rs=input('Accepted ripples in passband in dB : ');
Rp=input('Accepted ripples in stopband in dB : ');
Fs=input('Stopband frequncy in Hz : ');
Fp=input('Passband frequncy in Hz : ');
F=input('Enter sampling frequency : ');
Ws=2*Fs/F; %Normalized edge frequencies
Wp=2*Fp/F;
[n,Wn]=buttord(Wp,Ws,Rp,Rs);
[b,a] = butter(n,Wn,'high'); %[b,a] = butter(n,Wn,'ftype')
w=0:0.001:pi;
[h,w] = freqz(b,a,w);
Magh=abs(h);
MagdB=20*log10(Magh);
Phang=angle(h);
plot(w/pi,MagdB);
title('Magnitude Plot of Butterworth Filter (HPF) in dB');
xlabel('Normalized Frequency (Hz)');
ylabel('Magnitude (dB)');

Iir butterlowpass
Rs=input('Accepted ripples in passband in dB : ');
Rp=input('Accepted ripples in stopband in dB : ');
Fs=input('Stopband frequncy in Hz : ');
Fp=input('Passband frequncy in Hz : ');
F=input('Enter sampling frequency : ');
Ws=2*Fs/F; %Normalized edge frequencies
Wp=2*Fp/F;
[n,Wn]=buttord(Wp,Ws,Rp,Rs);
[b,a] = butter(n,Wn,'low'); %[b,a] = butter(n,Wn,'ftype')
w=0:0.001:pi;
[h,w] = freqz(b,a,w);
Magh=abs(h);
MagdB=20*log10(Magh);
Phang=angle(h);
%subplot(2,1,1);
plot(w/pi,MagdB);
title('Magnitude Plot of Butterworth Filter (LPF) in dB');
xlabel('Normalized Frequency (Hz)');
ylabel('Magnitude (dB)');
%subplot(2,1,2);
%plot(w/pi,Phang);
%title('Phase Angle Plot of Butterworth Filter');
%xlabel('Frequency (Hz)');
%ylabel('Magnitude (dB)');

Iir chebyshev high
Rs=input('Accepted ripples in passband in dB : ');
Rp=input('Accepted ripples in stopband in dB : ');
Fs=input('Stopband frequncy in Hz : ');
Fp=input('Passband frequncy in Hz : ');
F=input('Enter sampling frequency : ');
Ws=2*Fs/F; %Normalized edge frequencies
Wp=2*Fp/F;
[n,Wn]=cheb1ord(Wp,Ws,Rp,Rs);
[b,a] = cheby1(n,Rp,Wn,'high'); %[b,a] = butter(n,Wn,'ftype')
w=0:0.001:pi;
[h,w] = freqz(b,a,w);
Magh=abs(h);
MagdB=20*log10(Magh);
Phang=angle(h);
plot(w/pi,MagdB);
title('Magnitude Plot of Chebychev Filter Type I (HPF) in dB');
xlabel('Normalized Frequency (Hz)');
ylabel('Magnitude (dB)');

Iir chebyshev lowpass
Rs=input('Accepted ripples in passband in dB : ');
Rp=input('Accepted ripples in stopband in dB : ');
Fs=input('Stopband frequncy in Hz : ');
Fp=input('Passband frequncy in Hz : ');
F=input('Enter sampling frequency : ');
Ws=2*Fs/F; %Normalized edge frequencies
Wp=2*Fp/F;
[n,Wn]=cheb1ord(Wp,Ws,Rp,Rs);
[b,a] = cheby1(n,Rp,Wn); %[b,a] = butter(n,Wn,'ftype')
w=0:0.001:pi;
[h,w] = freqz(b,a,w);
Magh=abs(h);
MagdB=20*log10(Magh);
Phang=angle(h);
plot(w/pi,MagdB);
title('Magnitude Plot of Chebychev Filter Type I (LPF) in dB');
xlabel('Normalized Frequency (Hz)');
ylabel('Magnitude (dB)');

Linear conv
clc;
n= input(“Enter n:”);
x1= input(“ Enter First Sequence:”);
m= input(“Enter m:”);
x2=input(“Enter Second Sequence:”);
k= min(n)+min(m):max(n)+max(m);
x3=conv(x1,x2);
stem(k,x3);
title(“Linear Conv”);
xlabel(“Time”);
ylabel(“Amplitude”);

LPF using different window

N=7;
w=0:0.01:pi;
n=0:1:N-1;
eps=0.001;
alpha=(N-1)/2;
wc1=pi/3;
wc2=(2*pi)/3;
A=sin(wc1*(n-alpha+eps));
B=sin(wc2*(n-alpha+eps));
C=sin(pi*(n-alpha+eps));
D=pi*(n-alpha+eps);
hl=A./D;
hh=(C-B)./D;
hbs=(A-B+C)./D;
hbp=(B-A)./D;
wb=blackman(N);
whm=hamming(N);
whn=hanning(N);
wr=boxcar(N);%boxcar fn is which is zero over entire real line except an interval at which it equals A
hlb=hl.*transpose(wb);
hlhm=hl.*transpose(whm);
hlhn=hl.*transpose(whn);
hlhr=hl.*transpose(wr);
hlbf=freqz(hlb,1,w);
hlhmf=freqz(hlhm,1,w);
hlhnf=freqz(hlhn,1,w);
hlhrf=freqz(hlhr,1,w);
subplot(2,4,1)
plot(w,abs(hlbf));
title('lpf using blackman window');
xlabel('w');
ylabel('magnitude of lpf');
grid on
subplot(2,4,2)
plot(w,abs(hlhmf),'linewidth',2);
title('lpf using hamming window');
xlabel('w');
ylabel('magnitude of lpf');
subplot(2,4,3)
plot(w,abs(hlhnf),'linewidth',4);
title('lpf using hanning window');
xlabel('w');
ylabel('magnitude of lpf');
subplot(2,4,4)
plot(w,abs(hlhrf),'linewidth',6);
title('lpf using boxcar');
xlabel('w');
ylabel('magnitude of lpf');
subplot(2,4,5)
plot(w,angle(hlbf));
xlabel('w');
ylabel('phase of lpf');
grid on
subplot(2,4,6)
plot(w,angle(hlhmf),'linewidth',2);
xlabel('w');
ylabel('phase of lpf');
grid on
subplot(2,4,7)
plot(w,angle(hlhnf),'linewidth',4);
xlabel('w');
ylabel('phase of lpf');
grid on
subplot(2,4,8)
plot(w,angle(hlhrf),'linewidth',6);
xlabel('w');
ylabel('phase of lpf');
grid on



%calculate z transform
x=(0.5.^n)- (0.4.^n);
syms z n;
Hz=ztrans(x)

%obtain pole-zero plot
r=input('enter the coefficients:');
p=input('enter the poles:');
k=input('enter the constant:');
[num,den]=residuez(r,p,k)
zplane(num,den)


output




%accept numerator and denominator coefficients of z-transform
%calculate inverse transform & plot
num=input('Enter numerator coefficents');
den=input('Enter denominator coefficents');
[r,p,k]=residuez(num,den)
N=length(p);
n=0:1:25;
x=0;
for j=1:1:N
r_mag=abs(r(j)); r_phase=angle(r(j));
p_mag=abs(p(j)); p_phase=angle(p(j));
term=r_mag.*(p_mag.^n).*cos((p_phase.*n) + r_phase);
x=x+term;
end;
I=[k zeros(1,(26-length(k)))];%impulse terms
x=x+I;
stem(n,x);
grid on;
xlabel('n-->');
ylabel('x[n]');






%plot the inverse transform using 'impz' function
b=input('Enter numerator coefficents');
a=input('Enter denominator coefficents');
impz(b,a)