%Butterworth lowpass filter

clc;clear all;close all;

alphap=input('Enter the passband attenuation for lowpass filter (in dB):');

alphas=input('Enter the stopband attenuation(in dB) for lowpass filter:');

wp=input('Enter the passband digital frequency in rad/sample for lowpass filter:');

ws=input('Enter the stopband digital frequency in rad/sample for lowpass filter:');

[N,wc]=buttord(wp/pi,ws/pi,alphap,alphas);% returns the lowest order of the digital Butterworth filter

[b,a]=butter(N,wc);

w=0:.01:pi;

[h,w]=freqz(b,a,w);

figure,

M=20*log10(abs(h));

an=angle(h);

an=an*(180/pi);

figure,

subplot(2,1,1);

plot(w/pi,M,'r','LineWidth',2);

title('magnitude response of Butterworth lowpass filter:');

xlabel('(a) Normalized freq. -->');

ylabel('Gain in dB-->');

grid on;

subplot(2,1,2);

plot(w/pi,an,'r','LineWidth',2);

title('phase response of Butterworth lowpass filter is:');

xlabel('(b) Normalized freq. -->');

ylabel('Phase in degrees-->');

grid on;
%Butterworth highpass filter
clear all;

alphap=input('Enter the passband attenuation (in dB) for highpass filter:');

alphas=input('Enter the stopband attenuation(in dB) for highpass filter:');

wp=input('Enter the passband digital frequency in rad/sample for highpass filter:');

ws=input('Enter the stopband digital frequency in rad/sample for highpass filter:');

[N,wc]=buttord(wp/pi,ws/pi,alphap,alphas);% returns the lowest order of the digital Butterworth filter

[b,a]=butter(N,wc,'high');

w=0:.01:pi;

[h,w]=freqz(b,a,w);

M=20*log10(abs(h));

an=angle(h);

an=an*(180/pi);

figure,

subplot(2,1,1);

plot(w/pi,M,'g','LineWidth',2);

title('magnitude response of Butterworth highpass filter:');

xlabel('(a) Normalized freq. -->');

ylabel('Gain in dB-->');

grid on;

subplot(2,1,2);

plot(w/pi,an,'g','LineWidth',2);

title('phase response of Butterworth highpass filter is:');

xlabel('(b) Normalized freq. -->');

ylabel('Phase in degrees-->');

grid on;
%Butterworth bandpass filter
clear all;

alphap=input('Enter the passband attenuation (in dB) for bandpass filter:');

alphas=input('Enter the stopband attenuation(in dB) for bandpass filter:');

wp=input('Enter the passband digital frequency range in rad/sample for bandpass filter:');

ws=input('Enter the stopband digital frequency range in rad/sample for bandpass filter:');

[N,wc]=buttord(wp/pi,ws/pi,alphap,alphas);% returns the lowest order of the digital Butterworth filter

[b,a]=butter(N,wc);

w=0:.01:pi;

[h,w]=freqz(b,a,w);

figure,

M=20*log10(abs(h));

an=angle(h);

an=an*(180/pi);

figure,

subplot(2,1,1);

plot(w/pi,M,'b','LineWidth',2);

title('magnitude response of Butterworth bandpass filter:');

xlabel('(a) Normalized freq. -->');

ylabel('Gain in dB-->');

grid on;

subplot(2,1,2);

plot(w/pi,an,'b','LineWidth',2);

title('phase response of Butterworth bandpass filter is:');

xlabel('(b) Normalized freq. -->');

ylabel('Phase in degrees-->');

grid on;
%Butterworth bandstop filter
clear all;

alphap=input('Enter the passband attenuation (in dB) for bandstop filter:');

alphas=input('Enter the stopband attenuation(in dB) for bandstop filter:');

wp=input('Enter the passband digital frequency range in rad/sample for bandstop filter:');

ws=input('Enter the stopband digital frequency range in rad/sample for bandstop filter:');

[N,wc]=buttord(wp/pi,ws/pi,alphap,alphas);% returns the lowest order of the digital Butterworth filter

[b,a]=butter(N,wc,'stop');

w=0:.01:pi;

[h,w]=freqz(b,a,w);

figure,

M=20*log10(abs(h));

an=angle(h);

an=an*(180/pi);

figure,

subplot(2,1,1);

plot(w/pi,M,'k','LineWidth',2);

title('magnitude response of Butterworth bandstop filter:');

xlabel('(a) Normalized freq. -->');

ylabel('Gain in dB-->');

grid on;

subplot(2,1,2);

plot(w/pi,an,'k','LineWidth',2);

title('phase response of Butterworth bandstop filter is:');

xlabel('(b) Normalized freq. -->');

ylabel('Phase in degrees-->');

grid on;
Matlab Coding for HPF:
clc;

clear all;

close all;

disp('enter the IIR filter design specifications');

rp=input('enter the passband ripple');

rs=input('enter the stopband ripple');

wp=input('enter the passband freq');

ws=input('enter the stopband freq');

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

w1=2*wp/fs;

w2=2*ws/fs;

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

disp('Frequency response of IIR HPF is:');

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

w=0:.01:pi;

[h,om]=freqs(b,a,w);

m=20*log10(abs(h));

an=angle(h);

figure,subplot(2,1,1);

plot(om/pi,m);

title('magnitude response of IIR filter is:');

xlabel('(a) Normalized freq. -->');

ylabel('Gain in dB-->');

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

title('phase response of IIR filter is:');

xlabel('(b) Normalized freq. -->');

ylabel('Phase in radians-->');
MATLAB Coding for LPF and Output waveforms:
%FIR filter design
clc;
close all;clear all;
rp=.05;
rs=.04;
fp=1500;
fs=2000;
Fs=9000;
wp=2*fp/Fs
ws=2*fs/Fs
num=-20*log10(sqrt(rp*rs))-13;
dem=14.6*(fs-fp)/Fs;
N=ceil(num/dem)
W1=boxcar(N+1);
b1=fir1(N,wp,W1);
w=0:0.01:pi;
H1=freqz(b1,1,w);
subplot(1,2,1);
plot(w/pi,abs(H1),'r','LineWidth',2);
M1=20*log10(abs(H1));
grid on
hold on
W2=hanning(N+1);
b3=fir1(N,wp,W2);
w=0:0.01:pi;
H2=freqz(b3,1,w);
subplot(1,2,1);
plot(w/pi,abs(H2),'b','LineWidth',2);
M2=20*log10(abs(H2));
grid on
hold on
W3=blackman(N+1);
b3=fir1(N,wp,W3);
w=0:0.01:pi;
H3=freqz(b3,1,w);
subplot(1,2,1);
plot(w/pi,abs(H3),'g','LineWidth',2);
M3=20*log10(abs(H3));
hold on
legend('rectangular window','Hanning window','Blackman window');
xlabel('Frequency w/pi------>');
ylabel('magnitude of H(jw)------>');
hold off;
subplot(1,2,2);
plot(w/pi,M1,'r','LineWidth',2);
grid on;
hold on;
plot(w/pi,M2,'b','LineWidth',2);
hold on;
plot(w/pi,M3,'g','LineWidth',2);
legend('rectangular window','Hanning window','Blackman window');
xlabel('Frequency w/pi------>');
ylabel('magnitude of H(jw) in dB------>');

A) Convolution Using Overlap Save Method

clc;

clear all; close all;

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

h = input('Enter the sequence h(n) = ');

disp('Length of input sequence x(n)');

Ls = length(x)

disp('Length of impulse sequence h(n)');

M = length(h)

disp('Block size');

L=input('Enter the block size:');

Ns = Ls+M-1;
N=L+M-1; % the size of h and block of x should be made equal to size N for carrying out curcular convolution. The
size of the result of circular convolution is also N.

disp('Impulse sequence h(n) after padding zeros');

h = [h zeros(1,L-1)] % Size of h(n) is M. So, L-1 zeros padded so that size of h(n) becomes L+M-1=N

disp('Input sequence x(n) after padding M-1 zeros in the beginning');

nx=[zeros(1,M-1) x]

Ls1=length(nx)

for i = 1:L:Ls1

if ((Ls1-i+1)<N)

nx=[nx zeros(1,N-(Ls1-i+1))];

end

end

disp('Input sequence x(n) after padding zeros at the end');

nx

disp('Length of Input sequence x(n) after padding zeros at the end');

Ls1=length(nx)

R=mod(Ls1,L);

disp('number of blocks');

K=floor(Ls1/L)

y = zeros(1,Ls1) % size of the overall result

for i = 1:L:(K*L)

disp('Block starts at');

disp('Sequence of the block is');

x1 = nx(i:i+N-1)% x1 is the block

disp('circular convolution of block sequence with h(n) is');

y1 = round(cconv(x1,h,N))

disp('output sequence y(n) after adding result of circular convolution of block is')

y(i:(i+L-1)) = y1(M:N)

end

subplot(3,1,1);

stem(x(1:Ls),'fill');

grid on;
title('Input Sequence x(n)');

xlabel('Time --->');

ylabel('Amplitude --->');

subplot(3,1,2);

stem(h(1:M),'fill','k');

grid on;

title('Input Sequence h(n)');

xlabel('Time --->');

ylabel('Amplitude --->');

subplot(3,1,3);

disp('Convolution Using Overlap Save Method = ');

disp(y(1:Ns));

stem(y(1:Ns),'fill','r');

grid on;

title('Convolution Using Overlap Save Method');

xlabel('Time --->');

ylabel('Amplitude --->');

B) Convolution Using Overlap Add Method

clc;

close all;clear all;

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

h = input('Enter the sequence h(n) = ');

disp('Length of input sequence x(n)');

Ls = length(x)

disp('Length of impulse sequence h(n)');

M = length(h)

L=input('Enter the block size:');

Ns = Ls+M-1; %this is the length of the final sequence after adding all blocks

disp('Impulse sequence h(n) after padding zeros');

h = [h zeros(1,L-1)] % L-1 zeros appended because for doing circular convolution the length of each sequence should
be L+M-1
disp('Number of blocks of x(n)');

K=floor(Ls/L)

R=mod(Ls,L);

N=L+M-1;

if R==0

nx=x;

else

nx=[x zeros(1,L-R)];

end

disp('Input sequence x(n) after padding zeros');

nx

disp('Length of sequence x(n) after padding zeros');

Ls1=length(nx)

Ns1 = Ls1+M-1;

disp('Number of blocks of x(n) after padding zeros');

K1=floor(Ls1/L)

y = zeros(1,Ns1); % size of the overall result

for i = 1:L:(K1*L)

disp('Block starts at');

disp('Sequence of the block is');

x1=[nx(i:i+L-1) zeros(1,M-1)]

disp('circular convolution of block sequence with h(n) is');

yblock = round(cconv(x1,h,N))

disp('output sequence y(n) after adding result of circular convolution of block is');

y(i:i+N-1) = y(i:i+N-1)+yblock(1:N)

end

subplot(3,1,1);

stem(x(1:Ls),'fill');

grid on;

title('Input Sequence x(n)');

xlabel('Time --->');

ylabel('Amplitude --->');

subplot(3,1,2);
stem(h(1:M),'fill','k');

grid on;

title('Input Sequence h(n)');

xlabel('Time --->');

ylabel('Amplitude --->');

subplot(3,1,3);

disp('Convolution Using Overlap Add Method = ');

disp(y(1:Ns));

stem(y(1:Ns),'fill','r');

grid on;

title('Convolution Using Overlap Add Method');

xlabel('Time --->');

ylabel('Amplitude --->');