Prove all Convolution Properties by the following, (Use x[n] and h1[n] for Commutative property) 𝒙[𝒏] = 𝟐[𝒖[𝒏 + 𝟐] − 𝒖[𝒏 − 𝟏𝟐]]. And the impulse response is given by, 𝒉𝟏 [𝒏] = 𝟎. 𝟗[𝒖[𝒏 − 𝟐] − 𝒖[𝒏 − 𝟏𝟑]]. 𝒉𝟐 [𝒏] = 𝟎. 𝟒[𝒖[𝒏 + 𝟏𝟐] + 𝒖[𝒏 − 𝟐]].
Source code
t=0:5;% here i use the time 0 to 5
x=2*[heaviside(t+2)-heaviside(t-12)]
h1=0.9*[heaviside(t-2)-heaviside(t-3)]
h2=0.4*[heaviside(t+12)+heaviside(t-2)]
m=length(x);
n=length(h1);
l=length(h2);
a1=m+n-1;
a2=n+l-1;
a3=a2+m-1;
a4=l+a1-1;
a5=m+l-1;
a6=a5+a1-1;
a=t:1:t+m-1;
b=t:1:t+n-1;
c=t:1:t+l-1;
c1=b+c;
c2=a+b;
c3=a+c1;
c4=c+c2;
c5=a+c;
c6=c2+c5;
d=c2:1:c2+a1-1;
d1=c1:1:c1+a2-1;
d2=c3:1:c3+a3-1;
d3=c4:1:c4+a4-1;
d4=c5:1:c5+a5-1;
d5=d4+d;
%commutative property: x[n]*h1[n]=h1[n]*x[n]
%L.H.S=x[n]*h1[n];
w=conv(x,h1)
subplot(3,2,1)
stem(d,w)
xlabel('Time');
ylabel('Magnitude');
title('commutative property x[n]*h1[n]');
%R.H.S=h1[n]*x[n]
w1=conv(h1,x)
subplot(3,2,2)
stem(d,w1)
xlabel('Time');
ylabel('Magnitude');
title('commutative property h1[n]*x[n]');
%ASSOCIATIVE PROPERTY:
x[n]*(h1[n]*h2[n])=(x[n]*(h1[n])*h2[n]
%L.H.S=x[n]*(h1[n]*h2[n])
w2=conv(h1,h2) %h1[n]*h2[n]
w3=conv(x,w2) %x[n]*(h1[n]*h2[n])
subplot(3,2,3)
stem(d2,w3)
xlabel('Time');
ylabel('Magnitude');
title('ASSOCIATIVE PROPERTY x[n]*(h1[n]*h2[n])');
%R.H.S=(x[n]*(h1[n])*h2[n]
w4=conv(w,h2)
subplot(3,2,4)
stem(d3,w4)
xlabel('Time');
ylabel('Magnitude');
title('ASSOCIATIVE PROPERTY (x[n]*(h1[n])*h2[n])');
% DISTRIBUTIVE PROPERTY: x[n]+(h1[n]*h2[n])=x[n]*h1[n]+x[n]*h2[n])
%L.H.S=x[n]+(h1[n]*h2[n])
h3=h1+h2;
w5=conv(x,h3);
subplot(3,2,5)
stem(d5,w5)
xlabel('Time');
ylabel('Magnitude');
title('DISTRIBUTIVE PROPERTY x[n]+(h1[n]*h2[n])');
%R.H.S=x[n]*h1[n]+x[n]*h2[n])
w6=conv(x,h1); %x[n]*h1[n]
w7=conv(x,h2); %x[n]*h2[n]
w8=w6+w7;
subplot(3,2,6)
stem(d5,w8)
xlabel('Time');
ylabel('Magnitude');
title('DISTRIBUTIVE PROPERTY x[n]*h1[n]+x[n]*h2[n])');
Result:
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