Question

Circular vs. Linear Convolution

Consider sequences

(x[0], x[1], x[2], x[3], x[4], x[5], x[6], x[7])=(1,1,1,1,0,0,0,0)

and

(h[0], h[1], h[2], h[3], h[4], h[5], h[6], h[7])=(1,2,3,4,3,2,1,0)

where x[n]=0 for n ∉\{0, …, 7\} and h[n]=0 for n ∉\{0, ..., 7\}.

(a) Find the convolution of these two signals, and sketch the result.

(b) Find the 8-point circular convolution of these two signals, and sketch the result.

(c) Assume that each of the signals has been zero padded up to a length 16. Find the 16 -point circular convolution of these two zero-padded signals, and sketch the result.

Answer 1

close all,
clear all,
clc,

x = [1,1,1,1,0,0,0,0];
h = [1,2,3,4,3,2,1,0];

ConvLinear = conv(x,h);

figure,
subplot(3,1,1); stem(x,'filled'); title('X-Plot');
subplot(3,1,2); stem(h,'filled'); title('h-Plot');
subplot(3,1,3);
stem(ConvLinear,'filled')
ylim([0 (max(ConvLinear)+1)]);
title('Linear Convolution of x and h')

ConvCirc = cconv(x,h,8);
figure,
subplot(3,1,1); stem(x,'filled'); title('X-Plot');
subplot(3,1,2); stem(h,'filled'); title('h-Plot');
subplot(3,1,3);
stem(ConvCirc,'filled')
ylim([0 (max(ConvCirc)+1)]);
title('8-Point Circular Convolution of x and h')

x = [1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0];
h = [1,2,3,4,3,2,1,0,0,0,0,0,0,0,0,0];
ConvCirc = cconv(x,h,16);
figure,
subplot(3,1,1); stem(x,'filled'); title('X-Plot');
subplot(3,1,2); stem(h,'filled'); title('h-Plot');
subplot(3,1,3);
stem(ConvCirc,'filled')
ylim([0 (max(ConvCirc)+1)]);
title('16-Point Circular Convolution of x and h')

X-Plot 0.5 h-Plot 2 Linear Convolution of x and h 10

X-Plot 0.5 h-Plot 2 8-Point Circular Convolution of x and h 10

X-Plot 0.5 14 h-Plot 8 10 14 16-Point Circular Convolution of x and h 10 10 14

Output

x =   1 1 1 1 0 0 0 0

h =   1 2 3 4 3 2 1 0

Linear Conv. of X and H

ConvLinear =   1 3 6 10 12 12 10 6 3 1 0 0 0 0 0

8-Point Circular Conv. of X and H

ConvCirc =   4 4 6 10 12 12 10 6

16-Point Circular Conv. of X and H

ConvCirc =   1.0000 3.0000 6.0000 10.0000 12.0000 12.0000 10.0000 6.0000

3.0000 1.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000

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