%%
function convolution_test
close all
set(0, 'DefaultAxesFontSize', 18)
%
t1 = linspace(-3, 3, 50);
x1 = ones(size(t1));
x1(t1 < -1) = -1;
x1(t1 > 1) = -2;
%
t2 = linspace(-2.5, 2, 20);
x2 = sin(t2);
%
figure
plot(t1, x1, 'p-', t2, x2, 'p-')
title('signals before synchronization')
legend('x1', 'x2', 'Location', 'best')
%
[T1, X1, T2, X2, T, d] = prepareX(t1, x1, t2, x2);
%
figure
plot(T1, X1, 'p-', T2, X2, 'p-')
title('signals after synchronization')
legend('x1', 'x2', 'Location', 'best')
% computing covolution using conv function
tic
X = d * conv(X1, X2);
toc
% computing covolution using fft function
tic
X1 = [X1 zeros(1, size(T, 2) - size(X1, 2))];
X2 = [X2 zeros(1, size(T, 2) - size(X2, 2))];
X1_f = fft(X1);
X2_f = fft(X2);
X_f = X1_f .* X2_f;
Xf = d * real(ifft(X_f));
toc
%
figure
plot(T, X, T, Xf, 'o')
title('convolution of two signals')
legend('using conv', 'using fft', 'Location', 'best')
end
%%
function [T1, X1, T2, X2, T, d] = prepareX(t1, x1, t2, x2)
d1 = t1(2) - t1(1);
d2 = t2(2) - t2(1);
d = min(d1, d2);
%
t1s = Quan(t1(1), d);
t1s = t1s + d * onestep(t1(1) - t1s);
t1e = Quan(t1(end), d);
t1e = t1e - d * onestep(t1e - t1(end));
%
t2s = Quan(t2(1), d);
t2s = t2s + d * onestep(t2(1) - t2s);
t2e = Quan(t2(end), d);
t2e = t2e - d * onestep(t2e - t2(end));
%
T1 = t1s : d : t1e;
X1 = interp1(t1, x1, T1);
%
T2 = t2s : d : t2e;
X2 = interp1(t2, x2, T2);
%
Ts = t1s + t2s;
Te = t1e + t2e;
T = Ts : d : Te;
end
%%
function y = Quan(x, delta)
y = delta * round(x / delta);
end
%%
function y = onestep(x)
y = 0.5 * (sign(x) + 1);
end