Fourier transform using Convolution

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Nurhan Aydinalp
Nurhan Aydinalp 2020-12-23
编辑： Paul 2020-12-24
I have two signals x(t) = sin(2.*pi.*t)/(pi.*t) and y(t) = x(t) I want to calculate z(t) = x(t)*y(t) and z(JW).I should plot x(t), x(JW), y(t), y(JW) and z(t), z(JW) using subplot. z(JW)=(1/(2pi))*(convolution(x(t),y(t))), I have the following code:w = [-6.*pi 6.*pi];
syms x(t)
x(t) = sin(2.*pi.*t)./(pi.*t);
subplot(3,2,1)
fplot(t,x(t));
title('x(t) vs t');
xlabel('time');
ylabel('x(t)')
X_J_W = fourier(x(t));
subplot(3,2,2)
fplot(X_J_W,w);
title('X(JW) vs w')
ylabel('X(JW)')
xlabel('W')
syms y(t)
y(t) = sin(2.*pi.*t)./(pi.*t);
subplot(3,2,3)
fplot(t,y(t));
title('y(t) vs t');
xlabel('time');
ylabel('y(t)')
Y_J_W = fourier(y(t));
subplot(3,2,4)
fplot(Y_J_W,w);
title('Y(JW) vs w')
ylabel('Y(JW)')
xlabel('W')
syms z(t)
z(t) = x(t).*y(t);
subplot(3,2,5)
fplot(z(t));
C_X_Y = conv(X_J_W,Y_J_W,'full');
Z_J_W = (1./(2.*pi)*(C_X_Y));
subplot(3,2,6)
fplot(Z_J_W,w)
in the convolution part I get
Error using conv2
Invalid data type. First and second arguments must be numeric or logical.
Error in conv (line 43)
c = conv2(a(:),b(:),shape);
and I do not know how to fix it.

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回答（1 个）

Matt J
Matt J 2020-12-23
You must use int to implement a symbolic convolution integral. conv is for numeric convolution.
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Matt J
Matt J 2020-12-24
Ran in:
Truncating the convolution seems to help:
syms x(t) y(t) z(t) c(t) X(w) Y(w) tau
x(t) = sin(2.*pi.*t)./(pi.*t);
y(t) = sin(2.*pi.*t)./(pi.*t);
z(t)=x(t).*y(t);
X(w) = fourier(x(t));
Y(w) = fourier(y(t));
c(t)=int(x(tau).*y(t-tau),tau,-100,+100);
fplot(c(t))
Paul
Paul 2020-12-24
编辑：Paul 2020-12-24
Nurhan,
Why compute the convolution of x(t) and y(t)? I thought the problem at hand is related to the product of x(t) and y(t).
If z(t) = x(t)y(t), then
Z(w) = conv(X(w),Y(w))/2/pi:
>> syms u
>> Z(w)=int(X(u)*Y(w-u),u,-inf,inf)/2/pi;
>> Z(w)
ans =
-((heaviside(- w - 4*pi)*(w + 4*pi))/2 - w*heaviside(-w) + (heaviside(4*pi - w)*(w - 4*pi))/2)/pi
>> fplot(Z(w),[-20 20])
The result can be confirmed by numerically computing the Fourier transform of z(t):
>> fun=matlabFunction(z(t)*exp(-1j*w*t));
>> wr=-20:.1:20;
>> for ii=1:numel(wr),q(ii)=integral(@(t)fun(t,wr(ii)),-20,20);end
>> hold on
>> plot(wr,real(q),'ro'),grid

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