IFFT of Convolution equivalence

I was trying out the equation IFFT(x*y) = IFFT(x).IFFT(y), where both 'x' and 'y' are complex numbers. The 'x' and 'y' are 1x8 matrices. The result was different for both sides of the equation. However i was able to prove that x*y = IFFT[ FFT(x).FFT(y)] and FFT(x*y)=FFT(x).FFT(y). What did I miss out when I tried to solve the IFFT relation? Please let me know of any source that has helpful content regarding this relationship

Answer 1

[Ridwan Alam]

Sorry for the confusion. Here is what I tried:

a1 = randi(50,8,1);
b1 = randi(50,8,1);
x = complex(a1,b1)
a2 = randi(50,8,1);
b2 = randi(50,8,1);
y = complex(a2,b2)
xifft = ifft(x)
yifft = ifft(y)
mifft = xifft.*yifft
z = cconv(x,y,length(x))
zifft = ifft(z)./length(z)
% zifft-mifft is almost zero

Summary: you need circular convolution in frequency domain instead of linear convulation.

More details:

https://www.mathworks.com/matlabcentral/answers/59333-convolution-in-frequency-domain-not-convolution-in-time-domain?s_tid=answers_rc1-2_p2_MLT

https://www.mathworks.com/matlabcentral/answers/38066-difference-between-conv-ifft-fft-when-doing-convolution?s_tid=answers_rc1-3_p3_Topic

[JOB]

Thanks again, I equated it by using linear convolution with zero padding.

[Ridwan Alam]

sure. glad to help.

Answer 2

[David Goodmanson]

Hi JOB/Ridwan

Here is a small example where a and b are padded with zeros, so that regular convolution can be compared with convolution by fft and ifft. (The zeros ensure that for fft and ifft, which do circular convolution, the nonzero parts can't overlap by going 'the other way around the circle.'

a = [(1:3)+i*(2:4) zeros(1,4)]
b = [(3:5)+i*(4:6) zeros(1,4)]
n = length(a)
convab = conv(a,b)
convab1 = ifft(fft(a).*fft(b))
convab2 = n*fft(ifft(a).*ifft(b))

All of these results agree (not counting that the conv result is a longer vector and contains more zeros than the other two). For the convab2 result you have to multiply by an extra factor of n. This is because the Matlab ifft algorithm contains an overall factor of (1/n) and the fft does not.

[Ridwan Alam]

Hey David,

Thanks for the example. I agree, the circular convolution is the issue here.

I believe what JOB is trying to show is that:

z = ifft(convab);
m = ifft(a).*ifft(b);

But m is not equal to z, not even to z/length(a).

[David Goodmanson]

HI Ridwan,

what you say is true, but it's because a &b have different length than convab. One could do a circular convolution of a and b by a method other than ifft and then compare, but this example just pads out a and b to simulate a linear convolution as before, then pads them out again to be the same length as convab:

a = [(1:3)+i*(2:4) zeros(1,4)];
b = [(3:5)+i*(4:6) zeros(1,4)];
n = length(a);
convab = conv(a,b)
nconvab = length(convab);
apad = [a zeros(1,nconvab-n)]
bpad = [b zeros(1,nconvab-n)]
% the following two expressions are the same
ifft(convab)
nconvab*ifft(apad).*ifft(bpad)

[JOB]

Thanks for the lucid explanation.