Question about utilizing FFT, convolution and LTI systems

Hello all, I have a solution from an equation that is considered linear time invariant, such that i(t)*h(t)=o(t) where i,h,o, are the input, impulse response, and output respectively. If you take these to fourier domain, then you can get H(w)=o(w)/i(w), where H(w) is the frequency response. What I am trying to do is obtain h(t) and take the convolution of h(t) and i(t) to obtain o(t). Here are my steps.

step 1: i_t=data1 o_t=data2

step 2: iw=fft(iw) ow=fft(ow)

step 3: Hw=ow./iw

step 4: ht=ifft(Hw)

step 5: ot_new=conv(i_t,ht)

when i compare the new output to the original output they do not match. I tried using conv(i_t,ht,'same') but that also does not give me anything good. I know that my system satisfies the linear time invariant conditions. Am I doing something wrong? I've also tried playing around with the fft shift and ifftshift with no luck. Thanks!

Answer 1

[Honglei Chen]

The FFT corresponds to the circular convolution rather than the linear convolution. I would do the following

N = numel(o_t);
ot_compare = ot_new(1:N);
ot_compare(1:N-1) = ot_compare(1:N-1)+ot_new(N+1:end);

Then you can see ot_compare and o_t are the same.

HTH

[Nam Tran]

Wow it worked!! Thank you so much!