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I have a doubt about DFT matrix and channel impulse response.

Let $h$ is a channel impulse response , $F$ is a DFT matrix. IF I take a product. I got: $$ F* h=\tilde{h}$$

$\tilde{h}$ is a frequency response of the channel, isn't? or channel impulse response in frequnce domain

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  • $\begingroup$ If $h$ in time-domain, taking Fourier Transform will give its frequency response. $\endgroup$
    – jomegaA
    Feb 24, 2020 at 13:52
  • $\begingroup$ $\tilde{h}$ is the frequency response of the channel. How could frequency response of the channel and channel impulse response in frequency domain be different? ;-) $\endgroup$
    – jomegaA
    Feb 24, 2020 at 22:25
  • $\begingroup$ @jomegaA I have read a research paper where h was called as channel impulse response and product DFT matrix with h as channel coefficient in freq domain. " coefficient in freq domain" made me a question $\endgroup$
    – Jang Lee
    Feb 25, 2020 at 9:11
  • $\begingroup$ ...perhaps the channel impulse response and channel coefficients mean the same. Yes, the product of DFT matrix and in your case $F$ with $h$ is channel impulse response in frequency domain. $\endgroup$
    – jomegaA
    Feb 25, 2020 at 9:17
  • $\begingroup$ As in the answer given by Engineer you may understand what is the product of dftmtx and h is. Or you can take fft(h) which is also the channel impulse response in frequency domain. $\endgroup$
    – jomegaA
    Feb 25, 2020 at 9:21

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You are right. If you have MATLAB, you can do a little experiment to double check yourself:

h = randn(100, 1);     % make some random impulse response
F = dftmtx(length(h)); % make the DFT matrix (100-by-100)
abs(fft(h) - F*h)      % this number should be small
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  • $\begingroup$ Want to put an additional info here, F*h is actually the Fourier transform of h (frequency response). $\\$ From matlab documentation, "A discrete Fourier transform matrix is a complex matrix whose matrix product with a vector computes the discrete Fourier transform of the vector. dftmtx takes the FFT of the identity matrix to generate the transform matrix." $\endgroup$
    – jomegaA
    Feb 24, 2020 at 14:01
  • $\begingroup$ Nevertheless the faster approach could be fft(h) $\endgroup$
    – jomegaA
    Feb 24, 2020 at 14:12
  • $\begingroup$ if a frequency response is given, can I take ifft to get a channel matrix? $\endgroup$
    – Jang Lee
    Feb 25, 2020 at 9:14
  • $\begingroup$ If $\mathbf{h}$ is instead a channel matrix $\mathbf{H}$, you can still do $\mathbf{F}\mathbf{H}=\tilde{\mathbf{H}}$ to get the DFT of $\mathbf{H}$. You just need to make sure that the number of rows in $\mathbf{H}$ is the same as the $\mathbf{F}$ $\endgroup$
    – Engineer
    Feb 25, 2020 at 14:55

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