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I have a matrix $X$ of size .i.e $(8,8)$ with $M$ columns and $N$ rows. Suppose I took it's $iFFT$ column wise resulting matrix $x$ of size $(8,8)$ too,

$x = F^HX_{(:,m)}, $ ...... (1),

where $F^H$ is Fourier matrix which is complex numbers and $m = 0,1,....M-1$ the selected column.

After organizing signal $x$ in row-wise, $x' = [x_0, x_1, ....,x_N]$ where $x_k$ denotes the $k^{th}$-row of matrix $x$. Then, after convolution with channel $h$, the resulted signal can be written as :

$y = h⨂x'$ , ...... (2),

where $⨂$ denotes the convolution operation.

Then by converting $y$ into frequency domain:

$Y = HX'$ , ...... (3),

where $Y, H$ and $X'$ are the frequency-domain signal of $y,h$ and $x'$, respectively.

To estimate the channel $h$ based on $Y$, it's straightforward by performing $h = iFFT(Y/x')$, and $x'$ is equal to $x$ reshaped in row-wise.

The problem is I don't want to use all matrix $x$ as pilots.

I need to estimate $h$ based on some pilot vectors in matrix $x$, which means the pilots vectors in matrix $x$ are $x_p = x(1:4:N,:)$ and $x_p$ is the pilot vectors here.

So how can I get the relationship between pilots vectors in $x$ and their correspondent in $Y$ based on $Eq.(3)$? Which means how can I estimate the channel $h$ based on vectors row in matrix $X$?

Marcus Müller
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Fatima_Ali
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  • There is no notion of symbol in the above definition. So I ask, how would you tansmit x, do you transmit one coloumn at a time with N subcarriers? – Dsp guy sam May 01 '20 at 11:49
  • I detail how to do channel estimation using the Wiener-Hopf equations here: https://dsp.stackexchange.com/questions/31318/compensating-loudspeaker-frequency-response-in-an-audio-signal (with the received vector, transmit vector, you solve for the autocorrelation matrix and cross correlation vector of the two and from that get a least squared estimate of the channel) does that answer you question? – Dan Boschen May 01 '20 at 13:14
  • @Dspguysam $x$ is representing one symbol here with $K=NM$ subcarriers – Fatima_Ali May 01 '20 at 15:05
  • @DanBoschen Thank you for your reply. I have checked that estimation way you provided, that doesn't answer my question because my question requires to get the relationship between input and output vectors. As I mentions above, I start taking the $iFFT$ column wise and then collect the vector in row-wise, so how can I get the relationship between input and output pilot data first? – Fatima_Ali May 01 '20 at 15:09
  • I see- why can't you divide Y[k]/x'[k] for each bin that you have a pilot? Is it that you actually have multiple pilots per bin? – Dan Boschen May 01 '20 at 15:43
  • you mean dividing each row separately as $H_i = (Y[k]/x'[k])$ and then estimated channel will be $iFFT(H_0, H_1....H_{N/4})$, where $H_k$ is the vector resulted of dividing every pilot vectors? If you mean that way, I tried it but can't get the channel back !! However when performing $iFFT(Y./x')$, it gives the exacts channel $h$ – Fatima_Ali May 01 '20 at 16:41
  • @Fatima_Ali, The question is not clear. Do you have subsampled data you want to estimate the channel with? So the the input is subsampled differently from the output? – Royi Jul 20 '23 at 07:16
  • @DanBoschen, Solving the inverse problem by going to the correlation matrix only makes it less stable numerically. It is better to solve it directly. – Royi Jul 20 '23 at 07:17
  • @Royi yes good point - I like showing the solution with the correlation matrix as a more intuitive explanation. I’ve never run into any instability numerically in common applications where I would use it but I do see what you are referring to. – Dan Boschen Jul 20 '23 at 10:44
  • @Royi I set up a question for you to answer related to this – Dan Boschen Jul 20 '23 at 12:13
  • @Royi sorry I misunderstood by what you meant by solving directly. I showed here https://dsp.stackexchange.com/a/87352/21048 how using the Wiener Hopf equations led to a superior result rather than solving “directly” with the FFT if that is what you meant (in which case there is no new question) – Dan Boschen Jul 21 '23 at 04:09
  • @DanBoschen, We have talked about it. Solving in spectrum of time domain will result the same answers if the convolution model is the same. I was talking direct in the meaning no need to use the correlation matrix. – Royi Jul 21 '23 at 07:24
  • Yes understood and applicable to those cases. The link I gave shows the results for non-periodic cases. But I see now that I did not misunderstand- thanks for your answer and comments at: https://dsp.stackexchange.com/a/88797/21048 – Dan Boschen Jul 21 '23 at 11:07

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