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 $X$?

  • $\begingroup$ 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? $\endgroup$ May 1, 2020 at 11:49
  • $\begingroup$ I detail how to do channel estimation using the Wiener-Hopf equations here: dsp.stackexchange.com/questions/31318/… (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? $\endgroup$ May 1, 2020 at 13:14
  • $\begingroup$ @Dspguysam $x$ is representing one symbol here with $K=NM$ subcarriers $\endgroup$
    – Fatima_Ali
    May 1, 2020 at 15:05
  • $\begingroup$ @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? $\endgroup$
    – Fatima_Ali
    May 1, 2020 at 15:09
  • $\begingroup$ 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? $\endgroup$ May 1, 2020 at 15:43


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