# On the channel estimation based on vector

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 $$X$$?

• 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 1 '20 at 11:49
• 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? – Dan Boschen May 1 '20 at 13:14
• @Dspguysam $x$ is representing one symbol here with $K=NM$ subcarriers – Fatima_Ali May 1 '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 1 '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 1 '20 at 15:43