I have a signal $X$ with length of $N$, multiplied with any unitary matrix, i.e the transpose of DCT matrix as:
$x = D' X$
where $D'$ is the transpose of the of DCT matrix. Then let's add $CP$ guard interval into the signal $x$. The resultant signal after adding the cp into the head of the signal is $x_2$. We transmit $x_2$ through a channel $h$ leading to have the signal $y$ as follows:
$y = h*x_2$ ...... where $*$ is the convolution operation.
Hint: the convolution is not circular in that case as we used DCT instead of DFT.
At the receiver, after removing the delay of the channel and the CP guard interval, I used the following steps:
$x_3 = ifft(fft(y)./H)$ ..... I used ZF equalizer, $H$ is the frequency-domain channel found by $H = fft(h,N)$
Then, the signal is recovered by multiplying by the DCT matrix
$x_4 = D * x_3$
My question, how does it work the step when I get $x_3$?? is there mathematical expression for that, , I think it's called frequency domain equalizer, but I didn't get its mathematical expression. Most importantly, why does it work however we don't have circular convolution when getting $y$? !!