Mathematical expression of Multicarrier frequency equalizer

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 $$y$$? !!

thank you

Hint: the convolution is not circular in that case as we used DCT instead of DFT.

Luckily, that's not true! At the receiver, you're doing a DFT, and you've prepared the transmit signal such that a convolution with the channel looks like a circular one by prepending the CP.

The fact that the data inside has something to do with the DCT is irrelevant there.

Then, the signal is recovered by multiplying by the DCT matrix

Yep, because that inverts $$D'$$.

I think it's called frequency domain equalizer,

No, the frequency-domain equalizer is $$\text{IFFT}(\text{FFT}/H)$$.

• First thank you for your reply, but ... 1- You said I prepared the circular convolution, I didn't get what you mean you prepared the signal to look like a circular? The CP was added after the DCT multiplication, so I think there is not circular convolution between the transmitted signal and the channel, is that right?. could you please explain that mathematically if possible? 2- regarding the frequency-domain equalizer, why did you say No? I think what you added is similar to what I wrote to get $x_3$, is that right? Jan 5 '21 at 2:51
• Again: Ignore the fact that you used the DCT on your input data. It's irrelevant to this problem! What you did do is take some signal, put the end of it before the beginning (CP). When your channel convolves that with the channel impulse response, and you then just take the right part of it, it looks like circular convolution! It's exactly the same explanation as for CP-OFDM. Jan 5 '21 at 8:03
• I see .. so can we use circular Toeplitz matrix to equalize the signal $y$ ? it means in that case the frequency-domain equalizer won't be required anymore. Second, you didn't reply my second question in my previous comment. Jan 5 '21 at 8:13