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I have OFDM system, where the modulated signal $x$ with length $N$, and $N$ is the number of subcarriers in OFDM symbol. $x$ is multiplied with a well-known unitary matrix $G \in N \times N$ before performing the inverse Fourier transform $F^{-1}$ matrix to produce the time-domain signal $y$, so:

$y = F^{-1}Gx \ \ \ \ \ \ \ \ \ \ \ \ (1)$ ,

The cyclic prefix is added to $y$, and it is transmitted through the channel $h$, the received signal $r$ can be expressed, when ignoring the noise, as

$r = h © y \ \ \ \ \ \ \ \ \ \ \ \ (2)$ , where $©$ represent the circular convolution.

At the recieving end, when removing the CP and convert the received signal into frequency domain by multiplication with discrete Fourier transform, the resulting signal become

$R = Fr = HY = HGx \ \ \ \ \ \ \ \ \ \ \ \ (3)$ where $H$ is the frequency-domain channel impulse response.

Assuming we transmitted pilot data in $x$, it means a part of signal $x$ is transmitted as pilot which is well known, can we estimate the channel using $R$ ? for example, Is there any optimization algorithm which can estimate $H$ to produce the well-known pilots in $x$, then that $H$ will represent the estimated channel?

EDIT:

In short, I need to estimate the channel using pilots data inserted in frequency domain subcarriers in presence of the matrix $G$ used before performing iFFT operation. It means I want to use comb type pilots (It is not block type) where the pilots are inserted in frequency domain in each symbol and other subcarriers will carry data.

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  • $\begingroup$ Your question is answered in any decent wireless communications textbook. Could you narrow down your question to the specific aspect of estimation and/or equalization that you're struggling with? $\endgroup$
    – MBaz
    Commented Jan 11, 2023 at 18:48
  • $\begingroup$ @MBaz My process is different to the others systems existed in wireless communications textbooks by the matrix $G$, I need to estimate the channel using pilots but the data was multiplied with matrix $G$ before performing the iFFT operation. $\endgroup$
    – Sajjad
    Commented Jan 12, 2023 at 1:31
  • $\begingroup$ I saw that, so your pilots are actually $Gx$, right? Use $R$ to to estimate $H$, equalize, recover $Gx$, multiply by $G^{-1}$ and obtain $\hat{x}$. Alternatively, you may consider $HG$ to be the channel and estimate it directly. I fail to see the difficulty, but maybe I'm missing something? $\endgroup$
    – MBaz
    Commented Jan 12, 2023 at 2:15
  • $\begingroup$ @MBaz I need to use comb type pilot (not block type). I case of using block type, that becomes straightforward as you mentioned but what I am asking about is when using comb type. Thanks for your clarification; I added that note into the question. $\endgroup$
    – Sajjad
    Commented Jan 12, 2023 at 4:49
  • $\begingroup$ What's the problem with estimating $HG$ as the "effective" channel? $\endgroup$
    – MBaz
    Commented Jan 12, 2023 at 14:45

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