# OFDM channel estimation using the transmitted signal multiplied with a matrix

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.

• 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?
– MBaz
Jan 11 at 18:48
• @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. Jan 12 at 1:31
• 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?
– MBaz
Jan 12 at 2:15
• @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. Jan 12 at 4:49
• What's the problem with estimating $HG$ as the "effective" channel?
– MBaz
Jan 12 at 14:45