This is a follow up to the Cyclic Prefix (CP) I did here.

There I gave a graphical explanation and here I will discuss how it happens theoretically. I recently came across the derivation in a text [1]. So here goes.

Considering an N-point FFT system, a block of data points

,

is to be transmitted during a symbol time. The data vector is sent to an IFFT module. The operation of the FFT, i.e. the discrete Fourier Transformation (DFT) is given by the following matrix:

where denotes the -th entry in the DFT matrix .

Then the transmitted data vector after the IFFT operation is.

.

The normalization factor takes care of the total bit energy as it is the same as the original vector because .

Assuming the channel length (number of multipaths) is , the OFDM symbol requires a CP of length . Then the total symbol length is .

After the (linear) convolution through the channel and removal of the CP, the received signal (at the input to the FFT) can be written,

where is the AWGN noise vector. The channel matrix is a circulant matrix due to the insertion of the CP in the transmitted data vector and can be written as,

Here are the channel impulse responses and for .

Circulant matrices have an important property:

The eigenvectors of a circulant matrix of a given size are the columns of the discrete Fourier transform matrix of the same size[Wikipedia].

So we have,

Thus,

where,

Therefore, is just the frequency response of the -th subcarrier and hence, the eigenvalues of are the frequency response of the channel.

Then, going back to the received signal vector,

Sending this received data vector through the DFT,

Here and that the statistical properties of .

So you see how the insertion of the CP helps OFDM receivers easily decode the transmitted date vector by a simple inverse DFT operation (a single-tap equalization).

## 6 comments

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## ga

February 16, 2010 at 7:35 pm (UTC 0) Link to this comment

Thank you very much for this great explanations about cp. I have been looking for something like that in many books and web sites.

Which one is the reference [1] you mentioned?

Best regards

## admin

February 17, 2010 at 8:07 am (UTC 0) Link to this comment

Welcome.

The reference is:

“Adaptive and Iterative Signal Processing in Communications” by Jinho Choi.

Reason I wanted to publish this was also because not many texts on OFDM discuss all the purposes of CP (they only mention ISI elimination) and this is the only book I came across the theoretical derivation for the circular convolution.

## kal

March 26, 2010 at 6:41 pm (UTC 0) Link to this comment

thanks for the post. it give me clear insight another perspective of cyclic prefix. may i know what the reference that u refer in previous and this post. thanks.

## Arun

January 11, 2011 at 4:51 pm (UTC 0) Link to this comment

Thanks, made my understanding a lot better!

Arun

## Vishnu

March 24, 2011 at 11:37 pm (UTC 0) Link to this comment

Hello sir, I am doing Mtech thesis on SCFDMA.In that I have to implement Tomlinson Harashima precoding , I got a paper for simulation,but in that they had not mention about channel impulsse response, which is very much important in this case, what can I do ?Is there any method for calculating h(n) for a multipath fading channel,please help me

## pościel

April 16, 2011 at 5:42 pm (UTC 0) Link to this comment

Bardzo mnie to zaciekawiło,będę tu zaglądać