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 . 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,
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).