Cyclic Prefix - Use in OFDM

Use in OFDM

Cyclic Prefixes are used in OFDM in order to combat multipath by making channel estimation easy. As an example, consider an OFDM system which has subcarriers. The message symbol can be written as:

The OFDM symbol is constructed by taking the inverse discrete Fourier transform (IDFT) of the message symbol, followed by a cyclic prefixing. Let the symbol obtained by the IDFT be denoted by

.

Prefixing it with a cyclic prefix of length, the OFDM symbol obtained is:

.

Assume that the channel is represented using

.

Then, after convolution with the channel, which happens as

which is circular convolution, as becomes . So, taking the Discrete Fourier Transform, we get

.

where is the discrete Fourier transform of . Thus, a multipath channel is converted into scalar parallel sub-channels in frequency domain, thereby simplifying the receiver design considerably. The task of channel estimation is simplified, as we just need to estimate the scalar coefficients for each sub-channel and once the values of are estimated, for the duration in which the channel does not vary significantly, merely multiplying the received demodulated symbols by the inverse of yields the estimates of and hence, the estimate of actual symbols .