Cyclic prefix
In telecommunications, the term cyclic prefix refers to the prefixing of a symbol, with a repetition of the end. The receiver is typically configured to discard the cyclic prefix samples, but the cyclic prefix serves two purposes: For the cyclic prefix to serve its objectives, it must have a length at least equal to the length of the multipath channel. The concept of a cyclic prefix is traditionally associated with OFDM systems, however the cyclic prefix is now also used in single carrier systems to improve the robustness to multipath propagation. Principle. A cyclic prefix is often used  in conjunction with modulation to retain sinusoids' properties in multipath channels. It is well known that sinusoidal signals are eigenfunctions of linear, and time-invariant systems. Therefore, if the channel is assumed to be linear and time-invariant, then a sinusoid of infinite duration would be an eigenfunction. However, in practice, this cannot be achieved, as real signals are always time-limited. So, to mimic the infinite behavior, prefixing the end of the symbol to the beginning makes the linear convolution of the channel appear as though it were circular convolution, and thus, preserve this property in the part  of the symbol after the cyclic prefix. Use in OFDM. OFDM uses cyclic prefixes to combat multipath by making channel estimation easy. As an example, consider an OFDM system that has formula_1 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 formula_4, 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 formula_8 becomes formula_9. So, taking the discrete Fourier transform, we get where formula_11 is the discrete Fourier transform of formula_12. Thus, a multipath channel is converted into scalar parallel sub-channels in the frequency domain, thereby simplifying the receiver design considerably. The task of channel estimation is simplified, as we just need to estimate the scalar coefficients formula_13 for each sub-channel and once the values of formula_14 are estimated, for the duration in which the channel does not vary significantly, merely multiplying the received demodulated symbols by the inverse of formula_13 yields  the estimates of formula_16 and hence, the estimate of actual symbols formula_17.