Cyclic prefix

In telecommunications, the term cyclic prefix refers to the prefixing of a symbol with a repetition of the end. Although the receiver is typically configured to discard the cyclic prefix samples, the cyclic prefix serves two purposes.

• As a guard interval, it eliminates the intersymbol interference from the previous symbol.
• As a repetition of the end of the symbol, it allows the linear convolution of a frequency-selective multipath channel to be modelled as circular convolution, which in turn may be transformed to the frequency domain using a discrete Fourier transform. This approach allows for simple frequency-domain processing, such as channel estimation and equalization.

In order for the cyclic prefix to be effective (i.e. to serve its aforementioned objectives), the length of the cyclic prefix must be at least equal to the length of the multipath channel. Although the concept of cyclic prefix has been traditionally associated with OFDM systems, the cyclic prefix is now also used in single carrier systems to improve the robustness to multipath propagation.

Principle

Cyclic prefix is often used in conjunction with modulation in order 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

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 $N$ subcarriers.^[1] The message symbol can be written as:

$\mathbf{d} = [d_0, d_1, \ldots d_{N - 1}]^T$

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

$\mathbf{x}^\prime = [x[0], x[1], \ldots x[N - 1]]^T$.

Prefixing it with a cyclic prefix of length $L-1$, the OFDM symbol obtained is:

$\mathbf{x} = [x[N - L + 1], \ldots x[N - 2], x[N - 1], x[0], x[1], \ldots x[N - 1]]^T$.

Assume that the channel is represented using

$\mathbf{h} = [h_0, h_1, \ldots h_{L-1}]^T$.

Then, after convolution with the channel, which happens as

$y[m] = \sum_{l = 0}^{L - 1} h[l] x[m - l] \quad L-1 \le m \le N-1$

which is circular convolution, as $x[m - l]$ becomes $x^\prime [(m - l)\mod N]$. So, taking the Discrete Fourier Transform, we get

$Y[k] = H[k]\cdot X[k]$.

where $X[k]$ is the discrete Fourier transform of $\mathbf{x}$. 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 $H[k]$ for each sub-channel and once the values of $\{H[k]\}$ are estimated, for the duration in which the channel does not vary significantly, merely multiplying the received demodulated symbols by the inverse of $H[k]$ yields the estimates of $\{X[k]\}$ and hence, the estimate of actual symbols $[d_0, d_1, \ldots d_{N - 1}]^T$.

References

1. http://www.create.ucsb.edu/ATON/01.01/OFDM.pdf

• Fundamentals of Wireless Communication, by David Tse and Pramod Viswanath, Cambridge University Press (2005).
• A short tutorial on the significance of cyclic prefix in OFDM systems.