Cyclic prefix

Intelecommunications,the termcyclic prefixrefers to the prefi xing of asymbolwith a repetition of the end. The receiver is typically configured to discard the cyclic prefix samples, but the cyclic prefix serves two purposes:

• It provides aguard intervalto eliminateintersymbol interferencefrom the previous symbol.
• It repeats the end of the symbol so the linear convolution of a frequency-selective multipath channel can be modeled ascircular convolution,which in turn may transform to the frequency domain via adiscrete Fourier transform.This approach accommodates simple frequency domain processing, such as channel estimation and equalization.

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 withOFDMsystems, however the cyclic prefix is now also used insingle carriersystems to improve the robustness tomultipathpropagation.

Principle

A cyclic prefix is often usedin conjunction with modulation to retainsinusoids'properties inmultipathchannels. It is well known that sinusoidal signals areeigenfunctionsoflinear,andtime-invariantsystems. Therefore, if the channel is assumed to belinearandtime-invariant,then a sinusoid of infinite duration would be aneigenfunction.However, in practice, this cannot be achieved, as real signals are always time-limited. So, to mimic the infinite behavior, prefi xing the end of the symbol to the beginning makes the linearconvolutionof the channel appear as though it werecircular convolution,and thus, preserve this property in the part of the symbol after the cyclic prefix.

Use in OFDM

OFDMuses cyclic prefixes to combat multipath by making channel estimation easy. As an example, consider an OFDM system that has$N$subcarriers. The message symbol can be written as:

$\mathbf {d} ={\begin{bmatrix}d_{0}&d_{1}&\ldots &d_{N-1}\end{bmatrix}}^{\textsf {T}}$

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

$\mathbf {x} ={\begin{bmatrix}x[0]&x[1]&\ldots &x[N-1]\end{bmatrix}}^{\textsf {T}}$.

Prefi xing it with a cyclic prefix of length$L_{1}-1$,the OFDM symbol obtained is:

$\mathbf {z} ={\begin{bmatrix}x[N-L_{1}+1]&\ldots &x[N-2]&x[N-1]&x[0]&x[1]&\ldots &x[N-1]\end{bmatrix}}^{\textsf {T}}.$

Assume that the channel is represented using

$\mathbf {h} ={\begin{bmatrix}h_{0}&h_{1}&\ldots &h_{L_{2}-1}\end{bmatrix}}^{\textsf {T}}$.

Then, the convolution with this channel, which happens as

$y[m]=\sum _{l=0}^{L-1}h[l]z[m-l]\quad 0\leq m\leq N-L_{1}-2$

results in the received symbols$\mathbf {y}$. Now only if$L_{1}\geq L_{2}$,this is thecircular convolutionof$\mathbf {z}$and$\mathbf {h}$at the values$m\geq L_{1}$,since here$\mathbf {z}$becomes$x[(m-l)\mod N]$.Hence, taking the discrete Fourier transform of these values, we get

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

where$X[k]$is thediscrete Fourier transformof$\mathbf {x}$,i.e.$\mathbf {d}$.Thus, a multipath channel is converted into scalar orthogonal sub-channels in the frequency domain, thereby simplifying the receiver design considerably. The task of channel estimation is simplified, as we just need to have access to an estimate of the scalar coefficients$H[k]$,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${\begin{bmatrix}d_{0}&d_{1}&\ldots &d_{N-1}\end{bmatrix}}^{\textsf {T}}$.

See also

• Guard interval
• Interpacket gap
• Intersymbol interference

References

• Fundamentals of Wireless Communication,by David Tse and Pramod Viswanath, Cambridge University Press (2005).

External links

• A short tutorial onthe significance of cyclic prefix in OFDM systemsArchived2023-10-05 at theWayback Machine.