View source for Cyclic prefix

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In [[telecommunications]], the term '''cyclic prefix''' refers to the prefixing of a [[symbol (data)|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:
* It provides a [[guard interval]] to eliminate [[intersymbol interference]] from the previous symbol.
* It repeats the end of the symbol so the linear convolution of a frequency-selective multipath channel can be modeled as [[circular convolution]], which in turn may transform to the frequency domain via a [[discrete 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 with [[Orthogonal frequency-division multiplexing|OFDM]] systems, however the cyclic prefix is now also used in [[single carrier]] systems to improve the robustness to [[Multipath propagation|multipath]] propagation.

== Principle ==
A cyclic prefix is often used{{Citation needed|date=April 2019}} in conjunction with modulation to retain [[Sine wave|sinusoids']] properties in [[multipath propagation|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 <math>N</math> subcarriers. The message symbol can be written as:

:<math>\mathbf{d} = \begin{bmatrix}d_0 & d_1 & \ldots & d_{N-1}\end{bmatrix}^\textsf{T}</math>

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

:<math>\mathbf{x} = \begin{bmatrix}x[0] & x[1] & \ldots & x[N - 1]\end{bmatrix}^\textsf{T}</math>.

Prefixing it with a cyclic prefix of length <math>L_1-1</math>, the OFDM symbol obtained is:

:<math>\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}.</math>

Assume that the channel is represented using

:<math>\mathbf{h} = \begin{bmatrix}h_0 & h_1 & \ldots & h_{L_2-1}\end{bmatrix}^\textsf{T}</math>.

Then, the convolution with this channel, which happens as

:<math>y[m] = \sum_{l = 0}^{L-1} h[l] z[m - l] \quad 0 \le m \le N - L_1 -2</math>

results in the received symbols <math>\mathbf{y}</math>.
Now only if <math>L_1 \geq L_2</math>, this is the [[circular convolution]] of <math>\mathbf{z}</math> and <math>\mathbf{h}</math> at the values <math>m \geq L_1</math>, since here <math>\mathbf{z}</math> becomes <math>x [(m - l)\mod N]</math>. Hence, taking the discrete Fourier transform of these values, we get

:<math>Y[k] = H[k] \cdot X[k]</math>.

where <math>X[k]</math> is the [[discrete Fourier transform]] of <math>\mathbf{x}</math>, i.e. <math>\mathbf{d}</math>. 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 <math>H[k]</math>, for the duration in which the channel does not vary significantly, merely multiplying the received demodulated symbols by the inverse of <math>H[k]</math> yields the estimates of <math>\{X[k]\}</math> and hence, the estimate of actual symbols <math>\begin{bmatrix}d_0 & d_1 & \ldots & d_{N - 1}\end{bmatrix}^\textsf{T}</math>.

== See also ==
*[[Guard interval]]
*[[Interpacket gap]]
*[[Intersymbol interference]]

== References ==
{{Reflist}}
{{refbegin}}
*[http://www.ifp.uiuc.edu/~pramodv/wireless_book.html Fundamentals of Wireless Communication], by David Tse and Pramod Viswanath, Cambridge University Press (2005).
{{refend}}

==External links==
*A short tutorial on [http://sites.google.com/site/mdanishnisar/pubs/01_OFDM_Tutorial_Nisar.pdf the significance of cyclic prefix in OFDM systems] {{Webarchive|url=https://web.archive.org/web/20231005015452/https://accounts.google.com/v3/signin/identifier?continue=https%3A%2F%2Fsites.google.com%2Fsite%2Fsites%2Fsystem%2Ferrors%2FWebspaceNotFound%3Fpath%3D%2Fmdanishnisar%2Fpubs%2F01_OFDM_Tutorial_Nisar.pdf&followup=https%3A%2F%2Fsites.google.com%2Fsite%2Fsites%2Fsystem%2Ferrors%2FWebspaceNotFound%3Fpath%3D%2Fmdanishnisar%2Fpubs%2F01_OFDM_Tutorial_Nisar.pdf&ifkv=AYZoVhdRJ9GBHB6cgOLtinHy_6b1zB2NMaNvXGAKQyNqtVonHsSet2zAnxfz7-E-DgzgO4izumx80A&passive=1209600&service=jotspot&flowName=GlifWebSignIn&flowEntry=ServiceLogin&dsh=S-1777759296%3A1696470891541142&theme=glif |date=2023-10-05 }}.

[[Category:Quantized radio modulation modes]]