In
telecommunications
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
, the term cyclic prefix refers to the prefixing of a
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
, 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
In telecommunications, guard intervals are used to ensure that distinct transmissions do not interfere with one another, or otherwise cause overlapping transmissions. These transmissions may belong to different users (as in TDMA) or to the same ...
to eliminate
intersymbol interference
In telecommunication, intersymbol interference (ISI) is a form of distortion of a signal in which one symbol interferes with subsequent symbols. This is an unwanted phenomenon as the previous symbols have a similar effect as noise, thus making ...
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 Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
, which in turn may transform to the frequency domain via a
discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex- ...
. 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
OFDM
In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission and a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital commun ...
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
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
, and
time-invariant
In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
systems. Therefore, if the channel is assumed to be
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
and
time-invariant
In control theory, a time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is ...
, then a sinusoid of infinite duration would be an
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
. 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
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
of the channel appear as though it were
circular convolution Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
, and thus, preserve this property in the part of the symbol after the cyclic prefix.
Use in OFDM
OFDM
In telecommunications, orthogonal frequency-division multiplexing (OFDM) is a type of digital transmission and a method of encoding digital data on multiple carrier frequencies. OFDM has developed into a popular scheme for wideband digital commun ...
uses cyclic prefixes to combat multipath by making channel estimation easy. As an example, consider an OFDM system that has
subcarriers. The message symbol can be written as:
:
The OFDM symbol is constructed by taking the
inverse discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a compl ...
(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 Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discre ...
, as