The raised-cosine filter is a
filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...
frequently used for
pulse-shaping in digital
modulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...
due to its ability to minimise
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 ...
(ISI). Its name stems from the fact that the non-zero portion of the
frequency spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
of its simplest form (
) is a
cosine function, 'raised' up to sit above the
(horizontal) axis.
Mathematical description
The raised-cosine filter is an implementation of a low-pass
Nyquist filter
A Nyquist filter is an electronic filter used in TV receivers to equalize the video characteristics. The filter is named after the Swedish–US engineer Harry Nyquist (1889–1976).
VSB
In analogue TV broadcasting the visual radio frequency (RF ...
, i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
about
, where
is the symbol-period of the communications system.
Its frequency-domain description is a
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
-defined
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, given by:
:
or in terms of havercosines:
:
for
:
and characterised by two values;
, the ''roll-off factor'', and
, the reciprocal of the symbol-rate.
The
impulse response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
of such a filter is given by:
:
in terms of the normalised
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
. Here, this is the "communications sinc"
rather than the mathematical one.
Roll-off factor
The
roll-off
Roll-off is the steepness of a Transfer function, transfer function with frequency, particularly in network analysis (electrical circuits), electrical network analysis, and most especially in connection with filter (signal processing), filter cir ...
factor,
, is a measure of the ''excess bandwidth'' of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of
.
Some authors use
.
[ :de:Raised-Cosine-Filter German version of Raised-Cosine-Filter]
If we denote the excess bandwidth as
, then:
:
where
is the symbol-rate.
The graph shows the amplitude response as
is varied between 0 and 1, and the corresponding effect on the
impulse response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
. As can be seen, the time-domain ripple level increases as
decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.
''β'' = 0
As
approaches 0, the roll-off zone becomes infinitesimally narrow, hence:
:
where
is the
rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as
\operatorname(t) = \Pi(t) =
\left\{\begin{array}{r ...
, so the impulse response approaches
. Hence, it converges to an ideal or
brick-wall filter
In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given cutoff frequency, without affecting lower frequencies, and has linear phase response. The filter's impulse response is a sinc functi ...
in this case.
''β'' = 1
When
, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:
:
or
:
Bandwidth
The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero frequency-positive portion of its spectrum, i.e.:
:
As measured using a spectrum analyzer, the radio bandwidth B in Hz of the modulated signal is twice the baseband bandwidth BW (as explained in
, i.e.:
:
Auto-correlation function
The
auto-correlation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
function of raised cosine function is as follows:
:
The auto-correlation result can be used to analyze various sampling offset results when analyzed with auto-correlation.
Application
![Raised-cosine-ISI](https://upload.wikimedia.org/wikipedia/commons/e/ed/Raised-cosine-ISI.svg)
When used to filter a symbol stream, a Nyquist filter has the property of eliminating ISI, as its impulse response is zero at all
(where
is an integer), except
.
Therefore, if the transmitted waveform is correctly sampled at the receiver, the original symbol values can be recovered completely.
However, in many practical communications systems, a
matched filter
In signal processing, a matched filter is obtained by correlating a known delayed signal, or ''template'', with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal wi ...
is used in the receiver, due to the effects of
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
. For zero ISI, it is the
net response of the transmit and receive filters that must equal
:
:
And therefore:
:
These filters are called
root-raised-cosine filters.
Raised cosine is a commonly used
apodization
In signal processing, apodization (from Greek "removing the foot") is the modification of the shape of a mathematical function. The function may represent an electrical signal, an optical transmission or a mechanical structure. In optics, it is p ...
filter for
fiber Bragg gratings.
References
* Glover, I.; Grant, P. (2004). ''Digital Communications'' (2nd ed.). Pearson Education Ltd. .
* Proakis, J. (1995). ''Digital Communications'' (3rd ed.). McGraw-Hill Inc. {{ISBN, 0-07-113814-5.
* Tavares, L.M.; Tavares G.N. (1998) ''Comments on "Performance of Asynchronous Band-Limited DS/SSMA Systems" ''. IEICE Trans. Commun., Vol. E81-B, No. 9
External links
Technical article entitled "The care and feeding of digital, pulse-shaping filters"originally published in
RF Design
''Defense Electronics'' (formerly ''RF Design'') is a Penton Media trade magazine
A trade magazine, also called a trade journal or trade paper (colloquially or disparagingly a trade rag), is a magazine or newspaper whose target audien ...
, written by Ken Gentile.
Linear filters
Telecommunication theory