Raised-cosine filter
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The raised-cosine filter is a particular electronic filter, frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form (β = 1) is a cosine function, 'raised' up to sit above the f (horizontal) axis.
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[edit] Mathematical description
The raised-cosine filter is an implementation of a low-pass Nyquist filter, i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd symmetry about 
, where T is the symbol-period of the communications system.
Its frequency-domain description is a piecewise function, given by:
![H(f) = \begin{cases}
T,
& |f| \leq \frac{1 - \beta}{2T} \\
\frac{T}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left[|f| - \frac{1 - \beta}{2T}\right]\right)\right],
& \frac{1 - \beta}{2T} < |f| \leq \frac{1 + \beta}{2T} \\
0,
& \mbox{otherwise}
\end{cases}](../../../../math/6/3/a/63ac6ef7fb8a1be66eca186636f4ff4a.png)
and characterised by two values; β, the roll-off factor, and T, the reciprocal of the symbol-rate.
The impulse response of such a filter is given by:
, in terms of the normalized sinc function.
[edit] Roll-off factor
The roll-off factor, β, is a measure of the excess bandwidth of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of . If we denote the excess bandwidth as Δf, then:
where 
is the symbol-rate.
The graph shows the amplitude response as β is varied between 0 and 1, and the corresponding effect on the impulse response. As can be seen, the time-domain ripple level increases as β decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.
[edit] β = 0
As β approaches 0, the roll-off zone becomes infinitesimally narrow, hence:
where rect(.) is the rectangular function, so the impulse response approaches 
. Hence, it converges to an ideal or brick-wall filter in this case.
[edit] β = 1
When β = 1, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:
![H(f)|_{\beta=1} = \left \{ \begin{matrix}
\frac{1}{2}\left[1 + \cos\left(\pi fT\right)\right],
& |f| \leq \frac{1}{T} \\
0,
& \mbox{otherwise}
\end{matrix} \right.](../../../../math/9/6/1/961df8f581c33549a42343953bf300df.png)
[edit] Bandwidth
The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero portion of its spectrum, i.e.:
[edit] Application
When used to filter a symbol stream, a Nyquist filter has the property of eliminating ISI, as its impulse response is zero at all nT (where n is an integer), except n = 0.
Therefore, if the transmitted waveform is correctly sampled at the receiver, the original symbol values can be recovered completely.
However, in most practical communications systems, a matched filter must be used in the receiver, due to the effects of white noise. This enforces the following constraint:
i.e.:
To satisfy this constraint whilst still providing zero ISI, a root-raised-cosine filter is typically used at each end of the communication system. In this way, the total response of the system is raised-cosine.
[edit] References
- Glover, I.; Grant, P. (2004). Digital Communications (2nd ed.). Pearson Education Ltd. ISBN 0-13-089399-4.
- Proakis, J. (1995). Digital Communications (3rd ed.). McGraw-Hill Inc. ISBN 0-07-113814-5.
[edit] External links
- - Technical article entitled 'The care and feeding of digital, pulse-shaping filters' originally published in RF Design.
