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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 ($beta\; =\; 1$) is a cosine function, 'raised' up to sit above the $f$ (horizontal) axis.
## 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 $frac\{1\}\{2T\}$, where $T$ is the symbol-period of the communications system.### Roll-off factor

The roll-off factor, $beta$, is a measure of the excess bandwidth of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of $frac\{1\}\{2T\}$. If we denote the excess bandwidth as $Delta\; f$, then:#### $beta\; =\; 0$

As $beta$ approaches 0, the roll-off zone becomes infinitesimally narrow, hence:#### $beta\; =\; 1$

When $beta\; =\; 1$, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:### 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.:## Application

## References

## External links

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 + cosleft(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}

- $0\; leq\; beta\; leq\; 1$

The impulse response of such a filter is given by:

- $h(t)\; =\; mathrm\{sinc\}left(frac\{t\}\{T\}right)frac\{cosleft(frac\{pibeta\; t\}\{T\}right)\}\{1\; -\; frac\{4beta^2\; t^2\}\{T^2\}\}$, in terms of the normalized sinc function.

- $beta\; =\; frac\{Delta\; f\}\{left(frac\{1\}\{2T\}right)\}\; =\; frac\{Delta\; f\}\{R\_S/2\}\; =\; 2TDelta\; f$

where $R\_S\; =\; frac\{1\}\{T\}$ is the symbol-rate.

The graph shows the amplitude response as $beta$ 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 $beta$ decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.

- $lim\_\{beta\; rightarrow\; 0\}H(f)\; =\; mathrm\{rect\}(fT)$

where $mathrm\{rect\}(.)$ is the rectangular function, so the impulse response approaches $mathrm\{sinc\}left(frac\{t\}\{T\}right)$. Hence, it converges to an ideal or brick-wall filter in this case.

- $H(f)|\_\{beta=1\}\; =\; left\; \{\; begin\{matrix\}$

0,& mbox{otherwise} end{matrix} right.

- $BW\; =\; frac\{1\}\{2\}R\_S(1+beta)$

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 many practical communications systems, a matched filter is used in the receiver, due to the effects of white noise. For zero ISI, it is the __net__ response of the transmit and receive filters that must equal $H(f)$:

- $H\_R(f)cdot\; H\_T(f)\; =\; H(f)$

And therefore:

- $|H\_R(f)|\; =\; |H\_T(f)|\; =\; sqrt$
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These filters are called root-raised-cosine filters.

- 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.

- - Technical article entitled 'The care and feeding of digital, pulse-shaping filters' originally published in RF Design.

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Last updated on Sunday September 14, 2008 at 11:44:07 PDT (GMT -0700)

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Last updated on Sunday September 14, 2008 at 11:44:07 PDT (GMT -0700)

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