- Raised-cosine filter
The raised-cosine filter is a particular
electronic filter , frequently used forpulse-shaping in digitalmodulation due to its ability to minimiseintersymbol interference (ISI). Its name stems from the fact that the non-zero portion of thefrequency spectrum of its simplest form (eta = 1) is acosine 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.Its frequency-domain description is a
piecewise function, given by::H(f) = egin{cases} T, & |f| leq frac{1 - eta}{2T} \ frac{T}{2}left [1 + cosleft(frac{pi T}{eta}left [|f| - frac{1 - eta}{2T} ight] ight) ight] , & frac{1 - eta}{2T} < |f| leq frac{1 + eta}{2T} \ 0, & mbox{otherwise}end{cases}:0 leq eta leq 1 and characterised by two values; eta, the "roll-off factor", and T, the reciprocal of the symbol-rate.
The
impulse response of such a filter is given by::h(t) = mathrm{sinc}left(frac{t}{T} ight)frac{cosleft(frac{pieta t}{T} ight)}{1 - frac{4eta^2 t^2}{T^2, in terms of the normalized
sinc function .Roll-off factor
The roll-off factor, eta, 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:
:eta = frac{Delta f}{left(frac{1}{2T} ight)} = 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 eta 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 eta decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.
=eta = 0=As eta approaches 0, the roll-off zone becomes infinitesimally narrow, hence:
:lim_{eta ightarrow 0}H(f) = mathrm{rect}(fT)
where mathrm{rect}(.) is the
rectangular function , so the impulse response approaches mathrm{sinc}left(frac{t}{T} ight). Hence, it converges to an ideal or brick-wall filter in this case.
=eta = 1=When eta = 1, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:
:H(f)|_{eta=1} = left { egin{matrix} frac{1}{2}left [1 + cosleft(pi fT ight) ight] , & |f| leq frac{1}{T} \ 0, & mbox{otherwise}end{matrix} ight.
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.:
:BW = frac{1}{2}R_S(1+eta)
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 many practical communications systems, a
matched filter is used in the receiver, due to the effects ofwhite 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
These filters are called root-raised-cosine filters.
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.External links
* [http://images.industryclick.com/files/4/0402Gentile50.pdf - Technical article entitled 'The care and feeding of digital, pulse-shaping filters'] originally published in RF Design.
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