Raised-cosine filter

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The raised-cosine filter is a particular electronic filter, frequently appearing in telecommunications systems due to its ability to minimise inter-symbol interference (ISI). It is so called due to the non-zero portion of the frequency spectrum of its simplest form (β = 1) being 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 \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)| = \begin{cases} 1.0, & |f| \leq \frac{1 - \beta}{2T} \\ \frac{1}{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}
0 \leq \beta \leq 1

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 (assuming linear phase) is given by:

h(t) = \mathbf{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\frac{\pi\beta t}{T}\right)}{1 - \frac{4\beta^2 t^2}{T^2}}, in terms of the normalized sinc function.
Amplitude response of raised-cosine filter with various roll-off factors
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Amplitude response of raised-cosine filter with various roll-off factors
Impulse response of raised-cosine filter with various roll-off factors
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Impulse response of raised-cosine filter with various roll-off factors

[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 \frac{1}{2T}. If we denote the excess bandwidth as Δf, then:

\beta = \frac{\Delta f}{\left(\frac{1}{2T}\right)} = \frac{\Delta f}{R_S/2} = 2T\Delta f

where R_S = \frac{1}{T} 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:

\lim_{\beta \rightarrow 0}|H(f)| = \mathbf{rect}(fT)

where 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.

[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.

[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.:

BW = \frac{1}{2}R_S(1+\beta)

[edit] Application

Consecutive raised-cosine impulses, demonstrating zero-ISI property
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Consecutive raised-cosine impulses, demonstrating zero-ISI property

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:

H_R(f) = H_T^*(f)

i.e.:

|H_R(f)| = |H_T(f)| = \sqrt{|H(f)|}

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.

Note that the phase response of the receiver filter must be the negative of that of the transmitter filter.

[edit] Phase response

To successfully eliminate ISI, the implementation of the (root-)raised-cosine filter should be phase-linear.

This filter is often implemented as a digital FIR filter, which allows the phase-linearity constraint to be easily achieved.

[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

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