Raised-cosine filter

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). Electronic filters are Electronic circuits which perform Signal processing functions specifically intended to remove unwanted signal components and/or enhance wanted In digital telecommunication pulse shaping is the process of changing the waveform of transmitted pulses In Telecommunications, modulation is the process of varying a periodic Waveform, i In Telecommunication, intersymbol interference ( ISI) is a form of Distortion of a signal in which one symbol interferes with 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. Familiar concepts associated with a Frequency are colors musical notes radio/TV channels and even the regular rotation of the earth

1 Mathematical description
- 1.1 Roll-off factor
  - 1.1.1 β = 0
  - 1.1.2 β = 1
- 1.2 Bandwidth
2 Application
3 References
4 External links

Mathematical description

The raised-cosine filter is an implementation of a low-pass Nyquist filter, i. In communications the Nyquist ISI criterion describes the conditions which when satisfied by a communication channel, result in no Intersymbol interference or ISI 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. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or

Its frequency-domain description is a piecewise function, given by:

$H(f) = \begin{cases} T, & |f| \leq \frac{1 - \beta}{2T} \\ \frac{T}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left[|f| - \frac{1 - \beta}{2T}\right]\right)\right], & \frac{1 - \beta}{2T} < |f| \leq \frac{1 + \beta}{2T} \\ 0, & \mbox{otherwise} \end{cases}$

$0 \leq \beta \leq 1$

and characterised by two values; $β$ , the roll-off factor, and $T$ , the reciprocal of the symbol-rate. In Mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of the Independent The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

The impulse response of such a filter is given by:

$h(t) = \mathrm{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\frac{\pi\beta t}{T}\right)}{1 - \frac{4\beta^2 t^2}{T^2}}$ , in terms of the normalized sinc function. The impulse response of a system is its output when presented with a very brief input signal an impulse In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes

Amplitude response of raised-cosine filter with various roll-off factors

Impulse response of raised-cosine filter with various roll-off factors

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 $Δ f$ , 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. The impulse response of a system is its output when presented with a very brief input signal an impulse 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.

$β = 0$

As $β$ approaches 0, the roll-off zone becomes infinitesimally narrow, hence:

$\lim_{\beta \rightarrow 0}H(f) = \mathrm{rect}(fT)$

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

$β = 1$

When $β = 1$ , the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:

$H(f)|_{\beta=1} = \left \{ \begin{matrix} \frac{1}{2}\left[1 + \cos\left(\pi fT\right)\right], & |f| \leq \frac{1}{T} \\ 0, & \mbox{otherwise} \end{matrix} \right.$

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. :

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

Application

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 $n T$ (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. In Telecommunications a matched filter is obtained by correlating a known signal, or Template, with an unknown signal to detect the White noise is a random signal (or process with a flat Power spectral density. 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.

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

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

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