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| The '''raised-cosine filter''' is a [[filter (signal processing)|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 (<math>\beta = 1</math>) is a [[cosine]] function, 'raised' up to sit above the <math>f</math> (horizontal) axis.
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| ==Mathematical description==
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| [[Image:Raised-cosine filter.svg|thumb|right|300px|Frequency response of raised-cosine filter with various roll-off factors]]
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| [[Image:Raised-cosine-impulse.svg|thumb|300px|right|Impulse response of raised-cosine filter with various roll-off factors]]
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| The raised-cosine filter is an implementation of a low-pass [[Nyquist ISI criterion|Nyquist filter]], i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd [[symmetry]] about <math>\frac{1}{2T}</math>, where <math>T</math> is the symbol-period of the communications system.
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| Its frequency-domain description is a [[piecewise]] [[function (mathematics)|function]], given by:
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| :<math>H(f) = \begin{cases}
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| T,
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| & |f| \leq \frac{1 - \beta}{2T} \\
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| \frac{T}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left[|f| - \frac{1 - \beta}{2T}\right]\right)\right],
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| & \frac{1 - \beta}{2T} < |f| \leq \frac{1 + \beta}{2T} \\
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| 0,
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| & \mbox{otherwise}
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| \end{cases}</math>
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| :<math>0 \leq \beta \leq 1</math>
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|
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| and characterised by two values; <math>\beta</math>, the ''roll-off factor'', and <math>T</math>, the reciprocal of the symbol-rate.
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| The [[impulse response]] of such a filter <ref>[http://www.commsys.isy.liu.se/TSKS04/lectures/3/MichaelZoltowski_SquareRootRaisedCosine.pdf Michael Zoltowski - Equations for the Raised Cosine and Square-Root Raised Cosine Shapes]</ref> is given by:
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| :<math>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}}</math>, in terms of the normalised [[sinc function]].
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| ===Roll-off factor===
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| The [[roll-off]] factor, <math>\beta</math>, is a measure of the ''excess bandwidth'' of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of <math>\frac{1}{2T}</math>. If we denote the excess bandwidth as <math>\Delta f</math>, then:
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| :<math>\beta = \frac{\Delta f}{\left(\frac{1}{2T}\right)} = \frac{\Delta f}{R_S/2} = 2T\Delta f</math>
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| where <math>R_S = \frac{1}{T}</math> is the symbol-rate.
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| The graph shows the amplitude response as <math>\beta</math> 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 <math>\beta</math> decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.
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| ====<math>\beta = 0</math>====
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| As <math>\beta</math> approaches 0, the roll-off zone becomes infinitesimally narrow, hence:
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| :<math>\lim_{\beta \rightarrow 0}H(f) = \mathrm{rect}(fT)</math>
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| where <math>\mathrm{rect}(.)</math> is the [[rectangular function]], so the impulse response approaches <math>\mathrm{sinc}\left(\frac{t}{T}\right)</math>. Hence, it converges to an ideal or [[brick-wall filter]] in this case.
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| ====<math>\beta = 1</math>====
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| When <math>\beta = 1</math>, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:
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| :<math>H(f)|_{\beta=1} = \left \{ \begin{matrix}
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| \frac{T}{2}\left[1 + \cos\left(\pi fT\right)\right],
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| & |f| \leq \frac{1}{T} \\
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| 0,
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| & \mbox{otherwise}
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| \end{matrix} \right.</math>
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| ===Bandwidth===
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| The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero portion of its spectrum, i.e.:
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| :<math>BW = \frac{1}{2}R_S(\beta+1)</math>(0<T<1)
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| ===Auto-correlation function===
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| The [[auto-correlation]] function of raised cosine function is as follows:
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| <math>R\left(\tau\right) = T \left[\mathrm{sinc}\left( \frac{\tau}{T} \right) \frac{\cos\left( \beta \frac{\pi \tau}{T} \right)}{1 - \left( \frac{2 \beta \tau}{T} \right)^2} - \frac{\beta}{4} \mathrm{sinc}\left(\beta \frac{\tau}{T} \right) \frac{\cos\left( \frac{\pi \tau}{T} \right)}{1 - \left( \frac{\beta \tau}{T} \right)^2} \right]</math>
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| The auto-correlation result can be used to analyze various sampling offset results when analyzed with auto-correlation.
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| ==Application==
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| [[Image:Raised-cosine-ISI.png|thumb|500px|Consecutive raised-cosine impulses, demonstrating zero-ISI property]] | |
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| When used to filter a symbol stream, a Nyquist filter has the property of eliminating ISI, as its impulse response is zero at all <math>nT</math> (where <math>n</math> is an integer), except <math>n = 0</math>.
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| Therefore, if the transmitted waveform is correctly sampled at the receiver, the original symbol values can be recovered completely.
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| 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 <u>net</u> response of the transmit and receive filters that must equal <math>H(f)</math>''':'''
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| :<math>H_R(f)\cdot H_T(f) = H(f)</math>
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| And therefore''':'''
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| :<math>|H_R(f)| = |H_T(f)| = \sqrt{|H(f)|}</math>
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| These filters are called [[root-raised-cosine filter|root-raised-cosine]] filters.
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| ==References==
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| {{Reflist}}
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| * Glover, I.; Grant, P. (2004). ''Digital Communications'' (2nd ed.). Pearson Education Ltd. ISBN 0-13-089399-4.
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| * Proakis, J. (1995). ''Digital Communications'' (3rd ed.). McGraw-Hill Inc. ISBN 0-07-113814-5.
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| * Tavares, L.M.; Tavares G.N. (1998) ''Comments on "Performance of Asynchronous Band-Limited DS/SSMA Systems" ''. IEICE Trans. Commun., Vol. E81-B, No. 9
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| ==External links==
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| *[http://www.nonstopsystems.com/radio/article-raised-cosine.pdf Technical article entitled "The care and feeding of digital, pulse-shaping filters"] originally published in RF Design, written by Ken Gentile.
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| [[Category:Linear filters]]
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| [[Category:Telecommunication theory]]
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Greetings. Let me begin by telling you the author's name - Phebe. Years in the past we moved to North Dakota. To gather coins is what his family and him appreciate. Managing individuals has been his day job for a whilst.
Feel free to surf to my website - http://www.animecontent.com