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In [[signal processing]], a '''root-raised-cosine filter''' ('''RRC'''), sometimes known as '''square-root-raised-cosine filter''' ('''SRRC'''), is frequently used as the transmit and receive filter in a [[digital communication]] system to perform [[Matched filter|matched filtering]]. This helps in minimizing [[intersymbol interference]] (ISI). The combined response of two such filters is that of the [[raised-cosine filter]].  It obtains its name from the fact that its frequency response, <math>H_{rrc}(f)</math>, is the square root of the frequency response of the raised-cosine filter, <math>H_{rc}(f)</math>:


:<math>H_{rc}(f) = H_{rrc}(f)\cdot H_{rrc}(f)</math>
or:


:<math>|H_{rrc}(f)| = \sqrt{|H_{rc}(f)|}</math>
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== Why is it required ==
To have minimum ISI ([[Intersymbol interference]]), the overall response of transmit filter, channel response and receive filter has to satisfy [[Nyquist ISI criterion]]. [[Raised-cosine filter]] is the most popular filter response satisfying this criterion. Half of this filtering is done on the transmit side and half of this is done on the receive side. On the receive side, the channel response, if it can be accurately estimated, can also be taken into account so that the overall response is [[Raised-cosine filter]].
 
== Mathematical Description ==
[[File:Srrc.png|thumb|500px|The impulse response of a root-raised cosine filter for three values of ''β'': 1.0 (blue), 0.5 (red) and 0 (green).]]
The RRC filter is characterised by two values; ''β'', the ''[[roll-off]] factor'', and ''T<sub>s</sub>'', the reciprocal of the symbol-rate.
 
The [[impulse response]] of such a filter can be given as:
 
:<math>h(t) = \begin{cases}
1-\beta+4\dfrac{\beta}{\pi},
      & t = 0 \\
 
\dfrac{\beta}{\sqrt{2}}
\left[
\left(1+\dfrac{2}{\pi}\right)\sin\left(\dfrac{\pi}{4\beta}\right) +
\left(1-\dfrac{2}{\pi}\right)\cos\left(\dfrac{\pi}{4\beta}\right)
\right],
      & t = \pm \dfrac{T_s}{4\beta} \\
 
\dfrac{\sin\left[\pi \dfrac{t}{T_s}\left(1-\beta\right)\right] + 4\beta\dfrac{t}{T_s}\cos\left[\pi\dfrac{t}{T_s}\left(1+\beta\right)\right]}{\pi \dfrac{t}{T_s}\left[1-\left(4\beta\dfrac{t}{T_s} \right)^2 \right]},
      & \mbox{otherwise}
\end{cases}</math>,
though there are other forms as well.
 
Unlike the raised-cosine filter, the impulse response is not zero at the intervals of ±''T<sub>s</sub>''. However, the combined transmit and receive filters form a raised-cosine filter which does have zero at the intervals of ±''T<sub>s</sub>''. Only in the case of ''β''=0 does the root raised-cosine have zeros at ±''T<sub>s</sub>''.
 
== References ==
 
* S. Daumont, R. Basel, Y. Lout, "Root-Raised Cosine filter influences on PAPR distribution of single carrier signals", ISCCSP 2008, Malta, 12-14 March 2008.
* Proakis, J. (1995). ''Digital Communications'' (3rd ed.). McGraw-Hill Inc. ISBN 0-07-113814-5.
[[Category:Linear filters]]
[[Category:Telecommunication theory]]

Latest revision as of 20:50, 8 January 2015


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