Mathematical Description
The RRC filter is characterised by two values; β, the roll-off factor, and Ts, the reciprocal of the symbol-rate.
The impulse response of such a filter can be given as:
![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 + 4\beta\dfrac{t}{T_s}\cos\left}{\pi \dfrac{t}{T_s}\left}, & \mbox{otherwise}
\end{cases}](http://upload.wikimedia.org/math/3/a/8/3a869aeb0b5355038363296d70fd0fef.png)
,
though there are other forms as well.
Unlike the raised-cosine filter, the impulse response is not zero at the intervals of ±Ts. However, the combined transmit and receive filters form a raised-cosine filter which does have zero at the intervals of ±Ts. Only in the case of β=0 does the root raised-cosine have zeros at ±Ts.
Read more about this topic: Root-raised-cosine Filter
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