User:Disselboom/SRRC

Square Root Raised Cosine Filters (SRRC), or root raised-cosine filters, are used in transmit and receive filtering to perform matched filtering. Their combined response is that of the raised-cosine filter. They obtain their names from the fact that their frequency response, $$H_{srrc}(f)$$, is the square root of the frequency of the raised-cosine filter, $$H_{rc}(f)$$:


 * $$H_{rc}(f) = H_{srrc}(f)\cdot H_{srrc}(f)$$

or:


 * $$|H_{srrc}(f)| = \sqrt{|H_{rc}(f)|}$$

Mathematical Description
The SRRC filter is characterised by two values; $$\beta$$, the roll-off factor, and $$T_s$$, the reciprocal of the symbol-rate.

The impulse response of such a filter can be given as:


 * $$h(t) = \begin{cases}

1-\beta+4\frac{\beta}{\pi}, & t = 0 \\ \frac{\beta}{\sqrt{2}}\left[\left(1+\frac{2}{\pi}\right)\sin\left(\frac{\pi}{4\beta}\right) + \left(1-\frac{2}{\pi}\cos\left(\frac{\pi}{4\beta}\right)\right)\right], & t = \pm \frac{T_s}{4\beta} \\ \frac{\sin\left[\pi \frac{t}{T_s}\left(1-\beta\right)\right] + 4\beta\frac{t}{T_s}\cos\left[\pi\frac{t}{T_s}\left(1+\beta\right)\right]}{\pi \frac{t}{T_s}\left[1-\left(4\beta\frac{t}{T_s} \right)^2 \right]}, & \mbox{otherwise} \end{cases}$$, though there are other forms as well.

It should be noted that unlike the raised-cosine filter, the impulse response is not zero at the intervals of $$\pm T_s$$. However, the combined transmit and receive filters form a raised-cosine filter which does have zero at the intervals of $$\pm T_s$$. Only in the case of $$\beta = 0$$, does the root raised-cosine have zeros at $$\pm T_s$$