I recently asked a question here regarding limiting the bandwidth of a digital baseband signal and was introduced to pulse-shaping using a "raised-cosine filter". At the time, I mistakenly assumed this to be a sinc function combined with the Hann window. This is because the Hann window is also known as the raised cosine window, and that seemed logical to me. However, after reading on Wikipedia, I see that the "raised-cosine filter" is actually another beast entirely. This function appears to be a sinc function windowed by another sinc-like function, with two parameters, β and T. The extent of this function, like sinc, appears to be infinite, for all values of β and T. This thoroughly confuses me, as I thought the whole point of windowing the sinc function was to give it finite support. What am I missing?
An ideal raised cosine impulse is band-limited with cut-off frequency $f_c=(1+\beta)/(2T)$, where $\beta$ ($0\le\beta\le 1$) is the roll-off factor. Since it is band-limited, it must have infinite support in the time domain. The important difference with a sinc impulse is that its envelope decays much faster (for $\beta>0$, of course).
Note that even though an ideal raised cosine filter has infinite support, in a practical implementation you would use a filter with finite support (implemented as an FIR filter) that approximates the ideal raised cosine impulse in some interval around its maximum.
“ Ideal raised cosine filters have an infinite number of taps. Therefore, practical raised cosine filters are windowed. The window length is controlled using the FilterSpanInSymbols property. In this example, we specify the window length as six symbol durations, i.e., the filter spans six symbol durations.”