Introduction

The raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI) as it fulfills the 1st Nyquist criterion. Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form ($\beta = 1$) is a cosine function, 'raised' up to sit above the f (horizontal) axis.

Its frequency-domain description is a piecewise function, given by: $$ H(f) = \begin{cases} T, & |f| \leq \frac{1 - \beta}{2T} \\ \frac{T}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left[|f| - \frac{1 - \beta}{2T}\right]\right)\right], & \frac{1 - \beta}{2T} < |f| \leq \frac{1 + \beta}{2T} \\ 0, & \mbox{otherwise}\end{cases} $$