Raised-cosine filter

The raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form ( β = 1 {displaystyle eta =1} ) is a cosine function, 'raised' up to sit above the f {displaystyle f} (horizontal) axis. The raised-cosine filter is a filter frequently used for pulse-shaping in digital modulation due to its ability to minimise intersymbol interference (ISI). Its name stems from the fact that the non-zero portion of the frequency spectrum of its simplest form ( β = 1 {displaystyle eta =1} ) is a cosine function, 'raised' up to sit above the f {displaystyle f} (horizontal) axis. The raised-cosine filter is an implementation of a low-pass Nyquist filter, i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd symmetry about 1 2 T {displaystyle {frac {1}{2T}}} , where T {displaystyle T} is the symbol-period of the communications system. Its frequency-domain description is a piecewise-defined function, given by: or in terms of havercosines: