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 \beta =1} ) is a cosine function, 'raised' up to sit above the f {\displaystyle f} (horizontal) axis.
Mathematical description
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:
H ( f ) = { 1 , | f | ≤ 1 − β 2 T 1 2 [ 1 + cos ( π T β [ | f | − 1 − β 2 T ] ) ] , 1 − β 2 T < | f | ≤ 1 + β 2 T 0 , otherwise {\displaystyle H(f)={\begin{cases}1,&|f|\leq {\frac {1-\beta }{2T}}\\{\frac {1}{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,&{\text{otherwise}}\end{cases}}}
or in terms of havercosines:
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