Raised Cosine Filters exist primarily to shape pulses for use in communications systems. (Excellent background information on this subject may be found in Ken Gentile's article, 0402Gentile50.pdf, published by RF Design in April, 2002).

The ideal raised cosine filter frequency response consists of unity gain at low frequencies, a raised cosine function in the middle, and total attenuation at high frequencies. The width of the middle frequencies is defined by the roll-off factor constant, Alpha, (0<Alpha<1).

In FilterSolutions, the passband frequency is defined as the 50% signal attenuation point.

When the pass band frequency of a raised cosine filter is set to half the sample rate, then the impulse response, Nyquist's first criteria, is satisfied, i.e.: the impulse response is zero, for T = NTs, where N is an integer, and T is the sample rate.

Mathematically, the frequency response may be written as:

The ideal raised cosine filter frequency response is shown below: