In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ... and
statistics , the raised cosine distribution is a continuous
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ... supported on the interval
mu-s,\mu+s /math>. The probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ... f(x;\mu,s)=\frac
\left +\cos\left(\frac\,\pi\right)\right ,=\frac\operatorname\left(\frac\,\pi\right)\,
for \mu-s \le x \le \mu+s and zero otherwise. The cumulative distribution function (CDF) is
:F(x;\mu,s)=\frac\left +\frac + \frac \sin\left(\frac \, \pi \right) \right /math>
for \mu-s \le x \le \mu+s and zero for x<\mu-s and unity for x>\mu+s .
The moments of the raised cosine distribution are somewhat complicated in the general case, but are considerably simplified for the standard raised cosine distribution. The standard raised cosine distribution is just the raised cosine distribution with \mu=0 and s=1 . Because the standard raised cosine distribution is an even function , the odd moments are zero. The even moments are given by:
:
\begin
\operatorname E(x^) & = \frac\int_^1 +\cos(x\pi) ^\,dx = \int_^1 x^ \operatorname(x\pi)\,dx \\ pt & = \frac+\frac\,_1F_2 \left(n+\frac; \frac, n+\frac; \frac \right)
\end
where \,_1F_2 is a generalized hypergeometric function
In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, whic ... See also
* Hann function
The Hann function is named after the Austrian meteorologist Julius von Hann. It is a window function used to perform Hann smoothing. The function, with length L and amplitude 1/L, is given by:
:
w_0(x) \triangleq \left\.
For digital sign ...
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