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Raised cosine
Probability density function
Cumulative distribution function
Parameters
μ
{\displaystyle \mu \,}
real )
s
>
0
{\displaystyle s>0\,}
real ) Support
x
∈
[
μ
−
s
,
μ
+
s
]
{\displaystyle x\in [\mu -s,\mu +s]\,}
PDF
1
2
s
[
1
+
cos
(
x
−
μ
s
π
)
]
=
1
s
hvc
(
x
−
μ
s
π
)
{\displaystyle {\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,}
CDF
1
2
[
1
+
x
−
μ
s
+
1
π
sin
(
x
−
μ
s
π
)
]
{\displaystyle {\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}
Mean
μ
{\displaystyle \mu \,}
Median
μ
{\displaystyle \mu \,}
Mode
μ
{\displaystyle \mu \,}
Variance
s
2
(
1
3
−
2
π
2
)
{\displaystyle s^{2}\left({\frac {1}{3}}-{\frac {2}{\pi ^{2}}}\right)\,}
Skewness
0
{\displaystyle 0\,}
Excess kurtosis
6
(
90
−
π
4
)
5
(
π
2
−
6
)
2
=
−
0.59376
…
{\displaystyle {\frac {6(90-\pi ^{4})}{5(\pi ^{2}-6)^{2}}}=-0.59376\ldots \,}
MGF
π
2
sinh
(
s
t
)
s
t
(
π
2
+
s
2
t
2
)
e
μ
t
{\displaystyle {\frac {\pi ^{2}\sinh(st)}{st(\pi ^{2}+s^{2}t^{2})}}\,e^{\mu t}}
CF
π
2
sin
(
s
t
)
s
t
(
π
2
−
s
2
t
2
)
e
i
μ
t
{\displaystyle {\frac {\pi ^{2}\sin(st)}{st(\pi ^{2}-s^{2}t^{2})}}\,e^{i\mu t}}
In probability theory and statistics , the raised cosine distribution is a continuous probability distribution supported on the interval
[
μ
−
s
,
μ
+
s
]
{\displaystyle [\mu -s,\mu +s]}
probability density function (PDF) is
f
(
x
;
μ
,
s
)
=
1
2
s
[
1
+
cos
(
x
−
μ
s
π
)
]
=
1
s
hvc
(
x
−
μ
s
π
)
for
μ
−
s
≤
x
≤
μ
+
s
{\displaystyle f(x;\mu ,s)={\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right){\text{ for }}\mu -s\leq x\leq \mu +s}
and zero otherwise. The cumulative distribution function (CDF) is
F
(
x
;
μ
,
s
)
=
1
2
[
1
+
x
−
μ
s
+
1
π
sin
(
x
−
μ
s
π
)
]
{\displaystyle F(x;\mu ,s)={\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}
for
μ
−
s
≤
x
≤
μ
+
s
{\displaystyle \mu -s\leq x\leq \mu +s}
x
<
μ
−
s
{\displaystyle x<\mu -s}
x
>
μ
+
s
{\displaystyle 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
μ
=
0
{\displaystyle \mu =0}
s
=
1
{\displaystyle s=1}
even function , the odd moments are zero. The even moments are given by:
E
(
x
2
n
)
=
1
2
∫
−
1
1
[
1
+
cos
(
x
π
)
]
x
2
n
d
x
=
∫
−
1
1
x
2
n
hvc
(
x
π
)
d
x
=
1
n
+
1
+
1
1
+
2
n
1
F
2
(
n
+
1
2
;
1
2
,
n
+
3
2
;
−
π
2
4
)
{\displaystyle {\begin{aligned}\operatorname {E} (x^{2n})&={\frac {1}{2}}\int _{-1}^{1}[1+\cos(x\pi )]x^{2n}\,dx=\int _{-1}^{1}x^{2n}\operatorname {hvc} (x\pi )\,dx\\[5pt]&={\frac {1}{n+1}}+{\frac {1}{1+2n}}\,_{1}F_{2}\left(n+{\frac {1}{2}};{\frac {1}{2}},n+{\frac {3}{2}};{\frac {-\pi ^{2}}{4}}\right)\end{aligned}}}
where
1
F
2
{\displaystyle \,_{1}F_{2}}
generalized hypergeometric function .
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
![{\displaystyle x\in [\mu -s,\mu +s]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/021cb61824dc30c9ce4228710410d45d7b8ea2dd)
![{\displaystyle {\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8fe6565ff842d25cf9ac9946e3454f278992d8)
![{\displaystyle {\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5fe6b908cecf264d0bc4a34c554b027ad3bb88)
![{\displaystyle [\mu -s,\mu +s]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4092914759a1beeec30258141ccc43c0686f8459)
![{\displaystyle f(x;\mu ,s)={\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\,\pi \right)\right]\,={\frac {1}{s}}\operatorname {hvc} \left({\frac {x-\mu }{s}}\,\pi \right){\text{ for }}\mu -s\leq x\leq \mu +s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10e881d815068ad8b253740178c997fe2d569289)
![{\displaystyle F(x;\mu ,s)={\frac {1}{2}}\left[1+{\frac {x-\mu }{s}}+{\frac {1}{\pi }}\sin \left({\frac {x-\mu }{s}}\,\pi \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d946fb3c3452f89b48341393ced089a0699fdffd)
![{\displaystyle {\begin{aligned}\operatorname {E} (x^{2n})&={\frac {1}{2}}\int _{-1}^{1}[1+\cos(x\pi )]x^{2n}\,dx=\int _{-1}^{1}x^{2n}\operatorname {hvc} (x\pi )\,dx\\[5pt]&={\frac {1}{n+1}}+{\frac {1}{1+2n}}\,_{1}F_{2}\left(n+{\frac {1}{2}};{\frac {1}{2}},n+{\frac {3}{2}};{\frac {-\pi ^{2}}{4}}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd88646853daa97101c07fa637ef17568602b698)
