在概率论和定向统计中，冯·米塞斯分布（von Mises distribution ）指一种圆上连续概率分布模型。它也被称作循环正态分布（circular normal distribution ）和缠绕正态分布（wrapped normal distribution ）的一种近似，因为它正是正态分布的循环模拟。冯·米塞斯分布可以用于模拟多径衰落的MIMO收发天线的AoA和AoD的统计特性，对实际的信道环境具有较好的拟合效果。
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f ( x ∣ μ , κ ) = e κ c o s ( x − μ ) 2 π I 0 ( κ ) f(x|\mu, \kappa)=\dfrac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)} f ( x ∣ μ , κ ) = 2 π I 0 ( κ ) e κ c o s ( x − μ )
也可以表达为级数形式
f ( x ∣ μ , κ ) = 1 2 π ( 1 + 2 I 0 ( κ ) ∑ j = 1 ∞ I j ( κ ) cos [ j ( x − μ ) ] ) f(x \mid \mu, \kappa)=\frac{1}{2 \pi}\left(1+\frac{2}{I_{0}(\kappa)} \sum_{j=1}^{\infty} I_{j}(\kappa) \cos [j(x-\mu)]\right) f ( x ∣ μ , κ ) = 2 π 1 ( 1 + I 0 ( κ ) 2 j = 1 ∑ ∞ I j ( κ ) cos [ j ( x − μ ) ] )
x x x 2 π 2\pi 2 π
I j ( κ ) I_j(\kappa) I j ( κ ) j j j
μ \mu μ μ \mu μ
κ \kappa κ κ > 0 \kappa > 0 κ > 0
当κ \kappa κ ∼ \sim ∼ f ( x ) = 1 2 π f(x)=\dfrac{1}{2\pi} f ( x ) = 2 π 1
当κ \kappa κ ∞ \infty ∞ μ \mu μ 1 κ \frac{1}{\kappa} κ 1 f ( x ) = κ 2 π exp ( − κ ( x − μ ) 2 2 ) f(x)=\frac{\sqrt{\kappa}}{\sqrt{2 \pi}} \exp \left(-\frac{\kappa(x-\mu)^{2}}{2 }\right) f ( x ) = 2 π κ exp ( − 2 κ ( x − μ ) 2 )
N o n e None N o n e
μ \mu μ
v a r ( x ) = 1 − I 1 ( κ ) I 0 ( κ ) var(x)=1-\dfrac{I_1(\kappa)}{I_0(\kappa)} v a r ( x ) = 1 − I 0 ( κ ) I 1 ( κ )
− κ I 1 ( κ ) I 0 ( κ ) + ln [ 2 π I 0 ( κ ) ] -\kappa \frac{I_{1}(\kappa)}{I_{0}(\kappa)}+\ln \left[2 \pi I_{0}(\kappa)\right] − κ I 0 ( κ ) I 1 ( κ ) + ln [ 2 π I 0 ( κ ) ]
