第2ç« +随机信å&middo

第2ç« +随机信å&middo

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1 第二章 随机信号和噪声 2.8 正弦信号加 窄带噪声 2011 10 8 8
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信息传输与处理实验室 蒋铃鸽 2.8 正弦信号加窄带噪声的包络和相位 0 () cos ( ) st a t ω θ = + 是常数, 是 [0 2 π ] 间均匀分布的随机变量 00 ()cos ()s in cs nt n t t n t t = ☆则混合信号为: ( 正交表示 ) 0 cos( ) x ts t n ta t n t At t At t ωω = += + + =− 其中 cc A n t θ+ ± s ss A n t ± a
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信息传输与处理实验室 蒋铃鸽 正弦信号加窄带噪声的包络和相位 [ ] 0 () ()cos x tA t t t ωψ =+ 22 () 0 cs At A t = +≥ 1 () tan 0 () 2 s c tt ψ ψπ =≤ x(t) 还可以写成包络 - 相位形式 :
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信息传输与处理实验室 蒋铃鸽 合成信号的包络与相位的联合分布 z 1 先求 分布,假设 给定 , 则均值与 方差为 : () , () cs A tA t θ { } cos c EA a = { } sin s a = { } { } 2 DA θσ = = 22 cos 10 (, ) ( ,) 1 sin 01 ( , ) p.d.f 11 exp ( cos ) ( sin ) cos ) , sin 1 ( , ) exp 2 cos cc c s ss c s nA a fAA fnn J J a fnn Aa AA fA J J A A A a a A θθ πσ σ ψ ψθ =− = == ⎧⎫ ⎡⎤ + ⎨⎬ ⎣⎦ ⎩⎭ = = + 其中 是二维独立高斯 上式 其中 变换为 θψ ⎛⎞ ⎜⎟ ⎝⎠
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信息传输与处理实验室 蒋铃鸽 合成信号的包络与相位的联合分布 z 2 、求 的一维 p.d.f () A t [] 2 0 2 22 2 0 2 0 0 0 2 ( , ) 11 exp ( ) exp cos( ) 1 exp cos 2 1 e x p ( ) ( ) 2 fA d Aa A A ad Ix x d A A a I π θψ θ ψ ψψ σσ σ = ⎫⎡ =− + ⎨⎬ ⎩⎭ = ⎧⎫ =−+ = ⎜⎟ 引入零阶修正贝塞尔函数: A 为莱斯分布
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信息传输与处理实验室 蒋铃鸽 零阶修正贝塞尔函数 () 2 0 22 0 2( ! ) n n n x Ix n = = 01 x ± 2 24 4 0 1 46 4 x xx e =+ + + " 1 x ² 0 2 x e x π ( ) 0 {} 2 0 0 1 exp cos 2 x d φ ³ 级数表示: 近似式:
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信息传输与处理实验室 蒋铃鸽 小信噪比时 2 22 () e x p 2 AA fA σ =− ⎩⎭ 2 da ± 定义 2 0 2 e x p 2 d A I d σσ ⎛⎞ ⎜⎟ ⎝⎠ 小信噪比的情况下, a
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第2ç« +随机信å&middo

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