青岛网站推,wordpress主题切换,seo实战视频,公众号文章存储wordpress文章目录AWGN信道向量模型后验均值与协方差的关系从实数域拓展到复数域小结AWGN信道向量模型
考虑一个随机向量x∼pX(x)\boldsymbol x \sim p_{\boldsymbol X}(\boldsymbol x)x∼pX(x)#xff0c;信道模型为 qxv,v∼N(0,Σ)\boldsymbol q \boldsymbol x \boldsymbol v, \…
文章目录AWGN信道向量模型后验均值与协方差的关系从实数域拓展到复数域小结AWGN信道向量模型
考虑一个随机向量x∼pX(x)\boldsymbol x \sim p_{\boldsymbol X}(\boldsymbol x)x∼pX(x)信道模型为
qxv,v∼N(0,Σ)\boldsymbol q \boldsymbol x \boldsymbol v, \ \ \ \boldsymbol v \sim \mathcal N(\boldsymbol 0, \boldsymbol \Sigma)qxv, v∼N(0,Σ)
已知观测值q\boldsymbol qq将后验估计的均值表示为Fin(q,Σ)E[x∣q]F_{in}(\boldsymbol q,\boldsymbol \Sigma)\mathbb E[\boldsymbol x| \boldsymbol q]Fin(q,Σ)E[x∣q]协方差表示为Ein(q,Σ)Cov[x∣q]\mathcal E_{in}(\boldsymbol q, \boldsymbol \Sigma)\text{Cov}[\boldsymbol x| \boldsymbol q]Ein(q,Σ)Cov[x∣q]。
后验均值与协方差的关系
后验均值Fin(q,Σ)F_{in}(\boldsymbol q,\boldsymbol \Sigma)Fin(q,Σ)与协方差Ein(q,Σ)\mathcal E_{in}(\boldsymbol q, \boldsymbol \Sigma)Ein(q,Σ)满足如下关系式
∂∂qFin(q,Σ)Ein(q,Σ)Σ−1\frac{\partial}{\partial \boldsymbol q} F_{in}(\boldsymbol q, \boldsymbol \Sigma) \mathcal E_{in}(\boldsymbol q,\boldsymbol \Sigma) \boldsymbol \Sigma^{-1}∂q∂Fin(q,Σ)Ein(q,Σ)Σ−1
证明对Σ0\boldsymbol \Sigma \boldsymbol 0Σ0正定定义函数
A0(q)∫pX(x)ϕ(q−x;Σ)dxA1(q)∫xpX(x)ϕ(q−x;Σ)dxA2(q)∫xxTpX(x)ϕ(q−x;Σ)dx\begin{aligned} A_0(\boldsymbol q) \int p_{\boldsymbol X}(\boldsymbol x) \phi(\boldsymbol q-\boldsymbol x; \boldsymbol \Sigma) \mathrm{d} \boldsymbol x \\ A_1(\boldsymbol q) \int \boldsymbol x p_{\boldsymbol X}(\boldsymbol x) \phi(\boldsymbol q-\boldsymbol x; \boldsymbol \Sigma) \mathrm{d} \boldsymbol x \\ A_2(\boldsymbol q) \int \boldsymbol {xx}^T p_{\boldsymbol X}(\boldsymbol x) \phi(\boldsymbol q-\boldsymbol x; \boldsymbol \Sigma) \mathrm{d} \boldsymbol x \\ \end{aligned} A0(q)A1(q)A2(q)∫pX(x)ϕ(q−x;Σ)dx∫xpX(x)ϕ(q−x;Σ)dx∫xxTpX(x)ϕ(q−x;Σ)dx
其中ϕ(q−x;Σ)\phi(\boldsymbol q-\boldsymbol x; \boldsymbol \Sigma)ϕ(q−x;Σ)表示似然分布pQ∣Xp_{\boldsymbol Q|\boldsymbol X}pQ∣X均值为x\boldsymbol xx协方差为Σ\boldsymbol \SigmaΣ的高斯分布即
ϕ(q−x;Σ)≡N(x,Σ)\phi(\boldsymbol q-\boldsymbol x; \boldsymbol \Sigma) \equiv \mathcal {N}(\boldsymbol x, \boldsymbol \Sigma)ϕ(q−x;Σ)≡N(x,Σ)
特殊地先考虑A0(q)A_0(\boldsymbol q)A0(q)
A0(q)∫pX(x)ϕ(q−x;Σ)dx∫pX(x)pQ∣X(q∣x)dxpQ(q)\begin{aligned} A_0(\boldsymbol q) \int p_{\boldsymbol X}(\boldsymbol x) \phi(\boldsymbol q-\boldsymbol x;\boldsymbol \Sigma) \mathrm{d} \boldsymbol x \\ \int p_{\boldsymbol X}(\boldsymbol x) p_{\boldsymbol Q|\boldsymbol X}(\boldsymbol q| \boldsymbol x) \mathrm{d} \boldsymbol x \\ p_{\boldsymbol Q}(\boldsymbol q) \end{aligned} A0(q)∫pX(x)ϕ(q−x;Σ)dx∫pX(x)pQ∣X(q∣x)dxpQ(q)
根据期望的定义可以写出
Fin(q,Σ)A1(q)A0(q)F_{in}(\boldsymbol q,\boldsymbol \Sigma) \frac{A_1(\boldsymbol q)}{A_0(\boldsymbol q)}Fin(q,Σ)A0(q)A1(q)
根据Cov[w]E[wwT]−E[w]E[wT]\text{Cov}[\boldsymbol w] \mathbb E[\boldsymbol w \boldsymbol w^T] - \mathbb E[\boldsymbol w] \mathbb E[\boldsymbol w^T]Cov[w]E[wwT]−E[w]E[wT]可以写出
Ein(q,Σ)A2(q)A0(q)−A12(q)A02(q)\mathcal E_{in}(\boldsymbol q,\boldsymbol \Sigma) \frac{A_2(\boldsymbol q)}{A_0(\boldsymbol q)} - \frac{A^2_1(\boldsymbol q)}{A^2_0(\boldsymbol q)}Ein(q,Σ)A0(q)A2(q)−A02(q)A12(q)
对高斯分布求导可得
∂∂qϕ(q−x;Σ)ϕ(q−x;Σ)⋅(x−q)TΣ−1\frac{\partial}{\partial \boldsymbol q} \phi(\boldsymbol q- \boldsymbol x; \boldsymbol \Sigma) \phi(\boldsymbol q- \boldsymbol x; \boldsymbol \Sigma) \cdot {(\boldsymbol x- \boldsymbol q)}^T \boldsymbol \Sigma^{-1}∂q∂ϕ(q−x;Σ)ϕ(q−x;Σ)⋅(x−q)TΣ−1
基于此我们可以得到 ∂∂qFin(q,Σ)∂∂qA1(q)A0(q)∂A1(q)∂qA0(q)−A1(q)∂A0(q)∂qA02(q)A2(q)Σ−1A0(q)−A1(q)A1T(q)Σ−1A02(q)Ein(q,Σ)Σ−1\begin{aligned} \frac{\partial}{\partial \boldsymbol q} F_{in}(\boldsymbol q, \boldsymbol \Sigma) \frac{\partial}{\partial \boldsymbol q} \frac{A_1(\boldsymbol q)}{A_0(\boldsymbol q)} \\ \frac{ \frac{\partial A_1(\boldsymbol q)}{\partial \boldsymbol q} A_0(\boldsymbol q) - A_1(\boldsymbol q) \frac{\partial A_0 (\boldsymbol q)}{\partial \boldsymbol q} } { A^2_0(\boldsymbol q)} \\ \frac{A_2(\boldsymbol q) \boldsymbol \Sigma^{-1}}{A_0(\boldsymbol q)} - \frac{A_1(\boldsymbol q) A^T_1(\boldsymbol q) \boldsymbol \Sigma^{-1}}{A^2_0(\boldsymbol q)} \\ \mathcal E_{in}(\boldsymbol q, \boldsymbol \Sigma) \boldsymbol \Sigma^{-1} \end{aligned} ∂q∂Fin(q,Σ)∂q∂A0(q)A1(q)A02(q)∂q∂A1(q)A0(q)−A1(q)∂q∂A0(q)A0(q)A2(q)Σ−1−A02(q)A1(q)A1T(q)Σ−1Ein(q,Σ)Σ−1
证毕。
从实数域拓展到复数域
考虑一个复随机向量x∼pX(x)\boldsymbol x \sim p_{\boldsymbol X}(\boldsymbol x)x∼pX(x)信道模型为
qxv,v∼CN(0,Σ)\boldsymbol q \boldsymbol x \boldsymbol v, \ \ \ \boldsymbol v \sim \mathcal {CN}(\boldsymbol 0, \boldsymbol \Sigma)qxv, v∼CN(0,Σ)
对于上述推导过程实数域和复数域的差别于一下两个方面
转置-共轭转置只是notation的转换实高斯分布-复高斯分布主要关注求导
求导主要体现在 ∂∂q∗ϕ(q−x;Σ)ϕ(q−x;Σ)⋅(x−q)HΣ−1\frac{\partial}{\partial \boldsymbol q^{*}} \phi(\boldsymbol q- \boldsymbol x; \boldsymbol \Sigma) \phi(\boldsymbol q- \boldsymbol x; \boldsymbol \Sigma) \cdot {(\boldsymbol x- \boldsymbol q)}^H \boldsymbol \Sigma^{-1}∂q∗∂ϕ(q−x;Σ)ϕ(q−x;Σ)⋅(x−q)HΣ−1
类似地可以得到复数域的关系表达式为 ∂∂q∗Fin(q,Σ)Ein(q,Σ)Σ−1\frac{\partial}{\partial \boldsymbol q^{*}} F_{in}(\boldsymbol q, \boldsymbol \Sigma) \mathcal E_{in}(\boldsymbol q,\boldsymbol \Sigma) \boldsymbol \Sigma^{-1}∂q∗∂Fin(q,Σ)Ein(q,Σ)Σ−1
小结
AWGN信道向量模型为 qxv,x∼pX(x)v∼N(0,Σ)\boldsymbol q \boldsymbol x \boldsymbol v, \ \ \ \boldsymbol x \sim p_{\boldsymbol X}(\boldsymbol x) \boldsymbol v \sim \mathcal {N}(\boldsymbol 0, \boldsymbol \Sigma)qxv, x∼pX(x)v∼N(0,Σ)
MMSE估计均值与协方差的关系为 实数域 ∂∂qE[x∣q]Cov[x∣q]Σ−1\frac{\partial}{\partial \boldsymbol q} \mathbb E[\boldsymbol x| \boldsymbol q] \text{Cov}[\boldsymbol x| \boldsymbol q] \boldsymbol \Sigma^{-1}∂q∂E[x∣q]Cov[x∣q]Σ−1 复数域v∼CN(0,Σ)v \sim \mathcal {CN}(\boldsymbol 0, \boldsymbol \Sigma)v∼CN(0,Σ) ∂∂q∗E[x∣q]Cov[x∣q]Σ−1\frac{\partial}{\partial \boldsymbol q^{*}} \mathbb E[\boldsymbol x| \boldsymbol q] \text{Cov}[\boldsymbol x| \boldsymbol q] \boldsymbol \Sigma^{-1}∂q∗∂E[x∣q]Cov[x∣q]Σ−1
退化到标量时令ν∼N(0,σ2)\nu \sim \mathcal{N}(0, \sigma^2)ν∼N(0,σ2)则 实数域 ∂∂qE[x∣q]1σ2var[x∣q]\frac{\partial}{\partial q} \mathbb E[ x| q] \frac{1}{\sigma^2} \text{var}[ x| q] ∂q∂E[x∣q]σ21var[x∣q] 复数域v∼CN(0,σ2)v \sim \mathcal {CN}(0, \sigma^2)v∼CN(0,σ2) ∂∂q∗E[x∣q]1σ2var[x∣q]\frac{\partial}{\partial q^{*}} \mathbb E[ x| q] \frac{1}{\sigma^2} \text{var}[ x| q]∂q∗∂E[x∣q]σ21var[x∣q]
注意上述结论不对x\boldsymbol xx的先验分布pX(x)p_{\boldsymbol X}(\boldsymbol x)pX(x)做任何要求。
