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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
q=x+v,v∼N(0,Σ)\boldsymbol q = \boldsymbol x + \boldsymbol v, \ \ \ \boldsymbol v \sim \mathcal N(\boldsymbol 0, \boldsymbol \Sigma)q=x+v, 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Σ>0\boldsymbol \Sigma > \boldsymbol 0Σ>0（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)dx=pQ(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)dx=pQ(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−1=Ein(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
q=x+v,v∼CN(0,Σ)\boldsymbol q = \boldsymbol x + \boldsymbol v, \ \ \ \boldsymbol v \sim \mathcal {CN}(\boldsymbol 0, \boldsymbol \Sigma)q=x+v, 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
q=x+v,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)q=x+v, 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。