考虑一个随机向量 x ∼ p X ( 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 q q,将后验估计的均值表示为 F i n ( 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 E i n ( 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 F i n ( q , Σ ) F_{in}(\boldsymbol q,\boldsymbol \Sigma) Fin(q,Σ)与协方差 E i n ( q , Σ ) \mathcal E_{in}(\boldsymbol q, \boldsymbol \Sigma) Ein(q,Σ)满足如下关系式
∂ ∂ q F i n ( q , Σ ) = E i n ( 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 0 Σ>0(正定),定义函数
A 0 ( q ) = ∫ p X ( x ) ϕ ( q − x ; Σ ) d x A 1 ( q ) = ∫ x p X ( x ) ϕ ( q − x ; Σ ) d x A 2 ( q ) = ∫ x x T p X ( x ) ϕ ( q − x ; Σ ) d x \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 p Q ∣ X p_{\boldsymbol Q|\boldsymbol X} pQ∣X,均值为 x \boldsymbol x x协方差为 Σ \boldsymbol \Sigma Σ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 A 0 ( q ) A_0(\boldsymbol q) A0(q)
A 0 ( q ) = ∫ p X ( x ) ϕ ( q − x ; Σ ) d x = ∫ p X ( x ) p Q ∣ X ( q ∣ x ) d x = p Q ( 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
F i n ( q , Σ ) = A 1 ( q ) A 0 ( 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 [ w w T ] − E [ w ] E [ w T ] \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
E i n ( q , Σ ) = A 2 ( q ) A 0 ( q ) − A 1 2 ( q ) A 0 2 ( 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
∂ ∂ q ϕ ( q − x ; Σ ) = ϕ ( q − x ; Σ ) ⋅ ( x − q ) T Σ − 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
∂ ∂ q F i n ( q , Σ ) = ∂ ∂ q A 1 ( q ) A 0 ( q ) = ∂ A 1 ( q ) ∂ q A 0 ( q ) − A 1 ( q ) ∂ A 0 ( q ) ∂ q A 0 2 ( q ) = A 2 ( q ) Σ − 1 A 0 ( q ) − A 1 ( q ) A 1 T ( q ) Σ − 1 A 0 2 ( q ) = E i n ( 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。
考虑一个复随机向量 x ∼ p X ( x ) \boldsymbol x \sim p_{\boldsymbol X}(\boldsymbol x) x∼pX(x),信道模型为
q = x + v , v ∼ C N ( 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:
求导主要体现在
∂ ∂ q ∗ ϕ ( q − x ; Σ ) = ϕ ( q − x ; Σ ) ⋅ ( x − q ) H Σ − 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 ∗ F i n ( q , Σ ) = E i n ( 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
AWGN信道向量模型为
q = x + v , x ∼ p X ( 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
∂ ∂ 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
复数域( v ∼ C N ( 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 , σ 2 ) \nu \sim \mathcal{N}(0, \sigma^2) ν∼N(0,σ2),则
实数域
∂ ∂ q E [ x ∣ q ] = 1 σ 2 var [ 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 ∼ C N ( 0 , σ 2 ) v \sim \mathcal {CN}(0, \sigma^2) v∼CN(0,σ2))
∂ ∂ q ∗ E [ x ∣ q ] = 1 σ 2 var [ 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 x x的先验分布 p X ( x ) p_{\boldsymbol X}(\boldsymbol x) pX(x)做任何要求。