高斯相除

The distribution of Complex Gaussian is

\begin{aligned} (1) & N_{c} (x | a, A) \propto \exp [- (x - a)^{H} A^{- 1} (x - a)] \end{aligned}

$\begin{align} \mathcal{N}_c(\boldsymbol{x}|\boldsymbol{a},\boldsymbol{A})\propto \exp\left[{-(\boldsymbol{x}-\boldsymbol{a})^H\boldsymbol{A}^{-1}(\boldsymbol{x}-\boldsymbol{a})}\right] \end{align}$
Division of two complex Gaussian distribution

\begin{aligned} (2) & \frac{N_{c} (x | a, A)}{N_{c} (x | b, B)} & \propto \exp [- (x - a)^{H} A^{- 1} (x - a) + (x - b)^{H} B^{- 1} (x - b)] \\ (3) & \propto \exp [- (x^{H} (A^{- 1} - B^{- 1}) x - x^{H} (A^{- 1} a - B^{- 1} b) - (a^{H} A^{- 1} - b^{H} B) x)] \\ (4) & \propto \exp {- [x - (A^{- 1} - B^{- 1})^{- 1} (A^{- 1} a - B^{- 1} b)]^{H} (A^{- 1} - B^{- 1}) [x - (A^{- 1} - B^{- 1})^{- 1} (A^{- 1} a - B^{- 1} b)]} \end{aligned}

$\begin{align} \frac{\mathcal{N}_c(\boldsymbol{x}|\boldsymbol{a},\boldsymbol{A})}{\mathcal{N}_c(\boldsymbol{x}|\boldsymbol{b},\boldsymbol{B})}&\propto\exp\left[{-(\boldsymbol{x}-\boldsymbol{a})^H\boldsymbol{A}^{-1}(\boldsymbol{x}-\boldsymbol{a})+(\boldsymbol{x}-\boldsymbol{b})^H\boldsymbol{B}^{-1}(\boldsymbol{x}-\boldsymbol{b})}\right]\\ &\propto \exp\left[{-\left({\boldsymbol{x}^H(\boldsymbol{A}^{-1}-\boldsymbol{B}^{-1})\boldsymbol{x}-\boldsymbol{x}^H(\boldsymbol{A}^{-1}\boldsymbol{a}-\boldsymbol{B}^{-1}\boldsymbol{b})-(\boldsymbol{a}^H\boldsymbol{A}^{-1}-\boldsymbol{b}^H\boldsymbol{B})\boldsymbol{x}}\right)}\right]\\ &\propto \exp \left\{{-[\boldsymbol{x}-(\boldsymbol{A}^{-1}-\boldsymbol{B}^{-1})^{-1}(\boldsymbol{A}^{-1}\boldsymbol{a}-\boldsymbol{B}^{-1}\boldsymbol{b})]^{H}(\boldsymbol{A}^{-1}-\boldsymbol{B}^{-1})[\boldsymbol{x}-(\boldsymbol{A}^{-1}-\boldsymbol{B}^{-1})^{-1}(\boldsymbol{A}^{-1}\boldsymbol{a}-\boldsymbol{B}^{-1}\boldsymbol{b})]}\right\} \end{align}$
Based on above derivation, we have

\begin{aligned} (5) & N_{c} (x | a, A) N_{c} (x | b, B) \propto N (x | (A^{- 1} - B^{- 1})^{- 1} (A^{- 1} a - B^{- 1} b), (A^{- 1} - B^{- 1})^{- 1}) \end{aligned}

$\begin{align} \mathcal{N}_c(\boldsymbol{x}|\boldsymbol{a},\boldsymbol{A})\mathcal{N}_c(\boldsymbol{x}|\boldsymbol{b},\boldsymbol{B})\propto \mathcal{N}(\boldsymbol{x}|(\boldsymbol{A}^{-1}-\boldsymbol{B}^{-1})^{-1}(\boldsymbol{A}^{-1}\boldsymbol{a}-\boldsymbol{B}^{-1}\boldsymbol{b}),(\boldsymbol{A}^{-1}-\boldsymbol{B}^{-1})^{-1}) \end{align}$

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