Ephraim and Malah suppression rule [3]

This involves deriving the MMSE STSA estimator using a complex Gaussian model of the a priori probability distribution of speech and noise Fourier expansion coefficients. If $y[n]=x[n]+b[n]$ and $X(k)=A_k exp(j\alpha_k)$, then the MMSE estimator of $A_k$ is

$$\begin{array}{lll} \hat{A}_k & = & \mathcal{E}[A_k\vert Y_... ...\alpha_k) P(a_k, \alpha_k) d \alpha_k d a_k }\\ \end{array}$$

With the assumption of Fourier coefficients having a Gaussian distribution, the polar form of the coefficients have the following marginal distribution

$$P(a_k)=\left\{ \begin{array}{ll} \frac{2a_k}{\lambda_x(k... ...0,\infty)\\ 0 & \textrm{ otherwise } \end{array} \right.$$

and

$$p(\alpha_k)=\left\{ \begin{array}{ll} \frac{1}{2\pi} & \t... ...[-\pi,\pi)\\ 0 & \textrm{ otherwise } \end{array} \right.$$

The prior pdf is

$$P(Y_k\vert a_k,\alpha_k) = \frac{1}{\pi \lambda_b(k)} exp\l... ...{1}{\lambda_b(k)} \vert Y_k-a_k e^{j\alpha_k}\vert^2 \right\}$$

The joint pdf is

$$P(a_k,\alpha_k)=\frac{a_k}{\pi \lambda_x(k)} exp(-\frac{a_k^2}{\lambda_x(k)})$$

The posterior density can be worked out to be

$$P(a_k\vert Y_k)=\frac{a_k}{\sigma_k^2} exp\left(-\frac{a_k^... ... \sigma_k^2}\right) I_0\left(\frac{a_ks_k}{\sigma_k^2}\right)$$

where

$$\begin{array}{lll} \frac{1}{\lambda(k)} & = & \frac{1}{\la... ...k & = & \frac{\epsilon_k}{1+\epsilon_k} \gamma_k \end{array}$$

The authors use the first moment of the posterior distribution giving

$$\hat{A}_k=\lambda(k)^{1/2}\Gamma(1.5)\Phi(-0.5,1;-v_k)$$

They also extend the amplitude estimator under signal presence uncertainty (see for example Maximum Likelihood estimator) but this is beyond the scope of this summary.