Maximum-Likelihood spectral amplitude estimator [6]

Let $y[n]=x[n]+b[n]$ and $X(k)=A_k exp(\alpha_k)$ with the noise having a Gaussian distribution. The prior density function is (also see Ephraim and Malah suppression rule):

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

The maximum-likelihood approach attempts to choose the parameter value that maximizes the parameterized pdf, that is the parameter value which is most likely to have caused the observation. The ML estimation is used for estimating an unknown parameter of a given pdf when no a priori information about it is available.

$$\begin{array}{lll} \hat{A}_k & = & \max_{a_k} P(Y_k\vert a... ...mbda_x(k)+\lambda_b(k)}} \right\} \vert Y_k\vert \end{array}$$

The performance of the algorithm during silent frames is not adequate because the starting assumption is that the signal is always present. The authors suggest a two-state soft-decision approach by using the binary hypothesis model:

$$\begin{array}{lll} H_0: & \textrm{ speech absent: } & Y_k=... ... speech present: } & Y_k=A_k e^{j\alpha_k} + B_k \end{array}$$

The MMSE solution is

$$\begin{array}{lll} \hat{A}_k & = & \mathcal{E}[A_k\vert Y_... ...^2-\lambda_b(k)} \right\} \times P(Y_k\vert H_1) \end{array}$$

since $\mathcal{E}[A_k\vert Y_k,H_1]$ is the minimum variance estimate of A and the Maximum Likelihood estimate is asymptotically efficient.

$$P(Y_k\vert H_0)=\frac{2Y_k}{\lambda_b} exp(-\frac{Y_k^2}{\lambda_b}) \textrm{ (Rayleigh pdf for complex Gaussian noise) }$$

$$P(Y_k\vert H_1)=\frac{2Y_k}{\lambda_b} exp(-\frac{Y_k^2+A_k... ...\right) \textrm{ (Rician pdf for $A_k e^{j\alpha_k} + B_k$) }$$

Assuming $P(H_0)=P(H_1)=1/2$ and using Bayes' theorem,

$$P(Y_k\vert H_1)= \frac {exp(-\epsilon)I_0\left[2\sqrt{\ep... ...]} {1+exp(-\epsilon)I_0\left[2\sqrt{\epsilon \gamma}\right]}$$

denoting $\epsilon=A_k^2 / \lambda_b$ to be the a priori SNR and $\gamma=Y_k^2 / \lambda_b$ to be the a posteriori SNR.