Generalization of Maximum A Posteriori Amplitude Estimator Under Speech Presence Uncertainty for Speech Enhancement

Abstract

In this paper, we focus on the estimation-based frequency-domain speech enhancement methods under speech presence uncertainty. Through the minimization of an average risk function, a generalization of maximum a posteriori spectral amplitude estimator is derived. By adjusting the cost parameters, we can control the error caused by noise falsely detected as speech. Our experimental results show that the proposed system can be a simple alternative to Abramson’s simultaneous detection and estimation approach for speech enhancement since it involves merely estimation under speech presence uncertainty and does not require any detector. Moreover, the proposed estimator takes advantage of a more straightforward implementation, since there is no need for the computation of Bessel functions.

Appendix

We want to find $\hat{A}$ that minimizes the phrase,

$$\begin{aligned} J&= qb_{11} f(Y\vert H_1 )(1-\int \limits _{\hat{A}-\Delta }^{\hat{A}+\Delta } {\int \limits _0^{2\pi } {f(Y\vert A,\alpha )f(A,\alpha \vert H_1 )/f(Y\vert H_1 )d\alpha dA} } ) \nonumber \\&+\,(1-q)b_{01} f(Y\vert H_0 )( {1+U(G_f R-\hat{A}-\Delta )-U(G_f R-\hat{A}+\Delta )}).\qquad \quad \end{aligned}$$

(22)

In an optimization problem, for finding $\hat{\theta }$ that minimizes $J(\hat{\theta }), \, J(\hat{\theta })$ is differentiated according to $\hat{\theta }$ and set equal to zero. The answer would be $\hat{\theta }=\theta _{opt}$. Alternatively, to solve the optimization problem, we can consider $\hat{\theta }-\theta _{opt}$ to be equal to $\frac{\partial }{\partial \hat{\theta }}J$, i.e., $\hat{\theta }-\theta _{opt} \equiv \frac{\partial }{\partial \hat{\theta }}J$. Based on this remark, we follow the derivation of Eq. (13).

Differentiating (22) with respect to $\hat{A}$ yields

$$\begin{aligned} \frac{\partial }{\partial \hat{A}}J&= qb_{11} f(Y\vert H_1 )\frac{\partial }{\partial \hat{A}}(1-\int \limits _{\hat{A}-\Delta }^{\hat{A}+\Delta } {\int \limits _0^{2\pi } {f(Y\vert A,\alpha )f(A,\alpha \vert H_1 )/f(Y\vert H_1 )d\alpha dA} } ) \nonumber \\&+\,(1-q)b_{01} f(Y\vert H_0 )\frac{\partial }{\partial \hat{A}}(1+U(G_f R-\hat{A}-\Delta )-U(G_f R-\hat{A}+\Delta ).\nonumber \\ \end{aligned}$$

(23)

Since our purpose is finding $\hat{A}$ that minimizes the phrase,

$$\begin{aligned} \left( {1-\int \limits _{\hat{A}-\Delta }^{\hat{A}+\Delta } {\int \limits _0^{2\pi } {f(Y\vert A,\alpha )f(A,\alpha \vert H_1 )/f(Y\vert H_1 )d\alpha dA} } }\right) . \end{aligned}$$

(24)

As known from estimation theory, the optimum $\hat{A}$ that minimizes (24) is one that can maximize the posterior distribution $f(A,\alpha \vert Y,H_1 )$ if $\Delta$ is small enough. In other words, the optimum $\hat{A}$ is the MAP spectral amplitude estimator as derived in [10]

$$\begin{aligned} \hat{A}=\frac{\xi +\sqrt{\xi ^2+2\frac{\xi (1+\xi )}{\gamma }} }{2(1+\xi )}R \triangleq G_\mathrm{MAP} R. \end{aligned}$$

(25)

So, based on the above-mentioned remark, we have

$$\begin{aligned} \hat{A}-G_\mathrm{MAP} R\equiv \quad \frac{\partial }{\partial \hat{A}}\left( {1-\int \limits _{\hat{A}-\Delta }^{\hat{A}+\Delta } {\int \limits _0^{2\pi } {f(Y\vert A,\alpha )f(A,\alpha \vert H_1 )/f(Y\vert H_1 )d\alpha dA} } }\right) \!.\qquad \end{aligned}$$

(26)

Also, in finding $\hat{A}$ that minimizes the phrase:

$$\begin{aligned} \left( {1+U(G_f R-\hat{A}-\Delta )-U(G_f R-\hat{A}+\Delta )}\right) \!, \end{aligned}$$

(27)

the optimum estimator is $\hat{A}=G_f R$, if $\Delta$ is small enough. Based on the above-mentioned remark, we have

$$\begin{aligned} \hat{A}-G_f R\equiv \frac{\partial }{\partial \hat{A}}\left( {1+U(G_f R-\hat{A}-\Delta )-U(G_f R-\hat{A}+\Delta )}\right) . \end{aligned}$$

(28)

By substituting Eqs. (26) and (28) into Eq. (23), we obtain

$$\begin{aligned} \frac{\partial }{\partial \hat{A}}J=qb_{11} f(Y\vert H_1 )(\hat{A}-G_\mathrm{MAP} R)+(1-q)b_{01} f(Y\vert H_0 )(\hat{A}-G_f R)\!. \end{aligned}$$

(29)