# A Supervised Speech Enhancement Approach with Residual Noise Control for Voice Communication
## 3 Proposed Algorithm
### Trade-Off Criterion in Subband  
In traditional speech enhancement approaches, speech distortion and noise reduction in the subband can be considered separately. The subband square error of the speech and the subband residual noise can be, respectively, given by  
\[J_{s}\left[M_{l}\left(k\right)\right]=\left|f\left(S_{l}\left(k\right) \right)-g\left(S_{l}\left(k\right),D_{l}\left(k\right),M_{l}\left(k\right) \right)\right|^{2}\text{,} \tag{5}\]  
and  
\[J_{d}\left[M_{l}\left(k\right)\right]=\left|h\left(S_{l}\left(k\right),D_{l} \left(k\right),M_{l}\left(k\right)\right)\right|^{2}\text{,} \tag{6}\]  
where \(h(a,b,c)\) is a transform function. When \(f(a)=\left|a\right|\), \(g(a,b,c)=\left|ac\right|\), and \(h(a,b,c)=\left|bc\right|\), \(E\left\{J_{s}\left[M_{l}\left(k\right)\right]\right\}\) and \(E\left\{J_{d}\left[M_{l}\left(k\right)\right]\right\}\) become the MSE of the speech magnitude and the residual noise power in the subband, respectively, which are identical with [27] (8.31) and (8.32).  
By minimizing the subband MSE of the speech with a residual noise control, an optimization problem can be given to derive the nonlinear spectral gain, which is given by  
\[\min\limits_{M_{l}\left(k\right)}E\left\{J_{s}\left[M_{l}\left(k \right)\right]\right\}\text{,} \tag{7}\] \[\text{s.t.}\text{ }E\left\{J_{d}\left[M_{l}\left(k\right) \right]\right\}=\left|\mathfrak{K}\left(\beta_{l}\left(k\right),D_{l}\left(k \right)\right)\right|^{2}\text{,}\]  
where \(\mathfrak{\lambda}\left(\beta,b\right)\) is a transform function. \(\beta_{l}\left(k\right)\in\left[0\text{ }1\right]\) could be both a frequency and frame-dependent factor that can be introduced to control the residual noise flexibly.  
The optimal spectral gain in (7) can be solved theoretically by the Lagrange multiplier method, which is  
\[\min_{M_{l}\left(k\right)}\left\{\begin{array}{c}E\left\{J_{s}\left[M_{l}\left(k \right)\right]\right\}+\mu E\left\{J_{d}\left[M_{l}\left(k\right)\right]\right\} \\ -\mu\Re\left(\beta_{l}\left(k\right),D_{l}\left(k\right)\right)|^{2}\end{array} \right\}, \tag{8}\]  
where \(\mu\geq 0\) is a Lagrange multiplier. When \(f(a)=|a|,g(a,b,c)=|ac|,h(a,b,c)=|bc|\), and \(\Re\left(\beta,b\right)|=\beta E\{|b|^{2}\}\), the optimal spectral gain can be derived from (8) and the constraint in (7), which can be given by  
\[M_{l}\left(k\right)=\xi_{l}\left(k\right)\big{/}(\xi_{l}\left(k\right)+\mu_{l }\left(k\right)), \tag{9}\]  
where \(\xi_{l}\left(k\right)=E\{|S_{l}\left(k\right)|^{2}\}/E\{|D_{l}\left(k\right)| ^{2}\}\) is the _a priori_ SNR. It is not always possible to derive \(M_{l}\left(k\right)\) mathematically, especially when \(f(a),g(a,b,c)\), \(h(a,b,c)\), and \(\lambda\left(\beta,b\right)\) have very complicated expressions. Moreover, it is difficult to accurately estimate the noise power spectral density in non-stationary noise environments [28; 29; 30]. However, it seems that this optimization can be easily solved by supervised approaches. To transfer this problem, we need to define the fullband square error of the speech and the fullband residual noise power to derive the loss function for supervised approaches.