# A Supervised Speech Enhancement Approach with Residual Noise Control for Voice Communication
## 2 Problem Formulation  
In the time domain, the noisy signal can be modelled as  
\[x\left(n\right)=s\left(n\right)+d\left(n\right), \tag{1}\]  
where \(s\left(n\right)\) is the clean speech and \(d(n)\) is the additive noise. In the frequency domain, (1) can be written as  
\[X_{l}\left(k\right)=S_{l}\left(k\right)+D_{l}\left(k\right), \tag{2}\]  
where \(X_{l}\left(k\right)\), \(S_{l}\left(k\right)\), and \(D_{l}\left(k\right)\) are, respectively, discrete Fourier transforms (DFT) of \(x(n)\), \(s(n)\), and \(d(n)\) with the frame index \(l\) and the frequency bin \(k\).  
For practical applications, we only have the time-domain noisy signal \(x(n)\) or its frequency-domain version \(X_{l}(k)\), the problem becomes how to estimate \(s(n)\) or \(S_{l}(k)\) from its noisy signal. It is common to use MMSE as a criterion in unsupervised speech enhancement approaches. Before introducing MMSE, we first define the square error as  
\[J_{x}\left[M_{l}\left(k\right)\right]=\left|f\left(S_{l}\left(k\right)\right)- g\left(S_{l}\left(k\right),D_{l}\left(k\right),M_{l}\left(k\right)\right) \right|^{2}, \tag{3}\]  
where \(M_{l}\left(k\right)\) is a nonlinear spectral gain function, \(f(a)\) and \(g(a,b,c)\) are the transform functions. When \(f(a)=\left|a\right|\) and \(g(a,b,c)=\left|(a+b)c\right|\), \(\underset{M_{l}\left(k\right)}{\text{min}}E\left\{J_{x}\left[M_{l}\left(k \right)\right]\right\}\) results in MMSE spectral amplitude estimator in [2], where \(E\left\{\bullet\right\}\) is the expectation operator. When \(f(a)=\log(\left|a\right|)\) and \(g(a,b,c)=\log(\left|(a+b)c\right|)\)\(\min\limits_{M_{l}\left(k\right)}E\left\{J_{x}\left[M_{l}\left(k\right)\right]\right\}\) leads to MMSE log-spectral amplitude estimator in [3]. More complicated forms of \(f(a)\) and \(g(a,b,c)\) can be chosen; for example, many perceptually-weighted error criteria can be included, which can be referred to [10].  
For supervised approaches, the square error in the subband is often defined as the loss function in the fullband, which is  
\[\mathcal{J}_{x}=\sum\limits_{k\in\mathcal{K}}\sum\limits_{l\in\mathcal{L}}J_{ x}\left[M_{l}\left(k\right)\right]\text{.} \tag{4}\]  
One can get that, when \(f(a)=\log(\left|a\right|)\) and \(g(a,b,c)=\log(\left|(a+b)c\right|)\), \(\min\limits_{M_{l}\left(k\right)}\left\{\mathcal{J}_{x}\right\}\) is to minimize the MSE of log-spectral amplitude between the clean speech and the estimated speech, which is the training target in [6].  
Note that (3) and (4) are quite similar and the most obvious difference between them is that \(J_{x}\left[M_{l}\left(k\right)\right]\) is the subband square error, while \(\mathcal{J}_{x}\) is the fullband square error. The other difference is that the nonlinear spectral gain can be derived theoretically by minimizing \(E\left\{J_{x}\left[M_{l}\left(k\right)\right]\right\}\) when the probability density function (p.d.f.) of the speech and that of the noise are both given, while it is difficult to derive the nonlinear spectral gain by minimizing \(\mathcal{J}_{x}\), where this gain can often be mapped from the input noisy features after training the supervised machine learning model. In all, it seems that all subband square error functions can be generalized to the fullband ones as supervised training targets.  
In the above formulation, MMSE is utilized to optimize the speech spectrum recovery in the T-F domain. However, due to the stochastic characteristic of noise components and the performance limit of the network, the residual noise tends to be unnatural and may severely degrade the speech quality. Moreover, noise suppression and speech distortion are not separately considered in an explicit way, which motivates us to reformulate the optimization towards the trade-off between noise suppression and speech distortion and derive a new type of generalized loss function in Section 3.