# Options for Performing DNN-Based Causal Speech Denoising Using the U-Net Architecture +
## 4 Experiment and Performance Evaluation
### Further Considerations of Loss Function in the STFT and STDCT Domains  
The results in Table 1 demonstrate that the U-Net architecture is practical and efficient for speech enhancement. The performance discussed in the previous section is based on training U-Nets using MSE as the loss function. In DNN-based speech denoising, loss functions that integrate magnitude constraints with complex spectral optimization are commonly used to enhance speech quality and intelligibility. Additionally,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\ \begin{tabular}{c} **Input** \\ **Type** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Initial** \\ **SNR (dB)** \\ \end{tabular}  } & \multicolumn{6}{c}{**Classical U-Net with CCABs**} & \multicolumn{6}{c}{**Advanced U-Net with GLFBs**} \\ \cline{2-13}  & & \multirow{2}{*}{**CSIG**} & \multirow{2}{*}{**CBAK**} & \multirow{2}{*}{**COVL**} & \multirow{2}{*}{**PESQ**} & \multirow{2}{*}{**STOI**} & \multirow{2}{*}{**Resulting**} & \multirow{2}{*}{**CSIG**} & \multirow{2}{*}{**CBAK**} & \multirow{2}{*}{**COVL**} & \multirow{2}{*}{**PESQ**} & \multirow{2}{*}  
power compression has been employed to improve the estimated speech quality further. Given the proposed U-Nets' superior performance in the STFT and STDCT domains, we apply techniques in [10,23] to refine the loss function, potentially improving speech quality further:  
\[\mathbf{\mathcal{L}}_{comp}\big{(}\hat{Y}_{DFT}[k],X_{DFT}[k]\big{)}=\alpha\mathbf{ \mathcal{L}}_{MSE\_Mag}\big{(}|\hat{Y}_{DFT,\beta}[k]|,\big{|}X_{DFT,\beta}[k ]\big{|}\big{)}+(1-\alpha)\mathbf{\mathcal{L}}_{MSE\_RI}\big{(}\hat{Y}_{DFT,\beta}[ k],X_{DFT,\beta}[k]\big{)} \tag{13}\]  
with \(\hat{Y}_{DFT,\beta}[k]\) and \(X_{DFT,\beta}[k]\) defined as  
\[\hat{Y}_{DFT,\beta}(k)=\big{|}\hat{Y}_{DFT}[k]|^{\beta}\frac{\hat {Y}_{DFT}[k]}{|\hat{Y}_{DFT}[k]|}=\big{(}\hat{Y}_{DFT}[k]\hat{Y}_{DFT}^{*}[k] \big{)}^{\beta/2}\frac{\hat{Y}_{DFT}[k]}{\sqrt{\hat{Y}_{DFT}[k]\hat{Y}_{DFT }[k]}};\] \[X_{\beta}(k)=|X_{DFT}[k]|^{\beta}\frac{X_{DFT}[k]}{|X_{DFT}[k]|}=(X _{DFT}[k]X_{DFT}^{*}[k])^{\beta/2}\frac{X_{DFT}[k]}{\sqrt{X_{DFT}[k]\hat{X}_{DFT }[k]}}.\]  
It is worth noting that the second term on the right-hand side of Equation (13) also contains the phase information, as the calculation of power spectra (namely, \(\hat{Y}_{DFT}[k]\hat{Y}_{DFT}^{*}[k]\) and \(X_{DFT}[k]X_{DFT}^{*}[k]\)) involves both the real and imaginary components. \[\hat{Y}_{DFT}[k]\hat{Y}_{DFT}^{*}[k]=\big{(}\text{Re}\big{\{}\hat{Y}_{DFT}[k ]\big{\}}\big{)}^{2}+\big{(}\text{Im}\big{\{}\hat{Y}_{DFT}[k]\big{\}}\big{)}^ {2}=Y_{DFT-R}^{2}[k]+Y_{DFT-I}^{2}[k]\] \[X_{DFT}[k]X_{DFT}^{*}[k]=(\text{Re}\big{\{}X_{DFT}[k]\big{\}})^{2 }+(\text{Im}\big{\{}X_{DFT}[k]\big{\}})^{2}=X_{DFT-R}^{2}[k]+X_{DFT-I}^{2}[k]\]  
In the above context, the subscript \(\beta\) attached to a DFT coefficient indicates the exponent used to modify the magnitude. A \(\beta\) value between 0 and 1 not only aligns with human auditory perception of sound intensity but also reduces the dynamic range of spectral coefficients, thus enhancing network estimation. Parameter \(\alpha\) represents a mixing ratio for combining the magnitude loss \(\mathbf{\mathcal{L}}_{MSE\_Mag}(\cdot)\) and the STFT-RI loss \(\mathbf{\mathcal{L}}_{MSE\_RI}(\cdot)\). The resulting loss function, termed \(\mathbf{\mathcal{L}}_{comp}\big{(}\hat{Y}_{DFT}(k),X_{DFT}(k)\big{)}\) and referred to as composite mean squared error (CMSE), is formulated on the understanding that human auditory perception aligns more closely with a logarithmic scale, and the sensitivities to spectral magnitudes and phases differ. Values of \(\alpha=0.5\) and \(\beta=0.5\) have been reported to achieve satisfactory results [10,23]. [PERSON] et al. [24] employed a similar approach with STDCT coefficients. They used a composite loss function comprising two loss values: the MSE loss calculated from absolute STDCT values and the MSE loss derived from the original polar values.