# DeCGAN: Speech Enhancement Algorithm for Air Traffic Control
## 2 Materials and Methods
### Loss Function  
In the present work, a generator loss function and a metric discriminator loss function, based on the output complex spectrum \(\left[\hat{X}_{r},\hat{X}_{i}\right]\), are designed to improve the post-training quality of speech. The generator loss function consists of the combined loss, the adversarial loss, and the time-domain loss:  
\[L_{G}=\alpha L_{\text{TF}}+\beta L_{GAN}+\gamma L_{\text{Time}}\, \tag{7}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) represent the weight coefficients of the three loss terms. For convenience, we set these weight coefficients as \(\alpha=1,\beta=0.05,\gamma=0.2\). After that, we normalize the  
Figure 3: Network structure of the metric discriminator.  
loss function using softmax. \(L_{TF}\) represents the combined loss of the magnitude loss \(L_{Mag}\) and the phase-aware loss \(L_{RI}\):  
\[\left\{\begin{array}{l}\hat{X}_{m}=\sqrt{\hat{X}_{F}^{2}+\hat{X}_{i}^{2}}\\ L_{Mag}=E_{X_{m}\hat{X}_{m}}\big{[}\big{\|}X_{m}-\hat{X}_{m}\big{\|}_{2}\big{]}\\ L_{RI}=E_{X_{i}\hat{X}_{i}}\big{[}\big{\|}X_{r}-\hat{X}_{r}\big{\|}_{2}\big{]}+E_ {X_{i}\hat{X}_{i}}\big{[}\big{\|}X_{i}-\hat{X}_{i}\big{\|}_{2}\big{]}\\ L_{TF}=mL_{Mag}+(1-m)L_{RI}\end{array}\right., \tag{8}\]  
where \(\hat{X}_{m}\) and \(X_{m}\) represent the magnitude spectrograms of the denoised and clean speech samples, respectively; \(m\) is the weight coefficient, which was found to allow the model to reach the best speech denoising performance when it was equal to 0.7. \(X_{r}\) and \(X_{i}\) represent the real and imaginary parts of the complex spectrum of clean speech samples, respectively.  
The adversarial loss of the generator is defined as \(L_{GAN}\):  
\[L_{GAN}=E_{X_{m}\hat{X}_{m}}\big{[}\big{\|}D\big{(}X_{m},\hat{X}_{m}\big{)}-1 \big{\|}_{2}\big{]}\, \tag{9}\]  
where \(D\) denotes the discriminator. Accordingly, the adversarial loss expression for the discriminator is:  
\[L_{D}=E_{X_{m}}[\big{\|}D(X_{m},X_{m})-1\big{\|}_{2}]+E_{X_{m}\hat{X}_{m}} \big{[}\big{\|}D\big{(}X_{m},\hat{X}_{m}\big{)}-Q_{PESQ}\big{\|}_{2}\big{]}\, \tag{10}\]  
where \(Q_{PESQ}\) represents the normalized PESQ score. In this algorithm, the PESQ score is normalized to the range \([0,1]\). Additionally, it has been reported that introducing a time-domain loss can significantly optimize the speech enhancement performance, which can be expressed as \(L_{Time}\):  
\[L_{Time}=E_{x,\hat{x}}[\big{\|}x-\hat{x}\big{\|}_{1}]\, \tag{11}\]  
where \(\hat{x}\) represents the enhanced time-domain signal of speech, and \(x\) represents the target signal, or the clean time-domain signal. In this context, \(\|\cdot\|_{1}\) denotes the \(\updownarrow\)-norm, while \(\|\cdot\|_{2}\) denotes the \(\updownarrow^{2}\)-norm. We intentionally did not incorporate alternative loss functions such as \(L1/L2\) loss or perceptual loss. On the one hand, while \(L1/L2\) losses are commonly used in GAN frameworks, the loss functions tend to measure only pixel-level or signal-level differences and may fall short of capturing the semantic content and perceptual characteristics of complex speech signals. This limitation is particularly critical in air traffic control communications, where preserving intelligibility and subtle speech nuances is paramount. On the other hand, although perceptual loss can potentially offer a more human-aligned evaluation by comparing high-level features, its computational complexity makes it less suitable for real-time processing requirements. Therefore, we designed our loss function to balance adversarial training with reconstruction and frequency-domain consistency, ensuring both enhanced speech quality and operational efficiency under the specific conditions of our study.