# A Dual-Channel End-to-End Speech Enhancement Method Using Complex Operations in the Time Domain
## 2 Algorithm Description
### Analysis of the Signal Characteristics at the Input  
In this section, we provide the details to show how to generate the complex inputs in time domain.  
The received signals by each microphone in time domain are  
\[x_{0}(k)[t]=s_{0}(k)[t]+n_{0}(k)[t], \tag{1}\]  
\[x_{1}(k)[t]=s_{1}(k)[t]+n_{1}(k)[t], \tag{2}\]  
where \(t\in\{0,1,\ldots,N\}\) denotes the sample index in each frame and \(k\in\{0,1,\ldots,M\}\) denotes the frame index. The clean speech captured by the two microphones, denoted by \(s_{0}(k)[t]\) and \(s_{1}(k)[t]\), respectively, is given by  
\[s_{0}(k)[t]=s(k)[t]*h_{0}(k)[n], \tag{3}\]  
\[s_{1}(k)[t]=s(k)[t]*h_{1}(k)[n], \tag{4}\]  
where \(h_{0}(k)[n]\in\mathbb{R}^{K\times 1}\) and \(h_{1}(k)[n]\in\mathbb{R}^{K\times 1}\) indicate the room impulse responses (RIRs) corresponding to the two microphones; \(n\in\{0,1,\ldots,K-1\}\) denotes the sampling point of the filters; and \(s(k)[t]\in\mathbb{R}^{N\times 1}\) is the sound source. In this study, \(s_{0}(k)[t]\) corresponds to the direct path signal in the reference microphone and is utilized as the target [15]. Since the distance between the two microphones is small and they are positioned on the same  
Figure 1: CNABCFCN network.  
horizontal line, we select \(c=0\) as the reference microphone. While noise is also affected by the RIR, it must be eliminated in speech noise reduction tasks, regardless of its form. Therefore, the effect of RIR on noise is generally not explicitly expressed.  
In frequency-domain processing, the RI parts are naturally present in spectrograms. However, in the time domain, only real-time speech is available, which poses the first challenge of generating its corresponding imaginary part. In this study, we used the Hilbert transform to construct the analytic signal. The Hilbert transform constructs the analytic function \(x_{c}^{a}(k)[t]\in\mathbb{C}^{N\times 1}\) of a discrete signal, where \(\mathbb{C}\) is the complex number field. For convenience and because the network operates in a frame-by-frame manner, we omit the frame number (\(k\)) and denote \(x_{c}^{a}(k)[t]\) as \(x_{c}^{a}[t]\) hereafter. For the negative frequency part of the signal spectrum, we apply the following correction:  
\[X_{c}^{a}(f) =\begin{cases}2X_{c}(f),&f>0\\ X_{c}(f),&f=0\\ 0,&f<0\end{cases} \tag{5}\] \[=X_{c}(f)+sgn(f)X_{c}(f),\]  
where \(X_{c}^{a}(f)\in\mathbb{R}^{L\times 1}\) and \(X_{c}(f)\in\mathbb{R}^{L\times 1}\) are the spectra of \(x_{c}^{a}[t]\) and \(x_{c}[t]\), respectively, and subscript \(c\) represents the microphone index. The function \(sgn(\cdot)\) represents the sign function. Thus, the definition of the analytic function \(x_{c}^{a}[t]\) is  
\[x_{c}^{a}[t] =\mathcal{F}^{-1}[X_{c}(f)+sgn(f)X_{c}(f)] \tag{6}\] \[=\mathcal{F}^{-1}[X_{c}(f))+\mathcal{F}^{-1}(sgn(f))*\mathcal{F}^ {-1}(X_{c}(f))]\] \[=x_{c}[t]+j(\frac{1}{\pi t}*x_{c}[t])\] \[=x_{c}[t]+jx_{c}[t],\]  
where \(\mathcal{F}^{-1}\) is the inverse Fourier transform. Equation (6) shows that the imaginary part \(\hat{x}_{c}[t]\in\mathbb{R}^{N\times 1}\), which can be obtained with the inverse Fourier transform of the positive half of the frequency spectrum, \(X_{c}(f)\). The one-dimensional real signal \(x_{c}[t]\in\mathbb{R}^{N\times 1}\) is convolved with the impulse response of \(\frac{1}{\pi t}\) to produce its corresponding imaginary part, and a two-dimensional complex signal \(x_{c}^{a}[t]\in\mathbb{C}^{N\times 1}\) finally serves as input to the proposed network.