## 3 Proposed Methods  
This part is divided into three parts to introduce the proposed PLC methods based on phase correction and DNN. First, the baseline method of the PLC based on the DNN is described. Second, the basic principle of the phase correction method in the time domain is presented. Third, the frequency domain phase correction method based on the GLA is introduced.  
Let \(x(t)\) be the decoded speech signal without packet loss in front of a lost packet, where \(x(t)\) may include several packets as the input signal of the PLC. Let \(y(t)\) be the  
Figure 2: Probabilities of packet loss in different channel states.  
concealed signal of that lost packet as the output signal of the PLC. The short-time Fourier transform (STFT) of these two signals are represented as \(X_{k}(\omega)\) and \(Y_{k}(\omega)\), respectively, that is,  
\[X_{k}(\omega) =A_{X,k}(\omega)e^{j\angle\varphi_{X,k}(\omega)},A_{X,k}(\omega)=|X _{k}(\omega)| \tag{8}\] \[Y_{k}(\omega) =A_{Y,k}(\omega)e^{j\angle\varphi_{Y,k}(\omega)},A_{Y,k}(\omega)=|Y _{k}(\omega)| \tag{9}\]  
where \(\varphi_{X,k}(\omega)\) and \(\varphi_{Y,k}(\omega)\) denote the phase spectra of \(x(t)\) and \(y(t)\), respectively, \(A_{X,k}(\omega)\) and \(A_{Y,k}(\omega)\) denote the amplitude spectra of \(x(t)\) and \(y(t)\), respectively, \(k\) and \(\omega\) indicate frame index and frequency.  
Considering the case of a packet loss in continuous transmission of the packets, the speech signal included in this lost packet needs to be recovered by PLC through those received packets in front of the lost packet. Here, the content of a packet consists of one frame speech signal. Inspired by [21], the baseline DNN-based PLC method can be divided into two parts: amplitude prediction and phase prediction. Since the amplitude spectrum obtained by the STFT has a large dynamic range, and most of the values are concentrated in the low-frequency region, if amplitude spectrum is directly used by DNN, it is not conducive for the training of the neural network.  
Therefore, it is necessary to employ the log-spectrum to compress the dynamic range of the amplitude spectrum and improve the learning effect of the network [19; 20]. The block diagram of the baseline DNN-based PLC method is shown in Figure 3. The input signal comes from some normal packets consecutively received at the receiver in front of the lost packet. \(\mathbf{A}_{Y,D}\) and \(\varphi_{Y,D}\) are amplitude and phase spectrum vectors of the output signal \(y(t)\), which are estimated by the DNN, respectively. The output signal \(y(t)\) in the time domain is derived from inverse short-time Fourier transform (ISTFT) of the estimated complex spectrum.  
Similar to the DNN-based amplitude spectrum prediction in some speech processing, the log-power spectra of \(T\) frames in front of the lost frame are taken as the input features of the DNN, and the log-power spectrum of the lost frame is taken as the training target of the DNN. As shown in Figure 4, once the DNN is well trained, the log-power spectrum set \(\boldsymbol{\
abla}(\omega)\) including \(T\) previous frames without packet loss can be used to estimate the log-power spectrum \(\Psi_{Y}(\omega)\) of the lost frame, where \(\boldsymbol{\
abla}(\omega)=[\Psi_{X,Y-T}(\omega),\cdots,\Psi_{X,Y-2}(\omega),\Psi_{X,Y-1}(\omega)]\).  
For the phase spectrum, the DNN with the same structure is used. The phase spectra of previous \(T\) frames are used as the input of the DNN to predict the phase spectrum of the lost frame. Since the range of phase spectrum of the speech signal is from \(-\pi\) to \(\pi\) and approximately obeys uniform distribution, it does not have the same harmonic structure as the amplitude spectrum. Using DNN in the baseline PLC method may cause estimation error of the phase spectrum for the lost frame [31]. To solve this problem, two kinds of phase correction methods are proposed in this paper to improve the PLC performance.  
Figure 3: Block diagram of baseline DNN-based PLC method.