## 3 Proposed Methods
### Frequency Domain Phase Correction  
In Section 3.1., the PLC method based on WSOLA and DNN was established to correct the phase in the time domain for preserving periodic characteristics of the speech signal. In this subsection, the phase correction method in the frequency domain based on the GLA [11] is described. Given the signal \(c(t)\) in the time domain, it can be represented as follows in the frequency domain by performing an STFT:  
\[C_{k}(\omega)=A_{C,k}(\omega)e^{j\angle_{C,k}(\omega)}\quad A_{C,k}(\omega)=|C _{k}(\omega)|, \tag{18}\]  
where \(C_{k}(\omega)\) indicates the STFT of \(c(t)\) at frequency \(\omega\) of the \(k\)th frame. According to [31], as shown in Figure 7, when STFT is performed, STFT\((c(t))=C_{k}(\omega)\). Normally, it must have \(\text{ISTFT}(C_{k}(\omega))=c(t)\). If the estimation of the \(C_{k}(\omega)\) is not accurate, the above normal time-frequency relationship will be destroyed, that is, the estimated spectrum \(\hat{C}_{k}(\omega)\) is no longer equal to the spectrum \(P\big{(}\hat{C}_{k}(\omega)\big{)}\) obtained after ISTFT and STFT, where \(P\big{(}\hat{C}_{k}(\omega)\big{)}=\text{STFT}\big{(}\text{ISTFT}\big{(}\hat{ C}_{k}(\omega)\big{)}\big{)}\). This phenomenon is called the inconsistency of the STFT and ISTFT. Considering this problem, the GLA was proposed in [26; 27]. A typical  
Figure 6: Illustration of time domain PLC method. application of the GLA is to correct the phase spectrum according to the amplitude spectrum in speech enhancement, where the enhanced amplitude spectrum of speech signal and the phase spectrum of noisy speech result in the inconsistency in speech reconstruction since the phase spectrum of clean speech is unknown. Therefore, in this paper, the goal of using the GLA is to make the phase spectrum match the estimated amplitude spectrum. In order to improve PLC performance, in this paper, the GLA is used to correct the phase spectrum estimated by DNN based on the estimated amplitude spectrum by DNN in the lost frame. Let \(\mathbf{A}_{Y,D}\) be the amplitude spectrum of the lost frame estimated by DNN, and \(\mathbf{A}_{X,D}=[\mathbf{A}_{X,Y-T},\,\ldots,\,\mathbf{A}_{X,Y-2},\,\mathbf{ A}_{X,Y-1}]\) be the amplitude spectrum set of the previous \(T\) frames without packet loss. Combining \(\mathbf{A}_{X,D}\) and \(\mathbf{A}_{Y,D}\), we have the following set:  
\[\mathbf{A}_{D}=\big{[}\mathbf{A}_{X,Y-T}\;\;\;\cdots\;\;\;\mathbf{A}_{X,Y-1}\; \;\;\;\mathbf{A}_{Y,D}\big{]}. \tag{19}\]  
Similarly, the initial phase set of the GLA can be represented as  
\[\boldsymbol{\varphi}_{D}^{[0]}=\big{[}\boldsymbol{\varphi}_{X,Y-T}\;\;\;\cdots \;\;\;\boldsymbol{\varphi}_{X,Y-1}\;\;\;\boldsymbol{\varphi}_{Y,D}\big{]} \tag{20}\]  
where \(\boldsymbol{\varphi}_{Y,D}\) denotes the phase of the lost frame, which is estimated by DNN, and \(\boldsymbol{\varphi}_{X,Y-t}\), \(t=1,\,\ldots,\,T\), are the phase set of the previous \(T\) frames without packet loss. Thus, the complex spectrum of the signal can be initialized as  
\[\mathbf{D}^{[0]}=\mathbf{A}_{D}\cdot e^{j\boldsymbol{\varphi}_{D}^{[0]}}. \tag{21}\]  
Assume that \(\mathrm{P}_{\varphi}\{\cdot\}\) is an operation taking the phase after ISTFT and STFT, denoted as:  
\[\mathrm{P}_{\varphi}\{\cdot\}=\angle\{\mathrm{P}(\cdot)\} \tag{22}\]  
where \(\angle\{\cdot\}\) is the operation extracting the phase, \(\mathrm{P}(\cdot)=\mathrm{STFT}[\mathrm{ISTFT}[\cdot]]\), which means to first perform ISTFT and then perform STFT. The implementation process of the GLA is given in Algorithm 1. In Algorithm 1, \(N\) is the iteration number, according to the results given by [26]. In this experiment, \(N\) is set to 1000, and \(d(t)\) consists of two parts; one is the input signal \(x(t)\), the other is the concealed signal \(y(t)\) by the PLC. Figure 7: Schematic diagram of the inconsistency.