## 4 Evaluation
### Comparison with Other Methods  
One of the objectives of the present study is to improve the quality of the reproduced speech signals by maintaining a harmonic structure in the voiced segments. A degree of harmonicity can be measured by the autocorrelation function, which is computed in the time domain. In this study, the average autocorrelation value over the voiced frames was also adopted to evaluate the quality of the reproduced speech signals. The average autocorrelation of the speech samples \(y(0),\ldots,y(N-1)\) is given by  
\[\bar{R}_{yy}=\frac{1}{N_{v}}\sum_{k=1}^{N_{f}}w(k)\Big{[}\frac{1}{W_{L}}\sum_{ n=1}^{W_{L}}y(kW_{R}+n)y(kW_{R}+n-P_{k})\Big{]} \tag{22}\]  
where \(N_{v}\), \(N_{f}\), \(W_{L}\), and \(W_{R}\) denote the number of the voiced frames, the total number of frames, the frame length, and the frame interval, respectively. \(w(k)\) is given by  
\[w(k)=\left\{ \begin{array}{ll}1&\text{if $k$-th frame is voiced frame.}\\ 0&\text{otherwise.}\end{array} \right. \tag{23}\]  
\(P_{k}\) is the pitch period for the \(k\)-th frame that is estimated by the speech signal. For the speech signal, the average of \(\bar{R}_{yy}\) was obtained as 0.881. This indicates that voiced speech samples are strongly correlated at pitch intervals. The performance of the proposed method was compared with other methods that were designed for estimating speech parameters. The list of the methods for comparison is shown in Table 4. These methods were already employed in SSI (e.g., Gaussian mixture model (GMM) [16; 21; 22; 25; 26; 27; 30], long short term memory (LSTM) [27; 46], and multi-layer perceptron (MLP) [10; 24; 25; 27]). For the methods RNN, LSTM, and MLP (without HE), the hyperparameters were tuned to minimize the overall costs. The resultant architecture of the RNN and LSTM consisted of three hidden layers, 160 hidden nodes, and four time steps. The same dataset was commonly used for all methods to obtain the results. The results are presented in Table 4. The superiority of the proposed method (MLP with PD and HE) is remarkable in terms of the average PESQ and STOI. The MLP + PD + HE is the only method that produces an average PESQ greater than 2.0. The common property of the methods LR, MLR, GMM is that speech parameters are given by a linear combination of some representative vectors (e.g., mean vectors of each Gaussian component). This resulted in ambiguous and unclear voices, due to the averaging effects. These are the possible reasons for lowering the PESQ and STOI. Such problems were partially alleviated by applying the neural networks (RNN, LSTM, MLP), and hence the average PESQ and STOI were increased. The quality of the reproduced speech signals was further improved by applying PD. The average LSD, however, was increased in the case where the PD was adopted. This is due to the fact that the loss function of the MLP + PD method is given by the weighted sum of the two costs (MSE and PD), resulting in increasing MSE compared with RNN, LSTM, and MLP methods. Although the average LSDs of the PD-based methods are greater than those of other methods, perceptually more preferred results can be achieved by the PD-based methods. This is confirmed by the fact that the two PD-based methods (MLP + PD and MLP + PD + HE) yielded a higher average PESQ, compared with other methods. The feasibility of harmonic enhancement (HE) is also confirmed by the results shown in Table 4. The average PESQ and STOI of the MLP + PD + HE method are 2.108 and 0.567, which are the highest among all methods. The average PESQ was increased by 0.107 when HE was adopted. Such improvements in terms of PESQ are due mostly to the preservation of a harmonic structure in the voiced segments. This is also confirmed by the autocorrelation results where the MLP + PD + HE method revealed a remarkably high value compared with other methods. Although the maximum of the average autocorrelation obtained from the reproduced speech signals is almost half that from the voiced speech signals, this was sufficient to increase the average PESQ.