# Whispered Speech Conversion Based on the Inversion of Mel Frequency Cepstral Coefficient Features
## 2 Whispered Speech Conversion Process
### Inversion of Speech Features  
The MFCC analysis is based on the mechanism of human hearing characteristics, i.e., the analysis of the speech spectrum based on the results of human hearing experiments. For this process, it is a common practice to pass the speech signal through a series of filter banks, which are called Mel filter banks. When the speech signal passes through the filter banks, the output is the power spectrum at the Mel frequency. By performing a discrete cosine transform (DCT) on all the log operation of filter bank outputs, we obtain the MFCC as follows:  
\[mfcc=DCT\{\log(z)\} \tag{8}\]  
where \(Z\) is the energy spectrum of the Mel filter as follows:  
\[z=\Phi y \tag{9}\]  
The weight matrix \(\Phi\in R^{K\times(N/2+1)}\) of the Mel frequency scale is the tone frequency based on the perception by the human ear. The value of the Mel frequency scale exhibits a roughly logarithmic distribution corresponding to the actual frequency. The speech frequency can be divided into a series of triangular filter sequences, i.e., the Mel filter banks, in which \(K\) is the spectral line of the Mel filter banks, \(y\) is the power spectrum of speech sound \(x\) and \(X\) is the frequency domain data of speech by STFT, as follows:  
\[y=|X|^{2}=|F\{x\}|^{2} \tag{10}\]To obtain the reconstructed speech by the inverting MFCC, the process of obtaining MFCC features was inverted. Since the DCT, log, and squaring operations are all reversible, when \(Z\) and \(\Phi\) are known, it is possible to more accurately estimate the energy spectrum \(y\) of the speech from the MFCC features and its phase spectrum [26; 27] to obtain the inverted speech frame. 1. Estimation of energy spectrum Y Because of the sparse characteristics of speech signals, the use of sparse characteristics in their decomposition can effectively avoid interference from some noises, which is of great significance for signal reconstruction. [PERSON] et al. [28] conducted an in-depth study on the sparse decomposition of signals and demonstrated the advantages of the L1/2 norm in the sparseness of signal decomposition, which has achieved good results when applied to the reconstruction of sparse signals. In this study, we propose an L1/2 + 2 double sparse constraint-based signal reconstruction model and a solution for \(y\) with known \(Z\) and \(\Phi\). The objective function model is as follows: \[\min_{y}\{||\Phi y-z||_{2}^{2}+\sum_{i}^{N}\left(\lambda_{z}||y_{i}||_{1/2}^{1/ 2}+\lambda_{1}||y_{i}||_{2}^{2}\right)\}\] (11) where \(\lambda_{1}\) and \(\lambda_{2}\) are the regularization parameters that control the sparseness and smoothness, respectively, of the coefficient vector, satisfying \(\lambda_{1}>0\) and \(\lambda_{2}>0\). From Equation (4), we obtain the updated formula for y as follows: \[ \begin{array}{l}y=\left(\mathrm{I}+\frac{\mu\lambda_{2}}{2}||\cdot||_{1/2}^{ 1/2}\right)^{-1}((1-\lambda_{1}\mu)y+\mu\Phi^{T}(z-\Phi y))\\ \Rightarrow y=H_{\lambda_{2}\mu,1/2}(B_{\mu}(y))\end{array} \] (12) where \(\
abla(||\cdot||_{1/2}^{1/2})\) is the gradient of \(||y||_{1/2}^{1/2};\Phi^{T}\) is the transpose of \(\Phi\); \((1+\frac{\mu\lambda_{2}}{2}||\cdot||_{1/2}^{1/2})^{-1}\) is the operator of y; \(H_{\lambda_{2}\mu,1/2}(\cdot)=\left(\mathrm{I}+\frac{\lambda_{2}\mu}{2}\
abla( ||\cdot||_{1/2}^{1/2})\right)^{-1}\); \(B_{\mu}(y)=(1-\lambda_{1}\mu)y+\mu\Phi^{T}(z-\Phi y)\). According to the literature [29], we set the following: \[ \begin{array}{l}y=\left\{\begin{array}{l}\frac{4}{3}[B(y)]_{i}*(\cos(\frac {\pi}{3}-\frac{\phi_{\lambda_{2}\mu}(y_{i})}{3}))^{2},\big{|}[B(y)]_{i}\big{|} >\frac{3/54}{4}(\lambda_{2}\mu)^{\frac{2}{3}}\\ 0,\end{array}\right.\\ \text{otherwise}\end{array} \] (13) where \(\phi_{\lambda_{2}\mu}(y_{i})=\arccos\left(\frac{\lambda_{2}\mu}{8}(\frac{|y _{i}|}{3})^{-\frac{2}{3}}\right)\), and \(i\) is the number of iterations (\(i=1,2,3\cdots\)). By the method, the energy spectrum can be directly calculated from the MFCC spectrum without fundamental frequency estimation. 2.