## 3 Continuous Speech Analysis/Synthesis System
### Harmonic-to-Noise Ratio  
The main goal of vocoders is to achieve high speech intelligibility. It has been shown previously that the mixed excitation source model yields sufficiently good quality in the synthesized speech by reducing buzziness and breathiness [51]. Such an analysis/synthesis system may also suffer from some degradations: (1) loss of the high-frequency harmonic components, (2) high-frequency noise components, and (3) noise components in the main formants. As the degree of these losses increases, more noise appears, consequently degrading the speech quality greatly [52].  
In this work, we propose adding a continuous harmonic-to-noise ratio (HNR) as a new excitation parameter to our vocoder in order to alleviate previous problems. Consequently, the excitation model in the proposed vocoder is represented by three continuous parameters: F0, MVF, and HNR. There are various methods of time and frequency domain algorithms available in the literature to estimate HNR in speech signals (for a comparison, see [53]). As we are dealing here with time domain processing, we want to follow the algorithm by [54] to estimate the level of noise in human voice signals for the following reasons: (1) the algorithm is very straightforward, flexible and robust, (2) it works equally well for low, middle, and high pitches, and (3) it is correctly tested for periodic signals and for signals with additive noise and jitter.  
For a time signal \(x(t)\), the autocorrelation function \(r_{x}(\tau)\) can be defined as  
\[r_{x}(\tau)\cong\int x(t)x(t+\tau)dt \tag{10}\]  
This function has a global maximum for \(\tau=0\). The fundamental period \(T_{0}=1/F_{0}\) is defined as the value of \(\tau\) corresponding to the highest maximum of the \(r_{x}(\tau)\), and the normalized autocorrelation is  
\[r_{x}(\tau)=\frac{r_{x}(\tau)}{r_{x}(0)} \tag{11}\]  
We could make such a signal \(x(t)\) by taking a harmonic signal \(H(t)\) with a period \(T_{0}\) and adding a noise \(N(t)\) to it. We can now write Equation (10) as  
\[r_{x}(\tau)=r_{H}(\tau)+r_{N}(\tau) \tag{12}\]  
Because the autocorrelation of a signal at 0 equals the power in the signal, Equation (11) at \(\tau_{max}\) represents the relative power of the harmonic component of the signal, and its complement represents the relative power of the noise component:  
\[r_{x}(\tau_{max})=\frac{r_{H}(0)}{r_{x}(0)} \tag{13}\]  
\[1-r_{x}(\tau_{max})=\frac{r_{N}(0)}{r_{x}(0)} \tag{14}\]  
Thus, the HNR is defined at \(\tau_{max}>0\)  
\[HNR\triangleq\frac{r_{x}(\tau_{max})}{1-r_{x}(\tau_{max})} \tag{15}\]  
Accordingly, the HNR is positively infinite for purely harmonic sounds, while it is very low for noise (see Figure 4). In a continuous vocoder, our approach here is to use the HNR to weight the excitation signal in both voiced and unvoiced frames. If we define the generation of the voiced excitation frame \(v[k]\) as  
\[v[k]=p[k]*w_{v} \tag{16}\]  
then, the weighted voice \(w_{v}\) value can be determined by  
\[w_{v}=\sqrt{\frac{hnr[i]}{hnr[i]+1}},i=\frac{K}{F_{slift}*f_{s}} \tag{17}\]  
where \(p[k]\) is the residual PCA voiced signal, \(F_{slift}\) is 5 ms frame shift, \(f_{s}\) is the sampling frequency, and \(K\) is the location of impulse in original impulse excitation. Similarly, the unvoiced excitation frame \(u[k]\) and the unvoiced weight \(w_{u}\) value can also be computed by  
\[u[k]=n[k]*w_{u} \tag{18}\]  
\[w_{u}=\sqrt{\frac{1}{hnr[i]+1}},i=k \tag{19}\]  
where \(n[k]\) is the additive Gaussian noise. As a result, the voiced and unvoiced speech signal are added in the ratio suggested by the HNR, and then used to excite the MGLSA filter as illustrated in the bottom part of Figure 3.