## 3 Modification of MCRA Algorithm
### Detection of Voxel Frames  
A harmonic spectrum distributes in the frequency ranging from 50 to 500 Hz for a vowel. Performing low-pass filtering on noisy speech with cut-off frequency of 500 Hz to obtain the low-pass signal \(\phi(\eta,v)\) can be applied to estimate a pitch period by reducing the inferring of high-frequency signals. In turn, we compute the auto-correlation function and the average magnitude difference function (AMDF) of the low-passed signal \(R_{\phi}(\eta,\tau)\), given as  
\[R_{\phi}(\eta,\tau)=\frac{1}{N}\sum_{v=0}^{N-1}\phi(\eta,v)\cdot\phi(\eta,v+| \tau|) \tag{18}\]  
\[AMDF(\eta,\tau)=\frac{1}{N}\sum_{v=0}^{N-1-|\tau|}\left|\phi(\eta,v)-\phi(\eta,v+|\tau|)\right| \tag{19}\]  
In the position of the pitch period, the value of the AMDF is small; meanwhile, the value of \(R_{\phi}(\eta,\tau)\) given in (18) is large. The ratio between \(R_{\phi}(\eta,\tau)\) and AMDF is enlarged, yielding the increasing of the discriminability. It is beneficial to improve the accuracy in estimating the pitch period. A weighted autocorrelation function (WAC) is then computed to improve the discriminability at the pitch position, given as [12]  
\[WAC(\eta,\tau)=\frac{R_{\phi}(\eta,\tau)}{AMDF(\eta,\tau)+\varepsilon} \tag{20}\]  
where \(\varepsilon\) (\(\varepsilon\) = 5) is a constant value to prevent the denominator being zero. A modified pitch period \(T^{\prime}_{0}(\eta)\) is employed to improve the pitch estimation, given as [13]  
\[T^{\prime}_{0}(\eta)=\left\{ \begin{array}{ll}T_{0}(\eta)&,\text{if}\left|T_{ 0}(\eta)-T_{0}(\eta-1)\right|\leq T^{\prime}_{0}\\ 0&,\text{otherwise}\end{array} \right. \tag{21}\]  
Figure 4: Contour of the estimated noise power for a sub-band of speech (spoken by a female speaker, interfered by factory noise with an average SegSNR = 10 dB). Blue line: Power of noisy speech; green dotted line: true noise; red dotted line: estimated by MCRA method; black solid line: proposed method. where \(T_{0}^{r}(\eta)\) is the maximum allowed value for pitch variation in adjacent frames and empirically chosen to be 6. A vowel continues for some successive frames. The detected pitch period \(T_{0}^{\prime}(\eta)\) can be further refined by rejecting the vowel candidates with a short period. The refined pitch \(T_{0}^{ref}(\eta)\) can be expressed by [13]  
\[T_{0}^{ref}(\eta)=\left\{ \begin{array}{ll}T_{0}^{\prime}(\eta)&,\text{if }\eta^{E}(l)-\eta^{S}(l)\geq M^{T_{0}}\\ 0&,\text{ otherwise}\end{array} \right. \tag{22}\]  
where \(M^{T_{0}}\) (\(M^{T_{0}}\) = 5) represents the minimum period for a vowel segment. \(\eta^{E}(l)\) and \(\eta^{S}(l)\) denote the ending and the starting frames of the \(l^{\text{th}}\) vowel segment. A harmonic spectral bin is estimated by the fundamental frequency \(\Omega_{0}(\eta)\) obtained by  
\[\Omega_{0}(\eta)=N/T_{0}^{ref}(\eta) \tag{23}\]  
where \(\Omega_{0}(\eta)\) and \(T_{0}^{ref}(\eta)\) are represented in terms of spectral bin and sample indices in the experiments. The fundamental frequency obtained by (23) is refined and shifted with an offset \(\Omega_{0}^{Bias}(l)\), given as  
\[\Omega_{0}^{s}(\eta)=\Omega_{0}(\eta)-\Omega_{0}^{Bias}(l) \tag{24}\]  
where \(\Omega_{0}^{Bias}(l)\) can be computed by  
\[\Omega_{0}^{Bias}(l)=\frac{1}{l_{e}-l_{i}}\cdot\sum_{\eta=l_{i}}^{l_{e}-1} \Omega_{0}(l,\eta)-\Omega_{0}{}^{\prime}(l,\eta) \tag{25}\]  
where \(l_{i}\) and \(l_{e}\) represent the start and end frames for the \(l^{th}\) segment, respectively. \(\Omega_{0}{}^{\prime}(l,\eta)\) denotes the frequency near \(\Omega_{0}(l,\eta)\) with the spectral peak. In (25), the positions of \(l_{i}\) and \(l_{e}\) can be well defined by the estimation of onset and offset for a vowel in slight noise interference. These two positions are difficult to estimate accurately when the level of interference noise increases. Accordingly, we employ robust harmonics, which contain strong speech energy, to detect vowel frames in an utterance. Robust harmonics appear at the neighbor sub-bands of the multiple fundamental frequencies, i.e., \(k\Omega_{0}\).