## 2 Method
### Multi-band spectral subtraction method  
The multi-band spectral subtraction method is based on the assumption that the additive noise will be stationary and uncorrelated with the clean speech signal. If \(y(n)\), the noisy speech, is composed of the clean speech signal \(s(n)\) and the uncorrelated additive noise signal \(d(n)\), then:  
\[y(n)=s(n)+d(n) \tag{1}\]  
The power spectrum of the corrupted speech can be approximately estimated as:  
\[\left|Y(\omega)\right|^{2}\approx\left|S(\omega)\right|^{2}+\left|D(\omega) \right|^{2} \tag{2}\]  
where \(\left|Y(\omega)\right|^{2}\), \(\left|S(\omega)\right|^{2}\) and \(\left|D(\omega)\right|^{2}\) represent the noisy speech short-time spectrum, the clean speech short-time spectrum, and the noise power spectrum estimate, respectively. Most of the subtractive-type algorithms have different variations allowing for flexibility in the variation of the spectral subtraction. [PERSON] _et al._[20] proposed the generalized spectral subtraction scheme is described as follows:  
\[\left|\hat{S}(\omega)\right|^{r}= \begin{cases}\left|Y(\omega)\right|^{r}- \alpha\left|\hat{D}(\omega)\right|^{r},&\text{if}\quad\dfrac{\left|\hat{D}( \omega)\right|^{r}}{\left|Y(\omega)\right|^{r}}<\dfrac{1}{\alpha+\beta}\\ \beta\left|\hat{D}(\omega)\right|^{r},&\text{otherwise,}\end{cases}  \tag{3}\]  
where \(\alpha(\alpha>1)\) is the over-subtraction factor[16][20], which is a function of the segmental SNR. \(\beta(0\leq\beta\leq 1)\) is the spectral floor, and \(\gamma\) is the exponent determining the transition sharpness. Here we set \(\gamma=2\), and \(\beta=\)0.002. This implementation assumes that the noise affects the speech spectrum uniformly, the over-subtraction factor \(\alpha\), furthermore, subtracts an over-estimate of the noise over the whole spectrum. However, the noise in the EL speech maybe colored and does not affect the speech signal uniformly over the entire spectrum. Figure 1 shows the estimated segmental SNR for five frequency bands [0\(\sim\)300 Hz (Band 1), 300\(\sim\)1 KHz (Band 2), 1K\(\sim\)2K (Band 3), 2K\(\sim\)3K (Band 4), 3K\(\sim\)5K (Band 5)] of EL speech corrupted by radiated noise. It can be seen from Figure 1 that the SNR of the low frequency bands (Bands 1, 2) was significantly higher than the SNR of the high frequency bands (Bands 3\(\sim\)5). The largest SNR difference among the SNR was about 25 dB, a large difference. This phenomenon suggests that the noise signal does not affect the speech signal uniformly over the whole spectrum, therefore, subtracting a constant factor of noise spectrum over the whole frequency spectrum may remove speech also. In order to take into account the fact that colored noise affects the EL speech spectrum differently at various frequencies, the multi-band spectral subtraction technique [16] is used in this proposed approach, which estimates a suitable factor that will subtract just the necessary amount of the noise spectrum from each frequency sub-band. In this study, the speech spectrums was divided into N (N=5) non-overlapping bands, and spectral subtraction was performed independently in each band. Hence the estimate of the clean speech spectrum in the \(i\)th band is obtained by:  
\[\left|\hat{S}_{i}(\omega)\right|^{2}=\left|Y_{i}(\omega)\right|^{2}-\alpha_{i} \delta_{i}\left|\hat{D}_{i}(\omega)\right|^{2},\ \ \ \ \ b_{i}\leq\omega\leq e_{i} \tag{4}\]  
where \(\alpha_{i}\) is the over-subtraction factor of the \(i\)th frequency band, and \(\delta_{i}\) is a tweaking factor that can be individually set for each frequency band to customize the noise removal properties. \(b_{i}\) and \(e_{i}\)are the beginning and ending frequency of the \(i\)th frequency band. The whole algorithm is as shown in Figure 2.  
Figure 1: The segmental SNR of five frequency bands of EL speech. Figure 2: The proposed speech enhancement scheme.