## 3 The Proposed Multiresolution Sub-band-APA for the Speech Enhancement  
Speech is a wideband and non-stationary signal, where each frequency band has different information. This feature for the speech signal provides the conditions to evaluate the speech spectrum components by considering different frequency resolution. For example, speech information is condensed at the lower part of the spectrum. Therefore, the accuracy of the speech enhancement algorithm is increased by a focus to low frequency components. In this article, a specific sub-band processing along with a filter bank was proposed for paying more attention to lower frequencies by the use of filters with narrower bandwidths. Table 2 shows the information to design and implement this analysis filter bank. There is still not any certain rule for selecting the number of frequency bands. Of course, by having narrower band filters in low frequencies, we have more frequency resolution, but the concern is the computational complexity. In other hands, adding each more filter means entering more microphone pairs and more calculations. Based on the experiments, this number of frequency bands prepares enough performance and acceptable level of complexity. As seen, the filter bandwidth is smaller in low frequencies in comparison with high frequencies. This property increases the frequency resolution for low frequencies. The most important benefit in sub-band processing is the noise estimation from the silent part of the speech signal in each sub-band. Since in the proposed denoising method, the noise estimation is required as an input for the enhancement algorithm. Therefore, the more accurate and stationary noise estimation is obtained by sub-band processing of the speech signal, which increases the denoising algorithm performance. If \(x_{m}[n]\) is considered as an input signal for the \(m\)-th microphone, the analysis filter output for the CNMA is expressed as:  
\[x_{m,i}[n]=x_{m}[n]*h_{i}[n]\text{ where }\{m=1,\ldots,8\text{ and }i=1,\ldots,4\}, \tag{4}\]  
where \(x_{m,i}[n]\) is the analysis filter output and \(h_{i}[n]\) is the impulse response for this filter. Therefore, the spatial aliasing is eliminated from each microphone pairs of CNMA by the use of analysis filters, which are designed specifically for each subarray. In the following, the microphone signals are entered to the proposed analysis filter bank for the sub-band processing. As shown in Table 2, each microphone signal is divided into 10 sub-bands. These numbers of sub-bands were selected based on our experiments in order to provide a proper efficiency and with low computational complexity, by preparing a high frequency resolution in low frequencies. Therefore, the output of the proposed analysis filter bank is expressed as:  
\[y_{m,i,j}[n]=x_{m,i}[n]*F_{j}[n]\text{ where }\{j=1,\ldots,10\text{ },i=1,\ldots,4\text{ },m=1,\ldots,8\}, \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Filters** & **Bandwidth (Hz)** & **\(\mathbf{f_{min}}\) (Hz)** & **\(\mathbf{f_{max}}\) (Hz)** & **Filter Length (Samples)** \\ \hline \(F_{1}[n]\) & 462 & 50 & 512 & 93 \\ \(F_{2}[n]\) & 462 & 512 & 975 & 115 \\ \(F_{3}[n]\) & 450 & 975 & 1425 & 102 \\ \(F_{4}[n]\) & 525 & 1425 & 1950 & 124 \\ \(F_{5}[n]\) & 975 & 1950 & 2925 & 109 \\ \(F_{6}[n]\) & 975 & 2925 & 3900 & 118 \\ \(F_{7}[n]\) & 975 & 3900 & 4875 & 131 \\ \(F_{8}[n]\) & 975 & 4875 & 5850 & 140 \\ \(F_{9}[n]\) & 975 & 5850 & 6825 & 146 \\ \(F_{10}[n]\) & 975 & 6825 & 7800 & 151 \\ \hline \end{tabular}
\end{table} 
Table 2: The required information to design the analysis filter bank for sub-band processing in the proposed CNMA-SBAPA algorithm. where \(F_{j}[n]\) is the impulse response for each sub-band filter in the analysis filter bank and \(y_{m,i,j}[n]\) is the output of the analysis filter bank for the \(j\)-th sub-bands and \(m\)-th microphone. The signals \(y_{m,i,j}[n]\) are the sub-band microphone signals for the proposed sub-band-APA algorithm.