## 2 The Microphone Model and Proposed Nested Microphone Array
### The Proposed Uniform Circular Nested Microphone Array  
The microphone array increases the accuracy of the speech enhancement algorithms due to increasing the information. However, the spatial aliasing based on the inter-microphone distances destroys the recorded speech signals, and in the following, the performance of the speech enhancement algorithms. Nested microphone array has the capability to eliminate the spatial aliasing [29]. In this section, a uniform CNMA is proposed where by having a symmetrical shape, provides the same probability for all speakers around the array, and the quality of the enhanced signals are not dependent on the position of the speakers. Additionally, its small structure helps to be applicable in most of the conditions in comparison with other big arrays. Figure 1 shows the block diagram of the proposed speech enhancement algorithm, where the NMA part with its analysis filters and down-sampler blocks are shown in the left side. The speech signal has a frequency range of [0-8000] Hz with a sampling frequency \(F_{s}=16,000\) Hz. The proposed CNMA is designed for the frequency range [50-7800] Hz, which covers the wideband speech spectrum. The CNMA is structured by four subarrays. The first subarray is designed for the range B1 = [3900-7800] Hz, of central frequency \(f_{\rm{C1}}=5850\) Hz. The inter-microphone distance (\(d_{\rm{lim}}\)) should be \(d_{\rm{lim}}<\lambda/2\)(\(\lambda\) is the wavelength of the highest frequency component in the related sub-band) to avoid the spatial aliasing, this is \(d_{\rm{lim}(1)}<2.2\) cm for the first subarray. The second subarray covers the frequency range B2 = [1950-3900] Hz with a central frequency of \(f_{\rm{C2}}=2925\) Hz, therefore \(d_{\rm{lim}(2)}=2d_{\rm{1}}<4.4\) cm. The third subarray is defined for the frequency range B3 = [975-1950] Hz  
Figure 1: The block diagram of the proposed circular nested microphone array in combination with the sub-band affine projection algorithm (CNMA-SBAPA) for the speech enhancement. with a central frequency of \(f_{\rm{G3}}=1462\) Hz and \(d_{\rm{lim}(3)}=4d_{1}<8.8\) cm. Finally, the forth subarray is designed for the frequency range B4 = [50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75] Hz with a central frequency of \(f_{\rm{G4}}=512\) Hz and the inter-microphone distance is \(d_{\rm{lim}(4)}=8d_{1}<17.6\) cm. For a more complexity system, a higher number of microphones could be considered to design a larger nested microphone array. Table 1 shows the summarized information to design the uniform CNMA. The microphone array was structured to have the closest microphone distances as \(d_{sim(1)}=2.2\) cm (for the simulated data) based on the designed CNMA. Therefore, the first subarray included the microphone pairs {1,2}, {2,3}, {3,4}, {4,5}, {5,6}, {6,7}, {7,8}, and {8,1}. The microphone pairs {1,3}, {3,5}, {5,7}, {7,1}, {2,4}, {4,6}, {6,8}, and {8,1} were selected for the second subarray with an inter-microphone distance of \(d_{sim(2)}=4.2\) cm. The third subarray has the inter-microphone distance of \(d_{sim(3)}=5.6\) cm. Then, the microphone pairs {1,4}, {2,5}, {3,6}, {4,7}, {5,8}, {6,1}, {7,2}, and {8,3} wereconsidered for this subarray. For the last subarray, the inter-microphone distance is \(d_{sim(4)}=6\) cm and the microphone pairs {1,5}, {2,6}, {3,7}, and {4,8} were selected for the implementation. Given our actual microphone array, the minimum inter-microphone distance that we could have was 2.7 cm (for the real data). For this reason, we did two evaluations, one for simulated data with \(d_{sim(1)}=2.2\) cm, as dictated by the theory, and one for real data with \(d_{real(1)}=2.7\) cm, to match our hardware. All subarrays are shown in Figure 2, which shows the designed CNMA with its small shape. Each subarray needs an analysis filter bank to avoid the spatial aliasing and imaging. Figure 1 (left and right sides) shows the analysis and synthesis filter banks along with the up-sampler and down-sampler blocks.