## 5 Conclusions
The microphone array increased the accuracy in the enhanced algorithms based on the increasing of information, but the spatial aliasing decreased the efficiency because of inter-microphone distances. In this article, a uniform circular nested microphone array was proposed for the speech enhancement algorithms. This nested array was designed in a way that the microphones were located at specific distances to eliminate the spatial aliasing, in combination with analysis filters to provide the proper information for the speech enhancement algorithms. In addition, the speech information is different in various frequency bands. Therefore, the specific sub-band processing was proposed to have especial attention to the speech spectrum components. The frequency bands were designed to have the maximum resolution in low frequency components. In the following, the APA was implemented on all frequency bands, which was obtained by the sub-band processing and circular nested microphone array. The projection factor (\(N\)=4) was considered for the CNMA-SBAPA in order to keep the computational complexity in an acceptable range along with the superior accuracy. Finally, the synthesis filter bank was implemented on the sub-band signals and the enhanced signal was generated by the summation through all sub-bands. The proposed algorithm was compared with the LMS, traditional APA, RLS, DB-MWF, and MNMF-MVDR methods on the real and simulated data for white and colored noises under the SNRs range [-10,-5,0,5,10, and 15]dB. In all conditions the proposed method had a superior accuracy in comparison with previous works. In addition, the proposed method was compared based on the speed of convergence with previous works, which it was much faster among all the other algorithms. Since the proposed enhancement algorithm was implemented on stationary signals, where its benefit was increasing the speed of convergence in adaptive filters. One of the future works is reducing the size of the array and decreasing the number of microphones (without having a high effect on the quality) to be applicable for smartphone applications. Even the type of the microphones is important. In this article, we used a high quality microphone, which provides the signals with proper amplitude from the environment. The use of normal microphones in smartphones is another challenge, which could be an area for future work. Another area for future work is to find the best numbers of sub-bands to provide the maximum performance and lowest computational complexity, where the numbers of sub-bands will not be fixed and it should be adaptive based on the speech components. Conceptualization, A.D.F. and P.A. and D.Z.-B. ; methodology, A.D.F. and P.A. ; software, A.D.F., P.A. and H.D. ; validation, M.S., P.P., D.Z.-B. and C.A.-M.; formal analysis, A.D.F. and P.A. ; investigation, A.D.F. and P.A. ; resources, A.D.F., P.A., D.Z.-B. and P.I. ; data curation, A.D.F. ; writing--original draft preparation, A.D.F., P.A. and D.Z.-B. ; writing--review and editing, P.P.-J., C.A. and D.Z.-B. ; supervision, P.I. ; project administration, P.A., H.D., M.S. and D.Z.-B. ; funding acquisition, P.A. and A.D.F. All authors have read and agreed to the published version of the manuscript. This research was funded by FONDECYT Postdoctorado No. 3190147, FONDECYT No. 11180107 and ANID PFCHA/Beca de Doctorado Nacional/2019 21190489. This work was supported by the Vicerrectoria de Investigacion y Postgrado of the Universidad Tecnologica Metropolitana, the Vicerrectoria de Investigacion y Postgrado, and Faculty of Engineering Science of the Universidad Catolica del Maule. The authors declare no conflict of interest. The following abbreviations are used in this manuscript:  
AP & Affine projection  
APA & Affine projection algorithm  
AR & Auto-regressive  
ARHMM & Auto-regressive hidden Markov model  
CNMA & Circular nested microphone array  
CNMA-SBAPA & Circular nested microphone array in combination with sub-band affine projection algorithm