## 3 The Proposed Multiresolution Sub-band-APA for the Speech Enhancement
In the following, the sub-band-APA (SBAPA) algorithm along with the circular nested microphone array (CNMA-SBAPA) is proposed for the speech enhancement. Adaptive filters as an important tool in digital signal processing have been utilized for many years in such application as: speech signal enhancement, system identification, localization and tracking, etc. In adaptive filters, the coefficients change periodically to be adapted based on the time varying features of the noise, and this property increases the performance of the denoising system in comparison with normal methods. In addition, these filters are non-linear and homogeneous since their features are dependent on the input signal. The adaptive filters have the following advantages: low delay and better tracking in non-stationary conditions [30]. These advantages are very important in dereverberation, denoising, time delay estimation, channel equalization, and speaker tracking applications. In these applications, low delay and robustness against of non-stationary noisy and reverberant conditions are important parameters to improve the performance of the proposed systems. The existence of the reference signal, which is hidden in the filter coefficient estimations, defines the system performance. Figure 5 shows the general structure of the adaptive filter in denoising applications. We change the notation for input signal in adaptive filter (\(y_{m,i,j}[n]\)) to \(y[n]\) for simplifying the mathematical expressions. An adaptive filter is expressed as follows [31]:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]  
where \(n\) is the time index, \(z[n]\) is the adaptive filter output, and \(w_{L}[n]\) is the adaptive filter coefficients with length \(L\). The update algorithm in Figure 5 is considered as a principal part for an adaptive filter, which is the APA in this article. The main idea for an adaptive filter is to minimize the error signal \(e[n]\) to make the output of the filter as similar as the desired signal. The input signal \(y[n]\) for the adaptive filter is considered as the summation of the noise (\(v[n]\)) and desired signal (\(d[n]\)), which is described as:  
\[y[n]=d[n]+v[n]. \tag{7}\]  
The adaptive filter has a FIR structure, namely the filter is designed based on the limited number of coefficients in the time domain. For a filter with order of \(L\), the filter coefficients are defined as:  
\[w_{L}[n]=[w[0],w[1],\ldots,w[L-1]]. \tag{8}\]  
Figure 5: The general structure of the adaptive filter for denoising applications. The error signal or cost function is defined as the difference between estimated and desired signal, namely:  
\[e[n]=d[n]-z[n]. \tag{9}\]  
As shown in Equation (6), the output of the adaptive filter \(z[n]\) is defined as the convolution between the filter coefficients \(w_{L}[n]\) and the input signal \(y[n]\), where \(y[n]\) is considered as the input of the adaptive filter, namely:  
\[y[n]=[y[n],y[n-1],\ldots,y[n-L]]. \tag{10}\]  
In addition, the adaptive filter coefficients change during the time, which is written as:  
\[w_{L}[n]=w_{L}[n-1]+\Delta w_{L}[n], \tag{11}\]  
where \(\Delta w_{L}[n]\) is defined as the correction factor for the filter coefficients. The adaptive filter produces the correction factor based on the input and error signal. In Figure 5, several algorithms can be considered for updating the filter coefficients. The APA is one of the fastest and most efficient methods for this purpose. The AP algorithms were introduced to improve the speed of convergence in the gradient-based algorithms, especially when the input signal has a non-stationary spectrum. It is because the speed of convergence is decreased in the case of non-stationary and constraint spectrum [30]. Filter update equation is one of the most important features in the AP algorithms, which uses \(N\) vectors of the input data to update the filter coefficients instead of using one vector of the input data, i.e., the normalized least mean square (NLMS). Therefore, more information was considered in the time for accurately updating the filter coefficients.