## 3 The Proposed Multiresolution Sub-band-APA for the Speech Enhancement
Thus, the AP algorithm is known as an improved and extended version of the NLMS method or it can be expressed mathematically as a constraints minimization problem, which is expressed as follows. The variation for \(L\) filter coefficients during the two consecutive times is given by:  
\[\Delta w_{L}[n]=w_{L}[n]-w_{L}[n-1]. \tag{12}\]  
We minimized Equation (13) under \(N\) constraints, which are shown in Equation (14) to extend the adaptive filter algorithm. \[\|\Delta w_{L}[n]\|^{2}=\Delta w_{L}^{\mathsf{T}}[n]\Delta w_{L}[n], \tag{13}\]  
where \(N\) constraints are defined as follows:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]0  
where \(y[n-k]\) is the vector of \(N\) last sample from the input signal and \(d[n]\) is the desired signal, see \[z[n]=w_{L}[n]*y[n], \tag{6}\]1 The proposed solution formulates the update algorithm for AP, which is expressed as:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]2  
where:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]3  
and \(e_{N}[n]\) is a vector of size \(N\times 1\), which is written as:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]4  
The vector \(d_{N}[n]\) is the desired signal with size \(N\times 1\), namely:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]5The general format for AP algorithm is obtained by rewriting Equation (15) as:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]6  
If \(e_{N\tau}[n]\) is considered as \(e_{N\tau}[n]=d_{N\tau}[n]-A_{\tau}w_{L}[n-1-\alpha(N-1)]\), then:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]7  
and the signal \(d_{N\tau}[n]\) is expressed as:  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]8  
As shown in Equation (19), the \(N\) required vectors to update the adaptive filter are not necessarily to be the last data vectors. Therefore, several versions of AP algorithms are defined based on the way to select the input data and parameters in Equation (19). There are some developed algorithms based on these parameters selections such as: the NLMS along with the orthogonal correction factor (OCF-NLMS) [32], the partial rank affine projection algorithm (PRAPA) [33], and the standard APA [34] whose parameters are \(\alpha=0,\,\delta=0,\,\tau=1\). If \(\delta\) parameter differs to 0, the APA algorithm is extended to APA with regularization (R-APA) [35], where the update equation for the filter coefficients is a specific case of the Levenberg Marquardt regularized APA (LMR-APA) algorithm [36]. The introduced AP algorithm contains one input signal. Since pairs of microphones are used in the proposed CNMA, the AP algorithm is generalized to a two-microphone version [37]. Firstly, the generalization of a two-microphone structure is defined, where each microphone contains the mixing speech and noise signal, which is expressed as (see Figure 6a):  
\[z[n]=w_{L}[n]*y[n], \tag{6}\]9  
where \(s_{i}[n]\) represents the source signals, \(q_{m}[n]\) is the microphone signals, \(L\) is the impulse response length, and \(\mathbf{p}_{im}[r]\) are the impulse responses between the microphone and sources. These impulse responses are considered as linear time-invariant (LTI) systems. Two source signals \(s_{i}[n]\) are selected as the speech signal \(s[n]\) and noise signal \(b[n]\). It is assumed that the speech and noise signals are independent, which means \(E[s[n]b[n-m]]=0,\,\forall m\), where \(E\) denotes to expected value. Then, the noise and speech signals are uncorrelated. Based on the general structure, which is shown in Figure 6a, the microphone signals \(q_{1}[n]\) and \(q_{2}[n]\) are expressed as follows:  
\[y[n]=d[n]+v[n]. \tag{7}\]0 \[y[n]=d[n]+v[n]. \tag{7}\]1  
\[y[n]=d[n]+v[n]. \tag{7}\]2 (**a**) The general structure of the proposed denoising system, (**b**) the simplified presented model for the two-microphone system, and(**c**) the affine projection algorithm (APA) structure for the two-microphone. In addition, \(\mathbf{p}_{11}\) and \(\mathbf{p}_{22}\) represent the impulse responses for direct path, and \(\mathbf{p}_{12}\) and \(\mathbf{p}_{21}\) are cross-coupling for the channels between the sources and microphones.