New Sub-Band Proportionate Variable NLMS Algorithm for the Identification of Acoustical Dispersive-and-Sparse Impulse Responses

Digital signal processing is a very important field on communication systems, enabling smooth information exchange across vast distances and diverse environments, serving as the foundation of our interconnected world. Adaptive signal processing, noise reduction and echo cancellation are crucial for the efficiency and reliability of modern communication systems [1]. They involve processing and analyzing signals to extract useful information from noisy environments, leading to enhanced data transmission and reception [2]. Advanced adaptive filtering algorithms used in communication, such as acoustic system identification [3] and beamforming [4, 5], allow systems to adapt in changing channel conditions, ensuring continuous connectivity.

Acoustic impulse response identification is a fundamental aspect of adaptive filtering algorithms, which is integrated to diverse fields like telecommunications, audio engineering, and control systems [6-8]. The Least Mean Squares (LMS) and its normalized version (NLMS), utilize input signals and desired outputs to estimate the acoustical impulse response of a system [6]. By iteratively adjusting filter coefficients, the algorithm minimizes the error between predicted and actual outputs. This process allows us to adapt to changing environments in real-time, making it incredibly useful for applications like echo cancellation in telephony and reducing noise in audio processing. Identifying impulse responses using adaptive filtering enables systems to accurately capture the dynamic behavior of complex systems, leading to improved performance, improved signal quality, and more accurate responses in cases of varying conditions [9-11]. The basic proportionate NLMS algorithm and its variants are used in sparse impulse response identification applications [12-15]. To improve the convergence rate, the sub-band decomposition is very useful, but the final steady-state error is large [16-19]. To solve this problem, the addition of a variable step size parameter is crucial [20, 21].

In this paper, we propose a new sub-band implementation of the proportionate variable-step-sizes NLMS algorithm. The utilization of the µ-law compression in the PNLMS algorithm allows for enhanced handling of signals with varying dynamic ranges, effectively preserving the fine details while adapting to larger variations. By adopting a sub-band structure, the algorithm can segment the input signal into different frequency components, enabling precise adaptation within specific frequency ranges. Moreover, the incorporation of independent variable parameters improves the adaptability by optimal step-sizes to match the unique characteristics of each sub-band.

Section 2 presents the impulse response identification system using the basic NLMS algorithm. The new implementation of the proposed sub-band proportionate algorithm with new separated variable-step-sizes parameters, is presented and discussed in Section 3. Section 4 presents a comprehensive discussion of simulation results, incorporating diverse objective criteria that highlight the performance of the proposed SPV-NLMS algorithm. Section 5 presents the conclusion of this study.

For some applications [3], the length of acoustic impulse responses can be high, often ranging from 100 to 400 milliseconds and characterized by sparse impulse responses. An essential aspect is the individual adjustment of adaptation step-sizes for these sub-filters. This signifies that the variable parameter assigned to each independent sub-filter is personalized and fine-tuned autonomously in comparison to the others.

In this section, we present the proposed sub-band proportionate variable-step-sizes NLMS algorithm, noted SPV-NLMS. In SPV-NLMS, the µ-law proportionate enhances signal handling for varying dynamic ranges, preserving details while adapting to larger variations. The proposed algorithm introduces an innovative approach by employing proportionate variable step sizes specifically tailored to each individual sub-band frequency. While existing proportionate algorithms, such as PNLMS, excel in improving the convergence rate during the initial stages of adaptation, their performance significantly deteriorates thereafter. Sub-band decomposition-based algorithms, on the other hand, offer the advantage of achieving high convergence rates, particularly when input sequences exhibit strong correlation. To ensure minimal misalignment in the steady state, various variable step-size (VSS) algorithms have been developed in the literature.

The key contribution of the proposed algorithm lies in leveraging these advantages to deliver superior performance compared to existing methods. It achieves precise adaptation across diverse frequency components, with the unique incorporation of distinct variable step-size parameters for each sub-band further enhancing the algorithm's adaptability and efficiency. The general diagram of the proposed SPV-NLMS algorithm is presented in Figure 3.

As an initial stage, as presented in Figure 3, (see two blocs 1 and 2), two analysis filter banks combined with decimation parts to partition $x(n)$ and $d(n)$ into $N$ separate frequency sub-bands are employed The decimation parts are used to down-sample the $N$-sub-signals. The process of analysis filtering involves the decomposition of the input signal into $N$. sub-bands, wherein each of these sub-bands corresponds to a distinct frequency range. Each filter is responsible for isolating a particular set of frequencies, thereby generating an array of sub-band signals. The output sub-signals of two analysis filter banks can be described as follows:

$\left\{\begin{array}{c}x_1(n)=x(n)^* h_1(n) \\ x_2(n)=x(n)^* h_2(n) \\ \vdots \\ x_N(n)=x(n) * h_N(n)\end{array}\right.$               (10)

$\left\{\begin{array}{c}d_1(n)=d(n)^* h_1(n) \\ d_2(n)=d(n)^* h_2(n) \\ \vdots \\ d_N(n)=d(n) * h_N(n)\end{array}\right.$                  (11)

Each analysis filter directly relates to one of the sub-signals. The comprehensive expressions, both in general and vector form, for decomposing the input and desired sub-signals are as follows:

$x_i(n)=x(n)^* h_i(n)$           (12)

$d_i(n)=d(n) * h_i(n)$              (13)

$x_i(n)=\left[\mathbf{x}_l(n)\right]^T \mathbf{h}_i(n)$          (14)

$d_i(n)=[\mathbf{d}(n)]^T \mathbf{h}_i(n)$          (15)

with $i=1,2, \ldots, N, L$ is the length of analysis filter and:

$\begin{gathered}\mathbf{h}_i(n)=\left[h_{(i, 1)}(n), h_{(i, 2)}(n), \ldots, h_{(i, L)}(n)\right]^T, \\ \mathbf{x}_l(n)=[x(n), x(n-1), \ldots, x(n-L+1)]^T, \\ \mathbf{d}(n)=[d(n), d(n-1), \ldots, d(n-L+1)]^T .\end{gathered}$

The down-sampling factor is denoted as $D$, and it governs the extent of reduction in the sample rate, specifically $N=D$. In this process, the timing index " $n$ " is modified to " $k$ " to reflect the decimated timing. The result of this decimation applied to the input sub-signals are defined as:

$\left\{\begin{array}{c}x_1(k)=x_1(n D) \\ x_2(k)=x_2(n D) \\ \vdots \\ x_N(k)=x_N(n D)\end{array}\right.$          （16）

$\left\{\begin{array}{c}d_1(k)=d_1(n D) \\ d_2(k)=d_2(n D) \\ \vdots \\ d_N(k)=d_N(n D)\end{array}\right.$              (17)

The general formulation of the decimated input and the desired sub-signals can be presented as follows:

$x_i(k)=x_i(n D)$          (18)

$d_i(k)=d_i(n D)$         (19)

where, $x_i(n)$ represent the original input sub-signals prior to decimation, $x_i(k)$ denote the decimated input subsignals, $d_i(n)$ represent the original desired sub-signals prior to decimation, and $d_i(k)$ denote the decimated desired subsignals. Using the sub-band methodology provides an efficient adaptation procedure, leading to faster convergence rates when identifying acoustic impulse responses within long sparse environments (the third block of Figure 3).

3.jpg

Figure 3. The detailed scheme of the proposed adaptive algorithm for impulse response identification

All adaptation formulas of the proposed SPV-NLMS algorithm, which relies on distinct proportionate and variable-step-size parameters, are defined as follows:

$\left\{\begin{array}{c}\tilde{\boldsymbol{h}}_1(k)=\tilde{\boldsymbol{h}}_1(k-1)+\mu_1(k) \frac{\boldsymbol{P}_1(k-1) \boldsymbol{x}_1(k) e_1(k)}{\alpha_1(k)} \\ \widetilde{\boldsymbol{h}}_2(k)=\widetilde{\boldsymbol{h}}_2(k-1)+\mu_2(k) \frac{\boldsymbol{P}_2(k-1) \boldsymbol{x}_2(k) e_2(k)}{\alpha_2(k)} \\ \vdots \\ \widetilde{\boldsymbol{h}}_N(k)=\widetilde{\boldsymbol{h}}_N(k-1)+\mu_N(k) \frac{\boldsymbol{P}_N(k-1) \boldsymbol{x}_N(k) e_N(k)}{\alpha_N(k)}\end{array}\right.$           (20)

with $\alpha_1(k), \alpha_2(k), \ldots, \alpha_N(k)$ represent the modified normalization factors linked to each input sub-signals $x_1(k)$, $x_2(k), \ldots, x_N(k)$, respectively. These modified factors are given by:

$\left\{\begin{array}{c}\alpha_1(k)=\left[\mathbf{x}_1(k)\right]^T \mathbf{P}_1(k-1) \mathbf{x}_1(k)+\varepsilon \\ \alpha_2(k)=\left[\mathbf{x}_2(k)\right]^T \mathbf{P}_2(k-1) \mathbf{x}_2(k)+\varepsilon \\ \vdots \\ \alpha_N(k)=\left[\mathbf{x}_N(k)\right]^T \mathbf{P}_N(k-1) \mathbf{x}_N(k)+\varepsilon\end{array}\right.$            (21)

In this context, the M × M size matrices $\mathbf{P}_1(k-1), \mathbf{P}_2(k-$ $1), \ldots, \mathbf{P}_N(k-1)$ are used to govern the step-sizes associated with each coefficient of the individual sub-filters. $\mu_1(k)$, $\mu_2(k), \ldots, \mu_N(k)$ represent the individual variable-step-size parameters assigned respectively to the sub-filters $\tilde{\mathbf{h}}_1(k)$, $\tilde{\boldsymbol{h}}_2(k), \ldots, \tilde{\boldsymbol{h}}_N(k)$.

Next, all control diagonal matrices $\mathbf{P}_i(k-1)$ and individual variable-step-size parameters $\mu_i(k)$ that are used to ensure a rapid and efficient adaptation of all sub-filters $\widetilde{\boldsymbol{h}}_i(k)$ are detailed. The general updating equation by the proposed SPV-NLMS algorithm is given by:

$\tilde{\mathbf{h}}_i(k)=\tilde{\mathbf{h}}_i(k-1)+\mu_i(k) \frac{\mathbf{P}_i(k-1) \mathbf{x}_i(k) e_i(k)}{\alpha_i(k)}$           （22）

with, $\begin{aligned} & \tilde{\mathbf{h}}_i(k)=\left[\tilde{h}_{i, 1}(k), \tilde{h}_{i, 2}(k), \ldots, \tilde{h}_{i, M}(k)\right]^{\mathrm{T}}, \quad \mathbf{x}_i(k)=\left[x_i(k),\right. \left.x_i(k-1), \ldots, x_i(k-M+1)\right]^{\mathrm{T}}, \quad \alpha_i(k)=\left[\mathbf{x}_i(k)\right]^T \mathbf{P}_i(k- { 1) } \mathbf{x}_i(k)+\varepsilon .\end{aligned}$

Block 3 of Figure 3 presents the adaptation filters in sub-band form using N individual proportionate and variable-step-size parameters. We note that the decimated error sub-signals in the output are given by:

$\left\{\begin{array}{c}e_1(k)=d_1(k)-y_1(k) \\ e_2(k)=d_2(k)-y_2(k) \\ \vdots \\ e_N(k)=d_N(k)-y_N(k)\end{array}\right.$            (23)

The general formula for the sub-signal $y_i(k)$ of the all-output filters can be defined as:

$y_i(k)=x_i(k) * \tilde{h}_i(k)$             (24)

The general vectorial equation is the following:

$y_i(k)=\left[\mathbf{x}_i(k)\right]^T \widetilde{\mathbf{h}}_i(k)$            (25)

By substituting the last equation into Eq. (23) for estimated sub-signals, the next expressions are obtained:

$\left\{\begin{array}{c}e_1(k)=d_1(k)-\left[\mathbf{x}_1(k)\right]^T \tilde{\mathbf{h}}_1(k) \\ e_2(k)=d_2(k)-\left[\mathbf{x}_2(k)\right]^T \tilde{\mathbf{h}}_2(k) \\ \vdots \\ e_N(k)=d_N(k)-\left[\mathbf{x}_N(k)\right]^T \tilde{\mathbf{h}}_N(k)\end{array}\right.$              (26)

Considering the general equation for error sub-signals, it can be expressed as follows:

$e_i(k)=d_i(k)-\left[\mathbf{x}_i(k)\right]^T \tilde{\mathbf{h}}_i(k)$           (27)

The utilization of the proportionate adaptive algorithm proves to be particularly effective in the context of identifying sparse impulse responses (see Block 4). This adaptive filtering technique enhances the precision of signal processing tasks in scenarios where the impulse responses are sparsely distributed. The algorithm capitalizes on the inherent sparsity by assigning varying step-sizes to individual filter coefficients as presented in fourth step. Larger coefficients, which are more influential in contributing to the response, are assigned proportionately larger step-sizes. In the proportionate adaptive approach, each filter coefficient is assigned a unique step-size [14].

This means that larger coefficients are granted greater increments, resulting in an amplified convergence rate for those specific coefficients.

The µ-law compression function employed in our algorithm is instrumental in emphasizing small signal variations while compressing larger amplitudes, making it particularly effective for signals with large dynamic ranges. For larger values of $\left|\tilde{h}_{i, m}(k)\right|$, the logarithmic nature of the compression reduces the relative magnitude, mitigating the impact of dominant coefficients.

Conversely, for smaller coefficients, the compression function behaves approximately linearly, allowing for more precise adjustments. This balance ensures improved steady-state precision and enhances convergence speed, as smaller coefficients are not neglected during adaptation, and large variations are effectively controlled.

Additionally, µ-law compression inherently provides robustness to outliers and better handling of non-stationary environments, making it a critical feature for adaptive filtering algorithms in dynamic scenarios.

The values of the diagonal control matrix $\mathbf{P}_i(k)$ are determined by the following formulas:

$\left\{\begin{array}{c}\mathbf{P}_1(k)={diag}\left[p_{1,1}(k), p_{1,2}(k), \ldots, p_{1, M}(k)\right] \\ \mathbf{P}_2(k)={diag}\left[p_{2,1}(k), p_{2,2}(k), \ldots, p_{2, M}(k)\right] \\ \vdots \\ \mathbf{P}_N(k)={diag}\left[p_{N, 1}(k), p_{N, 2}(k), \ldots, p_{N, M}(k)\right]\end{array}\right.$              (28)

The diagonal elements of $\mathbf{P}_i(k)$, as represented in Eq. (28), are denoted by $p_{i, m}(k)$ with $m=1,2, \ldots, M$. These elements are determined through the following expressions:

$\left\{\begin{aligned} p_{1, m}(k)= & \frac{\gamma_{1, m}(k)}{\sum_{l=1}^M\left[\gamma_{1, l}(k)\right]} \\ p_{2, m}(k)= & \frac{\gamma_{2, m}(k)}{\sum_{l=1}^M\left[\gamma_{2, l}(k)\right]} \\ & \vdots \\ p_{N, m}(k)= & \frac{\gamma_{N, m}(k)}{\sum_{l=1}^M\left[\gamma_{N, l}(k)\right]}\end{aligned}\right.$               (29)

where, the general diagonal elements of $\mathbf{P}_i(k)$ is given by:

$p_{i, m}(k)=\frac{\gamma_{i, m}(k)}{\sum_{l=1}^M\left[\gamma_{i, l}(k)\right]}$              (30)

and

$\left\{\begin{array}{c}\gamma_{1, m}(k)={Max}\left\{\rho \times F\left(\left|\tilde{h}_{1, m}(k)\right|\right), F_{Log }\left(\left|\tilde{h}_{1, m}(k)\right|\right)\right\} \\ \gamma_{2, m}(k)={Max}\left\{\rho \times F\left(\left|\tilde{h}_{2, m}(k)\right|\right), F_{Log }\left(\left|\tilde{h}_{2, m}(k)\right|\right)\right\} \\ \vdots \\ \gamma_{N, m}(k)={Max}\left\{\rho \times F\left(\left|\tilde{h}_{N, m}(k)\right|\right), F_{Log }\left(\left|\tilde{h}_{N, m}(k)\right|\right)\right\}\end{array}\right.$              (31)

with the general formula is:

$\gamma_{i, m}(k)={Max}\left\{\rho \times F\left(\left|\tilde{h}_{i, m}(k)\right|\right), F_{{Log }}\left(\left|\tilde{h}_{i, m}(k)\right|\right)\right\}$           (32)

with $\rho=5 / M$, to prevent any inactivity during the adaptation phase.

The functions $F\left(\left|\tilde{h}_{1, m}(k)\right|\right), F\left(\left|\tilde{h}_{2, m}(k)\right|\right), \ldots, F\left(\left|\tilde{h}_{\mathrm{N}, m}(k)\right|\right)$ are given by:

$\left\{\begin{array}{l}F\left(\left|\tilde{h}_{1, m}(k)\right|\right)={Max}\left\{\begin{array}{c}\delta, F_{Log}\left(\left|\tilde{h}_{1,1}(k)\right|\right), F_{Log}\left(\left|\tilde{h}_{1,2}(k)\right|\right), \\ \ldots, F_{log }\left(\left|\tilde{h}_{1, M}(k)\right|\right) .\end{array}\right\} \\ F\left(\left|\tilde{h}_{2, m}(k)\right|\right)={Max}\left\{\begin{array}{c}\delta, F_{Log}\left(\left|\tilde{h}_{2,1}(k)\right|\right), F_{Log}\left(\left|\tilde{h}_{2,2}(k)\right|\right), \\ \ldots, F_{Log}\left(\left|\tilde{h}_{2, M}(k)\right|\right) .\end{array}\right\} \\ \vdots \\ F\left(\left|\tilde{h}_{N, m}(k)\right|\right)={Max}\left\{\begin{array}{c}\delta, F_{Log}\left(\left|\tilde{h}_{N, 1}(k)\right|\right), F_{Log}\left(\left|\tilde{h}_{N, 2}(k)\right|\right) \\ \ldots, F_{Log}\left(\left|\tilde{h}_{N, M}(k)\right|\right) .\end{array}\right\}\end{array}\right.$               (33)

The general function $F\left(\left|\tilde{h}_{\mathrm{I}, m}(k)\right|\right)$ is:

$F\left(\left|\tilde{h}_{i, m}(k)\right|\right)={Max}\left\{\begin{array}{c}\delta, F_{log }\left(\left|\tilde{h}_{i, 1}(k)\right|\right), F_{log }\left(\left|\tilde{h}_{i, 2}(k)\right|\right), \\ \ldots, F_{log }\left(\left|\tilde{h}_{i, M}(k)\right|\right)\end{array}\right\}$          (34)

where, $\delta$ is the regularization component that is typically set at 0.01, and introduces a slight non-zero value, contributing to the regulation and stabilization of the updating process. The $N$ logarithmic functions are given by:

$\left\{\begin{aligned} F_{ {Log }}\left(\left|\tilde{h}_{1, m}(k)\right|\right)= & \frac{ln \left(\mu \times\left|\tilde{h}_{1, m}(k)\right|+1\right)}{ln (\mu+1)} \\ F_{ {Log }}\left(\left|\tilde{h}_{2, m}(k)\right|\right)= & \frac{ln \left(\mu \times\left|\tilde{h}_{2, m}(k)\right|+1\right)}{ln (\mu+1)} \\ & \vdots \\ F_{ {Log }}\left(\left|\tilde{h}_{N, m}(k)\right|\right)= & \frac{ln \left(\mu \times\left|\tilde{h}_{N, m}(k)\right|+1\right)}{ln (\mu+1)}\end{aligned}\right.$            (35)

with the general logarithmic function is given by:

$F_{L o g}\left(\left|\tilde{h}_{I, m}(k)\right|\right)=\frac{ln \left(\mu \times\left|\tilde{h}_{i, m}(k)\right|+1\right)}{\ln (\mu+1)}$            (36)

The determination of the positive quantity $\mu$ depends on the noise level. For identifying sparse impulse responses, a practical selection for $\xi$ is 0.001 . Additionally, $\mu$ stands for the inverse of $\xi, \mu=1 / \xi$.

An important contribution presented in Block 5 is the distinct optimal step-size relationships designed for faster convergence and very low error. In adaptive filtering, the significance of the step-size lies in its efficient role during updates of coefficients. Our focus in optimizing these distinct step-size parameters is dual: enhancing the convergence rate and lessening the difference between desired various impulse responses and both estimated and actual impulse responses. It's significant that our primary attention centers on $N$ error sub-signals, denoted as $e_i(k)$, which derive from the process of adaptive sub-filtering. We define $\boldsymbol{\epsilon}_i(k)$ as weight-error vectors used to measure the variance between real $\boldsymbol{h}$ and estimated sub-filter, $\widetilde{\boldsymbol{h}}_i(k)$.

$\left\{\begin{array}{c}\boldsymbol{\epsilon}_1(k)=\boldsymbol{h}-\widetilde{\boldsymbol{h}}_1(k) \\ \boldsymbol{\epsilon}_2(k)=\boldsymbol{h}-\widetilde{\boldsymbol{h}}_2(k) \\ \vdots \\ \boldsymbol{\epsilon}_N(k)=\boldsymbol{h}-\widetilde{\boldsymbol{h}}_N(k)\end{array}\right.$             (37)

The formula for the weight-error vectors $\boldsymbol{\epsilon}_i(k)$ is:

$\boldsymbol{\epsilon}_i(k)=\boldsymbol{h}-\widetilde{\boldsymbol{h}}_i(k)$          (38)

Based on Eq. (38), the minimization of mean square deviation (MSD) is used and determined by the next system of equations:

$\left\{\begin{aligned} c_1(k)=E\left[\left\|\boldsymbol{\epsilon}_1(k)\right\|^2\right] & =E\left[\left\|\mathbf{h}-\tilde{\mathbf{h}}_1(k)\right\|^2\right] \\ c_2(k)=E\left[\left\|\boldsymbol{\epsilon}_2(k)\right\|^2\right] & =E\left[\left\|\mathbf{h}-\tilde{\mathbf{h}}_2(k)\right\|^2\right] \\ \vdots & \\ c_N(k)=E\left[\left\|\boldsymbol{\epsilon}_N(k)\right\|^2\right] & =E\left[\left\|\mathbf{h}-\tilde{\mathbf{h}}_N(k)\right\|^2\right]\end{aligned}\right.$            (39)

After replacing Eq. (20) into Eq. (39), the next system of equations is obtained:

$\left\{\begin{array}{c}c_1(k)-c_1(k-1)=\mu_1^2 E\left[\frac{\left(e_1(k)\right)^2}{\alpha_1(k)}\right]- 2 \mu_1 E\left[\frac{\epsilon_1^T(k-1) \mathbf{P}_1(k-1) \mathbf{x}_1(k) e_1(k)}{\alpha_1(k)}\right] \\ c_2(k)-c_2(k-1)=\mu_2^2 E\left[\frac{\left(e_2(k)\right)^2}{\alpha_2(k)}\right]- 2 \mu_2 E\left[\frac{\epsilon_2^T(k-1) \mathbf{P}_2(k-1) \mathbf{x}_2(k) e_2(k)}{\alpha_2(k)}\right] \\ \vdots \\ c_N(k)-c_N(k-1)=\mu_N^2 E\left[\frac{\left(e_N(k)\right)^2}{\alpha_N(k)}\right]- 2 \mu_N E\left[\frac{\epsilon_N^T(k-1) \mathbf{P}_N(k-1) \mathbf{x}_N(k) e_N(k)}{\alpha_N(k)}\right]\end{array}\right.$          (40)

The formula for the system of Eq. (40) is:

$\begin{gathered}c_i(k)-c_i(k-1)=\mu_i^2 E\left[\frac{\left(e_i(k)\right)^2}{\alpha_i(k)}\right]- 2 \mu_i E\left[\frac{\epsilon_i^T(k-1) \mathbf{P}_i(k-1) \mathbf{x}_i(k) e_i(k)}{\alpha_i(k)}\right]\end{gathered}$          (41)

Based on the last equation, the relationship $c_i(k)-$ $c_i(k-1)=f\left(\mu_i\right)$ is obtained. It's important to allow that each separated fixed step-size $\mu_i$ carries an influence on the respective function $f\left(\mu_i\right)$, subsequently affecting the MSD. When we select for specific values of $\mu_i$ that maximize the function $f\left(\mu_i\right)$, we are effectively ensuring that the MSD experiences the most substantial reduction between successive iterations (from $k-1$ to $k$ ), i.e., $c_i(k)<c_i(k-1)$. By using the principal selection of $c_i(k)<c_i(k-1)$, it can be inferred that $f\left(\mu_i\right)<0$.

$\mu_i^2 E\left[\frac{\left(e_i(k)\right)^2}{\alpha_i(k)}\right]-2 \mu_i E\left[\frac{\epsilon_i^T(k-1) \mathbf{P}_i(k-1) \mathbf{x}_i(k) e_i(k)}{\alpha_i(k)}\right]<0$           (42)

By employing Eq. (42) the obtained general expression of all optimal values of step-size is the following:

$\mu_{i, o p t}<2\left\{\frac{E\left[\frac{\epsilon_i^T(k-1) \mathbf{P}_i(k-1) \mathbf{x}_i(k) e_i(k)}{\alpha_i(k)}\right]}{E\left[\frac{\left(e_i(k)\right)^2}{\alpha_i(k)}\right]}\right\}$            (43)

We note that all these optimal values are bounded as, $0<$ $\mu_{i, \text { opt }}<2$, and indicates that the optimal values of adapted step-size should fall within the range of 0 to 2 . This is a common guideline in adaptive filtering to ensure stability and convergence of the algorithm. Too small values might lead to slow convergence, while too large values might cause instability or divergence.

We define $\Delta_i(k)$ as a small quantity associated with the adaptive sub-filter $\tilde{h}_i(k)$, with $\mu_{i, \text { opt }}<2 \Delta_i(k)$, and $\Delta_i(k)$ is defined by:

$\Delta_i(k)=\left\{\frac{E\left[\frac{\epsilon_i^T(k-1) \mathbf{P}_i(k-1) \mathbf{x}_i(k) e_i(k)}{\alpha_i(k)}\right]}{E\left[\frac{\left(e_i(k)\right)^2}{\alpha_i(k)}\right]}\right\}$            (44)

To verifies the Eq. (43) and $0<\mu_{i, \text { opt }}<2$, apply that $\Delta_i(k)$ must be less to 1 . In the following, we will introduce our innovative approach to determine separated Variable-stepsizes parameters $\mu_i(k)$ using recursive estimations. Through these novel adaptations, we establish recursive formulas for calculating $\mu_{i, o p t}$, where $i$ takes values from 1 to $N$. In the last equation, it is important to highlight that all $\Delta_i(k)$ quantities are small sub-unitary scalar values. However, the $N$ small $\Delta_i(k)$ quantities can be approximated through an analysis of cross-correlation function between input signal $x_i(k)$ and estimated output sub-signal $e_i(k)$ of each adaptive sub-filter $\tilde{h}_i(k)$. We introduce a new approach to estimate $\Delta_i(k)$ using newly derived estimates denoted as $\tilde{\Delta}_i(k)$ that is equal to

$\tilde{\Delta}_i(k)=\frac{\left\|\boldsymbol{K}_i(k)\right\|^2}{\rho_i+\left\|\boldsymbol{K}_i(k)\right\|^2}$          (45)

We note that $0<\rho_i$ and $\left\{\rho_i+\left\|\boldsymbol{K}_i(k)\right\|^2\right\}>\left\|\boldsymbol{K}_i(k)\right\|^2$, it follows that $\tilde{\Delta}_i(k)<1$. Also, they are calculated as follows:

$\left\{\begin{array}{c}\tilde{\Delta}_1(k)=\frac{\left\|\boldsymbol{K}_1(k)\right\|^2}{\rho_1+\left\|\boldsymbol{K}_1(k)\right\|^2} \\ \tilde{\Delta}_2(k)=\frac{\left\|\boldsymbol{K}_2(k)\right\|^2}{\rho_2+\left\|\boldsymbol{K}_2(k)\right\|^2} \\ \vdots \\ \tilde{\Delta}_N(k)=\frac{\left\|\boldsymbol{K}_N(k)\right\|^2}{\rho_N+\left\|\boldsymbol{K}_N(k)\right\|^2}\end{array}\right.$          (46)

We notice that:

$\left\{\begin{array}{c}\mu_{1,  {opt }}<2 \times \tilde{\Delta}_1(k) \\ \mu_{2, { opt }}<2 \times \tilde{\Delta}_2(k) \\ \vdots \\ \mu_{N,  {opt }}<2 \times \tilde{\Delta}_N(k)\end{array}\right.$            (47)

In the last equations, we change 2 by the maximum value step-size denoted $\mu_{{max}}$, which is selected to achieve a faster convergence rate, under the condition that this maximum value is less than 2 . Now, we note that the optimal separated stepsizes are given by $\mu_{i, \text { opt }}(k)=\mu_{\max } \tilde{\Delta}_1(k)$. We then propose a method to independently and iteratively estimate the $N$ optimal step-size parameters using the following formulas:

$\left\{\begin{array}{c}\tilde{\mu}_{1, {opt}}(k)=\mu_{max} \times \frac{\left\|\boldsymbol{K}_1(k)\right\|^2}{\rho_1+\left\|\boldsymbol{K}_1(k)\right\|^2} \\ \tilde{\mu}_{2, {opt}}(k)=\mu_{max} \times \frac{\left\|\boldsymbol{K}_2(k)\right\|^2}{\rho_2+\left\|\boldsymbol{K}_2(k)\right\|^2} \\ \vdots \\ \tilde{\mu}_{N, {opt}}(k)=\mu_{max } \times \frac{\left\|\boldsymbol{K}_N(k)\right\|^2}{\rho_N+\left\|\boldsymbol{K}_N(k)\right\|^2}\end{array}\right.$           (48)

The general formula for $N$ optimal Variable-step-sizes parameters is the following:

$\tilde{\mu}_{i, o p t}(k)=\mu_{max } \times \frac{\left\|\boldsymbol{K}_i(k)\right\|^2}{\rho_i+\left\|\boldsymbol{K}_i(k)\right\|^2}$           (49)

The $N$ vectors $\boldsymbol{K}_i(k)$ are calculated taking in consideration the diagonal proportionate step-size $\mathbf{P}_i(k-1)$ and normalized by input sub-signal energy $\alpha_i(k)$. The estimation of these vectors follows the procedure outlined below:

$\left\{\begin{array}{c}\boldsymbol{K}_1(k)=\lambda_1 \boldsymbol{K}_1(k-1)+\left(1-\lambda_1\right) \frac{\mathbf{P}_1(k-1) \mathbf{x}_1(k) e_1(k)}{\alpha_1(k)} \\ \boldsymbol{K}_2(k)=\lambda_2 \boldsymbol{K}_2(k-1)+\left(1-\lambda_2\right) \frac{\mathbf{P}_2(k-1) \mathbf{x}_2(k) e_2(k)}{\alpha_2(k)} \\ \vdots \\ \boldsymbol{K}_N(k)=\lambda_N \boldsymbol{K}_N(k-1)+\left(1-\lambda_N\right) \frac{\mathbf{P}_N(k-1) \mathbf{x}_N(k) e_N(k)}{\alpha_N(k)}\end{array}\right.$           (50)

with $0<\lambda_1<1,0<\lambda_2<1, \ldots$, and $0<\lambda_N<1$. Generally, we can write:

$\boldsymbol{K}_i(k)=\lambda_i \boldsymbol{K}_i(k-1)+\left(1-\lambda_i\right) \frac{\boldsymbol{P}_i(k-1) \boldsymbol{x}_i(k) e_i(k)}{\alpha_i(k)}$               (51)

We note here that $\alpha_i(d)=\left[\mathbf{x}_i(k)\right]^T \mathbf{P}_i(k-1) \mathbf{x}_i(k)+\varepsilon$.

Finally, to ensure the convergence of the proposed SPV-NLMS algorithm. The values of all independent variable parameters must be verified using the following conditions:

$\left\{\begin{array}{l}\mu_1(k)= \begin{cases}\mu_{min } & { if } \tilde{\mu}_{1, {opt }}(k)<\mu_{min } \\ \tilde{\mu}_{1, { opt }}(k) & { otherwise }\end{cases} \\ \mu_2(k)= \begin{cases}\mu_{min } & { if } \tilde{\mu}_{2, { opt }}(k)<\mu_{min } \\ \tilde{\mu}_{2, {opt}}(k) & {otherwise }\end{cases} \\ \mu_N(k)= \begin{cases}\mu_{min } & {if} \tilde{\mu}_{N, {opt }}(k)<\mu_{\min } \\ \tilde{\mu}_{N, {opt}}(k) & { otherwise }\end{cases} \end{array}\right.$           (52)

The general separated Variable-step-sizes parameters can be managed through the following condition:

$\mu_i(k)=\left\{\begin{array}{lc}\mu_{min } & \text { if } \tilde{\mu}_{i, { opt }}(k)<\mu_{min } \\ \tilde{\mu}_{i, { opt }}(k) & { otherwise }\end{array}\right.$           (53)

While taking into account the good quality level, $\mu_{min }$ represents the possible minimum value that ensures and attains an optimal performance.

3.1 Optional reconstruction block

As an additional or optional stage, our attention is directed towards the synthesis filter bank that is placed in the output of the proposed algorithm. In some applications, such as acoustic noise reduction, speech enhancement and acoustic echo cancellation, the full-band output signal is required and must be calculated. For this aim, in our methodology, an interpolation block as illustrated in Figure 4 is integrated. The process of precisely reconstructing the entire error signal across all frequencies involves interpolating the $N$ output error sub-signals and using $G_1(z), \ldots, G_N(z)$.

4.jpg

Figure 4. The detailed bloc for creating full-band signal

The interpolated sub-errors are calculated as follows:

$\begin{cases}e_1(n)= \begin{cases}e_1\left(\frac{k}{I}\right) & n=0, \pm I, \pm 2 I, \ldots \\ 0 &   { otherwise }\end{cases} \\ e_2(n)= \begin{cases}e_2\left(\frac{k}{I}\right) & n=0, \pm I, \pm 2 I, \ldots \\ 0 &  { otherwise }\end{cases} \\ e_N(n)= \begin{cases}e_N\left(\frac{k}{I}\right) & n=0, \pm I, \pm 2 I, \ldots \\ 0 &   { otherwise }\end{cases} \end{cases}$              (54)

The general formula for the N interpolated error sub-signals and full-band output signal is the following:

$e_i(n)=\left\{\begin{array}{cl}e_i\left(\frac{k}{I}\right) & n=0, \pm I, \pm 2 I, \ldots \\ 0 & { otherwise }\end{array}\right.$               (55)

$e(n)=\sum_{i=1}^N \boldsymbol{g}_i^T(n) \boldsymbol{e}_i(n)$            (56)

where, $\mathbf{g}_i(n)=\left[g_i(n), g_i(n-1), \ldots, g_i(n-L+1)\right]^{\mathrm{T}}$, and $\boldsymbol{e}_i(n)=\left[e_i(n), e_i(n-1), \ldots, e_i(n-L+1)\right]^{\mathrm{T}}$.

This research paper provides a detailed investigation into the performance and effectiveness of the proposed SPV-NLMS algorithm for identifying various acoustic impulse responses. The algorithm is designed based on a sub-band structure, with each sub-filter utilizing individual proportionate step sizes. The study begins by establishing the novelty of the proposed algorithm in comparison to existing classical NLMS variants. A comprehensive derivation of the variable step size is presented, followed by an in-depth mathematical analysis of the algorithm's stability.

Through extensive simulations and analyses, the SPV-NLMS algorithm demonstrates robust performance, superior stability, a fast convergence rate, and remarkably low MSE values across diverse acoustic scenarios, including More-Dispersive, Dispersive, Sparse, and More-Sparse environments. The research examines various aspects of the algorithm, including stability under white noise conditions using MSE criteria. Notably, the proposed algorithm consistently outperforms other NLMS variants, highlighting its adaptability, accuracy, and efficiency. It effectively manages challenges introduced by USASI noise, maintains stable convergence, and achieves low final MSE values. By integrating sub-band processing with separate variable step-size adjustments, the algorithm excels in handling complex and dynamic acoustic systems, making it an ideal solution for applications requiring high stability, precise convergence, and dependable results.

Furthermore, the algorithm’s strong ERLE performance in suppressing residual signals underscores its effectiveness in echo cancellation tasks, proving its suitability for practical applications. As a potential future direction, the implementation of the proposed algorithm on digital signal processors and its application in real-world acoustic echo cancellation scenarios is recommended.