New Advanced Deep Learning Variable Step-Size Adaptive Feed-Forward Algorithm for Two-Sensor Acoustic Noise Reduction

Acoustic noise remains a critical challenge in modern telecommunication systems such as hands-free telephony, teleconferencing systems, and hearing aids. Effective noise reduction in these applications requires accurate modeling and adaptive estimation of the acoustic channel. While single-channel noise reduction techniques have shown reasonable performance due to their simplicity and ease of deployment [1], they are often inadequate in environments with highly non-stationary noise or complex acoustic reverberations. Classical approaches such as minimum mean square error (MMSE) estimators [2], spectral subtraction [3], and Wiener filters [4] have been widely adopted but still face trade-offs between noise suppression and speech distortion. To overcome these limitations, adaptive filtering algorithms have been extensively studied in both fullband [5] and subband [6] forms based on adaptive filtering algorithms [5-9]. Nonetheless, single-channel approaches often struggle with non-stationary signals due to the difficulty in robustly estimating the noise characteristics.

Multi-channel techniques, particularly blind source separation (BSS) approaches [10-12], have emerged as powerful alternatives for acoustic noise reduction, leveraging spatial diversity to improve speech intelligibility and signal quality. However, systems with more than two microphones are often impractical for mobile or embedded applications due to increased hardware and computational complexity. As a compromise, two-channel BSS systems offer a balanced trade-off between performance and implementation complexity, and they have demonstrated superior results in terms of speech enhancement and convergence rate [13-15].

In this context, various two-channel feed-forward and backward BSS structures have been proposed, with the feed-forward structure offering better noise suppression but introducing some signal distortion, and the backward structure providing cleaner output signals at the cost of lower noise reduction. To mitigate these issues, post-filtering techniques and symmetric adaptive decorrelation (SAD) algorithms have been introduced [16, 17]. Furthermore, several adaptive filtering algorithms tailored to two-channel BSS in sub-band form have been proposed to further improve convergence behavior and robustness in reverberant environments [16, 18, 19]. More recently, two-channel variable step-size (VSS) NLMS algorithms have been developed to address the long-standing trade-off between fast convergence and low steady-state misadjustment [20]. A forward-and-backward two-channel VSS NLMS framework was introduced, where step-size adaptation was guided by a signal decorrelation function [15]. However, these conventional VSS methods still exhibit suboptimal performance in sparse acoustic environments, especially under complex noise conditions.

To address these limitations, in this paper, we propose a novel deep learning-based variable step-size estimation mechanism implemented on two-sensor noise reduction technique based on NLMS adaptive filtering. A deep neural network is trained on multiple types of acoustic noise signals to learn the relationship between noisy-signal power characteristics and optimal step-size values. Our novelty is an MSD-supervised, VAD-gated step-size and its integration in a two-sensor FF-NLMS system, with policy-aligned features computed on only noisy frames. This adaptive mechanism enables the algorithm to dynamically adjust the step-size parameters, resulting in faster convergence and improved speech quality in sparse and dispersive impulse response situations. The proposed method is rigorously evaluated through extensive experiments using objective performance criteria. Experimental results confirm the effectiveness and robustness of the deep learning-guided approach for variable step-size estimation and two-sensor acoustic noise reduction.

This paper is presented as follows: in Section 2, the two-channel convolutive mixing problem between speech and noise signals is detailed. Section 3 is reserved for the presentation of the proposed two-sensor Feed-forward technique adapted by new deep learning-based variable step-size mechanism. The comparative simulation results are presented in Section 5, and finally, the conclusion of this paper is presented in Section 6.

In this sub-section, we present the proposed two-sensor feed-forward structure based on deep learning variable step-sizes approach and we give the optimal solutions of adaptive filters, as we present the formulations of the DL-VSS-FNLMS algorithm in the time-domain implemented on this structure. The global model of the proposed two-sensor Feed-forward algorithm is presented in Figure 4.

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Figure 4. Global structure of proposed DL-VSS-FNLMS

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Figure 5. Detailed presentation of proposed DL-VSS-FNLMS algorithm

Noting that, all two-sensor Feed-forward techniques are based on the assumptions that the sources signals $sp(n)$ and $ns(n)$ are mutually independent, i.e., $E[s p(n) n s(n-m)]=0, \forall m$, or alternatively, this implies that they are uncorrelated. The proposed DL-VSS-FNLMS algorithm introduces a deep learning-assisted variable step-size adaptive filtering framework for acoustic noise reduction, a database of noise signals is first combined with clean speech through a convolutive mixing model to simulate real-world noisy environments. A deep learning model predicts the optimal adaptation parameters for the adaptive filter $w(n)$. The proposed algorithm integrates a voice activity detector (VAD), which identifies speech-active and noise-only segments to control the filter adaptation. This selective update mechanism enhances noise suppression during active speech and reduces computational overhead during silent or noise-only periods, resulting in more efficient and targeted filter adaptation. The detailed structure of proposed DL-VSS-FNLMS algorithm is presented in Figure 5.

3.1 Voice activity detection mechanism

The proposed DL-VSS-FNLMS algorithm incorporates a voice activity detection (VAD) mechanism to guide the adaptation of its filtering process. VAD systems are commonly used to distinguish between segments of speech that contain only background noise or silence, as presented in Figure 6. This information is then used to control how and when filter coefficients are updated.

In the DL-VSS approach, adaptive filter w(n) processes the incoming noisy signal. A key feature of this method is that the filter w(n) is updated only during noise-only intervals or inactive speech periods. This selective adaptation strategy reduces the computational load and improves overall efficiency. Figure 7 shows a schematic of the VAD system, detailing how it manages the filter updates within the two-sensor Feed-forward configuration of the DL-VSS algorithm.

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Figure 6. Example of speech signal segmentation using VAD

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Figure 7. The control of adaptive filter by VAD system

To enhance the adaptability of the proposed two-sensor Feed-forward algorithm, the step-size parameter $\mu_{D L}(n)$ is dynamically estimated using a deep learning model. These parameters are crucial for ensuring both convergence and efficient tracking performance of the adaptive filter $\boldsymbol{w}(n)$. The prediction of the variable step-size is achieved by training a neural network using relevant input features derived from the noisy speech signal during noise periods, as identified by the VAD system.

3.2 Adaptive filter formulation

For updating the adaptive filter $\boldsymbol{w}(n)$, we propose to use two-sensor Feed-forward structure adapted by the NLMS algorithm but controlled in this case by the new deep learning variable step-sizes parameters. The update formula of adaptive filter by this new algorithm are adjusted according the VAD given as follows:

$\boldsymbol{w}(n)=\left\{\begin{array}{cl}\boldsymbol{w}(n-1)+\mu_{D L}(n) \frac{\widetilde{s p}(n) \boldsymbol{m} \boldsymbol{s}_2(n)}{\varepsilon+\left\|\boldsymbol{m} \boldsymbol{s}_2(n)\right\|^2} & \text { if } V A D=0 \\ \boldsymbol{w}(n-1) & \text { if } V A D=1\end{array}\right.$    (5)

where, $\mu_{D L}(n)$ represents the adaptive step-size estimated by deep learning model, with the necessary and sufficient condition to guarantee the convergence and the stability of this algorithm in the MSE sense is $0<\mu_{D L}(n)<2$, and $\varepsilon$ presents a very small positive constant. The tap-weight vectors of the adaptive filters $\boldsymbol{w}(n)$ is defined respectively by:

$\boldsymbol{w}(n)=\left[w_1(n), w_2(n), \ldots, w_M(n)\right]^T$

We also define vector (last M values) of inputs noisy signal, $m s_2(n)$ as:

$\boldsymbol{m} \boldsymbol{s}_2(n)=\left[m s_2(n), m s_2(n-1), \ldots, m s_2(n-M+1)\right]^T$

where, M is the length of adaptive filter.

3.3 Proposed minimization of variable step-size

The adaptation step-size is a key parameter in the performance of adaptive filter $w(n)$. To ensure rapid convergence and accurate tracking of the optimal filter coefficients, we define the step-size as the result of a minimization process that targets the mean-square deviation (MSD) between the ideal impulse response and the current filter weights. We aim to find the optimal step-size parameter $\mu_{D L}^{o p t}(n)$ for adaptive filter $w(n)$ such that the deviation from the ideal filter $h_{21}$ is minimized:

$\xi(n)=\boldsymbol{h}_{21}-\boldsymbol{w}(n)$     (6)

We define the expressions of MSD for adaptive filter $\boldsymbol{w}(n)$ as follows:

$M S D(n)=E\left[\|\xi(n)\|^2\right]$  (7)

We derive a relation showing how MSD evolves over time as a function of the step-size parameter. After deriving from the previous equation and computing the squared Euclidean model, we arrive at the following equation:

$\begin{aligned} \operatorname{MSD}(n)-\operatorname{MSD} & (n-1) =\mu_{D L}^2 E\left[\frac{(\widetilde{s p}(n))^2}{\sigma_2(n)}\right] -2 \mu_{D L} E\left[\frac{\xi^T(n-1) \widetilde{\mathbf{m s}}_{\mathbf{2}}(n) \widetilde{s p}(n)}{\sigma_2(n)}\right]\end{aligned}$   (8)

To ensure improvement, $\operatorname{MSD}(n)-\operatorname{MSD}(n-1)<0$, given that $\mu_{D L}^{\text {opt }}(n)<2\, \nabla(n)$. Here, $\nabla(n)$ is small value calculated from the cross-correlation between the input and output signal of adaptive filter. To avoid direct computation of expectations, we propose the recursive formulas, $\mu_{D L}^{o p t}(n)= \mu_{\max } \widetilde{\nabla}(n)$, with the estimated quantities $\widetilde{\nabla}(n)$ is given by,

$\widetilde{\nabla}(n)=\mu_{\max }\left[\frac{\|\mathbf{Q}(n)\|^2}{\rho+\|\mathbf{Q}(d)\|^2}\right]$   (9)

where, $\mathbf{Q}(n)$ captures the cross-correlation dynamics and is updated over time, using the next recursive estimation,

$\mathbf{Q}(n)=\lambda \mathbf{Q}(n-1)+\frac{(1-\lambda)}{\sigma_2(n)+\varepsilon} \widetilde{s p}(n) \widetilde{\mathbf{m s}}_2(n)$    (10)

with $0<\lambda_i<1$. This theoretical optimal value $\mu_{D L}^{\text {opt }}(n)$ is then used as the training target for a deep neural network. The network learns to predict the step-size dynamically from combined features, replicating this minimization behavior without needing explicit MSD calculations during inference.

3.4 Deep learning VSS parameters estimation

To enable effective prediction of the variable step-size parameter $\mu_{D L}(n)$ in the two-sensor adaptive Feed-forward NLMS process, it is important to provide the deep learning model with input features that accurately reflect the dynamic properties of the second noisy speech signal $m s_2(n)$. This section outlines the feature extraction methodology used to construct informative input vectors for training the neural network. This part is divided in three important parts, (i) noisy speech database, (ii) audio parameters extraction, and (iii) variable step-sizes deep learning prediction, as presented in Figure 8.

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Figure 8. Three parts of deep learning VSS parameters extraction

3.4.1 Noisy speech database generation

In this part, the convolutive mixing model used in this work simulates the real-world interaction between a speech source and its acoustic environment. It is characterized by two types of acoustic impulse responses, dispersive and sparse [23-25], which reflect the physical behavior of sound propagation in closed spaces. The dispersive impulse responses are characterized by long-duration, densely populated coefficients that model environments with rich reverberation and multiple reflections, such as large rooms or halls, where the energy spreads over time. In contrast, the sparse impulse responses contain only a few significant nonzero coefficients, representing environments with limited reflections or directional propagation paths, such as small or acoustically treated spaces. This realistic modeling is essential for generating training data that challenges the adaptive filtering system and allows the deep learning model to learn context-aware step-size adaptation strategies. To train and evaluate the deep learning model for variable step-size prediction in adaptive filtering, we constructed a dataset of noisy speech signals composed of clean speech signals combined with various environmental noises using a two-channel convolutive mixing model. The clean speech signals were taken from publicly available speech corpora, i.e., TIMIT [26], which provide high-quality recordings from multiple speakers under clean conditions. These signals serve as the primary content to be enhanced by the adaptive filtering process.

After preparing the clean speech recordings, we mixed them with several different noise types selected from the NOISEX-92 database [27]. These included a variety of real-world acoustic environments such as white noise, babble noise, F16 aircraft noise, factory1 noise, HF channel noise, and buccaneer noise. These diverse noise environments were chosen to evaluate the robustness and generalization capability of the proposed algorithm under both stationary and non-stationary interference. Each noise signal was convolved with the clean speech using a two-channel convolutive mixing model, which simulates the propagation of noise through an acoustic environment such as a room or enclosure. This model allows the creation of realistic mixed signals that closely replicate the challenging conditions encountered in practical applications.

The resulting dataset includes a wide variety of speech-in-noise examples that reflect both temporal and spectral variability (see Figure 9). These mixtures are essential for training the neural network to predict step-size parameters that respond appropriately to dynamic and nonstationary acoustic environments.

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Figure 9. Generation of noisy speech database

3.4.2 Audio parameters extraction

The procedure begins with signal normalization to reduce amplitude variability, followed by the extraction of Mel-Frequency Cepstral Coefficients (MFCCs), which are commonly used in speech and audio analysis for their ability to capture phonetic and spectral content. To complement these, additional perceptual and cepstral features, including Gammatone Cepstral Coefficients (GTCCs), their first and second temporal derivatives, and energy-based spectral descriptors from the ERB, Bark, and Mel scales, are computed using the audio Feature Extractor. All extracted features are then concatenated into unified vectors representing each time frame of the speech signal. To further refine the training data and improve the model’s ability to distinguish between active and inactive speech regions, a silence detection step based on frame energy is applied. The resulting feature set provides a rich and discriminative representation of the acoustic signal.

(i) Second noisy speech signal preprocessing

The second input noisy speech signal $m s_2(n)$ of the twosensor Feed-forward structure, is first normalized to ensure that its amplitude remains within the range $[-1,1]$. The normalization was performed using the following operation:

$m s_{2, n}(n)=\frac{m s_2(n)}{\max \left|m s_2(n)\right|}$    (11)

Its main objective of the noisy speech normalization is to improve numerical stability during signal processing by scaling the signal amplitude within a predictable range, thus preventing computational errors or loss of precision. Additionally, it helps maintain the quality and comparability of extracted features, making them less sensitive to variations in recording volume. This ensures that the deep learning model can generalize more effectively across different speech recordings, regardless of their original amplitude levels.

(ii) MFCC coefficient configuration

The MFCCs are widely used acoustic features in speech and audio processing due to their ability to compactly represent the short-term power spectrum of a signal in a perceptually meaningful way. In this work, MFCCs were extracted as part of the input features for the deep learning model, following a carefully designed segmentation and preprocessing procedure. Given that speech is a non-stationary signal, it must be analyzed over short segments where it can be approximated as quasi-stationary. To achieve this, the signal was divided into overlapping frames using the three important parameters:

Window length: A duration of 25 milliseconds was selected, corresponding to $L_W=\operatorname{round}\left(0.025 \times \mathrm{f}_s\right)$, where f_s is the sampling rate. This window size strikes a balance between temporal resolution (to capture rapid transitions such as phonemes) and frequency resolution (to preserve spectral content).

Hop length (overlap): A shift of 10 milliseconds was applied between successive frames: $L_H=\operatorname{round}\left(0.01 \times \mathrm{f}_s\right)$ resulting in a frame overlap of 15ms. This overlap ensures continuity across frames, reduces the risk of missing transient information, and provides smoother transitions in the extracted features.

$L_{\text {overlap }}=L_W-L_H$     (12)

Hamming window: To minimize spectral leakage due to discontinuities at the frame boundaries, a Hamming window was applied to each frame. This window gradually attenuates the signal at the edges of each segment, improving the accuracy and robustness of the spectral analysis.

$\begin{gathered}H w(n)=0.54-0.46 \times \cos \left(\frac{2 \pi n}{L_W-1}\right), \\ 0 \leq n<L_W\end{gathered}$   (13)

(iii) MFCC coefficient extraction

Once the signal was windowed and preprocessed, 13 MFCCs were computed per frame. The process follows these main steps: Short-Time Fourier Transform (STFT) to convert each frame into the frequency domain, Mel-filterbank processing, where the magnitude spectrum is passed through a series of triangular filters spaced according to the Mel scale to mimic human auditory perception, Logarithmic compression to emulate the non-linear perception of loudness, and finally the Discrete Cosine Transform (DCT) to decorrelate the filterbank outputs and retain only the most relevant coefficients.

The extraction of MFCCs is based on a series of mathematical transformations applied to audio signal segments, aiming to capture relevant spectral characteristics for acoustic analysis. The mathematical formulation is given by:

$\operatorname{MFCC}(t)=\operatorname{DCT}\left(\log \left(\operatorname{MelSpectrum}\left(m x_{2, n}(t)\right)\right)\right.$      (14)

where, $m x_{2, n}(t)$ is the time-domain signal segment centered at time n , MelSpectrum represents the application of a triangular filterbank based on the Mel scale to the signal's spectrum, $\log$ simulates the nonlinear human perception of loudness, and $D C T$ is used to decorrelate the $\log _{-} M e l$ features and compact the information.

The output of the Mel_filterbank Energy is defined by:

$E_m=\log \left(\sum_k\left|M X_{2, n}[k]\right|^2 \times H_m[k]\right)$    (15)

with, $M X_{2, n}[k]$ is the FFT of the windowed noisy signal $m x_{2, n}(n), H_m[k]$ presents Mel filterbank, $\left|M X_{2, n}[k]\right|^2$ the power spectral density of the noisy signal $m x_{2, n}(n)$, and finally the Logarithm is applied to match human auditory perception. Then, the cepstral coefficients $c_n$ are computed as,

$\begin{gathered}c_n=\sum_{m=1}^M E_m \times \cos \left(n \times \frac{\pi}{M} \times\left(m-\frac{1}{2}\right)\right), 0 \leq n <N_{\text {coeffs }}\end{gathered}$    (16)

$M$ is the number of Mel filters, $N_{\text {coeffs }}$ is the number of desired cepstral coefficients (typically 12 or 13), and it converts the M_logMel energies into $N_{\text {coeffs }}$ decorrelated cepstral coefficients.

(iv) Extended spectral feature extraction

In addition to the standard MFCCs, a broader and more perceptually-informed set of audio features was extracted. This step aims to capture complementary spectral characteristics that enhance the robustness and expressiveness of the features fed into the deep learning model, particularly in the presence of noise. The following spectral representations were computed, each based on different psychoacoustic models of human hearing:

Equivalent Rectangular Bandwidth Spectrum (ERB): This feature emulates the frequency resolution of the human cochlea by analyzing the signal across perceptually equivalent frequency bands. It provides a fine-grained spectral decomposition aligned with auditory filter bandwidths.

Given a short-time Fourier-transformed magnitude spectrum $\left|M X_{2, n}(f)\right|^2$, the ERB Spectrum at frame n is computed by filtering this spectrum through a bank of ERBspaced filters $H_m^{E R B}(f)$:

$E R B_m(n)=\log \left(\sum_f\left|M X_{2, n}(f)\right|^2 \times H_m^{E R B}(f)\right)$    (17)

with $M X_{2, n}(f)$ is the Fourier transform of the signal frame centered at time $\mathrm{t}, H_m^{E R B}(f)$ is the frequency response of the $m^{\text {th }}$ Gammatone filter in the ERB-scaled filterbank, $m= 1,2, \ldots, M$ where $M$ is the number of ERB bands.

Bark Spectrum: This representation divides the frequency axis into critical bands that reflect how the human ear groups frequencies. It is especially useful in capturing perceptually relevant changes in the spectral envelope.

Mel Spectrum: The Mel scale approximates how humans perceive pitch. It is linear in the low-frequency range and logarithmic in the high-frequency range, thereby giving more resolution to lower frequencies where speech energy is concentrated.

These spectral features provide multidimensional insights into the energy distribution across the perceptual frequency space, helping the neural model better interpret phonetic and prosodic content.

Cepstral Features: GTCC and Derivatives

GTCC: Similar to MFCCs but derived from a Gammatone filterbank, which is believed to more accurately model the frequency selectivity of the human auditory system. GTCCs are particularly effective in noisy environments and provide an alternative spectral representation of the audio signal.

Steps for GTCC extraction

Let $m x_{2, n}(n)$ be a time-domain frame centered at time t, and apply the windowed STFT:

$M X_{2, n}(k)=\operatorname{SFTT}\left(m x_{2, n}(n)\right)$  (18)

Power Spectrum

$P_t(k)=\left|M X_{2, n}(k)\right|^2$   (19)

Apply a Gammatone filterbank $G_m[k]$ (with $M$ filters):

$E_m(t)=\sum_k P_t(k) \times G_m[k]$     (20)

Logarithmic Compression

$\widetilde{E}_m(t)=\log \left(E_m(t)+\varepsilon\right)$  (21)

with $\varepsilon$ is a small constant to avoid $\log (0)$

Discrete Cosine Transform (DCT)

$\operatorname{GTCC}_n(t)=\sum_{m=1}^M \tilde{E}_m(t) \times \cos \left(\frac{n \pi}{M} \times\left(m-\frac{1}{2}\right)\right), 0 \leq n<N_{\text {coeffs }}$      (22)

Delta Coefficients ($\Delta G T C C$) and Delta-Delta Coefficients $\left(\Delta^2 G T C C\right)$: These temporal derivatives of GTCCs represent the first and second-order changes over time analogous to velocity and acceleration. By capturing how spectral features evolve, they add important dynamic context that is essential for modeling time-varying signals like speech.

$\Delta G T C C$ represent the first-order temporal derivatives (rate of change):

$\Delta G T C C_n(t)=\frac{\sum_{l=0}^L l \times\left(G T C C_n(t+l)-G T C C_n(t-l)\right)}{2 \times \sum_{l=0}^L l^2}$  (23)

where, L is the window size for derivative calculation (usually 2).

$\Delta^2$GTCC is the second-order temporal derivatives (acceleration):

$\begin{aligned}\Delta^2 \operatorname{GTCC}_n(t) =\frac{\sum_{l=0}^L l \times\left(\Delta G T C C_n(t+l)-\Delta G T C C_n(t-l)\right)}{2 \times \sum_{l=0}^L l^2}\end{aligned}$    (24)

The final extended spectral feature vector for each time frame t is defined as:

$\operatorname{AFE}(t)=\left[\begin{array}{c}G T C C_n(t), \Delta G T C C_n(t), \Delta^2 G T C C_n(t), \\ \operatorname{Mel}(t), \operatorname{Bark}(t), E B R(t)\end{array}\right]$    (25)

This rich, multidimensional representation significantly enhances the descriptive power of the input features, enabling the neural network to more accurately model and adapt to complex acoustic environments.

(v) Features combination

Once the spectral and cepstral features—including MFCCs, spectral representations (Mel, Bark, ERB), and GTCCs with their temporal derivatives—have been extracted, they are concatenated to form a single combined feature matrix. This process brings together complementary aspects of the speech signal, enriching the representation for downstream learning tasks.

By merging features with different perceptual and spectral perspectives, the combined matrix captures both the static and dynamic properties of the speech content. This multidimensional feature set serves as a robust and informative input to the deep learning model, enabling it to better distinguish between speech and noise, and to adapt effectively in complex acoustic environments.

$F_t=[M F C C(t), A F E(t)]$    (26)

The feature set is chosen to support step-size control rather than phonetic recognition. Frame energy stabilizes $\mu_{D L}(n)$ during bursts; MFCC/$\Delta$ capture near-end speech leakage that risks speech distortion if the step size is too large; GTCC/$\Delta$ are more noise-robust and track narrowband/colored interferers; ERB/Bark/Mel provide coarse, low-variance spectral envelopes that help at very low SNR.

(vi) Automatic silence detection

To enhance the training dataset for the step-size prediction model, Automatic silence detection was applied. Frame-level energy was computed as the squared norm of each feature vector:

$E_t=\sum_{i=1}^D F_{t, i}^2$   (27)

where, $D$ is the dimension of the feature vector. A threshold based on the $10^{{th }}$ percentile of the energy distribution was used to identify silent frames:

$Silence\quad Frames =\left\{t \mid E_t<\right.$ percentile $\left.(E, 10)\right\}$    (28)

These silent frames are particularly informative, as they typically correspond to regions where the optimal adaptation step-size should be small or even zero. This enables the neural model to better generalize across both speech-active and inactive segments.

(vii) Steps of audio features extraction

In Figure 10, we present all steps used for audio feature extraction based on all subparts presented previously.

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Figure 10. Proposed audio features extraction for proposed deep learning model

3.5 Deep learning model for variable step size estimation

This study introduces a deep learning-based model designed to estimate the adaptive step-size parameters, specifically $\mu_{D L}(n)$, for the dual-microphone NLMS algorithm. The model's objective is to dynamically predict this parameter directly from acoustic features extracted from noisy speech. The chosen architecture is a Recurrent Neural Network (RNN) employing stacked Long Short-Term Memory (LSTM) layers, which are particularly effective at capturing temporal dependencies inherent in sequential audio features.

(1) Data Preparation and Normalization

The input Features presented previously in subsection 3.4. The network's input is a comprehensive feature matrix derived from preceding stages, integrating MFCCs, GTCCs, and various other spectral representations (ERB, Bark, Mel). These features are concatenated to form a rich representation for each time frame of the noisy speech signal. The corresponding labels for the network are the step-size parameter $\mu(n)$. These are computed for each time frame using a predefined analytical method presented previously in subsection 3.3. The entire dataset is randomly partitioned into training and testing subsets, maintaining a 70:30 ratio respectively.

To ensure numerical stability and consistency during the training phase, z-score normalization is applied to the input features. This normalization uses the mean and standard deviation calculated exclusively from the training set. This step ensures equal contribution from all feature dimensions during learning and accelerates convergence. The training features are transposed and converted into a sequence format suitable for time-series modeling by the RNN.

(2) Network Architecture

The architecture of the proposed Recurrent Neural Network (RNN), as presented in Figure 11, is specifically designed to model the temporal evolution of acoustic features and to predict the variable step-size parameters of the dual-microphone NLMS algorithm. The network leverages a stack of Long Short-Term Memory (LSTM) layers, which are well-suited for sequence learning due to their ability to capture long-range dependencies.

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Figure 11. Detailed deep learning model used for acoustic noise reduction

The architectural components are detailed below:

Input Layer: The model begins with a sequenceInputLayer, which accepts sequential input data. This layer is configured to match the dimensionality of the input feature vector (i.e., the number of concatenated acoustic features per time frame).

First LSTM Block: The first recurrent block consists of an LSTM layer with 128 memory units. This layer processes the temporal sequence and captures short- to medium-term dependencies across frames. A dropout layer with a dropout rate of 20% is applied immediately after this layer to reduce overfitting and improve generalization.

Second LSTM Block: A second LSTM layer with 64 units is stacked atop the first to enable deeper sequence representation learning. This layer further refines the model’s capacity to understand longer-term temporal structures. Again, a 20% dropout is applied post-activation.

Third LSTM Block: The third and final LSTM layer contains 32 units and is configured with OutputMode ='last', ensuring that only the final hidden state is propagated forward. This design choice enables the network to summarize the entire input sequence into a compact latent representation, capturing the most relevant temporal information for the prediction task.

Fully Connected Layers: The compressed temporal representation is passed through two fully connected (dense) layers: (i) The first dense layer has 64 neurons, followed by a ReLU activation function to introduce non-linearity and enable the learning of complex mappings. (ii) the second dense layer comprises a single neuron, responsible for outputting the predicted continuous-valued step-size coefficient for the current frame.

Output Layer: A regression layer is used as the final output component. It maps the scalar output from the dense layer to the real-valued target, allowing the model to perform a frame-level regression task.

(3) Training Strategy

To effectively optimize the proposed RNN architecture for variable step-size prediction, a carefully designed training protocol was adopted. This strategy ensures robust convergence, generalization, and efficient utilization of computational resources. The key components of the training process are described as follows:

Optimization Algorithm: The network parameters are optimized using the Adam optimizer, a widely adopted stochastic gradient-based method that combines the advantages of Adaptive Gradient Algorithm (AdaGrad) and Root Mean Square Propagation (RMSProp). Adam provides efficient and stable convergence through adaptive learning rates and momentum terms for each parameter.

Training Duration: The training process is executed over 100 full epochs, allowing sufficient opportunity for the model to learn complex temporal patterns and minimize prediction error. Empirical tuning confirmed that this duration balances model performance and training time.

Mini-Batch Configuration: A mini-batch size of 32 samples is employed during training. This batch size provides a good trade-off between convergence stability and computational efficiency, especially for time-series data where memory consumption can become a constraint.

Data Shuffling: To enhance the model's ability to generalize beyond the training dataset and prevent overfitting to sequence order, shuffling is applied to the training data at the end of each epoch. This disrupts any spurious correlations due to ordering in the training sequences.

Loss Monitoring and Custom Callbacks: A custom callback function is integrated into the training loop to record and monitor the evolution of the training loss after each epoch. This enables continuous assessment of the learning progress and early detection of issues such as stagnation or overfitting.

Software Implementation: The entire model training pipeline, including network definition, loss tracking, and data preprocessing, is implemented using Deep Learning Toolbox. Specifically, the trainNetwork function is utilized to manage the iterative optimization process.

This RNN-based predictor provides a robust, data-driven mechanism to adaptively determine optimal step-size parameter for the NLMS algorithm, thereby improving its robustness and convergence characteristics across diverse acoustic environments.

3.6 Enhanced speech signal estimation

In this structure, we use an adaptive filter $\boldsymbol{w}(n)$ to identify the two acoustical impulse response $\boldsymbol{h}_{21}$ of two-channel convolutive mixture system. In the last part of proposed algorithm and after convergence, the optimal solution is given by some studies [11, 15, 21, 22]:

$\boldsymbol{w}(n)=\boldsymbol{h}_{21}$  (29)

With this optimal solution of the adaptive filters, the estimated speech signal can be rewritten as follows:

$\begin{aligned} \widetilde{s p}(n)=s n(n) & *\left[\boldsymbol{h}_{21}-\boldsymbol{w}(n)\right]+s p(n) *\left[\delta(n)-\boldsymbol{h}_{12} * \boldsymbol{w}(n)\right]\end{aligned}$    (30)

$\widetilde{s p}(n)=s p(n) * D_s$  (31)

where, $D_s$ represents small distortion.

In this paper, we have proposed a novel deep learning-based variable step-size estimation method integrated into a two-sensor Feed-forward NLMS algorithm, effectively enhancing adaptive filtering performance in complex acoustic environments. Utilizing an RNN with stacked LSTM layers and a diverse set of acoustic features (MFCCs, GTCCs, ERB, Bark, Mel), the DL-VSS model demonstrated its ability to dynamically and accurately predict optimal step sizes. The model’s robustness was further supported by reliable energy-based silence detection across varied noise types, which provided critical contextual cues for accurate prediction. Objective criteria (MAE, MSE, R²) validated the predictive strength of the approach, especially in sparse conditions, confirming its efficacy as a data-driven solution for adaptive noise reduction in challenging environments, dispersive and sparse situations. Compared to classical FNLMS with fixed step sizes, the proposed DL-VSS-FNLMS algorithm consistently achieved faster convergence, lower steady-state system mismatch, and improved segmental SNR, reflecting superior noise suppression and speech clarity in both dispersive and sparse conditions. Moreover, it ensured more stable step-size control, particularly in cases where classical methods falter due to large step-size instability.

This study shows that our deep learning model effectively performs simultaneous noise reduction and dereverberation on the NOISEX-92 benchmark, enabling direct comparison with prior work. Nonetheless, the methodology is designed to generalize, and future work will evaluate the model across broader, more diverse datasets to fully assess real-world robustness.