Adaptive regularized spectral reduction for stabilizing ill-conditioned bone-conducted speech signals

Overview

LP is a fundamental tool in speech signal processing due to its effectiveness in modeling vocal tract dynamics. However, traditional ACR methods suffer from severe numerical instability in ill-conditioned environments such as BC speech, where spectral dynamic range expansion causes eigenvalue spread and matrix ill-conditioning (Zhang, Sugiura & Shimamura, 2022; Rahman & Shimamura, 2013; Prasad, Jyothi & Velmurugan, 2021). ACR often results in condition numbers exceeding 80 dB and a mean squared error of 0.12, as shown in Table 2, degrading performance especially in noisy environments where the signal-to-noise ratio is below 10 dB.

Covariance-based and spectral regularization methods

EC techniques improve LP stability by integrating forward-backward prediction errors with adaptive regularization (Edraki et al., 2024), reducing the condition number to 50 dB and offering moderate robustness (Wang et al., 2022b). However, they rely on static regularization heuristics and struggle in extreme noise. RMC approaches (Ohidujjaman et al., 2023, 2024) explicitly model spectral attenuation using IIR filters with coefficients $\varphi =0.32$ and $\psi =0.82$, achieving 20 dB attenuation in the 2–4 kHz range. Although RMC improves spectral stability and achieves 50 dB condition numbers, its fixed regularization parameter limits adaptability across diverse spectral scenarios. MVDR offers better noise suppression by minimizing power while preserving spectral features (Kabal, 2003). However, it suffers from high computational cost and is sensitive to model mismatch in highly variable BC signals.

Statistical and spectral filtering techniques

Statistical denoising methods, such as spectral subtraction (Vaseghi, 1996) and Wiener filtering (Abd El-Fattah et al., 2014; Cheng et al., 2023), achieve moderate signal to noise ratio (SNR) improvements (5–10 dB) under stationary noise assumptions. However, they are ineffective against spectral distortion due to dynamic range expansion and cranial filtering in BC speech, often yielding condition numbers >80 dB and higher spectral bias.

Advanced computational approaches

Deep learning models, including U-Net architectures and multimodal BC-AC fusion systems (Li, Yang & Yang, 2024; Wang, Zhang & Wang, 2022a), achieve strong gains in perceptual evaluation of speech quality (PESQ) (2.5–2.8) and SNR (12–15 dB). Yet, they lack interpretability, suffer from overfitting on unseen data (10–20% accuracy loss), and exhibit high computational cost (1–2 s per frame), limiting their use in real-time or edge scenarios.

Subspace and adaptive filtering techniques, such as SVD-based LP (Kumaresan & Tufts, 1981) and Kalman filtering (Millidge et al., 2021), improve robustness by isolating dominant spectral components or dynamically adapting to noise. Still, they require accurate noise models and runtime resources (0.5–0.8 s/frame), constraining their scalability and applicability in embedded systems. Overall, existing methods offer partial solutions to the ill-conditioning problem in BC speech. EC and RMC improve stability but lack frame-wise adaptability. Deep learning methods provide end-to-end enhancement but fail to control spectral dynamics explicitly. Subspace and adaptive methods reduce spectral bias but are computationally intensive. This leaves a clear gap for spectrally adaptive and computationally efficient methods.

To address these limitations, we propose the RSR framework, which dynamically tunes the regularization parameter $\lambda$ per frame based on spectral dynamic range estimation. RSR effectively compresses the eigenvalue spectrum, achieving an improved condition number of 35.54 dB, a lower mean squared error of 0.08, and a reduced runtime of 0.35 s, as detailed in Table 2. By combining adaptive regularization with orthogonal decomposition, RSR maintains numerical stability even under extreme ill-conditioning, surpassing prior methods in robustness, spectral control, and real-time feasibility.

Computational efficiency and robustness

The RSR method introduces additional computational complexity due to its regularization and iterative updates. However, its primary advantage lies in stabilizing solutions in ill-conditioned scenarios. Empirical evidence demonstrates that the RSR method significantly outperforms traditional methods such as LS and RLS in environments characterized by high noise levels and singular matrices, offering superior stability and accuracy.

Table 2 compares the computational cost, error rate, stability, and noise tolerance of RSR with LS and RLS methods. Although RSR incurs a higher computational cost, its enhanced stability and noise resilience justify the additional processing time. Furthermore, using orthogonal decomposition to solve optimization equations ensures computational efficiency, making the method viable for real-time applications.

The originality of the proposed RSR method lies in its ability to achieve a balanced trade-off between computational complexity and performance metrics such as error rate and numerical stability. Unlike traditional LS and RLS methods, which either compromise stability or incur high computational costs without explicit control over spectral dynamic range, the RSR framework introduces an adaptive regularization mechanism that significantly compresses eigenvalue expansion. This improves error rate performance while maintaining computational time within practical limits, as evidenced in Table 2. Such a methodological advancement ensures the RSR method is theoretically novel and practically valuable, especially for ill-conditioned environments encountered in BC speech processing.

It is important to clarify that while regularization introduces a controlled bias term into the least squares framework, it does not inject random noise. Instead, the regularization term $\lambda ||x|{|}^2$ penalizes excessively large solution components, thereby improving numerical stability and mitigating the ill-conditioning caused by spectral.

Noise robustness and parameter sensitivity

To evaluate the robustness of the RSR method, we conducted experiments under varying noise levels, assessing its sensitivity to the regularization parameter $\lambda$. Figure 1 illustrates that the RSR method consistently achieves lower error rates than simpler methods, even as noise levels increase. We mitigate the computational overhead typically associated with iterative parameter tuning by selecting a fixed $\lambda$ based on empirical studies.

The RSR method, while computationally intensive, demonstrates clear advantages in stability and accuracy, particularly in challenging, ill-conditioned, and noisy environments. Its ability to deliver real-time performance, enabled by orthogonal decomposition, positions it as a practical and effective solution for applications requiring robust and stable outcomes. By building on foundational principles such as those established in the EC method, the RSR framework leverages advanced mathematical techniques to achieve significant improvements in performance across diverse speech processing applications (Fulop, 2011; Ezzine & Frikha, 2017; Li, Yang & Yang, 2024). Figure 2 illustrates the synthetic BC speech generation process, where synthetic AC vowel signals are transformed into synthetic BC vowels. This transformation is achieved using a low-pass IIR filter, which replicates the spectral attenuation characteristics of BC speech, enabling controlled analysis of spectral transformations.

Synthetic BC vowel

Synthetic BC vowels were derived from synthetic AC vowels to assess the performance of the proposed method. The generation of synthetic AC vowel signals involved the excitation of an all-pole filter using a periodic impulse train, as outlined in Lawrence Marple (1991). The mathematical formulation of the all-pole filter’s transfer function is given by:

(12) $$H(z)=\frac{K_0}{1+{\sum }_{m=1}^n\beta (m)z^{-m}+{\sum }_{l=1}^p\gamma (l)z^{-2l}},$$where $K_0$ ( $K_0=0.1106$) denotes the gain factor, $\beta (m)$ and $\gamma (l)$ represent the first and second sets of filter coefficients and $n$ and $p$ indicate the filter orders. These parameters were specifically selected to replicate the spectral characteristics of AC vowels. Each vowel has a distinct fundamental frequency $F_0$; hence, using a constant $K_0$ may not fully capture energy differences. We refined $K_0$ using:

$$K_0=\frac{G}{{\sum }_{i=1}^p{\alpha }_i^2}$$where G is the target gain and ${\alpha }_i$ are LP coefficients. This formulation aligns with recent perspectives on spectral analysis and energy normalization in speech processing (Ohidujjaman et al., 2024). Table 4 presents distinct $K_0$ values computed for each vowel.

Table 5 provides the filter coefficients for generating the synthetic AC vowels. The AC vowels were processed through a low-pass infinite impulse response (IIR) filter to create synthetic BC vowels, adhering to the methodology outlined in Zhang, Sugiura & Shimamura (2022). The IIR filter emulates the attenuation characteristics of BC speech by transforming the synthetic AC speech signal, $a(n)$, into the synthetic BC speech signal, $\hat{b}(n)$ according to the following relationship:

(13) $$\hat{b}(n)=\psi \hat{b}(n-1)+\varphi a(n),$$where $\psi$ and $\varphi$ are the filter coefficients set to 0.82 and 0.32, respectively. The attenuation level achieved by the IIR filter is analytically computed as:

$$\mathrm{A}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}(\mathrm{d}\mathrm{B})=20{\log }_{10}\left(\frac{\varphi }{1-\psi }\right)$$

With $\psi =0.82$ and $\varphi =0.32$, the resulting attenuation is approximately 5 dB:

$$\mathrm{A}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=20{\log }_{10}\left(\frac{0.32}{0.18}\right)\approx 5\mathrm{d}\mathrm{B}$$

This design moderately attenuates high-frequency components while preserving sufficient spectral energy for analysis. Although some literature models up to 20 dB attenuation (Weber-Wulff et al., 2023), our selected parameters balance attenuation and speech signal integrity for algorithm evaluation. The synthetic BC vowels generated through this process enable a controlled analysis of the spectral transformations inherent in BC speech, facilitating the systematic evaluation of the proposed method under simulated conditions. Figure 3 shows the amplitude response of the IIR filter. The filter is designed to attenuate high-frequency components while preserving low-frequency energy, effectively mimicking the spectral profile of bone-conducted speech. This figure highlights the reduction in spectral energy above a certain frequency threshold, consistent with BC speech properties.

Figure 4 compares the spectral distributions of AC and BC speech signals for the vowel /a/ case. The BC speech spectrum demonstrates a pronounced concentration of energy in the low-frequency range, in contrast to the broader spectral distribution observed in AC speech. This figure underscores the spectral differences between the two modalities, illustrating the unique characteristics of BC speech.

Deriving the regularization parameter μ

From the experimental results, we derived the constant K as 0.00170. Using the formula $\mu =KA$, where A is the positive amplitude of the BC speech signal, the value of $\mu$ for different amplitude levels is shown in Table 8. This approach helps determine the appropriate regularization parameter $\mu$ based on the amplitude level of the input BC speech signal.

For asymmetric amplitude dynamic ranges (e.g., $[-50,+100]$), we use the positive maximum amplitude $A=100$ for the rule-based calculation of $\mu$. This ensures the regularization adapts to the maximum spectral intensity, maintaining model stability. Therefore, $\mu =K\cdot 100$ is applied in such cases. This rule holds for all asymmetric SDRs, where only the positive maximum is considered in the computation to ensure robustness across varying dynamic conditions.

Performance evaluation

The condition number K, measured in decibels (dB), is a widely accepted metric for quantifying ill-conditioning in numerical computations. For this study, the condition number is computed as follows:

(16) $$K=10{\log }_{10}\left(\frac{||\mathbf{B}|{|}_{\mathrm{F}}\cdot ||{\mathbf{B}}^{-1}|{|}_{\mathrm{F}}}{{\sum }_{i=1}^n{w}_i{\lambda }_i}\right),$$where $||\mathbf{B}|{|}_{\mathrm{F}}$ and $||{\mathbf{B}}^{-1}|{|}_{\mathrm{F}}$ denote the Frobenius norms of the matrix $\mathbf{B}$ and its inverse, respectively. The terms $w_i$ and ${\lambda }_i$ represent the weighting coefficients and eigenvalues. This formulation incorporates a weighted eigenvalue summation, allowing for a more nuanced analysis of matrix stability. A lower condition number K indicates better eigenvalue compression, directly translating to improved numerical stability in ill-conditioned systems. The proposed RSR method was evaluated against conventional methods, including ACR, SC, and the EC approach across synthetic BC vowels. The results, summarized in Table 9, demonstrate the superior performance of the RSR method, which achieved significantly lower condition numbers. This improvement highlights its ability to suppress eigenvalue expansion more effectively than competing techniques, thereby enhancing model stability under diverse spectral conditions.

While speech is inherently non-stationary, the RSR method operates on short analysis frames where the signal can be assumed quasi-stationary. Within each frame, adaptive regularization is applied based on spectral dynamic range, which ensures numerical stability without the need for iterative learning or global training.

The computational complexity of the RSR framework remains dominated by the linear prediction matrix inversion step ( $\mathcal{O}(p^3)$), with a negligible additional cost for adaptive $\mu$ estimation. Our experiments (Table 2) confirm that the method remains practical for real-time processing.

Impact of regularization parameter μ on condition number

The regularization parameter $\mu$ plays a pivotal role in stabilizing the linear prediction process under ill-conditioned spectral scenarios. In the RSR framework, the autocorrelation matrix $\mathbf{R}$ is modified as $\mathbf{B}=\mathbf{R}+\mu \mathbf{I}$, where $\mathbf{I}$ is the identity matrix. This transformation directly impacts the eigenvalue distribution of $\mathbf{R}$, lifting smaller eigenvalues and compressing the spectral dynamic range. Consequently, the matrix condition number—quantified by Eq. (16)—is reduced, enhancing numerical robustness. Although $\mu$ does not appear explicitly in the condition number formula, its effect is embedded through the modified matrix $\mathbf{B}$. As shown in Tables 9 and 10, appropriately tuning $\mu$ leads to significant gains in matrix stability while preserving critical spectral information. This justifies the adaptive per-frame selection strategy employed in RSR, where $\mu$ is aligned with spectral energy to achieve a trade-off between over-smoothing and instability.

Computational considerations and practical feasibility

While speech signals are inherently non-stationary, the RSR method operates on short analysis frames (20–30 ms), where the signal is considered quasi-stationary—a common assumption in speech processing.

Within each frame, the adaptive regularization parameter $\mu$ is determined based on the spectral dynamic range (SDR), allowing the method to stabilize ill-conditioned scenarios without iterative optimization or global training.

The computational complexity of the RSR method is primarily governed by the linear prediction matrix inversion, which has a complexity of $\mathcal{O}(p^3)$. The additional cost for computing $\mu$ is negligible, involving simple frame-level SDR estimation.

Our experiments confirm that the RSR method is computationally efficient and suitable for real-time BC speech processing systems.

Evaluation with real BC vowels

To validate the RSR method under practical conditions, we conducted experiments using real BC vowels derived from the RASC-863 corpus and a 30k daily dialogue corpus. These datasets collectively provide extensive phonetic coverage and topic diversity, enabling a rigorous evaluation of speech processing methods. The experimental setup adhered to ISO 3745 standards for anechoic chambers, ensuring high-fidelity recordings. A SabineTek-designed headset with BC microphones and a Zoom H1n recorder was utilized to simultaneously acquire AC and BC speech. The dataset comprises recordings from 100 native Chinese speakers (ages 20–35) who speak standard Mandarin. Postprocessing included manual segmentation and cleaning, yielding 42 h of labeled utterances. The finalized database is publicly accessible at https://github.com/wangmou21/abcs (Wang et al., 2022c).

Condition numbers for real BC vowels were calculated using Eq. (16), and the results are presented in Table 10. The RSR method consistently outperformed conventional approaches, achieving the lowest condition numbers across all tested vowels. These findings corroborate the synthetic vowel results, underscoring the robustness and generalizability of the RSR approach.

Performance evaluation and findings

The analysis reveals the robust performance of the proposed RSR method compared to conventional approaches such as ACR, SC, and EC. RSR consistently achieves superior condition number compression, which enhances numerical stability and spectral fidelity. This improvement results from its dynamic regularization mechanism, which adapts to frame-wise spectral conditions.

Figure 5 illustrates the inverse relationship between condition number and spectral bias. As the condition number decreases, spectral bias also reduces, affirming the effectiveness of RSR in maintaining spectral structure. RSR achieves the lowest values for both metrics, outperforming other methods in mitigating ill-conditioning.

Figure 6 compares condition numbers across vowels (A, I, U, E, O). The RSR method consistently yields the lowest condition numbers across all cases, demonstrating strong suppression of eigenvalue expansion. While deep learning models offer strong end-to-end performance, they lack explicit spectral control, making them less effective in ill-conditioned environments. RSR directly addresses this limitation by enforcing numerical stability through eigenvalue regularization.

Table 11 provides a detailed evaluation of each method’s numerical characteristics and regularization strategy. The RSR method demonstrates superior performance in error minimization and condition number reduction while uniquely supporting frame-level adaptive control—surpassing traditional and heuristic-based techniques in robustness and adaptability.

Key insights and contributions

RSR addresses a critical challenge in BC speech processing: spectral ill-conditioning during linear prediction analysis. This issue occurs when the autocorrelation matrix becomes nearly singular due to non-uniform spectral energy distribution, particularly in BC speech, where energy is concentrated at low frequencies. A key symptom of this is eigenvalue expansion, where the wide spread between eigenvalues leads to large condition numbers and unstable LP coefficient estimation. RSR mitigates this issue by introducing a regularization term adaptively tuned per frame. This dynamic approach allows stable estimation of LP coefficients while preserving important spectral structure, outperforming traditional LP methods and fixed-parameter regularization techniques.

Broader sources of Ill-conditioning

Although eigenvalue expansion is the dominant cause of ill-conditioning in BC speech, other factors also contribute. These include poor data scaling, rank deficiencies in the autocorrelation matrix, and limitations imposed by finite numerical precision. While RSR indirectly mitigates some of these effects by normalizing amplitude dynamic range, it does not explicitly address rank deficiency or precision-aware processing. Future improvements could incorporate rank-revealing matrix decompositions or solvers optimized for low-precision environments.

Alternative methods, such as subspace approaches based on singular value decomposition (SVD) or adaptive filtering techniques like Kalman filtering, could offer additional pathways to address ill-conditioning. These strategies may provide enhanced stability in rank-deficient or dynamically varying acoustic conditions. Future research should compare RSR with these approaches across spontaneous speech and multilingual datasets to fully characterize stability and computational complexity trade-offs.

Perceptual implications and human-centric applications

Since BC speech predominantly carries low-frequency information, RSR’s emphasis on preserving this region may enhance perceptual phenomena such as speaker identification or self-voice recognition. Previous work suggests that playback of self-voice through BC pathways enhances auditory self-awareness (Orepic et al., 2023). RSR could further support this by maintaining spectral fidelity.

Nonetheless, overly aggressive regularization may attenuate high-frequency components critical for consonant clarity, such as fricatives and plosives. These phonemes play a key role in speech intelligibility. Future perceptual experiments, including sentence-based intelligibility tests, will be needed to determine the perceptual impact of RSR in user-facing scenarios like hearing aids, smart headsets, and VR audio systems.

Speculative applications and broader impact

The improvements in numerical stability offered by RSR open possibilities for applications beyond BC speech processing. In secure speech systems, spectral compression enabled by RSR may reduce signal leakage and improve privacy. In biomedical signal processing, RSR could help stabilize the analysis of weak and noisy signals such as electrocardiograms or electroencephalograms. Additionally, in augmented and virtual reality platforms, RSR could support real-time voice augmentation by providing consistent and perceptually stable spectral shaping under variable acoustic conditions. Integrating RSR with perceptually informed models could enhance its utility in human-centered applications by aligning numerical processing with auditory system characteristics.

Despite its advantages, RSR currently assumes quasi-stationarity within analysis frames. This assumption may not hold during rapid transitions in conversational speech, potentially limiting performance. Also, the matrix inversion step in the algorithm remains computationally intensive. It may constrain deployment in real-time or embedded platforms.

The proposed RSR method represents a significant advancement in BC speech processing. Directly addressing the ill-conditioning problem improves numerical stability while maintaining spectral fidelity. The framework’s adaptability, real-time compatibility, and potential for perceptual benefit position it as a promising solution for various communication, security, health, and immersive media applications.