Speech Signal Modeling: Methods and Advances

• Speech Signal Modeling is a framework that uses mathematical models to capture and analyze the structure and dynamics of human speech for various applications.
• It integrates classical techniques such as source–filter decomposition and LPC with modern approaches like deterministic plus stochastic models and deep generative architectures.
• Practical implementations include MFCCs, time-varying harmonic models, and deep BLSTM-RNNs, which improve performance in analysis, synthesis, and speaker identification.

Speech signal modeling encompasses the development and application of mathematical frameworks that capture the underlying structure, dynamics, and variability of human speech waveforms for purposes spanning analysis, coding, enhancement, synthesis, and recognition. Core modeling paradigms include source–filter decomposition, explicit excitation models, high-dimensional feature extraction, statistical generative models, and deep representation learning frameworks. Progress in speech signal modeling has been driven by advances in both domain-specific signal processing and data-driven machine learning, with renewed emphasis on models that balance interpretability, statistical tractability, and direct applicability to real-world speech processing tasks.

1. Foundational Principles and Classical Models

At its foundation, speech signal modeling is historically rooted in the source–filter paradigm, which treats speech as the output of an excitation source (e.g., quasi-periodic glottal pulses for voiced sounds, noise for unvoiced sounds) filtered by the resonant properties of the vocal tract. Linear predictive coding (LPC) operationalizes this via an all-pole model: $s[n] = \sum_{k=1}^{K} a_k\, s[n-k] + e[n]$ where $s[n]$ is the speech waveform, $\{a_k\}$ are autoregressive coefficients modeling the vocal tract, and $e[n]$ is the excitation (residual) signal. This yields tractable statistical models for analysis and parametric synthesis (Young, 2013).

Classical approaches assume minimum-phase filtering, but real speech is inherently mixed-phase due to the anticausal (maximum-phase) component associated with the glottal open phase. To address this, maximum-phase modeling decomposes the speech signal using preemphasis, time reversal, and sequential LP analysis to separate minimum-phase and maximum-phase components, resulting in significantly sparser residuals (Drugman, 2020).

The spectral representation via Mel-frequency cepstral coefficients (MFCCs) is ubiquitous for parametric modeling in recognition. Frames of speech are windowed, Fourier transformed, passed through a mel filterbank, log-compressed, and subjected to a discrete cosine transform for decorrelation (Young, 2013).

2. Excitation Modeling Beyond the Source–Filter Assumption

Conventional all-pole excitation models inadequately capture the structure of the LPC residual, especially for applications such as vocoding and speaker characterization. The deterministic plus stochastic model (DSM) provides a two-band decomposition: $r(n) = d(n) + s(n)$ where $d(n)$ (deterministic component) models low-frequency, periodic structure via an orthonormal (PCA-derived) basis, and $s(n)$ (stochastic component) models high-frequency, noise-like content shaped both spectrally and temporally. DSM parameters—first eigenresidual $u_1(n)$, noise filter $h(n)$, and envelope $e(n)$—are extracted pitch-synchronously per speaker (Drugman et al., 2019).

The DSM framework yields improved parametric synthesis quality (HMM-based vocoding with DSM outperforms traditional pulse excitation and matches STRAIGHT), and enables robust speaker identification via “glottal signatures” constructed from deterministic and stochastic components. On TIMIT, joint use of these features yields a 96.35% identification rate, surpassing prior glottal-feature methods (Drugman et al., 2019).

Sparse excitation models, such as pole–zero models with block-sparse (glottal pulse-like) and Gaussian excitations, further enhance spectral fidelity and excitation recovery. Variational EM procedures enforce block sparsity, outperforming traditional all-pole and single-component models in capturing antiformants and complex spectral detail (Shi et al., 2017).

3. Statistical, Time-Varying, and Non-Stationary Speech Models

Short-time stationarity is often assumed, but speech is inherently non-stationary due to prosody, coarticulation, and speaker variability. Feature-based parametric models address this by tailoring representations to signal type:

• Voiced phonemes: complex amplitude-modulated (AM) sinusoids,
• Unvoiced/fricative phonemes: complex frequency-modulated (FM) structures,
• Transients: sums of damped complex exponentials (Sircar, 2018).

Parameter estimation leverages spectral peak analysis and linear least squares. For instance, the complex AM model of a vowel /ooo/ can be estimated via autocorrelation spectrum analysis and fitted to within a few percent RMS error over 50 ms segments (Sircar, 2018).

Time-varying harmonic models extend periodic representations to accommodate amplitude and frequency modulation: $$s(t) = \frac{a_0}{2} + \sum_{p=1}^P A_p(t) \cos\bigl(p \Phi_0(t) + \Phi_p(t)\bigr)$$ with $A_p(t)$ and $\Phi_0(t)$ modeled as low-order polynomials over the window. Estimation proceeds via alternating minimization over amplitude and phase coefficients, decoupling slow modulations from fast aperiodicity (e.g., subharmonics, diplophonia). These models yield robust harmonics-to-noise ratio (HNR) estimates even for severely modulated or dysphonic voices (Ikuma et al., 2022).

Waveform representation frameworks have advanced statistical parametric speech synthesis by embedding a full phase vector (smoothed as group delay) alongside the magnitude spectrum, jointly modeled via deep BLSTM-RNNs. This produces superior time- and phase-domain accuracy relative to systems that ignore phase or assume minimum-phase filters (Fan et al., 2015).

4. Deep Generative, Factorized, and Symbolic Speech Models

Modern approaches leverage deep generative models to factorize speech into underlying sources of variation—phonetic content, speaker characteristics, paralinguistic factors—within invertible or variational frameworks. The factorial discriminative normalization flow (DNF) introduces: $$x = f(z) = f_K \circ f_{K-1} \circ \cdots \circ f_1([z^{(Q)}, z^{(S)}])$$ where independent subcodes $z^{(Q)}, z^{(S)}$ encode phone and speaker identity, enabling disentangled manipulation and analysis. Exact-likelihood training and invertibility yield perfect reconstruction and selective transformation of speech factors with minimal cross-factor distortion (Sun et al., 2020).

Symbolic sequential models, implemented via vector-quantized VAEs, extract discrete, phoneme-like latent sequences from speech. These representations—integrated into speech enhancement architectures through multi-head attention—function as implicit linguistic constraints, improving perceptual (PESQ) and intelligibility (STOI) metrics over standard U-Net and multitask baselines. The assigned discrete symbols correlate with phoneme classes and cluster phonetically similar sounds, facilitating interpretable enhancement (Liao et al., 2019).

Unsupervised text-like “unit language” representations generated via n-gram modeling over discrete HuBERT embeddings enable denoising and cross-lingual alignment in speech-to-speech translation systems. Task-prompt strategies disentangle cross-modal and cross-lingual signals, yielding BLEU improvements competitive with transcript-based upper bounds (Zhang et al., 21 May 2025).

5. Multichannel, Multimodal, and Device-Specific Speech Modeling

Far-field and device-specific scenarios require models that accommodate spatial, spectral, and multimodal variability. Multichannel speech modeling exploits the correlations among time, frequency, and array channel via multivariate autoregressive (MAR) processes. These models jointly infer features in all three dimensions, with 3-D CNNs subsuming spatial filtering (e.g., beamforming) and spectro-temporal feature extraction, leading to 9–10% relative word error rate reductions over best conventional pipelines across reverberant benchmarks (Purushothaman et al., 2019).

Device-specific transfer function modeling is important for emerging sensor platforms. For instance, phoneme- and talker-dependent transfer characteristics between the outer-face and in-ear microphones in hearables are modeled via time-invariant FIR filters per phoneme. This speech-dependent relative transfer function (SD-RTF) approach substantially reduces spectral distortion compared to speech-independent models, particularly in talker/utterance-mismatched conditions, and is computationally feasible for real-time implementation (Ohlenbusch et al., 2023).

Millimeter-wave radar-based voice sensing is analytically connected to classical speech acoustics through models that relate neck-surface displacement (Δ(t)) to the acoustic pressure signal (p(t)) via an inverse vocal tract filter, time integration, and tissue delay. The radar transfer function is given by: $$H_{\mathrm{radar}}(\omega) = \frac{k_d}{K} \frac{e^{-j\omega\tau_d}}{j\omega H_{\mathrm{vt}}(\omega)}$$ Statistical validation over N=66 speakers shows that model-predicted vibrations align with radar measurements significantly better than the acoustic signal alone (Lenz et al., 19 Mar 2025).

6. Key Evaluation Metrics and Methodological Considerations

Evaluation protocols for speech models employ matched and mismatched utterance/talker conditions and quantify performance using metrics such as log-spectral distance (LSD), mel-cepstral distance (MCD), PESQ, STOI, and speaker identification rates. For residual modeling, sparsity measures (kurtosis, Gini, Hoyer's) diagnose excitation accuracy (Drugman, 2020, Shi et al., 2017). For device models, speech-dependent RTFs achieve LSD ≈ 2–2.5 dB under talker-mismatch, outperforming universal RTFs by ~3 dB (Ohlenbusch et al., 2023).

Subjective (MOS, CMOS) and objective (e.g., DNSMOS, WER) measures are collectively used, and signal fidelity is further stressed in waveform-domain evaluations and end-to-end synthesis (Fan et al., 2015, Richter et al., 2023).

Methodological distinctions include:

• The importance of pitch-synchronous analysis for excitation modeling,
• The use of alternating optimization (e.g., in polynomial-coefficient estimation for nonstationary models),
• Sparse Bayesian frameworks for enforcing block structure (e.g., VEM for block-sparse excitation (Shi et al., 2017)),
• Cross-modal integration using attention or generative modeling for learning robust latent spaces (Sun et al., 2020, Liao et al., 2019).

7. Emerging Directions and Outlook

Active research is expanding model expressivity, physiological grounding, and multimodal integration:

• Deep generative architectures for unsupervised decomposition into latent factors,
• Robust modeling for far-field, device- and modality-specific scenarios (e.g., in-ear, radar),
• Task-oriented symbolic and discrete representations for denoising, translation, and code-switching (Liao et al., 2019, Zhang et al., 21 May 2025),
• End-to-end learnable frontend–backend pipelines leveraging rich raw waveform and cross-device information (Purushothaman et al., 2019, Richter et al., 2023, Lenz et al., 19 Mar 2025).

This suggests a convergence of physically interpretable models, statistical generative frameworks, and representational learning for comprehensive, high-dimensional, and application-adaptive speech signal modeling.

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References

• Deterministic plus Stochastic residual model and its applications (Drugman et al., 2019)
• Deep generative factorization for speech signal (Sun et al., 2020)
• Time-varying harmonic models for voice signal analysis (Ikuma et al., 2022)
• Parametric modeling of non-stationary signals (Sircar, 2018)
• Maximum phase modeling for sparse linear prediction (Drugman, 2020)
• 3-D feature and acoustic modeling for far-field recognition (Purushothaman et al., 2019)
• Speech-dependent transfer function models for in-ear hearables (Ohlenbusch et al., 2023)
• Speech production model for radar (Lenz et al., 19 Mar 2025)
• Waveform representation for phase-embedded parametric synthesis (Fan et al., 2015)
• Statistical modeling in continuous speech recognition (Young, 2013)
• Symbolic sequential modeling for speech enhancement (Liao et al., 2019)
• Factorial DNF-based deep speech factorization (Sun et al., 2020)
• VEM method for sparse pole-zero voice excitation (Shi et al., 2017)
• Causal generative diffusion models for speech improvement (Richter et al., 2023)
• Unit language for textless S2ST modeling (Zhang et al., 21 May 2025)