Stochastic signal representation via harmonic parameter factorization

1 Introduction

Stochastic variability is intrinsic to many physical and telecommunication signals. Thermal and shot noise, multipath fading, amplifier noise, oscillator phase noise, polarization-mode fluctuations, and time-varying interference all induce randomness that cannot be eliminated without sacrificing bandwidth, latency, or sensitivity. In practice, the question is rarely whether a signal is stochastic, but how to represent and process it efficiently while preserving the statistics that matter for inference and decision-making.

Conventional digital signal processing (DSP) pipelines typically address stochasticity by estimating features from quantized samples (e.g., envelopes, band energies, spectral moments, instantaneous frequency indicators) and then applying filtering, detection, or classification. These approaches are effective, yet they remain tightly coupled to the continuous-amplitude representation and inherit implementation constraints (analog-to-digital converter (ADC) precision, estimator bias, calibration drift, windowing choices). In parallel, stochastic computing encodes real-valued scalars as Bernoulli bitstreams, enabling low-complexity arithmetic and error-tolerant aggregation (Gaines and Tou, 1969; Alaghi and Hayes, 2013). However, most stochastic-computing workflows implicitly assume that the values to be encoded are already available as explicit scalars, leaving open a structured, reconstructable, physics-informed mapping from a stochastic waveform window to an interpretable set of independent bitstreams. At the same time, recent studies in multisensor and information fusion have emphasized that heterogeneous measurements often require uncertainty-aware and event-oriented integration strategies rather than direct comparison in native physical units (Xue et al., 2021; Roheda et al., 2021; Guti et al., 2022; Li et al., 2024a; Liu et al., 2025; Li et al., 2024b). This is particularly relevant when sensor outputs must be transformed into a common representation suitable for subsequent probabilistic or stochastic processing. In parallel, recent developments in stochastic computing have continued to refine bitstream-based arithmetic and representation schemes, highlighting both their efficiency and their sensitivity to representation design choices (Liu et al., 2024; Chen et al., 2025; Wang et al., 2024a).

In this work, we propose and validate a stochastic signal representation via harmonic parameter factorization, which bridges waveform-domain modeling and bitstream-domain processing. The core idea is to approximate each analysis window by a small set of dominant harmonic components and then encode each component’s interpretable parameters—amplitude, frequency, and phase represented in a bounded and numerically stable form cos ⁡ φ k , sin ⁡ φ k—into mutually decorrelated Bernoulli bitstreams. This parameterization avoids phase-wrap ambiguities and supports stable decoding via simple bitstream statistics. Recent decomposition- and phase-estimation studies also indicate that compact signal representations should be evaluated not only by nominal reconstruction accuracy, but also by robustness of parameterization, especially for phase variables near discontinuity boundaries and for signals with structure distributed across multiple temporal scales (Oliveira et al., 2024; Zhou et al., 2022). The approach is aligned with a broader trend toward structured, differentiable, and reproducible signal-processing interfaces that treat transforms and parameterizations as first-class computational objects (Hayes et al., 2024).

In contrast to recent multisensor fusion studies that focus on uncertainty-aware or event-driven integration at the decision level (Xue et al., 2021; Roheda et al., 2021; Guti et al., 2022; Li et al., 2024a; Liu et al., 2025; Li et al., 2024b), and in contrast to recent stochastic-computing studies centered on arithmetic primitives or hardware architectures (Liu et al., 2024; Chen et al., 2025; Wang et al., 2024a), the present work addresses the signal-representation layer itself. It is also distinct from modern decomposition-oriented approaches, where the primary emphasis is often placed on adaptive mode extraction or time-frequency separation rather than on stochastic parameter encoding and bitstream-domain representation of interpretable harmonic components (Oliveira et al., 2024; Zhou et al., 2022).

A key aspect of the proposed framework is that it enables controlled validation of the representation independently of any particular downstream stochastic processor. To isolate representational error (finite K, finite bitstream length N, probability mapping/calibration) from algorithmic distortions, we employ a NOT–NOT identity pass through the bitstreams. This protocol verifies that the encode–process–decode loop preserves intended probabilities, while explicitly monitoring inter-stream correlations—known to bias stochastic computations and to degrade accuracy if bitstreams are not decorrelated (Alaghi and Hayes, 2013). In addition to reconstruction-level fidelity metrics, including root-mean-square error (RMSE) and correlation, we test whether the reconstructed signal preserves key process statistics of the original stochastic waveform, including power spectral density (PSD), autocorrelation function (ACF), and amplitude distributions.

The proposed interface is intended as a general and reproducible bridge between noisy waveform measurements and bitstream-domain processing, including telecommunication contexts where noise and phase dynamics are fundamental rather than incidental. By grounding the factorization in a harmonic parameterization of each window, the method retains interpretability and exposes a principled trade-off: increasing K reduces truncation error, while increasing N reduces stochastic encoding error. This separation of error sources provides a transparent route to designing representations that meet a target fidelity under latency and complexity constraints. More broadly, the work complements Frontiers-reported evidence that stochasticity and stochastic/binary abstractions can be an asset (not only a limitation) when building emerging computing paradigms (Morshed et al., 2023) and can be explicitly leveraged in hardware-level switching dynamics used for stochastic or probabilistic computation (Bengel et al., 2021).

Contributions of this paper:

• We formulate a harmonic parameterization of a stochastic signal window using a small set of K dominant components and a phase representation based on (cos ⁡ φ k , ⁡ sin ⁡ φ k) to avoid phase-wrap ambiguities.
• We propose a factorized encoding of (A k , f k , ⁡ cos ⁡ φ k , ⁡ sin ⁡ φ k) into mutually decorrelated Bernoulli bitstreams with a reproducible pseudo-random number generator (PRNG)/seeding policy.
• We define an end-to-end validation protocol based on NOT–NOT identity processing, correlation control, and multi-level fidelity metrics (parameter RMSE; reconstruction RMSE/correlation; PSD/ACF/distribution similarity).
• We provide a fully reproducible implementation and metadata logging enabling repetition across K ∈ 3 , 6 , 9 , 12, S N R grids, and bitstream lengths N.

2 Materials and methods

2.1 Input signals, sampling, and windowing protocol

We consider a discrete-time signal x n sampled at rate f s and processed in windows of duration T w, containing L = T w f s samples. Windows are non-overlapping by default (overlap is optional and reported when used). Each window is treated as a locally stationary realization of a stochastic process, enabling stable spectral peak selection and consistent parameter estimation. Prior to spectral analysis, a taper (e.g., Hann) is applied to reduce spectral leakage in finite-length observations and to improve the robustness of peak picking under noise (Harris, 1978; Oppenheim and Schafer, 2010).

For validation, we use synthetic signals with controlled signal-to-noise ratio (SNR) and known harmonic structure, as well as real digitized traces serving as practical case studies. Noise is injected as additive white Gaussian noise (AWGN) to achieve target SNR ∈ {0,5,10,15} dB, with the signal-to-noise ratio defined per window by Equation 1: where P s is the average power of the windowed clean signal and P n is the AWGN power.

S N R = 10 ⁡ log 10 P s / P n , (1)

2.2 Harmonic decomposition within a finite window

Within each window, the signal is decomposed into a finite set of sinusoidal components using FFT/DFT-based spectral analysis (Harris, 1978; Oppenheim and Schafer, 2010). From the magnitude spectrum, we select K dominant components according to a deterministic rule, where K denotes the number of retained harmonic components used to represent the current analysis window. This selection is performed reproducibly according to the following rules:

• Peak-based selection: choose the K largest local maxima in ∣ X k ∣ within the analysis band.
• Band restriction (telecom setting): restrict search to a predefined band [f min , f max] to avoid out-of-band artifacts.
• Tie-breaking: if peaks are equal within tolerance, select the lower-frequency component first.

For real-valued x n, only positive-frequency bins are used, and each selected bin corresponds to a sinusoidal harmonic characterized by amplitude, frequency, and phase. This yields the parameter set A k , f k , φ k k = 1 K per window. For each selected frequency bin k, the initial frequency estimate is given by Equation 2:

f k = k f s L . (2)

The complex FFT coefficient X k is used to estimate phase as φ k = ∠ X k. The amplitude estimate A k is obtained from ∣ X k ∣ using a deterministic scaling rule; when a taper is used, an amplitude correction based on the window coherent gain is applied to ensure consistent amplitude interpretation across windows (Harris, 1978).

2.3 Harmonic parameter set and phase representation via cos ⁡ φ k and sin ⁡ φ k

Each selected harmonic is described by:

• Amplitude A k: magnitude-related contribution of the k-th component.
• Frequency f k: FFT-bin frequency mapped into Hz.
• Phase φ k: relative phase of the component within the window.

Instead of encoding φ k directly, we encode the phase on the unit circle via the two bounded parameters defined in Equation 3:which avoids discontinuities at 0/2π and provides a bounded, wrap-free representation of phase on the unit circle. Phase is recovered according to Equation 4.

c k = cos ⁡ φ k , s k = sin ⁡ φ k , (3)

φ ^ k = atan ⁡ 2 s ^ k , c ^ k . (4)

This representation is particularly convenient for probability mapping because c k , s k ∈ − 1 , 1 admit a direct linear mapping into [0,1]. This avoids discontinuities at 0/2π and supports stable error metrics for phase-like variables (Fisher, 1993).

This wrap-free phase parameterization is consistent with correlation-optics and optical-singularity frameworks, where phase structure and correlation properties are treated as primary descriptors of random optical fields and their diagnostics (Angelsky et al., 2022a; Angelsky et al., 2022b; Angelsky et al., 2021).

Accordingly, throughout this work we evaluate phase-related errors using R M S E computed separately for cos ⁡ φ k and sin ⁡ φ k, rather than R M S E of φ k itself, which would be ill-defined due to phase wrapping. In the revised manuscript, this expected advantage is further verified by a dedicated phase-wrap ambiguity experiment comparing direct stochastic phase encoding with the proposed bounded (cosφ, sinφ)-based representation.

2.4 Optional polarization factor p o l k

If polarization is included, each harmonic may additionally carry a scalar polarization descriptor p o l k ∈ 0 , 1 (e.g., the fraction of energy in a reference polarization channel, or another bounded polarization indicator consistent with the measurement setup). In the present validation, p o l k is treated as an optional extension: it can be fixed (control case) or estimated from dual-polarization measurements if available. When enabled, it is encoded into a Bernoulli stream identically to other bounded parameters.

2.5 Mapping “parameter → probability” with calibration metadata

Each harmonic parameter is mapped to a probability p ∈ 0 , 1 via a monotone calibrated transform. We use parameter-specific normalization ranges (either fixed from calibration or estimated robustly from training windows using quantiles), then apply clipping to guarantee bounds. The mapping is defined per parameter type:

• Amplitude: A k ↦ p A k. We use either linear normalization in [A min , A max] or log-compressed mapping for wide dynamic range (telecom/optical scenarios) (Gaines and Tou, 1969; Alaghi and Hayes, 2013).
• Frequency: f k ↦ p f k using normalization in [f min , f max] (analysis band).
• Phase pair: c k = cos ⁡ φ k ↦ p c k , s k = sin ⁡ φ k ↦ p s k through a direct linear map from [−1,1] to [0,1]: values near +1 map close to 1, values near −1 map close to 0.
• Polarization (optional): p o l k ↦ p p o l k by direct identity mapping if already in [0,1], or by linear normalization otherwise.

All mapping parameters (ranges, transform type, clipping policy) are stored in metadata to ensure exact reproducibility across runs and to make decoding unambiguous.

The mapping is strictly monotone within the calibrated range so that decoding is uniquely defined by the stored inverse transform (metadata).

2.6 Bernoulli bitstream generation and decorrelation policy

For each probability p θ, we generate a Bernoulli bitstream S θ i ∈ 0 , 1 of length N using the standard “PRNG + comparator” scheme in stochastic computing (Gaines and Tou, 1969; Alaghi and Hayes, 2013). In each clock tick, a pseudo-random number r i ∼ U 0 , 1 is drawn and compared to p θ; the output bit is 1 if r i < p θ, else 0.

Critical requirement: all streams must be statistically independent (or at least strongly decorrelated). We enforce this by:

• Using independent PRNG instances per stream, or
• Using a single PRNG with distinct, non-overlapping seeds per stream, derived from (trial id, k, parameter type).

As an ablation (Section 2.10), we also test the failure mode where multiple streams share the same seed or PRNG state, which increases inter-stream correlation and can bias downstream statistics.

2.7 Bitstream length N and experiment grid

To characterize the accuracy–latency tradeoff, we vary the bitstream length

N ∈ 2 8 , 2 9 , 2 10 , 2 11 , 2 12 , 2 13 .

Each window produces K harmonics and, per harmonic, M p parameter streams. In the main configuration we use M p = 4 (amplitude, frequency, cosine, sine), and optionally M p = 5 if polarization is enabled. Thus, the total number of streams per window is (4 K) or (5 K). For the illustrative modeling case K = 3 with polarization enabled, this yields 15 Bernoulli streams per window, matching the intended “compact but non-trivial” demonstration.

We run a full grid across:

• K ∈ 3 , 6 , 9 , 12 ,
• S N R ∈ 0 , 5 , 10 , 15 d B ,
• N ∈ 28 , … , 213 ,

with multiple Monte Carlo trials per point (e.g., 30–100) to stabilize reported averages and confidence intervals.

2.8 Test stochastic processor: NOT–NOT identity check

To validate “input → bitstreams → processing → bitstreams → reconstruction” without introducing algorithm-specific distortions, we employ a deliberately neutral processing block: two successive NOT operations applied bitwise to each stream. Since NOT–NOT is the identity, any mismatch between the reconstructed output and the reference signal can be attributed to harmonic truncation (K), probability estimation noise due to finite N, mapping and quantization effects, or correlation artifacts in PRNG/seeding, rather than to the processing logic itself. This makes NOT–NOT a controlled pass-through test for the proposed representation.

2.9 Decoding: probability estimation, parameter recovery, and signal reconstruction

For each stream S θ i we estimate the probability by the fraction of ones over the length-N sequence. The recovered probability p ^ θ is then converted back to the normalized parameter via the inverse mapping recorded in metadata, yielding A ^ k , f ^ k , c ^ k , s ^ k and optionally p o l ^ k. Phase is recovered according to Equation 4. The reconstructed time-domain signal x ^ [n] in the same window is obtained as the sum of the K recovered harmonics using {A ^ k , f ^ k, φ ^ k}. Because only K components are retained, reconstruction is generally approximate; therefore, the evaluation is performed not only in the time domain but also at the level of process statistics (Section 2.10).

2.10 Evaluation metrics: parameter-, signal-, and statistics-level reproducibility

We evaluate reproducibility at three levels:

Parameter-level accuracy. For each K, S N R, and N, we compute R M S E (and optionally bias/mean absolute error (MAE)) for recovered harmonic parameters:

• Amplitude R M S E across harmonics and trials,
• Frequency R M S E,
• R M S E for cos ⁡ φ and sin ⁡ φ (preferred over R M S E of φ due to angular wrap).

Signal-level reconstruction quality. We compute time-domain measures between x n and x ^ [n] within each window:

• R M S E/normalized R M S E,
• Pearson correlation coefficient.

Statistics-level reproducibility (primary target). Since the goal is to reproduce the stochastic process characteristics, we compare:

• PSD: difference between the reference and reconstructed power spectral densities, estimated by Welch’s method for stable spectral estimation (Welch, 1967), and reported as an integrated error over the analysis band (e.g., L 2 norm in linear scale or dB-weighted error).
• ACF: distance between reference and reconstructed autocorrelation functions over a chosen lag range, reported as an L 2 norm or normalized error.
• Amplitude distribution: distance between amplitude (or sample) distributions, using a one-dimensional metric such as Kolmogorov–Smirnov (KS) distance or Jensen–Shannon divergence (one selected metric is sufficient for the main text; others may be placed in Supplementary).

These three families of metrics jointly quantify whether the method preserves (a) harmonic parameters, (b) waveform similarity, and (c) process statistics.

2.11 Baselines and ablation studies

To contextualize performance, we use two reference comparisons:

Baseline 1 (DFT-only truncation bound). Reconstruction from the same N harmonics without Bernoulli streams (pure floating-point parameter pipeline). This isolates the irreducible loss due to harmonic truncation and windowing.

Ablation (correlated streams). We intentionally reuse the same seed (or PRNG state) across multiple streams to induce inter-stream correlation, then quantify degradation in parameter and statistics-level metrics. This directly demonstrates the necessity of bitstream decorrelation for factorized multi-stream representations (Morshed et al., 2023; Bengel et al., 2021).

All implementation settings (window length L, window function, FFT size, peak-picking rule, mapping ranges, PRNG type and seed policy, trial counts) are recorded in a metadata bundle to ensure strict reproducibility and straightforward peer verification.

2.12 Implementation in Google Colab (reproducible computational workflow)

All experiments were implemented in a self-contained Google Colab notebook (Python 3.x) using standard scientific libraries (NumPy/SciPy/Matplotlib). The notebook executes a single end-to-end pipeline: windowing and harmonic selection (top-K), probability mapping with stored metadata, Bernoulli bitstream generation with a defined seeding policy, NOT–NOT identity processing, decoding and signal reconstruction, and export of all figures/tables at publication settings (including 300 DPI raster output). The full executable code, configuration, and reproducibility instructions are provided in the Supplementary Code package.

To clarify the practical computational structure of the method, Table 1 summarizes the main processing stages, including spectral analysis, dominant-harmonic selection, parameter estimation, stochastic encoding, decoding, and reconstruction. This breakdown highlights how the overall computational cost depends on the signal length, the retained harmonic count K, and the bitstream length N. In particular, increasing K enriches the representational model, whereas increasing N improves stochastic estimation precision at the cost of longer bitstreams and higher computational load.

TABLE 1. Main processing stages and computational structure of the proposed method. Summary of the principal stages of the workflow, including spectral analysis, dominant-harmonic selection, parameter estimation, stochastic encoding, decoding, and reconstruction. The table clarifies the role of each stage and indicates the dominant complexity trends with respect to signal length, retained harmonic count K, and bitstream length N.

To clarify the practical computational structure of the method, Table 1 summarizes the main processing stages, including spectral analysis, dominant-harmonic selection, parameter estimation, stochastic encoding, decoding, and reconstruction. This breakdown highlights how the overall computational cost depends on the signal length, the retained harmonic count K, and the bitstream length N. In particular, increasing K enriches the representational model, whereas increasing N improves stochastic estimation precision at the cost of longer bitstreams and higher computational load.

The overall workflow of the proposed stochastic harmonic representation scheme is summarized in Supplementary Figure SR1.

3 Results

3.1 Model truncation in the harmonic representation (clean case)

To separate representational loss from stochastic encoding effects, we first evaluated the truncation error of the harmonic model in the absence of bitstream encoding. A clean reference waveform was synthesized from K t r u e = 12 harmonics with randomly drawn parameters (Supplementary Figure S1). We then reconstructed the signal by retaining only the lowest-frequency components, demonstrating progressive loss of high-frequency details as K decreases from 12 to 9, 6, and 3 while preserving the low-frequency envelope and overall structure (Supplementary Figure S2).

Quantitatively, truncation error decreases monotonically with the number of retained harmonics, providing a baseline “model-only” fidelity curve that is independent of stochastic effects (Supplementary Figure S3). This step is critical because it defines the irreducible approximation error floor for any downstream bitstream-domain processing, even under infinite bitstream length.

3.2 Parameter-level stochastic encoding accuracy

We next assessed the accuracy of stochastic encoding/decoding at the parameter level. Each harmonic parameter was mapped to a calibrated probability and encoded into an independent Bernoulli bitstream of length N, followed by decoding via empirical bitstream statistics.

The results show the expected scaling: parameter recovery error decreases with increasing bitstream length N, while higher SNR improves estimation across all tested conditions (Supplementary Figure S4). A complete Monte Carlo grid summary over K , N , S N R is reported in Supplementary Table S1, enabling reproducible comparison of encoding precision under different latency/complexity budgets.

These findings confirm that stochastic encoding introduces a controlled, quantifiable error component whose magnitude can be reduced by increasing N and by operating under higher S N R, independent of truncation effects.

To provide a more explicit point of reference, we also added a comparison against two baseline representations: a floating-point harmonic reconstruction baseline without Bernoulli encoding, and a stochastic harmonic baseline using direct phase encoding instead of the proposed bounded (cosφ, sinφ) representation. These comparisons separate truncation-limited fidelity from stochastic-encoding effects and clarify the specific contribution of the proposed phase representation.

The corresponding comparative results are provided in Supplementary Figure SR2.

3.3 Reconstruction fidelity from decoded stochastic parameters

We then evaluated end-to-end reconstruction: after encoding and decoding the harmonic parameters, the waveform was reconstructed by summing the estimated harmonics. The resulting reconstruction error combines two contributions: truncation error due to finite K and stochastic encoding error due to finite N and parameter probability quantization.

Reconstruction R M S E decreases as K increases, demonstrating that retaining more harmonics systematically improves signal-level fidelity (Supplementary Figure S5). In the main text we highlight this dependence as the primary performance trade-off of the framework (Figure 1, derived from Supplementary Figure S5). A full grid summary of reconstruction metrics over K , N , S N R is provided in Supplementary Table S2.

Importantly, the separation between the clean truncation baseline (Supplementary Figure S3) and the stochastic reconstruction curve (Supplementary Figure S5) demonstrates that the framework makes it possible to distinguish representational loss (finite K) from encoding-induced uncertainty (finite N) in a transparent way.

In addition, a dedicated phase-wrap ambiguity analysis confirms that the proposed bounded cos ⁡ φ , sin ⁡ φ-based encoding consistently reduces wrapped phase RMSE relative to direct stochastic phase encoding and improves stability in the vicinity of the − π / π boundary.

3.4 Phase-wrap ambiguity analysis

To verify the expected robustness of the proposed phase representation, we added a dedicated phase-wrap ambiguity experiment comparing direct stochastic phase encoding with the proposed bounded representation based on cos ⁡ φ , sin ⁡ φ. The comparison was performed in a wrap-sensitive regime, with particular attention to the vicinity of the − π / π boundary.

The results show that the proposed representation consistently reduces wrapped phase RMSE across the tested bitstream lengths. In addition, the bounded phase-pair representation exhibits improved stability near the angular discontinuity, whereas direct phase encoding is more vulnerable to boundary-related distortion. These findings confirm that representing phase through cos ⁡ φ , sin ⁡ φ is not merely a formal reformulation, but a practically advantageous strategy for stochastic encoding of phase-like parameters, especially in view of the broader importance of stable phase estimation and synthesis in harmonic signal representations (Oliveira et al., 2024).

The phase-wrap ambiguity results are summarized in Figure 2, while additional visual examples and boundary-sensitive illustrations are provided in the revision addendum supplementary package.

3.5 Sensitivity to bitstream correlation (ablation)

A known limitation of stochastic computing is bias induced by inter-stream correlation. To quantify robustness against this effect, we performed a correlation stress test comparing independent PRNG/seeding (decorrelated streams) against a shared-RNG condition.

The results show a clear degradation of reconstruction fidelity under correlated bitstreams, and the gap increases at shorter bitstream lengths where sampling noise and correlation effects compound (Supplementary Figure S6). The complete numerical ablation outcomes are summarized in Supplementary Table S3.

This experiment supports the practical requirement that stochastic parameter streams should be generated with explicit decorrelation policies (e.g., independent seeds per parameter and per harmonic) whenever unbiased downstream processing is required.

3.6 Real-signal case studies

To complement the synthetic validation, we added two real-signal case studies based on a digitized ECG fragment and a digitized earthquake seismogram fragment. For each case, we also analyzed smoother derived representations in the form of envelope- and average-based curves.

The real-signal results confirm that the required harmonic budget depends strongly on the structural complexity of the representation. In the ECG case, the raw signal required a moderate increase in K, while the envelope- and average-based representations were reproduced more compactly, which is consistent with the broader use of ECG signals as a benchmark domain for structured time-frequency and decomposition-based analysis (Pradhan et al., 2023). In the seismogram case, the difference was much more pronounced: the raw seismogram required on the order of hundreds of retained harmonics to achieve the target fidelity, whereas the smoother derived representations required substantially smaller K, supporting the practical relevance of seismogram-like signals as structurally demanding reconstruction targets (Wang et al., 2024b).

These findings indicate that K should not be interpreted as a universally fixed parameter. Instead, it must be selected according to signal class, representation level, and the target fidelity criterion. The required-harmonic-budget summary across the tested real-signal variants is shown in Figure 3, while the seismogram-specific reconstruction trend is presented in Figure 4. Additional representative reconstruction examples for both real-signal cases are provided in the revision addendum supplementary package.

4 Discussion

The proposed framework introduces a reproducible interface that converts a stochastic waveform within a finite window into a structured set of independent Bernoulli bitstreams by factorizing the window into a small number of harmonic components and encoding interpretable parameters. The key conceptual advantage is that processing can occur at the probability/parameter level without requiring direct manipulation of dense continuous-amplitude samples.

A central outcome of the study is the clear separation of error sources. Clean-case truncation experiments (Supplementary Figures S2,S3) define the representational error floor imposed by finite K, while parameter-level and reconstruction-level experiments (Supplementary Figure S4; Figure 1; Supplementary Tables S1,S2) quantify the stochastic encoding uncertainty due to finite bitstream length N and noise conditions. This separation is practically valuable: it enables engineering trade-offs where K controls model fidelity and N controls stochastic precision under latency constraints.

The correlation ablation (Supplementary Figure S6; Supplementary Table S3) highlights a critical implementation requirement: even when marginal probabilities are correct, inter-stream correlations can systematically bias downstream stochastic computations and degrade reconstruction. Therefore, reproducible decorrelation policies (independent seeds across parameters and harmonics) are not an optional detail but a necessary part of any robust harmonic-parameter bitstream representation.

At the statistics level, we demonstrate that the representation can preserve key descriptors of stochastic processes (PSD, ACF, amplitude distributions), not merely pointwise waveform similarity (Figure 5; Supplementary Figure S7). This supports the intended use-case of parameter/statistics-level processing, where the objective is to maintain the statistical content relevant for inference and decision-making.

From a practical perspective, the proposed representation provides a compact and interpretable interface between stochastic waveforms and bitstream-based processors: a window of samples is replaced by a small set of physically meaningful parameters carried by independent Bernoulli streams. This positioning at the representation layer complements recent progress in uncertainty-aware multisensor fusion and recent advances in stochastic-computing primitives and architectures, while remaining distinct from both lines of work (Xue et al., 2021; Roheda et al., 2021; Guti et al., 2022; Li et al., 2024a; Liu et al., 2025; Li et al., 2024b; Liu et al., 2024; Chen et al., 2025; Wang et al., 2024a). This enables direct integration with stochastic computing architectures while preserving control over fidelity through two explicit knobs—K, which governs model truncation error, and N, which governs stochastic encoding noise. As a result, accuracy, latency, and computational cost can be traded off in a transparent and application-driven manner, without entangling representation error with downstream processing logic.

The present validation was intentionally designed around an identity processing block (NOT–NOT) to isolate representational effects from algorithm-specific distortions. While this provides a clean end-to-end verification of the encoding–decoding loop, future work should evaluate non-trivial stochastic operations (e.g., arithmetic or logic primitives and their compositions) to quantify how parameter-level errors and residual inter-stream correlations propagate through deeper stochastic pipelines.

A second limitation is that the harmonic parameterization is evaluated within finite windows under the assumption of local stationarity. Rapidly time-varying processes, non-sinusoidal transients, and strongly nonstationary regimes may require adaptive windowing, overlap strategies, or hybrid dictionaries (e.g., harmonic + transient atoms), as well as improved peak-selection criteria beyond top-K magnitude peaks. Relatedly, the current mapping from parameters to probabilities uses calibrated ranges and clipping; extending this step toward formally optimized or data-driven calibration (with explicit guarantees on bias and variance) is an important direction.

The revised study substantially extends the original validation by adding broader signal and noise conditions, explicit comparison against baseline representations, a dedicated phase-wrap ambiguity analysis, and two real-signal case studies. Nevertheless, further work is still needed to explore more diverse real measurement domains, alternative decomposition families under matched fidelity criteria, and implementation-oriented computational optimization under platform-specific constraints.

Limitations of the current study include the finite-harmonic model assumption and dependence on windowing and peak selection in the frequency domain. Highly nonstationary signals or signals with significant broadband components may require either larger K, adaptive window strategies, or alternative basis expansions. Nevertheless, the presented validation protocol and reproducible code provide a transparent benchmark for extending the method and comparing it against baseline representations under equal fidelity constraints.

Generative AI statement

The author(s) declared that generative AI was used in the creation of this manuscript. Generative AI was used to support the preparation of this manuscript by improving English language clarity and style, assisting with wording and structure, and generating and refining Python code used for simulations and figure/table generation. All scientific content, methodological decisions, interpretations, and final edits were made by the authors, who reviewed and verified the AI-assisted text and code and take full responsibility for the manuscript.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frsip.2026.1816476/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Johan De Villiers, University of Pretoria, South Africa

Reviewed by: Megha S. Kumar, Amrita Vishwa Vidyapeetham University, India

Long Chen, Northwestern Polytechnical University, China