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Physically-Constrained Harmonic Separation for Robust Heart and Respiratory Rate Estimation from Wrist Photoplethysmography
Nouhaila Fraihi, Ouassim Karrakchou, Mounir Ghogho
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 98%
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Summary
The paper proposes the Physically-Constrained Harmonic Separation (PCHS) framework, a deep learning approach for robust heart rate (HR) and respiratory rate (RR) estimation from wrist-worn photoplethysmography (PPG) during physical activity. PCHS treats signal decomposition as an analysis-by-synthesis problem, using a cross-modal encoder that incorporates accelerometer data via Feature-wise Linear Modulation (FiLM) to condition the PPG representation. A physics-guided harmonic decoder then synthesizes the physiological signal using time-varying parameters (fundamental frequency, harmonic amplitudes, gain, and baseline) while capturing motion artifacts in a residual branch. The framework utilizes a multi-objective loss function including reconstruction fidelity, physiological-motion separation (via push-pull constraints), and parameter smoothness. The method was validated on the PPG-DaLiA dataset, demonstrating superior performance in disentangling physiological dynamics from non-stationary motion artifacts compared to existing methods.
Entities (9)
Relation Signals (5)
Empatica E4 → collects → Photoplethysmography
confidence 100% · Wrist data are collected using an Empatica E4 on the non-dominant wrist, providing PPG sampled at 64 Hz
Physically-Constrained Harmonic Separation → evaluatedon → PPG-DaLiA
confidence 100% · Experiments on the motion-intensive PPG-DaLiA dataset demonstrate that PCHS outperforms state-of-the-art methods
Physically-Constrained Harmonic Separation → uses → Feature-wise Linear Modulation
confidence 100% · enabling motion-informed artifact attribution via Feature-wise Linear Modulation (FiLM)
Accelerometer → conditions → Physically-Constrained Harmonic Separation
confidence 90% · where accelerometer measurements condition artifact separation
RespiBAN Professional → providesgroundtruthfor → Heart Rate
confidence 90% · Reference signals are acquired using a chest-worn RespiBAN Professional device, including ECG and respiratory inductive plethysmography
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Abstract
Abstract:Wrist-worn photoplethysmography (PPG) enables continuous monitoring of cardiopulmonary physiology, but reliable heart rate (HR) and respiratory rate (RR) estimation in free-living conditions remains challenging due to non-stationary motion artifacts that spectrally overlap with physiological dynamics. Existing signal-processing methods degrade under strong motion, while unconstrained deep learning approaches often lack physiological interpretability and identifiable structure. We propose a Physically-Constrained Harmonic Separation (PCHS) framework that formulates HR and RR estimation from wrist PPG as an analysis-by-synthesis problem, where accelerometer measurements condition artifact separation rather than directly regressing vital signs. A physics-guided harmonic generator decomposes the observed signal into quasi-periodic physiological components and a motion-related residual, enabling HR recovery from the fundamental frequency and RR prediction from respiratory-driven modulations of the harmonic parameters. Robust reconstruction objectives, separation constraints, and uncertainty-aware weighting stabilize the decomposition under motion. Experiments on the motion-intensive PPG-DaLiA dataset demonstrate that PCHS outperforms state-of-the-art methods while yielding interpretable signal decompositions that effectively disentangle physiological activity from motion artifacts.
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- Source: https://arxiv.org/abs/2606.30156v1
- Canonical: https://arxiv.org/abs/2606.30156v1
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Physically-Constrained Harmonic Separation for Robust Heart and Respiratory Rate Estimation from Wrist Photoplethysmography †thanks: This research was partially funded by Mohammed VI Polytechnic University through the i-Respire research project. †thanks: © 2026 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Nouhaila Fraihi1, Ouassim Karrakchou1, and Mounir Ghogho2,3 Abstract Wrist-worn photoplethysmography (PPG) enables continuous monitoring of cardiopulmonary physiology, but reliable heart rate (HR) and respiratory rate (R) estimation in free-living conditions remains challenging due to non-stationary motion artifacts that spectrally overlap with physiological dynamics. Existing signal-processing methods degrade under strong motion, while unconstrained deep learning approaches often lack physiological interpretability and identifiable structure. We propose a Physically-Constrained Harmonic Separation (PCHS) framework that formulates HR and R estimation from wrist PPG as an analysis-by-synthesis problem, where accelerometer measurements condition artifact separation rather than directly regressing vital signs. A physics-guided harmonic generator decomposes the observed signal into quasi-periodic physiological components and a motion-related residual, enabling HR recovery from the fundamental frequency and R prediction from respiratory-driven modulations of the harmonic parameters. Robust reconstruction objectives, separation constraints, and uncertainty-aware weighting stabilize the decomposition under motion. Experiments on the motion-intensive PPG-DaLiA dataset demonstrate that PCHS outperforms state-of-the-art methods while yielding interpretable signal decompositions that effectively disentangle physiological activity from motion artifacts. I INTRODUCTION Photoplethysmography (PPG), widely used in wrist-worn wearables, enables continuous monitoring of cardiovascular and respiratory activity; however, reliable heart-rate (HR) and respiratory-rate (R) estimation in free-living conditions remains challenging due to motion artifacts that obscure physiological information [3, 2]. The wrist PPG signal consists of a pulsatile cardiac component, slow respiratory-driven baseline variations, and coupled amplitude and frequency modulations [13]. In free-living conditions, motion artifacts are typically additive, highly non-stationary, and spectrally overlapping with both cardiac (≈0.5≈ 0.5–44 Hz) and respiratory (<0.5<0.5 Hz) dynamics [12], rendering fixed filtering and simple noise suppression ineffective. To mitigate motion corruption, prior work has relied on signal-processing pipelines that use inertial measurements as motion references for artifact suppression, followed by spectral rate estimation. Representative methods employ accelerometer-guided sparse spectral reconstruction or adaptive time–frequency filtering to attenuate motion-related components before extracting HR from dominant spectral peaks [4, 5, 7]. Respiratory rate (R) estimation typically exploits respiration-induced amplitude, baseline, or frequency modulations of the PPG waveform, sometimes combined with inertial sensing to suppress motion-correlated energy in the respiratory band [14, 15]. While effective under controlled conditions, these approaches assume separability between motion and physiological dynamics, an assumption that breaks down in free-living settings due to non-stationary and spectrally overlapping motion artifacts. More recently, deep learning (DL) approaches have been proposed to jointly perform artifact suppression and vital-sign estimation from PPG within unified frameworks. Some DL methods primarily operate on raw PPG signals, formulating HR and R estimation as end-to-end regression tasks [9, 18, 17]. Building on these PPG-only formulations, subsequent DL frameworks incorporate inertial measurements to explicitly account for motion effects. These approaches fuse PPG and acceleration signals in a supervised manner to improve robustness under movement [1, 8, 16]. Despite recent accuracy gains, most deep learning–based approaches still formulate vital-sign estimation from PPG as a black-box regression problem. Even when incorporating accelerometer signals through supervised fusion, such models offer limited physiological interpretability and may rely on spurious correlations between motion patterns (e.g., activity type, movement intensity, or periodic arm swing) and HR or R. This does not guarantee recovery of underlying physiological dynamics and can promote shortcut learning from motion cues [6]. These limitations have motivated knowledge-informed learning approaches that integrate physiological constraints to improve robustness and interpretability. Recent methods, such as KID-PPG, incorporate explicit motion-aware processing and probabilistic inference to enhance heart-rate estimation under motion [10]. Such formulations typically adopt simplified coupling assumptions between inertial signals and photoplethysmography. While effective in many scenarios, ensuring fully identifiable and physiologically disentangled representations remains challenging in the presence of non-stationary and spectrally overlapping motion artifacts. These limitations motivate approaches that treat motion as an explanatory source of corruption rather than a predictive shortcut, and that enforce identifiable, physiology-driven modeling. Motivated by this perspective, we propose a Physically-Constrained Harmonic Separation (PCHS) framework for robust HR and R estimation from wrist PPG under motion. PCHS formulates denoising as an analysis-by-synthesis problem in which a deep learning model, conditioned on accelerometer data, predicts time-varying physiological parameters, such as the cardiac fundamental frequency and harmonic amplitudes, that synthesize a pulse-synchronous PPG component, while remaining content, including motion artifacts and unmodeled distortions, is captured by an additive residual and a time-varying reliability estimate. This formulation yields interpretable cardiac and respiratory representations. The main contributions of this work are: • Cross-modal motion-conditioning encoder: We introduce a motion-aware dual-stream encoder in which accelerometer measurements condition PPG representations via feature-wise linear modulation (FiLM), enabling motion-informed artifact attribution without directly regressing vital signs. • Physically constrained analysis-by-synthesis with robust objectives: We formulate PPG decomposition as an analysis-by-synthesis problem that predicts time-varying physiological parameters and synthesizes a pulse-synchronous PPG component using a harmonic model. Robust reconstruction and separation objectives stabilize the decomposition under motion while isolating non-physiological content in an additive residual. • Uncertainty-aware vital-sign inference: We estimate HR from the recovered cardiac fundamental frequency and R from physiologically meaningful low-frequency modulations, using a learned time-varying reliability signal to downweight motion-corrupted intervals during inference. The remainder of this paper is organized as follows. Section I presents the proposed framework, detailing the model architecture, physiological decomposition, and training objective. Section I describes the experimental protocol, including the dataset, ablation studies, and comparative results. Finally, Section IV concludes the paper and outlines directions for future work. I Methods I-A Overview and Design Principles We propose a Physically-Constrained Harmonic Separation (PCHS) framework for robust HR and R estimation from wrist-worn photoplethysmography (PPG) under motion. As illustrated in Fig. 1, PCHS comprises three components: (i) a cross-modal motion-conditioning encoder, in which a compact PPG encoder extracts multiscale, morphology-aware temporal features while a lightweight accelerometer (ACC) encoder captures motion; ACC features modulate PPG representations via feature-wise linear modulation (FiLM) [11] at multiple stages to enable motion-adaptive feature extraction; (i) a physically constrained harmonic decoder, which maps motion-conditioned features to time-varying physiological parameters, including cardiac fundamental frequency, harmonic amplitudes, gain, and baseline, synthesizing a pulse-synchronous physiological waveform via a harmonic model, while a residual branch captures non-harmonic, motion-consistent artifacts and an uncertainty head predicts local signal reliability; and (i) an outputs and inference module, which performs reliability-aware vital-sign estimation, inferring HR from the recovered fundamental frequency and predicting R from physiologically meaningful parameter modulations, with uncertainty used to downweight motion-corrupted intervals. Overall, PCHS formulates signal decomposition and vital-sign estimation as an analysis-by-synthesis problem, in which physiological structure is enforced by construction through a harmonic generative model, improving identifiability, interpretability, and robustness in free-living conditions. Figure 1: Overview of PCHS. A cross-modal encoder uses FiLM for motion adaptation, while a physics-constrained decoder synthesizes physiological parameters and residuals for reliability-aware inference. I-B Problem Formulation Let x(t)∈ℝx(t) denote a wrist-worn PPG measurement at time t/fst/f_s, with fsf_s being the sampling frequency, and let (t)∈ℝ3u(t) ^3 denote the synchronized tri-axial ACC signal. To obtain a rotation-invariant motion representation, we compute the accelerometer magnitude u(t)=‖(t)‖2.u(t)=\|u(t)\|_2. (1) Following prior wrist-PPG formulations that model motion artifacts as non-stationary additive components [15, 10], we express the observed signal as x(t)=μphys(t)+r(t)+ϵ(t),t=1,⋯,Tx(t)= _phys(t)+r(t)+ε(t), t=1,·s,T (2) where μphys(t) _phys(t) denotes the clean, quasi-periodic physiological component dominated by cardiac and respiratory activity, r(t)r(t) represents motion-induced artifacts, ϵ(t)ε(t) accounts for sensor noise and unmodeled disturbances, and T is the number of samples in the observation window. Given paired inputs x(t),u(t)\x(t),u(t)\, the objective is to decompose the observation into physiologically interpretable and motion-related components while robustly estimating HR and R. To this end, we adopt an encoder–decoder architecture in which a motion-conditioned encoder extracts separation-aware representations and a physically constrained decoder reconstructs a structured physiological signal decomposition from these representations. I-C Cross-Modal Conditioning Encoder The encoder extracts separation-relevant representations from wrist PPG by jointly processing optical and inertial signals in a dual-stream architecture. This cross-modal conditioning enables adaptive feature modulation across temporal scales and reduces spurious coupling between motion and physiological dynamics. I-C1 PPG Encoder The PPG stream is processed by a lightweight residual network tailored for 1D biosignals. The design prioritizes parameter efficiency while preserving morphological detail critical for respiratory analysis. It consists of a stem layer (kernel size 7, stride 2) followed by three residual stages, each comprising two Basic Residual Blocks with 1D convolutions, Batch Normalization, and SiLU activations. Dilated convolutions (d∈1,2,3,4d∈\1,2,3,4\) are employed to capture multiscale temporal features without increasing model complexity, preserving the temporal resolution required to model subtle respiratory modulations. I-C2 Motion-Aware Conditioning via FiLM The accelerometer signal u(t)u(t) is processed by a lightweight 1D CNN comprising three convolutional blocks with kernel sizes k∈7,5,3k∈\7,5,3\ and strided pooling, progressively increasing the channel dimension to Cacc=64C_acc=64. Global average pooling produces multi-scale motion embeddings early,mid,late\z_early,z_mid,z_late\, which condition the corresponding stages of the PPG encoder via Feature-wise Linear Modulation (FiLM) [11]. Given a PPG feature map H and a motion embedding vector z, FiLM applies an affine modulation: FiLM(∣)=()⊙+(),FiLM(H )= γ(z) + β(z), (3) where () γ(z) and () β(z) are channel-wise scaling and shifting vectors predicted by learned linear projection heads. This conditioning enables motion-adaptive feature modulation without explicitly subtracting motion-related waveforms. The resulting motion-conditioned representations serve as inputs to a physically constrained decoder that reconstructs a structured physiological signal through harmonic synthesis. I-D Physically-Constrained Harmonic Decoder We employ a physics-informed harmonic decoder inspired by Differentiable Digital Signal Processing [22] and classical models of PPG signal composition [12]. The decoder is built around an explicit generative model of the physiological PPG signal, enforcing quasi-periodic structure by construction and enabling identifiable signal decomposition. I-D1 Harmonic synthesis The clean physiological component is modeled as a harmonic signal with time-varying parameters. Given an instantaneous fundamental frequency trajectory f0(t)f_0(t), the phase is computed as θ(t)=2π∑τ=0tf0(τ)fs.θ(t)=2π _τ=0^t f_0(τ)f_s. (4) The physiological signal is synthesized as μphys(t)=(1+g(t))∑k=1KAk(t)ksin(kθ(t))+b(t), _phys(t)=(1+g(t)) _k=1^K A_k(t)k (kθ(t))+b(t), (5) where Ak(t)A_k(t) are the harmonic amplitudes controlling pulse morphology, the gain g(t)g(t) models slow amplitude modulation, and the baseline b(t)b(t) captures low-frequency variations due to respiration and sensor drift. I-D2 Multi-head parameter prediction Motion-conditioned features from the encoder are processed by a shared refinement block consisting of two 1D convolutional layers (C=256C=256, kernel size k=3k=3) with Batch Normalization and SiLU activation. The refined representation is then routed to multiple parallel heads that predict the parameters of the harmonic synthesis model, along with a residual artifact term and a time-varying uncertainty estimate. All convolutional layers use stride 11 with appropriate padding, ensuring that all predicted trajectories preserve the temporal resolution T of the input PPG signal. Fundamental frequency The cardiac fundamental frequency f0(t)∈[0.5,3.0]f_0(t)∈[0.5,3.0] Hz is predicted using a pointwise 1D convolution and bounded with a sigmoid activation. This constraint stabilizes phase integration and improves identifiability. Harmonic amplitudes Time-varying harmonic amplitudes Ak(t)∈ℝ+A_k(t) ^+ for k=1,2,3k=1,2,3 are predicted using a pointwise 1D convolution. Using three harmonics captures the dominant pulse morphology while limiting overfitting to high-frequency noise. Global gain and baseline The global gain g(t)∈[0,1]g(t)∈[0,1] and baseline term b(t)∈ℝb(t) are predicted using pointwise 1D convolutional heads. The gain models slow pulse amplitude variations due to contact changes and hemodynamic effects, while the baseline captures low-frequency respiratory effects and sensor drift. A fixed moving-average filter is applied to the baseline to enforce spectral separation from cardiac content. Residual artifact To capture signal components not explained by the harmonic model, a residual artifact r(t)r(t) is predicted using a three-layer convolutional head with channel dimensions 64,32,1\64,32,1\ and kernel size k=3k=3. This branch models non-harmonic, motion-induced distortions. Uncertainty The predicted uncertainty reflects local signal unreliability due to irrecoverable motion. A time-varying uncertainty estimate σ2(t)σ^2(t) is predicted as the log-variance logσ2(t) σ^2(t) by a two-layer convolutional head. Higher variance indicates segments corrupted by unrecoverable motion and is used to downweight unreliable intervals during reconstruction and inference. The final reconstructed signal is given by s(t)=μphys(t)+r(t),s(t)= _phys(t)+r(t), (6) where μphys(t) _phys(t) and r(t)r(t) now denote the predicted physiological component, obtained from the harmonic synthesis in Eq. (5), and the residual produced by the decoder, respectively. The decoded harmonic parameters are physiologically interpretable: the recovered fundamental frequency directly enables HR estimation, while slower temporal modulations of the amplitudes and frequency encode respiratory dynamics. HR and R estimation are learned jointly through a multi-objective training formulation that balances reconstruction fidelity, physiological separation, and rate estimation accuracy. I-E Multi-Objective Optimization Training employs a composite objective that enforces reconstruction fidelity, identifiable physiological–motion separation, temporal smoothness of decoded parameters, and accurate vital-sign estimation. Reconstruction and spectral consistency Waveform morphology and frequency content are constrained using a negative Pearson correlation loss [23] and a multi-resolution spectral loss [24]: ℒρ _ρ =1−Cov(x,s)σxσs, =1- Cov(x,s) _x _s, (7) ℒspec _spec =m[‖log|STFTm(x)|−log|STFTm(s)|‖1], =E_m\! [ \| \! |STFT_m(x) |- \! |STFT_m(s) | \|_1 ], (8) where x∈ℝTx ^T is the observed PPG signal, s∈ℝTs ^T is its reconstruction, σx _x and σs _s denote the corresponding standard deviations, and STFTm(⋅)STFT_m(·) is computed with window sizes m∈32,64,128,256,512m∈\32,64,128,256,512\. The reconstruction loss is ℒrec=ℒρ+ℒspec.L_rec=L_ρ+L_spec. (9) Push–pull separation constraints Physiological–motion disentanglement is enforced via ℒsep=ℒpull+λorthℒorth+ℒnoise.L_sep=L_pull+ _orth\,L_orth+L_noise. (10) Physiology pull To prevent collapse of the physiological component, we enforce sufficient energy in the cardiac band: ℒpull=max(0,α−Pband(μphys)),L_pull= \! (0,α-P_band( _phys) ), (11) where μphys∈ℝT _phys ^T is denotes the reconstructed physiological component (pulse-synchronous cardiac signal), Pband(⋅)P_band(·) denotes the proportion of power spectral density within [fHRmin,fHRmax][f_HR ,f_HR ], and α is a fixed minimum energy threshold. Orthogonality constraint Statistical independence between the physiological signal and the residual is encouraged via cosine similarity. For a mini-batch of size B, ℒorth=1B∑i=1B|phys(i)⋅(i)|‖phys(i)‖2‖(i)‖2.L_orth= 1B _i=1^B | μ_phys^(i)·r^(i) |\| μ_phys^(i)\|_2\,\|r^(i)\|_2. (12) where (i)∈ℝTr^(i) ^T is the residual signal for the i-th sample. Orthogonality is enforced as a soft regularization; separation can degrade under non-stationary motion that spectrally overlaps with cardiac dynamics; the complementary spectral and uncertainty-aware terms mitigate the resulting residual leakage. Residual noise constraint To discourage periodic structure in the residual, cardiac-band energy is penalized: ℒnoise=Pband().L_noise=P_band(r). (13) Here, Pband()P_band(r) denotes the normalized proportion of spectral power of the residual signal r contained within the cardiac frequency band [fHRmin,fHRmax][f_HR ,f_HR ], computed relative to its total spectral energy. Parameter smoothing Temporal smoothness of decoded physiological parameters is imposed using total-variation regularization: ℒsm=ℒdiff(f0)+ℒdiff(g)+∑k=1Kℒdiff(Ak),L_sm=L_diff(f_0)+L_diff(g)+ _k=1^KL_diff(A_k), (14) with ℒdiff(y)=1T−1∑t=1T−1(yt+1−yt)2,L_diff(y)= 1T-1 _t=1^T-1(y_t+1-y_t)^2, (15) where f0(t)f_0(t), g(t)g(t), and Ak(t)A_k(t) denote the predicted fundamental frequency, gain, and harmonic amplitudes. Heart- and respiratory-rate estimation Heart rate is obtained from the recovered fundamental frequency via reliability-weighted temporal averaging, HR^=60⋅∑tw(t)f0(t)∑tw(t),w(t)=exp(−logσ2(t)), HR=60· _tw(t)\,f_0(t) _tw(t), w(t)= \! (- σ^2(t) ), (16) where σ2(t)σ^2(t) denotes the time-varying uncertainty predicted by the decoder. Respiratory rate is predicted from a fused representation of decoded physiological modulations by concatenating the estimated gain (amplitude modulation), baseline variation, fundamental frequency, and harmonic amplitude modulations into a feature vector, which is mapped to a scalar estimate R R using a lightweight 1D convolutional sub-network and bounded to [RRmin,RRmax][R_ ,R_ ] via a sigmoid. Direct ℓ1 _1 supervision is applied to both vital signs: ℒhr=|HR^−yHR|,ℒr=|R^−yRR|,L_hr= | HR-y_HR |, _r= | R-y_R |, (17) where yHRy_HR and yRRy_R denote the ground-truth heart and respiratory rates, respectively. Loss weighting The total objective combines all losses using homoscedastic uncertainty weighting [25]: ℒtotal=∑ℓ∈rec,sep,sm,hr,r12τℓ2ℒℓ+logτℓ2,L_total= _ ∈\rec,\,sep,\,sm,\,hr,\,r\ 12\, _ ^2\,L_ + _ ^2, (18) where τℓ2∈ℝ+ _ ^2 ^+ are learnable scalars that balance the reconstruction, separation, smoothness, HR, and R losses. I Experiments I-A Data Acquisition PPG-DaLiA: We evaluate our method on the PPG-DaLiA dataset [1], which is designed to benchmark robust vital-sign estimation from wrist-worn sensors under motion-intensive, free-living conditions. The dataset comprises approximately 36h36\,h of synchronized recordings from 15 subjects performing eight daily-life activities, including sitting, walking, cycling, stair ascent/descent, driving, and working, with labeled transition periods. Wrist data are collected using an Empatica E4 on the non-dominant wrist, providing PPG sampled at 64Hz64\,Hz and tri-axial accelerometer data sampled at 32Hz32\,Hz. Reference signals are acquired using a chest-worn RespiBAN Professional device, including ECG and respiratory inductive plethysmography sampled at 700Hz700\,Hz. Ground-truth heart rate is derived from the chest-worn ECG signal, while respiratory rate ground truth is obtained from the chest-worn respiration waveform using peak detection on the clean excursion signal, following standard procedures [16, 28]. I-B Experimental Setup Preprocessing All signals are segmented into 8 s windows with a 2 s stride, following established protocols for wrist PPG analysis [10, 8]. PPG and accelerometer signals are temporally aligned and resampled to a common sampling rate of 64 Hz. PPG windows are normalized to zero mean and unit variance, while accelerometer signals are kept in their original scale and used solely for motion conditioning. Evaluation protocol We adopt a leave-one-subject-out (LOSO) evaluation protocol, consistent with prior studies on PPG-DaLiA [1, 10]. In each fold, data from one subject are held out for testing while the remaining subjects are used for training. Performance is reported as the mean across all subjects using mean absolute error (MAE). For R estimation, the proposed model produces an R prediction for each 8 s window. To enable fair comparison with prior methods that rely on longer temporal contexts (e.g., 32 s windows [16]), we report results for both the raw 8 s predictions, reflecting low-latency estimation, and aggregated 32 s estimates obtained by averaging four consecutive windows. Training details All models are implemented in PyTorch and trained end-to-end using the AdamW optimizer with a learning rate of 6×10−46× 10^-4 and weight decay of 10−410^-4. Training is conducted for 60 epochs with a batch size of 64, using cosine annealing for learning-rate scheduling. The proposed PCHS model contains approximately 1.97M trainable parameters, comparable to recent PPG-based deep learning approaches. I-C Experimental Results To better understand the contribution of each design component, we conduct ablation studies to isolate the effect of the key architectural and modeling choices in the proposed framework. I-C1 Ablation Study Table I quantifies the contribution of the key components of the proposed Physically-Constrained Harmonic Separation (PCHS) framework through targeted ablations. We evaluate the impact of (i) motion-aware conditioning via FiLM, (i) explicit residual modeling with separation regularization, (i) physics-guided harmonic synthesis, and (iv) uncertainty-aware inference. Motion-aware conditioning Removing FiLM-based conditioning prevents accelerometer-derived motion context from modulating physiological features, leading to a clear degradation in performance (HR MAE increases from 3.20 to 3.75 bpm; −17.2%-17.2\%). This confirms that explicit motion conditioning enables adaptive artifact suppression at the feature level, beyond what implicit learning alone can achieve. Residual modeling and separation In the full model, the signal is decomposed as x(t)=μphys(t)+r(t)x(t)= _phys(t)+r(t), with a separation loss enforcing orthogonality and suppressing cardiac-band energy in the residual. Removing the residual pathway yields the largest performance drop (HR MAE 4.10 bpm; −28.1%-28.1\%), indicating that forcing the harmonic model to absorb broadband noise compromises identifiability. Disabling only the separation constraints also degrades accuracy (HR MAE 3.70 bpm), highlighting the necessity of explicit push–pull regularization to prevent physiological leakage. Physics-guided harmonic synthesis Replacing the harmonic analysis-by-synthesis generator with direct waveform regression significantly reduces accuracy (HR MAE 3.92 bpm; −22.5%-22.5\%). This demonstrates that enforcing quasi-periodicity and smooth amplitude modulation through a harmonic prior is critical for stable frequency estimation in low-SNR, motion-corrupted conditions. Uncertainty-aware inference Disabling reliability-guided temporal aggregation results in a consistent performance drop (HR MAE 3.55 bpm; −10.9%-10.9\%). By down-weighting segments with high predicted uncertainty, the model mitigates transient corruption that cannot be resolved by structural constraints alone. Overall, the ablation results show that PCHS derives its robustness from the complementary interaction of motion conditioning, physics-guided decomposition, explicit residual modeling, and uncertainty-aware inference, rather than from any single component in isolation. TABLE I: Ablation study on PPG-DaLiA. Accuracy change denotes relative performance drop compared to the full PCHS model. Variant HR MAE ↓ (bpm) Accuracy Drop (%) Ours (Full PCHS) 3.20 – w/o FiLM conditioning 3.75 −17.2-17.2 w/o Residual pathway 4.10 −28.1-28.1 w/o Separation loss 3.70 −15.6-15.6 w/o Harmonic synthesis 3.92 −22.5-22.5 w/o Reliability weighting 3.55 −10.9-10.9 Adaptive Loss Balancing Analysis As part of the ablation analysis, we examine the effect of adaptive loss balancing using homoscedastic uncertainty weighting [25], as defined in eq.18 Sec. I, which allows the model to dynamically prioritize tasks based on their estimated uncertainty. Table I illustrates the evolution of the learned loss weights during training. Once the physics-constrained decoder stabilizes, a clear reallocation of task importance emerges. In particular, the Respiratory Rate objective receives the highest final weight (λ≈1.32λ≈ 1.32), indicating reduced aleatoric uncertainty when respiratory features are extracted from the recovered physiological signal μphys _phys. In contrast, the separation loss (ℒsepL_sep remains relatively low-weighted (λ≈0.20λ≈ 0.20), reflecting its role as a regularizer rather than a primary optimization target. This adaptive weighting enables the model to prioritize clinically relevant rate estimation without destabilizing the learning of harmonic representations. TABLE I: Evolution of learned loss weights (λ) during training. Bold indicates the highest final weight. Loss Term Epoch 1 Epoch 10 Epoch 20 Ep 30 Reconstruction (ℒrecL_rec) 0.10 0.22 0.35 0.41 Separation (ℒsepL_sep) 0.10 0.15 0.18 0.20 Smoothness (ℒsmL_sm) 0.10 0.25 0.38 0.45 Heart Rate (ℒhrL_hr) 0.10 0.55 0.82 0.95 Respiratory Rate (ℒrrL_r) 0.10 0.60 1.15 1.32 I-C2 Comparison with the state-of-the-art Heart Rate estimation Table I shows that the proposed PCHS framework achieves the lowest average HR mean absolute error on the PPG-DaLiA dataset (3.20±1.213.20± 1.21 bpm), outperforming state-of-the-art signal-processing and deep learning approaches across all 15 subjects. Performance gains are consistent under both low- and high-motion conditions, with particularly strong improvements for subjects affected by severe motion (e.g., S5, S8, and S9). In comparison, existing signal-processing and learning-based methods exhibit reduced robustness under strong motion, reflecting the challenges posed by non-stationary and spectrally overlapping artifacts in free-living conditions. While recent deep learning and knowledge-informed approaches improve overall accuracy, variability remains for heavily corrupted recordings. By treating motion as a conditioning signal rather than a predictive shortcut, PCHS enables motion-aware physiological inference without suppressing pulse-synchronous cardiac content. This physiology-driven analysis-by-synthesis formulation isolates cardiac structure from motion-related corruption, yielding interpretable and identifiable representations that support robust heart-rate estimation. TABLE I: Per-subject HR MAE (bpm) on PPG-DaLiA. Best results are highlighted in bold. Method S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 Mean Signal Processing SpaMaPlus [1] (PPG,ACC) 8.86 9.67 6.40 14.10 24.06 11.34 6.31 11.25 16.04 6.17 15.15 12.03 8.50 7.76 8.29 11.06 TAPIR [19] (PPG) 4.50 4.50 3.20 6.00 5.00 3.40 2.80 6.30 8.00 2.90 5.10 4.70 3.10 5.00 4.10 4.57 CurToSS [7] (PPG) 5.40 4.30 3.00 8.00 2.20 2.80 3.30 8.50 12.60 3.60 3.60 6.10 3.00 5.50 3.70 5.04 Deep Learning DeepPPG [1](PPG,ACC) 7.73 6.74 4.03 5.90 18.51 12.88 3.91 10.87 8.79 4.03 9.22 9.35 4.29 4.37 4.17 7.65 TEMPONet [20](PPG,ACC) 4.37 3.74 2.43 5.49 13.48 5.71 2.23 7.86 8.94 3.32 5.34 7.71 2.03 2.94 3.58 5.27 AugmentPPG [20](PPG,ACC) 4.37 3.74 2.43 5.49 9.41 3.63 2.23 7.86 8.94 3.32 5.34 7.64 2.03 2.94 3.58 4.86 PULSE [21](PPG,ACC) 3.78 3.04 2.20 4.41 6.95 3.71 2.39 8.17 6.19 2.60 3.85 5.22 1.98 3.13 2.79 4.03 KID-PPG [10](PPG,ACC) 4.27 3.46 2.07 5.61 3.01 2.74 1.39 7.13 9.53 2.77 3.58 4.52 1.48 2.48 2.84 3.79 EnhancePPG [8](PPG,ACC) 3.35 3.15 2.20 4.38 5.64 2.35 1.93 5.16 6.38 2.87 3.30 5.49 1.84 2.43 2.53 3.54 PCHS(OURS) (PPG,ACC) 3.52 2.92 2.40 5.41 3.23 3.52 1.75 5.07 5.65 2.07 3.13 3.60 1.24 2.20 2.27 3.20 Respiratory-rate estimation on PPG-DaLiA Table IV compares the proposed PCHS model with recent state-of-the-art methods for R estimation on PPG-DaLiA. Because R has a low-frequency content (periods of roughly 2–5 s), reliable estimation favors longer temporal context spanning multiple respiratory cycles, which motivates evaluation at two operating points. At Raw 8 s (low-latency inference), PCHS achieves an MAE of 3.54 bpm, competitive with deep learning baselines such as RRWaveNet [17] (3.41 bpm) and substantially better than hybrid heuristic approaches like A–P Synthesis [26] (8.12 bpm). At Agg. 32 s, matching the temporal resolution of prior work, PCHS reaches 2.15 bpm, outperforming the transfer-learning-based approach of [16] (2.29 bpm) and the MultiScale CNN of [27] (2.70 bpm). TABLE IV: Comparison of Average R MAE (bpm) on the PPG-DaLiA dataset. Best results are highlighted in bold. Method Input Avg MAE CNN [18] PPG 3.16 A-P Synthesis [26] PPG, ACC 8.12 MultiScale CNN [27] PPG, ACC, Gyr 2.70 RRWaveNet [17] PPG 3.41 Transfer Learning [16] PPG, ACC 2.29 PCHS(OURS) (Raw 8 s) PPG, ACC 3.54 PCHS(OURS) (Agg. 32 s) PPG, ACC 2.15 I-C3 Qualitative interpretability Beyond quantitative HR/R accuracy, PCHS provides interpretable intermediate variables for qualitative inspection of the learned decomposition. The decoder explicitly separates the physiologically structured component μphys(t) _phys(t), synthesized from the predicted parameters, from the additive residual r(t)r(t) capturing motion-related distortions. Figure 2 shows a representative example from subject S05, including the raw PPG signal, reconstructed physiological component, and harmonic evolution. The reconstructed harmonic structure remains smooth and exhibits a stable amplitude hierarchy, indicating that cardiac morphology is explained through the physics-constrained analysis-by-synthesis pathway. Periods of elevated accelerometer activity coincide with increased residual energy and higher predicted uncertainty, consistent with the intended artifact modeling and reliability-aware inference. In contrast, the estimated fundamental frequency remains stable and physiologically plausible throughout the window, demonstrating robust cardiac frequency tracking under motion. Together, these observations support the interpretability and physiological validity of the proposed PCHS framework. Figure 2: Qualitative visualization of the PCHS decomposition for a representative window from subject S05, showing the raw PPG signal, reconstructed physiological component μphys(t) _phys(t), fundamental frequency, harmonic amplitudes Ak(t)\A_k(t)\, and motion-related indicators. IV Conclusion This work presented a Physically-Constrained Harmonic Separation framework for robust heart-rate and respiratory-rate estimation from wrist photoplethysmography under motion. By casting vital-sign estimation as an analysis-by-synthesis problem, the proposed approach explicitly decomposes the observed PPG into an interpretable physiological component and a motion-driven residual, treating accelerometer measurements as explanatory cues for corruption rather than predictive shortcuts. A physics-guided harmonic pulse model enables direct recovery of the cardiac fundamental frequency, while slow envelope and baseline modulations capture respiratory dynamics. A time-varying reliability signal further improves robustness by downweighting corrupted intervals during inference. Experiments on the PPG-DaLiA dataset demonstrate improved performance under motion-heavy conditions and yield interpretable decompositions that disentangle pulse structure from motion-induced corruption. Future work will extend the framework to additional datasets and downstream tasks, and investigate efficient deployment strategies for resource-constrained wearable devices. Validation on arrhythmic populations, where the quasi-periodicity assumption may not hold, is a particularly important open direction. More broadly, this study highlights the value of combining physiological structure with learning-based models to achieve robust and interpretable vital-sign estimation in unconstrained wearable sensing scenarios. References [1] A. Reiss et al., “Deep PPG: Large-scale heart rate estimation with convolutional neural networks,” Sensors, vol. 19, no. 14, p. 3079, 2019. [2] K. B. Kim et al., “Photoplethysmography in wearable devices: a comprehensive review of technological advances, current challenges, and future directions,” Electronics, vol. 12, no. 13, p. 2923, 2023. [3] P. H. Charlton et al., “Wearable photoplethysmography for cardiovascular monitoring,” Proc. IEEE, vol. 110, no. 3, p. 355–381, 2022. [4] Z. Zhang et al., “TROIKA: A general framework for heart rate monitoring using wrist-type photoplethysmographic signals during intensive physical exercise,” IEEE Trans. Biomed. Eng., vol. 62, no. 2, p. 522–531, 2015. [5] R. W. C. G. R. Wijshoff et al., “Reduction of periodic motion artifacts in photoplethysmography,” IEEE Trans. Biomed. Eng., vol. 64, no. 1, p. 196–207, 2017. [6] E. K. Naeini et al., “A Deep Learning-based PPG Quality Assessment Approach for Heart Rate and Heart Rate Variability,” ACM Trans. Comput. Healthcare, vol. 4, no. 4, p. 1–22, 2023. [7] M. Zhou and N. Selvaraj, “Heart rate monitoring using sparse spectral curve tracing,” in Proc. IEEE EMBC, p. 5347–5352, 2020. [8] L. Benfenati et al., “EnhancePPG: Improving PPG-based heart rate estimation with self-supervision and augmentation,” in Proc. IEEE Int. Conf. AI Circuits Syst. (AICAS), 2025. [9] B. U. Demirel et al., “Temporal cardiovascular dynamics for improved PPG-based HR estimation,” IEEE J. Biomed. Health Inform., 2025. [10] C. Kechris et al., “KID-PPG: Knowledge-informed deep learning for extracting heart rate from a smartwatch,” IEEE Trans. Biomed. Eng., 2024. [11] E. Perez et al., “FiLM: Visual reasoning with a general conditioning layer,” in Proc. AAAI Conf. Artif. Intell., vol. 32, no. 1, 2018. [12] P. H. Charlton et al., “The 2023 wearable photoplethysmography roadmap,” Physiol. Meas., vol. 44, no. 11, p. 111001, 2023. [13] J. Allen, “Photoplethysmography and its application in clinical physiological measurement,” Physiol. Meas., vol. 28, no. 3, p. R1, 2007. [14] P. H. Charlton et al., “An assessment of algorithms to estimate respiratory rate from the electrocardiogram and photoplethysmogram,” Physiol. Meas., vol. 37, no. 4, p. 610, 2016. [15] D. Jarchi et al., “Estimation of respiratory rate from motion contaminated photoplethysmography signals incorporating accelerometry,” Healthcare Technol. Lett., vol. 6, no. 1, p. 19–26, 2019. [16] K. Kazemi et al., “Respiration rate estimation via smartwatch-based photoplethysmography and accelerometer data: A transfer learning approach,” Proc. ACM IMWUT, vol. 9, no. 1, p. 1–24, 2025. [17] P. Osathitporn et al., “RRWaveNet: A compact CNN for PPG respiratory rate estimation,” IEEE Internet Things J., vol. 10, no. 18, p. 15943–15952, 2023. [18] D. Bian et al., “Respiratory rate estimation using PPG: A deep learning approach,” in Proc. IEEE EMBC, p. 5948–5952, 2020. [19] N. Huang et al., “Robust PPG-based ambulatory heart rate tracking algorithm,” in Proc. IEEE EMBC, p. 5929–5934, 2020. [20] A. Burrello et al., “Improving PPG-based heart-rate monitoring with synthetically generated data,” in Proc. IEEE BioCAS, p. 153–157, 2022. [21] P. Kasnesis et al., “Feature-level cross-attentional PPG and motion signal fusion for heart rate estimation,” in Proc. IEEE COMPSAC, p. 1731–1736, 2023. [22] J. Engel et al., “DDSP: Differentiable digital signal processing,” in Proc. Int. Conf. Learn. Represent. (ICLR), 2020. [23] X. Li et al., “Remote heart rate measurement from face videos under realistic situations,” in Proc. IEEE CVPR, p. 4264–4271, 2014. [24] R. Yamamoto et al., “Parallel WaveGAN: Fast waveform generation with GANs,” in Proc. IEEE ICASSP, p. 6199–6203, 2020. [25] A. Kendall et al., “Multi-task learning using uncertainty to weigh losses for scene geometry and semantics,” in Proc. IEEE CVPR, p. 7482–7491, 2018. [26] N. Huang et al., “Novel continuous respiratory rate monitoring using an armband wearable sensor,” in Proc. IEEE EMBC, p. 7470–7475, 2021. [27] K. Kazemi et al., “Robust CNN-based respiration rate estimation for smartwatch PPG and IMU,” in Proc. Int. Conf. Bioinformatics Res. Appl. (ICBRA), p. 94–100, 2023. [28] A. Schäfer et al., “Estimation of breathing rate from respiratory sinus arrhythmia: Comparison of various methods,” Ann. Biomed. Eng., vol. 36, no. 3, p. 476–485, 2008.