Cycle-Level Fetal S1/S2 Event Synthesis

• The paper introduces a dynamic parametric synthesis model for fetal S1/S2 events using asymmetric damped-sinusoid kernels to replicate individual cardiac cycles.
• It calibrates cycle-specific parameters from real fPCG recordings and models beat-to-beat variability through controlled RR interval adjustments.
• Validation using temporal and spectral metrics confirms that the simulator reliably mimics key characteristics of authentic fPCG signals for algorithm benchmarking.

Cycle-Level Fetal S1/S2 Event Synthesis refers to a dynamic, parametric approach for simulating fetal phonocardiogram (fPCG) signals at the resolution of individual cardiac cycles. It addresses the challenges of signal scarcity, maternal interference, and transmission-induced attenuation by modeling the generation, calibration, transmission, and validation of fetal heart sound events (S1/S2) in a reproducible and physiologically realistic manner. This methodology enables the creation of synthetic abdominal fPCG datasets that support rigorous, controlled benchmarking of fPCG processing algorithms under diverse acquisition conditions (Zhou et al., 23 Jan 2026).

1. Parametric Modeling of S1/S2 Events

At the foundation is the construction of S1 and S2 heart sound events using an "asymmetric damped-sinusoid" kernel. For each event, the kernel is defined as:

$$\phi(t;A,f_{0},T_{a},\tau) = A\,\sin(2\pi f_{0}t)\; a(t;T_{a},\tau)$$

where the envelope $a(t;T_{a},\tau)$ employs a swift linear attack ($T_{a}$, commonly 8 ms), followed by exponential decay ($\tau$):

$$a(t;T_{a},\tau)= \begin{cases} \frac{t}{T_{a}}, & 0 \leq t < T_{a} \ \exp\left(-\frac{t-T_{a}}{\tau}\right), & t \geq T_{a} \ 0, & t < 0 \end{cases}$$

The adjustable parameters per event are amplitude ($A$), decay time ($\tau$), and carrier frequency ($f_{0}$). Each cardiac cycle is synthesized by summing S1 and S2 kernels placed according to identified beat onsets ($t_k$) and the cycle-specific systolic interval ($\Delta T^{(k)}$):

$$x_{f}(t) = \sum_{k} \left[\, \phi(t - t_k;\theta_{\mathrm{S1}^{(k)}}) + \phi(t - t_k - \Delta T^{(k)};\theta_{\mathrm{S2}^{(k)}}) \,\right]$$

The per-cycle parameter vectors $$\boldsymbol{\theta}^{(k)} = [A_{\mathrm{S1}}, A_{\mathrm{S2}}, \tau_{\mathrm{S1}}, \tau_{\mathrm{S2}}, \Delta T]^{(k)}$$ encode amplitude, decay, and interval structure, capturing essential variability.

2. Beat-to-Beat Variability and Heart Rate Dynamics

Fetal heart rhythm is synthesized using a variable RR interval series to reflect physiological heart rate variability (HRV):

$$RR_k = \overline{RR} + \alpha\,d_{k} + \eta_{k},\qquad t_{k+1} = t_k + RR_k$$

Here, $d_k$ models low-frequency drift, $\eta_k$ provides stochastic jitter, and the variability scale $\alpha$ is user-controlled. Per-cycle parameters $\boldsymbol{\theta}^{(k)}$ are sampled from empirically derived admissible ranges, introducing cycle-wise fluctuations in amplitude and duration, consistent with real fetal fPCG patterns.

3. Cycle-Wise Parameter Calibration from Real Data

Parameter calibration involves extracting cycle-level information from abdominal fPCG recordings:

• Bandpass filtering (20–150 Hz) isolates relevant frequency content.
• The Hilbert envelope is computed, smoothed at 8 Hz, and normalized.
• Envelope peaks detect beat onsets $t_k$ with minimum spacing reflecting physiological heart periods.
• Segments around each onset are carved and zero-mean normalized.

For each observed cycle, per-event parameters are fitted to the noise-free waveform $y^{(k)}(t)$ by minimizing

$$\underset{\boldsymbol\theta^{(k)}}{\mathrm{minimize}}\; \sum_{t} \left| y^{(k)}(t) - \left[ \phi(t;\theta_{\mathrm{S1}^{(k)}}) + \phi(t-\Delta T^{(k)};\theta_{\mathrm{S2}^{(k)}}) \right] \right|^2$$

using nonlinear least-squares. Empirical minima and maxima of $\{\boldsymbol{\theta}^{(k)}\}$, clipped to physiological bounds, define the cycle-wise parameter box $[\mathbf{l}_s, \mathbf{u}_s]$. During synthesis, new cycles sample $\boldsymbol{\theta}^{(k)}$ either uniformly or via truncated-Gaussian methods within this box. Optionally, Markov chain Monte Carlo (emcee) sampling can maintain more detailed joint-distribution structure without exceeding physiological limits.

4. Transmission Modeling and Addition of Interference

Synthesized fetal and maternal sources ($x_f + x_m$) are propagated through a time-invariant convolutional filter, modeling abdominal transmission effects:

$$h(t) = \frac{(h_1 * h_2)(t)}{\int_{0}^{\infty} (h_1 * h_2)(u)\, du}$$

where $h_1(t) = A_1 e^{-\beta_1 t} \mathbb{I}(t\ge0)$ and $h_2(t) = A_2 e^{-\beta_2 t} \mathbb{I}(t\ge0)$ are exponentials. This normalized filter imparts both attenuation and low-pass smoothing, crucial for matching observed abdominal fPCG characteristics.

Noise is added as colored AR(1) interference:

$$n_{t} = \rho\, n_{t-1} + \sqrt{1 - \rho^2}\, \varepsilon_{t}, \quad \varepsilon_{t} \sim \mathcal{N}(0,1)$$

Optionally, a slow gain envelope modulates amplitude:

$$g(t) = 1 + \gamma\, \mathrm{LP}\{w(t)\}, \quad \tilde n(t) = g(t)\,n(t)$$

The total output signal $x(t)$ merges the convolved source and noise at a specified signal-to-noise ratio (SNR):

$$x(t) = x_c(t) + \sigma_n \tilde n(t), \quad \sigma_n = \frac{\mathrm{RMS}(x_c)}{10^{\mathrm{SNR}/20}}$$

5. Validation Against Real Recordings

Validation is conducted via temporal and spectral metrics:

• Envelope-based metrics: Simulated vs. real Hilbert-smoothed envelopes demonstrate matching beat-to-beat peaks and amplitude modulation. Cycle-averaged envelope autocorrelation (ACF) curves are closely aligned, with the simulator reproducing intra-cycle spacing and smoothness.
• Frequency-domain metrics: Power spectral density (PSD) comparison using Welch’s method in the 20–150 Hz band reveals that simulated signals match the roll-off and spectral peak distribution of real recordings, except for marginally lower mid/high band power due to kernel compactness.

These validations confirm that the cycle-level synthesis and convolutional propagation reliably reconstruct both the envelope structure and spectral signature of authentic fPCG data.

6. Significance and Applications

This cycle-level event synthesis paradigm, integrated with data-driven calibration and explicit transmission modeling, enables the reproducible generation of long abdominal fPCG sequences. The resulting simulator provides a benchmark platform for the evaluation of fPCG processing methods, particularly in circumstances where real recordings are scarce or confounded by noise and interference. The approach supports rapid, controlled, and physiologically realistic benchmarking for algorithm development, comparative studies, and parameter sensitivity analyses (Zhou et al., 23 Jan 2026).

7. Methodological Considerations and Performance Implications

Key attributes include the low-dimensional yet flexible kernel representation, empirically bounded sampling of cycle-wise parameters, and the ability to impose joint-distribution structures. The convolutional model is explicit and two-stage, accommodating abdominal transmission effects. Configurable colored noise supports robust simulation of environmental and maternal interference. A plausible implication is that controlled access to adversarial and noise-rich sequences can accelerate the development and validation of fPCG extraction techniques. The modular workflow and open software release further facilitate reproducibility and adoption for research purposes.