1. Overview & Biomedical Principles
In physiological signal monitoring, selection of the discrete sampling rate $f_s$ represents a fundamental trade-off between signal reconstruction fidelity, temporal resolution for fiducial point extraction, power consumption, memory footprint, and wireless transmission bandwidth. Biomedical sensors such as Electrodermal Activity (EDA) sensors and Photoplethysmography (PPG) optical sensors capture biopotentials and microvascular hemodynamics governed by vastly different autonomic frequency spectra.
Photoplethysmography ($PPG$) measures non-invasive light absorption changes in microvascular tissue beds corresponding to cardiac blood volume pulse variations during each ventricular ejection phase. A high-fidelity PPG pulse waveform consists of a sharp systolic rise time (driven by arterial compliance), a primary systolic peak, a dicrotic notch reflecting aortic valve closure, and a secondary diastolic wave. The high-frequency spectral components of the PPG systolic ascent gradient $\left.\frac{dS}{dt}\right|_{\max}$ routinely exceed $10\text{ Hz}$ to $20\text{ Hz}$. Standard clinical pulse rate variability ($PRV$) metrics demand microsecond to millisecond timing accuracy of systolic peak fiducial points.
Electrodermal Activity ($EDA$), formerly known as Galvanic Skin Response ($GSR$), quantifies changes in skin electrical conductance ($\mu\text{S}$) resulting from sudomotor sweat gland activation driven by the sympathetic nervous system. EDA signals comprise two distinct physical sub-components:
- Tonic Skin Conductance Level ($SCL$): A slow-varying baseline conductance shifting over tens of seconds to minutes ($0\text{ Hz} - 0.05\text{ Hz}$).
- Phasic Skin Conductance Response ($SCR$): Transient impulse responses triggered by discrete emotional or cognitive stimuli, characterized by rise times of $1\text{ s} - 3\text{ s}$ and decay half-times of $2\text{ s} - 10\text{ s}$ ($0.05\text{ Hz} - 2\text{ Hz}$).
Under the Nyquist-Shannon Sampling Theorem, complete reconstruction of a continuous bandlimited signal $x(t)$ requires the discrete sampling frequency $f_s$ to strictly exceed twice the highest non-zero frequency component present in the spectrum:
$$f_s > 2 \cdot f_{\max} = 2 \cdot f_{Nyq}$$
Sampling below this theoretical limit induces spectral aliasing, where high-frequency signal energy and noise fold over into lower baseband frequencies, creating non-existent artificial spectral peaks, distorting pulse wave fiducial markers, and corrupting heart rate variability ($HRV$) calculation.
3. Mathematical Foundations & Technical Architecture
The synthetic continuous baseline signal $S_{analog}(t)$ is generated continuously within a high-precision physics frame loop. The Photoplethysmography wave $S_{PPG}(t)$ is modeled as a sum of dual asymmetric Gaussian functions parameterized by heart rate period $T = \frac{60}{HR}$:
$$S_{PPG}(t) = A_1 \exp\left(-\frac{(t \bmod T - \tau_1)^2}{2\sigma_1^2}\right) + A_2 \exp\left(-\frac{(t \bmod T - \tau_2)^2}{2\sigma_2^2}\right)$$
Where $A_1, \tau_1, \sigma_1$ govern the systolic main ejection peak, and $A_2, \tau_2, \sigma_2$ control the dicrotic wave reflecting systemic vascular reflection.
The discrete sampling operation performs uniform ideal impulse train multiplication followed by mid-tread uniform linear quantization across bit depth $B$:
$$x[n] = S_{analog}(n \cdot T_s) + \mathcal{N}(0, \sigma_{noise}^2)$$
$$x_q[n] = \Delta \cdot \left\lfloor \frac{x[n] - V_{\min}}{\Delta} + 0.5 \right\rfloor, \quad \Delta = \frac{V_{\max} - V_{\min}}{2^B - 1}$$
Signal reconstruction $\hat{S}(t)$ via Whittaker-Shannon interpolation evaluates the sinc convolution sum across sliding discrete windows:
$$\hat{S}_{sinc}(t) = \sum_{n=-\infty}^{\infty} x_q[n] \cdot \operatorname{sinc}\left(\frac{t - n T_s}{T_s}\right), \quad \operatorname{sinc}(u) = \frac{\sin(\pi u)}{\pi u}$$
When $f_s < 2 f_{\max}$, continuous spectral components above $f_{Nyq} = \frac{f_s}{2}$ fold into the baseband spectral domain at alias frequency $f_{alias}$:
$$f_{alias} = \left| (f_{in} + f_{Nyq}) \bmod f_s - f_{Nyq} \right|$$
This visualizer computes continuous discrete time dynamic frame calculations, calculating dynamic device pixel ratio normalization ($DPR$) inside decoupled absolute canvas wrappers to prevent infinite DOM expansion loops while maintaining absolute zero-scroll layout stability.