Experimental PN-PSD Characterization

• Experimental PN-PSD characterization is a suite of methods that employ pseudorandom noise and pulse-shape discrimination to extract high-resolution spectral and temporal information.
• Techniques utilize cross-correlation, Fourier transforms, and advanced filtering (e.g., Wiener and kernel ridge regression) to mitigate artifacts and enhance measurement fidelity.
• Applications include channel sounding, laser phase noise metrology, and particle discrimination, with rigorous calibration ensuring reliable quantitative performance.

Experimental PN-PSD characterization refers to the suite of measurement and analysis techniques aimed at determining power spectral density (PSD) or phase noise power spectral density (PN-PSD) using pseudorandom noise (PN) sequences or via experimental pulse-shape discrimination methodologies. PSD and PN-PSD characterization underpin a wide array of applications in signal integrity, photonic metrology, and particle discrimination, demanding rigorous calibration and artifact suppression for reliable quantitative interpretation.

1. Fundamentals of PN-PSD Characterization

PN-PSD characterization leverages the correlation properties of pseudorandom noise sequences and/or pulse-shape waveform analysis to extract spectral or temporal system information. In channel measurements, maximal-length linear PN sequences with near-ideal autocorrelation facilitate high-resolution impulse response estimation, enabling conversion to channel PSD via Fourier analysis (Ate et al., 2024). In phase noise metrology, experimental PN-PSD denotes the direct measurement and digital correction of the phase noise spectrum, notably within optical self-heterodyne setups where interference fringes can severely bias the PSD estimate if unmitigated (Riebesehl et al., 21 May 2025). In particle detection, PSD characterizes the ability to distinguish neutron-like from gamma-like events by exploiting differences in temporal scintillation waveform features (Teh et al., 2020, Hong et al., 2024).

2. Methodologies: Experimental PN-PSD Workflows

2.1 Correlative PN Sounding for Channel Characterization

• Maximal-length PN sequence of degree $m$ (e.g., $m=13$, $N=8191$ chips) is generated and transmitted at a chip rate sufficient to Nyquist-sample the bandwidth of interest (e.g., $5\,\mathrm{MHz}$ chip rate for $0$–$2.5\,\mathrm{MHz}$) (Ate et al., 2024).
• The receive path records the system output, and cross-correlation of the received signal with the known PN yields an impulse response estimate $\hat{h}[n]$.
• Windowing (e.g., Hann) and averaging over multiple ($\geq 30$) realizations reduce sidelobes and statistical variance.
• Discrete Fourier transform of $\hat{h}[n]$ produces the channel transfer function $H[f]$ and power spectral density $m=13$0.

2.2 Experimental Phase Noise PSD Extraction

• Asymmetric Mach–Zehnder self-heterodyne configuration with delay $m=13$1 and frequency offset $m=13$2 is employed (Riebesehl et al., 21 May 2025).
• The phase-difference PSD $m=13$3 is distorted by $m=13$4, inducing spectral notches (fringes).
• Naïve inversion diverges at fringe nulls; robust compensation uses a Wiener-type equalization filter incorporating $m=13$5, constructed from regions with high SNR via kernel ridge regression (KRR) on log-transformed spectral data.
• Final artifact-free PN-PSD is synthesized via a combination of KRR-based spectral interpolation and Wiener filtering.

2.3 Experimental PSD in Particle Discrimination

• Digitized scintillation waveforms are acquired at $m=13$6 or higher, often using dual-end photomultiplier tube (PMT) readout (Teh et al., 2020, Hong et al., 2024).
• Integration gates on the waveform (Q₁: short/fast, Q₂: long/total) yield correlated charge pairs, with template-based or empirical fits defining gamma- and neutron-like ridges in $m=13$7–$m=13$8 space.
• Value-assigned PSD (VPSD) and position-corrected projections (PPSD) assign normalized continuous PSD scores to each pulse, correcting for longitudinal attenuation and maximizing separation as measured by the figure-of-merit (FoM).
• Performance is quantified via analytic or simulated evaluation of likelihood discrimination, with efficiency curves as a function of photoelectron count (NPE).

3. Experimental Setups and Key Parameters

Method/Domain	Signal/Source	Instrumentation
PN-based channel sounding	PN sequence (m=13)	Arbitrary waveform generator, oscilloscope, optocoupler isolation (Ate et al., 2024)
Short-delay self-heterodyne	CW laser, AOFS	Mach–Zehnder, AOFS, fiber delay, balanced photodetector, ≥5 GS/s scope (Riebesehl et al., 21 May 2025)
Particle PSD	NE-213/Gd-LS + PMTs	FADC ≥500 MHz, dual PMT, cubic spline interp. (Teh et al., 2020, Hong et al., 2024)

Parameterization is directly tied to system response characteristics:

• Sequence length ($m=13$9) and zero-padding in PN sounding define delay spread and frequency resolution (Ate et al., 2024).
• Delay-line length $N=8191$0 in self-heterodyne systems sets fringe periodicity and must be chosen relative to spectral analysis window (Riebesehl et al., 21 May 2025).
• Integration gate timings and charge extraction parameters are tuned to waveform shape and sampling interval; spline interpolation is favored for peak timing and charge resolution (Teh et al., 2020).

4. Data Analysis Strategies and Artifact Mitigation

Robust PN-PSD characterization necessitates meticulous treatment of windowing effects, statistical averaging, and systematic artifacts:

• In PN channel sounding, zero-padding avoids circular correlation artifacts; windowing the CIR suppresses spectral sidelobes, and optocoupler isolation cancels ground-loop-induced noise (Ate et al., 2024).
• In self-heterodyne PN-PSD, digital power spectrum equalization based on KRR-trained SNR models retains measurement fidelity across fringe nulls, mitigating divergence seen in naïve inversion (Riebesehl et al., 21 May 2025).
• In scintillation PSD experiments, longitudinal light attenuation is analytically modeled and corrected via coordinate transformations; ridge fitting to $N=8191$1–$N=8191$2 data allows intrinsic separation of pulse species, and gate optimization ensures near-optimal FoM (<10% loss with $N=8191$3 gate variance) (Teh et al., 2020).

5. Quantitative Results and Performance Metrics

5.1 Channel Sounding and Intrabody Links

• For galvanic intrabody communication using chicken-breast tissue as phantom, the measured channel gain is approximately $N=8191$4 at $N=8191$5, rolling on at the $N=8191$6 cutoff and improving to $N=8191$7 by $N=8191$8 (Ate et al., 2024).
• The measured PSD after averaging over 40 acquisitions is noise-limited below $N=8191$9; experimental curves are within $5\,\mathrm{MHz}$0–$5\,\mathrm{MHz}$1 of FEM simulation.

5.2 Self-Heterodyne PN-PSD

• Simulation and experimental validation show that the KRR+PSE method yields residual bias $5\,\mathrm{MHz}$2 for SNR $5\,\mathrm{MHz}$3; spectral spikes $5\,\mathrm{MHz}$4–$5\,\mathrm{MHz}$5 at fringe positions are fully suppressed (Riebesehl et al., 21 May 2025).

5.3 Particle PSD

Neutron Rejection ($5\,\mathrm{MHz}$6)	Gamma Retention ($5\,\mathrm{MHz}$7)	Required NPE
90%	97.8%	49
95%	99.4%	79
99%	99.9%	150

Separation sharpens with increasing NPE, and FoM scales roughly as $5\,\mathrm{MHz}$8; logistic functional forms empirically capture the rapid rise in $5\,\mathrm{MHz}$9 versus NPE (Hong et al., 2024). In NE-213 bar systems with dual PMTs, FoM improves by up to 75% at bar ends (position-corrected VPSD versus geometric-mean method), with benefits extending to analog readout at modest gate timing offsets ($0$0% FoM loss) (Teh et al., 2020).

6. Practical Guidelines and Limitations

• For PN-based channel sounding, degree $0$1 and chip rate $0$2 the max frequency of interest are recommended; at least $0$3 zeros padding, optocoupler isolation, and averaging over $0$4–$0$5 runs suppress artifacts (Ate et al., 2024).
• In short-delay PN-PSD, select $0$6 such that $0$7–$0$8 fringes span the SNR-rich region, use high-bandwidth balanced photodetectors and ≥5 GS/s digitization, and restrict KRR training set to high-SNR frequencies ($0$9–$2.5\,\mathrm{MHz}$0 dB) (Riebesehl et al., 21 May 2025).
• In PSD of scintillation signals, digitization at $2.5\,\mathrm{MHz}$1 MHz with cubic splines and separate ridge fitting by particle type are favored; NPE thresholds should be matched to application-driven rejection/retention targets with a minimum of $2.5\,\mathrm{MHz}$2 PE for $2.5\,\mathrm{MHz}$3, $2.5\,\mathrm{MHz}$4 (Hong et al., 2024).
• The KRR interpolation assumes smooth lineshapes; sharp spectral spurs $2.5\,\mathrm{MHz}$5 kHz may be insufficiently captured. Computational cost in KRR scales as $2.5\,\mathrm{MHz}$6, limiting training sets to a few hundred points (Riebesehl et al., 21 May 2025).

7. Extensions and Applications

Experimental PN-PSD characterization underpins quantitative modeling of communication channels (e.g., implantable intrabody links), laser phase noise metrology (yielding artifact-free spectral estimation over six or more decades of Fourier frequency), and advanced scintillator-based particle discrimination. Methodological advances in artifact rejection and position correction are directly generalizable to large detector arrays and emerging high-density readout applications (Ate et al., 2024, Teh et al., 2020, Hong et al., 2024, Riebesehl et al., 21 May 2025). Further integration of machine learning for model-free spectral compensation, as demonstrated for PN-PSD equalization, is a promising direction for future work in this domain.