Performance Bounds of Ranging Precision in SPAD-Based dToF LiDAR

TL;DR

This work derives the Cramér-Rao lower bound for SPAD-based dToF LiDAR while explicitly modeling dead-time pile-up and extends the analysis to photon-number-resolving SPADs. The CRLB expressions, validated by Monte Carlo simulations and maximum-likelihood estimation, reveal that pile-up not only reduces per-bin information but also couples the distance estimate to the received photon flux, creating an optimal operating point for photon flux and pulse width. Photon-number-resolving SPADs offer dynamic-range benefits but yield only modest improvements in fundamental ranging precision, with substantial gains requiring recording both TDC triggers and subpixel counts. The findings provide theoretical bounds and practical guidelines for selecting laser power, pulse width, and TDC/PNR configurations to approach the best achievable precision in SPAD-based dToF LiDAR systems.

Abstract

Lidar with direct time-of-flight (dToF) technology based on single-photon avalanche diode detectors (SPAD) has been widely adopted in various applications. However, a comprehensive theoretical understanding of its fundamental ranging performance limits--particularly the degradation caused by pile-up effects due to system dead time and the potential benefits of photon-number-resolving architectures--remains incomplete. In this work, the Cramer-Rao lower bound (CRLB) for dToF systems is theoretically derived accounting for dead time effects, generalized to SPAD detectors with photon-number-resolving capabilities, and are further validated through Monte Carlo simulations and maximum likelihood estimation. Our results reveal that pile-up not only reduces the information contained within individual ToF but also introduces statistical coupling between distance and photon flux rate, further degrading ranging precision. The derived CRLB is used to determine the optimal optical photon flux, laser pulse width, and ToF quantization resolution that yield the best achievable ranging precision. The analysis further quantifies the limited performance improvement enabled by increased photon-number resolution, which exhibits rapidly diminishing returns. These findings provide theoretical guidance for the design of dToF systems and the selection of their optimal operating points.

Figures (9)

• Figure 1: Theoretical histogram expectation and pile-up effect. Black dashed line: the theoretical laser pulse waveform $f(t-t_0)$ corresponding to a ToF of $t_0=5$. Solid lines: expected single-pulse histograms $Q_i$ under different peak received photon flux rates with $R=0.1$ (blue), 1 (orange), 10 (green), and 100 (red).
• Figure 2: Theoretical ranging performance $\delta t_0$ normalized to TDC resolution $\tau$ for estimating the ToF $t_0$ as a function of the peak received photon flux rate $R$ under three system architectures. Solid lines represent theoretical calculations, and crosses denote the variances of 100 sets of $t_0$ estimates obtained via maximum likelihood estimation from histograms generated by Monte Carlo simulations at different values of $R$. $f(t)$ is assumed as a gaussian-shaped laser pulse with full width at half maximum (FWHM) $w=4\tau$. Case 1 (blue): 1-PNR SPAD with $N=100$, $t_0 = 10\tau$ and $R_n=0.02$; Case 2 (range): 4-PNR SPAD with Type I system, $N=100$, $t_0 = 10.2\tau$ and $R_n = 0$; Case 3 (green): 4-PNR SPAD with Type II system, $N=1000$, $t_0 = 10.5\tau$ and $R_n = 0.01$.
• Figure 3: System ranging performance for a gaussian-shaped laser pulse with FWHM $w=3\tau$ and ToF with $t_0 = 10\tau$ as a function of $R$. Variation of system ranging precision $\delta t_0$ (blue solid, left axis), ranging precision with known $R$, $\delta_R t_0$ (orange dashed, left axis), ranging precision without accounting for dead time (green dashed, left axis), and correlation coefficient $\rho^2$ (red dotted line, right axis) are shown as functions of $R$ for comparison.
• Figure 4: System ranging performance as a function of background noise intensity $R_n$ with $w=3\tau$ and $t_0 = 10\tau$. (a) Variation of system ranging precision $\delta t_0$ (solid line, left axis), ranging precision with known $R$, $\delta_R t_0$ (dashed line, left axis), and correlation coefficient $\rho^2$ (dotted line, right axis) as functions of $R$, under different $R_n$ levels. (b) $\delta t_0$ as a function of $R_n$ for representative values of $R$. Black solid: optimal ranging precision under each $R_n$; black dotted: corresponding optimal $R=R_\mathrm{opt}$; Orange dashed: $R=1$; Green dashed: $R=20$; Red dashed: $R=100$; Points A, B, and C in both subfigures mark the optimal ranging performance points for different $R_n$ values.
• Figure 5: Ranging precision $\delta t_0$ as a function of $R$ for different values of $t_0$ varying from 0 (purple) to $\tau$ (yellow) with laser pulse width of $0.6\tau$. The worst-case ranging precision over $t_0 \in [0, \tau)$, denoted as $\Delta t_0$, is shown as a red dashed line. The minimum standard deviation $\Delta_\mathrm{m} t_0$ and the corresponding optimal received photon flux rate $R_\mathrm{opt}$ are marked by a black star.