## Abstract

This paper presents an analytical model and experimental validation for the detection performance and false-alarm rates for phase-encoded random modulation continuous-wave (RMCW) LiDAR. Derivation of the model focuses on propagating the effects of relevant noise sources through the system to determine an analytical expression for the detection rate, expressed by the probability of detection. The model demonstrates that probability of detection depends only on three factors: i) the mean signal-to-noise ratio (SNR) of the measurement; ii) the measurement integration time; and iii) speckle-induced intensity noise. The predicted analytical relationship between measurement SNR and probability of detection was validated by numerical simulations and experimental demonstrations in both a controlled fiber channel and under fully-developed speckle conditions in an uncontrolled free-space channel.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Light Detection and Ranging (LiDAR) is a class of technology that uses light to measure distance [1] with applications in geospatial mapping [2], wind velocity profiling [3], and autonomous vehicle navigation [4]. LiDAR works by emitting light with a time varying attribute towards a target and using the round trip time-of-flight, $\Delta \tau$, to infer distance: $d = c\Delta \tau /2$ (where $c$ represents the speed of light).

LiDAR sensors can be broadly classified as one of two detection techniques: 1) incoherent (or *direct*) detection; and 2) coherent (or *interferometric*) detection. Incoherent detection directly measures the intensity of the received light at a photodetector, from e.g., a pulsed laser source, and is sensitive only to the light’s intensity. For incoherent LiDAR sensors, the time-varying attribute used to measure distance is exclusively optical power. In contrast, coherent detection combines the received light with a reference local oscillator field at a photodetector, creating an interferometer. The local oscillator applies a coherent gain to the signal of interest and acts as an optical phase reference, making the measurement sensitive to both intensity and phase/frequency. For coherent LiDAR, the time-varying attribute used to measure distance can be, for example, optical power [5], phase, or frequency [6–8].

Coherent LiDAR’s sensitivity to optical phase enables measurement of relative radial velocity between the sensor and an illuminated object by measuring Doppler-induced frequency shifts. This capability is particularly important in applications concerned with sensing relative movement (in addition to absolute distance), such as wind velocimetry [9,10] and autonomous vehicle perception [11,12]. Despite its advantages over incoherent detection, there are a number of challenges associated with coherent detection that can degrade performance including frequency offsets due to Doppler [13], phase noise due to laser frequency noise, and speckle-induced phase and intensity fluctuations [14,15].

Random modulation continuous-wave (RMCW) LiDAR is a technique, first demonstrated in 1983 by Takeachi et al. [16], in which a pseudo-random bit sequence (PRBS) is modulated onto the amplitude or phase of the sensor’s outgoing light [16]. Light scattering from a distant object returns to the sensor after a time delay corresponding to its round-trip time-of-flight. The received light’s time-of-flight is estimated by correlating the received signal with a local template of the PRBS using, for example, a matched filter [13,17]. One class of PRBS recognised for its excellent correlation properties and commonly used in RMCW LiDAR are maximal-length sequences (m-sequences) [18]. Since its first demonstration, few examples of *coherent* RMCW LiDAR exist in the literature, possibly due to the technical challenges associated with its sensitivity to relative velocity, laser phase noise and speckle.

One challenge relevant to all forms of LIDAR, particularly in the context of safety critical autonomous driving, is determining the range and reliability at which objects of interest may be detected. In 2001, Gatt et al. [19] presented a comparison of incoherent (i.e., direct) and coherent receivers, concerned specifically with the detection statistics for both techniques. Whilst Gatt et al.’s work serves as an important reference for incoherent and coherent LiDAR, it is general in nature and does not explore how noise and bandwidth-limiting effects propagate through a matched filter.

In this paper, we present a comprehensive analysis of how different sources of noise, loss, and bandwidth-limiting effects influence the performance of coherent RMCW LiDAR. We evaluate performance in terms of two primary metrics: Probability of False Alarm (PFA), which describes the likelihood of a sensor returning a detection given that there is no target in sight; and Probability of Detection (PD), which describes how likely the sensor is to detect a target at the correct distance. This paper derives an analytical model for probability of detection for a coherent phase-encoded RMCW LiDAR sensor, taking into consideration realistic sources of loss (e.g., due to beam divergence and speckle), and noise (e.g., due to shot noise and speckle-induced intensity noise).

The paper is structured as follows: Section 2 presents an overview of the optical architecture of a coherent phase-encoded RMCW LiDAR sensor. Section 3 explores how the propagation of detected optical signals through the sensor’s electronics and digital signal processing algorithms influences measurement performance. Section 4 derives analytical solutions for measurement signal-to-noise ratio, probability of false alarm and probability of detection. Section 5 presents numerical simulations and experiments used to validate the analytical models presented in section 4. Section 6 compares the results from the numerical simulation and experiments against the analytical model. A conclusion is presented in section 7.

## 2. System architecture

Figure 1 shows the optical configuration for a coherent phase-encoded random modulation continuous-wave LiDAR sensor.

A laser is split into two paths. The first (lower) path serves as a reference local oscillator (LO) for coherent detection. The second (upper) path is phase-encoded with a pseudo-random bit sequence (PRBS), $c(t)$, which is the time-varying attribute required for absolute ranging. The phase-encoded light, referred to as the *probe*, passes through an optical circulator before propagating into free-space using a telescope. A portion of the light scattering off a distant object travels back towards the sensor and is re-coupled into the co-axial (mono-static) telescope. The circulator guides the received probe light towards a 90-degree optical hybrid. The 90-degree optical hybrid coherently combines two input fields using a particular combination of half and quarter wave-plates such that the resulting output signals are 90 degrees out of phase [20]. At the input to the 90-degree optical hybrid, the probe electric field can be modelled as

When $\beta = \pi$ (i.e., at full $\pi$ modulation depth), Eq. (1) can be represented as

where $\rho (t) = 1-2c(t) \in [-1,1]$ represents the antipodal form of $c(t)$. The phase of the received signal, $\phi (t-\tau )$, consists of several components, including: 1) the phase of the probe signal; 2) laser phase noise; 3) environmental phase noise (e.g., due to speckle and atmospheric turbulence); and 4) Doppler-induced phase noise caused by relative motion between the sensor and illuminated object. Phase noise affects measurement by redistributing energy outside of the correlation bandwidth of the matched filter, degrading measurement SNR [13].The received probe and local oscillator fields are combined within the 90-degree optical hybrid and interfered at a pair of balanced photodetectors, producing orthogonal projections of the probe electric field relative to the local oscillator [21]. The voltage waveforms produced by the two balanced photodetectors are

where $P_{LO}$ is the optical power in the local oscillator, $R_P$ is the photodetector responsivity ($R_P =q D_{QE}/h\nu$), $q$ is the charge of an electron (coulombs), $D_{QE}$ is the detector quantum efficiency, $h$ is Planck’s constant, $\nu$ is the frequency of the light (Hz), $G_P$ is the trans-impedance amplifier gain, and $\phi _\tau$ is the phase difference between the probe and local oscillator at the point of detection, $\phi _\tau = \phi (t-\tau ) - \phi (t)$.The degree to which phase noise affects measurement is highly dependent on the characteristics and environment of the system (i.e., the linewidth of the laser, the atmospheric conditions, and the target surface qualities) and must be considered accordingly. The impact of laser phase noise on SNR is explored in [22] in terms of target distance relative to the laser’s coherence length. In this analysis, we focus specifically on the regime where the effect of phase noise is negligibly small compared to other sources of noise within the system. This occurs when: 1) the coherence time of the laser is significantly larger than the maximum ranging requirements of the system; 2) the temporal coherence of the speckle and atmospheric turbulence is significantly longer than the time over which a measurement is taken; and 3) that the relative velocity between the target and system is sufficiently small as to not degrade the measurement [13]. While this reads as a strict set of requirements to impose on the system, the experimental validation presented in section 5 shows that the analytical model agrees with experimental results captured in an uncontrolled environment. Analysis taking into account these sources of phase noise would be relevant to industry but large in scope and deserving of its own publication.

## 3. DSP and statistical propagation of signal & noise

#### 3.1 Electrical component

Figure 2 illustrates the electrical and digital signal processing components of a phase encoded RMCW LiDAR system, beginning with balanced photo-detection [23]. Balanced detection introduces shot noise $(\eta _s)$ and dark noise (i.e., photodetector thermal noise) $(\eta _d)$ as additive white Gaussian noise processes, with the following normal distributions

where $I_D$ is the dark current of the photodetector (amps) and $f_s$ is the frequency at which the signal is digitized (Hz). Balanced photodetectors also apply some gain, $G_P$, in the form of transimpedance amplification (TIA) and filter the input signal with some frequency response, $H(f)$, due to internal bandwidth limiting effects. The effect of filtering on the signal is explored in section 3.2. For now, a filtered time-series function will be expressed with the form $S_{H}(t)$ where the subscript $H$ represents the frequency response of the filter $H(f)$.After the detectors, low-noise amplifiers apply some fixed gain, $G_A$, to the signal and introduce Johnson-Nyquist noise [24], $\eta _{A}\sim N(0, \sigma _A^2)$. Sufficiently large gain $G_A$, assures that the ADC noise floor and the effect of quantization error are negligible. The resulting in-phase and quadrature signals digitized at the ADC are

#### 3.2 Digital signal processing

The digital signal processing component of the system determines the delay of the received signal in two steps. First, the in-phase and quadrature digitized signals are correlated with a reference template signal to produce a complex correlation profile. Second, the magnitude of the resulting profile is taken to produce the final, phase-invariant correlation profile. The correlation between the digitized signals, $V_{I_{dig}}$ and $V_{Q_{dig}}$, and the reference PRBS template, $\rho$, are given by

In Eqs. (10) and (11), there are two distinct operations of interest: the correlation of a filtered m-sequence with a reference PRBS, $C_\rho (n) = (\rho \star \rho _{H}(t+\tau ))$, and the correlation of the noise terms with a reference PRBS, $C_\eta (n)=(\rho \star \eta _{i})$.

### 3.2.1 Correlation of a filtered signal

Cross-correlation is performed efficiently in the Fourier domain using a combination of fast Fourier transforms (FFTs) and inverse FFTs [25]

Low pass filtering the correlation profile of an m-sequence signal results in: 1) a decrease in peak correlation magnitude; 2) a constant shift in delay of the correlation output due to the filter’s phase response; and 3) asymmetric distortion of the correlation profile. The latter two effects are stationary and can thus be calibrated. As such, delay shifts and correlation distortion due to filtering are ignored in our model.

The decrease in peak height can be well approximated by the square root of the change in signal power due to filtering. For the case of a low pass filter with cut-off frequency of $f_c$, the resulting correlation profile, $C_\rho$, can be approximated as

### 3.2.2 Correlation with filtered noise

The correlation of unfiltered white noise, $\eta _i(t)\sim N(0, \sigma _i^2)$, with an m-sequence code, $\rho$, can be expressed in the time domain as

This operation multiplies each random deviate, $\eta [t]$, by either 1 or -1 and sums over the entire signal, which does not affect the underlying distribution of the random deviates [27]. The noise in the resulting correlation profile follows the normal distribution

For the case of filtered white noise, the change in variance is calculated using the equivalent noise bandwidth of the resulting spectral profile. Correlating low-pass filtered noise, $\eta _{i_H}$, with a polar m-sequence, $C(t)$, follows a Gaussian distribution defined by

where $\sigma _i$ is the standard deviation of the noise profile $\eta _i$ before filtering and $r_{f_c}$ is the ratio between power before and after filtering (as in Eq. (14)).### 3.2.3 In-phase and quadrature correlation profiles

Substituting the expressions for the correlated components of the two signals (Eqs. (13) and (17)), into the in-phase and quadrature correlation profile (Eqs. (10) and (11)) and combining the noise terms using the normal sum theorem [27] yields the following expressions

It is important to note that $C_{N_1}(n)$ and $C_{N_2}(n)$ terms above are independent, identically distributed random vectors, defined as $C_N(n) \sim N\left (0, \sigma _N^2 \right )$, where

In focusing specifically on the weak signal regimen, we assume the contribution of the noise correlation profile, $C_N$, is much larger than that of the signal, $C_\rho$, for values other than the correlation peak. Hence, the correlation profiles can be approximated as

### 3.2.4 Combination of I and Q correlation profiles

The square sum and square root of the two orthogonal correlation profiles corrects for the distribution of signal power between the in-phase and quadrature signal components. The final correlation profile, $C_f$, is given by

A statistical description of $C_f$ is determined by propagating $C_I$ and $C_Q$ through Eq. (23). This operation is equivalent to calculating the magnitude of a non-zero mean normal distribution. It can be shown that the value of the correlation profile at the correct measurement delay, $n=d$, follows the Rice distribution

Hence, the statistical description of the final correlation profile is

## 4. Analytical performance models

With a statistical description of the final correlation profile, Eq. (26), it is possible to construct analytical models for the signal-to-noise ratio (SNR) and the detection statistics of the system. For the detection statistics, we first consider the probability of false alarm, and then explore the probability of detection for two distinct cases: the case of a glint target in which the return optical power is constant (e.g. a mirror or reflective surface), and the case of a diffuse target in which the return optical power fluctuates due to speckle (e.g. a lambertian surface).

#### 4.1 Signal-to-noise ratio

The signal-to-noise ratio of a correlation profile is given by [28]

where $s$ is the height of the correlation profile ($s^2$ being the power in the signal of interest) and $E[\eta ^2]$ is the mean squared value of the correlation profile noise floor. In the case of Rayleigh distributed noise only, the mean squared value is equal to $2\sigma _N^2$ where $\sigma _N$ is the scale parameter of the distribution. The measurement SNR is thenNoise fluctuations cause variation in the measurement peak height and hence SNR. To account for this, we focus our analysis on the mean SNR of the system, replacing the numerator of Eq. (28) with $E[s^2]$. For the case where the mean peak height is significantly larger than the variation in the measurement peak, $\sqrt {r_{f_c}}G_\rho N_L/\sigma _N \gg 1$, the squared mean value of the Rice distribution can be approximated as $E[s^2] \approx (\sqrt {r_{f_c}}G_\rho N_L)^2 + \sigma _N^2$. Applying this approximation , the theoretical signal-to-noise ratio of the system is found as

In the shot noise limit, where the contributions of photodetector dark noise and amplifier voltage noise are negligible relative to shot noise, this simplifies to the familiar form

as previously derived in [19]. In this regime, SNR is analogous to photon counting and the correlation profile length, $N_L$, is analogous to the measurement integration time. The extra factor of $1/2$ relative to the result in Gatt et al. is an artifact introduced by the combination of the I and Q correlation profiles.#### 4.2 Probability of false alarm

A false alarm will occur when an outlier sample in the noise is measured with high SNR (relative to the noise variance). The probability of false alarm indicates the likelihood that the system will report a measurement above some threshold correlation value, $\gamma _T$, when there is no return signal. For this study, we consider a Constant False Alarm Rate threshold model [29], but the analysis can be readily generalised to other threshold schemes.

For the case of an independent and identically distributed (i.i.d) noise floor, the PFA is the probabilistic inverse of the likelihood that all points in the correlation profile are below the threshold value. The latter part of which is given by one minus the distributions CDF to the power of the number of points in the correlation profile, $N_L$. Therefore the probability of false alarm is given by

In the case where the noise equivalent bandwidth (NEB) of the dominating noise source is larger than the chip frequency of the encoded PRBS, the noise becomes locally correlated and no longer i.i.d. This can be accounted for by introducing a numerical scaling factor, $\kappa _{EB}$, in Eq. (31) to scale the number points in the correlation profile, $N_L$, to reflect the effective number of independent degrees-of-freedom in the resulting (over-sampled) system noise. Substituting the Rayleigh distribution CDF and rewriting the expression in units of SNR (using Eq. (28)), we arrive at the following expression

where $S_T = \gamma _T^2/(2\sigma _N^2)$. The corresponding inverse expression for $S_T$ isEquation (33) allows us to determine tolerance thresholds for the system’s sensitivity to noise. For example, for a correlation profile length of $N_L \kappa _{EB}= 2^{10}$, a threshold SNR value of 11.41dB equates to a tolerance level of 0.1%. This means that if the system reports a measurement above this threshold SNR, there is a 0.1% chance that it is due to noise and hence not a true measurement.

#### 4.3 Probability of detection for a glint reflector

The probability of detection describes the likelihood that the measurement peak is both the maximum value of the correlation profile and above the threshold for detection, $\gamma _T$. It is calculated by determining the likelihood that all criteria are filled for a given signal strengths, $x$, and then integrating over all possible strengths, weighted by the PDF of the measurement peak. This is expressed as

The solid curves in Fig. 3 display the probability of detection curves for the case of a glint target under a range of tolerance thresholds.

#### 4.4 Probability of detection for a diffuse reflector

Many real world surfaces do not behave as glint reflectors and are better approximated as diffuse. For the case of a diffuse target, the return optical power will fluctuate due to speckle and the expected ’mean SNR’ will vary on a measurement-to-measurement basis. We assuming the speckle coherence time is greater than the measurement integration time. Generalization to time-vaying speckle requires a significant change in signal processing strategy, particularly relating to implementation of a matched filter for a non-stationary waveform. Speckle can be included into the model through the introduction of a multiplicative random variable, $\alpha _I$, representing the fluctuation in return signal power and is sampled once per measurement.

The intensity fluctuation, $\alpha _I$, follows a negative exponential distribution [30]

The dashed lines in Fig. 3 display a range of diffuse PD curves for a diffuse target with different noise tolerance thresholds.

## 5. Numerical and experimental validation

The analytical results derived in section 4 were verified using both numerical simulation and experimental data. The simulation modeled all noise sources as stochastic random variables in a Monte-Carlo style. Two experiments were performed to separately verify the glint and diffuse models, one in optical fiber and the other in a free space outdoor ranging to a paper target.

#### 5.1 Monte-Carlo simulation

The simulation was written in Matlab. Figure 4 illustrates a block diagram form of the Monte-Carlo simulation. The simulated output of a laser was propagated through numerical models of each component of the optical, electrical and digital system. Time delay was simulated by digitally delaying the probe signal. The effect of bandwidth-limited electronics was simulated using Butterworth filters with cutoffs and orders that were varied to model particular electronics.

All noise sources were introduces as random variables. Each additive white Gaussian noise source (being shot noise, dark noise and Johnsen-Nyquist noise) was scaled and added into the signal chain at its respective location. Laser phase noise was simulated as a Gaussian random walk in phase, with a laser linewidth dependent step size [31]. Speckle was simulated by multiplying the return probe signal by a constant variable drawn from an exponential distribution (per Monte-Carlo iteration). The rate parameter of the negative exponential distribution was calculated based on the input target distance, reflectivity and beam characteristics of the system. Speckle was turned off when simulating a glint target.

All system parameters were set to match those of the experimental setup, outlined below. The mean SNR was swept across a range of values, and for each SNR an ensemble of 1000 ranging measurements were simulated. Each measurement was analysed to determine whether the correct delay was detected or a misdetection had occurred.

#### 5.2 Glint reflector model validation

A glint return is characterized by a constant signal power, which can be well approximated using an attenuated signal in optical fiber, forming a Mach-Zehnder interferometer. Figure 5 displays the optical setup used to experimentally verify the analytical expression for the probability of detection with a glint reflector.

Light from a 1550nm laser (Koheras Basik Mikro E15) was split down two paths using a $70/30$ beam splitter to produce a probe ($70\%$) and a local oscillator ($30\%$) field. This particular splitting ratio ensured the system was operating in a shot noise limited regime. The phase of the probe beam was encoded with a 10-bit m-sequence code at a chip rate of 200MHz using a iXblue MPZ-LN10 electro-optic modulator (EOM). This code length and frequency resulted in a maximum unambiguous measurement range of 767 meters. The PRBS was generated using a Centellax T2GP1A PRBS generator. The probe and local oscillator were then both attenuated.

The probe was passed through a $99/1$ splitter, where the $99\%$ arm was used to infer the optical power of the probe arm at the 90-degree optical hybrid. The power in the $99\%$ arm was measured using an AQ-2140 optical powermeter. The probe ($1\%$ arm) was passed through a 50-meter spool of single mode fiber to impart a delay on the signal. The probe beam then travelled through a set of polarization control paddles before being coherently combined with the local oscillator in the 90-degree optical hybrid (Kylia COH28).

The optical signals at the output of the 90-degree optical hybrid were detected at a pair of balanced photodetectors (Insight BDP-1). After, the photodetector outputs were amplified (+14 dB gain) and digitized for processing using a NI-5785 ADC.

The optical power of the probe at the 90-degree optical hybrid was initially fixed to $\sim 6$pW and a set of 480 consecutive measurements were taken. Upon conclusion, the optical power was lowered and the measurement process was repeated at twenty separate optical power levels, ranging down to the sensitivity limit of the powermeter (which occurred at an equivalent probe power level of $\sim 40$fW). Twelve further sets of measurements were taken below this power level, using a femtoWatt sensitive photodetector to guide the relative optical power. Each set of measurements was processed to determine the probability of detection and the observed mean measurement SNR, with no noise tolerance thresholding applied (meaning the system is operating with a threshold SNR of 0dB and at a tolerance level of 100%).

#### 5.3 Diffuse target model validation

For the case of a diffuse target, the fiber probe arm from the previous experiment was replaced with a circulator-telescope combination, as illustrated in Fig. 5. A collection of the telescope/free space beam characteristics can be found in Table 2. The system was used to range a 60 cm by 120 cm paper target at distances varying from 50 to 400 meters in 50 meter intervals in an outdoor, uncontrolled environment. Paper was used as it is a reasonable approximation for a diffuse surface. A large silver steering mirror and a visible red laser were used to aim the system at the target, which was mounted on a frame with adjustable pitch to minimise the beam’s angle of incidence.

A baseline measurement was taken with the beam angled towards the sky to determine and calibrate for the telescope’s prompt reflection. At each range, a set of 480 measurements were taken over a time frame of 30 seconds to attain a sufficiently large sample of speckle realizations due to system movements and atmospheric effects. The probability of detection and observed mean measurement SNR were determined for each target distance, with no noise tolerance thresholding applied.

## 6. Results

Figure 6 displays the analytical (Eqs. (35) and (38)), numerical and experimentally observed relationships between the mean signal-to-noise ratio and the probability of detection for both the glint and diffuse targets. The experimental relationships displays high levels of agreement with both the simulation results and the analytical solutions. There is slight discrepancy in the diffuse target case as the experimental data seems to indicate the model is underestimating the probability of detection. This artifact is likely due to the signal processing techniques used to remove the prompt telescope reflection slightly misrepresenting the statistics of the final correlation profile and making the measurement SNR appear slightly lower than it actually was. These results serve as validation that the analytical model derived in this paper accurately reflects the behaviour of RMCW LiDAR measurement.

## 7. Conclusions

This paper has presented an analytical model that describes the signal-to-noise ratio and probability of detection of a coherent phase-encoded RMCW LiDAR system for the case of both a glint and diffuse target. The model was achieved by statistically propagating the effect of signal attenuation and noise through each component of the system and arriving at a statistical description for the final correlation profile from which measurement is made. Through the model, it was observed that while the signal-to-noise ratio of the system depended on a large number of system parameters, the probability of detection only depended on the mean SNR of the signal, the integration time of measurement and the statistical nature of signal strength fluctuations. Using both numerical simulation (utilizing a Monte-Carlo style approach) and experimentation, the analytical model was shown to encapsulate the behaviour of the system.

## Funding

Australian Research Council Centre of Excellence for Gravitational Wave Discovery (CE170100004); Australian Research Council Centre of Excellence for Engineered Quantum Systems (CE170100009).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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