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Abstract
We report on a laser frequency sweep linearization method by iterative learning pre-distortion for frequency-modulated continuous-wave (FMCW) light detection and ranging (LiDAR) systems. A pre-distorted laser drive voltage waveform that results in a linear frequency sweep is obtained by an iterative learning controller, and then applied to the FMCW LiDAR system. We have also derived a fundamental figure of merit for the maximum residual nonlinearity needed to achieve the transform-limited range resolution. This method is experimentally tested using a commercial vertical cavity surface-emitting laser (VCSEL) and a distributed feedback (DFB) laser, achieving less than 0.005% relative residual nonlinearity of frequency sweep. With the proposed method, high-performance FMCW LiDAR systems can be realized without expensive linear lasers, complex linearization setups, or heavy post-processing.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Light detection and ranging (LiDAR) technologies have many applications ranging from scientific researches, industrial manufacturing to robotics and autonomous vehicles [1,2]. Compared to the conventional pulsed time-of-flight method, frequency-modulated continuous-wave (FMCW) technology is able to achieve high resolution without requiring fast electronics or high optical power, and is immune to direct sun light and interference from other LiDAR transmitters, thanks to coherent detection [1,3]. It is also capable of simultaneously detecting target position and velocity in a single measurement through the Doppler frequency shift [4].
FMCW LiDAR employs a frequency-swept laser and coherent detection. The reflected light from the target is combined with the local reference in a photodetector, converting the roundtrip distance to a beat note. If the laser frequency sweep is linear, the beat frequency for a stationary target is time-invariant and can be extracted using a Fourier Transform (FT). Unfortunately, most tunable lasers do not have an intrinsically linear relation between the optical frequency and the tuning signal. Various linearization methods are needed to achieve a linear frequency sweep [5]. Residual nonlinearity will degrade the resolution and signal-to-noise-ratio (SNR) of LiDAR detection [6].
There are two main approaches to address this issue: (1) post-processing of sampled data and (2) linearization of the laser frequency sweep. The resampling method has been reported to successfully deal with nonlinear frequency sweep in post-processing [7–9], which reduces the hardware complexity of FMCW LiDAR at the expense of computation power, processing delay and latency. In addition, the resampling method is not compatible with velocity detection since the uneven resampling in time domain obscures the Doppler signal. On the other hand, many research works have focused on real-time active frequency sweep linearization using the optical phase locked loop (OPLL), which results in good frequency sweep linearity [10–12]. However, OPLL requires fast electronics with short loop delay, increasing the complexity and cost. Other methods to generate a linear frequency sweep include sideband-modulation by external modulators such as a Mach-Zehnder modulator [13] or an electro-optic modulator [14], and stretching an ultra-fast optical pulse [15], but these methods generally have limited frequency excursion and require complex setups.
Digital pre-distortion has been investigated for linearization problems in various fields such as optical communications [16,17], RF power amplifiers [18,19], and laser frequency sweep linearization [5,10,20]. Conventionally, to obtain a proper pre-distorter, a numerical model of the system behavior is first assumed and then experiments are performed to optimize the parameters. For example, Karlsson and Olsson assumed a linear time-invariant model of the laser, measured the laser’s response at different modulation frequencies, and compensated for nonlinearity from the 1st to the 31st order harmonics of modulation signal using a pre-distorter [5]. Satyan et al. used a current dependent gain model and solved for a pre-distorted waveform as the initial input of an OPLL [10]. Minissale et al. modeled the relation of laser frequency sweep and drive voltage by a power function and determined the optimized index number for a linear frequency sweep by experiments [20]. However, the linearity of these methods is fundamentally limited by the assumed models.
Iterative learning control (ILC) has been proven effective in controlling repetitive dynamic systems and has been successfully applied to a large number of areas [21,22]. It does not require any model for the system, making it an ideal candidate for frequency sweep linearization in FMCW LiDARs. In this paper, we propose a laser frequency sweep linearization method by ILC pre-distortion. As a proof of concept, we experimentally demonstrated an FMCW LiDAR using a commercial vertical-cavity surface-emitting laser (VCSEL) and a distributed feedback (DFB) laser. Simultaneous measurements of distance and velocity are successfully performed. This method does not require complex feedback control or post-processing, and is straightforward to implement with any kind of tunable lasers. It is suitable for laser frequency sweep linearization for FMCW LiDAR, particularly for systems aiming at high performance with low cost and simple setup.
2. Method and analysis
A schematic of ILC pre-distortion frequency sweep linearization is shown in Fig. 1(a). The goal of the ILC process is to find a pre-distorted laser drive voltage waveform ud(t) that leads to the desired laser frequency sweep νd(t), a triangle waveform with period 2Tramp, without a priori knowledge of the dynamic behavior of the laser. The ILC process starts from an initial input u1(t) and iteratively updates the voltage waveform uk(t) at the kth iteration by the learning controller such that the laser frequency variation νk(t) converges to the desired νd(t). A triangle voltage waveform is used for the initial input u1(t) in this work though the ILC process will converge even faster if u1(t) is closer to ud(t).
Fig. 1 Schematic of ILC pre-distortion of laser frequency sweep linearization. (a) Block diagram of the ILC process. (b) Detailed setup for laser frequency sweep measurement.
At the kth iteration, with laser drive voltage uk(t), the iterative controller measures the laser frequency sweep νk(t), calculates the error ek(t) = νd(t) – νk(t), and then generates the drive voltage for the next iteration uk+1(t) according to uk(t) and ek(t). There are many different types of update algorithms for ILC, among which the simplest one, uk+1(t) = uk(t) + p·ek(t) where p is a constant, called linear update algorithm, is used in this work. It has been proven that νk(t) will converge to νd(t) using the linear update algorithm with a sufficiently small p [18,23].
Figure 1(b) shows the experimental setup for measuring the laser frequency sweep. The laser output is sent to a Mach-Zehnder interferometer (MZI) with a relative delay time τ between the two arms. A pair of balanced photodetectors record the beat signal from the MZI. A Hilbert Transform (HT) is used to extract the phase of the beat signal [7]. Assuming the laser frequency is ν(t), the beat signal frequency is fb(t) = ν(t + τ) – ν(t). If τ is significantly small, fb(t) can be approximated by the first term in its Taylor series, i.e. fb(t) = ν'(t)·τ, thus the beat signal phase extracted by the Hilbert Transform is φb(t) = 2πτ·∫ν'(t)dt = 2πτ·ν(t). Therefore, the laser frequency sweep can be directly calculated from the extracted beat signal phase.
Once ud(t) is obtained by ILC, it is used as the laser drive voltage waveform for the FMCW LiDAR. The schematic of an FMCW LiDAR is shown in Fig. 2(a). Reflected light from the target interferes with the light from the reference arm and the beat signal is recorded by a detector. Since the laser frequency sweep is linearized, the beat signal frequency fb is proportional to the target distance D, i.e. D = cτt/2 = cfb/(2γ), where c is the speed of light, γ is the laser frequency sweep rate and τt = fb/γ is the relative delay time between the reference and the probe arms. The distance detection resolution is
where δfb is the spectral peak width of the beat signal. If the target is not stationary, the Doppler frequency shift fDoppler, which is proportional to the target velocity, can be detected by the difference of the beat frequencies between the up and the down ramps, respectively fb,up and fb,down (Fig. 2(b)). The target velocity iswhere λ is the laser wavelength.Fig. 2 (a) Schematic of an FMCW LiDAR. The drive voltage ud(t) obtained by the ILC pre-distorter is applied to the laser. (b) The variation of optical frequency versus time for the reference and probe light paths with a Doppler frequency shift due to target motion.
3. Figure of merit for residual nonlinearity
To better understand the effect of frequency sweep nonlinearity in the FMCW LiDAR, we model the laser frequency sweep in one ramp by ν(t) = ν0 + γt + νnl(t), where ν0 is the optical carrier frequency, γt represents an ideal linear frequency sweep, and νnl(t) represents the nonlinear part of frequency sweep. Similar to the previous analysis that approximates the beat signal frequency by the first term of its Taylor series, the phase of the beat signal can be represented by
In general, Eq. (3) represents a modulated phase, where γτt is the carrier frequency and 2πτtνnl(t) is the modulation term. Though the modulation term is generally not a sinusoidal function, the bandwidth of the beat signal can still be estimated by the Carson bandwidth rule [24,25]
where β is the modulation index or the maximum phase deviation from the carrier phase, and fm is the modulation frequency. The exact expressions of β and fm require detailed knowledge of the phase modulation term. Here, we use simple approximations to derive a concise yet useful expression. For the modulation term 2πτtνnl(t) in Eq. (3), β can be approximated by the root mean square (rms) value of frequency sweep nonlinearity νnl,rms, i.e. β = 2πτtνnl,rms. The residual nonlinearity νnl(t) is typically dominated by the low-frequency component for thermally tuned lasers and ILC does not focus on any specific frequency, therefore fm can be roughly approximated by the lowest frequency corresponding to one ILC period, i.e. fm = 1/(2Tramp). Therefore, Eq. (4) can be written asEven in the case where νnl(t) is not low-frequency dominated, Eq. (5) still provides an estimation of the best achievable resolution. Plugging into Eq. (1), the FMCW LiDAR distance resolution is
where Δν = γTramp is the maximum frequency sweep excursion. When 2πτtνnl,rms <<1, the nonlinearity of frequency sweep is negligible and Eq. (6) reduces towhich is consistent with the regular transform-limited or bandwidth-limited resolution of FMCW LiDAR [26].On the other hand, when 2πτtνnl,rms >>1, the nonlinearity dominates and Eq. (6) gives a nonlinearity-limited range resolution of
The relative nonlinearity of laser frequency sweep can also be represented by the linear regression coefficient of determination r2 = 1 – SSres/SStot, where SSres is the residual sum of squares, and SStot is the total sum of squares [27]. For a measured frequency sweep with N samples, SSres = N(νnl,rms)2 and SStot = N × variance of all samples (VAR). Since the nonlinearity νnl,rms is significantly smaller than the total frequency excursion Δν even before applying ILC linearization, for the purpose of calculating the VAR the frequency samples can be treated as uniformly distributed, i.e. VAR = (Δν)2/12 [28]. Therefore 1 – r2 = 12(νnl,rms/Δν)2. Plug into Eq. (8), the nonlinearity-limited range resolution is
Equation (6) suggests that 1/(2πτt) is a good figure of merit of nonlinearity νnl,rms. If νnl,rms< 1/(2πτt) after ILC pre-distortion linearization, the linearity is sufficient to achieve the transform-limited resolution. This figure of merit also highlights the need to have lower residual nonlinearity for longer range targets.
4. Experimental results of ILC pre-distortion
As a proof of concept, we applied the ILC pre-distortion to a commercial VCSEL (RC32xxx1-FFAimst, Raycan) with 1550 nm wavelength. The optical frequency is thermally tuned by an AC voltage u(t) superimposed on a 3 V DC bias. The beat signal from MZI with relative delay τ = 5 ns is recorded by an analog-to-digital converter (ADC) (PXIe-5114, National Instruments) with 125 MS/s sample rate. The HT and ILC voltage update are processed by a laptop computer, and the laser drive voltage is produced by an arbitrary waveform generator (33250A, Keysight Technologies). The laser drive voltage data is down-sampled to 10 MS/s sample rate when transferred to the arbitrary waveform generator due to the data length and transfer time limitation. The initial input u1(t) and the desired output νd(t) are both triangle waveforms with 4 kHz frequency (Tramp = 125 μs) for both up and down ramps. Each ILC iteration is based on one period (both up and down) of voltage and frequency ramp. Theoretically, the entire frequency sweep will be linearized by ILC. However, in a real system, the transition between the up and down ramps cannot be perfectly recovered due to the limited bandwidth and slew rate of the voltage generator. To minimize the impact of the non-ideal transition, a 100 μs region of interest (ROI) (80% of the ramp) is used for evaluating the linearity.
Experimentally, the peak-to-peak voltage is set to 150 mV, resulting in a laser frequency excursion of Δν = 49 GHz in each ROI. For each iteration, the frequency sweep data is averaged over 1000 periods to reduce noise. A least squares linear fitting is applied to the averaged laser frequency sweep, and (1 – r2) is plotted in Fig. 3(a). It clearly shows that the ILC process gradually improves the sweep linearity, until it saturates after 200 iterations due to the limited resolution of our waveform generator as well as the noises from the laser and the frequency measurement setup. The speed of convergence can be increased by optimizing the p coefficient in the ILC algorithm, which will be discussed later in this paper. The smallest single-ramp nonlinearity νnl,rms among all iterations is 1.5 MHz at the down ramp of the 223rd iteration. According to Eq. (6), the transform-limited resolution is achieved at target distances of up to 16 m, longer than the coherence length of the laser (~10 m). We believe better linearity and longer detection range can be achieved using a laser with a narrower linewidth.
Fig. 3 Experimental results of laser frequency sweep linearization of VCSEL by ILC. (a) residual nonlinearity versus the number of iterations. (b) Laser frequency sweep and the corresponding drive voltage waveforms at the 256th iteration. The ROI is labeled by red color. (c), (d) The down- and up-ramp laser frequency sweeps and residual errors in the ROIs of the 256th iteration. (e), (f) The down- and up-ramp laser frequency sweeps and residual errors of the 1st iteration for comparison.
The linearized frequency sweep and the corresponding drive voltage waveform of the 256th iteration are shown in Fig. 3(b), and the detailed waveforms in the ROIs and their residual errors are shown in Figs. 3(c) and 3(d). For comparison, the initial laser frequency sweep waveforms are plotted in Figs. 3(e) and 3(f). The ILC pre-distortion method reduces the nonlinearity by 259x and 388x, leaving a residual nonlinearity of only 0.0043% and 0.0049% of the frequency excursion in the up and down ramps, respectively. By increasing the amplitude of the drive voltage to 500 mV, frequency sweep with a larger excursion (Δν = 163 GHz) is also successfully linearized by the ILC pre-distortion method with similar performance, as shown in Fig. 4.
Fig. 4 Experimental results of linearized VCSEL frequency sweeps and residual errors of (a) down and (b) up ramps by ILC for 163 GHz frequency excursion.
In order to achieve a stable FMCW LiDAR resolution, it is necessary to test the long-term stability of laser frequency sweep linearity with the pre-distorted, but fixed, voltage waveform. The residual frequency sweep nonlinearity νnl,rms for Δν = 49 GHz is monitored over 2 hours, with 4 samples/min, as shown in Fig. 5. The linearity of both up and down ramps are stable without degradation. This test is performed in the laboratory. As the environment changes or the laser ages, ILC may need to be applied periodically to compensate for the change. In our experiments shown in Fig. 3, an ILC starting from triangle voltage waveform takes about 10 min, mainly limited by the data transfer rate from the ADC to the PC. It potentially can be completed within 1 min with an optimized setup. Moreover, an ILC compensation will not start from the triangle waveform but from an intermediate waveform close to the desired one, thus taking only several seconds to complete.
To demonstrate the effectiveness of the ILC linearization on different types of lasers, it was also applied on a commercial DFB laser (DFB-1550, Optilab) using the same setup. The DFB laser is biased at 200 mA and modulated at 4 kHz rate with ± 70 mA amplitude, resulting in a frequency excursion of approximately 36 GHz. The linearization results are shown in Fig. 6. Slightly better linearity is achieved with the DFB laser.
Fig. 6 Experimental results of linearized DFB laser frequency sweeps and residual errors of (a) down and (b) up ramps by ILC for 36 GHz frequency excursion.
As mentioned in the introduction section, the linear update algorithm, uk+1(t) = uk(t) + p·ek(t) where p is a constant, is used for ILC in this work. Faster convergence can be achieved with a larger p coefficient. On the other hand, small p is necessary to ensure convergence of the ILC process. Figure 7 shows the experimental results with different p coefficients when linearizing the up-ramp of the DFB laser. Here, p is dimensionless as the frequency nonlinearity ek(t) is normalized by the total frequency excursion and the laser drive voltage uk(t) is normalized by its amplitude. The iteration number needed to reach the linearity threshold of 1 – r2 = 6 × 10−8 decreases from 71 to just 5 when p is increased from 0.1 to 1.5. As p increases over 2.5, the ILC algorithm fails to converge. Other advanced algorithms may further reduce the required iteration number.
Fig. 7 Residual nonlinearity of the up-ramp of the DFB laser versus the number of iterations for various values of the p coefficient.
5. FMCW LiDAR imaging results
To demonstrate the transform-limited resolution achieved by ILC laser frequency sweep linearization, single point FMCW LiDAR distance and velocity measurements are first tested using a target mounted on a motorized linear translation stage (UTS100PP, Newport Corporation). The pre-distorted drive voltage ud(t) is applied to the VCSEL. We used a frequency excursion of Δν = 155 GHz in the experiments, corresponding to 0.97 mm transform-limited resolution for distance measurements. The transform-limited velocity measurement resolution is 3.9 mm/s, corresponding to the 0.1 ms ROI length of each ramp. The beat signal is recorded with the same ADC and sample rate (125 MS/s) as in the ILC linearization. Figure 8(a) shows the beat signal spectra measured by one single up or down ramp of laser frequency sweep, where both spectral linewidths are clearly transform-limited. Figure 8(b) shows the LiDAR distance measurement results of a stationary target by one single up or down ramp, which agree with the input displacement of the stage controller within the 0.97 mm resolution. Figure 8(c) shows the LiDAR velocity measurement results with and without averaging, both agree with the input velocity of the stage controller. Averaging over multiple measurements helps increase the resolution.
Fig. 8 Single point FMCW LiDAR ranging results. (a) Beat signal spectra during the up- and down-ramp of the frequency sweep. (b) Stationary target distance measurement results over 100 mm stage displacement with 0.25 mm step. The down-ramp results are artificially offset by 10 mm for clarity. (c) Target velocity measurement results over ± 20 mm/s range with 1 mm/s step.
For FMCW LiDAR 3D imaging experiment, the DFB laser is used because it has higher optical power. The monostatic LiDAR system has a 25.4 mm transmitting/receiving aperture at approximately 3 m distance from the objects. The laser beam is scanned by a 2-axis mechanical scanning galvo system (GVS012, Thorlabs) with a field of view of 24° (azimuth) × 20° (elevation) and 0.02° increments in both axes. The output optical power at the transmitting aperture is 15 mW, and the beam is focused by a lens to approximately 1 mm diameter at 3 m distance. The beat signal is recorded by an ADC and post-processed (Fast Fourier Transform) by a laptop computer, and the point cloud is visualized by CloudCompare software. Figure 9 shows 3D scan results of a stationary scene in the lab. Details of a cup, a stack of books, a toy, a keyboard, a monitor, and the front side of a desk and the wall are recovered with good fidelity. The top of the desk and mouse are not well detected due to the specular reflection and shallow incident angles of the laser beam.
Fig. 9 3D imaging using the FMCW LiDAR. (a) Camera image of the scene (not in the same view angle of LiDAR). (b), (c) Measured 3D point clouds (same point cloud in two view angles) visualized by CloudCompare software with eye-dome lighting [29]. The color of the points represents depth.
6. Conclusion
In this paper, we show that laser frequency linearization by ILC pre-distortion can be successfully applied to FMCW LiDAR and achieve transform-limited resolution of distance and velocity measurement without post-processing. This method eliminates the need for expensive linear lasers, feedback loops for laser linearization or heavy post-processing, thus reducing the complexity and cost of FMCW LiDAR systems, a key requirement for commercial applications such as autonomous vehicles and robotics. We have also introduced a figure of merit for the required residual nonlinearity to achieve transform-limited range resolution.
Funding
Berkeley Sensor and Actuator Center (BSAC).
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