Xiao Ai,* Richard Nock, John G. Rarity, and Naim DahnounDepartment of Electrical and Electronic Engineering, University of Bristol,Merchant Venturers Building, Woodland Road, BS8 1UB, Bristol, UK
Lidar systems can be divided into two categories:(i) pulsed time-of-flight, in which the distance is cal-culated by the elapsed time between the transmittedand received pulses and (ii) continuous wave (CW)time-of-flight, in which distance is evaluated usingeither the phase delay or frequency change of thebackscattered continuous radiation. The cost androbustness of the pulsed methods are limited by thehigh peak power of the pulsed laser and the fast re-ceiver response requirements, thus CW methods areoften preferred for low-cost applications. Among ex-isting CW lidar techniques, the random-modulationcontinuous wave (RM-CW) lidar systems have someunique benefits compared to other techniques suchas phase shifting, heterodyne, and frequency-modulated CW [1]. These benefits include long un-ambiguous range, low mutual interference betweendevices, and the ability to resolve multiple targetsin a single measurement.
In the RM-CW lidar, a pseudorandom binary se-quence (PRBS) is transmitted, backscattered bythe target, and received as an attenuated and de-layed sequence, the level of attenuation following
the inverse square law to the distance. The delaytime used to calculate the target distance is evalu-ated by observing the peak in the digital cross-correlation function (CCF) between the receivedsequence and delayed versions of the transmitted se-quence. Previous studies [2–6] have demonstratedthe RM-CW lidar systems over kilometer ranges. Forthe far limits of the desired detection range undervery low signal-to-noise ratio (SNR), they are typi-cally designed for a resolution of meters.
Conventional thinking says that the resolution ofthe RM-CW lidars is limited by the analog-to-digitalconverter (ADC) sampling rate and the PRBS bitwidth (chip time). For instance, a chip time of 40nsand an ADC operating at a 25MHz sampling ratewould normally produce a system with resolutionlimited to 6m. Here, we show that under high SNRassumption for medium-range (<200m) operations,this resolution limitation can be reduced by ordersof magnitude (from meters to centimeters) using asimple resolution enhancement technique. The re-sultant resolution is only limited by the quantumshot noise and bandwidth of the receiver. In previousstudies [7–11], some RM-CW based systems withcentimeter resolution have been developed throughvarious techniques. However, they have somelimitations such as long acquisition time, lack of
parallelism, high signal processing complexity, orhigh cost.
Our proposed technique is based on the originalRM-CW lidar principle with the addition of ananalog integrator installed prior to the digital cross-correlation process. Therefore, we call this mixedrandom-modulation continuous wave (MRMCW)lidar. The basic operating principle of MRMCW isillustrated in Fig. 1 and compared to the originalRM-CW. For the RM-CW technique, the transmittedsignal is synchronized to the clock, it is transmittedto, and reflected by the target incurring a delay. Adigital cross-correlation on the transmitted and re-ceived signals yields a delta function with a resolu-tion of one chip time. For the MRMCW technique thereceived signal is integrated and sampled on eachclock rising edge leading to a digital CCF which isa set of samples on a triangle. Thus, one can then ex-tract the distance information by interpolation, lead-ing to a high distance resolution not limited by theADC sampling rate. Interpolation can be performedeither geometrically or by optimization techniques.With the gradient descent optimization technique,multiple nearby targets can be detected.
With centimeter resolution, the MRMCW lidarcould be applied to numerous medium-range applica-tions. One example is automotive collision avoidancesystems [12] that warn the driver when the vehicle ison a collision course with other vehicles or obstacles.Such systems need to sense the environment in thethird dimension with centimeter accuracy in millise-cond integration time (to reduce range smearing attypical approach speeds). It is also essential thatthere is low interference between systems mountedon different vehicles. It could also be applied in three-dimensional (3D) multispectral imaging [13], wherea series of lasers at different wavelengths is modu-lated with unique PRBSs, thus allowing a single re-ceiver (RX) to decode all the spectral information and
distance to the targets. Another potential applicationcould be inter-spacecraft ranging as demonstratedin [8].
The paper is organized as follows. Section 2 de-scribes the MRMCW lidar principle and presentsits mathematical model. In Section 3, interpolationmethods used for distance calculation are explainedand the distance accuracy is calculated. Some reali-zation details are discussed in Section 4. Section 5describes the demonstrator developed in our labora-tory and presents the analytical and experimentalresults. Section 6 concludes the paper.
2. MRMCW Lidar Principle
The distance from the transmitter to the target canbe calculated by the single-trip time delay multipliedby the speed of light. It is obvious that an improvedtime resolution would lead to a better distanceresolution. In the MRMCW lidar developed, thetime delay is estimated accurately by interpolation.Therefore, a low-cost ADC can be used to achievehigh resolution. The schematic of the MRMCW lidarprinciple is shown in Fig. 2, in which the transmitted(TX) laser beam is modulated with the PRBS gener-ated by the linear feedback shift register (LFSR),which is then backscattered by the target. Thereceived light is delayed and attenuated is then col-lected by the RX. The received signal is integrated bythe free running integrator (INT). Then the inte-grated signal is digitized by the ADC, which thenforms the integral sequence. The digital cross-correlation (XCORR) carried out between theintegral sequence and the original PRBS results insamples on a triangle with its peak position indicat-ing the round-trip time delay. Hence, the distance tothe target can be calculated by interpolating thedigital CCF. The proposed system is described follow-ing the guidelines of [6,11] for a comparison.
Among several popular PRBSs, the M-sequence isselected due to its well-defined triangular autocorre-lation property [11] which allows for a piecewise lin-ear interpolation. An nth order M-sequence can begenerated by an n-stage LFSR producing a PRBSwithN ¼ 2n − 1 bits. AM-sequence signal in the time
Fig. 1. Principle of traditional RM-CW systems and the proposedMRMCW technique. Accuracy of the RM-CW system is given bythe period of the system clock, and MRMCWexhibits subclock per-iod accuracy via a process of interpolation on the digital CCF.
Fig. 2. Schematic of the MRMCW lidar principle. See text in firstparagraph of Section 2 for component denomination.
domain aðtÞ can then be represented by the summa-tion of a series of step functions ψðtÞ [6].
aðtÞ ¼XNi¼1
aiψðt − iTcÞ; ð1Þ
in which
ψðtÞ ¼�1 0 ≤ t < Tc
0 otherwise ; ð2Þ
where ai is the bipolar binary M-sequence, in whicheach bit takes the value of 1 or −1 with a bit duration(chip time) of Tc. A 15 bit M-sequence is depictedin Fig. 3.
The autocorrelation function (ACF) Rðt0Þ is illu-strated in Fig. 4. For a long sequence, it can beapproximated with [11]
Rðt0Þ ¼Z
NTc
0aðtÞaðtþ t0Þdt; ð3Þ
≈
(NTc
����1 −t0Tc
���� L− < t0 < Lþ
0 otherwise; ð4Þ
which denotes a triangle centered around the originwith the width of 2Tc and period of NTc; the lowerbound is L− ¼ TcðNl − 1Þ and the upper bound isLþ ¼ TcðNlþ 1Þ, l ∈ Z. The use of a long PRBS is as-sumed, making it plausible to approximate the offsetto zero which is otherwise −Tc. The ACF is normal-ized to R̂ðt0Þ ¼ 1=ðNTcÞRðt0Þ for notation consistencyin further equations.
In the MRMCW lidar, the transmission beam ismodulated with the M-sequence (on-off rectangularpulses) represented by PtðtÞ ¼ P0½aðtÞ þ 1�, in whichP0 is the laser emitted average power (approximatelyhalf the peak power).
The receiving light PsðtÞ is attenuated and delayedby τ given as
PsðtÞ ¼ GPtðt − τÞ: ð5Þ
Assuming Lambertian-surface targets (followsLambert cosine law) and the laser spot is smallerthen the field-of-view of the detector, the attenuationratio G defining the percentage of received light fromtransmitted light due to the fixed light collecting op-tics size follows the inverse square law with respectto the distance. G can be represented by Eq. (6),which is simplified from the Eq. (11) in [6]:
G ¼ ηtηrα cos θYrAr
πD2 ; ð6Þ
where ηt and ηr are the optical efficiencies of thetransmitter and RX; α represents the target Albedo;θ is the laser beam incidence angle to the target; Yr isthe range dependent overlap function, which is de-fined by the fraction of the laser beam covered bythe RX field-of-view, dependent on the optical setup;Ar denotes the RX aperture area; D is the distance tothe target.
The ambient light Pb, which introduces additionalshot noise to the system, can be described by Eq. (7),which is modified from Eq. (15) in [6]:
Pb ¼ ηrαEλω tan2 βAr; ð7Þ
where Eλ is the ambient light spectral irradiance, ωdenotes the bandwidth of the optical filter prior tothe detector, and β represents half of the angle ofview of the RX.
The total received optical power at the detector isthus PrðtÞ ¼ PsðtÞ þ Pb. The integral sequence Ii ofPrðtÞ sampled at Ts is then defined byFig. 3. 15bit M-sequence. Tc denotes the chip time.
Fig. 4. 15bit M-sequence ACF. NTc represents the totalintegration time; the offset is −Tc, negligible only for long PRBS.
The digital CCFCn between the sequence Ii and anoversampling version of the transmittedM-sequenceAi is described by Eq. (9) derived in Appendix A.
Cn ¼XrcsNi¼1
IiAi−n ≈ EsR̂ðnTs − τÞ; ð9Þ
in which the maximum received signal energy is
Es ¼ GP0NTc; ð10Þ
where the oversampling factor rcs is equal to thechip time to the sampling time factor Tc=Ts, forinstance, rcs ¼ 2 and a ¼ ½−1; 1; 1� result in A ¼½−1;−1; 1; 1; 1; 1�. In this paper, rcs is selected to be2 as increasing this would result in more Cn samples,which would improve the interpolation accuracy atthe expense of a higher sampling rate ADC being re-quired, which would in turn increase overall systemcost. The resultant Cn contains a shifted and attenu-ated version of the ACF [Eq. (4)]. Therefore, Cn ismade of samples on a triangle.
Assuming an ideal interpolation of Cn leads to atriangle Cðt0Þ defined by
Cðt0Þ ¼ EsR̂ðt0 − τÞ; ð11Þ
where the peak of the triangle indicates the round-triptime delay τ leading to the distance D to the target
D ¼ 12cτ: ð12Þ
In the MRMCW lidar, the statistics of the detectedsignal photons and the ambient photons will follow aPoissonian distribution (assuming a shot-noise lim-ited system). Hence, The maximum SNR SNRmaxor SNRðt0 ¼ τÞ is determined by
SNRmax ¼ξEs
½μeξðEs þ EbÞ�1=2¼
� ξμe
�12 Es
ðEs þ EbÞ12;
ð13Þ
in which the conversion coefficient ξ from energy tophoton number is
ξ ¼ ðhf Þ−1; ð14Þ
and the ambient light energy Eb collected at thetarget is
Eb ¼ PbNTc; ð15Þ
where μe is the excess noise factor of the avalanchephotodiode (APD) detector; h represents Planck con-stant; f is the frequency of light.
3. Interpolation Methods and Distance AccuracyAnalysis
Similar to the interpolating time-interval meter [14],which utilizes interpolation to evaluate the time-of-arrival, the MRMCW lidar achieves high resolu-tion by interpolation in the correlation domain. Inthis section, we discuss two methods of interpolation:(i) a geometry based method presented by [11], whichutilizes the triangular shape of the CCF and (ii) acurve fitting based method using the gradientdescent technique.
As discussed, the digital CCF Cn is made of sam-ples on a triangle. With rcs ¼ 2, utilizing the trianglegeometry, the round-trip time delay τ can be deter-mined by Eq. (16) which is illustrated in Fig. 5:
τ ¼ ðm − 1ÞTs þCmþ1
Cm−1 þ Cmþ1Tc; ð16Þ
wherem is the sample number when the digital CCFis maximal, and Cm−1 and Cmþ1 are samples aroundthe maximum.
The exact distance standard deviation σD iscomplex. An approximation analysis shown inAppendix B simplifies it to
σD ≈
ffiffiffi2
p
4cTc
SNRmax: ð17Þ
Combining Eqs. (13) and (17) and assuming Yr ¼ 1for simplicity (transmission and receiving opticscompletely overlapped), results in
σD ≈
�πhc2D2
8αfμeTc
cos θηrηtArNP0
�1
þ πEλω tan2 βD2
cos θηtP0
��1=2
: ð18Þ
Fig. 5. CCF. m represents the sample where the digital CCF ismaximal, τ denotes the round-trip time delay, Cm−1 and Cmþ aresamples around the maximum, Cmax ¼ Cðt0 ¼ τÞwill be referencedin Appendix B.
This equation shows the dependence of the standarddeviation and the systems parameters. One canobserve that in order to reduce σD, thus improvingthe measurement accuracy, the system parametersrequired to maximize are: ηr, ηt, Ar, N, and P0.Conversely, those that are required to minimizeare: f , μe, Tc, ω, and β.
Problems with the geometry-based technique arisewhen measuring multiple closely spaced targets. Forinstance, when the laser beam spot is at the bound-ary of two targets with different depths, a mixture oftwo triangles will be observed in the CCF. As illu-strated in Fig. 6, when two targets are spaced morethan cTc apart, they can be individually resolved bythe geometric method. When the two targets arespaced less than cTc away from each other, their re-spective triangles cannot be identified, as illustratedin Fig. 7. The gradient descent technique [15] isselected to provide the solution.
The gradient descent technique searches for themost suitable values of the parameters Es and τ thatminimize the cost function Ω, which quantifies thedifference between the sampled cross-correlationSn and the mathematically modeled values CnðEs; τÞ.The cost function is represented by
Ω ¼XNn¼1
½CnðEs; τÞ − Sn�2: ð19Þ
The gradient descent is an iterative technique. De-noting the estimating parameters as a column vectorxk ¼ ½Es; τ�T , where k is the iteration step. In eachiteration, xk moves along the local gradient directionof the cost function with the step length representedby Γ. The process is terminated when an acceptableerror is reached. It is graphically illustrated in Fig. 8,and the process is described by
xkþ1 ¼ xk − Γ∇ΩðxkÞ: ð20ÞWhen Cn is extended to Eq. (21), a second target
return is included. In this case, four parameters xk ¼½Es0; τ0;Es1; τ1�T are subsequently estimated usingthe same process, leading to two individually re-solved round-trip time delays, thus distances to thetargets.
Cn ¼ Es0R̂ðnTs þ τ0Þ þ Es1R̂ðnTs þ τ1Þ: ð21Þ
4. Realization Issues of the MRMCW Lidar
In previous sections, the MRMCW lidar was analyzedtheoretically. Through preliminary experiments, itwas observed that the integrator was prone to satura-tion, due to the high low-frequency gain of the integra-tor and the large low-frequency components of thereceived PRBS signal. A traditional integrator designwould typically include a reset mechanism to preventthe saturation. However, information is lost duringthe reset process. In our system a square wave (sym-metric about zero volts representing�1) with a period
Fig. 6. Digital CCF of two spaced targets. The dark dashed tracerepresents the first surface samples; the light dashed trace repre-sents the second surface samples.
Fig. 7. Digital CCF of two nearby targets. The dark dashed tracerepresents the first surface samples; the light dashed trace repre-sents the second surface samples.
Fig. 8. Gradient descent technique. The minimum of the costfunction Ω is approached iteratively, in this example, initiallyx0 ¼ ½2; 0�T , at the 100th iteration x100 ¼ ½1; 4:5�T , results in a mini-mized cost function with an error of 2 × 10−6.
of a chip time is mixed with (multiplies) the receivedsignal prior to the integrator. This shifts the receivedsignal spectrum higher, effectively moving the nearDC components upward to prevent saturation. For re-covery, if a signal is multiplied by −1 at the mixer inthe analog domain, it is subsequentlymultiplied by −1in the digital domain. After this process, the currentlysampled ADC data is subtracted from the previousone to obtain the sectional integral for each of the halfchip time wide slices. Hence, for our system, a resetmechanism is not needed.
As shown in Fig. 9, the voltage range of the inte-gration is much smaller when the mixer is applied,noting that it is essential to match the ADC samplingtrigger with the transition edge of the square wave.In the demonstrator, the FPGA I/O delay primitiveswere used to achieve an adjustable delay in steps of78 ps, otherwise the sectional integral will be inaccu-rate, leading to a skewed correlation triangle.
In traditional RM-CW lidar systems [2–6], thecross-correlation operation is postprocessed on per-sonal computers. In the case of long PRBSs, evenwith the help of the fast Fourier transform [16], thecomputational cost and memory requirement arestill high. Because of the targeted 100m maximumdistance of the MRMCW lidar, a FPGA based real-time (synchronizing to the signal reception) cross-correlator can be used. Its schematic is illustratedin Fig. 10, which is modified from the cross-correlatorproposed in [17]. It is highly efficient compared to thetraditional methods, for instance, with a maximumdistance of 200m, a round-trip time delay of 1:3 μs,using the chip time of 40ns, and the chip time to sam-pling time ratio of 2, a 68-tap correlator (68 pairs ofmixer and accumulators) is required. Furthermore,the ADC samples are only multiplied by 1 and −1.Therefore, the multiplications are reduced to two’scomplement negation operators.
5. MRMCW Lidar Demonstrator and Results
A low-cost and real-time MRMCW lidar demonstra-tor has been designed and constructed for validation.The experiments were carried out in our laboratoryunder fluorescent lighting conditions (ambient lightspectral irradiance approximated to be 1Wm−2 nm−1
at 635nm). Because of laboratory space restrictions,the maximum range evaluated is limited to 12m.The applied components are shown in Fig. 11, withthe layout diagram shown in Fig. 12 and componentslisted in Table 1.
The transmitter consists of two visible semi-conductor laser diodes (LA1 and LA2), with λ ¼635nm. Both laser diodes are modulated by varyingthe drive current, with each laser diode transmittinga different PRBS (PRBS1 andPRBS2), one for the dis-tance measurement and the other one being used tointroduce an interference signal for mutual interfer-ence characterization. The modulation currents arebetween 21 (logic 0) and 36mA (logic 1), with 20mAbeing the lasers threshold current. Therefore, the la-ser is always on, this is to avoid excessive on-timeshrinking due to the time the laser diode takes to lasefrom spontaneous emission, also this reduces thelaser diode relaxation oscillation. The transition be-tween current levels has a rise time of approximately
Fig. 9. Integrator output voltage range with amixer applied priorto the ADC, shown as the solid trace which exhibits a small voltagerange. Without the mixer is shown as the dashed trace which ex-hibits large voltage range. The mixer stops potential integrator sa-turation and reduces the dynamic range requirement of the ADC.
Fig. 10. Block diagram of the XCORR and the M-sequence gen-erator (LFSR). Seven taps with the digital CCF are illustrated.
Fig. 11. Assembly of the MRMCW demonstrator, showing the(1) APD module, (2) LNA, (3) mixer, (4) integrator, (5) FPGA boardwith an (6) ADC board underneath the FPGA, (7) laser drivers,(8, 9) laser modules, and (10) lens tube RX.
2ns which indicates a transmission bandwidth of170MHz [0:35=ð2nsÞ]. The laser beams are colli-mated (divergence of 0:4mR) and have an averagetransmission power of 1:4mW. Beams are aimed atzero incidence angle to a white diffusive screen with90% albedo. The modulation chip time is set to 40ns.
The RX is a 1 in:. (1 in: ¼ 2:54 cm) lens tube enclos-ing a convex lens (L1) with a focal length of 100mmand an optical bandpass filter (F) with 5nm FWHM,centered to the laser wavelength to filter out theambient light. The returned light is then coupled toa multimode fiber with a core diameter of 600 μmwhich is connected to the APD module (APD). TheAPD module has a bandwidth of 120MHz keepingspectral content up to the 5th harmonic of thereceived PRBS waveform, which enables the system
to maintain a fairly triangular CCF. The signal isthen amplified by a low-noise amplifier (LNA)matching the APD module bandwidth. Aiming toprevent the integrator from saturation, a 25MHz(40ns period) square wave is mixed with the ampli-fied signal. The mixed signal is then integrated by anINT. The INT is constructed via the use of an opera-tional amplifier with an operational frequency rangeof 100KHz–250MHz. This is wide enough to containthe spectral contents of the signal (120MHz band-width of the APD plus 25MHz frequency up-shiftdue to the mixing process).
The 12 bit ADC digitizes the integrated signal at50MS=s, which is synchronized to the mixer and istriggered by the FPGA. Decreasing the ADC resolu-tion would reduce the dynamic range of the RX,which would limit the maximum and minimum op-erative range of the device. Increasing the samplingrate would not increase the distance accuracy signif-icantly, because the accuracy is provided by the ana-log principle. However, as shown in Eq. (18), areduction in chip time combined with increasing thesampling rate will improve the distance resolution,but this sets higher requirements on the APD andintegrator bandwidth.
In the FPGA, a 16-stage LFSR is implemented togenerate an 65; 535bit PRBS. The FPGA alsocontrols the laser driver and executes the cross-correlation between the sampled signal and the de-layed versions of the PRBS. In principle, the distancecalculation can also be implemented in the FPGA.However, for experimental convenience this task isoff-loaded to the computer via a serial port. The sys-tem parameters can be configured via a serial portcontrolled by the personal computers’s side software.The distance can be calculated either by the geome-trical method or the gradient descent technique.
To calibrate out the propagation delay through theelectronic components, a measurement performed at1m was used as a reference. Measurements are thencarried out by placing the target screen at variouslocations within the 12m range from the lidar re-stricted by the laboratory space. For each distancemeasurement, 200 trials per location are carriedout, with each trial having a 2:6ms acquisition time.These datasets are used to obtain themean distancesas shown in Fig. 13 and the distance standard devia-tions as shown in Fig. 14.
Also shown in Fig. 14 is a set of distance standarddeviations with mutual interference from the second-ary laser beam (the same received power as the mea-surement laser beam at the detector), which hasdemonstrated the MRMCW lidar’s ability to copewith mutual interference well. No major distance er-rors was observed in the measurements, becausePRBS1 and PRBS2 have a very low cross-correlation.However, the distance standard deviation deterio-rates slightly, mainly due to the shot noise of PRBS2being added to the system. This effect can be calcu-lated using Eq. (18), where Pb now needs to considerthe average received energy of the interference laser.
In Fig. 15, the shape of the CCF was obtained byfixing the target at 4m, then applying a time delayfor the ADC sampling trigger in the range of 0–130ns(to include the whole triangle) with 1ns steps. Thefollowing nonlinear defects can be observed: (i) thefinite bandwidth of the APD (120MHz) results inrounded edges and (ii) the area outside the triangleis asymmetric, with the area to the right hand side ofthe triangle being nonzero due to the nonlinearity ofthe detector’s low-frequency phase response. As pre-viously discussed, both geometrical and curve fittingdistance evaluation assume that the CCF is triangu-lar. Nonlinearity of the CCF results in a centimeterlevel distance nonlinearity, as shown in Fig. 16. Thisnonlinearity is independent of amplitude, and thuscan be calibrated out via a look-up table. However,temperature variation may influence the bandwidth
of the APD leading to a centimeter level nonsyste-matic error.
When two hard targets are present in the scenery,each object will create a triangular shape in the CCF.Assuming the targets are far (≥cTc) apart such thatthe triangles are not merged, the distance to theseobjects can be calculated via the geometric method.Otherwise, if objects are nearby (<cTc), the gradientdescent technique can be applied. An example isdemonstrated in Fig. 17, in which the solid trace dis-plays a digital CCF and contains two merged trian-gles produced by targets located at 369 and 1292 cmaway from the lidar. Their single-trip time delays arecalculated to be 12:8ns and 42:6ns, which are 29:2nsin separation (≤Tc). Utilizing the gradient descenttechnique, we have calculated their distances to be385 and 1278 cm, respectively.
Fig. 13. MRMCW distance measurements are compared with anindustrial laser range finder (BOSCH DLE 70 with 1mm distanceaccuracy). The white diffusive screen is placed in various locations,distances are measured with a 0° incidence angle, laboratory fluor-escent lighting conditions, and an acquisition time of 2:6ms.
Fig. 14. Distance standard deviation with mutual interference isshown as the light dashed trace. Without mutual interference isshown as the dark dashed trace. Theoretical shot noise limitedstandard deviation evaluated by Eq. (18) with andwithout the con-sideration of the overlap function (obtained experimentally) areshown as the solid trace and dashed trace.
Fig. 15. CCF with respect to the ADC trigger delay. The mea-sured result is shown as the solid trace which is obtained by delay-ing the ADC sampling trigger by 0–130ns in 1ns steps. Thetheoretical triangle is shown as the dashed trace obtained byEq. (11).
Fig. 16. Distance nonlinearity due to distortion in the CCF.
In this paper, a RM-CW based lidar aiming toovercome the resolution limitation restricted by theADC sampling rate has been demonstrated. In theproposed (MRMCW) lidar system, the enhanced dis-tance resolution is achieved by utilizing an integra-tor prior to the ADC and an interpolation process onthe digital CCF. Mathematical models are used toshow the MRMCW lidar principle and to derivethe theoretical distance accuracy. For validation, ademonstrator was designed and built in our labora-tory. The experimental distance standard deviationcorresponds well with the theoretical prediction,proving the system is shot noise limited with the sys-tem exhibiting accuracy of less than 2 cm up to 12meters. Such a resolution would require a 10GHzADC and a suitably wideband APD for conventionalRM-CW lidar systems. Hence, the proposed techni-que offers considerable cost advantages in compari-son to the RM-CW lidar systems for medium-rangesurface detection. Restricted by the current systemsetup, we did not evaluate the system in an outdoorenvironment, hence atmospheric attenuation factorswere not evaluated; restricted by laboratory size, themaximummeasured distance is limited to 12 meters.Under rainy and cloudy environments, the trianglecorrelation waveform may be distorted, thus the pro-posed technique is best suited to surface detection ina clear atmosphere.
The low mutual interference capability was veri-fied by introducing an interference beam modulatedwith a secondary PRBS, which only reduced the dis-tance accuracy by 0:8 cm at 12 meters. We have alsoobserved centimeter level distance nonlinearity,which was caused by phase nonlinearity and band-
width limitation of the RX. However, in principle,this could be calibrated with a look-up table. Further-more, preliminary experiments showed that multiplenearby targets could be resolved to 10 s of centimeterprecision by using the gradient descent technique.The proposed MRMCW lidar has the benefits ofhigh resolution, real time, and low cost, making itplausible for its applications in automotive collisionavoidance, multispectral imaging, inter spacecraftranging, etc.
The range accuracy of the MRMCW lidar system isshot-noise limited to the ambient light level and la-ser power. The usefulness of the proposed method re-sides under the assumption of high SNR. Accordingto Eq. (18), for the demonstrator to operate at 200mfrom the target, the distance accuracy for a white tar-get would exhibit an accuracy of 3m. In comparisonto the to the intrinsic RM-CW resolution of 3m (20nschip time and 50MHz ADC), there is no resolutiongain by using the proposed method on the demon-strator above 200m.
The current demonstrator was designed for theoryvalidation, it could be improved in a number of ways:first, the chip time could be reduced further to in-crease the distance accuracy. Second, a linear phaseamplifier could be applied which reduces the low-frequency phase nonlinearity of the RX. Third, thebandwidth of the APD and integrator could be im-proved. Lastly, a compact scanning system couldbe applied to achieve 3D imaging.
Appendix A: CCF Analysis
After the integrator, the ADC digitize the integratedsignal forming the integral sequence Ii, which isrepresented by
Ii ¼Z ðiþ1ÞTs
iTs
PrðtÞdt; ðA1Þ
thus equivalent to
Ii ¼Z
rscNTs
0PrðtÞψðt − iTsÞdt: ðA2Þ
Cn is described by
Cn ¼XrscNi¼1
IiAi−n: ðA3Þ
The substitution of Eq. (A2) into Eq. (A3) leads to
Cn ¼XrcsNi¼1
�ZrcsNTs
0PrðtÞψðt − iTsÞdt · Ai−n
�; ðA4Þ
in which, the integration and summation are finite.Hence, it can be rearranged to
Fig. 17. Preliminary experimental result for the gradient descenttechnique. The measured digital CCF shown as the solid trace areestimated as circles, which are a combination of the dark dashedtrace representing the first surface samples and the light dashedtrace representing the second surface samples. The targetsare placed at 369 cm and 1292 cm away from the lidar, trianglesare separated with calculated results of 385 cm and 1280 cm,respectively.
When a long PRBS (balanced sequence) is used,the second integral can be eliminated. PsðtÞ ¼GP0aðt − τÞ leading to
Cn ≈ GP0
ZNTc
0aðt − τÞaðt − nTsÞdt; ðA7Þ
with Es ¼ GP0NTc.
Cn ≈ EsR̂ðτ − nTsÞ; ðA8Þ
because R̂ is symmetrical to zero time
Cn ≈ EsR̂ðnTs − τÞ: ðA9Þ
Appendix B: Distance Standard DeviationApproximation
As discussed in Section 3,
τ ¼ ðm − 1ÞTs þCmþ1
Cm−1 þ Cmþ1Tc; ðB1Þ
approximating the Poisson distributed errors of Cm−1and Cmþ1 to Gaussian errors enables us to calculatethe round-trip time delay standard deviation στusing the Gaussian error propagation denoted by
στ ¼ ½ðδCm−1τÞ2σ2Cm−1
þ ðδCmþ1τÞ2σ2Cmþ1
�1=2; ðB2Þ
where the partial derivative of τ (δCm−1) w.r.t Cm−1 is
calculated as
δCm−1τ ¼ −
Cmþ1Tc
ðCm−1 þ Cmþ1Þ2; ðB3Þ
and the partial derivative of τ (δCmþ1) w.r.t Cmþ1 as
δCmþ1τ ¼ Tc
Cm−1 þ Cmþ1−
Cmþ1Tc
ðCm−1 þ Cmþ1Þ2: ðB4Þ
Assuming the magnitude of both samples around thepeak is equal to half of the CCF peak (Cm−1 ¼Cmþ1 ¼ Cmax=2), where Cmax ¼ Cðt0 ¼ τÞ. The stan-
dard deviation of the samples on the digital CCFis typically dominated by ambient shot noise ratherthan signal shot noise (because Eb ≫ Es), thus wehave σCm−1
¼ σCmþ1¼ σCmax
, where σCmax¼ σCðt0 ¼ τÞ,
hence, a simplified round-trip time standard devia-tion στ is given by
στ ≈ffiffiffi2
p
2Tc
σCmax
Cmax≈
ffiffiffi2
p
2Tc
SNRmax; ðB5Þ
thus the distance standard deviation σD can beapproximated as
σD ≈
ffiffiffi2
p
4cTc
SNRmax: ðB6Þ
This minimum bound was selected as the theoreticaldistance standard deviation. Following the same cal-culation, when Cm−1 ¼ 0, Cmþ1 ¼ Cmax, and σCm−1
¼σCmþ1
¼ σCmax, a larger approximation will be obtained
as shown in Eq. (B7).
σD ≈cTc
2SNRmax: ðB7Þ
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