Model mismatch and systematic errors in an optical FMCW distance measurement system

Transcription

1 Model msmatch and systematc errors n an optcal FMCW dstance measurement system ROBERT GROSCHE ept. of Electrcal Engneerng Ruhr-Unverstät Bochum Unverstätsstrasse 50, Bochum GERMANY Abstract: - In ths paper the problem of optcal FMCW (frequency modulated contnuous wave dstance estmaton s consdered. A measurement acquston system s realzed from whch the real data are collected. A smple sgnal model s tested and verfed usng both smulated and measured data. Accordng to ths sgnal model a dstance estmaton procedure s ntroduced. A performance comparson of the dstance measurements obtaned from real and synthetc data reveals that the acheved measurement accuracy of the real system can be precsely predcted from smulatons for the system specfcatons under consderaton and that the dstance sensor s therefore accurately modeled. Further we nvestgate the effects of systematc errors on the dstance estmaton performance n the acquston system. It s shown that systematc errors are neglgble compared to stochastc errors and the varance of the measurements s close to optmal. Key-Words: - FMCW, laser radar, dstance measurement, optcal sensor Introducton Optcal tme-of-flght (TOF dstance measurement systems are used n many ndustral and mltary applcatons. One advantage of optcal systems s ther ablty to obtan measurements wthout contact. In contrast to mcrowave radar systems where the carrer frequency s modulated, n optcal systems the ntensty of the laser beam s modulated. ependng on the modulatng sgnal, pulse- or contnuous wave (CW modulaton, varous methods exst for estmatng the TOF []. Most common for optcal dstance sensors are the pulse method and the CW phase dfference method. In ths paper a FMCW-system (frequency modulated contnuous wave s nvestgated. In contrast to the prevous mentoned methods, FMCW systems are not often used for optcal dstance measurements yet. One advantage of FMCW s ts mult target capablty, what means two or more axal targets can be detected and ther absolute locatons or ther dsplacement can be measured, respectvely. A verfcaton of the theoretcal descrpton for FMCW-systems by the correspondng real measurements has not been prevously accomplshed. In [] an FMCW-system s ntroduced and nvestgated concernng the stochastc dstance estmaton errors. Whle systematc errors are neglected, t s shown that the stochastc error s close to optmal and lmted by the Cramer-Rao bound. In ths paper the determnstc errors n the FMCWsystem are analyzed, that have ther reasons e.g. n frequency dependng hardware components, calbraton errors or fnte samplng effects. In secton the prncple of dstance measurements va FMCW s descrbed. After dervng the sgnal model, the algorthms for dstance estmaton are dscussed. For smplcty the focus of our nvestgatons les on the sngle target scenaro. The data acquston by real measurements wth an expermental setup developed at the Ruhr-Unverstät Bochum as well as by generatng synthetc data s presented n secton 3. Based on the sgnal spectrum the model msmatch s nvestgated. Furthermore, the estmaton bas of both expermental and smulated measurements s compared for seres of measurements and analyzed concernng systematc errors. FMCW dstance measurements TOF dstance measurement systems explot the wellknown relaton between dstance and TOF τ c = n τ ( va the speed of lght c. The factor ½ occurs due to lght propagaton towards the target and back. The refractve ndex n s assumed to be one n ths paper, what s almost correct for measurements n ar. Wth ( the dstance measurement bols down to a tme-delay measurement. ependng on the modulatng sgnal, dfferent algorthms exst to obtan the tme-delay τ.

2 . Sgnal model In ths paper the FMCW method s appled, where a lnear chrp s used as modulatng sgnal. Usng a homodyne detector reduces the bandwdth of the sgnal that has to be measured. Furthermore, the chrp lnearty s of great mportance for the dervaton of the sgnal model. The lnear chrp s descrbed by t s( t = rect exp j π f0t+ π t ( Tc Tc wth the chrp duraton T c, the chrp bandwdth B c and the center frequency f 0. The receved sgnal rt ( = Ast ( τ (3 s tme shfted by τ. The ampltude A of the sgnal depends on the reflecton propertes of the target and s assumed to be constant over the measurement tme. By usng a homodyne detector and a low pass flter, and under the assumpton τ << T c the nstantaneous frequency (IF sgnal gt ( = st ( rt ( m = A cos π τt+ π f0τ π τ (4 = Tc Tc t rect + nt ( Tc s obtaned. Generally, the transmtted sgnal s reflected at m dfferent axal targets. Hence the IF-sgnal s the superposton of multple snusodal functons wth dfferent tme-shft. In (4 n(t descrbes the addtve measurement nose, whch s nevtable n real systems. It s assumed to be Gaussan whte nose wth varance σ n and zero mean. By establshng the nstantaneous frequency f f and the correspondng phase ϕ f ff = τ, ϕf = π f0τ π τ (5 Tc Tc the sgnal model for the IF-sgnal durng the measurement tme can be wrtten as m g( t = A cos( π fft+ ϕ ( f + n t. (6 = The dstance estmaton problem reduces to the classc harmonc retreval problem or estmatng the frequences and correspondng phases of a mult tone sgnal [3]. For sngle tone sgnals one possblty s the estmaton n tme-doman by lnear regresson, as proposed n [4]. Ths estmaton s senstve to chrp nonlneartes and msses the potental of mult-tone estmaton, respectvely. In the case of a nonlnear chrp the frequency of the IF-sgnal s no longer constant over tme and the estmaton problem gets much more complcated. Of course the frequency f f as well as the correspondng phase ϕ f depend on the target dstance. Therefore the estmaton of frequency and phase of the IF-sgnal combned wth (5 and ( yelds smple dstance measurements.. Algorthm As the sgnal model s very smple, the correspondng algorthms should be reduced to a few necessary steps, too. For smplcty the followng algorthm s ntroduced for a sngle tone estmaton. Later t wll be easy to extend for multple tone estmaton. The IF-sgnal for a sngle target scenaro s gven by (6 wth m = : g( t = A cos( π ff t+ ϕf + n( t. (7 Computng the spectrum of the IF-sgnal leads to the frequency f f of g(t at the poston of the maxmum n the magntude of the spectrum. The phase ϕ f s the correspondng phase of the spectrum at f f. The selected estmaton procedure works as follows. To reduce leakage effects, whch occur n the FFT because of the fractonal relaton between the known samplng frequency and the unknown sgnal frequency, t s essental to use a wndow functon [5]. In ths case a hannng-wndow s appled. Furthermore, the Hlberttransform s performed to obtan the complex base-band representaton. In the deal case the sgnal frequency corresponds to the maxmum of the spectrum. Thus, an teratve maxmum search s appled, based on FFT. Interpolaton s used to obtan sub-sample resoluton. The poston of the maxmum gves an estmator for the sgnal frequency f f. The correspondng phase of the spectrum at f f yelds an estmator for the sgnal phase ϕ f. Accordng to the descrbed procedure the estmaton s processed n the followng steps. step : g'( t = g( t wndow functon (8 step : gh = hlbert ( g'( t (9 step 3: G( f = fft ( gh ( t (0 step 4: G( f ˆ f = max ( G( f ( step 5: ˆ ϕ angle ( ( ˆ f = G ff ( Usng (5 and ( two dstance estmators are obtaned. The coarse estmator, based on the frequency estmaton, s ˆ c Tc ˆ coarse = ff (3 whle the fne estmator, based on the phase estmaton, s ˆ c Tc Tc Tc 0 0 ( fne f f f k π B ϕ π =. c (4 The coarse estmator ˆ coarse avods the ambguty problem and s used to determne the unknown nteger k

3 n (4. The advantage of the fne estmator ˆ fne s the smaller varance [6]. 3 System Smulatons and expermental results In the followng secton the data acquston and the estmaton errors are descrbed. The data are collected from a real measurement system as well as synthetcally generated. By comparng the results of the two dfferent estmatons, conclusons about the model msmatch and systematc errors n the estmaton procedure are ponted out. and s chosen accordng to SNR values estmated from the real measurements. Fg. shows the power densty spectrum of the smulated IF-sgnal, whle Fg. 3 shows the power densty spectrum of the measured IF-sgnal. Comparng these two fgures shows well accordance between the two sgnals n frequency doman. In both fgures a sngle maxmum s clearly vsble. We conclude that the measured sgnal s well descrbed by (7 and the synthetc data. 3. ata acquston The realzed dstance measurement system has the followng specfcatons. The chrp generator s based on a fractonal dvder PLL [7]. Its parameters are T c = 3 ms, B c = 00 MHz and f 0 = 750 MHz. The chrp sgnal s splt n a 3-dB coupler. A laser dode wth a wavelength of λ = 635 nm s used n the transmtter. The recever conssts of an avalanche photo dode. The transmtted and the receved sgnal are mxed and low pass fltered. Ths ant-alasng flter has a cut-off frequency of 00 khz. Afterwards the sgnal s dgtalzed wth an analog-to-dgtal converter. Its samplng rate s f s = MHz and t has a resoluton of bt. The recorded measurement data are processed on a PC. MATLAB s used for evaluatng the algorthms and dsplayng the results. Fg. shows the block dagram of ths system. Fg. : Power densty spectrum of the smulated IF-sgnal wth SNR = 0 db, A = 0 mv and f s = MHz Fg. 3: Power densty spectrum of the the measured IF-sgnal wth SNR = 0 db, A = 0 mv and f s = MHz Fg. : Block dagram of the measurement system Several nternal errors are avoded by usng an nternal reference path that s measured after each dstance measurement. The target s placed on a ral wth a stepper motor, whch has a poston accuracy of 0.0 mm. In the smulatons the synthetc data are generated accordng to (7 usng MATLAB wth respect to the parameters of the realzed sensor. The sgnal-to-nose rato (SNR s gven by A SNR = (5 σ n However t remans unclear at ths pont weather there s a model msmatch that results n dfferent postons of the maxmum or devatng correspondng phases for a gven dstance. 3. Error comparson To analyze the performance of the measurement system two knds of errors are dstngushed. Systematc errors, also called determnstc errors, are represented by the estmaton bas e ˆ ˆ = E( (6 and stochastc errors are represented by the varance of the dstance estmaton σ ˆ ˆ ˆ = E( E( (7

4 where E(. denotes the statstcal expectaton, whch s calculated by the mean value of N measurements at one dstance: N = ˆ. (8 N = Thus n the followng the bas s calculated wth eˆ ˆ = (9 The true dstance s known from the poston of the stepper motor. Therefore (9 gves the absolute dstance error. The varance of a seres of measurements at M dfferent dstances s calculated wth M N ˆ σ ˆ ˆ = (. (0 NM j= = (0 shows, that the varance does not depend on the true dstance, but s gven by the devaton of the sngle measurements from ther mean value. Stochastc errors occur due to nose. For a gven SNR the stochastc error s lmted by the Cramer-Rao bound (CRB, whch s a lower bound for the varance of unbased estmators [8]. The performance of the measurement system concernng stochastc errors s nvestgated n []. The CRB of the dstance estmaton for ths sensor s ˆ c CRB( =, ( 4 π f0 fs Tc SNR whch s not dependent on the dstance tself. It s shown that the varance of the dstance estmaton by real measurements, as well as by smulated measurements follows the CRB very accurate, what shows that the estmaton procedure s close to optmal concernng stochastc errors. Whle we showed n secton 3. that the spectrum of the real IF-sgnal s well characterzed by the smulated sgnal, here we nvestgate systematc errors n dstance estmaton. To analyze the appearance of systematc errors the dstance between transmtter and target s vared n steps of cm between 3. m and 4.8 m, so that M = 6. At each dstance 50 measurements are performed, that means N = 50. At frst smulated measurements wth synthetc data are nvestgated. In Fg. 4 the estmaton error e ˆ ˆ vs. target dstance s presented, obtaned by the fne estmaton wth (4. Fg. 4: stance error versus target dstance for smulated measurements wth synthetc data wth SNR = 0 db, 50 averages per dstance; -+- lne s the respectve standard devaton The average dstance error for the smulaton of 50 measurements s approxmately +/ mm. The standard devaton of the entre seres of measurements, calculated wth the square root of (0, s 0. mm. For an equal scenaro a real measurement s accomplshed. The result s shown n Fg. 5. Fg. 5: stance error versus target dstance for real measurements wth SNR = 0 db, 50 averages per dstance; -+- lne s the respectve standard devaton For the real measurements the bas s about +/- 0. mm and the standard devaton s 0.34 mm. The bas has a perodc nature caused by systematc errors. Wth the results of Fg. 4 we conclude, that the algorthm does not produce a sgnfcant bas. The estmaton error decreases wth an ncreasng number of averages. Therefore, the bas n Fg. 5 must be an effect of the system hardware. A possble reason s a slghtly nonlnear chrp. Otherwse the standard devatons of both smulated and real measurements yelds good agreement wth the correspondng CRB as demonstrated n Tab.. standard devaton smulated measurements real measurements Cramer- Rao bound 0.0 mm 0.34 mm 0.6 mm Tab. : Standard devaton of the dstance measurements

5 As a result of these studes the estmaton bas of the real measurements s smaller than the respectve standard devaton. Thus the bas does not nfluence the measurement accuracy, f no averagng s appled. In consequence a model msmatch exsts between (7 and the IF-sgnal measured wth the real system. On the other hand t s neglgble for the consdered settngs, especally for SNR = 0 db. Therefore, the smple sgnal model ntroduced n secton. well descrbes the measured sgnal. In addton the proposed dstance estmaton procedure does not affect the estmaton accuracy. Only for hgher SNR a sgnfcant model msmatch exsts, so that n ths case the model must be adjusted or the system hardware must be revsed n order to reduce the systematc error even more. [6]. C. Rfe, R. R. Boorstyn, Sngle-Tone Parameter Estmaton from screte-tme Observatons, IEEE Trans. on Informaton Theory, Vol. t-0, No. 5, 974, pp [7] T. Musch, I Rolfes, B Schek, A Hghly Lnear Frequency Ramp Gernerator Based on a Fractonal vder Phase-Locked-Loop, IEEE Trans. on Instrumentaton And Measuremen, Vol. 48, No., 999, pp [8] A. H. Quaz, An overvew on the tme delay estmate n actve and passve systems for target localzaton, IEEE Trans. on Acoustcs, Speech, Sgnal Processng, Vol. 9, No. 3, 98, pp Concluson In ths paper a FMCW system s developed as a possble alternatve soluton to the most common systems for optcal dstance measurements. A sensor based on the FMCW-method has been nvestgated both from smulated and real measurements. The paper descrbes the smple sgnal model for whch a dstance estmaton procedure s proposed. The FMCW-method extracts the dstance nformaton from the nstantaneous frequency sgnal n frequency doman. Smulaton results obtaned from synthetc data generated accordng to the sgnal model are compared to measurements recorded from a real FMCW dstance sensor. For both cases, the measurement accuracy s analyzed. It s shown that the used sgnal model accurately represents the system for the settngs n consderaton. In ths dstance sensor systematc errors are neglgble due to proper desgn of hardware and algorthms and stochastc errors yeld measurement varance close to optmal. References: [] J. M. Rüeger, Electronc stance Measurement, Sprnger, 996 [] R. Grosche, Performance analyss of optcal FMCW- Radar systems, Proceedngs of OIMAP IV, 4 th Top. Meetng on Optoelectronc stance/splacement Measurement and Applcatons, Oulu, Fnnland, June , pp [3] P. Stoca, R. Moses, Introducton to Spectral Analyss, Prentce Hall, 997 [4] S. A. Tretter, Estmatng the Frequency of a Nosy Snusod by Lnear Regresson, IEEE Trans. on Informaton Theory, Vol. t-3, No. 6, 985, pp [5] E. O. Brgham, FFT-Anwendungen, Oldenbourg, 997