VIBRATION PARAMETER ESTIMATION USING FMCW RADAR. Lei Ding, Murtaza Ali, Sujeet Patole and Anand Dabak

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1 VIBRATION PARAMETER ESTIMATION USING FMCW RADAR Lei Ding, Murtaza Ali, Sujeet Patole and Anand Dabak Texas Instruments, Dallas, Texas University of Texas at Dallas, Rihardson, Texas ABSTRACT Vibration sensing is essential in many appliations. Traditional vibration sensors are ontat based. With the advane of low-ost and highly integrated CMOS radars, another lass of non-ontat vibration sensors is emerging. In this paper, we present detailed analysis on obtaining vibration parameters using frequeny modulated ontinuous wave (FMCW) radars. We establish the Cramer Rao lower bounds () of the parameter estimation problem and propose an estimation algorithm that ahieves the bounds in simulations. These analysis show that vibration sensing using FMCW radars an easily ahieve sub-hertz frequeny auray and mirometer level amplitude auray. Index Terms frequeny modulated ontinuous wave (FMCW) radars, vibration sensors, Cramer Rao lower bound, parameter estimation. 1. INTRODUCTION Vibration monitoring and analysis is an important tool in many appliations. In rotary mahinery [1], suh as motors and engines, it an help to ahieve optimum performane and provide early indiations of faulty omponents. In struture health monitoring [2], it an be used to monitor the health of bridges and buildings to ensure the safety of these strutures. In health are, the movement of human hests is also a vibration signal that provides information about respiration and heart rates [3]. Traditionally, most vibration monitoring appliations use ontat-based vibration sensors, suh as aelerometers. With the development of low ost and highly integrated CMOS radars [4, 5], another lass of vibration sensors is emerging, whih allows non-ontat measurements of vibration signals. The non-ontat approah is more flexible and avoids any oupling issues between sensors and vibration objets. Among different arhitetures for radar based vibration sensors, the frequeny-modulated ontinuous wave (FMCW) [6] arhiteture is one of the most flexbile. In a FMCW system (see Fig. 1), a periodi hirp signal is generated by a frequeny synthesizer and transmitted to the target. The mixing of refleted signal and the transmitted signal results in a low frequeny beat signal, whose frequeny diretly orresponds to the range of the target. The phase of the beat signal aross Waveform Generation Radar Proessing ADC Synthesizer PA LNA Vibrating Motor Fig. 1. Simplified blok diagram of a FMCW radar system. multiple hirps an be traked to provide vibration or Doppler information of the target. The arhiteture is also well-suited for low-ost CMOS implementation due to the low peak-toaverage power ratio of the transmitted signal, whih allows higher effiieny power amplifier operation. Beause of these advantages, we fous on vibration parameter estimation using the FMCW approah in this paper. In Setion 3, we present the details of FMCW radar signal proessing. New Cramer- Rao lower bounds () for vibration parameter estimates are derived in Setion 4. We then set up simulations and present a new estimation algorithm that ahieves the bounds in Setion 5. Finally, we onlude in Setion RELATION TO PRIOR WORK Vibration signal extration using FMCW radar has been presented in [7] in the ontext of human vital sign detetion. But no performane bounds are available to understand the theoretial limits of the estimation approah. Previous bounds derived for single tone parameter estimation [6] and motion parameter estimation for dual-frequeny radar [8] annot be applied here beause of different signal models. The main ontribution of this paper is the new s for these type of estimation problems and a new algorithm that ahieves the performane bounds in simulations. 3. FMCW RADAR SIGNAL PROCESSING Fig. 2 shows typial transmitted and reeived hirp signals in an FMCW system. Denoting the time within eah hirp ramp period as t, i.e., 0 < t < T r, a single transmitted hirp signal is defined as s(t) = exp [ j(2πf t + πkt 2 ) ], K = B T r, (1) /16/$ IEEE 2224 ICASSP 2016

2 Frequeny B T r t d ω b Chirp 0 Chirp 1 Chirp L-1 Time Fig. 2. A sequeny of L transmitted (solid line) and reeived (dashed line) hirp signals. where f is the arrier frequeny, and B is the bandwidth of the hirp signal. Assume that the distane between the target and the radar is where R 0 is the nomial position, and R(t) = R 0 + x(t), (2) x(t) = m sin(ω m t) (3) is the vibration movement of the target with amplitude m and frequeny ω m. The round trip delay of the radar signal bouning off the target is given by t d = 2R(t) = 2[R 0 + x(t)], (4) whih results in the following reeived signal. r(t) = A exp [ j(2πf (t t d ) + πk(t t d ) 2 ) ], (5) where A aounts for the amplitude hange of the signal aused by the round trip propagation and refletion. For eah hip, only the time period between t d and T r is used for mixing sine the time period between 0 and t d generates a high frequeny signal that is usually filtered out. Within the valid period, the beat signal for a single hirp after mixing beomes y(t) = s (t)r(t) A exp {j [2πKt d t + 2πf t d ]}, (6) where the term assoiated with t 2 d is omitted beause t2 d << t d t. From (6), we an see that the round trip delay is diretly related to the beat frequeny and phase. However, the variation of x(t) is very small within a single hirp. To obtain vibration signatures, we need to trak the phase hange of the beat signal aross a sequene of L hirps. The round trip delay for the l-th hirp within the hirp sequene is t d = 2[R 0 + x(lt r + t)]. (7) Close inspetion of x(lt r + t) generates the following approximation, onsidering that ω m t term is small. x(lt r + t) = m sin[ω m (lt r + t)] = m {sin(ω m lt r )os(ω m t) + os(ω m lt r )sin(ω m t)} m sin(ω m lt r ). (8) Exept for t d, the beat signal for the l-th hirp still has the same expression as in (6) beause of the periodiity of the hirp sequene, i.e., s(lt r + t) = s(t), and r(lt r + t) = r(t). Therefore, we an substitute (7) and (8) into (6) and obtain { [ 4πKR0 y(lt r + t) = A exp j t + 4πf R 0 ( 4πKt + + 4πf ) m sin(ω m lt r ) ]}.(9) After sampling and omitting 4πKt (Kt << f in typial FMCW radars for 0 < t < T r ), we have y(lt r + nt s ) = A exp [j (ω b nt s + ψ l )], (10) where T s is the sampling period and ω b = 4πKR 0 (11) ψ l = 4πf R 0 + 4πf m sin(ω m lt r ). (12) In FMCW radar terms, the beat signal is a 2-dimentional funtion of the fast time n and slow time lt r. The goal of the estimation algorithm is to find the vibration amplitude and frequeny. In order to do that, we have to find estimates for ω b and ψ l from the sampled beat signal. Those an be obtained by running fast Fourier transform () on the fast time, usually referred to as the range [9]. To illustrate the proess, we take a N-point disrete-time Fourier transform (DTFT) on the beat signal, i.e., Y (e jω, l) = n=0 exp[j(ω b nt s + ψ l )]exp( jωn) = exp(jψ l )exp[ (ω ω b T s )(N 1)/2] sin [(ω ω bt s )N/2] (13) sin [(ω ω b T s )/2] If we selet ˆω = ω b T s, i.e., the frequeny orresponding the the maximum of DTFT output, then the DTFT output at that frequeny only has the phase term, e jψ l. In reality, has limited resolution, the residue phase aused by imperfet frequeny estimation an have an impat on the estimation of ψ l term. Interestingly, if the DTFT is taken using index from N/2 to N/2 1 in (13), then the effet of the residue term is muh smaller sine the DTFT introdues muh less phase shift for ˆω that deviates from the true ω b T s. 4. CRAMER-RAO LOWER BOUND In this setion, we establish the s of vibration parameter estimation. The parameter estimation problem in (10) an be simplified into the estimation of the parameters of the following model given a blok of N by L samples of y(n, l). y(n, l) = A 1 exp {j [ω 1 n + φ 1 + A 2 sin(ω 2 l + φ 2 )]} + w(n, l), (14) 2225

3 where w(n, l) is assumed to be i.i.d. random variables following the omplex Gaussian distribution with variane. All model parameters, A 1, ω 1, φ 1, A 2, ω 2, and φ 2 are assumed to be real. To derive the bounds, we start with the probablity density funtion (PDF) of the noise, whih is p [w(n, l)] = 1 2πσ 2 exp { 1 w(n, l) 2 2σ2 } (15) Thus, the PDF of a blok of N L measured data given a set of model parameters is = p [Y; θ] L 1 l=0 n=0 { 1 2πσ 2 exp 1 } y(n, l) A 1e jα 2 (16) where α is used to denote ω 1 n+φ 1 +A 2 sin(ω 2 l+φ 2 ) and θ = [A 1, ω 1, φ 1, A 2, ω 2, φ 2 ]. With the PDF funtion in (16), we an derive the Fisher information matrix for θ, whih is given in (22). Note that in the Fisher information matrix, the terms that involve linear summation of sinusoidal funtions of ω 1 and ω 2 are approximated by zero, whih is valid for large N and L. The s of the estimators of the model parameters an now be found by examining the diagonal values of the inverse of the Fisher information matrix, whih are var(a 1 ) = σ2 NL 1 var(ω 1 ) = A 2 1 NL(N 2 1) (17) (18) var(φ 1 ) = 2σ2 (2N 1) A 2 1 NL(N + 1) (19) var(a 2 ) = var(ω 2 ) = var(φ 2 ) = A 2 1 NL (20) 24σ 2 A 2 1 A2 2 NL(L2 1) (21) 4σ 2 (2L 1) A 2 1 A2 2 NL(L + 1) (23) The variane for A 1, ω 1, and φ 1 are the same as those shown in [6] by using NL as the data length and setting the n 0 parameter in [6] to zero. The additional unknown parameters, A 2, ω 2, and φ 2 do not affet the bounds of those parameters. They are essentially deoupled exept that the amplitude A 1 affets estimation auray of A 2, f 2, and φ 2. To express the bounds diretly in terms of FMCW parameters, we an utilize the following relationships obtained by omparing (14) and (10). SNR = A2 1, ω 2 = 2πf m T r, A 2 = 4πf m. (24) After substituting (24) into (20) and (21), the s for vibration frequeny and amplitude are given by var(m) var(f m ) 2 16π 2 N L f 2 SNR (25) π 4 T 2 r m 2 N L (L 2 1)f 2 SNR (26) As an example, for a typial set of FMCW parameters used in the automotive appliations, i.e., SNR = 10dB, f = 76GHz, N = 875, L = 1000, the standard deviation bound of the amplitude is 0.11 um. For a vibration amplitude of 10 um, the standard deviation bound of the vibration frequeny is 0.04 Hz. From these numbers, we an see that the FMCW radar approah is apable of providing very aurate estimates of the parameters of a vibration signal. 5. PARAMETER ESTIMATION ALGORITHM In this setion, we set up simulations to verify the s presented in Setion 4. The reeived data in the simulations were generated aording to the signal model in (14) with the following parameters. A total of 500 random runs were performed to obtain variane of estimated parameters. N = 128, L = 128, A 1 = 1, A 2 = 0.5 f 1 and f 2 uniformly distributed from 0.05 to 0.45 φ 1 and φ 2 uniformly distributed from π/2 to π/2 Parameter estimation is done in two steps. In the first step, for eah l, we run on index n and determine the frequeny and phase orresponding to the maximum amplitude, reorded as ˆf 1 (l) and ˆφ 1 (l), respetively. The estimates for eah l are then averaged aross all hirps to generate the final estimates, ˆf 1 and ˆφ 1. The simulation results for different I(θ) = NL σ A NL()(2) A 2 6σ 2 1 NL() 2σ A NL() A 2 1 NL σ A NL 2σ A A2 2 NL(L 1)(2L 1) A 2 1 A2 2 NL(L 1) 4σ 2 A 2 1 A2 2 NL(L 1) 4σ 2 A 2 1 A2 2 NL (22) 2226

4 10-9 f 1 variane vs f 2 variane vs 10-8 Demod/Least-squares Variane Variane Fig. 3. Comparison between varianes obtained in simulations and the for f 1. SNRs are shown in Fig. 3. The results math well with the theoretial bounds. In the seond step, we estimate parameters f 2 and φ 2. A simple approah for ahieving this is to unwrap the phase, ˆφ 1 (l), reorded in the first step, remove its mean, and run another on the resulting sequene. Then the frequeny and phase assoiated with the maximum output beome the estimates of f 2 and φ 2. However, as shown in red irle markers in Fig. 4, the varianes of the estimates obtained using this approah are signifiantly higher than the bounds. The differene is aused by two main fators. The first one is the demodulation of ˆf 1. When we obtain ˆφ 1 (l), we essentially demodulate eah hirp by ˆf 1 (l) through the proess and then find the resulting phase. Although we have a more aurate estimate of ˆf 1, whih is obtained through averaging ˆf 1 (l), it is not used in the proess of estimating ˆφ 1 (l). This inauray auses larger varianes in estimating f 2 and φ 2. A more aurate approah to obtain ˆφ 1 (l) is ˆφ 1 (l) = 1 N n=0 [ y(n, l)exp( j2π ˆf ] 1 n) (27) The seond fator is the residue orrelation in the basis funtions of the. This is not problem in f 1 estimation sine f 1 is a omplex tone, in whih ase the maximum likelihood (ML) solution does not rely on perfet orthogonality [6]. The vibration signal here is a real sinusoidal funtion. An aurate parameter estimation requires perfet orthogonality [10, p. 193] between the osine and sine basis funtions used in the proess, whih is not possible for limited number of samples. To be more aurate, a least-square based grid searh algorithm [10, p. 193] is required at the expense of signifiantly more omputation than. With more aurate frequeny demodulation and least-squares, we an ahieve variane lose to the as shown in the green diamond markers in Fig. 4 for both f 2 and φ 2. Another soure of errors in the variane of φ 2 using omes from phase wrapping Variane (a) Variane vs. for f phi2 variane vs Demod/Least-squares 10-7 (b) Variane vs. for φ 2 Fig. 4. Comparison between varianes obtained in simulations and the s for f 2 and φ 2. around for a few inaurate phase estimates, whih auses very large errors. With the new estimation approah, this issue no longer affets results. The two step approah presented in this setion an also be shown to be the ML solution to the estimation problem in (13) by notiing the deoupling of the tone parameters in (22) and using arguments similar to those used in [6] and [10]. 6. CONCLUSIONS This paper presents detailed analysis on using FMCW radar for vibration parameter estimation. New s are established to understand the theoretial auray that an be ahieved by the FMCW radar approah. The s are verified in simulations and an be ahieved by the estimation algorithm developed in this paper. The analysis in this paper shows that vibration parameters measured with FMCW radar an ahieve very good auray, whih have the great potential to be used as preision non-ontat vibrations sensors. 2227

5 7. REFERENCES [1] K. G. MConnell and P. S. Varoto, Vibration testing: theory and pratie, John Wiley & Sons, Hoboken, N.J, 2nd edition, [2] J. A. Rie, C. Li, C. Gu, and J. C. Hernandez, A wireless multifuntional radar-based displaement sensor for strutural health monitoring, in Pro. SPIE, Sensors Smart Strutures Tehnol. Civil, Meh. and Aerosp. Syst., Apr. 2011, vol [3] C. Li, V. M. Lubeke, O. Bori-Lubeke, and J. Lin, A review on reent advanes in Doppler radar sensors for nonontat healthare monitoring, IEEE Trans. Mirow. Theory Tehn., vol. 61, no. 5, pp , May [4] B. P. Ginsburg, S. M. Ramaswamy, V. Rentala, E. Seok, S. Sankaran, and B. Haroun, A 160 GHz pulsed radar transeiver in 65 nm CMOS, IEEE J. Solid-State Ciruits, vol. 49, no. 4, pp , Apr [5] T.-Y. J. Kao, A. Y.-K. Chen, Y. Yan, T.-M. Shen, and J. Lin, A flip-hip-pakaged and fully integrated 60 GHz CMOS miro-radar sensor for heartbeat and mehanial vibration detetions, in Pro. IEEE Radio Freq. Integr. Ciruits Symp., June 2012, pp [6] D. Rife and R. R. Boorstyn, Single tone parameter estimation from disrete-time observations, IEEE Trans. Inf. Theory, vol. 20, no. 5, pp , Sept [7] G. Wang, J.-M. Munoz-Ferreras, C. Gu, C. Li, and R. Gomez-Garia, Appliation of linear-frequenymodulated ontinuous-wave (LFMCW) radars for traking of vital signs, IEEE Trans. on Mirow. Theory Tehn., vol. 62, no. 6, pp , June [8] P. Setlur, M. Amin, and F. Ahmad, Cramer-Rao bounds for range and motion parameter estimations using dual frequeny radars, in Pro. IEEE Int. Conf. Aoust. Speeh Signal Proess., Apr. 2007, vol. 3, pp. III 813 III 816. [9] D. E. Barrik, FM/CW radar signals and digital proessing, Teh. Rep. ERL 283-WPL 26, NOAA, July [10] S. M. Kay, Fundamentals of statistial signal proessing, Prentie Hall signal proessing series. Prentie-Hall PTR, Englewood Cliffs, N.J,