VIBRATION signatures associated with objects such as

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1 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER SAR-Based Vibration Estimation Using the Disrete Frational Fourier Transform Qi Wang, Student Member, IEEE, Matthew Pepin, Member, IEEE, Ryan J. Beah, Ralf Dunkel, Tom Atwood, Balu Santhanam, Senior Member, IEEE, Walter Gerstle, Armin W. Doerry, and Majeed M. Hayat, Senior Member, IEEE Abstrat A vibration estimation method for syntheti aperture radar (SAR) is presented based on a novel appliation of the disrete frational Fourier transform (DFRFT). Small vibrations of ground targets introdue phase modulation in the SAR returned signals. With standard preproessing of the returned signals, followed by the appliation of the DFRFT, the timevarying aelerations, frequenies, and displaements assoiated with vibrating objets an be extrated by suessively estimating the quasi-instantaneous hirp rate in the phase-modulated signal in eah subaperture. The performane of the proposed method is investigated quantitatively, and the measurable vibration frequenies and displaements are determined. Simulation results show that the proposed method an suessfully estimate a twoomponent vibration at pratial signal-to-noise levels. Two airborne experiments were also onduted using the Lynx SAR system in onjuntion with vibrating ground test targets. The experiments demonstrated the orret estimation of a 1-Hz vibration with an amplitude of 1.5 m and a 5-Hz vibration with an amplitude of 1.5 mm. Index Terms Frational Fourier transform, joint time frequeny analysis (JTFA), miro-doppler effet, subaperture, syntheti aperture radar (SAR), vibration. I. INTRODUCTION VIBRATION signatures assoiated with objets suh as ative strutures (e.g., bridges and buildings) and vehiles an bear vital information about the type and integrity of these objets. The ability to remotely sense minute strutural Manusript reeived September 28, 2010; revised August 1, 2011; aepted January 24, Date of publiation April 17, 2012; date of urrent version September 21, This work was supported in part by the U.S. Department of Energy under Award DE-FG52-08NA28782, by the National Siene Foundation under Award IIS , by the National Consortium for MASINT Researh, and by Sandia National Laboratories. Q. Wang, M. Pepin, and M. M. Hayat are with the Center for High Tehnology Materials and the Department of Eletrial and Computer Engineering, University of New Mexio, Albuquerque, NM USA ( qwang@ee.unm.edu; pepinm@ee.unm.edu; hayat@ee.unm.edu). R. J. Beah is with the Department of Mehanial Engineering, University of New Mexio, Albuquerque, NM, USA ( rbeah41@unm.edu). R. Dunkel is with General Atomis Aeronautial Systems, In., San Diego, CA USA ( Ralf.Dunkel@ga-asi.om). T. Atwood and A. W. Doerry are with Sandia National Laboratories, Albuquerque, NM USA ( tdatwoo@sandia.gov; awdoerr@ sandia.gov). B. Santhanam is with the Department of Eletrial and Computer Engineering, University of New Mexio, Albuquerque, NM, USA ( bsanthanam@ee.unm.edu). W. Gerstle is with the Department of Civil Engineering, University of New Mexio, Albuquerque, NM USA ( gerstle@unm.edu). Color versions of one or more of the figures in this paper are available online at Digital Objet Identifier /TGRS vibrations persistently and with high auray is extremely important for a number of reasons. First, it avoids the ost of aquiring and installing aelerometers on remote strutures. Seond, it alleviates the high ost of maintaining these sensors, and third, it enables sensing vibrations of strutures that are not easily aessible to engineers and maintenane personnel (e.g., pedestrian and train bridges over anyons, strutures and vehiles in a hostile land, et.). While light detetion and ranging (LIDAR) tehnology has been proposed and used for remote sensing of vibrations, it has failed to overome a number of persisting hallenges. First, due to the short wavelength of the standard illumination in LIDAR, loss and aberration due to laser propagation through air and vapor make LIDAR vibration sensing highly dependent on weather onditions. This makes it partiularly problemati when it is desirable to probe vibrating objet at a large distane (tens of kilometers or more). Seond, LIDAR systems are not typially easily mounted on small moving platforms due to the omplexity of the system. Syntheti aperture radar (SAR) is a well-established tehnique for high-resolution imaging of the Earth s surfae through measurement of its eletromagneti refletivity [1] [3]. The relatively long wavelengths, ompared with those of optial sensors, make SAR systems apable of remote imaging over thousands of kilometers regardless of weather onditions. In addition, small vibrations in the imaged surfaes introdue phase modulation in the refleted SAR signals, a phenomenon often referred to as the miro-doppler effet [4] [9]. As suh, in addition to imaging, SAR an also have the added benefit of enabling us to remotely measure surfae vibrations by estimating the orresponding miro-doppler effet. In standard SAR imaging, vibrations from strong satterers result in ghost targets around the satterers in the SAR image (in the azimuth diretion) that are generally diffiult to distinguish from images of stati satterers [10]. This ghosting is due to the fat that the returned SAR signals, even after they are preproessed (range ompressed and autofoused), still bear the vibration-indued time-varying phase, and the standard Fourier-transform-based analysis used in SAR proessing is inadequate to resolve suh nonstationary signals. Indeed, the SAR returned eho from a vibrating satterer after preproessing is a nonstationary signal whose instantaneous frequeny (IF) is linearly proportional to the vibration veloity [11]. To address this limitation of standard SAR, joint time frequeny analysis (JTFA) has been proposed to analyze the miro-doppler effet [4]. Different time frequeny transforms have been used, inluding the short-time Fourier transform [12], Cohen s lass /$ IEEE

2 4146 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER 2012 transform [13], and the adaptive time frequeny transform [14]. More omprehensive reviews on time frequeny methods are available in the literature; see, for example, [4]. Nonetheless, the JTFA merely provides a qualitative illustration of the vibration-indued frequeny modulation in the time frequeny representation, and it does not provide an estimation of the vibration amplitude and frequeny. Additional estimation proedures, suh as retrieving the IF trak from the time frequeny representation, are needed in order to estimate the vibration. This step is not trivial when the signal-to-noise ratio (SNR) is low. Moreover, beause the existing JTFA stops at the analysis stage, the apability and performane of the SAR-based vibration estimation are left uninvestigated. In this paper, a vibration estimation method using SAR is presented based on a novel appliation of the disrete frational Fourier transform (DFRFT). The proposed method provides a omplete estimation of the vibration signature by offering the history of the instantaneous aeleration and the spetrum of the vibrating objet. In this method, the onventional SAR proessing proedure is performed to obtain a nonstationary signal from the vibrating target. First, the returned SAR signals are demodulated, and the polar-to-retangular resampling is applied to the SAR phase history to orret the range ell migration. Seond, autofous is performed, and range ompression is applied to the reformatted SAR phase history. Next, the signal from a vibrating target is foused on a range line, and it is the aforementioned nonstationary signal. After the preproessing, the nonstationary signal is approximated by a hirp signal in a small time window, alled the subaperture. The DFRFT is then applied to estimate the vibration aeleration in sliding subapertures. The performane of the proposed method is quantified in terms of the measurable frequenies and displaements, and the effiay of the approah is demonstrated by experiments using the Lynx SAR system built by General Atomis Aeronautial Systems, In. (GA-ASI) [15]. The remainder of this paper is organized as follows. In Setion II, we provide a theoretial analysis of the vibrationindued frequeny modulation. In Setion III, the DFRFTbased vibration estimation method is introdued, follow by performane analysis in Setion IV. Simulations and experiments are provided in Setions V and VI, respetively. Setion VII ontains our onlusions. II. MODEL A. Motion Model Fig. 1 shows a 3-D SAR flight geometry, with a vibrating target loated at the origin. The nominal line-of-sight distane from the target to the radar sensor is r 0, with the radar sensor loated at polar angles ψ and θ to the target. Let r d (t) denote the projetion of the vibration displaement onto the line of sight from the target to the SAR sensor; the range of the vibrating target beomes r(t) r 0 r d (t). (1) Due to the hange of aspet angle of the target during the SAR data-olletion proess, the range r 0 hanges a little. However, Fig. 1. Three-dimensional SAR flight geometry. The vibrating target is loated at the origin, and the radar sensor is loated at (r 0,ψ,θ). modern SAR ompensates for the hange via proper modeling and post-signal-proessing tehnique [1] [3]. The projetion r d (t) is also modulated by the hange of aspet angle. For broadside SAR, the projet an be approximated by r d (t) r d0 os θ(t) (2) where r d0 represents the projetion of the vibration displaement for θ =0. The hange of aspet angle θ(t) due to the SAR geometry is known; therefore, we an estimate r d0 (t) from r d (t). For spotlight-mode SAR, the hange of aspet angle is usually small [3]. In this ase, we have r d (t) r d0 (t). Inthis paper, we onsider the ase of broadside spotlight-mode SAR for whih the aforementioned approximation is valid. B. Signal Model The small range perturbation of the vibrating target modulates the olleted SAR phase history. Consider a spotlightmode SAR whose sent pulse is a hirp signal, with arrier frequeny f and hirp rate K. Eah returned SAR pulse is demodulated by the sent pulse delayed appropriately by the round-trip time to the enter of the illuminated path. A demodulated pulse an be written as [3, Ch. 1] r(t) = i [ σ i exp j 4π(r ( ( i r ) f + K t 2r ))] (3) where σ i is the refletivity of the ith satterer, is the propagation speed of the pulse, and r is the distane from the path enter to the antenna. The polar-to-retangular resampling is then applied to the SAR phase history [3, Se. 3.5] to orret for range ell migration. The autofous is also performed at this stage. For small vibrations, the vibration-indued phase modulation in range diretion is very small [4], [5], [16]; therefore, it is ignored. Range ompression is applied to the phase history to separate the satterers in range. Fig. 2 shows the magnitude of the range-ompressed SAR phase history ontaining one stati point target and one vibrating point target. Assuming that all satterers at a speifi range are stati, the

3 WANG et al.: SAR-BASED VIBRATION ESTIMATION 4147 Fig. 2. Magnitude of the range-ompressed SAR phase history ontaining one stati point target and one vibrating point target. The two point targets are separated in range after range ompression. range-ompressed phase history at this speifi range an be written as [ ( σ i [n]exp j x[n] = i f y y i n 4πf )] r i + φ i + w[n] (4) for 0 n<n I, where n is the index of the olleted returned pulses, N I represents the total number of olleted returned pulses, y i is the ross-range position of the ith target, and φ i represents all additional (onstant) phase terms. The imaging fator f y is known and used to estimate the ross-range of the target. For spotlight-mode SAR, f y an be written as [2], [3] f y = 4πf V R 0 f prf (5) where V is the nominal speed of the SAR antenna, R 0 is the distane from the path enter to the midaperture, and f prf is the pulse-repetition frequeny (PRF). The SAR integration time is given by T I = f prf N I, and w[n] is additive noise. The signal x[n] in (4) is a stationary signal if all satterers are stati. The azimuth ompression, aomplished by applying the disrete Fourier transform (DFT) to x[n], will fous the stati satterers on the orret ross-range positions. However, when a vibrating satterer is present, x[n] has a nonstationary omponent beause r i is now a funtion of n for the vibrating satterer. The ross-range y i is also hanging for the vibrating satterer. However, beause R 0 is very large (tens of kilometers), f y is usually muh smaller than 4πf /; therefore, the phase modulation indued by time-varying y i is ignored [4], [5]. As suh, we use ȳ i to denote the average rossrange position of the vibrating satterer. For the same reason, a small hange in r i auses a relatively large flutuation to the Doppler frequeny f y y i. We would like to emphasize that azimuth ompression annot fous the vibrating satterer on the orret ross-range position beause the DFT spetrum of the nonstationary omponent usually has signifiant sidelobes [10]. Fig. 3 shows the reonstruted SAR image by applying azimuth ompression to the phase history as shown in Fig. 2. The sidelobes near the vibration target are ommonly referred to as the ghost targets [10]. The vibration-indued phase modulation Fig. 3. Reonstruted SAR image using the SAR phase history in Fig. 2. Target vibration introdues ghost targets along the azimuth diretion. is referred to as the miro-doppler effet [4]. Analysis tools other than the DFT are required to estimate vibrations and nonstationary targets in general. We define the signal of interest (SoI) as the range line in the range-ompressed phase history ontaining vibrating targets. An example is shown in Fig. 2. In this paper, we onsider ases for whih the vibrating satterer is well separated from other satterers in range (e.g., this may be possible by hoosing a proper data-olletion orientation). In this ase, the SoI an be written as x[n] =σ[n]exp [ ( j f y ȳn 4πf r d [n]+φ )] +w[n] (6) for 0 n<n. In the next setion, the DFRFT-based method is desribed and used to estimate the vibration r d [n] from x[n]. III. ALGORITHM DEVELOPMENT For its key role in our estimation proess, we will first review germane aspets of the DFRFT drawing freely from the literature [17], [18]. The vibration estimation method is then developed. A. Review of the DFRFT The ontinuous-time frational Fourier transform, first introdued by Namias in 1980 [19], is a powerful time frequeny analysis tool for nonstationary signals and has been found to have several appliations in optis and signal proessing [20]. Santhanam and MClellan [21] were the first to introdue a formulation of the DFRFT. Other formulations of the DFRFT are desribed in the exellent review paper by Pei and Ding [22]. The DFRFT formation used in this paper is speifially referred to as the multiangle entered DFRFT (MA-CDFRFT) in the literature [17]. More details an be found in [17] and [18]. Without ambiguity, we refer to the MA-CDFRFT as the DFRFT throughout the remainder of this paper. Let W denote the transformation matrix of the entered DFT. The frational power of W is defined as W α = V G Λ 2α/π VG T, where V G is the matrix of Grünbaum eigenvetors of W

4 4148 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER 2012 Fig. 4. Three-dimensional angle frequeny spetrum of the signal using the DFRFT. The hirp omponent and the sinusoidal omponent are orresponding to different peaks in the spetrum. No ross-term is introdued due to linearity of the DFRFT. and Λ 2α/π is a diagonal matrix with the frational powers of the eigenvalues of W. Assume that x[n] is a sequene of N samples. The DFRFT of x[n] is the DFT of an intermediate signal ˆx k [p] for eah index k, i.e., X k [r] = N 1 p=0 ( ˆx k [p]exp j 2π ) N pr where r =0, 1,...,N 1 is the angular index and the orresponding α is equal to 2πr/N. The intermediate signal ˆx k [p] is alulated by N 1 ˆx k [p] =v p (k) x[n]v p (n) (8) n=0 where v p (k) is the kth element of v p and v p is the pth olumn vetor of V G. It has been shown [17], [18], [23] that the DFRFT has the ability to onentrate a linear hirp into a few oeffiients and that we obtain an impulselike transform analogous to what the DFT produes for a sinusoid. Fig. 4 shows the DFRFT of a omplex signal ontaining two omponents: a pure 150- Hz sinusoid and a hirp signal with a enter frequeny of 200 Hz and a hirp rate of 400 Hz/s. The frequeny axis is the same as the one of the DFT. The DFRFT introdues a new angular parameter α to desribe the linear time frequeny relation of the signal. For α = π/2, the result of the DFRFT is the same as that of the DFT. The two peaks that orresponded to the sinusoid and the hirp are well separated, whih indiates that the two omponents have different enter frequenies and hirp rates. Fig. 5 shows the 2-D view of the angle frequeny spetrum generated by the DFRFT. B. Vibration Estimation Method The SoI desribed in Setion II-B is an example of nonstationary signals. They an be analyzed by means of using sliding short-time windows. In a short-time window starting at m, a (7) Fig. 5. Vertial view of the angle frequeny spetrum shown in Fig. 4. Blak and white orrespond to the highest and lowest amplitudes, respetively. seond-order approximation an be applied to the vibration displaement r d, and the SoI in (6) beomes [ ( x[n] σ exp j φ 4πf ( r d [m]+ f y ȳ 4πf ) v d [m] n f prf 2πf a d [m] fprf 2 n )]+w[n],m n<m+n 2 w (9) where N w is the size of the window. We assume that the refletivity of the target σ does not hange within the time window. The length of the time window is T w = f prf N w. When T w is muh less than the duration of the vibration, the seondorder approximation in (9) is fairly aurate. Aording to (9), x[n] in a short-time window is approximately a hirp signal, and its hirp rate is linearly proportional to the instantaneous vibration aeleration a d [m]. By estimating the hirp rates of x[n] in suessive sliding short-time windows, the vibrationaeleration history is estimated. The DFRFT is used to estimate the hirp rates, and the details are shown as follows. Inorporating the CZT: Beause the vibration-indued hirp rates are usually very small, a resolution enhanement algorithm, alled the the hirp z-transform (CZT) algorithm, is inorporated into the DFRFT. With the CZT, a more finely spaed interpolation of the spetrum of interest an be obtained than that offered by the DFT [12], [24]. As shown in Setion III-A, the final step of the DFRFT an be interpreted as the DFT of ˆx k [p] in (8) for eah frequeny index k in (7). Therefore, we an implement the CZT algorithm in the final step of the DFRFT in order to obtain more exat peak loations with respet to angle α [25]. Fig. 6 shows the CZTinorporated DFRFT of the signal with a zoom-in fator of two. The resolution of the peak position with respet to the angle α is improved. Estimating Chirp Rates: There is a one-to-one mapping from the angular position of the peak in the DFRFT plane to the hirp rate of the signal [17]. This mapping is dependent on the size of the DFRFT and the zoom-in fator of the CZT. Currently, there is no analyti form to desribe the mapping. Fig. 7 shows a mapping from the peak loation to the hirp

5 WANG et al.: SAR-BASED VIBRATION ESTIMATION 4149 Fig. 6. CZT-inorporated DFRFT of the signal with a zoom-in fator of two. By inorporating the CZT, the resolution of the peak position with respet to angle α is improved. rate where the DFRFT size is 160 and the zoom-in fator is 10. The mapping is generated by first using the DFRFT to estimate signals with different hirp rates and then interpolating the estimation results with the spline funtion. In pratie, the mapping is generated with a parameter set that works best for the partiular appliation and is stored for later use in estimating hirp rates. One the hirp rate is estimated by the DFRFT, the estimated vibration aeleration is alulated via â d [m] = f 2 prf 2πf ĉ r [m] (10) where ĉ r [m] is the estimated hirp rate. By estimating the aeleration in sliding short-time windows, the history of the vibration aeleration is estimated. The estimated vibration spetrum an be obtained by applying the DFT to the estimatedvibration-aeleration history. The DFRFT-based vibration estimation method is summarized in Algorithm 1. Usually, the DFRFT-based method is applied to the SoI with several different window sizes to ahieve the best performane. Algorithm 1 Proedure for the proposed vibrationestimation method 1) demodulate and reformat the SAR phase history, perform autofous; 2) apply range ompression to the SAR phase history, identify the SoI; 3) hoose a proper window size N w 4) for all m =0to m = N N w +1do 5) apply the DFRFT to the SoI in eah time window and estimate the hirp rate ĉ r [m]; 6) end for 7) alulate the estimated instantaneous vibration aeleration via â d [m] = (f 2 prf /2πf )ĉ r [m]; 8) reonstrut the history of the vibration aeleration and alulate its DFT 9) spetrum; 10) repeat steps 2 8 for different window sizes. Fig. 7. One-to-one mapping from the peak loation in the angle frequeny spetrum to the hirp rate using the DFRFT. The DFRFT size is 160, and the zoom-in fator is 10. IV. PERFORMANCE ANALYSIS In real-world appliations, the performane of the proposed method is affeted by the presene of noise. In the extreme ase, when the SoI is highly orrupted by noise, the estimated vibration aeleration would not be reliable. Thus, we are interested in knowing the SNR threshold above whih the estimation error is aeptable. To this end, we have used Monte Carlo simulations to evaluate the performane of the proposed method in estimating the hirp rate under different SNR levels. The SoI in the presene of noise in a given time window starting at m an be written as s[n] =σ exp [ j ( φ + ω n + r n 2)] + w[n] (11) for m n<m+ N w. The noise term w[n] is modeled as a zero-mean omplex-valued white Gaussian noise. The SNR is define as SNR =10log σ (12) σ w where σ 2 w is the variane of the additive noise. Beause the DFRFT is evaluated on disrete angular values, the step size in angle α is limited to ρ = 2π (13) ηn D where N D is the size of the DFRFT and η is the zoom-in fator. This also yields a finite resolution for hirp rate estimates. A. SNR Requirements We have evaluated the performane of the estimator shown in Fig. 7. The estimator is used to estimate hirp rates within the range from to rad/samples 2. This estimator roughly yields a resolution of rad/samples 2 in estimating the hirp rate. The normalized root mean square errors (NRMSEs) for five hirp rates are plotted in Fig. 8. The five hirp rates are , , , , and When the SNR inreases to 20 db, the NRMSEs of

6 4150 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER 2012 Fig. 8. Normalized mean-squared error in estimating five different hirp rates using the DFRFT-based estimator shown in Fig. 7. most of the hirp rates (exept for r = ) droptoan aeptable level (roughly 0.05). However, the errors plateau as the SNR inreases. The residual errors are mainly from the quantization error due to the limited resolution. When the estimated hirp rate is on the same order of the resolution limit, a small estimation error auses large NRMSE. This is seen by observing the NRMSE of the hirp rate of When the SNR is 30 db, the NRMSE in estimating the hirp rate of is still about 10%. Note that it is important to hoose a proper setting for the estimator in terms of the DRFT size N D and the zoom-in fator η. The size of the DFRFT N D is usually determined by other fators that will be explained later in this setion, and it an be larger than N w in some ases. By using a large zoom-in fator, the resolution of the estimator is enhaned, and the residual error is redued. Based on the prior information of the SAR system in use, a DFRFT size and a relatively small zoom-in fator are hosen to build the first estimator. If the estimator does not fit with the hirp rates indued by the vibration, then the zoom-in fator is inreased aordingly. We avoid using a very high resolution estimator (by hoosing a large zoom-in fator) in the very beginning due to the following two reasons. First, if the estimator beomes too sensitive, then highfrequeny noise will be introdued to the estimated vibration frequeny. Seond, the vibration-indued hirp rates may be beyond the range of the estimator. In the remainder of this setion, we assume that we work under aeptable SNRs and disuss other aspets of the performane of the proposed method. B. Resolution and Range of Estimated Vibration Frequenies Let us assume that the SNR requirement of the SoI is met in N samples. We have N N I, where N I is the total number of the returned pulses in the SAR phase history. We define the effetive observation time as T = f prf N. The vibration is estimated over the effetive observation time. Therefore, the resolution with respet to vibration frequeny is given by 1/(f prf N) that is lower bounded by 1/(f prf N I ). Next, the maximum measurable vibration frequeny (MMVF) is defined to be the maximum frequeny that a SAR system an estimate without any aliasing. Theoretially, the Nyquist Shannon sampling theorem ditates that the MMVF is upper bounded by f prf /2. However, suh an upper bound annot be reahed using the proposed method. The length of the time window should be muh less than the period of the vibration in order to redue the error introdued by the seond-order approximation. On the other hand, a ertain number of samples in the time window are required to estimate the hirp rate robustly. Although it is expensive or sometimes impratial to inrease the PRF, the SoI an be up sampled in order to estimate high vibration frequenies. As a remedy, we an up sample the SoI prior to applying the DFRFT to it. With up-sampling the SoI, the DFRFT size N D is enlarged larger than the window size N w. Aording to our experiene, the length of the time window has to be at least half the period of the vibration, and N w is at least 20. This yields a pratial MMVF that is approximately equal to f prf /40. C. MMVA and MMVD The minimum measurable vibration aeleration (MMVA) an be alulated from the speified parameters. When the hirp rate is small, the hirp rate an be obtained via [18] r = π N D (α p π/2) (14) where α p is the angular position of the peak in the DFRFT spetrum. The minimum angular differene in α that an be differentiated by the DFRFT is 2π/(ηN D ). Therefore, the MMVA is given by a (min) d = πf2 prf ηnd 2 f. (15) In the speial ase when the vibration is a single-omponent harmoni osillation, the minimum measurable vibration displaement (MMVD) an be derived in a straightforward fashion. In this ase, we know that a d = 4π 2 fv 2 r d, where r d is the vibration displaement. The vibration frequeny an be estimated from the alulated DFT spetrum of the estimated vibration aeleration, and it is denoted by ˆf v. Therefore, the MMVD in this ase is given by r (min) fprf 2 d = 2πηND 2 ˆf. (16) v 2 f Finally, the performane limits of the proposed method are summarized in Table I. V. S IMULATION-BASED CASE STUDY A simulated example is provided to demonstrate the apability of the proposed method in estimating a multiomponent harmoni vibration under realisti SNRs (e.g., 20 db). The simulated SAR is a spotlight-mode SAR working in the K u - band. Table II lists all the key system parameters assoiated

7 WANG et al.: SAR-BASED VIBRATION ESTIMATION 4151 TABLE I PERFORMANCE LIMITS OF THE DFRFT-BASED SAR VIBRATION ESTIMATION METHOD TABLE II SAR SYSTEM PARAMETERS USED IN THE SIMULATION Fig. 10. Real part of the SoI from simulated data. The SoI is a nonstationary signal, and its IF is modulated by the vibration. Fig. 11. DFRFT spetra of the SoI, from simulated data, in four different time windows. The peak loations are measured and used to estimate the vibration aelerations. Fig. 9. Reonstruted image from simulated SAR data. The vibrating satterer is at the enter of the image. Note that vibrations introdue many ghost targets in the azimuth diretion near the vibrating satterer. One stati satterer is above the vibrating satterer, and the other is below the vibrating satterer. with the simulation. Fig. 9 shows the reonstruted SAR image generated by using the algorithm desribed in [3]. There are three satterers in the images: The one in the middle is the vibrating satterer, and the rest are stati satterers. The vibration has two omponents: a 1.0-Hz osillation with an amplitude of 1 m and a 3.0-Hz osillation with an amplitude of 2 mm. Several vibration-indued ghost targets appear around the vibration satterer. The SoI is identified as the range line in the range-ompressed phase history where the range is orresponding to that of the vibrating target. The real part of the SoI is plotted in Fig. 10. When the proposed method was applied to the simulated data, the method produed the best result when the window size N w was set to 20 with an up-sampling fator of four. The CZT was inorporated in the DFRFT, and a zoom-in fator of eight was used. Fig. 11 shows the DFRFT of the SoI in four different time windows. Note that the positions of the peaks are slightly different from window to window, whih onfirms a time-varying vibration aeleration. The estimated vibration aeleration in the sliding time windows and its spetrum are shown in Figs. 12 and 13, respetively. The proposed method suessfully estimated the two vibrating omponents with a frequeny resolution of roughly 0.3 Hz. Comparison to a JTFA Method As desribed in Setion I, the JTFA methods use time frequeny distributions to provide an analysis of the miro-doppler effet. For its well-aepted performane [4], we use the smoothed pseudo Wigner Ville distribution (SP- WVD) as a representative JTFA method in the simulated example desribed before. To this end, we implemented the SPWVD by utilizing the widely used time frequeny toolbox [26]. The

8 4152 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER 2012 Fig. 12. Estimated aeleration history of the vibration from simulated data and by using the DFRFT-based method. Fig. 14. approah. Time frequeny representation of the SoI using the SPWVD Fig. 13. Estimated spetrum of the vibration using the DFRFT-based method using simulated data. The proposed method suessfully estimates two vibration omponents: 1.0 and 3.0 Hz. SPWVD of the SoI is shown in Fig. 14. The time frequeny representation in Fig. 14 roughly reveals 1- and 3-Hz vibration omponents. However, this is only a qualitatively dedued observation. To obtain preise estimates of the instantaneous aeleration from the time frequeny representation, further estimation proedures are required. On the other hand, the proposed method provides diret quantitative estimates of the history of the vibration aeleration and the vibration frequeny, and no further proedure is required. VI. EXPERIMENTAL CASE STUDIES Through an ongoing ollaboration with GA-ASI, we onduted two experiments with the Lynx airborne K u -band SAR system. The system parameters of the Lynx math those used in our simulations in Setion V. Using flight test data from the Lynx system, the proposed method suessfully estimated two vibrations from two different targets: a 1.5-m 1.0-Hz vibration and a 1.5-mm 5.0-Hz vibration. The details of these ase studies are provided next. Fig. 15. Experiment I: Vibrating target on the test ground near Julian, CA. The target is an aluminum triangular trihedral with a lateral length of 21 in. The vibration frequeny and amplitude were 1.0 Hz and 1.5 m, respetively. A. Experiment I In the first experiment, the vibrating target was an aluminum triangular trihedral with a lateral length of 21 in, as shown in Fig. 15. The motion of the target was a single-frequeny harmoni motion, driven by a d motor attahed to a rank and a piston. The vibration amplitude was 1.5 m, and the frequeny was 1.0 Hz. The target was positioned suh that the harmoni motion is in the range diretion. Fig. 16 shows the reonstruted SAR image that ontains the vibrating target. The nominal resolution of the reonstruted SAR image is 0.3 m in eah diretion. The vibrating target is loated at the bottom right portion of the image, and it appears as a horizontal line of the target eho and ghost targets. The vibration auses the ghost targets along the azimuth diretion in the reonstruted image of the target. Note that there are also several wellseparated stati targets extending from the enter of the image to the top right orner whih are not subjet to our analysis. In this experiment, the arrier frequeny was 15 GHz, and the PRF was 306 Hz. Due to seemingly limited SNR (the exat value is unknown), we seleted the total observation time T of this target to be 2.6 s entered at the time losest to target broadside. The length of eah time window was 0.26 s. Fig. 17 shows the DFRFT spetra of the SoI in four different time

9 WANG et al.: SAR-BASED VIBRATION ESTIMATION 4153 Fig. 16. Experiment I: Reonstruted SAR image provided by the GA-ASI Lynx system. The vibrating test target is in the lower right portion of this image. There are a few stati targets extending from the enter of the image to the top right orner whih are not subjet to our analysis. Fig. 19. Experiment I: Estimated vibration spetrum using the proposed method. The estimated vibration frequeny is 0.9 Hz; the atual value of the vibration frequeny was 1 Hz. Fig. 17. Experiment I: The DFRFT spetra of the SoI in four different time windows. Fig. 20. Experiment II: Vibrating target on the test ground near Julian, CA. The target is an aluminum triangular trihedral with a lateral length of 15 in. The atual vibration frequeny and amplitude were 5.0 Hz and 1.5 mm, respetively. The estimated vibration frequeny is 0.9 Hz, whih is very lose to the atual vibration frequeny of 1.0 Hz. Fig. 18. Experiment I: Estimated-vibration-aeleration history over 2.6 s using the proposed method. windows. The proposed method is applied to the SoI, and the estimated-vibration-aeleration history and the orresponding vibration spetrum are shown in Figs. 18 and 19, respetively. B. Experiment II In the seond experiment, the vibrating target was an aluminum triangular trihedral with a lateral length of 15 in, as shown in Fig. 20. Compared to the first experiment, the size of the trihedral is redued by 40%. Aordingly, the SNR was redued substantially. In ontrast to the first target that had a pure harmoni osillation, the vibrations in the seond were indued by the rotation of an unbalaned mass that was driven by a motor. The vibration s atual amplitude and frequeny were 1.5 mm and 5 Hz, respetively. Fig. 21 shows the SAR image that ontains the vibrating target. The nominal resolution of the reonstruted SAR image is 0.3 m in eah diretion. The

10 4154 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER 2012 Fig. 23. Experiment II: Estimated-vibration-aeleration history over 1 s using the proposed method. Fig. 21. Experiment II: The reonstruted SAR image provided by the GA- ASI Lynx system. The vibrating target is in the lower right portion of this image. Fig. 24. Experiment II: Estimated vibration spetrum using the proposed method. The estimated vibration frequeny is 5.2 Hz; the atual value was 5Hz. VII. CONCLUSION Fig. 22. Experiment II: DFRFT spetra of the SoI in four different time windows. vibrating target is at the bottom right portion of the image. The region near the vibrating target is magnified and displayed in the inset below the SAR image. Several ghost targets appear along the azimuth diretion. In this experiment, the arrier frequeny was 15 GHz, and the PRF was 270 Hz. Due to limited SNR, we seleted the total observation time of this target to be 1 s, entered at the time losest to target broadside. The length of eah time window was hosen to be 0.1 s. Fig. 22 shows the DFRFT spetra of the SoI in four different time windows. The proposed method was applied to the SoI, and the estimated-vibration-aeleration history and the orresponding vibration spetrum are shown in Figs. 23 and 24, respetively. The estimated vibration frequeny is 5.2 Hz, whih is lose to the atual vibration frequeny of 5.0 Hz. In this paper, a DFRFT-based method has been devised to estimate low-level vibrations of ground targets using the SAR platform. Unlike the JTFA, whih merely provides a qualitative illustration of the miro-doppler effet, the proposed method provides quantitative estimates of the vibration signature diretly from the SAR phase history, thereby produing the history of the instantaneous aeleration and the spetrum of the vibration. The performane of the proposed method has been haraterized quantitatively in terms of measurable vibration frequeny and displaement, as well as the signals SNR (in the SAR phase history) and observation window. In experiments utilizing the GA-ASI Lynx system, the proposed method suessfully retrieved two low-level vibrations from two different targets (with different SNRs). The proposed SAR-based vibration estimation method adds a new apability to modern SAR imaging systems. As suh, it enhanes the diversity and utility of SAR in appliations suh as target feature extration and objet reognition and lassifiation. In our future work, the effets of lutter and multiple satterers will be arefully examined. Modeling of real-world

11 WANG et al.: SAR-BASED VIBRATION ESTIMATION 4155 vibrating objets, suh as buildings and vehiles, ould also enable a better understanding of the miro-doppler effets in the SAR phase history and ould improve the performane of the proposed method in speifi appliations. ACKNOWLEDGMENT The authors would like to thank GA-ASI (San Diego, CA) for making the Lynx system available to this projet. REFERENCES [1] I. G. Cumming and F. H. Wong, Digital Proessing of Syntheti Aperture Radar Data: Algorithms and Implementation. Norwood, MA: Arteh House, [2] M. Soumekh, Syntheti Aperture Radar Signal Proessing With MATLAB Algorithms. New York: Wiley, [3] C. V. Jakowatz, D. E. Wahl, P. H. Eihel, D. C. Ghiglia, and P. A. Thompson, Spotlight-Mode Syntheti Aperture Radar: A Signal Proessing Approah. New York: Springer-Verlag, [4] V. C. Chen, F. Li, S.-S. Ho, and H. Wehsler, Miro-Doppler effet in radar: Phenomenon, model, and simulation study, IEEE Trans. Aerosp. Eletron. 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Raney, Syntheti aperture imaging radar and moving targets, IEEE Trans. Aerosp. Eletron. Syst., vol. AES-7, no. 3, pp , May [11] T. Sparr and B. Krane, Miro-Doppler analysis of vibrating targets in SAR, Pro. Inst. Elet. Eng. Radar Sonar Navigat., vol. 150, no. 4, pp , Aug [12] A. V. Oppenheim, R. W. Shafer, and J. R. Buk, Disrete-Time Signal Proessing, 2nd ed. Upper Saddle River, NJ: Prentie-Hall, [13] L. Cohen, Time Frequeny Transforms for Radar Imaging and Signal Analysis. Upper Saddle River, NJ: Prentie-Hall, [14] S. Qian and D. Chen, Introdution to Joint Time Frequeny Analysis Methods and Appliations. Englewood Cliffs, NJ: Prentie- Hall, [15] S. I. Tsunoda, F. Pae, J. Stene, M. Woodring, W. H. Hensley, A. W. Doerry, and B. C. Walker, Lynx: A high-resolution syntheti aperture radar, in Pro. SPIE Radar Sensor Tehnol. IV, 1999, vol. 3704, p. 20. [16] Q. Wang, M. Pepin, B. Santhanam, T. Atwood, and M. M. Hayat, SARbased vibration retrieval using the frational Fourier transform in slow time, in Pro. SPIE Radar Sensor Tehnol. XIV, 2010, vol. 7669, pp , K. I. Ranney and A. W. Doerry, Eds.. [17] J. G. Vargas-Rubio and B. Santhanam, On the multiangle entered disrete frational Fourier transform, IEEE Signal Proess. Lett., vol. 12, no. 4, pp , Apr [18] J. G. Vargas-Rubio and B. Santhanam, The entered disrete frational Fourier transform and linear hirp signals, in Pro. 11th DSP Workshop, 2004, pp [19] V. Namias, The frational order Fourier transform and its appliations to quantum mehanis, IMA J. Appl. Math, vol. 25, no. 3, pp , [20] L. B. Almeida, The frational Fourier transform and time frequeny representations, IEEE Trans. Signal Proess., vol. 42, no. 11, pp , Nov [21] B. Santhanam and J. H. MClellan, The disrete rotational Fourier transform, IEEE Trans. Signal Proess., vol. 44, no. 4, pp , Apr [22] S. C. Pei and J. J. Ding, Relations between frational operations and time frequeny distributions and their appliations, IEEE Trans. Signal Proess., vol. 49, no. 8, pp , Aug [23] J. G. Vargas-Rubio and B. Santhanam, An improved spetrogram using the multiangle entered disrete frational Fourier transform, in Pro. ICASSP, 2005, vol. 4, pp [24] L. R. Rabiner, R. W. Shafer, and C. M. Rader, The hirp z-transform algorithm, IEEE Trans. Audio Eletroaoust., vol. AU-17, no. 2, pp , Jun [25] L. S. Reddy, B. Santhanam, and M. M. Hayat, Multiomponent hirp demodulation using disrete frational Fourier transform, in Pro. 12th Digital Signal Proess. Workshop, 2006, pp [26] F. Auger, O. Lemoine, P. Gonçalvès, and P. Flandrin, The Time Frequeny Toolbox, pp , Ot [Online]. Available: Qi Wang (S 08) reeived the B.S. degree in eletrial engineering from Shanghai Jiaotong University, Shanghai, China, in He is urrently working toward the Ph.D. degree in the Department of Eletrial and Computer Engineering, University of New Mexio, Albuquerque. His researh interests inlude imaging algorithms for high-resolution syntheti aperture radar (SAR), detetion and estimation of nonstati targets for SAR, and nonstationary signal proessing. Matthew Pepin (M 79) reeived the B.S. degree in eletrial engineering (with distintion) and the M.E. degree from the University of Virginia, Charlottesville, in 1981 and 1982, respetively, and the Ph.D. degree from the Air Fore Institute of Tehnology, Wright Patterson Air Fore Base, OH, in He is urrently a Postdotoral Fellow with the Department of Eletrial and Computer Engineering, University of New Mexio, Albuquerque. His researh interests inlude sensors, spetral estimation, array proessing, and imaging algorithms. Ryan J. Beah is a urrently working toward the B.S. degree in mehanial engineering with a minor in mathematis at the University of New Mexio (UNM), Albuquerque. His work inludes undergraduate researh in the field of vibrometry for the Department of Eletrial and Computer Engineering, UNM, under Dr. Majeed M. Hayat. He is also involved in onduting laboratory experiments. His researh interests inlude designing and fabriation of laboratory equipment. Ralf Dunkel is originally from Germany. He reeived the B.S., M.S., and Ph.D. degrees from Martin Luther University of Halle Wittenberg, Halle, Germany. He is urrently the Diretor of System Reonnaissane Systems Group, General Atomis Aeronautial Systems, In., San Diego, CA. His primary areas of fous are airborne sensor testing and data analysis.

12 4156 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER 2012 Tom Atwood reeived the Ph.D. degrees in Eletrial and Computer Engineering from the University of New Mexio, Albuquerque, NM, in He has more than 20 years experiene in designing, building and fielding systems, spae, airborne and terrestrial, for Radar, Communiations and Signals Intelligene. He is urrently with Sandia National Laboratories, Albuquerque, NM. His urrent researh areas inlude: Software Reprogrammable Payloads and Cognitive Radio/Radar. Armin W. Doerry reeived the Ph.D. degree in eletrial engineering from the University of New Mexio, Albuquerque. He is urrently a Distinguished Member of the tehnial staff with the Surveillane and Reonnaissane Department, Sandia National Laboratories, Albuquerque. Sine 1987, he has worked in numerous aspets of syntheti aperture radar and other radar system analysis, design, and fabriation. Balu Santhanam (S 92 M 98 SM 05) reeived the B.S. degree in Eletrial Engineering from Saint Louis University, Saint Louis, MO, in 1992 and the M.S. and Ph.D. degrees in Eletrial Engineering from the Georgia Institute of Tehnology, Atlanta, in 1994 and 1998 respetively. He ompleted a year as a leturer and postdotoral researher in the department of Eletrial and Computer Engineering at the University of California Davis, from 1998 to In 1999, he joined the department of Eletrial and Computer Engineering at the University of New Mexio, Albuquerque, where he is urrently an assoiate professor of Eletrial Engineering. His urrent researh interests inlude disrete Frational Fourier analysis and appliations, multiomponent amplitude modulation/frequeny modulation models and demodulation, signal representations for modulated signals, hybrid independent omponent analysissupport vetor mahine systems and their appliation to pattern reognition problems. Dr. Santhanam was the reipient of the ECE-UNM distinguished teaher award in 2000 and He is urrently the hair of the Albuquerque hapter of the IEEE signal proessing and ommuniation soieties. He served as the tehnial area hair for adaptive systems for the Asilomar Conferene on Signals, Systems, and Computers, He served on the international program ommittee for the IEEE international onferene on Systems, Man, and Cybernetis in 2005 and Majeed M. Hayat (S 89-M 92-SM 00) was born in Kuwait in He reeived the B.S. degree (summa um laude) in eletrial engineering from the University of the Paifi, Stokton, CA, in 1985 and the M.S. and Ph.D. degrees in eletrial and omputer engineering from the University of Wisonsin, Madison, in 1988 and 1992, respetively. He is urrently a Professor of eletrial and omputer engineering with the University of New Mexio, Albuquerque, where he is an Assoiate Diretor of the Center for High Tehnology Materials and the General Chair of the Optial Siene and Engineering Program. From 2004 to 2009, he was an Assoiate Editor of Optis Express. Hehas authored or oauthored over 70 peer-reviewed artiles (H-Index of 24) and has four issued patents, three of whih have been liensed. His researh ativities over a broad range of topis, inluding avalanhe photodiodes, signal and image proessing, algorithms for infrared spetral sensing and imaging, optial ommuniation, and modeling and optimization of distributed systems and networks. Dr. Hayat is a Fellow of the Soiety of Photo-optial Instrumentation Engineers and a member of the Optial Soiety of Ameria. He is urrently the Chair of the Topial Committee of Photodetetors, Sensors, Systems and Imaging within the IEEE Photonis Soiety. He was a reipient of the National Siene Foundation Early Faulty Career Award (in 1998) and the Chief Sientist Award for Exellene (in 2006) from the National Consortium for Measures and Signatures Intelligene Researh and the Defense Intelligene Ageny. Walter Gerstle reeived the Ph.D. degree in strutural engineering from Cornell University, Ithaa, NY. Sine 1986, he has been with the University of New Mexio (UNM), Albuquerque, where he is urrently a Professor of ivil engineering. Reently, he has been working on the strutural design of optial and radio telesopes as well as researh in strutural aoustis with groups from the Department of Eletrial and Computer Engineering and the Department of Physis and Astronomy, UNM. His researh interests are in omputational strutural engineering and omputational mehanis.