Limitations and Capabilities of the Slanted Spectrogram Analysis Tool for SAR-Based Detection of Multiple Vibrating Targets

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1 Limitations and Capabilities of the Slanted Spetrogram Analysis Tool for SAR-Based Detetion of Multiple Vibrating Targets Adebello Jelili, Balu Santhanam, and Majeed Hayat Department of Eletrial and Computer Engineering, University of New Mexio, Albuquerque, NM USA Abstrat The reently developed improved spetrograms that use the disrete frational fourier transform (DFrFT) are used to evaluate multiple vibration signatures that represent targets in syntheti aperture radar (SAR) data. Multiple ground target vibrations that introdue phase modulation in the SAR returned signals are examined using standard pre-proessing of the return waveform signal followed by the appliation of the DFrFT. In this paper, we study the apabilities of these spetrogram tools with the intent of extending their limits by varying the harateristi features assoiated with eah vibrating target whih result in different outputs. The performane of the frational spetrogram tools, under the effets of various parameters is used to larify the limitations and advantages of these modified frational spetrograms for SAR-based vibrometry appliations. Index Terms syntheti aperture radar, disrete frational Fourier transform, miro-doppler effet, vibration estimation, lutter. I. INTRODUCTION Radar has long been used for military and non-military purposes in a wide variety of appliations suh as imaging, guidane, remote sensing and global positioning. Vibration signatures assoiated with objets suh as ative strutures and vehiles an bear vital information about the type and integrity of these objets [3]. These vibrating objets ause phase modulation of the azimuth phase history for a SAR system and this phase modulation is seen as a time-dependent miro-doppler frequeny [8] whih is useful for analyzing suh signals with appropriate time-frequeny methods in order to preserve it superior resolution potential [2]. Implementation of the DFrFT, whih is a generalization of the Disrete Fourier Transforms (DFT), shows promise for multiomponent hirp parameter estimation as it generates a strong peak for eah hirp whose loation in the 2D transform plane orresponds to the speifi enter frequeny and hirp rate [6]. In this paper, we will desribe the appliation of previously developed improved DFrFT-based spetrograms [6] on SAR data in terms of three riteria in order to properly evaluate and larify the limitations and usefulness of these tools. Speifially, we explore the onnetion between vibration frequenies and amplitudes on the number of side-lobes present in the This work was supported by the United States Department of Energy (Award No. DE-NA ) and the National Nulear Seurity Administration (NA-221) spetrograms. We further examine the effets of the hirp rates and lutter on the ability of these spetrograms in deteting vibrating targets. A. Motion Model II. SYNTHETIC APERTURE RADAR Fig 1, depits a three-dimensional SAR flight geometry, with a vibrating target loated at the origin. The nominal lineof-sight distane from the target to the radar sensor is r o, with the radar sensor loated at polar angles ψ and ϕ 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 Fig. 1: Three dimensional SAR flight o-ordinate system for the desription of an osillating point target. The vibrating target is loated at (r o, Ψ, ϕ) r(t) r o r d (t). (1) In this paper, we onsider the ase of broadside spotlight-mode SAR for whih the aforementioned approximation is valid [2]. B. Signal Model The small range perturbation of the vibrating target modulates the olleted SAR phase history [7]. Consider a spotlightmode SAR whose sent pulse is a hirp signal, with arrier frequeny and the hirp rate f and K, respetively. 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 [2] r(t) = i [ σ i exp j 4π(r i r ) (f + K(t 2r ) ], (2) where σ i is the refletivity of the ith satterer, is the propagation speed of the pulse, and r is the distane from /14/$ IEEE 172 Asilomar 2014

2 the path enter to the antenna [3], [4]. 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 the vibrating point target. Assuming that all satterers at a speifi range are stati, the rangeompressed phase history at this speifi range an be written as x[n] = i σ i [n] exp[ j(f y y i n 4πf r i + ϕ i )] + w[n], (3) for 0 n < N I, where y i is the ross-range position, ϕ i represent additional phase terms, and w[n] is additive noise with f y as the known imaging fator aording to [7]. We define the signal of interest (SoI) as the range line in the range-ompressed phase history ontaining vibrating targets. 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 i [n] = σ i [n] exp[ j(f y yn 4πf r d [n] + ϕ)] + w[n]. (4) Fig. 3: DFrFT magnitude of a frame of the SAR signal, The piee-wise linear spetrogram (red) passes through all the peaks, the polynomial spetrogram (yellow) also passes all peaks, the slanted spetrogram (green) aommodates only one peak and ignores the other two, the onventional spetrogram (blue) also ignores two peaks. related. So the DFrFT analysis tool shows promise for the analysis of signals frequeny hirping (i.e. linearly hanging) [6] suh as radar as employed in this paper. The onventional spetrogram assumes a multiomponent sinusoidal model over the analysis frame while the slanted spetrogram replaes that sinusoidal assumption with a multiomponent harmonially related hirp model. However, the data used in this paper ontains non-harmonially related hirps, therefore two DFrFT based spetrograms; polynomial fitted and piee-wise linear approahes were employed, whih provide sharper features than the slanted and onventional spetrogram as desribed in [7]. IV. SIMULATION AND PERFORMANCE ANALYSIS Fig. 2: The reonstruted SAR Image for vibrating target that introdues ghost lines along the azimuth diretion. III. IMPROVED SPECTROGRAMS The generalizations of the DFT suh as the DFrFT and in partiular, the entered version of the DFrFT based on the Grunbaum tridiagonal ommutor [3] have been shown to possess the apability to onentrate a hirp signal in a few transform oeffiients [4]. The improved spetrogram assumes a multiomponent sinusoidal model over the analysis frame by using the CDFrFT and the MA-CDFrFT to deompose the signal frame into the superposition of harmonially related hirp signals. In this paper, we apply the slanted spetrogram framework to non-harmoni hirp omponents using both piee-wise linear and polynomial fitted methods aording to [10]. Simulation results on syntheti hirps indiate that these generalized slanted spetrograms provide sharper features when the hirp omponents are not harmonially We use a SAR signal ontaining three vibrating objets with varying amplitudes and vibrating frequenies respetively for system parameters desribed for both noise free ase and noise ase. Using an initial sampling frequeny of MHz, a down-sampler is applied on the data to a sampling rate of 40 KHz. We alulated the spetrograms of the SAR signal using 256 point DFrFT for all ases and a 256 point Hamming window with 255 samples of overlap. The polynomial orders were varied throughout the experiment and the radar signals ontained omplex signals with three hirp omponents. The harateristis of this tool and its appliation to SAR are dependent on different parameters whih inlude (a) the vibrating frequenies, the amplitude of targets, () the hirprate effets, and (d) lutter and noise. During this investigation and evaluation, the polynomial fitted and the piee-wise linear produe the same results for most part, onsidering that SAR signals are non-harmonially related. The line onneting the zero hirp-rate and zero frequeny oordinates to the largest peaks does not pass through all the spetral peaks, therefore the onventional spetrogram and the slanted spetrogram produed blurry results sine both spetrogram benefits only 173

3 one peak and ignore two peaks aording to Fig 3. The polynomial fitted and the piee-wise linear spetrum provide sharper features and learer spetrograms sine they both take into aounts all the peaks. We used SAR signals with different ombinations of vibrating frequenies aording to Table I. for different ases. A. Effets of vibration frequenies and amplitudes The frequeny and amplitude of the SAR return signal measure the vibration signatures of both the target and lutter in the return waveform. Therefore, onsidering three vibrating targets by varying and inreasing their respetive amplitudes and vibrating frequenies inreases the number of side lobes present in the spetrograms as shown in Fig 4-6. Following these results, the degree of vibration present in the input data an be estimated and evaluated at different values. From the onduted simulations, it an be dedued that the presene of side-lobes is proportional to the degree of non-stationarity reeived from the SAR return signals and hanges in the enter frequenies have no effet on the output results. Comparing the different spetrogram outputs, the onventional performs equally well as the slanted spetrogram beause the latter ignores most peaks while both of the improved DFrFT spetrograms loate all peaks present. Table I: Different ombinations of targets parameter values used for evaluating the improved spetrograms, slanted spetrogram and onvention spetrogram. Case I Case II Case III T arget A B C A B C A B C Amplitude(mm) F requeny(hz) Case I : The old-slanted spetrogram ignores two of the peaks. This results from the fat that different frequenies generate a non-harmonially related signal where the oldslanted spetrogram annot loate all three peaks as shown in Fig 4. Case II : Presene of larger side-lobes ours with eah of the spetrogram tool, the onventional and old-slanted ignore most of the peaks and produe larger side-lobes than the improved spetrogram as shown in Fig 5. Though, the polynomial spetrogram output does not math that of the piee-wise spetrogram, the peaks for eah target an still be seen learly. These results an be dedued from the fats that the presene of targets with large vibration amplitude produe blurry peaks for the slanted spetrogram and generate larger side lobes for the improved spetrograms. Case III : This omprises of the ombination of CASE II and III with greater differenes in the values of the frequenies and amplitudes respetively. These hanges in vibration frequenies and amplitudes result in the presene of side-lobes for spetrograms (I), (III) and (IV) sine the amplitudes have values of 10 mm, 20 mm and 1 mm. Estimating the frequeny and amplitude thresholds due to the results from the above ases, we an derive boundaries for the vibration frequeny that an be deteted using the spetrogram tools. We know from [7] that the vibration frequeny f m is linearly proportional to the nominal veloity of the moving or rotating target. Correlating this to the level of non-stationarity of the return signal, we an dedue that f m is proportional to the level of vibration via stationarity deviations: S df = 1 f st /f m and S da = 1 A st /A m, where f m is the vibration frequeny, f st is the stati objet frequeny, and A vb,a st are the amplitudes of the vibrating and stati targets. The stationarity deviation perentage S d was alulated and the threshold for detetion was found within S df [0, 0.85] for the improved spetrograms and S d = for both onventional and slanted spetrogram. Using the same relationship for the amplitude, the tools fails to detet all targets out of range of S da [0, 0.98]. Therefore, we an exploit this information to determine the level of non-stationarity a signal ontains from the width of side-lobes in the output graphs, whih in turn gives us the approximate value of f m, A vb and veloity of the moving target. B. Chirp-rate effet SAR signals an in general be omprised of nonharmonially related hirp omponents resulting in the peaks not onneted by a single line and not passing through the zero hirp-rate and zero frequeny oordinates as shown in Fig 3. In order to evaluate the effet of these hirp-rates on the improved spetrograms, we look at SAR hirp signals omprised of three non-harmoni omponents with hirp rates Cr = [0.01, 0.05, 0.1] rad/samples 2 having vibrating frequenies of 1 Hz, 2 Hz and 2 Hz for eah target respetively. The polynomial fitted and the piee-wise linear methods take into aount all the peaks regardless of the value of Cr as shown in Fig 7-8. Fig. 7: Extrated spetrograms for a frame of non-harmonially related SAR hirp signal using Cr = 0.05 rad/samples 2. Estimating the hirp-rate thresholds from the simulation results, we observed that the higher the value of the hirp 174

4 (a) Fig. 4: (a) Extrated spetrograms for a frame of a non-harmonially related SAR hirp signal. Spetrogram of a SAR signal ontaining three vibrating target signal using spetrogram tools (I), (II), (III) and (IV) for CASE I. (a) Fig. 5: (a) Extrated spetrograms for a frame of a non-harmonially related SAR hirp signal. Spetrogram of a SAR signal ontaining three vibrating target signal using spetrogram tools (I), (II), (III) and (IV) for CASE II. (a) Fig. 6: (a) Extrated spetrograms for a frame of a non-harmonially related SAR hirp signal. Spetrogram of a SAR signal ontaining three vibrating target signal using spetrogram tools (I), (II), (III) and (IV) for CASE III. rates K, the higher the degree of non-stationarity of the output signals. Fig 7-8 shows that as the value of K inreases the slanted and onventional spetrogram gradually fail to detet all targets. Therefore, we an onlude that if the minimum measurable hirp rate for the slanted spetrogram (K min = 0.001rad/samples 2 ) denotes the stationary target s hirp rate 175

5 and using K min as a lower limit measure for deteting the vibration aeleration, sine it had been established in [8] that the hirp rate K is diretly proportional to the vibration aeleration a v for a target signal. For values of K > K min, the ratio of detetion ρ k = K/K min was used to estimate the threshold for the slanted spetrogram to be within range 1 < ρ k 4.5 and the improved spetrogram detets all target regardless of the value of K. This an be used to estimate the value of the vibration aeleration a v of targets. Fig. 8: Extrated spetrograms for a frame of non-harmonially related SAR hirp signal using Cr = 0.1 rad/samples 2. C. Clutter Effets Using vibrating frequenies of 1 Hz eah and amplitude of (1, 1, 10 ) mm respetively with lutter orresponding to a gamma-theta value of 1 whih is fixed with pixel size of N M to derive the SCR salar as a normalizing fator, along SCR and SNR of 20dB eah. As is evident from Fig 9, all targets were deteted by the improved spetrograms even with low values of SNR and SCR, but the slanted and onventional exhibit some target suppression due to high interferene of the lutter and noise even after appropriate filtering. V. CONCLUSION In this paper, we arried out a qualitative evaluation of the limits and apabilities of the slanted spetrogram in omparison with the improved spetrograms by evaluating the performane of eah tool using parameter variations. From the results, we an onlude that varying the harateristis of the input signal, the performane of the improved spetrograms outperforms the old-slanted spetrogram in terms of its ability to learly detet the number of targets present in the SAR signals without distortions or elimination of targets. We also notied that a minimum hirp rate or vibration frequeny/amplitude is needed before eah of these improved frational spetrogram tools an provide improvement over the slanted spetrogram. Therefore, from the analysis of eah tool it an be onluded that the non-stationary harater of the return signal manifests as side-lobes in the spetrograms whih an be observed when an inrease in the hirp rate or vibration frequeny and amplitude of the returned signal ours. In general, we an infer that the non-stationarity of the signal is dependent on the values of A vb, K and f m, whih an be used for the haraterization of target variables for reognition purposes. This vital information will be used in future work to establish a detailed relationship between these parameters and to populate a database of different target environments. REFERENCES [1] I. G. Cumming and F. H. Wong, Digital proessing of syntheti aperture radar data: algorithm and implementation, Arteh House, Norwood, MA, [2] C. V. Jakowatz, and D. E. Wahl, and P. H. Eihel, and D. C. Ghiglia and P. A. Thompson, Spotlight-Mode syntheti aperture radar: A signal proessing approah, Springer-Verlag, New York, [3] V. C. Chen, and F. Li, and S. S. Ho, and H. Wehsler, Miro-Doppler effet in radar: Phenomenon, model, and simulation study, IEEE Trans. Aerosp. Eletron. Syst., Vol. 42, No. 1, pp. 2-21, January, [4] Q. Wang, and M. Xing, and G. Lu, and Z. Bao, High-resolution three-dimension radar imaging for rapidly spinning target, IEEE Trans. Geosi. Remote Sens.,Vol. 46, No. 1, pp , January, [5] L. S. Reddy, and B. Santhanam, and M. M. Hayat, Multiomponent hirp demodulation using disrete frational Fourier transform, IEEE Pro. 12th Digital Signal Proessing Workshop, pp , Sept., [6] J. G. Vargas-Rubio and B. Santhanam, An Improved Spetrogram Using The Multi-angle Centered Disrete Frational Fourier Transform, Pro. of ICASSP, pp , [7] Q. Wang, and B. Santhanam, and M. M. Hayat, and R. Dunkel, and T. Atwood, and A. W. Doerry and M. Pepin, and R. J Beah, and W. Gerstle, SAR-based vibration estimation using the disrete frational fourier transform, IEEE Trans. Geosi. Remote Sens., Vol. 50, No. 10, pp , Ot., [8] Q. Wang, and B. Santhanam, and M. M. Hayat, and R. Dunkel, and T. Atwood et. al., Demonstration of target vibration estimation in syntheti aperture radar imagery, Pro. of IEEE Int. Geosi. Remote Sens. Symp., pp , [9] Shen Chiu, Clutter effets on ground moving target veloity estimation with SAR Along-Trak Interferometry, Pro. IEEE Int. Geosi. Remote Sens. Symp., Vol. 2, pp , July, [10] O. Agaoglu, and B. Santhanam, and M. Hayat, Improved Spetrograms using Disrete Fration Fourier Transform, Pro. of IEEE DSP/SPED Workshop, pp , Aug., Fig. 9: Extrated spetrograms for the orrupted SAR signal with three targets having the same vibration frequenies of 1 Hz eah. 176