1. Demonstration of Target Vibration Estimation in Synthetic Aperture Radar Imagery Qi Wang1,2, Matthew Pepin1,2, Ryan J. Beach2, Ralf Dunkel3, Tom Atwood2,4, Armin W. Doerry4, Balu Santhanam2, Walter Gerstle5, and Majeed M. Hayat1,2 1Center for High Technology Materials, University of New Mexico, Albuquerque, NM, 87131 USA 2Depart. of Electrical and Computer Science, University of New Mexico, Albuquerque, NM, 87131 USA 3General Atomics Aeronautics Systems, Inc., San Diego, CA 92064 USA 4Sandia National Laboratories, Albuquerque, NM 87185 USA 5Depart. of Civil Engineering, University of New Mexico, Albuquerque, NM, 87131 USA

2. Content Motivation Signal model Algorithm development The discrete fractional Fourier transform Experimental results Performance analysis Conclusions

4. The ability of remote sensing target vibrations is important:

5. Avoids the cost of acquiring and installing accelerometers on remote structures

6. Enables sensing vibrations of structures that not easily accessible

8. Micro-Doppler effect RF pulse: X0 y (azimuth) Frequency (Hz) t (sec) Doppler frequency caused by both the plane’s motion and target vibration Doppler frequency caused by the plane’s motion

10. fc: carrier frequency

11. c: speed of the microwave

12. V: speed of the antenna

13. y: azimuth position of the vibrating scatterer

14. fprf: the pulse repetition frequency

15. rd[n]: time-varying range of the vibrating scatterer

16. φ: other (constant) phase terms

17. NI: total number of pulses

19. Concentrates a linear chirp into a few coefficients we obtain an impulse-like transform analogous to what the discrete Fourier transform produces for a sinusoid.Intersection for α = π/2 The DFRFT of a complex-valued signal containing two component: a pure 150 Hz sinusoid and a chirp with a center frequency of 200 Hz and chirp rate of 400 Hz/s.

20. The DFRFT (cont’d) Let W denote the transformation matrix of the centered-DFT, the fractional power of W is defined as The DFRFT of a signal x[n] is the DFT of an intermediate signal for each index k (k = 0,1, …,N-1), that is where r = 0,1,…,N-1 is the newly introduced angular index and α=2πr/N. The intermediate signal is calculated by where vpis the pth column vector of VG. J. G. Vargas-Rubio and B. Santhanam, “On the multiangle centered discrete fractional Fourier transform,” IEEE signal Processing Letters, vol. 12, pp. 273-276, 2005

24. The vibration is caused by the rotation of an unbalanced mass.

25. Vibration magnitude: 1.5 mm; vibration frequency: 5 HzSAR image of the test ground near Julian, CA. It was generated by the Lynx system on 2010.

26. Experiment: the DFRFT spectra The changing of position of the peak in the DFRFT plain indicates a time-varying vibration acceleration of the target. Angle (rad) Frequency (Hz)

27. Estimation results Estimated vibration acceleration (x: time (s); y: acceleration ( m/s2) Estimated DFT spectrum of the vibration (x: frequency (Hz); y: magnitude (AU) Actual vibration frequency: 5 Hz

28. Performance analysis The vibration frequency resolution is limited by the SAR observation time of the target For the Lynx system, it ranges from 0.3 Hz to 1.0 Hz depending on the data collection geometry and the radar cross section of the target. The maximum measurable vibration frequency is upper-bounded by fprf/2 theoretically In practice, vibration frequencies up to 25 Hz can be easily estimated when fprf = 1000 Hz;

29. Conclusions A DFRFT-based method is proposed for SAR vibration estimation In the experiment, the proposed method successfully estimated a 1.5 mm, 5-Hz vibration from a corner reflector Performance analysis of the proposed method is carried out in terms of vibration frequency resolution and maximum measurable vibration frequency

30. Acknowledgement This work was supported by the United States Department of Energy (Award No. DE-FG52-08NA28782), the National Science Foundation (Award No. IIS-0813747), National Consortium for MASINT Research, and Sandia National Laboratories. The authors also thank GA-ASI for making the Lynx system available to this project.