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Co-registered Vibrometry Imaging: A Combined Synthetic-Aperture Radar Fractional-Fourier Transform Approach Univer

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**1. **Co-registered Vibrometry & Imaging: A Combined Synthetic-Aperture Radar & Fractional-Fourier Transform Approach University of New Mexico FY2008 University Project

**2. **Project Information Title of project: Co-registered Vibrometry and Imaging: A Combined Synthetic-Aperture Radar and Fractional-Fourier Transform Approach
Lead organization: University of New Mexico, Electrical & Computer Engineering Department
Project lead: Professor Majeed M. Hayat
Personnel:
UNM Faculty:
Prof. Majeed Hayat (ECE, 15%)
Prof. Balu Santhanam (ECE,15%)
Prof. Walter Gerstle (CIVIL Engr,15%)
Sandia collaborators: Tom Atwood and Toby Townsend (10%)

**3. **Program Details Date of award ($190,959 for FY08): Aug. 1, 2008
Date of receipt of funds: Aug. 1, 2008
Date work actually started: May 15, 2008 (via Pre-award)
Percent of FY-08 funds spent to date: ~80%
Percent of total work completed (over three year period) to date: ~33%

**4. **Imagine a turbine inside a building. We want to image the building and estimate the vibrations that may be exhibited by the building. To do so, we can use traditional SAR, which is known for its active illumination nature—not requiring daylight or thermal emission, and good resolution (well bellow 0.5m), and use a different platform for sensing vibrations via optical means, for example, and then we need to fuse the data from the two sensors. This step, in addition to requiring two sensing platforms, requires registration. What we’re doing is to use the SAR platform to do both., thereby simplifying the process and avoiding multi-sensor registration. Imagine a turbine inside a building. We want to image the building and estimate the vibrations that may be exhibited by the building. To do so, we can use traditional SAR, which is known for its active illumination nature—not requiring daylight or thermal emission, and good resolution (well bellow 0.5m), and use a different platform for sensing vibrations via optical means, for example, and then we need to fuse the data from the two sensors. This step, in addition to requiring two sensing platforms, requires registration. What we’re doing is to use the SAR platform to do both., thereby simplifying the process and avoiding multi-sensor registration.

**6. **We have an integral equation that we must solve; there may not be a unique solution but we can look at physical constraint. We can also send multiple pulses, each with different chirp rate, for example, and then approximate the inversion. As a first step, we considered a practical model for the unknown where we assume that vibrating objects are discrete, each with its own characteristics. We have an integral equation that we must solve; there may not be a unique solution but we can look at physical constraint. We can also send multiple pulses, each with different chirp rate, for example, and then approximate the inversion. As a first step, we considered a practical model for the unknown where we assume that vibrating objects are discrete, each with its own characteristics.

**9. **DFRFT Estimates

**10. **Previous Work: Vibration Identification Methodology

**11. **We have an integral equation that we must solve; there may not be a unique solution but we can look at physical constraint. We can also send multiple pulses, each with different chirp rate, for example, and then approximate the inversion. As a first step, we considered a practical model for the unknown where we assume that vibrating objects are discrete, each with its own characteristics. We have an integral equation that we must solve; there may not be a unique solution but we can look at physical constraint. We can also send multiple pulses, each with different chirp rate, for example, and then approximate the inversion. As a first step, we considered a practical model for the unknown where we assume that vibrating objects are discrete, each with its own characteristics.

**13. **Single Look: Vibration Frequency and Direction

**14. **Single Look Approach Envelope Fit Fitting the phase change envelope uses the slight change in amplitude of the vibration over the synthetic aperture
This method is least accurate around zero degrees when the vibration is directly aligned with the electromagnetic direction of propagation

**15. **Multilook Approach : Frequency and Direction Estimates

**16. **Animated Demonstration

**17. **Summary: 2-D Methodology

**18. **Enhancing Resolution via Non-uniform Frequency Sampling DFRFT: DFT of the sequence zk[p]:
Non-uniform DFT:
Evaluates Z-transform at locations of interest in the set zk

**19. **Nonuniform Sampling: NDFT Provides better peak resolution for larger in-band/out-band ratios
(¼ 0.8-1).
Frequency domain samples can be concentrated around DFRFT peaks.
Sharper peak locations translate to better center-frequency & chirp-rate estimates.

**20. **Subspace Approach DFRFT peak detection & chirp parameter estimation akin to DFT -- based sinusoidal frequency estimation: location of peak gives frequency estimate
Periodogram approach is statistically inconsistent. Subspace approaches yield asymptotically consistent estimates.
Covariance matrix of zk[p] is full-rank & eigenvalue spectrum not separable into S+N and N subspaces.
Subspace approach ? rank reduction needed.

**21. **Modeling Electromagnetic Wave Interactions with Vibrating Structures Goals:
Construct full-Maxwell’s equations models of the interaction of specific synthetic aperture radar pulses with vibrating objects
Produce simulated Doppler shift information for single / multi-mode vibrating buildings encompassing a variety of geometrical and material features.
Methodology:
Employ the finite-difference time-domain (FDTD) method, a grid-based, wide-band computational technique of great robustness
(~ 2,000 FDTD-related publications/year as of 2006, 27 commercial/proprietary FDTD software vendors)

**22. **FDTD Modeling Details Model the structures using an advanced algorithm that accommodates both the surface perturbations1, as well as their internal density modulations2.
Perform a near-to-far-field (NTFF) transformation to obtain the unique signatures of vibrating objects as would be recorded by a remote antenna system.
Complete the model with the advanced convolutional perfectly matched layer (CPML) to terminate the grid and a total-field/scattered-field formulation (TFSF) to generate the plane wave illumination of objects.

**23. **Ongoing and Future FDTD Work Current status and ongoing work:
We have implemented a 2-D FDTD model incorporating the CPML boundary conditions, NTFF transformation, TFSF formulation and surface vibrating perturbations.
Next steps will be to use the validated code to model a variety of structural geometries (rough surfaces, edges, corners) and materials (concrete, etc.), vibrating at specific modes as specified by the civil engineers on our team.
Future Work:
Extend the 2-D model to a fully 3-D simulation of synthetic aperture radar signals interacting with vibrating structures.

**24. **Modeling Vibrations and Physical Structures

**26. **SAR Vibrometry Laboratory Planning Simple laboratory for the experimental demonstration SAR-based vibrometry
Initial equipment concept complete
UNM Space allocated

**27. **Summary of Effort Against Objectives

**30. **Project Self-Assessment

**31. **Patents, Publications, and Experiments Associated with Project Q. Wang, M. M. Hayat, B. Santhanam, and T. Atwood, “SAR Vibrometry using fractional-Fourier-transform processing,” SPIE Defense & Security Symposium: Radar Sensor Technology XIII (Conference DS304), Orlando, FL, April 2009.
B. Santhanam, S. L. Reddy, and M. M. Hayat, “Co-channel FM Demodulation Via the Multi Angle-Centered Discrete Fractional Fourier Transform,” 2009 IEEE Digital Signal Processing Workshop," Marcos Islands, Jan. 2009, FL, 2009.
M. Madrid, J. J. Simpson, B. Santhanam, W. Gerstle, T. Atwood, and M. M. Hayat, "Modeling electromagnetic wave interactions with vibrating structures," IEEE AP-S International Symposium and USNC/URSI National Radio Science Meeting, Charleston, SC, June 2009, accepted.

**32. **Summary