THE IMPACT OF SIGNAL MODEL DATA COMPRESSION FOR TDOA/FDOA ESTIMATION

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1 THE IMPACT OF SIGNAL MODEL DATA COMPRESSION FOR TDOA/FDOA ESTIMATION Mark L. Fowler & Xi Hu Department of Electrical & Computer Engineering State University of New York at Binghamton SPIE 2008 San Diego, CA August 12,

2 The Problem Source Compressed Compressed Estimate TDOA/FDOA for Pairs Use TDOA/FDOA to Locate 1970s 1980s 1990s 2000s Sonar-Driven Research Hann, Tretter, Knapp, Carter, Schultheis, Weinstein, Etc. Radar/Comm-Driven Research Stein, Chestnut, Berger, Blahut, Torrieri, Etc. Question: How much of the Sonar TDOA/FDOA estimation work can be carried over to the Radar/Comm arena?? Answer: Not as much as many Radar/Comm researchers/practitioners think! 2

3 Compression Framework Sensor #2 Our Focus Here Evaluate FIM for Task CRLB = FIM 1 Local Sensor Data Sensor #1 Remote Sensor Data T Q 1 Q 2 T -1 Estimation ˆ T-F Filter Bank Q N 3

4 Signals: Sonar vs. Radar/Comm Two sampled passively-received complex-valued baseband signals: rn snt e wn jν 1nT [ ] = ( τ ) + [ ] r n s nt e w n jν 2nT [ ] = ( τ ) + [ ] Noise Model Zero-mean WSS processes Gaussian Independent of each other This much is the same for each case At least when the narrowband approximation can be used which we assume here so we can focus on the impact of differences in the statistical model. 4

5 Models: Sonar vs. Radar/Comm Passive Sonar Signal = Sound from Boat Erratic signal behavior Model as Random Process Zero-mean WSS Gaussian Independent of Noise Expected values taken over signal + noise ensemble Estimation accuracy is average over all possible noises and signals Passive Radar/Comm Signal = Pulse Train Structured signal behavior Model as Deterministic Specific pulse shape Pulse width & spacing Expected values taken over only noise ensemble Estimation accuracy is average over all possible noises for one specific signal 5

6 PDFs: Sonar vs. Radar/Comm For both cases the received data vector is Gaussian. But how TDOA/FDOA is embedded is very different. This is the key it impacts significant differences in: Fisher Info Matrix (FIM) / Cramer-Rao Bound (CRB) ML Estimator Structure Passive Sonar PDF: p ac Passive Radar/Comm PDF p em 1 (; r ) = exp det ( πc) ( πc ) { H 1 r C r} 1 (; r ) = exp ( r s ) C ( r s ) det { H 1 } TDOA/FDOA in Covariance TDOA/FDOA in Mean 6

7 FIM/CRB: Sonar vs. Radar/Comm For general Gaussian case the elements of the FIM: H μ μ C C [ J gg ] ij 2Re C tr C C = + i j i j Leads to VERY different forms for the two cases: Passive Sonar FIM: [ J sonar ] ij tr C C C 1 1 = i j C Passive Radar/Comm FIM: H s 1 s [ J radar ] ij = 2Re C i j Difficult to assess usually use Whittle s Theorem Depends on Covariance Sensitivity to Parameter Easy to numerically assess Depends on Signal Sensitivity to Parameter 7

8 Impact of FIM: Sonar vs. Radar Because the forms are different any sonar-case result is unlikely to carry over to radar-case: Passive Sonar TDOA and FDOA Estimates are Uncorrelated Holds under mild assumption of large BT Passive Radar/Comm TDOA and FDOA Estimates are Correlated * * [ J ] em = 2Re 2 jnts ( nt τ1) s ( nt τ1) + jnts ( nt 2 τ2) s ( nt τ2) σ1 n σ2 n This has an impact on data compression 8

9 Compression: Sonar vs. Radar/Comm Doing data compression for radar case we need to account for the non-zero off-diagonal FIM elements The Correlation Issue 2 Fowler/Chen ICASSP 2005 Compressed FIM ellipse is inside the Uncompressed FIM ellipse 1 When estimates are highly correlated the compression can t perturb the correlation very much!!! Fowler/Chen Sbmt. to T-AES Theorem: For the transform coding framework outlined above, the postcompression FIM has an information ellipse that lies inside the original FIM ellipse. 9

10 R-D Viewpoints Three Views of Source Probabilistic Signal Class Specific Input Theoretical Bounds, Etc. (Classical R-D Theory) Compression that exploits typical behavior Compression optimized to exploit specific behavior ( Operational R-D Theory) Coding Scheme (framework): design based on typical features Coding Parameters (bit allocation): chosen on input-by-input basis to optimize to a particular input Operational R-D R-D methods don t care what other possible realizations might occur next time the the only only thing that that matters is is what does does the the data data actually collected look look like Deterministic Signal Model!!! 10

11 Comments The two signal models lead to important differences in the results for the FIM. We have argued regardless of the type of signal expected, when using the FIM as a distortion measure in an operational rate-distortion sense the signal should be viewed as deterministic. 11