1. Ambiguity in Radar and Sonar Paper by M. Joao D. Rendas and Jose M. F. Moura Information theory project presented by VLAD MIHAI CHIRIAC

2. Introduction Radar is a system that uses electromagnetic waves to identify the range, altitude, direction, or speed of both moving and fixed objects such as aircraft, ships, motor vehicles, weather formations, and terrain. The ambiguity is a two-dimensional function of delay and Doppler frequency showing the distortion of an uncompensated match filter due to the Doppler shift of the return from a moving target

3. Introduction (cont.) Ambiguity function for Barker code

4. Introduction (cont.) Ambiguity function from the point of view of information theory and is based on Kullback directed divergence Models: - radar/sonar with unknown power levels - passive in which the signals are random - mismatched

5. Kullback direct divergence The Kullback direct divergence is a measure of similarity between probability densities. The KDD between two multivariate Gauss pdf’s p and q, which have the same  and distinct covariance matrices R  and R  0

6. Types of probability distribution functions Exponential densities (Gauss, gamma, Wishart and Poisson). These distribution depends on unspecified parameter called natural parameter The subfamily of exponential pdfs that results by parametrizing the natural parameter is called the curved exponential family.

7. Estimation of the interest parameters Estimate the natural parameter from the measured samples by computing the unstructured maximum- likelihood (ML) Estimate the desired parameters by minimizing the KDD distance between the true pdf and the curved exponential family.

8. The two step principle

9. Generalized log-likelihood ratio

10. Model Source signal: Received signal: Channel model: Noise + interference:

11. Ambiguity: No nuisance parameters The ambiguity function when we estimate , conditioned on the occurrence of  0 is: where I ub (  0 ) is an upper bound of I(  0 :  )

12. Ambiguity: Unwanted parameters Two subfamilies: VS The generalized likelihood ratio: where

13. Ambiguity: Unwanted parameters (cont.)

14. Consider the problem of estimation of the parameter  from observations described by the model G , where  is an unknown nonrandom vector of parameters. Definition – Ambiguity: The ambiguity function in the estimation of  conditioned on the occurrence of  0 = (  0,  0 ) is:

15. Ambiguity: Modeling inaccuracies For this situation the model is: where  is a vector which contains parameters, approximately known associated with propagation

16. Ambiguity: Modeling inaccuracies (cont.) The generalized likelihood ratio: Consider the parameter estimation problem described by the curved exponential family G 000  using the probabilistic model G 001  at the receiver. The ambiguity function in the estimation of , given that  0 is the true value of the parameter is: