Ambiguity in Radar and Sonar

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Ambiguity in Radar and Sonar

• Paper by
• M. Joao D. Rendas and Jose M. F. Moura
• Information theory project
• presented
• by
• VLAD MIHAI CHIRIAC

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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

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Introduction (cont.)

• Ambiguity function for Barker code

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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

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Kullback direct divergence

• The Kullback direct divergence is a measure of
  similarity between probability densities.
• The KDD between two multivariate Gauss pdfs p
  and q, which have the same ? and distinct
  covariance matrices R? and R?0

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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.

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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.

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The two step principle
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Generalized log-likelihood ratio
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Model

• Source signal

• Received signal

• Channel model

• Noise interference

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Ambiguity No nuisance parameters

• The ambiguity function when we estimate ?,
  conditioned on the occurrence of ?0 is

where Iub(?0) is an upper bound of I(?0?)
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Ambiguity Unwanted parameters

• Two subfamilies

VS

• The generalized likelihood ratio

where
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Ambiguity Unwanted parameters (cont.)
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Ambiguity Unwanted parameters (cont.)

• 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

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Ambiguity Modeling inaccuracies

• For this situation the model is

where ? is a vector which contains parameters,
approximately known associated with propagation
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Ambiguity Modeling inaccuracies (cont.)

• The generalized likelihood ratio

• Consider the parameter estimation problem
  described by the curved exponential family G000?
  using the probabilistic model G001? at the
  receiver.
• The ambiguity function in the estimation of ?,
  given that ?0 is the true value of the parameter
  is