Ambiguity in Radar and Sonar

Paper byM. Joao D. Rendas and Jose M. F.

MouraInformation theory project

presentedby

VLAD MIHAI CHIRIAC

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

Introduction (cont.)

Ambiguity function for Barker code

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

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 R0

: lnp

pI p q E

q

1 10 0

1: ln

2I p q tr R R N R R

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.

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.

ˆ ˆarg min : |I p r p r a

The two step principle

G

G

ˆ|p r a

p̂ r*a

A

A ˆ 'a

ˆ ''a

Probabilistic ModelNatural

Parameter

Generalized log-likelihood ratio

1 1

0 0 1 1

0 0

1

0 1 0 10

0 1

maxˆ ˆ, ln min : min :

max

ˆ ˆ: :

p G

p G p Gp G

p rH H I p r p I p r p

p r

I p r p I p r p

G

G0 0

ˆ|p r a

p̂ r*a

A

A 0

ˆa

1ˆa

G1 1

ˆ|p r a

Natural parameter

Probabilistic model

Model

• Source signal:

( ) : ,r t C f t w t t T

: k kC f t a f t • Received signal:

• Channel model:

,f t

• Noise + interference: w t

Ambiguity: No nuisance parameters

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

0

0

0 0

1

0

0

:

:

: :

:

T T

T T

H r p p r

H r p p r

I p p

I

0

00

:, 1

ub

I

I

where Iub(0) is an upper bound of I(0:)

G

G0

0|p r a

p̂ r

*a

A

A a

Natural parameterProbabilistic model

Ambiguity: Unwanted parameters

• Two subfamilies:

00 , ,G p r

, ,G p r

00 :H p r G 1 :H p r GVS

• The generalized likelihood ratio:

01 1

0 0 1 0: min : :p G

I p p I

where 00 0 00 0 0 0: : , :

ppI I p p r I p q q G

0

0 0arg min : ,p

I p p r

Ambiguity: Unwanted parameters (cont.)

G

G1 1| ,p r

p̂ r

*a

A

A 0

ˆa

Natural parameter

Probabilistic model

G2

G0 0| ,p r

2| ,p r

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:

0

0

0

00

:, 1

ub

I

I

Ambiguity: Modeling inaccuracies• For this situation the model is:

,( ) ,r t C f t w t t T where is a vector which contains parameters,

approximately known associated with propagation

G0

p̂ r

*a

A

A 0ˆa

Natural parameter

Probabilistic model real one

G00

00|p r

G1

11|p r

Probabilistic model used at receiver

G10

10|p r

G11

Ambiguity: Modeling inaccuracies (cont.)

• The generalized likelihood ratio:

0 0 0

0 1 0 10 , : :I p p I p p

• 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:

0

0 1

00

:, 1

I p p

I