ambiguity in radar and sonar paper by m. joao d. rendas and jose m. f. moura information theory...

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