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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