GWDAW-Gini.ppt

Coherent Target Detection
in Heavy-Tailed
Compound-Gaussian Clutter:
Survey and New Results
GWDAW 2010 - January 26, 2010 – Rome, Italy
Kevin J. Sangston
Atlanta, Georgia (GA)
jsangston@mac.com
Fulvio Gini, Maria S. Greco
Dept. of Ingegneria dell’Informazione
University of Pisa
Via G. Caruso 16, I-56122, Pisa, Italy
f.gini@iet.unipi.it, m.greco@iet.unipi.it

K.J. Sangston, F. Gini, M. Greco, GWDAW 2010, Jan. 26, Rome, Italy
2
Introduction
Outline
of the
Talk
Radar
Clutter
Modelling
Coherent
Radar
Detection
Three
formulations
of the OD Sub-optimum
Detectors
Concluding
Remarks
1
2
3 4
5
7
“Coherent Target Detection in Heavy-Tailed
Compound-Gaussian Clutter: Survey and New Results”
GLRT-LTD
and its
Performance
6

3
Radar systems detect targets by
examining reflected energy,
or returns, from objects
Along with target echoes, returns come
from the sea surface, land masses,
buildings, rainstorms, and other sources
Much of this clutter is far stronger than
signals received from targets of interest
The main challenge to radar systems is
discriminating these weaker target
echoes from clutter
Coherent signal processing
techniques are used to this end
• The IEEE Standard Radar Definitions (Std 686-1990) defines coherent signal
processing as echo integration, filtering, or detection using the amplitude of the
received signals and its phase referred to that of a reference oscillator or to the
transmitted signal.
Key of the definition: use of phase information in the signal processing
Courtesy of SELEX S.I.
Introduction

4
Radar Clutter Modeling and the Detection Problem
The problem of coherent detection splits naturally into two fundamental
sub-problems:
In the first sub-problem, a model must be formulated for the underlying
disturbance (noise and/or clutter) from which the signals of interest are to be
discriminated.
The solution of this problem must achieve a balance between the
faithful representation of the actual disturbance and the mathematical
tractability of the resulting model.
In the second sub-problem, a detection structure which minimizes the loss
relative to the optimum detection structure, while at the same time being easy
to implement, must be obtained.
The mathematical tractability of the disturbance model is a significant factor
in determining the ease of implementation of the resulting detection structure.
The formulation of the optimal detector depends on the model that is
utilized to describe the non-Gaussian statistics of the disturbance.

5
In early studies [Gol51], the resolution capabilities of radar systems were
relatively low, and the scattered return from clutter was thought to comprise
a large number of scatterers
From the Central Limit Theorem (CLT), researchers in the field were led to
conclude that the appropriate statistical model for clutter was the Gaussian
model for the I & Q components, i.e., the amplitude R is Rayleigh distributed
Radar clutter modeling
R  Z  ZI
2
 ZQ
2
Z  ZI  jZQ
pZ (z) 
1
22 exp 
z2
22





 pR(r) 
r
2 exp 
r2
22





u(r)
2
, (0, ), I.I.D.
I Q
Z Z N 


6
In the quest for better performance, the resolution capabilities of radar systems
have been improved
For detection performance, the belief originally was that a higher resolution
radar system would intercept less clutter than a lower resolution system,
thereby increasing detection performance
However, as resolution has increased, the clutter statistics have no longer
been observed to be Gaussian, and the detection performance has not
improved directly
The radar system is now plagued by target-like “spikes” that give rise to
non-Gaussian observations
These spikes are passed by the detector as targets at a much higher false
alarm rate (FAR) than the system is designed to tolerate
The reason for the poor performance can be traced to the fact that the
traditional radar detector is designed to operate against Gaussian noise
New clutter models and new detection strategies are required to reduce
the effects of the spikes and to improve detection performance
Radar clutter modeling

7
Empirical studies have produced several candidate models for the
amplitude of spiky non-Gaussian clutter, the most popular being the Weibull
distribution, the K distribution, the log-normal, the generalized Pareto, etc.
Empirically observed models
Measured sea clutter data
(IPIX database)
The APDF parameters have
been obtained through the
Method of Moments (MoM)
the Weibull , K,
log-normal, etc.
have heavier tails
than the Rayleigh
10
-3
10
-2
10
-1
10
0
10
1
10
2
0 0.2 0.4 0.6 0.8 1
Weibull
Lognormal
K
GK
LNt
IG
Histogram
pdf
Amplitude (V)
r

8
The multidimensional compound-Gaussian model
In practice, radars process m pulses at time, thus, to determine the optimal radar
processor we need the m-dimensional joint PDF
Since radar clutter is generally highly correlated, the joint PDF cannot be derived
by simply taking the product of the marginal PDFs
The appropriate multidimensional non-Gaussian model for use in radar detection
studies must incorporate the following features:
1) It must account for the measured first-order statistics (i.e., the APDF
should fit the experimental data)
2) It must incorporate pulse-to-pulse correlation between data samples
3) It must be chosen according to some criterion that clearly distinguishes
it from the multitude of multidimensional non-Gaussian models
satisfying 1) and 2)

9
Conte and Longo [Con87] presented the idea of modelling the radar sea clutter
process as a spherically invariant random process (SIRP)
This idea leads to a multidimensional model of the clutter that includes
samples correlation and need not be Gaussian. By sampling a SIRP we obtain
a spherically invariant random vector (SIRV) whose PDF is given by
where z=[z1 z2 . . . zm]T is the m-dimensional complex vector representing
the observed data
A random process that gives rise to such a multidimensional PDF can be
physically interpreted in terms of a locally Gaussian process whose power
level t is modulated because of the radar's large scale sampling of the environment
The PDF of the local power t is determined by the fluctuation model of the number
N of scatterers
1
0
1
( ) { ( )} exp ( )
( )
H
m
p E p p d
t t
t t t t
t t
 
 
  
 
 

z z
z M z
z z
M

10
compound-Gaussian model
z is a SIRV
t is a positive r.v. (texture)
x is a complex Gaussian vector (speckle)
Yao’s representation theorem [Yao73]:
Sangston and Gerlach [San92, San94] derived the same multidimensional PDF by
an extension of the CLT, so providing a different physical interpretation and
justification of this multivariate model:
{ai}i.i.d., E{ai}= 0, E{aiai
H
} = M
z  t x
z 
1
N
ai
i1
N
 N
 

 z  t x
  ,
0, , 1, 1, ,
i i
CN i m
  
x M M
From a physical point of view, the major assumption of this model, that allows
the extension to the multidimensional setting, is that during the time that the m
radar pulses are scattered, the number N of scatterers remains fixed
We obtain the K distribution if N is a negative binomial r.v. and the Gaussian
distribution if N is deterministic, Poisson, or binomial)

11
Many known APDFs belong to the SIRV family:
Gaussian, Generalized Gaussian, Contaminated normal,
Laplace, Generalized Laplace, Chauchy, Generalized Chauchy,
K, Student-t, Chi, Generalized Rayleigh, Weibull, Rician,
Nakagami-m. The log-normal can not be represented as a SIRV
for some of them pt(t) is not known in closed form (see [Con95, Ran95])
The assumption that, during the time that the m radar pulses are scattered, the
number N of scatterers remains fixed, implies that the texture t is constant during
the coherent processing interval (CPI), i.e., completely correlated texture
A more general model is given by
Extensions to describe the clutter process, instead of the clutter vector,
investigated the cyclostationary properties of the texture process t[n]
(see [Gin01, Gin02, Gre04])
z[n] t[n] x[n], n 1,2, ,m.

12
s[n] A[n]ej[n]
p[n]
The detection problem: the radar transmits a coherent train of m
pulses and the receiver properly demodulates, filters and samples the
incoming narrowband waveform. The samples of the baseband complex
signal (in-phase and quadrature components) are:
Observed data:
Binary hypothesis test:
z  zI  jzQ [z[1] z[m]]T
H0: z  d
H1: z  s d



d = disturbance vector
(clutter and/or thermal noise)
s = target signal vector
 Perfectly known;
 Unknown:
 deterministic (unknown parameters, e.g., amplitude, initial
phase, Doppler frequency, Doppler rate, DOA, etc.)
 random (rank-one waveform, multi-dimensional waveform)
Target samples:
p is usually called “steering vector”
p  [p[1] p[2] p[m]]T
mx1 per each range cell

13
Clutter model for completely correlated texture: t[n]t during the CPI
(coherent processing interval), the non-Gaussian clutter vector is a SIRV:
The compound-Gaussian clutter model
Speckle
0
H t
 
z d x
  [ ],
0, , 1
i i
CN
 
x M M
  [ ]
2
,
0, , 1
G G i i
CN 
  
d c M M
If the clutter is Gaussian: 2
G
t 
 Texture
 
E
 t

The PDF of z under the hypothesis H0 is obtained by averaging the PDF of z|t
with respect to t
0 0
1
0 0
,
0
1
( ) { ( , )} exp ( )
( )
H
H H m
p H E p H p d
t t
t
t t t
t t
 
 
  
 
 

z z
z M z
z z
M

14
The optimum Neyman-Pearson detector
The optimum N-P detector is the LRT:
It depends on the target signal model:
Different models of s have been investigated
to take into account different degrees of a priori
knowledge on the target signal:
(1) s = perfectly known
(2) s=bp with b unknown deterministic and p perfectly known
(3) s=bp with , i.e., Swerling-I target model, and p perfectly known
(4) s=bp with and p unknown (known function of unknown parameters)
(5) s = Gaussian distributed random vector (known to belong to a subspace of dim. r<m)
 
2
0, S
CN
b 

 
2
0, S
CN
b 

0
0 1
0
( )
1
( ) exp ( ) , ( ) ?
( )m
q
p H p d p H
t t t
t t

 
  
 
 

z z
z
z z
M
1
0 ( ) H
q 

z z M z
1
1
0
0
1
0
( )
>
( ) e
<
( )
H
H
H T
H
p H
p H
 
z
z
z
z
z
 
1 0
1 0
( ) ( )
H H
p H E p H
 
s
z z
z z s

15
typically:
GLRT approach
(in this case a
Uniformly Most
Powerful (UMP) test
does not exist)
Case 2) Constant
complex amplitude
(b unknown deterministic)
Case 1) Perfectly known signal
 
2
0, S
CN
b 

1
0
1 >
Re{ }
<
H
H
H


s M z
1
0
2
1 >
<
H
H
H


p M z
2 D
j f n
n e 

p
The (coherent) whitening
matched filter (WMF)
Case 3) Constant
complex amplitude
(b random, Swerling-I)
LRT
Swerling-I:
[ ]
[ ] j n
A n e n

b
b
 

s p
The (incoherent) whitening
matched filter (WMF)
Coherent detection in Gaussian clutter: The Whitening Matched Filter (WMF)
ˆ
ˆML ML
b

s p
1
1
ˆ
H
ML H
b



p M z
p M p
is the maximum likelihood
(ML) estimate of b

16
The “estimator-correlator” interpretation
for Swerling-I target signal:
The mmse estimator of the target
signal is given by:
Alternative interpretation of the WMF: the Estimator-Correlator structure
SCR is the signal-to-clutter power
ratio at the input of the WMF:
The N-P detector for perfectly known signal
correlates the data with the known signal
vector, weighted by the inverse of the
disturbance covariance matrix
The N-P detector for Gaussian
random targets is obtained by
replacing the unknown signal
with its mmse estimate in the
N-P detector for perfectly
known signal
2 2
S G
SCR  

   
2
, 0, H
s S S
CN CN 
 
s m M pp
1
0
1 >
ˆ
Re{ }
<
H
H
H


s M z
1
0
2
1 >
<
H
H
H


p M z
G
b
   
z s d p c
1
1
1
ˆ
ˆ { } ( )
1
H
s SZ ZZ s MMSE
H
SCR
E
SCR
b




     
 
p M z
s s z m M M z m p p
p M p

17
Case 2). b unknown deterministic. A UMP test does not exist.
A suboptimal approach is the Generalized LRT (GLRT):
The test statistic is given by the LR for
known b, in which the unknown parameter
has been replaced by its maximum
likelihood (ML) estimate
When the number m of
integrated samples
increases ,
thus, we expect the
GLRT performance to
approach that of the
N-P detector for
perfectly known signal
where
1
0
0
1
0
( )
( )
1 >
exp exp ( ) 0
<
H
H
m
q
q
T p d
t t t
t t t

 
 
 
  
   
 
   
 

z
z
2
1
1 1
1 1
ˆ ˆ
( ) ( ) ( )
H
H H
ML ML H
q b b

 

    
p M z
z z p M z p z M z
p M p
1
1
ˆ
H
ML H
b



p M z
p M p
ˆ
ML
b b

1
0
0
0
0
0
ˆ
( )
>
ˆ
max ( ; ) ( ; )
<
( )
H
H
ML
H T
ML
H
p H
e
p H
b
b
b b

   
z
z
z p
z z
z
Coherent detection in compound-Gaussian clutter

18
The N-P optimum detector (OD) and the GLRT are difficult to implement in the LRT
form, since it requires a computational heavy numerical integration!
The LRT does not give insight that might be used to develop good suboptimum
approximations to the OD and GLRT
To understand better the operation of the OD and GLRT, reparametrize the conditional
Gaussian PDF by setting:
The key to understanding the operation of the OD and GLRT is to express it as a function
of the mmse estimate of a :
a is the reciprocal of the local
clutter power in the range cell
under test (CUT)
mmse estimate of a under
the hypothesis Hi (i=0,1)
after Sangston: [San92], [San99].
 
1
0
0
1
( )
( )
>
ln ( )
<
H
H
q
q
E x dx T
a
  
z
z
z
1
a
t

ˆ { ( )}
MMSE i
E q
a a
 z
Alternative interpretation of the OD : the Estimator-Correlator

19
In Gaussian disturbance with power we have (deterministic quantity)
This structure is of the form of an estimator-correlator (Schwartz, IEEE-IT, 1975)
The structure of the OD in compound-Gaussian clutter is the basic detection structure
of the OD in Gaussian disturbance with the quantity , which is known in the
case of Gaussian noise, replaced by the mmse estimate of the unknown random a
The quantity to be estimated is not the local clutter power t, but its inverse a =1/t
This formulation is also difficult to implement, but it is very important because
it suggests that sub-optimum detectors may be obtained by replacing the optimum
mmse estimator with sub-optimum estimators that may be simpler to implement
(e.g., MAP or ML)
 
1
0
0
1
2
1
( )
0 1
2 2 2 1
( )
( ) ( )
1 >
ln ( )
<
H
H
H
q
H
G G
q G
q q
dx T
  



   

z
z
p M z
z z
z
p M p
2
G

2
1 G
a 

2
1 G
a 

Alternative interpretation of the OD: the Estimator-Correlator

20
• First step: express the PDFs under the two hypotheses as:
where hm(q) is the
nonlinear monotonic
decreasing function:
The LRT can
be recast in
the form: fopt(q0,T) is the data-
dependent threshold, that
depends on the data only by
means of the quadratic statistic:
1
( ) ( ( )), 0,1
i m i
m
p H h q i

 
z z z
M
0
1
( ) exp ( )
m m
q
h q p d
t t t
t t

 
 
 
 

1
0 0 0
( , ) ( ( ))
T
opt m m
f q T q h e h q

  1
0 ( ) H
q 

z z M z
1
0
0 1 0
>
( ) ( ) ( , )
<
H
H
opt
q q f q T

z z
Alternative interpretation of the OD: the Data-Dependent Threshold

21
Perfectly known signal s (case 1):
the OD can be interpreted as the
classical whitening-matched filter
(WMF) compared to a data-
dependent threshold (DDT)
[San99], [Gin99]
1
0
1
0
2
0 1
0 1 0
>
Gaussian clutter: ( ) ( )
<
>
C-G clutter: ( ) ( ) ( , )
<
H
H
H
H
G
opt
q q T
q q f q T



z z
z z
In this formulation the LRT for
compound-Gaussian clutter has a
similar structure of the LRT in
Gaussian disturbance, with the
significant difference that now the
threshold of the test is not constant
but it depends on the data through q0
 
1
0
1 1
0
>
2Re ( , )
<
H
H
H H
opt
WMF
f q T
 

s M z s M s

22
Signal s with unknown complex
amplitude (Case 2): the GLRT again
can be interpreted as the classical
whitening-matched filter (WMF)
compared to the same
data-dependent threshold (DDT)
1
0
2
1 1
0
> ( , )
<
H
H
H H
opt
WMF
f q T
 

p M z p M p
• Example: K-distributed clutter.
In this case the texture is modelled as a Gamma r.v. with mean value  and order
parameter n. For n-m=0.5 we have:
In general, it is not possible to find a closed-form expression for the DDT, so it must be
calculated numerically.
2
0 0 0 0
( , )
4 4
opt
f q T q q T u q T
 
n n
   
   
   
   

23
This canonical structure suggests a practical way to implement the OD and GLRT
The DDT can be a priori tabulated, with the threshold T set according to the prefixed
PFA, and the generated look-up table saved in a memory
. . . . .
. . . . .
z
ESTIMATION ALGORITHMS
INPUT
DATA
CUT
. . . . .
z1 z 2 zK/
/2
z K
THRESHOLD
H1
H0
LOOK-UP
TABLE
H1
q0( z)
ˆ
M
ˆ
n, ˆ

f(q0
, )
WMF
x
Canonical structure of the Optimum Detector
This approach is highly time-saving, it is canonical for every SIRV, and it is useful
both for practical implementation of the detector and for performance analysis
by means of Monte Carlo simulation
This formulation provides a deeper insight into the operation of the OD and GLRT,
and suggests another approach for deriving good suboptimum detectors

24
Suboptimum approximations to the likelihood ratio test (LRT)
From a physical point of view, the difficulty in utilizing the LRT arises from the
fact that the power level t, associated with the conditionally Gaussian clutter is
unknown and randomly varying: we have to resort to numerical integration
The idea is: replace the unknown power level t with an estimate inside the LRT
Suboptimum detection structures
Candidate estimation
techniques:
MMSE, MAP, ML
The simplest is the ML
It is in the canonical form,
with the adaptive threshold
that is a linear function of q0
It has been
called the GLRT-
LQ (or NMF)
[Gin97]
1
0
2
1 1
0
> ( , )
<
H
H
H H
ML
f q T
 

p M z p M p
 
0 0
( , ) 1 T m
ML
f q q e
 
 
,
( )
ˆ , 0,1
i
ML i
q
i
m
t  
z
ln ˆ

(z)  mln
ˆ
t0
ˆ
t1






q0 (z)
2 ˆ
t0

q1(z)
2 ˆ
t1
>
H1
<
H0

T

25
Suboptimum approximations to the Likelihood Ratio
The Linear-Quadratic GLRT (GLRT-LQ), also called the Normalized Matched
Filter (NMF)
This detector is very simple to
implement
It has the constant false alarm rate
(CFAR) property with respect to
the clutter PDF
The same decision strategy was
also obtained, under different
assumptions, by Korado [Kor68],
Picinbono and Vezzosi [Pic70],
Scharf and Lytle [Sch71],
Conte et al. [Con95].
is the SCR at the
output of the WMF,
divided by m
   
1
0
2
1
1
1
> 1
<
H
H
H
T m H
H
e

 

 
p M z
p M p
z M z
2 1
H
S
m




 
p M p
 
E
 t

( 1)
T m
m
FA
P e



 
( 1)
0
1
1 ( )
m
T m
D
e
P p d
m
t
t
t t
t 
 
  

 
 
 

 


26
The mmse estimator of a=1/t may be difficult to implement in a practical detector.
e.g. for K-distributed clutter:
Suboptimum approximations to the Estimator-Correlator structure
Suboptimum detectors may be obtained by replacing the mmse estimator with
a suboptimal estimator (e.g., MAP or ML):
1
,
4 ( )
ˆ , 0,1
( ) 4 ( )
i
m
MMSE i
i i
m
q
K
i
q q
K
n
n
n

n
a
 n

 

 
 
 
  
 
 
 
z
z z
,
,
1
ˆ
ˆ
MMSE i
MMSE i
a
t

,
,
1
ˆ
ˆ
MAP i
MAP i
a
t

,
,
1
ˆ
ˆ
ML i
ML i
GLRT LQ
a
t
  
As the number m of samples becomes asymptotic large, the GLRT-LQ becomes equivalent
to the OD:
2
,
4 ( )
1 ( 1)
ˆ , 0,1
2 ( )
i
MAP i
i
q
m m
i
q
n
n n

a
     
 
z
z
1
, , ,
,
1
ˆ ˆ ˆ
, , 0,1
ˆ
( )
m
MMSE i MAP i ML i
i ML i
m
i
q
a a a
t

   
z

27
Suboptimum approximations to the Data-Dependent Threshold structure
The canonical structure in the form of a WMF
compared to a data dependent threshold is given by: the threshold fopt(q0,T)
depends in a complicated
nonlinear fashion on the
quadratic statistic q0(z)
The idea is to find a good approximation of fopt(q0,T ) that is easy to implement.
In this way, we avoid the need of saving a look-up table in the receiver memory
The approximation has to be good only for values of q0(z) that have a high
probability of occurrence
We looked for the best K-th order polynomial approximation in the MMSE sense:
fK(q0,T) is easy to compute from q0(z)
1
0
2
1 1
0
> ( , )
<
H
H
H H
opt
WMF
f q T
 

p M z p M p
0 0
0
( , )
K
i
K i
i
f q T c q

 
 
2
0 0
0
min ( , )
i
K
i
opt i
c
i
f q T c q

 
 

 
 
 


28
0
0.2
0.4
0.6
0.8
1
-4 -2 0 2 4 6 8 10
Opt. det.
Lin. approx.
Quad. approx.
GLRT-LQ det.
PD
SCR (dB)
m=16
m=4
m=4
m=4
Performance Analysis: Swerling-I target, K-distributed clutter
In all the cases we examined, the suboptimum detector based on the quadratic (2nd-order)
approximation has performance (i.e., PD) almost indistinguishable from the optimal
The detector based on 2nd-order
approximation represents a good
trade-off between performance and
computational complexity
It requires knowledge of the clutter
APDF parameters (n and )
As the number m of integrated
pulses increases, the detection
performance of the GLRT-LQ
approaches the optimal performance
The GLRT-LQ does not require
knowledge of n and 
It is CFAR w.r.t. texture PDF
5 3
10 , 0.5, 4.5, 10 ,
0.9 (1)
FA D
X
P f
AR
n 


   


29
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10
Opt. det.
Lin. approx.
Quad. approx.
GLRT-LQ det.
SCR (dB)
P
D
n
n
n
Swerling-I target: it was observed that in this case, PD increases much more
slowly as a function of SCR than for the case of known target signal
Clutter spikiness heavily affects
detection performance
n=0.1 means very spiky clutter
(heavy tailed)
n=4.5 means almost Gaussian clutter
Up to high values of SCR, the best
detection performance is obtained
for spiky clutter (small values of n),
i.e., it is more difficult to detect weak
targets in Gaussian clutter rather
than in spiky K-distributed clutter,
(provided that the optimal decision
strategy is adopted!)
5 3
10 , 0.5, 16, 10 ,
0.9 (1)
FA D
X
P f m
AR



   


30
n   (Gaussian clutter )
n  0.5 (spiky clutter )
0
0.2
0.4
0.6
0.8
1
-12 -10 -8 -6 -4 -2 0
OGD
GLRT-LQ det.
OKD
P
D
SCR (dB)
0
0.2
0.4
0.6
0.8
1
-20 -15 -10 -5 0
OGD
GLRT-LQ det.
PD
SCR (dB)
Optimum Detector in K-clutter (OKD)
Optimum Detector in Gaussian clutter (OGD) = Whitening Matched Filter (WMF)
This figure shows how a wrong
assumption on clutter model
(model mismatching) affects
5
3
10 , 0.5, 16,
10 , 0.9 (1)
FA D
X
P f m
AR
 

  
 

31
0
0.2
0.4
0.6
0.8
1
-30 -25 -20 -15 -10 -5 0
Opt. det.
GLRT-LQ
Opt. det. - real data
GLRT-LQ - real data
PD
SCR (dB)
0
0.2
0.4
0.6
0.8
1
n
n
n
-20 -15 -10 -5 0
GLRT -LQ
OGD
PD
SCR (dB)
OGD
OGD
GLRT -LQ
GLRT -LQ
Performance Analysis: Swerling-I, K-distributed clutter, real sea clutter data
The gain of the GLRT-LQ over the
mismatched OGD increases with
clutter spikiness (decreasing values of n)
Performance prediction have been
checked with real sea clutter data
The detectors make use of the
knowledge of , n, and M
(obtained from the entire set of data)
n  1.25,   4.44103
X  0.93 j0.24
m=16
m=16

32
Coherent radar detection in non-Gaussian clutter: Summary of the Results
 We have examined the problem of optimal and sub-optimal target detection
in Gaussian clutter and non-Gaussian clutter modelled by the compound-
Gaussian distribution
 Three interpretations of the optimum detector (OD) have been provided:
the likelihood ratio test (LRT),
the estimator-correlator (EC),
the whitening matched filter (WMF) and data-dependent threshold.
 With these reformulations of the OD, the problem of obtaining sub-optimal
detectors may be approached by either:
(i) approximating the LRT directly, utilizing an estimate of t
(ii) utilizing a sub-optimal estimate of a in the estimator-correlator structure,
(iii) utilizing a sub-optimal function to model the data-dependent threshold
in the WMF interpretation.
 Numerical results suggest that the GLRT-LQ [(i),(ii)] and the sub-optimal
detector based on quadratic approximation of the data-dependent threshold
[(iii)] represent the best trade-off between implementation complexity and

33
The Data-Dependent Threshold structure: New results
The LRT can be recast in the form:
fopt(q0,T) is the data-
dependent threshold,
that depends on the data
only by means of the
quadratic statistic:
1
0 0 0
( , ) ( ( ))
T
opt m m
f q T q h e h q

 
1
0 ( ) H
q 

z z M z
1
0
0 1 0
>
( ) ( ) ( , )
<
H
H
opt
q q f q T

z z
hm(q) is a nonlinear monotonic decreasing function,
which depends on the texture PDF:
 
0 0
1
( ) exp ( ) exp ( )
m
m m
q
h q p d q p d
t a
t t a a a a
t t
 
 
   
 
 
 
2
1 1 1
( )
p p
a t
a a
t a a
 
    
 
The left-hand term
depends on the target
signal model
Reparametrize in terms of the
inverse of the local clutter power:

34
The simplest extension beyond the Gaussian case would appear to be a linear
form for fopt(q0,T):
0 0
( , )
opt
f q T a bq
 
we will call a detector whit this form of the DDT: a linear-threshold detector (LTD).
The LTD was originally explored in [San99] as an approximation to fopt(q0,T): for the
case of K-distributed compound-Gaussian clutter, and then further investigated in
[Gin99] and [Gin02]. Now, we ask a much different question about the LTD:
Is there a compound-Gaussian model for
which the linear-threshold detector is the
optimum detector?
The Linear-Threshold Detector

35
If the answer is yes, the multivariate compound-Gaussian clutter model,
which is defined through the function hm(q0) must satisfy:
1
0 0 0
( ( ))
T
m m
q h e h q a bq

  
This problem can be reformulated as an Abel problem, which has a unique solution.
The solution provides a Gamma PDF for aand as a consequence the texture t has an
inverse-Gamma PDF:
1
1 1
( ) , 0
( )
p e
 
t
t 

t t
  t


 
 
 
  
0
( )
0 0
1 ( )
( ) ( )
( )
m
H m
m
p H q
 
  
   
 
   
 
    
   
    
z
z z
M
Consequently, the complex multivariate compound-Gaussian clutter model
is a complex m-variate t distribution with degrees of freedom  :
>0 is the shape
parameter and
>0 is the scale
parameter.

36
1
0
0 1 0
>
( ) ( ) 1 ( )
<
H
H
T
m
q q e q





   
   
   
 
 
z z z
The optimum detector in compound-Gaussian clutter with IG texture is given by:
1
0
2
0 1 2
>
Gaussian clutter: ( ) ( )
< 1
H
H
G
G G
T T
q q T

 a
  
z z
1
0
,
0 1
0 ,
0
>
CG clutter: ( ) ( )
< ˆ
ˆ [ ] , ( ) 1
H
H
m
opt
T
m
opt m
T
q q
m
E q T m e
q



a

a a 





 

    
 
 

z z

37
2
( 1)
2
0
( ) 2 ( ) 2 1 , 0
r
R
p r re p d r r r

a
a

a a a 

 

  
   
 
 

The clutter amplitude PDF that arises from the compound-Gaussian
clutter model with IG texture:
The clutter amplitude model
2
0
lim ( ) 2 , 0 (Rayleigh PDF)
2
lim ( ) ( ), 0
r
R
R
p r re r
p r u r r
r








 
 
the clutter amplitude statistics are extremely heavy-tailed in this limit
Interestingly, the
PDF of the clutter
intensity, W=R2, is
a generalized
Pareto
distribution

38
2
2
1
2 2
{ }
( )
if 2.
{ } { }
( 1)
k
k
k k
E R
k
E E R


 


t
 
   
   
   
     
   
 

 

10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
0 2 4 6 8 10
=1.01
1.2
2
3
pdf
Amplitude
: { } 1
E
 t 
Thus the multivariate clutter model with IG texture varies parametrically
between Gaussian clutter (when  goes to infinity) and extremely
heavy-tailed clutter (when goes to zero)
The clutter power
is finite only for >1
{ }
E t
The clutter amplitude model

39
1
0
0 1 0
>
( ) ( ) 1 ( )
<
H
H
T
m
q q e q

 
 
 
 
z z z
When the shape
parameter  goes
to zero:
Implications of the optimum detector
, 0
0 0
1 ( ) ( , )
ˆ
T
m m
ML
opt
T
e q f q T
 
a

  

  
 
 
z
which is the GLRT-LQ (or NMF)
detector previously explored [Gin97],
known to be the optimal detector as m
goes to infinity [Con95] and also known
to be a CFAR detector.
The remarkable observation here is that
the GLRT-LQ is the optimum detector for any m
for the CG clutter model with IG texture,
in the limit of extremely heavy-tailed clutter, i.e. as  goes to 0

40
 
1
0
2
1 1
0
> 1 ( )
<
H
H
T
H H m
e q




  
   
   
   
 
 
p M z p M p z
When the complex amplitude is deterministic unknown and we resort to the GLRT
approach (i.e. we replace b with its ML estimate in the LRT):
The GLRT-LTD and its Performance
2
1
0 1 1
( ) ( )
H
H
q q


 
p M z
z z
p M p
1
exp
FA
m
P T
m


 
 
 
   
 

 
 
( 1)
0
,
T m
m
FA
FA GLRT LQ
P e
P








41
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
0 2 4 6 8 10
=1.01
1.2
2
3
1.01
1.2
2
3
P
FA
Threshold (T)
m=4
m=16
Thus, for large m, the
GLRT-LDT detector is
in practice CFAR with
respect to both  and 
(the texture parameters)
1
exp
FA
m
P T
m


 
 
 
   
 

 
 
[ ]
1or 1
exp
(the Gaussian case)
FA
m
P T


 

42
In heavy-tailed clutter:
the GLRT-LTD largely
outperforms the
(mismatched) WMF
0
0.2
0.4
0.6
0.8
1
-20 -15 -10 -5 0 5 10 15 20
GLRT-LTD, =1.01
=1.2
=3
WMF, =1.01
=1.2
=3
P
D
SCR (dB)
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
0 100 200 300 400 500
=3
2
1.5
1.2
1.05
1.01
P
FA
Threshold
 
4 2
16, 10 ,
FA S
m P SCR E
 t

  
The False Alarm Rate (FAR)
increases when the clutter
becomes spikier (lower )

43
 
4 2
16, 10 ,
FA S
m P SCR E
 t

  
When the number of
pulses is large (say m>10),
the GLRT-LQ and the
optimal GLRT-LTD have
almost the same
performance
0
0.2
0.4
0.6
0.8
1
-20 -15 -10 -5 0 5 10 15 20
GLRT-LTD, =1.01
=1.2
=3
GLRT-LQ, =1.01
=1.2
=3
P
D
SCR (dB)
Detection performance of
the GLRT-LTD gets better
when the clutter becomes
spikier

44
Concluding Remarks
 Based on the interpretation of the optimum detector (OD) as a whitening
matched filter (WMF) compared to a data-dependent threshold (DDT), we
have derived a compound-Gaussian (CG) clutter model whose OD is the
optimum Gaussian WMF compared to a DDT that varies linearly with q0(z)
 The CG clutter model with inverse Gamma (IG) texture varies
parametrically from the Gaussian clutter model to a clutter model whose tails
are evidently heavier than any K model
 The GLRT-LQ, which is a popular suboptimum detector due to its CFAR
property, is an OD for the CG clutter model with IG texture, in the limit as
the tails get extremely heavy (small ), and the OD for any CG model for
large m
 The LTD is conceptually the simplest extension beyond the optimum
Gaussian detector (constant threshold). Although a quadratic-threshold
detector may be obtained [Gin99], such a detector will always be
suboptimum for any compound-Gaussian model

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