Plug-in Measure-Transformed Quasi Likelihood
Ratio Test for Random Signal Detection
Nir Halay and Koby Todros
Dept. of ECE, Ben-Gurion University of the Negev, Beer-Sheva, Israel
February 13, 2017
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Outline
I
Detection problem formulation
I
Existing detectors
I
Proposed detector
I
Simulation studies
I
Summary
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Detection Problem Formulation
H0 :
H1 :
(
Xn = Wn ,
n = 1, . . . , N
(s)
Wm ,
m = 1, . . . , M
Ym =
(
Xn = Sn a + Wn , n = 1, . . . , N
(s)
Y m = Wm ,
m = 1, . . . , M,
I
{Xn ∈ Cp }, {Ym ∈ Cp }: Primary and secondary data
I
{Sn ∈ C}: i.i.d. zero-mean signal with unknown distribution
I
a ∈ Cp : Known steering vector
I
{Wn }, {Wm }: i.i.d. zero-mean symmetrically distributed
homogeneous noise processes with unknown distribution
I
{Sn }, {Wn } and {Wm } are mutually independent
(s)
(s)
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Existing detectors
Gauss-Gauss detector [Jin & Friedlander 2005]
I
GLRT-based detector
I
Assumes normally distributed signal and noise
I
Simple implementation and tractable performance analysis
I
Sensitive to deviation from normality (e.g., in the case of
heavy-tailed noise that produces outliers)
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Existing detectors
CG-GLRT [Gerlach 1999, Shuai et. al. 2010]
I
Assumes elliptical compound-Gaussian noise
I
Resilient against heavy-tailed noise outliers
I
Iterative ML-estimation of the noise scatter matrix
I
I
Converges under some regularity conditions
I
Each iteration involves matrix inversion
I
Does not reject large norm outliers
Can be sensitive to non-elliptical noise
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Proposed Detector
General operation principle
Selects a Gaussian model that best empirically fits a transformed
probability distribution of the data
Advantages
I
Resilient to outliers (rejects large-norm outliers)
I
Involves higher-order statistical moments
I
Significant mitigation of the model mismatch effect
I
Computational and implementation simplicity
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Probability Measure Transform
Definition
Given a non-negative function u : X → R+ satisfying
0 < E [u (X) ; PX;H ] < ∞.
(u)
A transform Tu : PX;H → QX;H is defined as:
Z
(u)
Tu [PX;H ] (A) = QX;H (A) , ϕu (x; H)dPX;H (x) ,
A
where
ϕu (x; H) ,
u (x)
.
E [u (X) ; PX;H ]
The function u (·) is called the MT-function.
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Probability Measure Transform
The measure transformed mean and covariance
(u)
µX;H = E [Xϕu (X; H); PX;H ]
(u) (u)H
(u)
ΣX;H = E XXH ϕu (X; H); PX;H − µX;H µX;H
where
(u)
dQX;H
u (x)
ϕu (x; H) ,
=
E [u (X) ; PX;H ]
dPX;H
Conclusion
(u)
I
The mean and covariance under QX;H can be estimated using
only samples from PX;H .
I
u (x) non-constant & analytic ⇒ the mean and covariance
(u)
under QX;H involve higher-order statistical moments of PX;H .
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Probability Measure Transform
Proposition (Consistent empirical MT mean and covariance)
Let Xn , n = 1, . . . , N denote a sequence of i.i.d. samples from
PX;H , and define the empirical mean and covariance estimates:
(u)
µ̂X ,
N
X
Xn ϕ̂u (Xn )
n=1
(u)
Σ̂X ,
N
X
(u)
(u)H
Xn XH
n ϕ̂u (Xn ) − µ̂x µ̂x
n=1
where ϕ̂u (Xn ) ,
u(Xn )
PN
.
n=1 u(Xn )
If
h
i
E kXk2 u (X) ; PX;H < ∞,
(u) w.p.1
(u)
(u) w.p.1
(u)
then µ̂X −−−→ µX;H and Σ̂X −−−→ ΣX;H as N → ∞.
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Probability Measure Transform
Proposition (Robustness to outliers)
If the MT-function u(x) and u(x)kxk2 are bounded, then the
(u)
(u)
influence functions [Hampel, 1974] of µ̂X and Σ̂X are bounded.
Remark
Condition is satisfied when u (x) ∈ Gaussian family.
1
0.9
0.8
0.7
kIFΣ(u) (y) k
x
||IF||
0.6
kIFµ(u) (y) k
0.5
x
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
||y||
2.5
3
3.5
4
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Measure Transformed Gaussian Quasi LRT
MT-GQLRT [Todros & Hero, 2016 ]
(u)
Compares the empirical KLDs between QX;H and two Gaussian
(u)
(u)
(u)
(u)
measures Φ(µX;H 0 , ΣX;H 0 ) and Φ(µX;H 1 , ΣX;H 1 )
i h (u)
(u)
(u) 2
(u)
Tu = DLD Σ̂X ||ΣX;H 0 + µ̂X − µX;H 0 h (u)
i (u)
(u)
(u) 2
− DLD Σ̂X ||ΣX;H 1 − µ̂X − µX;H 1 (u)
Σx;H
0
−1
(u)
Σx;H
1
−1
H1
R τ,
H0
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Measure Transformed Gaussian Quasi LRT
MT-GQLRT for the considered detection problem
I
Class of MT-functions:
n
o
p
u (x) = v P⊥
x
,
v
:
C
→
R
,
v(x)
=
v(−x)
+
a
I
MT-mean and MT-covariance under H0 and H1 :
(u)
µX;H k = 0,
(u)
(u)
k = 0, 1
(u)
(u)
ΣX;H 0 = ΣW and ΣX;H 1 = σS2 aaH + ΣW
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Equivalent test statistic:
(u) −1 (u)
(u) −1
Tu0 = aH ΣW
ĈX ΣW
a
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Plug-in Measure Transformed Gaussian Quasi LRT
Plug-in MT-GQLRT
Plug-in the empirical MT-covariance of the noise obtained from
noise-only secondary data {Ym }
(u) −1
(u) −1 H1
(u)
Tu00 , aH Σ̂Y
ĈX Σ̂Y
a R t,
H0
Theorem (Asymptotic normality)
Assume that
(u)
1. σS2 > 0, ΣW is non-singular
i
h
2. E u2 (X) ; PX;H , E kXk4 u2 (X) ; PX;H are finite
Then
√
D
(u)
(u)
N Tu00 − ηH −−−−→ N 0, λH
for H = H0 , H1
N →∞
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Plug-in Measure Transformed Gaussian Quasi LRT
Threshold determination
(u)
t = η̂H 0 + Q
I
I
I
−1
q
(α)
(u)
λ̂H 0 ,
Guarantees asymptotic CFAR= α
(u) −1
(u)
η̂H 0 , aH Σ̂Y
a
(u)
λ̂H 0 ,
M
N
M
P
m=1
!2
2
−1
(u)
(u)
ϕ̂2u (Ym ) aH Σ̂Y
Ym − η̂H 0
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Plug-in Measure Transformed Gaussian Quasi LRT
Selection of the MT-function to induce outlier resilience
I
Gaussian MT-function:
2
2
uG (x; ω) = exp −kP⊥
xk
/ω
a
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Define the Asymptotic Relative Local Power Sensitivity to
change in σS2 (under nominal Gaussian distribution):
√
N G (ω, ΣW ) + Q−1 (α)
∂βuG ∂βLRT √
/
=
R (ω, ΣW ) ,
∂σS2
∂σS2 σ2 =0
N + Q−1 (α)
S
I
Select the lowest ω such that R (ω, ΣW ) ≥ r, 0 << r < 1
I
In practice ΣW is replaced by:
(u )
−1
G
2
Σ̂W (ω) = Σ̂Y (ω) − P⊥
/ω
a
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Simulation Studies
Setup
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BPSK signal with variance σS2
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Steering vector: a ,
p = 8, θ = 60◦
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Sample size in primary and secondary data: N = 1000,
M = 5000
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False alarm rate α = 0.05
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Asymptotic Relative Local Power Sensitivity: r = 0.9
√1
p
1, e−iπ sin(θ) , . . . , e−iπ(p−1) sin(θ)
T
,
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Simulation Studies
Detection in Gaussian noise
Zero-mean Gaussian noise with Toeplitz structured covariance Σ:
(
[Σ]i,j ,
2 j−i
σW
b ,
∗
2
σW bi−j ,
i≤j
,
i>j
|b| < 1,
1
Power
0.8
0.6
0.4
Omniscient LRT
Gauss-Gauss detector
CG-GLRT
Plug-in MT-GQLRT (empirical)
Plug-in MT-GQLRT (asymptotic)
0.2
0
-20
-18
-16
-14
SNR [dB]
-12
-10
-8
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Simulation Studies
Detection in elliptical CG noise
Elliptical t-distributed noise with λ = 0.2 degrees of freedom and
dispersion matrix Σ
1
Power
0.8
0.6
Omniscient LRT
Gauss-Gauss detector
CG-GLRT
Plug-in MT-GQLRT (empirical)
Plug-in MT-GQLRT (asymptotic)
0.4
0.2
0
-20
-18
-16
-14
-12
-10
SNR [dB]
-8
-6
-4
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Simulation Studies
Detection in non-elliptical noise
Elliptical t-distributed noise + BPSK interference (SIR = −5 [dB])
1
Power
0.8
0.6
0.4
Omniscient LRT
Gauss-Gauss detector
CG-GLRT
Plug-in MT-GQLRT (empirical)
Plug-in MT-GQLRT (asymptotic)
0.2
0
-20
-18
-16
-14
-12
-10
SNR [dB]
-8
-6
-4
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Conclusions
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Plug-in MT-GQLRT for detection of a random signal that lies
on a known rank-one subspace
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Simple implementation and tractable performance analysis
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Less sensitive to model mismatch as compared to other model
based detectors
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Extension to non-homogeneous noise environment
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