3rd international workshop on mathematical issues...
TRANSCRIPT
SPAC Lab, ECE
Signal Processing for MIMO and
Passive Radar
Hongbin Li Signal Processing and Communication (SPAC) Laboratory
Department of Electrical and Computer Engineering
Stevens Institute of Technology
Hoboken, NJ, USA
July 9, 2014
3rd International Workshop on Mathematical Issues in
Information Sciences (MIIS’2014)
SPAC Lab, ECE
Acknowledgement
• Collaborators
– Guolong Cui (UESTC)
– Braham Himed (AFRL)
– Jun Liu (Stevens)
– Muralidhar Rangaswamy (AFRL)
– Pu Wang (Schlumberger-Doll Research Center)
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Outline
• Part I: Moving target detection with distributed MIMO radars
– Non-homogeneous clutter
– Subspace based approach
– Parametric approach
• Part II: Waveform design for MIMO radar with constant modulus
and similarity constraints
– Design with practical constraints
– Two sequential optimization algorithms
• Part III: Passive detection with noisy reference
– Effect of noise in the reference signal
– Four different detectors
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Part I:
Moving Target Detection with Distributed
MIMO Radars
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MIMO Radars
• MIMO radar vs. Phased Array
– high spatial resolution
– more degrees of freedom
– better parameter identifiability
– flexible transmit beampattern
– increased spatial diversity
– detection diversity gain
Distributed MIMO Radar
with widely separated antennas
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Backscattering
• Radar target RCS is angle
selective
• Conventional radars experience
target fluctuation of 5-25 dB
• Distributed MIMO radar exploits
the angular spread of the target
backscatter in a variety of ways to
improve radar performance
Detection/estimation
performance improvement
through diversity gain
• Clutter response has similar
angular selectivity, causing non-
homogeneous clutter
Target Backscattering vs.
azimuth angle [Skolnik’03]
Angle-Selective Backscatterring
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Signal Model
• Systems Setup
– M transmit antennas (Tx)
– N receive antennas (Rx)
– K pulses in one coherent processing interval (CPI)
– Orthogonal probing waveforms from Tx
– M matched filters at each Rx
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Target Signal
• Target Signal:
– Doppler frequency (for a given target velocity)
– Doppler Steering Vector
– Amplitude
Complex-valued
Unknown but deterministic
Different for different Tx-Rx pairs
8
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Noise and Clutter
• Noise wm,n2 CK × 1 : zero-mean, spatially/temporally white
• Clutter cm,n2 CK × 1
– No clutter [Fishler et al.’05]
– Homogeneous clutter model: shares the same covariance
matrix [He-Lehmann-Blum-Haimovich’10]
– Subspace-based clutter model [Wang-Li-Himed’11]
Clutter is spanned by l Fourier bases
Different m,n for different TX-RX ) non-homogenous clutter power
Cutter covariance matrix structure is still homogeneous
Clutters fall within the column space of H
Buildings Uninterested slow-moving
vehicles
Plants (wind effects)
angle-selective backscattering non-homogeneous clutter
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The MTD Problem
• Moving target detection (MTD) is concerned with the following
composite hypothesis testing problem
• Target-free training signals drawn from neighboring resolution cells
may be available. Generally, they are non-homogeneous across
resolution cells
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Covariance Matrix Based Solutions
• The sample covariance matrix (SCM) based detector was introduced for MTD in homogeneous clutter [He-Lehmann-Blum-Haimovich’10]
• SCM require Kt ¸ 2K homogeneous training signals for each Tx-Rx pair
• A robust detector based on a compound Gaussian model [Chong-Pascal-Ovarlez-Lesturgie’10]
• Covariance is obtained by solving a fixed point equation (FPE)
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• GLRT based on the subspace clutter model [Wang-Li-Himed’11]
• Test variable has central/non-central beta distribution [Wang-Li-Himed’11]
• The SGLRT
– Achieves constant false alarm rate (CFAR)
– Needs no training signal
– Works if the clutter can be expressed using a few Fourier bases
Subspace Based GLRT
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Non-Homogeneous Clutter Modeling
A Parametric Approach
• Autoregressive models are capable of capturing the correlation of
radar clutter with a variety of power spectrums [Wang-Li-Himed’13]
• Clutter speckle is characterized by AR coefficients
• Clutter texture is characterized by the driving noise variance
• Different AR processes to model the disturbance observed at different
TX-RX pair ) truly non-homogeneous
• Parameter estimates can be obtained from test signal ) no need for
training signals
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Parametric MTD
• Recall the moving target detection (MTD) problem
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• The MIMO-PGLRT can be obtained by
• The second equality is due to statistical independence among different
Tx-Rx pairs
with i = 0, 1 denoting H0 and H1, respectively
MIMO Parametric GLRT
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• The simplified MIMO-PGLRT test statistic is [Wang-Li-Himed’13]
Parametric GLRT for MIMO Radar
• Local detector adaptively projects test signal into two different subspaces
– Orthogonal complement of regression matrix Ym,n
– Orthogonal complement of the target-suppressed Ym,n, using the
highlighted projection matrix
• Energy of projected signals are computed, compared, and integrated
over MN pairs
• PGLRT is an adaptive subspace detector, notably different from the
previous fixed subspace detector SGLRT
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• Asymptotic distribution of the MIMO-PGLRT test statistic is
where the non-centrality parameter is given by
• denotes the temporally whitened steering
vector
• Test statistic under H0 is independent of the disturbance
parameters ) asymptotically achieves CFAR
Asymptotic Performance
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CRB of Target Velocity
• CRB provides a lower bound on all unbiased estimate
• CRB is also useful for sensor placement/selection
• An general expression for the CRB is
– Geometry related terms
– Fisher information (FI) related term
Highlights:
Geoometry-related terms (cmn and smn ) are known in advance
Need to compute the Fisher information-related term ψmn
Both exact and asymptotic expressions for ψmn are obtained,
resulting in exact and asymptotic CRB
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CRB of Target Velocity
• The exact CRB is obtained by plugging the exact fisher
information-related term ψmn into the general CRB expression
– temporally whitened steering vector
– first derivatives of w.r.t.
– matrix consisting of first derivative of regressive steering vector
w.r.t.
– target amplitude
– driving noise variance for (m,n)-th AR model
– coefficient vector for (m,n)-th AR model
Observations:
Exact expression of the Fisher information-related term ψmn is a function of target amplitude, Doppler steering vector, AR coefficients, and AR driving noise variance
This expression is complicated and offers limited intuition
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CRB of Target Velocity
• The asymptotic CRB is obtained by plugging the asymptotic Fisher
information-related term ψmn into the general CRB expression
– is the power spectrum density of the (m,n)-th AR
interference at the (m,n)-th Doppler frequency
Observations:
Fisher information-related term ψmn is proportional to the SINR
|mn|2/mn(fmn), and inversely proportional to K3
The asymptotic CRB is simpler to compute
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Simulation Results
• Scenario --- 2 X 2 configuration
– M = 2 Tx
– N = 2 Rx
– Normalized target velocity
• Signal-to-noise ratio
• Clutter-to-noise ratio (subspace model)
• Signal-to-interference-plus-noise ratio
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Subspace GLRT
• Clutter is generated over
Fourier basis with non-
homogeneous power
• Two cases with
known/estimated target
velocity
• Results are averaged
over random target
velocity (direction) and
amplitude
• PA-AMF: phased-array
with adaptive matched
filter
• Two MIMO detectors:
GLRT and SCM
P. Wang, H. Li, and B. Himed, "Moving target detection using
distributed MIMO radar in clutter with non-homogeneous
power," IEEE-TSP, no.10, 2011
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A General Clutter Model
• Clutter temporal correlation function
• Covariance matrix
• Clutter covariance matrix for (m,n)th TX-RX pair
Clutter power spectrum density
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Parametric GLRT
• Clutter is from general
clutter model, non-
homogeneous across
different TX-RX pairs
• Two cases with
known/estimated target
velocity
• Two covariance matrix
based detectors are
included in comparison:
SCM and robust MIMO
• Results are averaged
over random target
velocity (direction) and
amplitude P. Wang, H. Li, and B. Himed, "A parametric moving target
detector for distributed MIMO radar in non-homogeneous
environment," IEEE-TSP, no.9, 2013
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Target Velocity Estimation
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Conclusions
• Examined the moving target detection (MTD) of distributed MIMO
radars in non-homogeneous clutter
• Proposed a subspace based GLRT
– Requires no training
– Can handle clutter with non-homogeneous power
– Works if the clutter can be expanded on a few Fourier bases (e.g.,
stationary platforms)
• Proposed a parametric GLRT
– No training needed
– Different AR models for different Tx-Rx transmit pairs
– Can handle fully non-homogeneous clutter
• Future directions
– Senor placement and optimization
– Non-orthogonal waveforms
– Moving platforms
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Part 2:
Waveform Design for MIMO Radar with
Constant Modulus and Similarity Constraints
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MIMO Waveform Design
• Transmitter only based designs: employ transmit beam pattern (BP) or
radar ambiguity function (AF) Fuhrmann and San Antonio, “Transmit beamforming for MIMO radar systems using signal cross-
correlation,” IEEE-AES, no.1, 2008
Stoica, Li, and Xie, “On probing signal design for MIMO radar,” IEEE-TSP, no.8, 2007
Wang, Wang, Liu, and Luo, “On the design of constant modulus probing signals for MIMO radar,”
IEEE-TSP, no.8, 2012
San Antonio, Fuhrmann, and Robey, “MIMO radar ambiguity functions,” IEEE-SP, no.1, 2007
Chen and Vaidyanathan, “MIMO radar ambiguity properties and optimization using
frequency–hopping waveforms,” IEEE-TSP, no.12, 2008
• Joint transmitter-receiver designs: based on mutual information or max
SINR criterion Yang and Blum, “MIMO radar waveform design based on mutual information and minimum mean –
square error estimation,” IEEE-AES, no.1, 2007
Leshem, Naparstek, and Nehorai, “Information theoretic adaptive radar waveform design for multiple
extended targets,” IEEE-TSP, no.1, 2007
Li, Xu, Stoica, Forsythe and Bliss, “Range compression and waveform optimization for MIMO radar: A
Cram´er–Rao bound based study,” IEEE-TSP, no.1, 2008
Friedlander, “Waveform design for MIMO radars,” IEEE-AES, no.3, 2007
Chen and Vaidyanathan, “MIMO radar waveform optimization with prior information of the extended
target and clutter,” IEEE TSP, no.9, 2009
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This Work
• Two constraints are imposed for practical MIMO radar
waveform design
– Constant modulus (CM) constraint: power amplifiers
often work in saturated mode, prohibiting amplitude
modulation in radar waveforms
– Similarity constraint: allows the designed waveform to
share some good ambiguity properties of a known
waveform
• We present a framework for joint TX-RX based MIMO radar
waveform design
– In the presence of signal-dependent interferences (e.g.,
clutter)
– Taking into account CM and similarity constraints
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• A co-located MIMO radar with NT TX antennas and NR RX antennas
• Let s(n) be the NT ×1 waveform vector and at(θ) the NT ×1 TX
steering vector. The signal seen at a location/angle θ is given by
• Let ar(θ) be the NR×1 RX steering vector. The received signal is given
by
• Stacking vectors x(n), s(n) and v(n) in time
• The problem of interest is to design the NR radar waveforms contained in the NRN £ 1 vector s
Signal Model
interference noise signal
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Waveform Design Criterion
• Pass the received signal x through a linear FIR receive filter w
• Output signal-to-interference-and-noise ratio (SINR)
where SNR = E[|0|2]/(v)
2 and INRk = E[|k|2 ]/(v)
2
• Constant modulus (CM) constraint
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Waveform Design Criterion
• Similarity Constraint: Let s0 be the reference waveform
where 𝜖 is a real parameter ruling the extent of the similarity
• The similarity constraint is equivalent to [De Maio et al.’09]:
• It is noted that 0 · · 2. For = 0, s is identical to s0 . For = 2, similarity
constraint vanishes and only the constant modulus constraint is in effect
• The constrained optimization problem (non-convex)
De Maio, Nicola, Huang, Luo, and Zhang, “Design of phase codes for radar performance
optimization with a similarity constraint,” IEEE-TSP, no. 2, 2009
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Sequential Optimization
Algorithm #1
Proposed SOA1
Luo, Ma, So, Ye, and Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE-SPM, no. 3, 2010
can be solved iteratively by semidefinite
relaxation (SDR)
Optimize ρ(s,w)
with respect to
w in terms of s
Substitute w
back into ρ(s,w)
and simplify
Fix (s) from last iteration,
iteratively update s by SDR
MVDR problem
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Sequential Optimization
Algorithm #1
• Relaxation by dropping the similarity and rank-one constraints
• Randomization to impose the rank-one and similarity constraints
– Generate L random vectors
– Construct feasible solutions to original problem
– Select the best solution among the L randomizations
De Maio, Nicola, Huang, Luo, and Zhang, “Design of phase codes for radar performance
optimization with a similarity constraint,” IEEE-TSP, no. 2, 2009
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Sequential Optimization
Algorithm #2
Proposed SOA 2
Optimize w by
maximizing the
SINR for a given s
Optimize s by
maximizing the
SINR for a given w
MVDR problem
Repeat above 2 steps till
convergence Solvable by SDR
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Sequential Optimization
Algorithm #2
• Relaxation
• Let X = yZ. Via Charnes-Cooper transform, above fractional problem
reduces to SDP
• Suppose that (X*,y*) is a solution to the SDP. Then, Z* = X*/y* is a
solution to the fractional problem
• Randomization can be applied in a similar way as in SOA1 to generate
solutions with rank-one and similarity constraints
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Simulation Results
Reference signal: orthogonal LFM
MIMO
antennas
NT 4
NR 8
Target 0
|0|2 20dB
Interferences
1 -50o
|1|2 30dB
2 -10o
|2|2 30dB
3 40o
|3|2 30dB
Noise v2 0dB
Parameters Set up
Beam pattern
optimal receive filter optimal waveform
SPAC Lab, ECE 0 50 10013
14
15
16
17
18
19
20
Iteration index
SIN
R (
dB
)
-50 0 50-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Angle ()
Magnitu
de (
dB
)
SOA1-EC
SOA2-EC
+ SOA1-CMC
SOA2-CMC
SOA1-EC
SOA2-EC
+ SOA1-CMC
SOA2-CMC
Simulation Results
• Consider waveforms obtained from the proposed algorithms with only
constant modulus constraint (i.e., SOA1-CMC and SOA2-CMC)
• SOA1-CMC and SOA1-EC increase with the iteration number, and both
are converge very fast (i.e., after 2-3 iterations). For SOA2-EC and
SOA2-CMC, the convergence speed is slower
• Optimal SINRs are nearly the same and, therefore, there is no
significant loss of SINR by imposing the constant modulus constraint
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Simulation Results
• The similarity constraint incurs an SINR loss. For example, with = 1.5,
the loss for SOA1-CMSC and SOA2-CMSC is 1.3 dB and, respectively,
2.4 dB
• In general, the smaller the value of , the higher the SINR loss. The
beampatterns show that as the similarity constraint becomes stronger,
the interference null also becomes higher
0 50 10013
14
15
16
17
18
19
20
Iteration index
SIN
R (
dB
)
-50 0 50-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Angle ()
Magnitude (
dB
)
SOA1-CMSC, =2
SOA2-CMSC, =2
+ SOA1-CMSC, =1.5
SOA2-CMSC, =1.5
SOA1-CMSC, =2
SOA2-CMSC, =2
+ SOA1-CMSC, =1.5
SOA2-CMSC, =1.5
0 50 10013
14
15
16
17
18
19
20
Iteration index
SIN
R (
dB
)
SOA1-CMSC, =2
SOA2-CMSC, =2
+ SOA1-CMSC, =0.5
SOA2-CMSC, =0.5
-50 0 50-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Angle ()
Ma
gn
itu
de
(d
B)
SOA1-CMSC, =2
SOA2-CMSC, =2
+ SOA1-CMSC, =0.5
SOA2-CMSC, =0.5
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Simulation Results
• As increases, the side lobe level becomes higher and higher. It is
important to recall from previous simulation results a larger generally
yields a higher output SINR. Hence, in practice, the choice of should
be made by an appropriate tradeoff between the range solution and
output SINR of the resulting waveform.
-500 -400 -300 -200 -100 0 100 200 300 400 500-90
-80
-70
-60
-50
-40
-30
-20
-10
0
IFFT bin index
Ma
gn
itu
de
(d
B)
LFM
SOA1-CMSC, =2
SOA1-CMSC, =1
SOA1-CMSC, =0.5
SOA1-CMSC, =0.1
-500 -400 -300 -200 -100 0 100 200 300 400 500-90
-80
-70
-60
-50
-40
-30
-20
-10
0
IFFT bin index
Magnitude (
dB
)
LFM
SOA2-CMSC, =2
SOA2-CMSC, =1
SOA2-CMSC, =0.5
SOA2-CMSC, =0.1
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Conclusions
• Addressed the problem of MIMO radar waveform design in an
environment with signal-dependent interference plus noise
• Proposed two sequential optimization algorithms, named SOA1
and SOA2, by maximizing the receiver output SINR, accounting
for the constant modulus constraint as well as a similarity
constraint
• Numerical results indicate that
the constant envelope constraint leads to waveforms with little SINR
loss compared with those obtained without the constraint. This
clearly motivates the use of our constant modulus waveforms which
can be used with efficient nonlinear power amplifiers.
the larger the similarity parameter, the larger the output SINR, but
the poorer the pulse compression performance. This suggests a
suitable tradeoff between the target detection probability and the
range resolution should be considered in practice
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Part 3:
Passive Detection with Noisy Reference
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Passive Radar
Target
Reference
Channel
Surveillance
Channel
Passive Radar
Non-cooperative
illuminators
Passive Radar: A class of radar systems that detect and tract objects
by processing reflections from non-cooperative sources of illumination
• Advantages
– Smaller, lighter, and cheaper over
active radars
– Less prone to jamming
– Resilience to anti-radiation
missiles
– Stealth operations
• Disadvantages
– Rely on third-party illuminators
– Waveforms out of control poor
spatial/doppler resolution
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• Cross-correlation: cross-correlates the received data from the
reference and surveillance channels – H. D. Griffiths and C. J. Baker, “Passive coherent location radar systems. Part 1: performance
prediction,” IEE RSN, 2005
– P. E. Howland, D. Maksimiuk, and R. Reitsma, “FM radio based bistatic radar,” IEE RSN, 2005
• Generalized canonical correlation: based on the largest eigenvalue of
the Gram matrix of the received data – K. S. Bialkowski, I. Vaughan L. Clarkson and S. D. Howard, “Generalized canonical correlation
for passive multistatic radar detection,” IEEE SSP, 2011
• Autocorrelation-based detection – K. Polonen and V. Koivunen, “Detection of DVB-T2 control symbols in passive radar system,”
IEEE 7th SAM, 2012
• Passive MIMO radar detection: employ multiple illuminators of
opportunity and multiple receivers – D. E. Hack, L. K. Patton, B. Himed and M. A. Saville, “Detection in passive MIMO radar
networks,” IEEE TSP, 2014
– D. E. Hack, L. K. Patton, B. Himed and M. A. Saville, “Centralized passive MIMO radar
detection without direct-path reference signals,” IEEE TSP, 2014
Related Work
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• Reference channel:
• Surveillance channel:
– s is the unknown transmitted signal, nr and nt are time delays
– d is a Doppler shift, and are propagation parameters
– v and w are i.i.d. Gaussian noise
• After delay and Doppler compensation
Signal Model
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• Cross-correlation is a widely used passive detector
– Simple to implement
– Need no prior knowledge about the transmitted signal
– Equivalent to the optimum MF used in active sensing when the
reference channel is noiseless
– Performance degrades significantly with noisy reference channel
• Noise always exists in RC ) need new passive detectors capable of
dealing with noise in RC
• We propose GLRT based detectors by taking into account the noise in
the RC for the following four cases: the signal model is deterministic or
stochastic, the noise power is known or unknown
Motivation of Proposed Solutions
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• Consider the GLRT with unknown noise power
– The likelihood function under hypothesis H1 is
– ML estimates of and are:
– Using these estimates, L1 becomes
– The ML estimate of is
Detectors in Deterministic Model
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– The likelihood function under hypothesis H0 is
– The ML estimate of is:
– Using the estimate, L0 becomes
– The ML estimate of is
• The GLRT detector with unknown noise power is
Detectors in Deterministic Model
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• The GLRT detector with known noise power can be obtained in a
similar way
– Equivalently, the test variable can be written in terms of
eigenvalues
Detectors in Deterministic Model
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• Stochastic model:
– transmitted signal s(n) are modeled as i.i.d. complex Gaussian with
zero-mean and unit variance
– Justified for sources with multiplexing techniques (e.g., DVB-T signal)
• With known noise power, the GLRT is
• With unknown noise power, the GLRT is
• The above two detectors are referred to as B-GLRT detectors
Detectors in Stochastic Model
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• As an example, consider B-GLRT detector with known noise power
• stimates of a and b can be obtained by numerically solving the
following equations:
where
• Use the Newton-Raphson iterative method to solve the equations, and
obtain the estimates of a and b
Detectors in Stochastic Model
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• For comparison, we consider two detectors
– cross-correlation (CC) detector:
– matched filter (MF) detector:
• Define the SNRs in the surveillance and reference channels as,
respectively,
Numerical Results
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Numerical Results
-20 -15 -10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Dete
ctio
n P
roba
bili
tyN = 100, SNR
r = -10 dB and P
fa = 0.01
GLRT, known
GLRT, unknown
Tcc
B-GLRT, known
B-GLRT, unknown
TMF
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Numerical Results
-20 -15 -10 -5 0 5 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SNR (dB)
Dete
ctio
n P
roba
bili
tyN = 100, SNR
r = 0 dB and P
fa = 0.01
GLRT, known
GLRT, unknown
TCC
B-GLRT, known
B-GLRT, unknown
TMF
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Conclusions
• Investigated passive detection with a reference channel and
a surveillance channel
• Proposed four GLRT detectors:
– Deterministic signal model, known noise power
– Deterministic signal model, unknown noise power
– Stochastic signal model, known noise power
– Stochastic signal model, unknown noise power
• The proposed four GLRT except the one developed with
unknown noise power in the stochastic model outperform the
CC detector, especially at low SNRr
• Detection performance of the proposed four detectors highly
depends on the SNRr in the reference channel: the higher
the SNRr, the better the detection performance
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Thank you!