radar adaptive detection and its applications · radar adaptive detection and its applications...

Radar Adaptive Detection and Its Applications

Presenter: Jun Liu

National Laboratory of Radar Signal ProcessingXidian University

2017.11.19

EIES 2017

Adaptive detection

BackgroundsA

Range cell

Guard cell

Cell under test

Training data

Covariance matrix estimation

Filter and detection

Adaptive detection

BackgroundsA

Range cell

Guard cell

Cell under test

Training data

Covariance matrix estimation

Filter and detection

① Adaptive detection: sufficient training data Detector design Statistical analysis

② Adaptive detection: limited training data Detector design Statistical analysis

③ Adaptive detection: no training data Detector design Statistical analysis

Contents

① Adaptive detection: sufficient training data

② Adaptive detection: limited training data

③ Adaptive detection: no training data

Contents

• The received data

– s is the steering vector of dimension N £ 1– a is a deterministic but unknown complex scalar– n is disturbance, and has Gaussian distribution with zero mean

and unknown covariance matrix αR, i.e.,– α = 1: homogeneous environment– α ≠ 1: partially homogeneous environment

• A set of training data: • Binary hypothesis testing

Adaptive Detection With Sufficient Data: Problem Formulation

• According to Neyman-Pearson criterion, the optimal detector is the likelihood ratio test given by

• Due to the unknown parameters, no uniformly most powerful test exists

• This motivates us to design detectors under various criteria

Adaptive Detection With Sufficient Data: Design Criteria

No uniformly most powerful test

• One-step generalized likelihood ratio test (GLRT)

• Two-step GLRT

• Rao test:

• Wald test:

Adaptive Detection With Sufficient Data: Design Criteria

• Generalized Likelihood Ratio Test– E. J. Kelly, “An adaptive detection algorithm,” IEEE Trans.

Aerosp. Electron. Syst., vol. 22, no. 1, pp. 115–127, Mar. 1986.• Adaptive Matched Filter

– F. C. Robey, D. R. Fuhrmann, E. J. Kelly, and R. Nitzberg, “A CFAR adaptive matched filter detector,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 1, pp. 208–216, Jan. 1992.

• Adaptive Coherence Estimator– S. Kraut, L. L. Scharf, and R. W. Butler, “The adaptive coherent

estimator: A uniformly most-powerful-invariant adaptive detection statistic,” IEEE Trans. Signal Process., vol. 53, no. 2, pp. 427–438, Feb. 2005.

• Rao Test– A. De Maio, “Rao test for adaptive detection in Gaussian

interference with unknown covariance matrix,” IEEE Trans. Signal Process., vol. 55, no. 7, pp. 3577–3584, Jul. 2007.

Adaptive Detection With Sufficient Data: State of the Art

• Homogeneous environment (α = 1):

• Partially homogeneous environment (α ≠ 1)

• These adaptive detectors are designed for rank-1 signal model10

Adaptive Detection With Sufficient Data: Conventional Adaptive Detectors

• Signal subspace model:

– S is the steering vector of dimension N £ p, and p < N– a is a deterministic but unknown complex vector of dimension p £ 1– n is disturbance, and has Gaussian distribution with zero mean and

unknown covariance matrix αR, i.e.,– α = 1: homogeneous environment– α ≠ 1: partially homogeneous environment

• A set of training data: • Advantages of signal subspace model

– Robustness– Polarization radar using multiple polarization channels

Adaptive Detection With Sufficient Data: Signal Subspace Model

Signal subspace model

• Binary hypothesis testing for the subspace model case:

• A uniformly most powerful test does not exist• Design detectors under various criteria:

– One-step GLRT– Two-step GLRT– Rao test– Wald test

Adaptive Detection With Sufficient Data: Signal Subspace Model

• Homogeneous environment (α = 1):

• Partially homogeneous environment (α ≠ 1)

Adaptive Detection With Sufficient Data: Detectors for Subspace Model

• The PFA of GLRT detector

• The PD of GLRT detector

Adaptive Detection With Sufficient Data: Analytical Performance-GLRT

• The PFA of AMF detector

• The PD of AMF detector

Adaptive Detection With Sufficient Data: Analytical Performance-AMF

• The PFA of Rao detector

• The PD of Rao detector

Adaptive Detection With Sufficient Data: Analytical Performance-Rao Test

• The PFA of ASD detector

• The PD of ASD detector

Adaptive Detection With Sufficient Data: Analytical Performance-ASD

• Double subspace model:

– X is the received data of N £ K– A is a known matrix of N £ J– C is a known matrix of M £ K– B is an unknown coordinate matrix of J £ M– N is noise matrix of N £ K, each column vector has IID Gaussian

distribution with zero mean and unknown covariance matrix αR• A set of training data:• This data model has many applications in

– Multichannel radar: sensor array, pulsed Doppler radar– MIMO radar– Distributed targets– Multiple-band radar

Adaptive Detection With Sufficient Data: Further Extension

• Special cases in homogeneous environments (α = 1):– N > 1 and J = M = K = 1

multichannel radar– N > 1, J > 1, and M = K = 1

signal subspace model– N > 1, J = 1, and M = K > 1

multiple-band or distributed target– N > 1, J > 1, and M = K > 1

Distributed target detection with uncertain steering vector– N > 1, J > 1, M > 1, K > 1

MIMO radar

• Special cases in partially homogeneous environments (α ≠ 1):– N > 1, J = 1 and M = K >1

Distributed target detection

• Adaptive detectors in homogeneous environments (α = 1):

• Adaptive detectors partially homogeneous environments (α ≠ 1):

• Adaptive detectors homogeneous environments (α = 1):

Adaptive Detection With Sufficient Data: Simulation Results

8 10 12 14 16 18 20 22 24 260

SNR (dB)

AODGLRTRaoWald2SDSNT1SNT2SNT3

19.9 19.92 19.94 19.96 19.98 20 20.02 20.04 20.06 20.08 20.10.992

• Adaptive detectors homogeneous environments (α = 1):

Adaptive Detection With Sufficient Data: Simulation Results

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

cos2(φ)

GLRTRaoWald2SDSNT1SNT2SNT3

• Adaptive detector design and performance analysis for the signal subspace model

• Many adaptive detectors are designed for the double subspace signal model

Adaptive Detection With Sufficient Data: Conclusions

1. Jun Liu, W. Liu, B. Chen, H. Liu, H. Li, and C. Hao, “Modified Rao test for multichannel adaptive signal detection,” IEEE Transactions on Signal Processing, vol. 64, no. 3, pp. 714—725, February 1, 2016.

2. W. Liu, W. Xie, Jun Liu, and Y. Wang, “Adaptive double subspace signal detection in Gaussian background—Part I: homogeneous environments,” IEEE Transactions on Signal Processing, vol. 62, no. 9, pp. 2345—2357, May 2014.

3. W. Liu, W. Xie, Jun Liu, and Y. Wang, “Adaptive double subspace signal detection in Gaussian background—Part II: partially homogeneous environments,” IEEE Transactions on Signal Processing, vol. 62, no. 9, pp. 2358—2369, May 2014.

4. Jun Liu, Z.-J. Zhang and Y. Yang, “Optimal waveform design for generalized likelihood ratio and adaptive matched filter detectors using a diversely polarized antenna,” Signal Processing, vol. 92, no. 4, pp. 1126—1131, Apr. 2012.

5. Jun Liu, Z.-J. Zhang, P.-L. Shui and H. Liu, “Exact performance of an adaptive subspace detector,” IEEE Transactions on Signal Processing, vol. 60, no. 9, pp. 4945—4950, Sep. 2012.

6. Jun Liu, Z.-J. Zhang, Y. Yang and H. Liu, “A CFAR adaptive subspace detector for first-order or second-order Gaussian signals based on a single observation,” IEEE Transactions on Signal Processing, vol. 59, no. 11, pp. 5126—5140, Nov. 2011.

Contents

• Reed, Mallett, and Brenann (RMB) rule:– the SNR loss is 3 dB when the amount of homogeneous

training data used to estimate the noise covariance matrix is approximately twice the dimension of the received signal

K ¼ 2N

• Problems: the number of homogeneous training data is limited

• Diagonal Loading– L. Du, and J. Li, “Fully Automatic Computation of Diagonal

Loading Levels for Robust Adaptive Beamforming,” IEEE Transactions on Aerospace and Electronic Systems, vol. 46, no. 1, January 2010

• Joint Domain Localized Processing– R. S. Adve, T. B. Hale and M. C. Wicks, “Practical joint domain

localised adaptive processing in homogeneous and nonhomogeneous environments. Part 1: Homogeneous environments,” IEE Proc.- Radar, Sonar Navig., vol. 147, No. 2, April 2000

• Knowledge-Aided Methods– J. R. Guerci and E. J. Baranoski, “Knowledge-aided adaptive

radar at DARPA,” IEEE Signal Processing Magazine, vol. 41, January 2006

Adaptive Detection With Limited Data: The State of Art

• Persymmetry exists, when a radar is equipped with a symmetrically spaced linear array for spatial domain processing or symmetrically spaced pulse trains for temporal domain processing

• Persymmetry means double symmetry, for example

• Our focus on convergence rate analysis in the persymmetric case

Adaptive Detection With Limited Data: Prior Structure on Covariance matrix

1222*23

132322*12

14131211

SSSSSSSSSSSSSSSS

Persymmetric matrix

11 12 13 14*12 22 23 24* *13 23 33 34* * *14 24 34 44

S S S SS S S SS S S SS S S S

Hermitian

Persymmetry

• The received data

– s is the steering vector of dimension N £ 1– a is a deterministic but unknown complex scalar– n is disturbance, and has Gaussian distribution with zero mean

and unknown covariance matrix R, i.e., • The minimum variance distortionless response (MVDR)

beamformer can be obtained by solving the optimization problem:

• The optimal weight vector is

Adaptive Detection With Limited Data: Convergence Rate

• Persymmetric structures– Persymmetric steering vector :– Persymmetric covariance matrix:

• The ML estimate of R with persymmetry (up to a scalar) is

• The persymmetric SMI beamformer in the matched case is given by

• The normalized output SNR of the persymmetric SMI beamformer in the matched case is

Adaptive Detection With Limited Data: Convergence Rate-Our Work

• In practice, mismatches occur in:– Steering vector: – Covariance matrix:

• The persymmetric SMI beamformer in the mismatched case is

• The normalized output SNR in the mismatched case is

• Problem: what are the average SNR losses in the matched and mismatched cases?

• Consider the matched case:• We aim to derive the distribution of

• Define

• The ML estimation can be rewritten as

• It is easy to show that

• Define a unitary transformation

• Two real Gaussian vectors:

• From complex domain to real domain

• The normalized output SNR of the persymmetric SMI beamformer can be recast as

• Using the Bartlett’s decomposition of a real Wishart matrix, we can obtain

• Then

• Recall RMB’s result:•

• The exploitation of persymmetry is equivalent to doubling the number of training data

• In the following we consider the mismatched case

• In practice, mismatches occur in:– Steering vector mismatch: – Covariance matrix mismatch:

• The persymmetric SMI beamformer in the mismatched case is

• The normalized output SNR in the mismatched case is

• Problem:

• It is very difficult to obtain an exact expression for • Instead, we seek to derive an approximate expression•

• Note that when

• Remark: for

• Hence,• Now, the problem of deriving the average SNR loss turns to

calculate• We prove that

• Result in the mismatched case:

• One special case:– When

– Recall Boroson’s result:

• Define• DOA of a target: 400

• DOAs of two interference signals: – 300 and 200

• Matched case:

4220 40 60 80 100 120 140 160

Sample Number K

ρpermatch, MC

ρpermatch, theory

ρmatch

N = 50

N = 30

Adaptive Detection With Limited Data: Convergence Rate-Simulation Results

• Mismatched case:

0 10 20 30 40 50 60 70 80 90 1000

Sample Number K

mismatch, MC

ρpermismatch, theory

ρmismatch

N = 30

N = 50

• Mismatched case:

0 10 20 30 40 50 60 70 80 90 1000

Sample Number (K)

N = 30, MCN = 30, theoryN = 50, MCN = 50, theory

• Persymmetric structures are exploited in adaptive filtering• A distribution of the normalized output SNR of the

persymmetric SMI beamformer is derived • An exact expression for the average SNR loss is obtained

in the matched case• An approximate expression for the average SNR loss is

obtained in the mismatched case:– Mismatch in the steering vector– Mismatched in the covariance matrix

Jun Liu, W. Liu, H. Liu, C. Bo, X.-G. Xia, and F. Dai, “Average SINR calculation of a persymmetricsample matrix inversion beamformer,” IEEE Transactions on Signal Processing, vol. 64, no. 8, pp. 2135—2145, April 15, 2016.

Adaptive Detection With Limited Data: Convergence Rate-Conclusions

• The received data:

– p is the steering vector of dimension m £ 1– a is a deterministic but unknown complex scalar– c is noise, and has Gaussian distribution with zero mean and

unknown covariance matrix M, i.e., • A set of training data for estimating the covariance matrix M

• Persymetric structures– Persymmetric steering vector :– Persymmetric covariance matrix:

Adaptive Detection With Limited Data: Persymmetric Adaptive Detection

• Binary hypothesis testing

• Use a unitary matrix T to transform the complex steering vector p and the complex covariance matrix M from complexdomain to real domain

• After transformation

• Note that s and R are both real

• The maximum likelihood estimate of R is

• The AMF using the persymmetric structures (i.e., P-AMF) becomes

• Note that s and are real, and x is complex

• The probability of false alarm is derived as

• It is not convenient to use this expression to calculate the detection threshold λ, since an integral is included

• We provide simpler expression to compute the detection threshold

• Case 1): when m is even and K = m/2, we have M =1/2, and

• Case 2): when m is odd and K = (m+1)/2, we have M =1, and

• Case 3): to facilitate setting the detection threshold in other cases, the probability of false alarm can be approximated as

• There exists a real orthogonal matrix U:

• Define

• Then

• It can be shown that

• The P-AMF detector can be equivalently written as

• Define , its PDF conditioned on ρ is

• When m is odd, the detection probability conditioned on ρ is

• Note that 56

• The unconditioned probability of detection for odd m is

• When m is even, M is not an integer. Intuitively, we can approximate the detection probability for m even as the arithmetic mean of the detection probabilities obtained by replacing m with m+1 and m-1

-10 -5 0 5 10 15 20 25 300

SNR (dB)

MFAMFPS-AMFMC

K = 7K = 35

Data dimension

Adaptive Detection With Limited Data: Numerical Results-Simulated Data

-10 -5 0 5 10 15 20 25 30 35 400

SNR (dB)

MFAMFPS-AMFMC

K =8K = 32

Data dimension

Adaptive Detection With Limited Data: Numerical Results-Simulated Data

0 5 10 15 20 25 30 35 4010

Detection Threshold λ

m=8, Range #15

MC,K=4MC,K=6MC,K=8theory K=4theory K=6theory,K=8

0 5 10 15 20 25 30 35 4010

m=8, Range# 27

0 5 10 15 20 25 30 35 4010

m=8, Range# 48

0 5 10 15 20 25 30 35 4010

m=8, Range #56

MC,K=4MC,K=6MC,K=8theoryK=4theory K=6theory,K=8

0 5 10 15 20 25 30 35 4010

m=8, Range #68

Adaptive Detection With Limited Data: Numerical Results-Real Data

• PFA = 0.01

-10 -5 0 5 10 15 20 25 30 35 400

SCR(dB)

m = 8, K = 8, Range#15, fd = 0.02

AMFP-AMFCA-CFAR

• PFA = 0.01

-10 -5 0 5 10 15 20 25 30 35 400

SCR(dB)

m = 8, K = 12, Range#15, fd = 0.02

AMFP-AMFCA-CFAR

• PFA = 0.01

-10 -5 0 5 10 15 20 25 30 35 400

SCR(dB)

m = 8, K = 16, Range#15, fd = 0.02

AMFP-AMFCA-CFAR

• Persymmetric structures are exploited in adaptive detection• Simpler expressions for the probability of false alarm of the

P-AMF are provided• Closed-form expression for the detection probability of the P-

AMF is derived• The P-AMF outperforms the conventional AMF, especially in

the case of limited training data

Jun Liu, G. Cui, H. Li, and B. Himed, “On the performance of a persymmetric adaptive matched filter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 4, pp. 2605—2614, October 2015.

Adaptive Detection With Limited Data: Persymmetric Adaptive Detection-Conclusions

Contents

• Distributed MIMO radar – Spatial diversity

• Collocated MIMO radar– Waveform diversity

Adaptive Detection Without Training Data: MIMO Radar Backgrounds

Center

• A collocated MIMO radar: M transmit antennas and N receive antennas

• The received data:

– X is the received data of N £ K– ar(θ) is the received steering vector of dimension N £ 1– at(θ) is the transmit steering vector of dimension M £ 1– S is the transmitted waveform matrix of M £ K– a is a deterministic but unknown complex scalar– V is disturbance, and each column vector has IID Gaussian

distribution with zero mean and unknown covariance matrix R• Binary hypothesis testing:

Adaptive Detection Without Training Data: Problem Formulation

• Known disturbance covariance matrix– I. Bekkerman and J. Tabrikian, “Target detection and localization

using MIMO radars and sonars,” IEEE Transactions on Signal Processing, vol. 54, no. 10, pp. 3873–3883, October 2006.

• Adaptive GLRT detector– L. Xu, J. Li, and P. Stoica, “Target detection and parameter

estimation for MIMO radar systems,” IEEE Transactions on Aerospace and Electronic Systems, vol. 44, no. 3, pp. 927–939, July 2008.

– No statistical analysis for the GLRT detector

Adaptive Detection Without Training Data: State of the Art

• Rao test:

• Wald test:

• Tunable detector:

Adaptive Detection Without Training Data: Proposed Detectors

• Probability of false alarm

• Probability of detection in the matched case

Adaptive Detection Without Training Data: Analytical Performance

• The nominal receive steering vector deviates from the actual one, and the mismatched angle is defined by

• Probability of detection

Adaptive Detection Without Training Data: Analytical Performance-Mismatched Case

• Matched case (cos2φ = 0, φ = 0o), M = N =10

Adaptive Detection Without Training Data: Numerical Results-Simulated Data

-15 -10 -5 0 5 10 15 20 25 300

SNR (dB)

(a) K = 1.5N

GLRTα = 0 (i.e., Wald)α = 0.5α = 1 (i.e., Rao)α = 5MC

-15 -10 -5 0 5 10 15 20 25 300

SNR (dB)

(b) K = 2N

• Mismatched case (cos2φ = 0.9412), M = N =10

-20 -15 -10 -5 0 5 10 15 20 25 300

SNR (dB)

(a) K = 1.5N

-20 -15 -10 -5 0 5 10 15 20 25 300

SNR (dB)

(b) K = 2N

• Mismatched case, M = N = 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

cos2φ

(a) SNR = 5 dB

GLRTα = 0 (i.e., Wald)α = 0.1α = 0.5α = 1 (i.e., Rao)α = 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

cos2φ

(b) SNR = 0 dB

GLRTα = 0 (i.e., Wald)α = 0.1α = 0.5α = 1 (i.e., Rao)α = 2

• M = N = 16

Adaptive Detection Without Training Data: Numerical Results-Real Data

0 0.02 0.04 0.06 0.08 0.1 0.12 0.1410

Threshold

GLRT, theoryGLRT, MCRao test, theoryRao test, MCWald test, theoryWald test, MC

• M = N = 16

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

Threshold

L = 10

GLRT, theoryGLRT, MCRao test, theoryRao test, MCWald test, theoryWald test, MC

• M = N = 16

0 4 8 12 16 20 24 28 320

SNR(dB)

N = 16, L = 10, fd=0.02

GLRTRaoWaldCA-CFAR

• M = N = 16

0 4 8 12 16 20 24 28 320

SNR(dB)

N = 16, L = 5, fd = 0.02

GLRTRaoWaldCA-CFAR

• Rao test, Wald test, and tunable detector are proposed in collocated MIMO radar

• Training data is not required for the adaptive detection• All three detectors exhibit CFAR against clutter covariance

matrix• Analytical expressions for the PFA and PD are derived for both

matched and mismatched cases

1. W. Liu, Y. Wang, Jun Liu, W. Xie, H. Chen, and W. Gu, “Adaptive detection without training data in collocated MIMO Radar,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 3, pp. 2469—2479, July 2015.

2. Jun Liu, S. Zhou, W. Liu, J. Zheng, H. Liu, and J. Li, “Tunable adaptive detection in collocated MIMO Radar,” IEEE Transactions on Signal Processing, Accepted for publication.

Adaptive Detection Without Training Data: Conclusions

Thank you!

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