Expected-Likelihood Covariance Matrix Estimation for ... - IEEE Xplore

Expected-Likelihood Covariance Matrix Estimation for Adaptive Detection Yuri I. Abramovich, ISRD, DSTO, Adelaide, Australia Nicholas K. Spencer, CSSIP, Adelaide, Australia Key Words: adaptive detection, covariance matrix estimation, maximum likelihood

SUMMARY & CONCLUSIONS We demonstrate that by adopting the new class of “expected-likelihood” (EL) covariance matrix estimates, instead of the traditional maximum-likelihood (ML) estimates, we can significantly enhance adaptive detection performance. These new estimates are found by searching within the properly parameterized class of admissible covariance matrices for the one that produces the likelihood ratio (LR) that is “closest possible” to the LR generated by the true (exact) covariance matrix.

1. INTRODUCTION We consider the generic adaptive detection problem, whereby a decision regarding the presence or absence of a target signal in a given single “primary” sample has to be derived based on some model regarding the target signal and interference, and given “secondary” (or “training”) data that contains the same interference as the primary “snapshot” [1, 2, 3, 4]. This training data is considered in lieu of missing information regarding the interference p.d.f., and, specifically for the Gaussian model considered in this paper, regarding the interference covariance matrix. Known adaptive detectors, such as the adaptive matched filter (AMF) detector [2] or the adaptive coherence estimator (ACE) detector [3] have been developed using the same approach, whereby one first derives a decision rule based on a given model for the known interference covariance matrix. Next, the a priori unknown covariance matrix is replaced by the maximum-likelihood (ML) estimate of this matrix, drawn from the training data. It was always recognized (see [2], for example) that this is an ad hoc approach, since the use of the ML estimate is not justified by detection performance considerations, especially in most practically interesting cases with modest or small sample support. Moreover, the ML criterion is inappropriate for any “nested” models regarding covariance matrix structure if one tries to correspondingly restrict the class of admissible matrices. A typical example is the finite-rank interference-only model, where the ML estimate of interference-only rank does not exist, while multiple additional criteria, such as informationtheoretic criteria (ITC) used for order selection, are even less obviously associated with the adaptive detection problem. Since there is no particular reason, apart from asymptotic considera-

tions that are invalid for small sample support, to support the ML estimate, the adaptive detection problem is open for different types of estimates to be investigated.

2. EXPECTED-LIKELIHOOD ADAPTIVE MATCHED FILTER (EL-AMF) FOR GAUSSIAN MODELS Let us suppose that, as usual, XN ≡ [x1 , . . . , xN ] ∈ C M×N ∼ CN (0, c2 R)

(1)

is the N-sample secondary (training) data (c2 > 0), with each independent sample described by an M -variate complex (circular) Gaussian distribution with covariance matrix c2 R (c2 > 0). Suppose the primary sample Y is also described by the complex Gaussian p.d.f. with ½ c1 R for H0 (no target) RY = (2) c1 R + σs SS H for H1 (target present) with σs > 0. In this paper, we consider two examples: 1. Homogeneous training case, where (3)

c1 = c2 and σs is an arbitrary unknown signal power. 2. Nonhomogeneous training case, where

(4)

c1 6= c2

and σs is an arbitrary unknown signal power. In the more general case, the training data XN and the primary sample Y are described by the p.d.f.’s: f (XN |η, χ12 ) f (Y |χ0 , χ12 : H0 ) f (Y |µ, χ12 : H1 )

η ∈ Ω2 χ0 ∈ Ω0 µ ∈ Ω1

(5) (6) (7)

where χ12 ∈ Ω12 is the set of common (interference) parameters that describe the p.d.f. for both the primary data Y and the secondary set XN , and η, χ0 and µ are parameters specific to each set and hypothesis. Thus the standard AMF (or ACE) technique, which we henceforth specifically refer to as ML-AMF, may be introduced as: ML-AMF:

(2)

Λ12 =

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sup f (Y |µ, χ12ML : H1 )

µ∈Ω1

sup f (Y |χ0 , χ12ML : H0 )

χ0 ∈Ω0

(8)

where (9)

χ12ML = arg max f(XN |η, χ12 ). η,χ12

In this case

However, in the cases considered in this paper, we can replace the likelihood function (LF) f(XN |η, χ12 ) by the likelihood ratio (LR) f(XN |η, χ12 ) ∈ (0, 1] LR(XN |χ12 ) = max η f0 (XN )

argχ12 max f(XN |η, χ12 ) = arg max LR(XN |χ12 ). χ12

(11)

The most important property of this LR is that, for the exact (0) (0) (true) χ12 , the p.d.f. does not depend on χ12 , ie. is scenariofree, depends only on the parameters M and N , and can be precalculated. Hence for a given probability P0 , the upper and lower bounds (αU and αL respectively) can be found such that Z

(0)

(12)

and so the ML solution χ12ML is far away from the true set of parameters in terms of the LR metric. Of course, very small LRs may be generated not only by the true parameters, but by a variety of completely erroneous solutions as well. For this reason, we propose an “expected likelihood” (EL) approach that is based on a certain parameterization of the estimate χ ˆ12 (β) such that (14) χ ˆ12 (β0 ) = χ12ML and the parameterization corresponds to some valid a priori assumptions regarding the class of covariance matrices. We can now introduce the EL-AMF detector, where we replace ML estimates by EL ones: (5) Λ12

=

sup f (Y |µ, χ ˆ12EL : H1 )

µ∈Ω1

sup f(Y |χ0 , χ ˆ12EL : H0 )

.

ˆ −1 (βEL )R ˆ N exp M det R = LR0 . ˆ −1 (βEL )R ˆN exp tr R

EL-AMF:

det N −1 Cˆ exp M exp tr N −1 Cˆ

(20)

where Cˆ ∼ CW(N, M, IM ), is described by a scenario-free (complex Wishart) p.d.f. LR0 is then determined from the equation Z 0

(1)

(1)

(21)

w(γ0 ) dγ0 = 0.5

(1)

by using either the analytical expression for f(γ0 ) or by direct Monte-Carlo simulations.

2..2 Arbitrary Scaling Factors for Interference Matrices; Fluctuating Target with Unknown Power Here we assume that the (total) power of the interference within the training data could be different to that in the primary data, so that the interference covariance matrix is the same up to an arbitrary scaling factor [4]. More specifically, ª ª © © H = c2 N R, E Y Y H |H0 = c1 R. (22) E XN XN Hence we have

Ω0 = c1 ,

Ω1 = (c1 , σs ),

Ω2 = c2 .

(23)

Following (15), we may introduce the traditional ML-AMF solution (that is the ACE detector [3]) as cos c 2ML =

χ0 ∈Ω0

This value LR0 is chosen by referring to the scenario-free (0) p.d.f. LR(XN |χ12 ). For example, the mean or median value of the p.d.f. could be selected.

(19)

Note that in this case

(15)

The EL estimate is the estimate which, for given training data XN , generates the specific precalculated likelihood ratio LR0 , ie. LR[XN |χ ˆ12 (βEL )] = LR0 . (16)

(18)

ˆ 0 ) should be chosen either in the standard way Here R(β ˆ N to get the well-known ML-AMF method, ˆ R(β0 = 0) = R or by (16) to get the EL-AMF rule:

LR0

with 0 < ε ¿ 1, so that with high probability (1−2ε), the exact (0) parameters χ12 generate the LR within the specified bounds. For most cases with relatively small sample size (N ' M ), we will show that (13) αU ¿ 1

EL-AMF:

ˆ −1 (βˆ0 )S|2 d |Y H R ≶d01 h∗ > 1. ˆ −1 (βˆ0 )S SH R

(0)

w[LR(XN |χ12 )] dLR = P0 = 1 − ε

(17)

Ω2 6= ∅.

Following (15), we derive the familiar decision rule

γ0 ≡ LR(XN |R0 ) =

w[LR(XN |χ12 )] dLR =

αU

0

Ω1 = σs ,

(1)

1

αL

Z

Ω0 = ∅,

(10)

that has the same ML solution for χ12ML : η,χ12

2..1 Homogeneous Interference Conditions; Fluctuating Target with Unknown Power

ˆ −1 S|2 |Y H R N ≶dd01 h∗ −1 −1 H H ˆ ˆ S R SY R Y N

(24)

N

and the EL-AMF solution as cos c 2ML =

ˆ −1 (βEL )S|2 |Y H R ≶d0 h∗ ˆ −1 (βEL )SY H R ˆ −1 (βEL )Y d1 SH R

(25)

whereby

h

ˆ −1 (βEL )R ˆ det R iM = LR0 . 1 ˆ −1 (βEL )R ˆ tr R M

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(26)

According to (16), LR0 is here specified by the scenario-free (2) p.d.f. that was derived in [5] for γ0 : (2)

γ0 = h

det Cˆ iM , 1 tr Cˆ

Cˆ ∼ CW(N, M, IM ).

(27)

M

Note that the above well-known models have been elucidated since they permit analytic solutions for the ML estimates σ ˆsML and cˆML . The same methodology can be applied for more complex models where such estimates are found numerically.

3. DETECTION PERFORMANCE ANALYSIS OF ML VS EL AMF DETECTORS To analyze the detection performance of our suggested technique, we first have to specify a scenario and the parametric family of covariance matrix estimates R(β). Consider an M = 12-sensor uniform linear antenna array, m = 6 independent Gaussian interference sources, each with 30 dB signal-to-white-noise ratio (SWNR), and observations comprising N = 2M = 24 training samples. The interference DOAs were chosen to be w6 ≡ sin θ6 = [−0.8, −0.4, 0.2, 0.5, 0.7, 0.9]

(28)

so that the eigenspectrum of the interference covariance matrix R0 = σn IM +

6 X

σj S(wj )S(wj )H

(29)

j=1

where σn is the white-noise power, and σn = 1,

σj = 1000,

wj = 2π

d sin θj λ

(30)

is similar to the eigenspectrum of the terrain-scattered spacetime covariance matrix in a side-looking airborne radar with three antenna sensors and four repetition periods [6]. Two target DOAs w0 have been selected to represent two extreme cases, namely ½ S0H [I − S6 (S6H S6 )−1 S6H ]S0 0.949 (w0H = −0.60) = 0.040 (w0L = 0.18). S0H S0 (31) In the first case (“fast target” in STAP terminology), total interference mitigation is not accompanied by a significant degradation in target SWNR, whereas in the second case (“slow target”), such interference “nulling” leads to a dramatic signal power reduction. Note that for the clairvoyant detector (R = R0 ), as well as for the standard AMF detectors, this distinction does not affect the receiver operational characteristic (ROC) if the output SNRs are identical: σsL S0LH R0−1 S0L = σsH S0HH R0−1 S0H .

(32)

Whether this property survives for the EL-AMF has yet to be explored.

For this purpose, we selected the two well-known families of covariance matrix estimates, ˆN , R(β) = βIM + R

R≥0

(33)

which is the diagonally loaded sample matrix family [7, 8], and M−m X 1 ˆ m+j U ˆM−m U ˆH λ M−m M −m j=1 (34) ˆm ; U ˆM−m ], ˆN ≡ U ˆm Λ ˆ mU ˆH ≡ [U R (35) m

ˆ mU ˆH + ˆm Λ R(m) = U m ˆM U

which is the finite interference subspace approximation or the “fast maximum likelihood” (FML) estimate family [9]. Unlike the traditional loaded sample matrix inversion (LSMI) and FML techniques, whereby the loading factor β in (33) or the interference subspace dimension m in (34) are based on some additional considerations, for our proposed ELAMF technique these parameters are to be optimized against ˆ Note that for N ≥ M, the above criteria for every given R. the unconstrained covariance matrix ML estimate leads to the solutions β = 0 in (33) and m = M − 1 in (34). Calculated over 107 Monte-Carlo trials, Fig. 1 presents the (1) scenario-free p.d.f. for the “general” test (20) w(γ0 ). Together (2) with the very similar p.d.f. for the “sphericity” test (27) w(γ0 ), they demonstrate that for 99% probability, each lower bound is αL = 0.008 while the upper bound is αU = 0.065. The median values are LR0 = 0.0257 and LR0 = 0.0262 respectively. Each of these LR values is orders of magnitude smaller than the ˆ N . For our case of LR = 1 generated by the ML estimate R M = 12, N = 24, these p.d.f.’s show that the probability of the exact covariance matrix generating a LR above 0.1 is about 10−4 ! Clearly, the ML estimate is completely inappropriate in terms of the w(γ0 ) metric. This was actually the most important factor in our development of the EL estimate. Let us first analyze the detection performance for our example scenario with unknown target power and homogeneous interference conditions. For the clairvoyant case R = R0 , the ROC has the well-known analytic expression Pd = exp[−

h ] 1 + σs S H R−1 S

(36)

where h is the threshold and σs S H R−1 S is the output SNR. Our analysis starts from a comparison of the traditional ML-AMF and clairvoyant methods with our EL-AMF method. We use the clairvoyant ROC function (36) to validate our 106 Monte-Carlo simulations with false-alarm rates set at PF A = 10−2 , 10−4 , and 104 trials for each signal scenario. The signal powers for the two (extreme) target scenarios in (31) were chosen so that the output SNRs are the same (32) (0 – 35 dB). Figs. 2 and 3 illustrate ROCs for the target w0H with the high ratio in (31) (“fast target”); figures calculated for the lowratio (“slow”) target are practically the same. We first note the ideal coincidence between the theoretically calculated ROCs for the clairvoyant case (curves labeled “theoretical”) and those obtained by Monte-Carlo simulations (“ideal”). This high accuracy validates the other Monte-Carlo results.

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The most important observation involves the comparison of the performance of the standard AMF (ML-AMF) technique with our EL-AMF method (separately optimizing loading factor and interference subspace dimension). We see that EL-AMF has a significant improvement; for example, for PF A = 10−4 and PD = 0.5, standard AMF suffers a 5 dB loss in SNR compared with the clairvoyant case, while the new EL-AMF has only 1 – 1.5 dB losses. As expected, over the entire range of false-alarm and detection probabilities analyzed, only a negligible difference is found between the optimum loading and optimum subspace methods, again proving the common nature of these two techniques [9]. Also, we demonstrated that the ROCs are exclusively specified by the output SNR, irrespective of the target and interference signal correlation in (31). Let us now turn our attention to the EL-AMF technique for the nonhomogeneous scenario (22); in this case, LR0 = 0.0262. Figs. 4–6 present the ROCs for the “fast” target; the ROCs for the “slow” target are practically indistinguishable. For this model, the original approximately 3 dB loss factor (for PF A = 10−4 , PD = 0.5) for ML-AMF with respect to the clairvoyant case (Fig. 4) is reduced to less than 1 dB by the optimum choice of loading factor or interference subspace dimension within the EL-AMF technique. While it may be argued that data-independent diagonal loading may result in similar performance, the most important distinction between EL-AMF and the standard LSMI algorithm is the direct data-dependent choice of the loading factor, based on the expected likelihood rather than on considerations typically used to justify the LSMI technique [10].

4. SUMMARY AND CONCLUSIONS We have introduced an approach called “expected likelihood” (EL), whereby we try to find the estimate that statistically generates the same LR as the exact covariance matrix. This is feasible in practice since the p.d.f. for the LR generated by the exact covariance matrix does not depend on the matrix itself, but only the parameters M and N , and so it can be precalculated. We have used two well-known families of covariance matrix estimates: the diagonally loaded sample matrix (ie. the loaded unconstrained ML solution) and the finite-subspace interference approximation of the ML solution. For these estimates respectively, the traditional ML criterion drives the loading factor to zero, and the interference subspace dimension to its maximum. For the EL-AMF (ACE) technique, we seek the loaded solution that generates the median LR value produced by the exact covariance matrix. For finite-rank approximations that have only a finite number of solutions, we simply find the one that is closest to the upper LR bound, if no solution within the bounds is available. The Monte-Carlo simulation results demonstrate that our EL criterion for the proper families (diagonally loaded, finite interference rank) gives a significant improvement in detection performance compared with the ML criterion, which for small sample support produces solutions far away from the exact ones. We emphasize that the introduced families include the standard (unconstrained) ML covariance matrix estimate, while the ma-

jor distinction stems from the attempt to get a statistically close LR to that of the exact covariance matrix, rather than just the (ultimate) maximum LR value. This is an important distinction from some optimum search over a restricted set of covariance matrices, such as the class of Toeplitz covariance matrices, for example. Any reliable a priori structural information on the covariance matrix should always lead to a detection improvement, however we chose the most generic families specifically to underline the difference in criteria (EL versus ML), rather than any possible difference in covariance matrix description.

REFERENCES [1] E. Kelly, “An adaptive detection algorithm,” IEEE Trans. Aero. Elect. Sys., vol. 22, no. 1, pp. 115–127, Mar. 1986. [2] F. Robey, D. Fuhrmann, E. Kelly, and R. Nitzberg, “A CFAR adaptive matched filter detector,” IEEE Trans. Aero. Elect. Sys., vol. 28, no. 1, pp. 208–216, Jan. 1992. [3] L. McWhorter, L. Scharf, and L. Griffiths, “Adaptive coherence estimation for radar signal processing,” in Proc. ASILOMAR-96, vol. 1, Pacific Grove, CA, USA, 1996, pp. 536–540. [4] S. Kraut and L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Sig. Proc., vol. 47, no. 9, pp. 2538–2541, Sept. 1999. [5] Y. Abramovich, N. Spencer, and A. Gorokhov, “Bounds on maximum likelihood ratio — Part I: Application to antenna array detection-estimation with perfect wavefront coherence,” IEEE Trans. Sig. Proc., vol. 52, no. 6, pp. 1524–1536, June 2004. [6] R. Klemm, Space-Time Adaptive Processing: Principles and Applications. UK: IEE, 1998. [7] Y. Abramovich, “A controlled method for adaptive optimization of filters using the criterion of maximum SNR,” Radio Eng. Electron. Phys., vol. 26 (3), pp. 87–95, 1981. [8] O. Cheremisin, “Efficiency of adaptive algorithms with regularised sample covariance matrix,” Radio Eng. Electron. Phys., vol. 27 (10), pp. 69–77, 1982. [9] K. Gerlach, “Outlier resistant adaptive matched filtering,” IEEE Trans. Aero. Elect. Sys., vol. 38, no. 3, pp. 885–901, July 2002. [10] O. Cheremisin, “Loading factor selection in the regularised algorithm for adaptive filter optimisation,” Radioteknika i Elektronika, vol. 30 (12), 1985, English translation should be found in Soviet Journal of Communication Technology and Electronics.

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−3

general non−sphericity test, ULA, M = 12, N = 24, 10000000 trials P (LR < 0.0142) = 10−1

P (LR < 0.0257) = 0.5

3

P (LR < 0.0437) = 1 − 10−1

P (LR < 0.0084) = 10−2 P (LR < 0.0055) = 10−3

2

P (LR < 0.0647) = 1 − 10−2 P (LR < 0.0842) = 1 − 10−3 P (LR < 0.1035) = 1 − 10−4

P (LR < 0.0038) = 10−4 1 0

−0.04

−0.02

0

0.02

0.04

0.06 LR

0.08

0.1

0.12

0.14

0.16

Figure 1: P.d.f. for the general test. matched, homog, EL−AMF, ULA, M = 12, m = 6, N = 24, 30dB SNR, 10000 trials, high−ratio target, P =10−4 FA

detection probability

1 0.8 0.6 0.4

EL−AMF (loading) ML−AMF ideal EL−AMF (subspace) theoretical

0.2 0

0

5

10

15 output SNR (dB)

20

25

30

Figure 2: EL-AMF ROC for the high-ratio target with a false-alarm probability of 10−4 . −2

matched, homog, EL−AMF, ULA, M = 12, m = 6, N = 24, 30dB SNR, 10000 trials, high−ratio target, PFA=10

1


sample probability

4

x 10

0.8 0.6 0.4


0.2 0

0

5

10

15 output SNR (dB)

20

25

Figure 3: EL-AMF ROC for the high-ratio target with a false-alarm probability of 10−2 .

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30

matched, nonhomog, EL−SMI, ULA, M = 12, m = 6, N = 24, 30dB SNR, 10000 trials, high−ratio target, PFA=10−4


1 0.8 0.6 0.4


0.2 0

0

5

10

15 output SNR (dB)

20

25

30

Figure 4: Nonhomogeneous EL-AMF ROC for the high-ratio target with a false-alarm probability of 10−4 . −3

matched, nonhomog, EL−SMI, ULA, M = 12, m = 6, N = 24, 30dB SNR, 10000 trials, high−ratio target, P =10 FA


1 0.8 0.6 0.4


0.2 0

0

5

10

15 output SNR (dB)

20

25

30

Figure 5: Nonhomogeneous EL-AMF ROC for the high-ratio target with a false-alarm probability of 10−3 . −2

matched, nonhomog, EL−SMI, ULA, M = 12, m = 6, N = 24, 30dB SNR, 10000 trials, high−ratio target, PFA=10


1 0.8 0.6 0.4


0.2 0

0

5

10

15 output SNR (dB)

20

25

30

Figure 6: Nonhomogeneous EL-AMF ROC for the high-ratio target with a false-alarm probability of 10−2 .

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