CFAR Matched Direction Detector Olivier Besson, Senior Member, IEEE, and
LouisL. Scharf, Fellow, IEEE
Abstract—In a previously published paper by Besson et al., we consid-ered the problem of detecting a signal whose associated spatial signature is known to lie in a given linear subspace, in the presence of subspace in-terference and broadband noise of known level. We extend these results to the case of unknown noise level. More precisely, we derive the general-ized-likelihood ratio test (GLRT) for this problem, which provides a con-stant false-alarm rate (CFAR) detector. It is shown that the GLRT involves the largest eigenvalue and the trace of complex Wishart matrices. The dis-tribution of the GLRT is derived under the null hypothesis. Numerical simulations illustrate its performance and provide a comparison with the GLRT when the noise level is known.
Index Terms—Array processing, detection, eigenvalues, Wishart matrices.
I. PROBLEMSTATEMENT
Detecting a signal in the presence of low-rank interference and broadband noise is an ubiquitous task in many array processing applications [2]. In the single-snapshot case, this problem has been studied in depth in [3], resulting in the so-called matched subspace detectors (MSDs). Adaptive versions of the MSD have been proposed and analyzed in [4] and referencestherein. In a recent paper [1], we considered the problem of detecting a signal whose steering vector is unknown, but known to lie in a subspace, using multiple snapshots from an array of sensors. More precisely, we used the following model for theL-dimensional received signal:
yyy(t) = aaas(t) + AAAuuu(t) + nnn(t)
aaa = HH:H (1)
In (1),aaa 2 L isthe unknown steering vector, which belongsto the p-dimensional subspace hHHi spanned by the columns of HH HH 2 L2p. In other words,aaa lies in a known subspace, but its orientation inhHHi isunknown. Thismodeling isrelevant in a number of applica-H tions (see the discussion in [1]), where there exists some uncertainty about the steering vector of interest. The columns ofAAA 2 L2Jform theJ-dimensional interference subspace hAAAi and uuu(t) denotesthe in-terference waveforms. Finally,nnn(t) isa zero-mean complex-valued Gaussian noise with covariance matrix2III. In contrast to [1], where 2 was assumed to be known, we consider it to be unknown in the present correspondence.
As in [1], we assume thatHH and AH AA are known full-rank matrices, and that the subspaceshHHi and hAH Ai are linearly independent. Thisim-A pliesthat no element ofhHHi can be written asa linear combination ofH vectorsinhAAAi, and that the composite matrix [ HHH AAA ] isfull rank. It is also assumed thats(t) and uuu(t) are deterministic sequences, as in [1]. Note that a stochastic framework could have been adopted, e.g., by as-suming thats(t) or/and uuu(t) are Gaussian random. This would lead to
Manuscript received February 1, 2005; revised September 5, 2005. The work of L. L. Scharf is supported by the Office of Naval Research under Contract N00014-01-1-1019. The associate editor coordinating the review of this manu-script and approving it for publication was Dr. Sven Nordebo.
O. Besson is with the Department of Avionics and Systems, ENSICA, 31056 Toulouse, France.
L. L. Scharf is with the Departments of ECE and Statistics, Colorado State University, Fort Collins, CO 80523-1373 USA.
Digital Object Identifier 10.1109/TSP.2006.874782
four possible models, each with a different generalized-likelood ratio test (GLRT). However, as observed in [5], these detectors would be ap-proximately equivalent when the interference-to-noise ratio is large.
II. GENERALIZED-LIKELIHOODRATIOTEST Our problem consists of deciding between the two hypotheses
H0: YYY = AAAUUU + NNN
H1: YYY = HHHsssT + AAAUUU + NNN (2) whereYYY = [ yyy(1) 1 1 1 yyy(N) ], sss = [ s(1) 1 1 1 s(N) ]T,UUU = [ uuu(1) 1 1 1 uuu(N) ], and NNN = [ nnn(1) 1 1 1 nnn(N) ]. In order to solve thisproblem, we consider the GLRT.
A. Derivation of the GLRT
In this section, we first derive the maximum-likelihood estimates (MLEs) of the unknown parameters under each hypothesis. The MLEs are then used to obtain the GLRT. Under the hypotheses made, the ob-servations are Gaussian distributed, and the likelihood function is given by [2], [6] `(YYY ) = exp 01 N t=1kyyy(t) 0 HHHs(t) 0 AAAuuu(t)k 2 (2)mN (3)
where = 0 under H0and = 1 under H1. When2isunknown, itsML estimate isreadily obtained as
2= 1 mN
N t=1
kyyy(t) 0 HHHs(t) 0 AAAuuu(t)k2: (4) Reporting (4) in (3), it followsthat the MLEsofsss, UUU, and are obtained by minimizing
Tr YYY 0 HHHsssT 0 AAUUAU YYY 0 HHsssH T 0 AAAUUU H (5) whereTr f:g stands for the trace of a matrix. At this stage, the problem isequivalent to that in [1], and we refer to [1] for detailsthat will be omitted here. The matrixUUU that minimizes(5) isgiven by
UU
U = AAAHAAA 01AAAH YYY 0 HHHsssT : (6) UnderH0, all unknown parametersare estimated, and the MLE of2 is
2
0= 1mNTr PPP?AAYYY YYYA H (7) wherePPPAAA denotesthe orthogonal projection onto hAAAi and PPP?AA =A III 0 PPPAAA the projection onto itsorthogonal complement. Under H1, inserting (6) in (5), one needs to minimize
Tr YYY 0 HHsssH T HPPP?AA YYY 0 HA HHsssT = HHHHHPPPAHAA?HH sss30 YYY HPPP? AAAHHH HHHHHPPP? A A AHHH 2 + Tr PPP?AAAYYY YYYH 0 HHHHHPPP? AA AYYY YYYHPPPA?AAHHH HHHHHPPP? AAAHHH : (8) The MLE of isthusgiven, up to a scaling factor, by the principal eigenvector of GGGHGGG 01GGGHYYY YYYHGG with GG GG = PPP?AAAHHH. The subspace 1053-587X/$20.00 © 2006 IEEE
hGGGi corresponds to the part of hHHHi in hAAiA?. The noise power estimate underH1is
2
1= 1mN Tr PPP?AAAYYY YYYH 0 max PPPGGGYYY YYYH (9) wheremaxf:g is the largest eigenvalue of the matrix between braces. Therefore, themN-root generalized-likelihood ratio (GLR) can be ob-tained as M2(YYY ) = `(YYY jH1) `(YYY jH0) 1=mN = 02 2 1 = Tr PPP ? A A AYYY YYYH Tr PPP?
AAAYYY YYYH 0 max PPPGGGYYY YYYH
: (10)
WhenM2(YYY ) is above some threshold, H1 isdecided to hold. The detector operatesinhAAAi?: there, it comparesthe energy of the most energetic component due tohHHHi to the total energy in hAAiA?. Note that M2(YYY ) isinvariant to transformationsthat rotate YYY within hGGi and toG scaling. When2isknown, the GLR isgiven by [1]
L1(YYY ) = 02max PPPGG
GYYY YYYH : (11)
Observe thatM2(YYY ) may also be replaced by the monotone function 1 0 M01 2 (YYY ), which is L2(YYY ) = 1 0 1 M (YYY) = PPPGGGYYYYYY Tr PPPAAAYYY YYY : (12) Thus, the known2in (11) isreplaced byTr PPPA?AAYYY YYYH , which is, within a factor1=mN, the MLE of 2underH0(see (7)).
Remark 1: In the single-snapshot case,YYY = yyy isan Lj1 vector, and
the GLR in (10) reducesto M2(yyy) = yyyHPPPA?AAyyy
yyyH PPP? A A A 0 PPPGGG yyy = yyyHPPP?AAyyyA yyyHPPP? AA APPP?GGGPPP?AAyyyA
where, to obtain the second equality, we made use of [3, eq. (3.4)–(3.7)]. Furthermore, using the fact thatPPPGGG = PPPA?AAPPPGGGPPPA?AA, it follows that the GLRT consists of comparing
M2(yyy) 0 1 = yyyHPPP?AAAPPPGGGPPP?AAAyyy yyyHPPP?
A A
APPPG?GGPPP?AAAyyy
(13)
to a threshold. The previous equation is the GLR for detecting a sub-space signal in subsub-space interference and noise of unknown level when a single snapshot is available (see [3, eq. (8.2)]). Note thatM2(yyy) 0 1 is the ratio of two chi-squared distributed random variables withr = p andq = L 0 J 0 p degreesof freedom, respectively. Therefore, it fol-lowsanF -distribution [2]. Accordingly, when p = 1, i.e., when there is no uncertainty about the steering vector of interest,GGG = PPP?AAAhhh = ggg isa vector andPPPGGGYYY YYYHhas a single eigenvalue. In this case
M2(YYY ) 0 1 = Tr PPP ?
AAAPPPGGGPPPA?AAYYY YYYH Tr PPP?
AAAPPP?GGPPGPA?AAYYY YYYH
(14)
is now the ratio of two chi-squared distributed random variables with r = N and q = N(L 0 J 0 1) degreesof freedom, respectively. WhenN = 1, it reducesto the GLRT for detecting a known signal
in subspace interference and noise of unknown level (see [3, equation (6.4)]).
B. Distribution of the GLR Under the Null Hypothesis
In order to set the threshold of the test for a given probability of false alarmPFA, we need to derive the probability density function (pdf) of the GLR under the null hypothesis. Although the derivation of the GLR for unknown2 is a straightforward extension of the GLR with known2, it turnsout that the derivation of itspdf ismuch more complicated in the present case, as is illustrated now. In order to obtain thispdf, we will write the GLR in a canonical from, i.e., asa function of independent random variables. To do so, let
UUU? AA
A = UUULjpG LjL0J0pUUU2 (15) denote an orthonormal basis for AAA? , whereUUUG2 L2pisa unitary basis forhGGGi and UUU2 2 L2L0J0pisa unitary basisfor the comple-ment ofhGGGi in AAA? , i.e., a unitary basis for PPPAPP?AAP?GGGPPP?AA . First,A note that underH0
PPP?
AAAYYY YYYH= UUU?AAAUUU?HAAA YYY YYYH= UUUA?AAUUUA?HAA NNNNNNH = UUUGUUUH
GNNNNNNH+ UUU2UUUH2NNNNNNH (16) so that
Tr PPP?
AAAYYY YYYH =Tr NNNGNNNHG +Tr NNN2NNNH2 (17a) max PPPGGGYYY YYYH =max NNNGNNNHG (17b) whereNNNG = UUUHGN [respectively, NNN NN2 = UUUH2NNN] is a p 2 N [respec-tively, aL0J0p2N] matrix whose columns are independent p-variate [respectively,L 0 J 0 p-variate] complex Gaussian vectors with co-variance matrix2III. Furthermore, NNNG andNNN2are uncorrelated and hence independent.
Let usdefinem = min(p; N), M = max(p; N), and let usdenote by1 > 2 > 1 1 1 > m 0 the first m eigenvaluesof WWWG = NNNGNNNH
G. Accordingly, let usdenote t2= Tr NNN2NNNH
2 ; t = Tr NNNGNNNHG = m k=1
k: (18) Then,M2(YYY ) can be rewritten as
M2(YYY ) = m k=1k+ t2 m k=2k+ t2 =t 0 1t + t2+ t2: (19)
From inspection of (19), it is clear that the GLR is invariant to scaling inNNN and is thus CFAR with respect to the noise level 2. Without loss of generality, we assume in the sequel that the columns ofNNN are independent complex Gaussian vectors with covariance matrixIII. As will become clearer below, it ismore convenient to consider
L2(YYY ) = MM22(YYY ) 0 1(YYY ) = t + t21 = t
1 +t t
= 1 + fa = ab (20)
instead ofM2(YYY ) since a = 1=t and b = (1 + t2=t)01are inde-pendent random variables, as will be shown next. First, we derive the pdf ofb. It iswell known [7] that t2has a central chi-squared distribu-tion withq = N(L 0 J 0 p) degreesof freedom. Accordingly, t has
a central chi-squared distribution withr = pN degreesof freedom. The pdf’soft2andt are thusgiven by
fT (t2) = 10(q)e0t tq01
2 ; t2 0
fT(t) = 10(r)e0ttr01; t 0: (21) Therefore,b = t=(t + t2) = 1=(1 + f) hasa beta distribution
fB(b) = 0(q + r)0(q)0(r)br01(1 0 b)q01; 0 b 1: (22) Next, we show thata isindependent of t and hence of b. Since WWWG= NN
NGNNNH
Ghas a complex Wishart distributionCWp(N; III), the joint pdf of itseigenvaluesisgiven by [8], [9] f3 ;3 ;...;3 (1; 2; . . . ; m) =c exp 0 m k=1 k m k=1 M0m k 1k<`m (k0 `)2 (23) wherec01 = mk=10(m 0 k + 1)0(M 0 k + 1). Let usmake the change of variablesfromfkgmk=1toz0= t; z1; . . . ; zm01, where
zk= tk; k = 1; . . . ; m; z1= t1 = a:
Note that mk=1zk= 1 and hence zm= 1 0 m01k=1 zk. The Jacobian of the transformation is easily seen to betm01. Therefore, the joint density function ofz0= t; z1= a; z2; . . . ; zm01factorsas
1 0(mM)tmM01e0t 2 c0(mM) m k=1 zkM0m 1k<`m (zk0 z`)2 which shows thatt isindependent of z1; . . . ; zm01and hence ofa = z1. Moreover, we find thatt is 2mNdistributed. The pdf ofz1could in principle be obtained as fZ (z1) = c0(pN) 1 1 1 m k=1 zM0m k 1k<`m (zk0z`)2dz21 1 1 dzm01 (24) where the integration isover the domain0 zm < zm01 < 1 1 1 < z1 1 and zm+zm01+1 1 1+z1= 1. However, it appearsquite com-plicated to obtain a closed-form expression for this integral for anyp. Indeed, it seems that there does not exist in the literature a closed-form and simple expression for the pdf ofa for any value of p. However, for the problem at hand,p is the dimension of the subspace, where aaa is expected to lie. Hence,p is typically small; otherwise, we have a very poor knowledge of the steering vector that the beamformer attempts to recover, which is contrary to common sense. Furthermore, as was illustrated in [1], choosingp > 2 doesnot result in any detection per-formance improvement, and hence the choicep = 2 or p = 3 appears to be the most appropriate. In the sequel, we derive closed-form expres-sions for the pdf ofa in the cases p = 2 and p = 3. We consider now thatN p: the case N = 1 is studied in [3], and considering p = 3, N = 2 isequivalent to considering N = 3, p = 2 by interchanging p andN in the expressions.
Whenp = 2, there isno integral in (24) and the pdf of a simply writes
fA(a) = c 0(2N)aN02(1 0 a)N02(2a 0 1)2; 1
2 a 1: (25) In this case, it is also possible to show that thenth-order moment of a isgiven by E fang = 0(2N) 22N02+n0(N) 2 n k=0 0 k+3 2 0(k + 1)0(n 0 k + 1)0 N + k+1 2 : (26) Whenp = 3, we need to integrate the function in (24) over the variablez2. Doing so, it can be shown that (for the sake of brevity, the detailed derivationsare omitted) the pdf ofa isgiven by
fA(a) = c1(1 0 a)2N+1aN03
2 (2N + 1)(2N 0 1)h2(a) 0 6(2N + 1)h(a) + 15 2 Bz(a) 32; N 0 2 + [1 0 z(a)]N02z3=2(a) 2 2(2N + 3)z2(a) 0 10 ; 13 a 1 (27) where c1 = c 0(3N)=22N(2N + 1)(2N 0 1), h(a) = ((3a 0 1)=(1 0 a))2, z(a) = min (1; h(a)) and Bz(a; b) isthe incomplete Beta function [10].
The pdf ofg = L2(YYY ) = ab isthusgiven by
fG(g) = fA(a)fB ag daa = fA gb fB(b)dbb (28) wherefB(:) isgiven by (22) and fA(:) isgiven by (25) when p = 2 and (27) whenp = 3. In the case p = 2, the integral reducesto
fG(g) = c2gr01 1 max(g;1=2)z N0q0r01(10z)N02(2z01)2(z0g)q01dz = c2gN02 min(1;2g) g (1 0 z) q01(2g 0 z)2(z 0 g)N02dz: (29)
For illustration purposes, Fig. 1 displays (29) for variousN, when L = 10, J = 2, and p = 2. It can be observed that the mean of fG(g) decreases as N increases and that the tail probabilities of fG(g) decrease more rapidly asN increases.
III. NUMERICALILLUSTRATIONS
In this section, we illustrate the performance of the CFAR-GLRT detector and compare it with the performance of the GLRT for known noise level. Similarly to [1], we consider a uniform linear array ofL = 10 sensors spaced a half-wavelength apart. We consider the case of a Ricean channel for which the steering vector can be written as [11]
aaa = aaa0+ 1pq q k=1
gkaaa( k) (30)
whereaaa0corresponds to the line-of-sight component, and the second term in the right-hand side of (30) stands for the contribution of scat-terers. Thegkare zero-mean, independent, and identically distributed
Fig. 1. Distribution of the GLR forN = 5, 10, 20. L = 10, J = 2, and p = 2.
Fig. 2. Probability of detection versus SNR.aaa 2 hHHHi. UR = 06 dB and P = 10 .
random variableswith power2g, and kare independent random vari-ableswith pdfp( ). The covariance matrix of the steering vector errors isgiven by [11]
CC Ca= 2
g aaa( )aaaH( )p( )d : (31)
When the angular spread of the scatterers is small, it is known thatCCCa has only a few significant eigenvalues; hence, subspace modeling of the steering vector becomes relevant. In the sequel, the actual steering vector isgenerated as
aaa = aaa0+ uuu1 (32)
whereaaa0 = aaa(0) isthe line-of-sight component, and uuu1isthe prin-cipal eigenvector ofCCCa: hence,p = 2 and HHH = [ aaa uuu1]. isdrawn from a proper complex-valued multivariate normal distribution with zero-mean and variance2 . In the simulations, we assume a Gaussian distribution for the scatterers with standard deviation = 15. We
Fig. 3. Probability of detection versus SNR.aaa 2 hHHHi. UR = 06 dB and N = 10.
Fig. 4. Probability of detection versus SNR. UR= 06 dB and P = 10 .
define the uncertainty ratio (UR) and the (array) signal-to-noise ratio (SNR) as UR= 10 log10 2 aaaH 0aaa0 (33) SNR= 10 log10 P aaa H 0aaa0+ 2 2 (34)
whereP is the power for the signal of interest. Finally, we assume that J = 2 interferencespresent, with DOAs020, 30and powers20 and 30 dB above the white noise level, respectively.
For each figure, one million Monte Carlo simulations are run with a differentaaa drawn from (32); thisenablesusto characterize the average behavior of the GLRTs. In Figs. 2 and 3, we display the probability of detection for various number of snapshotsN and various PFA, respec-tively. It can be observed that the CFAR-GLRT incurs only a 1-dB loss compared with the GLRT for known noise level, which is not an im-portant price to be paid given that we need not know the noise level.
Finally, we test the robustness of the detector whenaaa isgener-ated as in (30). In such a case, aaa does not belong to a subspace sinceCCaC isfull rank. However, the GLRT detectorsare used with
Fig. 5. Probability of detection versus SNR. UR= 06 dB and N = 10.
the assumption that aaa isgenerated asin (32). In thiscase, UR and SNR are defined asUR = 10 log10 Tr fCCCag=aaaH0aaa0 and SNR = 10 log10 P aaaH0aaa0+ Tr fCCag =C 2 . The detection perfor-mance is plotted in Figs. 4 and 5. Despite the fact thataaa doesnot belong tohHHHi, the detection performance isnot affected, and hence the detection scheme turns out to be rather robust.
IV. CONCLUSION
In this correspondence, we considered the problem of detecting a signal whose spatial signature is unknown but known to lie in a given linear subspace, in the presence of interferences and broadband noise. We have extended the results of [1] to the case of unknown noise level and derived the GLRT, which is CFAR with respect to the noise level. We showed that the GLRT detector involves the ratio of the largest eigenvalue of a complex Wishart matrix to its trace whereas, in the known noise level case, it involved the largest eigenvalue only. The distribution of the GLR was derived under the null hypothesis. Simu-lation results indicate that there is a 1-dB loss between the GLRT with known2and the CFAR-GLRT with unknown2. Furthermore, the detection test was shown to be rather robust when the spatial signature does not completely belong to a subspace.
REFERENCES
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[2] L. L. Scharf, Statistical Signal Processing: Detection, Estimation and Time Series Analysis. Reading, MA: Addison-Wesley, 1991. [3] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE
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[8] A. James, “Distributions of matrix variates and latent roots derived from normal samples,” Ann. Mathemat. Stat., vol. 35, pp. 475–501, Jun. 1964. [9] M. Kang and M.-S. Alouini, “Largest eigenvalue of complex Wishart matrices and performance analysis of MIMO MRC systems,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 418–426, Apr. 2003.
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Equalization of a MIMO Channel Using FIR Inverses K. Deergha Rao, Senior Member, IEEE
Abstract—In the modern method of equalization, the problem of mul-tiple-input multiple-output (MIMO) finite-impulse-response (FIR) channel equalization boils down to finding the MIMO FIR inverses. This correspon-dence proposes and proves a theorem that states the condition for the ex-istence of these inverses, which are also FIR. A numerical example is pro-vided to illustrate how the FIR inverses can be evaluated and used for equal-ization of a channel with known channel parameters.
Index Terms—Equalization, finite-impulse-response (FIR) inverses, mul-tiple-input multiple-output (MIMO) channel, multiuser system.
I. INTRODUCTION
Users in a wireless network share a common medium, and their transmissions may interfere with one another. A general model of a multiuser communication system is the multiple-input multiple-output (MIMO) channel, asshown in Fig. 1. Herexi(n) are transmitted sig-nalsfromM users, yi(n) are the received signals at N sensors, which can be antenna-array elementsor virtual receiversof temporal pro-cessing [1, vol. 1, ch. 8]. The number of sensors (N) must be at least equal to the number of source signals (M). The basic channel equal-ization problem is to design an estimator, such that multiple sources are extracted in an optimal fashion. It is unrealistic to assume that the receiver knowsthe channel parametersin a wirelessmobile network. Considerable research has been devoted to estimation of channel pa-rameters[2]–[4].
Generally, the data and channel responses may be represented as polynomialsin thez-transform domain, and the implementations are restricted to MIMO polynomial finite-impulse-response (FIR) filter. Here, a theorem is proposed and proved for finding the inverse of the MIMO FIR filter such that the inverse is also FIR. An example illus-trateschannel equalization with known channel parameters.
This correspondence is organized as follows. The channel model is described in Section II. In Section III, a theorem is proposed and proved for finding the inverse of the MIMO FIR filter such that the inverse is also FIR and illustrated for channel equalization with known channel and Section IV contains the conclusions.
Manuscript received May 24, 2005; revised August 14, 2005. The associate editor coordinating the review of thismanuscript and approving it for publica-tion wasProf. Yuri I. Abramovich.
The author iswith the Research and Training Unit For Navigational Electronics, Osmania University, Hyderabad 500 007, India (e-mail: [email protected]).
Digital Object Identifier 10.1109/TSP.2006.874793 1053-587X/$20.00 © 2006 IEEE
