Detection of Signals by Information Theoretic Criteria-OyE

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IEEERANSACTIONSNCOUSTICS,PEECH,NDIGNALROCESSING, VOL. ASSP-33,O. 2, APRIL 1985 387

MAT1 WAX AN D THOMAS KAILATH, FELLOW, IEEE

Abstract-A new approach is presented to the problem of detectingthe number of signals in a multichannel time-series, based n the appli-cation of the information theoretic criteria for model selection intro-duced by Akaike (AIC) and by Schwartz and Rissanen (MDL). Unlikethe conventional hypothesis testing based approach, the new approachdoes not require any subjective threshold settings; theumber of signalsis obtained merely by minimizing the AIC or theMDL criteria. Siula-tion results that illustrate the performance of the new method for thedetection of the number of signalsreceivedbya ensorarrayarepresented.

II. INTRODUCTION

N many problems in signal processing, the v ector of ob serva-tions can be modeled as a superposition of a finite num berof signals embedd ed in an additive noise. This is the case, forexample, in sensor array processing, in harmonic retrieval, inretrieving the poles of a system from the natural response, andin retrieving overlapping echoes rom radar backscatter. A ke yissue in theseproblems is thedetection of thenumberofsignals.One approach to this problem is based on the observationtha t the numb er of signals can be determined from the eigen-values of the covariancematrix of the observationvector.Bartlett [4] and Lawley [14]developedaprocedure, basedon a nFsted sequence of hypothesis tests, to imp lement this ap -proach.Fo reachhypothesis, the likelihoodratiostatistic iscomputed andcompared to a hreshold; hehypothesisac -cepted is the first one for which the threshold is crossed. Theproblem with this m ethod is the subjective judgment requiredfor deciding on the hresho ld levels.In this paper we present a new approach to the problem thatis based on the application of he nformation heoretic cri-teria for model selection introduced by A kaike (AIC) and bySchwartz nd Rissanen (MDL). The dvantage of this p-proach is tha t no subjective judgmen t is required in the d eci-sion process; the number of signals is determined as the valuefor which the AIC or the MDL criteria is minim ized.The paper is organized as follows. After the statem ent andformulation of th e problem nSection 11, the nformationtheoretic criteria for model selection are introduced in Section111. The ap plication of these criteria to th e problem of detect-ing the number of signals and the consistency of these criteria

Manuscript received May 19, 1983; revised June 1, 1984. This workwas supported n part by the Air Force Office of Scientific Research,Air Force Systems Command, under Contract AF 49-620-79-C-0058,the U.S. ArmyResearch Office, underContract DAAG29-79-C-0215,andby the Joint Services Program at Stanford University under Con-tract DAAG29-81-K-0057.The authors are with the Information Systems Laboratory, StanfordUniversity, Stanford,CA 94305.

are discussed in Sections IV and V, respectively. Simulationresults tha t illustrate the performance of the new meth od forsensor array processing are escribednSectionVI.Fre-quency domain extensions and some concluding remarks arepresented in SectionsVI1 and VIII, respectively.

11. FORMULATIONF THE PROBLEMThe observation vector in certain im portant p roblem s in sig-

nal processing such as sensor array processing [15] [20 ] [5][6 ] [lo] 121 [22] [25] harm onic retrieval [15] [12]pole retrieval from he natural response [ l l ] [24 ] , and re-trieval of overlapping echos from radar backscatter [7] de -noted by the p l vector x(t ), is successfully described by th efollowing model

x(t) A ( q ) n ( t )i = 1

wheresi( scalar complex waveform referred o as th e ith ignalA ( @ i )= a p 1 complex vector, parameterized by an un-known parameter vectorai ssociated w ith th e ith ignaln( a p 1 complex vector referred to as the additive

noise.We assume th at th e q ( q < p ) signals s l arecomp lex (analytic), stationary, and ergodic Gaussian randomprocesses, with zero mean and positive definite covariancem a-trix. The noise vector n( is assumed to be complex, station-ary, and ergodic Gaussian vector process, independent of thesignals, with zero mean and covariance matrix given bywhere a2 is an unk now n scalar constant and is the identitymatrix.A crucial problem associated with he, model described n(1) is tha t of determining the num ber of signals4 from a finiteset ofobservations x(tl), x(tN).A promising approach to this problem is based on the struc-ture of the covariance matrix of the observation vectorTo introdu ce this appro ach, e first rewrite (1) as

x(t) As(t) n(t) (2a)where A is the p 4 matrix

A [ A ( % ) N @ q > I (2b)and is the 4 1 vector

ST@) [s, (t) sq(t)]Because th e noise is zero mean and independent of the sig-nals, it follows hat th e covariance matrix of is given by

.OO 1985 IEEE

AlultIXDoM1a1UfIX Ra


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38 8 IEEE TRANSACTIONS O N ACOUSTICS, SPEECH, A N D S I G N A L PROCESSING, VOL. ASSP-33, NO. 2, A P R I L 1985

R t aZI (34where

ASAi (3b)with t denoting the conjugate transpose, and S enoting thecovarian ce matrix of he signals, i.e., E [ s ( . s( s ) ~ ]Assuming that the matrix A is of full column rank, i.e., thevectors A ( Q i ) (i 1 q) are linearly independent,and thatthe covariance matrix of the signals is nonsingular, t followsthat the rank of Ik is q, or equivalently, the p 4 smallesteigenvalues of are equal t o zero. Denoting the eigenvaluesof R by ha it follows, therefore, that the small-est p q eigenvalues of R are all equal to d .e.,

X q + 1 h, 0 2 .The number of signals q can hence be determined from the

muZtiplicity of the smallest eigenvalue of R . The problem isthat the covariance matrix R is unknown in practice. Whenestimated f rom a finite sample size, the resulting eigenvaluesare all different w ith probability one, thus making it difficultto determine the number of signals merely by observing theeigenvalues. Amoresophisticatedapproach to he problem ,developed by Bartlett [4] and Lawley [14] is based on a se-quence of hypothesis tests. The problems associated with thisapproach is the subjective judgm ent needed in the selection ofthe threshold levels for the different tests.In this paper we take a different approach. We pose the de-tection problem as a model selection problem and then applythe nform ation heoretic criteria formod el selection intro-duced by Akaike (AIC) and by Schwartz and Rissanen MDL).

111. INFORMATION THEORETICRITERIAThe informatio n theo retic criteria for model selection, in tro-duced by Akaike [ l ] [2 ], Schwartz [21], and Rissanen [17]address the following general p roblem. Given a set of N obser-vations X {x(l), x ( N ) } an d a family of models, that is,a parameterized family of prob ability densities f(XlO),electthe model thatbest fits the data.Akaikes proposal was to select the model which gives theminimum AIC, defined byAIC -2 logf(Xl6) 2 k (6)

where 6 is the maximu m likelihood estimate of the parametervector 0, an d k. is tlie number of free adjusted parameters in0. The first term is the well-known log-likelihood of lie max-imum ikelihood estimator of the parameters of th e mod el.The second erm is a bias correction term, inserted so as tomake the AIC an unbiasedestimate of the meanKulback-Liebler distance betwey the modeled density f(Xl0) nd theestimated den sityf(Xl0).Inspired by Akaikes p ioneering wo rk, Schw artz and Rissanenapproached the. problem from quite different points of view.Schw artzs appro ach is based Bayesian argum ents. He as-sumed that eachcompetingmodelcan beassigned apriorprobability, and proposed to select the model that yields themaximu m posterior probability. Rissanens approach is basedon information theoretic arguments. Since each model can be

used to encode th e observed data, Rissanen proposed to selectthe model that yields the minimum code length. It turns outthat in the large-sample limit, bo th Schwartzs and Rissanensapproaches yield he same criterion,given byMDL= -lOgf(XIS) t k og N . (7)

Note that apart froma factor of 2, the irst term is identical tothe corresponding one in the AIC, while the second term hasan extra factor of3 og N .IV. ESTIMATINGHE NUMBER F SIGNALS

To apply he nformation heoretic criteria to detect thenumb er of signals, or equivalently, to determ ine the rank ofthe matrix 9, e must first describe the family of competingmodels, or density functions that we are considering. Regard-ing the observations x ( t l x(tN) as identical and statisti-caily independ ent comp lex Gaussian random vectors of zeromean, the family of models is necessarily describedby the covariance matrix of Since our models covariance matrixis given by (3), it seems natural to consider the following faily of covariance matricesR ( k ) 02 1 (8)wheredenotesasemipositivematrix of rank k , an d u de-notes an unknown scalar. Note that k E (0, l , p l }ranges over the set of all possiblenum ber of signals.Using the well-known spectral representatio n theorem fromlinear algebra, we can express R ( k ) s

where XI, Xk an d v,, V k are the eigenvalues andeigenvectors, respectively, of R @ ) . Denoting by d k ) th eparameter v ector o f he mo del, t follows that

With this parameterizationwe now proceed to the derivationof the nformation heoretic criteria for he detection prob-lem. Since the o bservations are regarded as statistically inde-pendent complex Gaussian random vectors with zero mean,their joint probability densitys given by

Taking the logarithm and omitting terms that do not dependon th e parameter vector d k ) , we find that the log-likelihoodfunction L ( @ ) is given byI,(@@)) - N log det R@) t r [ R @ ) ] - l (12a)

where is the sample-covariance matrix defined by



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W A X A N D KAILATH: DETECTION OF SIGNALS BY INFORMATION THEORETIC CRI TERI A 389

Th emaximum-likelihoodestimate is the valueof d k ) hatmaxim izes (12). Following Anderson [3] these estimate s aregiven by

h i= l i i = l , . - . k (1 3 4

where la lp an d CI Cp are t he eigenvalues a,ndeigenvectors, respectively, of the sample covariance matrixR .Substituting the maxim um likelihood estimates (13) in thelog-likelihood (1 2) , with some straightforward manipulations,we obtain

Note that the term in the bracket is the ratio of the geom etricmean to the ar i thmeticmean of th e smallest p -.k eigenvalues.The number of free parameters ind k ) s obtained by count-ing the nu mber of degrees of freedom of he space span ned byd k ) . Recalling that the eigenvalues of a complex covariancematrix are real bu t tha t the eigenvectors are complex, it fol-lows that d k ) ha s k t 1 t ' 2 p k parameters. Howev er, not allof the parameters are independently adjusted; he eigenvectorsare constrained to have unit norm and to be mutually orthog-onal. This amounts o reduction of 2 k degrees of freedom dueto the normalization and 2&k(k 1) degrees of freedom dueto the mutua1.orthogonaIization. Thus, we o btain

(number 'of free adjusted parameters)k t 1 t 2pk [ i k ( k l) ] k(2p k ) t 1 . (15)

The form of AIC for thisproblem is therefore given byk )N

t 2k(2p k)

while the MDL criterion is given by

k(2p k ) og N .2

Th enumber of signals is determ ined as he value of k g(0, 1 , p I} for which either th e AIC or the MDL sminimized.

V. CONSISTENCYF THE C R I T E R I AWe have described tw o d ifferen t criteria for estimating num -

ber of signals. The natural questio n is which one should bepreferred. What can be said abo ut he goodness of the esti-mates ob tained b y these criteria. One possible benchmark testis the beh avior as the sample size increases. One would preferan estimator that yields the true num ber of signals with prob-ability one as th e sam ple size increases to in fin ity. An estima-tor with this prope rty is said to be consistent. By generalizinga meth od of proof given inRissanen [18] and Hannanan dQuinn [9 ], we shall show that he MDL yields a consistentestimate, while he AIC yields an ncon sistent estimate hattends, asymptotically, o overestimate the num ber of signals.The consistency of the MDL s proved by showing tha t inthe large-sample limit, MDL(k) is minim ized for k q . Takingfirst k 4 , t follows from (17) that

1 [MDL(q) MDL(k)]N

log i = k + l q - ki = k + l

t og

Since the eigenvalues of the samp le-covariance matrix j (i 1,, 4 ) are consistent estimates of the eigenvalues of the true

covariance matrix hi, it follows that in the large-sample limitthe eigenvalues li (i k t 1 , q ) are not ali eq ual withprobability one. Therefore, by the arithmetic-m ean eometric-mean inequality t follows that in the arge-sample imit

This implies that the first term in (18) is negative with prob-abil i ty one in' thearge-sample imit'. Similarly, by the general-ized arithmetic-mean geometric-mean inequalityw,A1 w 4 , AY'A,W2 w 1 t wa

it follows tha t in the arge sample limit



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390 I E E E TRANSACTIONS ON ACOUSTICS, S P E E C H , A N D SI GNAL PROCE SSI NG,O L. ASSP-33, NO . 2, APRIL 198

This implies that the second term in (1 8) is also negative withprobability one in the large-sample limit. Now , since the lastterm in (18) goes to zero as th e sample size increases, it fol-lows that the difference [MDL(q) MDYk)] is negative withprobability one in th e large-sample limit for k q .Taking now k 4 , t follows from (17) tha t

2 [MDL(k) MDL(q)](k 4 ) ( 2 p k log N

Note hat he terms in he curly bracket are twice the log-likelihoods of the maximum likelihood e stima tor und er hehypotheses that the rank of is q an d k , espectively. Thus,their difference is the likelihood-ratio for deciding betweenthese two hypotheses. From the general theory of likelihoodratios (see, e.g., C ox and H inkley [8]) it follows that heasym ptotic distribution of this statistic is x with number ofdegrees of freedom equal to the difference of the dimensionsof the parameter spaces under the two hyp othes is,.e.,

[ W P k ) 11 M 2 P 4 ) 11(k q ) ( 2 p k

Thus, as the sample size increase, the prob ability that the termin the curly bracket in ( 2 0 ) exceeds the first term in (20) isgiven by the area in the tail from (k 4 ) ( 2 p k q ) og N ofthe mentioned x2 distribution with (k q ) ( 2 p k 4 ) de-grees of freedom. Since th e area in th is tail approaches zero asthe sample size increases, it follows that in the large-samplelimit the difference [MDL(k) MDL(q)] is positive with pro b-ability one for k q. Combining this with theprevious resultfo r k q , t follows tha t MDL(k)has a minimum at k q .Repeating th e above arguments for he AIC, it followstha t in t he large-sample limit an d for k < q , the difference[AIC(q) AIC(k)] is negative w ith probability one. However,fo r k 4 , he differenc e [AIC(k) AIC(q)] has nonzero prob-ability to be negative even in the large-sample limit, since thetail from (k 4 ) ( 2 p k q t ) of the x 2 distribution with(k q ) ( 2 p k 4 t 1) degrees of freedom is defi nitely not

zero. Hence, the AIC ten ds o overestimate the number osignals q in th e large-sample limi t.We should n ote th at from the analysis above it follows thaany criteria of the form-log f(Xl4) tQ ( N ) (23

where or(N) and or(N)/N+ 0 yields a consistent estimatof the num ber of signals.

VI. SIMULATIONESULTSIn his section we present simulation results hat llustrate

the performance of our method when applied to sensor arrayprocessing.By the well-known dualitybetween spatial frequency an d temporal frequency, these examples can also beinterpreted in the con text of harmonic retrieval. The examples refer to a uniform linear array of p sensors with 4 incoherent sinusoidal plane waves impinging from directio ns { G 1 ,Assuming that the spacing between th e sensors s equato half the wavelength of th e impinging wavefronts, the vectoof the received signal at the array s then given by

2 ( G ~ ) t ( t ) (24awhere A(&) is the p 1 directionvector of the kthwavefront.

k = 1

A ( $ ~ ) T [le-jTk e - j ( q - l ) T k 1 (24b)with

7 k 77 sin q5k (24can d

random phase uniformly distributedon (0,277)n( vector of white noise with mean zero and covarianceNo te hat his model is a specialcaseof the genera

model presented in (1).In the first example, we considered an array with seven sensors ( p 7) and two sources ( 4 2) with directions-of-arrival20 an d 25 . The signal-to-noise ratio, defined as 10 log 1 2u2was 10 dB. Using N = 1 0 0 samples, the resulted eigenvalues ofthe sample-covariance matrix were 21.2359, 2.1717, 1.42791.0979, 1,05 44, 0.9432, and 0.7324. Observing thegradualdecrease of th e eigenvalues it is clear that the separation of th e

3 smallest eigenvalues from the 2 large ones is a difficulttask in which a naive approac h is likely to fail. However, applying the new approach we have presented above yielded thefollowing values for the AIC an d MDL.0 1 2 3 4 5

AIC 1180.800.5 71.4 75.5 86.83.26.0MD L 590.47.26.9 80.7 95.505.2 110.5.The minimum of b oth the AIC and the MDL is obtained, aexpected, for the 2.In the second example, we added another source at 10 toth e scenario described in the first ex ample. The eigenvalues ofthe sample-covariance in this c p e were 15.5891 , 8.18921.4715,1.1602,1.0172, 0.9210, and 0.6528. The resultingvalues of the AIC and theMDL were as follows.



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&AX A N D KAILATH: DETECTION OF SIGNALS BY I N F O R M A T I O N THEORETIC CRITERIA 391

AICMD LIn this case, the minimum value of both the AIC and theMDLis obtained, incorrectly, for 2 . The failure of th e AIC andMDL in this case is because of the inheren t limitation s of theproblem. ndeed, epeating the examplewith signal-to-noiseratio of 10 dByielded the following eigenvalues 14.5 595 ,6.7786,0.3786,0.1372,0.1109,0.0946, an d 0 .0687 , and con-sequentially th e following values of the AIC and theMDL.

5AICMDLNow the minimum of both the AIC and theMDL is obtained ,correctly, forq 3 .

VII. EXTENSIONO THE F R E Q U E N C YO M A I NThe starting point of our approach as the time domain ela-

tion (1). Howev er, in certain cases, especially in array process-ing, the freque ncy domain is more natural. As we shall show ,our approach is easily extended to handle frequency-domainobservations.Consider the frequency domain analog of the time domain

model (1); the observed vector is a Fourier coefficients vectorexpressed b y4

d w n ) A(mn ai) i(wn) n(wn> (25)i =where

s i ( w n ) = heFourier coefficient of the it h signal at re-quency o n ,A(w,, ai) a p 1complexvector,determinedby theparameter vector Qii associated with the it h signal

n ( w n ) a p 1 complex vector of the Fou rier coefficientsof th e additive noise.The problem, as in the time domain, s to estimate the num-be r of signals 4 from L samples of the Fourier-coefficients vec-By the well-knownanalogybetweenmultivariate analysis

an d time-series analysis (see, .g.,Wahba [ 2 3 ] ) , he time-domain approach carries over to the frequency domain withthe role of the sample-covariance matrix played by the peri-odog ram estimate of the pectral density matrix , given by

t o r x l ( o n ) , , x L ( a n ) -

Thus , the frequen cy-do main ersion of the AIC is givenbyf V

where Z l ( on ) lP(wn) re the eigenvalu? of the peri-odogram estimateof the spectral-density matrixK ( w n ) .

When the signals are wideband, namely,occupy severalfrequency bins, say, 01,. O Z + M , the information on th enumber of signals is contain ed in all the M frequen cy bins. As-suming that the observation time is much larger than the cor-relation times of the signals, it follows that the Fourier coeffi-cients corresponding to different frequencies are statisticallyindepe nden t (see, e.g., Whalen [ 2 6 , p. 811). The AIC for de-tecting the numb er of wide-band signals that occu py the fre-quency band is given by the sum of ( 2 7 ) overthe frequency range of interest

2 M [ k ( 2 p k)] (2 8)Th ecorrespondingexpression for the MDL criterion can besimilarly d erived.

V III. CO N CL U D I N G RE M A RK SA new approach to the detectionof the num ber o f signals in

a multichannel time series has been presented. The approachis based on the application of the AIC and MDL informationtheoretic criteria for mod el selection. Unlike the conven tionalhypothesis test based approach, thenew approach does not re-quire any su bjective threshold settings; the nu mber of signals isdetermined merely by minimizing either the AIC or the MDLcriterion.It has been shown that the MDL criterion yields a consistentestimate of the number of signals, while the AIC yields an in-consistent estimate tha t tends, asym ptotically, to overestimatethe nu mber of signals.The detection problem address ed in this paper is usually apart of a more complex comb ined detection-estimation prob -lem, where one wants to estimate the number as well as theparam eters of the signals in (1). Because of th ecomputationalcomplexity nvolved, heproblem is usuallysolved in two steps: first the num ber of signals is dete cted, andthen, with an est imateof the num ber of signals at hand , theparameters of the signals are estima ted. It shou ld be pointedout, however, that in principle the AIC and MDL criteria canbe applied t o he combined etection-estimation roblem[ 2 6 ] . Th eevaluationof he AIC and MDL in this case in-;elves th% computationof the maximum likelihood estimates

1 , @k, for every possible numb er of signals k Ep 1 ) . Solving this highly nonlinear problem for all Val-uesof k is comp utationally very expensive. Nevertheless, ifone is ready to p ay the price, the gain in term of performance,especially in difficult s ituations uch as low signal-to-noiseratio, sm all sample size, or closely spaced signals, may besignificant.

REFERENCESH.Akaike, Information theory and an extension of the maxi-m um likelihoodprinciple, in hoc. 2nd Int. Symp. Inform.



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Detection of Signals by Information Theoretic Criteria-OyE

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