ieee transactions on signal processing, vol. 52, no. …ieee transactions on signal processing, vol....

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004 23

Array Signal Processing in the Known Waveform andSteering Vector Case

Yi Jiang, Student Member, IEEE, Petre Stoica, Fellow, IEEE, and Jian Li, Senior Member, IEEE

Abstract—The amplitude estimation of a signal that is knownonly up to an unknown scaling factor, with interference and noisepresent, is of interest in several applications, including using theemerging quadrupole resonance (QR) technology for explosive de-tection. In such applications, a sensor array is often deployed forinterference suppression. This paper considers the complex ampli-tude estimation of a known waveform signal whose array responseis also known a priori. Two approaches, viz., the Capon and themaximum likelihood (ML) methods, are considered for the signalamplitude estimation in the presence of temporally white but spa-tially colored interference and noise. We derive closed-form expres-sions for the expected values and mean-squared errors (MSEs) ofthe two estimators. A comparative study shows that the ML es-timate is unbiased, whereas the Capon estimate is biased down-wards for finite data sample lengths. We show that both methodsare asymptotically statistically efficient when the number of datasamples is large but not when the signal-to-noise ratio (SNR) ishigh. Furthermore, we consider a more general scenario wherethe interference and noise are both spatially and temporally cor-related. We model the interference and noise vector as a multi-channel autoregressive (AR) random process. An alternating leastsquares (ALS) method for parameter estimation is presented. Weshow that in most cases, the ALS method is superior to the model-mismatched ML (M3

L) method, which ignores the temporal cor-relation of the interference and noise.

Index Terms—Asymptotic analysis, capon, interference suppres-sion, maximum likelihood, multichannel autoregressive randomprocess, parameter estimation, quadrupole resonance.

I. INTRODUCTION

E STIMATING the signal parameters in the presence of in-terference and noise via array processing is often encoun-

tered in practical applications (see, e.g., [1], [2], and the refer-ences therein). It is well known that temporal information aboutthe signal can be utilized to effectively suppress the interfer-ence and noise and, hence, to significantly improve the estima-tion accuracy. For example, [3] and [4] have studied the esti-mation of directions of arrival (DOAs) of signals with knownwaveforms and showed that significant improvements in accu-racy, interference suppression capability, and spatial resolutioncan be obtained. Through Cramér–Rao bound (CRB) analysis,[5] has studied the DOA estimation of a parametric signal and

Manuscript received August 30, 2002; revised March 19, 2003. This workwas supported in part by the National Science Foundation under Grant CCR-0104887 and the Swedish Science Council. The associate editor coordinatingthe review of this paper and approving it for publication was Dr. Alex B. Ger-shman.

Y. Jiang and J. Li are with the Department of Electrical and ComputerEngineering, University of Florida, Gainesville, FL 32611 USA (e-mail:[email protected]; [email protected]).

P. Stoica is with the Department of Systems and Control, Uppsala University,Uppsala, Sweden (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2003.820074

0 200 400 600 800 1000 1200 1400 1600--0.8

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--0.4

--0.2

0

0.2

0.4

0.6

0.8

1

exp(-- t/T2)

exp(-- t/T2* )

Te

Fig. 1. QR response of N in the TNT.

shown that exploiting temporal information about the signal canimprove the DOA estimation. Additional work dealing with theestimation of the parameters of known waveform signals can befound in [6]–[10].

All of the aforementioned papers concentrate on the estima-tion of signal parameters by exploiting the temporal informa-tion only. Exploiting both the temporal and spatial informationon the signal for interference suppression and signal parameterestimation has yet to be fully investigated in the previous litera-ture, yet it is of practical importance in some applications, suchas using the emerging quadrupole resonance (QR) technologyfor explosive detection [11], [12]. With reference to the QR ap-plication, the in the TNT, when stimulated by a sequenceof pulses, gives a characteristic response specific to the TNTconsisting of a sequence of echoes. We refer to this response asthe QR signal. Each echo of the QR signal is a back-to-back ex-ponentially damped sinusoid separated by an interval fromthe adjacent echoes and with a damping rate determined by aparameter (see Fig. 1). The echo amplitude also damps ex-ponentially as . The QR signal frequency and areall known quite accurately a priori, and the precise echo timing

is also available in practice. Hence, the QR signal is a prioriknown to within a multiplicative constant. The major challengeof using the QR technology for landmine detection is that theQR signal frequency falls within the frequency band for theAM/FM radio signals, and the QR signal frequency cannot bechanged. To suppress the radio interferences, an antenna arraycan be deployed with one of the sensors receiving the QR signalas well as the interference and noise (we refer to it as the main

1053-587X/04$20.00 © 2004 IEEE

24 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 52, NO. 1, JANUARY 2004

antenna), whereas the remaining sensors receive the interfer-ence and noise only (we refer to them as the reference antennas).Hence, one of the elements of the array steering vector for theQR signal is equal to one, and the remaining elements are zero.Thus, in this application, both temporal and spatial informationare available a priori.

Motivated by the QR application, we study herein theproblem of amplitude estimation of a signal with knownwaveform and steering vector since it is a mandatory stepfor detection. We consider two approaches [the Capon andthe maximum likelihood (ML) methods] that utilize both thetemporal and spatial information on the signal for amplitudeestimation in the presence of temporally white but spatiallycolored interference and noise. Based on the results in [13]and [14], we derive closed-form expressions for the expectedvalues and mean-squared errors (MSEs) of the two estimators.A comparative study shows that the ML estimate is unbiased,whereas the Capon estimate is biased downwards for finitedata sample lengths. We also show that both methods approachthe corresponding Cramér–Rao bound (CRB), i.e., they areasymptotically statistically efficient, when the number ofdata samples is large. However, they are not asymptoticallystatistically efficient when the signal-to-noise ratio (SNR)is high. Furthermore, we consider a more general scenariowhere the interference and noise vectors are both spatiallyand temporally correlated. We model the interference andnoise vector as a multichannel autoregressive (AR) randomprocess. An alternating least squares (ALS) method is proposedto tackle the amplitude estimation problem in this situation.We show that in most cases, the ALS method is superior tothe model-mismatched ML ( ) method that ignores thetemporal correlation of the interference and noise. Finally,numerical examples are presented to illustrate the theoreticalproperties and demonstrate the practical performance of theestimators.

The remainder of the paper is organized as follows. Sec-tion II formulates the problem of interest. The ML and Caponestimators are also given in that section. Section III gives theclosed-form expressions of the expected values and MSEsof the ML and Capon estimators and compares their statis-tical properties. In Section IV, we propose an alternating leastsquares (ALS) method to deal with the more general scenario ofboth spatially and temporally correlated interference and noise.Our theoretical findings are verified via numerical examples inSection V. Finally, Section VI gives our conclusions.

II. PRELIMINARIES

We consider the problem of estimating the complex ampli-tude of a known waveform signal in the presence of interferenceand noise:

(1)

where , denotes the th array outputvector (where is the number of sensors and is the numberof snapshots), the array steering vector of the signalof interest is known, and is the unknown complex ampli-tude of the signal whose temporal waveform is known.

First, we model the interference and noise termas a zero-mean temporally white but spatially colored circularlysymmetric complex Gaussian random process with an unknownand arbitrary, but fixed, spatial covariance matrix . We definethe SNR by lumping the interference and noise together in a“generalized noise” term:

SNRtr

(2)

where tr denotes the trace of a matrix, and

(3)

is the average power of the known waveform. We will discussthe extension to the case where the interference and noise termis both temporally and spatially correlated in Section IV.

The Capon and ML estimators are two widely used methodsin array processing [2]. The Capon method [also knownas the minimum variance distortionless response (MVDR)beamformer] estimates the signal amplitude via [15]

(4)

where

(5)

with denoting the conjugate transpose

(6)

and

(7)

It is easy to solve (5) and show that [2]

(8)

The ML method estimates the signal amplitude by maxi-mizing the likelihood function of the random vectors .We show in Appendix A that

(9)

where

(10)

Note that the only difference between the Capon and ML estima-tors is that the matrix in (8) is replaced by in (9). We will

JIANG et al.: ARRAY SIGNAL PROCESSING IN THE KNOWN WAVEFORM AND STEERING VECTOR CASE 25

show in the following sections that this seemingly minor dif-ference in fact leads to significant and interesting performancedifferences between the two estimators.

III. PERFORMANCE ANALYSIS OF ML AND CAPON

A. Performance Analysis of the ML Estimator

We present below a statistical performance analysis of the MLestimator. We prove that the ML estimator is unbiased. By de-riving the MSE of the ML estimate and comparing it with thecorresponding CRB, we show that the ML estimator is asymp-totically statistically efficient when the number of snapshots islarge but, in addition, that this is not the case for high SNR.

1) Bias Analysis: The matrix defined in (10) and thevector in (7) are both functions of the vectors ,which might suggest that they are correlated with each other.However, the lemma below somewhat surprisingly shows thatthey are in fact statistically independent of each other.

Lemma 1: Under the assumption made on the data model in(1), the vector and the matrix are statistically independentof each other.

Proof: See Appendix B.Utilizing this lemma and the conditional expectation rule, we

can easily show that the ML estimator is unbiased, i.e.,

(11)

where denotes calculating the expected value with re-spect to for a fixed .

Lemma 1 and its proof will also be helpful in the performanceanalyzes that follow.

2) Mean-Squared Error Analysis: Before calculating theMSE of the ML estimate, we first introduce the best possibleperformance bound for any unbiased estimator of , i.e., theCRB. Appendix C shows that the CRB has the followingcompact form:

CRB (12)

Note that

(13)

i.e., the error of the ML estimate of is

(14)

Hence, the MSE of the ML estimate is

MSE

(15)

(16)

To obtain (16) from (15), we have utilized the facts thatand are statistically independent of each other (seeLemma 1) and that are independently and identicallydistributed.

Let

(17)

According to (16) and (12)

MSECRB

(18)

As shown in the proof of Lemma 1

(19)

where are independently and iden-tically distributed. Hence, has a complex Wishart distribution[13], [14], [16], i.e., . In [13], theprobability density function (PDF) of has been pro-vided:

for

(20)Thus

(21)

(22)

Therefore, it follows from (18) and (22) that

MSE CRB (23)

The above equation shows that 1) the ML estimate isasymptotically statistically efficient for large snapshot lengths

, which was expected, and 2) the ML estimate is not asymp-totically statistically efficient for high SNR values whenis fixed. In fact, we can infer from (23) that the MSE (indecibels)-versus-SNR (in decibels) line is parallel to the


CRB-versus-SNR line, and thus, there is no “threshold effect”at low SNR. This theoretical result is verified via a numericalexample in Section V.

B. Performance Analysis of the Capon Estimator

We now establish the theoretical properties of the Capon esti-mator via an analysis that parallels the one in Section III-A. Ouranalysis relies on the results in [13], [14], and [17].

1) Bias Analysis: We have proved that the ML estimateis unbiased. We investigate the bias of the Capon estimate bystudying the relationship between the two estimators.

Lemma 2: Under the assumptions made on the data model in(1), we have

(24)

where

(25)

and the equality holds iff for some constant .Proof: See Appendix D.

Lemma 3: Under the assumption made on the data model in(1), any polynomial function of the defined in (24) is uncor-related with the ML estimate, i.e.,

where is an arbitrary polynomial.Proof: See Appendix E.

Based on Lemmas 2 and 3, we have

(26)

We derive below the PDF of . Recall that[cf. (126)] and . Following the tech-niques used in [13] and [14], we define

(27)

and

(28)

Then, and [16].Inserting (27) and (28) into (127) yields

(29)

where

(30)

We next introduce an unitary matrix such that

(31)

Note that and have the same PDFs as and ,respectively. Then, (29) can be further simplified as

(32)

We partition , , and as

(33)

(34)

(35)

where , , and are scalars, ,and are vectors, and both and are

matrices. Some useful relationships are asfollows [13], [14]:

(36)

(37)

(38)

(39)

To obtain (39) from (38), we have used the matrix inversionlemma.

Based on the partitions defined in (33)–(35), (32) can bewritten as

(40)

(41)

(42)

Hence, we have

(43)

where and , and theyare statistically independent of each other. It can been shown(see the [14, App.]) that the PDF of is

for (44)

and the first and second moments of this distribution are

(45)

(46)

Thus, it follows from (45) and (26) that

(47)

The above equation shows that the Capon estimate of is bi-ased downwards for a finite data sample number , whereas it


is asymptotically unbiased for large . We can also see that thebiasedness of the Capon estimate is not related to SNR. We notethat the result in (47) is not completely new. In [18] and [19],results similar to (47) when studying the convergence rate ofthe Capon beamformer, which was used to estimate the signalpower and waveform, are given. However, our assumptions aredifferent from those in [18] and [19], where the signals were as-sumed to be independently and identically distributed complexGaussian random vectors.

2) Mean-Squared Error Analysis: Based on Lemma 2, wecan obtain the MSE as

MSE

(48)

(49)

(50)

where to obtain (50) from (49), we have used Lemma 3 as wellas (11).

Combining (14) and (126) gives

(51)

Using the notations defined in (27) – (39), we can rewrite (51)as

(52)

(53)

(54)

Thus

CRB (55)

CRB (56)

CRB (57)

CRB (58)

CRB (59)

We obtained (57) from (56) using the fact that

(60)

It follows from (45) and (46) that the second term of (50) is

(61)

(62)

TABLE IM SENSORS, L SNAPSHOTS, AND SIGNAL AMPLITUDE �

Thus

MSE CRB (63)

We can conclude from (63) that the Capon estimate is asymptot-ically statistically efficient for a large data sample number . Inthe finite case, the MSE of the Capon estimate may be smallerthan the CRB for any unbiased estimator if the first term in (63)dominates the second one. However, the MSE of the Capon es-timate has an error floor equal to athigh SNR.

C. Summary of the Capon and ML Statistical Properties

• The ML estimate is unbiased.• The ML estimate is asymptotically statistically efficient

for large number of snapshots.• The ML estimate is not asymptotically statistically effi-

cient for high SNR.• The Capon estimate is biased downwards.• The Capon estimate is asymptotically unbiased and statis-

tically efficient for large number of snapshots.• The Capon estimate is neither asymptotically unbiased

nor asymptotically statistically efficient for high SNR.The above properties are summarized in Table I.

IV. EXTENSION TO MULTICHANNEL ARINTERFERENCE AND NOISE

The previous study assumed that the interference and noiseterm in (1) is spatially colored but temporally white, despite thefact that the interference and noise can be temporally correlated[1], [20]. In this section, we model the interference and noisevector as a multichannel autoregressive (AR) random processand propose an alternating least squares (ALS) method based onthe cyclic optimization approach [21]. For discussion purposes,we refer to the ML method in Section II that ignores the tem-poral correlation of the interference and noise as the model-mis-matched ML ( ) method.

A. Data Model

Consider the data model:

(64)

which is the same as the one in (1), except that the interferenceand noise term now satisfies the following AR equation [22]

(65)

where is the unit delay operator

(66)


and

(67)

where denotes the Kronecker delta function:

(68)

Note that if the interference component in is a multichannelAR process while the noise component in is white tempo-rally, then the interference and noise term will be a multichannelautoregressive and moving average (ARMA) random process,which can still be approximated by a multichannel AR process.The SNR for the data model in (64) is defined as

SNRtr

(69)

where is the covariance matrix of .

B. ALS Algorithm

Conditioned on the first data vectors , the log-like-lihood function is proportional to

tr

(70)

Maximizing the above function with respect to gives (see,e.g., [23])

(71)

After substituting (71) into (70), we need to minimize

(72)

with respect to both and . Hence,the optimization problem becomes more complicated than theone in Section II. Here, we propose an alternating least squared(ALS) approach to solve this problem.

To begin with, we obtain an initial estimate of by usingthe method [cf. (9)]. For a given estimate , let

. From (72), we get

(73)

where

...

Note that

(74)

Hence, the solution to (73) is

(75)

which is recognized as the multichannel Prony estimate of[22]. We assume that the order of the multichannel random

process AR is known. If is unknown, it can be estimated,for instance, by using the generalized Akaike informationcriterion (GAIC) [23].

For a given , we obtain an improved estimate of as fol-lows:

(76)

First, we consider the case of a known damped (or undamped)sinusoidal signal, i.e., with known frequency

and damping factor .Let

Note that the length of the new data sequenceis instead of . The solution to the above problem

is given by the ML estimator proposed in Section II:

(77)

where

(78)


and is the average power of the knownwaveform . The ALS approach maximizes the likeli-hood function cyclically. We set and obtain

. We then iterate the following two steps until the solu-tion converges, i.e., until the two consecutive estimates and

are sufficiently close:

(79)

which is given by (75) with replaced by , and

(80)

which is given by (77) with replaced by .Obviously, the likelihood function never decreases in any it-

eration. In the simulations reported in the next section, we foundthat ALS converges in two or three iterations, although there isno proof of convergence in general. Hence, the ALS estimatoris computationally quite efficient.

Next, we consider the case of an arbitrary known waveformsignal. Let

and

In addition, let be an orthogonal projection matrix defined as

(81)

where is the Moore–Penrose pseudo-inverse of [24],and let .

Then, (76) can be written concisely in a matrix form:

(82)

(83)

(84)

(85)

Note that minimizing the cost function in (85) requires a two-dimensional (2-D) search (since is complex-valued). To avoidthe search, we use Lemma 4 to obtain an approximate estimateof .

Lemma 4: For a large data sample number , minimizingis asymptotically

equivalent to minimizing

tr (86)

Proof: See Appendix F.It follows from (86) that

tr tr

tr tr (87)

101

102

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MS

E

Empirical MSE of ML Empirical MSE of Capon Theoretical MSE of ML Theoretical MSE of CaponCRB

SNR = 10 dB

Fig. 2. MSEs of �̂ and �̂ and the corresponding CRB versusLwhenSNR = 10 dB.

Minimizing with respect to yields

tr

tr(88)

where we remind the reader that . Because (88)is only an approximate solution to (85) in this more general case,ALS is not theoretically guaranteed to yield a more accurate so-lution than the method. However, in our numerical exam-ples, ALS outperforms in most cases, even for modest datasample lengths. To avoid any “divergence problem” in this casein which ALS is no longer an iterative maximizer, we simplypreimpose the number of iterations to be 3.

V. NUMERICAL AND EXPERIMENTAL EXAMPLES

We provide both simulated-data and real-life data examplesto demonstrate the performance the ML and Capon estimates.In all the simulated-data examples, we consider the case wherethe steering vector is given by and . Thiscorresponds to the case where the first of the sensorsreceives the signal as well as the interference and noise, whereasthe other three sensors receive the interference and noise only. Inall but the last example, we assume that , ,for simplicity. We obtain the empirical MSEs of the estimatesby using 500 Monte Carlo trials.

A. Spatially Colored but Temporally White Interference andNoise

We first consider simulated-data examples. We assume thatthe interference and noise term is a spatially colored but tempo-rally white Gaussian random vector with the spatial covariancematrix given by

(89)

where SNR.Fig. 2 shows the MSEs of the Capon and ML estimates ob-

tained from both theoretical predictions [based on (23) and (63)]and Monte Carlo trials as well as the corresponding CRB as afunction of when the SNR is 10 dB. As expected, both the


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Empirical MSE of ML Empirical MSE of Capon Theoretical MSE of ML Theoretical MSE of CaponCRB

L = 10

Fig. 3. MSEs of �̂ and �̂ and the corresponding CRB versus SNRwhen L = 10.

ML and Capon estimates approach the corresponding CRB asincreases since both methods are asymptotically statistically

efficient for large . Fig. 3 gives the MSEs of the Capon andML estimates as well as the corresponding CRB as a functionof SNR when . Note the error floor of the Capon esti-mate at high SNR due to its bias. Note also that the biased Caponestimate can have lower MSE than the unbiased ML estimate atlow SNR (yet this happens at MSE values that are too large tobe of practical value). As predicted by our theoretical analyses,for a fixed data length , MSE is parallel to the CRB,and no “threshold effect” occurs. Furthermore, the ML estimateis not asymptotically statistically efficient for high SNR.

Next, we present a real-life data example based on experi-mentally measured QR data. The main antenna of a QR land-mine detector receives a QR signal that consists of 40 echoes aswell as AM/FM interferences. We apply a fast Fourier transform(FFT) to each echo and only pick the value corresponding to theecho frequency . In this way, we compress the QR signal intoa signal with known waveform .Next, the data received at the three reference antennas is seg-mented into 40 blocks that occupy the same period of time asthe 40 echoes. We apply FFT to each block and pick three valuescorresponding to and the two adjacent frequency bins. Doingso, we get a virtual array with one main antenna and nine ref-erence antennas, i.e., . Although the aforemen-tioned preprocessing method might seem somewhat ad hoc, itworked well in our experiments. Fig. 4 shows the ML and Caponestimates of the QR signal amplitude in 30 experimental trials.Since we do not know the true value of the signal amplitude, wecannot compare the MSEs of the two estimators. Nevertheless,we let

(90)

where and are the empirical standard deviation andthe mean of (over the 30 trials). A small is desirable

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1300

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Am

plitu

de e

stim

ate

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Fig. 4. Amplitude estimates of a QR signal obtained via ML and Capon usingexperimentally measured data.

for signal detection. Based on the 30 trials, we get, which is smaller than .

B. Both Spatially and Temporally Colored Interference andNoise

We now consider the case of spatially and temporally corre-lated interference and noise. We generate a multichannel AR(2)random process with the method in [20]. The autocorrelationmatrices are given by

(91)

and

(92)

where SNR, controls the spatial correlation, partlydecides the temporal correlation, and defines the spectral peaklocation of the colored interference and noise in each channel.The data sample number is . When we use the true au-toregressive matrix in the ALS instead of the estimated one,we refer to the method as the known-AR ML (KML) approach.We include KML for comparison purposes only. Note that un-like the temporally white interference and noise case discussedpreviously, the performance of the and ALS estimators de-pends on the temporal frequency characteristics of the knownsignal. The following simulations are performed for both a con-stant signal and a known BPSK signal.

First, we consider the relationship between the cost functiondefined in (72) and . Because can be concentrated

out by using its estimate given in (75), is a function ofonly. Consider the constant signal case. Fig. 5 shows the meshplot of versus the real and imaginary part of . We can seethat there is only one local maximum around the true value of

.


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imag(β)

--C

2

Fig. 5. Mesh plot of�C (withC defined in (72)) for a constant signal versusthe real and imaginary part of � when SNR = �10 dB, � = 0:1, � = 0:6,! = 2, and the true value of � is 1.

For the constant signal case, our simulations show that thespatial correlation coefficient is not closely related to the gapbetween the performance of and ALS. However, the tem-poral correlation coefficient and the position of the spectralpeak have an impact on the relative performance of the twomethods (see Figs. 6 and 7). We summarize our observations asfollows:

A: Both ALS and work better for large and/orsmall .

B: ALS is slightly worse than for small and/orlarge .

C: ALS is significantly better than for large and/orsmall .

To explain these observations, we examine the signal as wellas the interference and noise term in the temporal frequency do-main. The signal is a constant and, hence, has power at zero fre-quency only. The power of the interference and noise is concen-trated around , especially for small . For large , the signalis separated from the interference and noise in the temporal fre-quency domain, which benefits both methods. Similarly, smaller

means higher correlation in the temporal domain or morepeaky spectra in the temporal frequency domain. Hence, bothestimators perform better for this case when is away fromzero. This explains Observation A. Next, we note that a large

means low correlation in the temporal domain, and hence,the interference and noise vector is approximately temporallywhite. For small , the signal and the interference and noiseterms are not well separated in the temporal frequency domain.KML behaves approximately as in either case. Since ALSis inferior to KML, ALS is also slightly worse than . Thisexplains Observation B. Observation C was expected since ALSestimates the temporal correlation of the interference and noiseand can suppress the interference and noise more efficiently inthis case in which the temporal correlation is significant.

Finally, we consider the known BPSK signal case. Because aBPSK signal is wideband in the temporal frequency domain, theimpact of the interference spectral peak location on the perfor-mance of the two methods is not as significant as in the constant

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E

M3LALS KML

Fig. 6. MSEs of ALS M L and KML estimates for a constant signal versusSNR when ! = 1 and � = 0:6.

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10--2

100

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SNR (dB)

MS

EM3LALS KML

Fig. 7. MSEs of ALS M L and KML estimates for a constant signal versusSNR when � = 0:1 and � = 0:6.

signal case, which was verified in our simulations. However,the temporal correlation parameter still controls the relativeperformance of the two methods as shown in Fig. 8. We also seefrom Fig. 8 that the ALS method significantly outperforms(by over 10 dB in SNR) even for modestly temporally correlatedinterference and noise ( ), although it performs simi-larly to when the temporal correlation of the interferenceand noise is weak ( ). Our simulations also suggest thata known wideband signal makes suppressing temporally corre-lated interference and noise easier than a narrowband one in thesense that better estimates of can be obtained in the widebandcase.

VI. CONCLUSIONS

We have investigated the problem of amplitude estimation fora signal with known waveform and steering vector in the pres-


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--20 --10 0 10 2010

--6

10-- 4

10--2

100

ρt = 0.1

SNR (dB)

MS

E

M3LALSKML

Fig. 8. MSEs of ALS M L and KML estimates for a known BPSK signalversus SNR when ! = 0 and � = 0:6.

ence of interference and noise. We first assumed that the inter-ference and noise vector was spatially colored but temporallywhite. The ML and Capon methods as well as the closed-formexpressions of the expected values and MSEs of the two esti-mators have been derived. We have shown that the ML estimateis unbiased as well as asymptotically statistically efficient forlarge data sample sets but, in addition, that it is not asymptot-ically statistically efficient for high SNR. We have also shownthat the Capon method is biased downwards, but it is asymp-totically unbiased and efficient for large data sample lengths.The bias of the Capon estimate dominates its variance for highSNR, which results in an error floor that does not decrease withSNR. At low SNR, however, Capon can outperform ML as wellas the CRB for any unbiased estimator. We then considered amore general scenario where the interference and noise vectorwas both spatially and temporally colored. We have proposed anALS method based on the idea of cyclic optimization. We haveshown that in most cases, ALS outperforms the estimator,which ignores the temporal correlation of the interference andnoise.

APPENDIX ADERIVATION OF THE ML ESTIMATOR

The normalized log-likelihood function of is propor-tional to

tr

(93)where denotes the determinant of a matrix. Maximizing theabove cost function with respect to gives (see, e.g., [1])

(94)

Hence, the ML estimate of is obtained as follows:

(95)

(96)

(97)

(98)

(99)

where from (97) to (98), we have used the fact that(see, e.g., [2]).

APPENDIX BPROOF OF LEMMA 1

In the following, a subscript is used to indicate the dimensionof a matrix for the sake of clarity and is dropped whenever con-venient. Consider an matrix

(100)

where , with denoting the transpose and

(101)

We can construct a unitary matrix

whose first column is chosen as.

Let

(102)

and

(103)


Inserting (100) into (102) gives

(104)

From (7), the first column of

(105)

and

(106)

where (or ) denotes the th column of (or ). Since,

(107)

It follows from (6), (10), and (107) that

(108)

Hence, is a function of , whereas.

We now show that are statistically independent ofeach other. The cross correlation matrix of any two columns of

has the form

(109)

which is an matrix with the element

tr (110)

where tr denotes the trace, and denotes the th row of .To obtain (110), we have used tr tr . Next, notethat

(111)

where is the element of the covariance matrix , andis the Kronecker delta function defined in (68). Substituting

(111) into (110) yields

tr (112)

for (113)

Therefore

(114)

Furthermore, based on the circularly symmetric property of, it is easy to show that for .

Since are Gaussian random vectors, they arestatistically independent of each other. Since is a function of

, and is a function of , we conclude thatand are statistically independent of each other.

APPENDIX CCRAMÉR–RAO BOUND

Let be a vector containing all of the real-valued unknownsin the data model in (1). Let . Then, the Fisher infor-mation matrix (FIM) for is [2]

FIM tr

Re (115)

where denotes the th element of . Because and dependon different elements of , FIM will be block diagonal withrespect to Re Im and the elements of , whereRe and Im denote the real and imaginary parts of a complexvariable, respectively. Hence, the first term of (115) does notaffect the CRB of , and

FIM Re Im

Re

(116)

(117)

Hence

CRB CRBRe CRBIm (118)

(119)

(120)

APPENDIX DPROOF OF LEMMA 2

We know from (10) that

(121)


Using the matrix inversion lemma on (121) gives

(122)

Substituting (122) into (8), we have

(123)

(124)

where is defined in (25) and by the Cauchy-Schwartz in-equality . Hence, the lemma is proved.

APPENDIX EPROOF OF LEMMA 3

We rewrite the in (7) as

(125)

where

(126)

and . Then, (25) can be reduced to

(127)

From (14) and (126), we get

(128)

According to the conditional expectation rule

(129)

Since and are statistically independent of each other, theconditional probability density function isan even function of . It follows from (127), (128), and (24) that

(130)

since is an odd function of . From (129), weobtain

for (131)

Hence, the lemma is proved.

APPENDIX FPROOF OF LEMMA 4

Let . Then, we have. Let denote the singular value

decomposition of . Since is an orthogonal projection ma-trix, the first ( rank ) diagonal elements ofare ones, and the rest are zeros. Hence, we get ,where consists of the first columns ofthe unitary matrix . Let denote the th row of . Then

(132)

Because as , where denotesthe th diagonal element of the defined in (67), we have

as . It follows from(132) that the diagonal elements of are bounded. Since

is a positive semi-definite matrix, we have. However, .

Hence, . Letdenote the eigenvalues of . Then

(133)

(134)

Since for large , (134) can be approximated as

(135)

[Note that if for , which is true for thecase of a known damped or undamped sinusoidal signal, (134)and (135) are exactly equal.] Thus, minimizing is asymptot-ically (for large ) equivalent to minimizing

tr (136)

The lemma is proved.

ACKNOWLEDGMENT

The authors are grateful to two anonymous reviewers whoprovided us with a large number of detailed suggestions for im-proving the submitted manuscript.

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Yi Jiang (S’02) was born in Jiangsu Province, China,in 1978. He received the B.Sc. degree from the Uni-versity of Science and Technology of China (USTC),Hefei, in 2001 and the M.Sc. degree from the Uni-versity of Florida (UF), Gainesville, in 2003, both inelectrical engineering. He is currently pursuing thePh.D degree with the Department of Electrical andComputer Engineering, UF.

His research interests include spectral estimation,array signal processing, and information theory.

Petre Stoica (F’94) received the D.Sc. degree inautomatic control from the Polytechnic Instituteof Bucharest (BPI), Bucharest, Romania, in 1979and an honorary doctorate degree in science fromUppsala University (UU), Uppsala, Sweden, in1993.

He is Professor of system modeling with the De-partment of Systems and Control at UU. Previously,he was a Professor of system identification and signalprocessing with the Faculty of Automatic Controland Computers at BPI. He held longer visiting posi-

tions with Eindhoven University of Technology, Eindhoven, The Netherlands;Chalmers University of Technology, Gothenburg, Sweden (where he held aJubilee Visiting Professorship); UU; The University of Florida, Gainesville;and Stanford University, Stanford, CA. His main scientific interests are in theareas of system identification, time series analysis and prediction, statisticalsignal and array processing, spectral analysis, wireless communications, andradar signal processing. He has published seven books, ten book chapters, andsome 500 papers in archival journals and conference records on these topics.The most recent book he co-authored, with R. Moses, is entitled Introductionto Spectral Analysis (Englewood Cliffs, NJ: Prentice-Hall, 1997). Recently, heedited two books on signal processing advances in wireless communicationsand mobile communications (Englewood Cliffs, NJ: Prentice-Hall, 2001). Heis on the editorial boards of five journals in the field: Journal of Forecasting;Signal Processing; Circuits, Signals, and Signal Processing; Digital SignalProcessing—A Review Journal; and Multidimensional Systems and SignalProcessing. He was a Co-Guest Editor for several special issues on systemidentification, signal processing, spectral analysis, and radar for some of theaforementioned journals, as well as for PROCEEDINGS OF THE IEEE.

Dr. Stoica was co-recipient of the IEEE ASSP Senior Award for a paper onstatistical aspects of array signal processing. He was also recipient of the Tech-nical Achievement Award of the IEEE Signal Processing Society for funda-mental contributions to statistical signal processing with applications in time-se-ries analysis, system identification, and array signal processing. In 1998, he wasthe recipient of a Senior Individual Grant Award of the Swedish Foundation forStrategic Research. He was also co-recipient of the 1998 EURASIP Best PaperAward for Signal Processing for a work on parameter estimation of exponentialsignals with time-varying amplitude, a 1999 IEEE Signal Processing SocietyBest Paper Award for a paper on parameter and rank estimation of reduced-rankregression, a 2000 IEEE Third Millennium Medal, and the 2000 W. R. G. BakerPrize Paper Award for a paper on maximum likelihood methods for radar. Hewas a member of the international program committees of many topical confer-ences. From 1981 to 1986, he was the Director of the International Time-SeriesAnalysis and Forecasting Society, and he has been a member of the IFAC Tech-nical Committee on Modeling, Identification, and Signal Processing since 1994.He is also a member of the Romanian Academy and a Fellow of the Royal Sta-tistical Society.

Jian Li (S’87–M’91–SM’97) received the M.Sc.and Ph.D. degrees in electrical engineering from TheOhio State University (OSU), Columbus, in 1987and 1991, respectively.

From April 1991 to June 1991, she was anAdjunct Assistant Professor with the Departmentof Electrical Engineering, OSU. From July 1991 toJune 1993, she was an Assistant Professor with theDepartment of Electrical Engineering, Universityof Kentucky, Lexington. Since August 1993, shehas been with the Department of Electrical and

Computer Engineering, University of Florida, Gainesville, where she iscurrently a Professor. Her current research interests include spectral estimation,array signal processing, and their applications.

Dr. Li is a member of Sigma Xi and Phi Kappa Phi. She received the 1994National Science Foundation Young Investigator Award and the 1996 Officeof Naval Research Young Investigator Award. She was an Executive CommitteeMember of the 2002 International Conference on Acoustics, Speech, and SignalProcessing, Orlando, FL, May 2002. She has been an Associate Editor of theIEEE TRANSACTIONS ON SIGNAL PROCESSING since 1999. She is presently amember of the Signal Processing Theory and Methods (SPTM) Technical Com-mittee of the IEEE Signal Processing Society.

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