Joint Space-Time Parameter Estimation for Wireless ... Space-Time Parameter Estimation for Wireless Communication ... by Watkins Johnson Company. ... Estimation for Wireless Communication

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 5, MAY 1998 1333

Joint Space-Time Parameter Estimationfor Wireless Communication Channels

Gregory G. Raleigh and Tibor Boros

Abstract—Joint space-time parameter estimation methods arediscussed for time-varying channels in a multipath environment.A spatiotemporal array manifold model is developed that makes itpossible to apply traditional parameter estimation techniques forextracting both the angle and delay parameters from the observeddata. The conditional Cramer–Rao bound (CRB) is derived forunbiased space-time parameter estimators.

I. INTRODUCTION

A NUMBER of methods are available in the literature forestimating the signal parameters in the classical model

(1)

where

describes the array manifold, is the observedsignal vector, is the unknown signal vector,

is the additive measurement noise, anddenotes the number of antennas. For prior work on direction ofarrival (DOA) estimation algorithms, see the conditional max-imum likelihood method [1]–[12] and the references therein,the unconditional maximum likelihood method [14]–[22], theMUSIC algorithm and its variations [23]–[24], theESPRITalgorithm [25], theMODE estimate [26]–[27], subspace fittingmethods [28]–[29], etc. The primary goal of this paper is toextend the above techniques to more complex signal modelsthat incorporate the effects of path delay in a rapidly fadingpropagation environment. We refer to the problem of jointlyestimating the angle of arrival and path delay parameters asspace-time parameter identification.

Although our primary application area can be termed “wire-less subscriber location,” the techniques developed in thispaper can be applied to other fields as well (e.g., radar andsonar).

In urban areas, radio signals can experience severe multipathreflection so that the energy transmitted by the portable unitarrives to the base station from many different directions. Insuch cases, classical direction of arrival estimation algorithmsare of limited use for three reasons. First, the coherent multi-path interference tends to degrade angle estimation accuracy,especially when the spatial separation between the propagation

Manuscript received March 1, 1996; revised February 13, 1997. Thiswork was sponsored by Watkins Johnson Company. The associate editorcoordinating the review of this paper and approving it for publication wasProf. Michael D. Zoltowski.

G. G. Raleigh is with Clarity Wireless, Belmont, CA 94002 USA.T. Boros is with ArrayComm, Inc., San Jose, CA 95134 USA.Publisher Item Identifier S 1053-587X(98)03249-8.

paths is small. Second, even if the incoming signals arewidely separated, there remains an inherent ambiguity aboutthe correct physical direction of the portable. Finally, thenumber of estimated angles must be smaller than the numberof antennas. Therefore, angle estimation in a severe multipathenvironment requires very large arrays.

Jointly estimating the angle and delay parameters of eachpropagation path has a number of advantages. First, therelative delay estimates between two or more synchronizedbase stations can be used in conjunction with the DOAinformation to estimate the location of the portable unit.Second, as we shall demonstrate in Section V, it is theoreticallypossible to exploit the difference in path delays to improveangle estimation accuracy and vice-versa (see Figs. 5 and 6in Example 5.2). Third, when several angles are detected, themost probable portable direction is the one that is associatedwith the smallest delay because the direct path has the shortestpropagation time. Finally, as shown in Section IV, in the space-time array manifold model, the number of propagation pathscan be significantly larger than the number of antennas.

Section II describes the time-varying multipath signal modelfor the received data. The time-varying array manifold conceptis extended to space-time manifolds in Section III. Various pa-rameter estimation techniques are described for the joint angleand delay estimation problem in Section IV. The Cramer-Rao bound (CRB) for space-time parameter estimators isderived in Section V. Finally, conclusions and further researchpossibilities are discussed in Section VI.

II. THE RECEIVED SIGNAL MODEL

Instead of the classical time-invariant model (1), in this pa-per, we shall use the more sophisticated time-varying multipathmodel developed in [30]–[32] (see [33]–[35] for earlier workon scalar channels). In the time-varying model, the receivedsignal is obtained as

(2)

where denotes the impulse response of thesingle-input multiple-output digital channel from the portableto the receiving array, and

contains the transmitted symbols that contribute to the receivedsignal at the th sampling instant. It is further assumed that thetransmitted waveforms arrive to the receiver viadominant

1053–587X/98$10.00 1998 IEEE

1334 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 5, MAY 1998

Fig. 1. Propagation geometry for the multipath channel model.

paths (see Fig. 1) so that can be expressed as

......

(3)

where is the time-varying amplitude of the RF signalpassing through theth propagation path, is the arrayresponse of the receiving antenna array as a function of theDOA, and the vector

(4)

denotes the truncated, delayed, and sampled impulse responseof the pulse shaping filter as a function of the path delay. In(4), denotes the lag, and denotes the length of thepulse shaping filter, whereas denotes the time-continuousNyquist pulse shape. The path delay is usually expressed inrelative units (i.e., normalized by the symbol period).

The inherent assumptions associated with the channel model(3) are that the receiver is perfectly synchronized1 (symbol-spaced sampling), moreover, that the participating quantitiesare invariant over the bandwidth of the transmitted signal(narrowband assumption).

The mathematical model displayed in (2) and (3) closelyapproximates the behavior of the propagation channel betweena portable unit located “below the clutter” and a base stationthat is located “above the clutter.” In practice, the portableis usually surrounded by a plurality of “small” objects thatoften block line-of-site propagation to the base station. Theseobjects are termed “local scatterers” since they are in the nearvicinity of the portable unit, and they scatter the energy of thetransmitted radio wave with a random amplitude and phase.

In contrast, the base station antenna array is located onhigh ground and does not experience local scattering. Instead,the base station receives energy from large features in theterrain such as large buildings, hillsides, etc. These features aretermed “dominant reflectors” since the bulk of the base station-received energy arrives from these objects. Each dominantreflector reradiates energy that arrives from the local scatteringmatrix surrounding the portable. The dominant reflectors arein the far field of the base station receiver so that the energy

1For the sake of simplicity, here, we assume that the clock signals at thetransmitter and at the receiver are perfectly synchronous. In practice, therelative timing error between the transmitter and the receiver must be takeninto account. In a multipath channel, the receiver generally is not synchronizedin such a manner that the path delay associated with the direct path becomesexactly zero.

RALEIGH AND BOROS: JOINT SPACE-TIME PARAMETER ESTIMATION FOR WIRELESS COMMUNICATION CHANNELS 1335

arriving from each dominant reflector can be modeled as aplane wave. A well-defined path delay can be associated witheach dominant path, provided that the inverse of the signalbandwidth is much smaller than the time required for thesignal to propagate across the local scattering matrix (typically30–300 ns).

As the portable unit moves, the phase of the transmittedwaveform to each local scatterer varies in time, which givesrise to time-domain fading phenomena along each dominantreflector path. Since the amplitude and phase of the propaga-tion path from each local scatterer to a dominant reflector israndom, the composite complex path amplitude is alsorandom. If the number of local scatterers is large, then overa short period of time, can be modeled as astationaryGaussian random process. In practice, this “quasistationary”assumption works well if the observation period is smallerthan 0.1–1 s (provided the portable unit travels with 60mi/hr). The time and frequency correlation behavior of therandom process is determined by the local scattering matrixand the Doppler shift that is caused by the relative motion ofthe portable unit. In particular, the absolute value offollows a Rayleigh distribution (see [33] on Rayleigh andlog-normal fading). It is important to note that the Gaussianfading model is not critical to the development of the spatio-temporal parameter estimation algorithms that are presentedin the following sections.

Assuming that the array response and the pulse shapingfunction are known, theth propagation path is describedby three parameters:

1) the angle of arrival ;2) the path delay ;3) the path strength

The common objective of space-time parameter estimationalgorithms is to extract the parameter vectors

from the received data

III. T HE SPACE-TIME ARRAY MANIFOLD MODEL

The main difficulty in extending the classical parameter esti-mation techniques to the joint angle-delay estimation problemarises from the fact that in (2), the impulse response matrix

varies rapidly in time. As a result, the received signalvector is highly nonstationary, even if and

are stationary random processes. In fact, these difficultieshave created a widespread interest in adaptive algorithms thatestimate both the unknown vector and the time-varyingchannel without making use of initial DOA estimates.

In what follows, we shall assume that a noisy channelestimate is already recovered from the received data.Recall that according to the Rayleigh model,is astationaryGaussian random process, and therefore, various standardtechniques (such as Kalman filtering, Wiener filtering, and

filtering) can be used to extract a reliable channelestimate. Channel estimation problems have recently attractedconsiderable attention in the research literature (see, e.g.,

TABLE ICORRESPONDINGQUANTITIES OF THE CLASSICAL AND SPATIO-TEMPORAL MODELS

[31]–[32], [36]–[39], and the references therein). The richnessof the available information prevents us from discussing thedetails in this paper.

Under the above assumptions, can be represented as

(5)

where models the estimation error. Equation (3) impliesthat can be further expressed as

(6)

where the rank-one matrix

(7)

denotes the normalized2 impulse response associated with theth propagation path.

In order to bring out the similarities between (6) and (1),we shall introduce the vector quantities

vec

vec (8)

where vec enumerates the entries of in a row-wiseorder. Thus, (6) can be written as

(9)

where

2Here, “normalized” means that�l(t) = 1:


Fig. 2. Surface plot of the inverse of the spatio-temporalMUSIC cost function (linear scale).

Furthermore, (7) and (8) imply that the generalized (space-time) array vector has the following special structure:

where

and denotes the Kronecker matrix product.3

In this form, the time-varying model (9) for the channelestimate coincides with the classical signal model (1). The onlydifference is that the columns of the space-time array mani-fold depend on two parameters. The correspondingquantities of (1) and (9) are displayed in Table I.

It is interesting to note that certain algorithms such asMUSICare statistically inefficient when the elements of inthe classical model (1) are correlated. In contrast, it is a well-accepted assumption that the various paths in the multipathchannel model fade independently. Thus, in practice, theelements of in (9) are uncorrelated, although it is notnecessary to make this restriction in order to proceed with thealgorithms.

IV. SPACE-TIME PARAMETER ESTIMATION TECHNIQUES

The common assumptions associated with the models (1)and (9) are as follows:

A1) The number of propagation pathsis known (see [14]and [40]–[41] on how to estimate from the inputdata

3AAABBB is a block matrix so that the(i; j)th block ofAAABBB is AijBBB:

A2) and the array manifold is un-ambiguous so that the steering vectors and

are linearly independent wheneveror Thus, is an upper bound onthe number of propagation paths that can be resolvedin this framework. The fact that an -element antennaarray is able to resolve more than angles and delaysis somewhat surprising (see Figs. 2 and 3).

A3) The additive noise is Gaussian distributed4 with

where denotes the Kronecker delta.5

Based on the channel model (9), most of the parameterestimation techniques can be extended to extract both the angleof arrivals and the path delays from theestimated channel data Here, we only mention a fewspecific examples.

A. The Conditional (Deterministic)Maximum Likelihood Method

The vector is assumed to be nonrandom, i.e., fixedover the ensemble of realizations. Under this assumption, theML estimates of and are obtained by minimizing the costfunction

tr

4More complicated noise models can be easily incorporated as long as thecovariance ofeee(t) is known in advance. For brevity and simplicity, we shallnot discuss colored noise models in this paper.

5�t;s = 1 whent = s, and�t;s = 0 whent 6= s:


Fig. 3. Contour plot of the inverse of the spatio-temporalMUSIC cost function (log scale).

where

(10)

denotes the estimated covariance matrix of The globalminimum of the cost function can be found by usingvarious search algorithms such as simulated annealing [7],alternating projection [9], and dynamic programming [12].

B. The Unconditional (Stochastic) MaximumLikelihood Method

The vector is assumed to be a stationary Gaussianrandom process with

Under these assumptions, the ML estimates ofand areobtained by minimizing the cost function

tr

where

If and are not known in advance, then they can beestimated from as

tr

(11)

where

C. The MUSIC Algorithm

Let be the eigenvectors of the esti-

mated covariance matrix arranged in a descending orderof the associated eigenvalues. TheMUSIC estimates of and

are obtained by finding the minima ofthe pseudo spectrum

where

(12)

denotes the estimated signal subspace.

D. Signal Subspace Fitting

Let be the estimated signal subspace as defined in (12).The estimates of and are obtained by minimizing the costfunction

(13)

over all vectors and all matricesExample 4.1: In order to illustrate a practical procedure

for space-time parameter estimation, let us consider a simpleexample for theMUSIC algorithm. We shall assume that amultiple antenna receiver collects blocks of data.Each data block contains symbol spaced samples ofthe array output. The time separation between each block is2000 symbols. Equation (2) is used to generate the data. Thepulse shaping function is truncated raised cosine with a symbolperiod of (PACS standard). The fading parameters

are generated from a modified Jakes model [33] with amaximum Doppler frequency shift of Hz. The


number of propagation paths is Two of the paths haveidentical angles and different delays, and two of them haveidentical delays and different angles. The array is anelement uniform linear array with 1/2 wavelength spacingand omnidirectional elements. The data symbols are generatedfrom a random uniform 4-QAM distribution. The additivenoise vector is white Gaussian, and the average signal-to-noiseratio (SNR) is 6 dB.6

The time-varying channel is assumed to be constant overthe symbols of a received data block, but it can varysignificantly between the consecutive blocks. In order to beable to identify , the condition

must hold (where denotes the length of the FIR channel).The linear least-squares channel estimate estimate canbe obtained as

vec vec

where

and the superscript denotes the Moore–Penrose pseudoinverse.

The signal subspace estimateis then determined by theeigenvectors associated with thelargest eigenvalues of thesample covariance matrix (10). In order to be able to identifythe signal subspace, the condition

must hold. Moreover, the channel must vary enough be-tween the data blocks to partially decorrelate thedominantpropagation paths. In a typical system in the 900-MHz cellularband, the portable should move only a few centimeters overthe observation period.

The estimated angles and delays are found by performing atwo-parameter search over the space-time array vectorand determining the minima of the two dimensionalMUSICcost function (13). The surface plot (linear scale) and thecontour plot (logarithmic scale) for the inverseMUSIC costfunction are depicted in Figs. 2 and 3, respectively. The trueangle and delay parameters for each path are marked on thecontour plot by ’s.

This example clearly demonstrates that an-element an-tenna array is able to resolve more thanincoming signals.

V. THE CRAMER-RAO BOUND

The complete statistical analysis of various space-time esti-mation algorithms is beyond the scope of this paper. Nonethe-

6Average SNR is defined here as the ratio of the average (fading) signalenergy to the average noise energy.

less, it will be useful to state the CRB that indicates thetheoretical limit on the performance of any unbiased parameterestimator.

For the classical model (1), the conditional CRB has beenthoroughly analyzed in [10] and [11] (see [1]–[6] for earlierresults). The performances of various conditional and un-conditional DOA estimators have been closely studied andcompared in [22] (see also [13]–[21]). In this section, weextend the results of [10] and [11] and obtain new formulasfor the conditional CRB that incorporate both angle and delayestimates.

Theorem 1: Under the assumptionsA1)–A3), the condi-tional CRB for and can be expressed as

Re

(14)

where

diag

and

Proof: See Appendix A.The asymptotic behavior of the CRB for large can be

characterized by assuming that is a second-order ergodicsequence so that the limit

(15)

exists. Under this assumption, the following statement is valid.Theorem 2: For sufficiently large , the conditional CRB

can be approximated as

Re

(16)where

and denotes the Hadamard (element-wise) matrix product.7

Proof: See Appendix B.

7The definition of�: AAA;BBB 2 m�n 7! AAA � BBB 2 m�n so that[AAA�BBB]i;j = [AAA]ij � [BBB]ij :


The characteristic features of the general formula (14) aredemonstrated via specific examples below.

Example 5.1:Let so that the space-timearray manifold depends on one angle parameterandone delay parameter Assume that is given by

Then

......

and

...

......

where Thus

and we have the equation shown at the bottom of the page.Moreover, the CRB matrix is given by

(17)

SNR(18)

and

SNR(19)

where SNR , and is thetotal energy of the pulse shaping filter. Note that in this case,

(a)

(b)

Fig. 4. Dependence of the CRB on the delay parameter�:

the CRB matrix does not depend on In Fig. 4, the squareroots of and are plotted versus for the mostfrequently used raised cosine pulse shape

sinc (20)

The family of curves in Fig. 4 is parametrized in terms of theroll-off factor

Fig. 4 clearly demonstrates that is minimal for, i.e., when theenergy of the pulse shaping

filter is the largest. On the other hand, is minimal for


Fig. 5. CRB for the angle estimate(� = 0:35):

Fig. 6. CRB for the delay estimates(� = 0:35):

, i.e., when thesensitivityof the pulseshaping function is the largest. For , the raisedcosine function collapses to thesinc function. In this case,

and become independent ofExample 5.2:Let be the raised cosine pulse shaping

function (20) with roll-off factor Assume that ,

and let and be given by

In this case, the array manifold depends on two angleparameters and two delay parameters In the


following experiment, and were fixed, and theasymptotic CRB formula (16) was evaluated for several valuesof and The results are depicted below in Figs. 5 and 6.

Fig. 5 displays as a function of and Thefollowing observations are in order.

Observation 1.1:It is interesting to note that even a smalldelay is sufficient to improve the CRB for the angle estimates.It is somewhat counterintuitive that the improvement is par-ticularly large when the angle separation issmall.

Observation 1.2:For large delays, becomes indepen-dent of the angle separation and uniformly converges to aconstant value determined by (18).

Observation 1.3:For small delays, converges to theclassical curve shown in [11, pp. 2145–2146]. In the specialcase when , the CRB formula (14) is equivalentto (4.1) in [10, p. 725].

Observation 1.4:For large angle separations, each curve inFig. 5 approaches the constant value given by (18).

Observation 1.5:For small angle separations, the CRBdoes notnecessarily tend to infinity. In particular, as

the CRB again converges to (18). Thismakes sense since the space-time array manifold and,thus, the CRB should be a smooth periodic function of

(21)

In particular, for uniform linear arrays the relationis valid as well.

In a similar manner, Fig. 6 displays the CRB for thedelay estimate as a function of and We need to makethe following observations.

Observation 2.1:The plots demonstrate that even a smallangle separation can significantly improve the accuracy of thedelay estimates. The improvement is particularly large whenthe delay separation is small.

Observation 2.2:For large angle separations, ap-proaches the delay estimate (19) that is valid for a singleincoming path [see Fig. 4(b)].

Observation 2.3:For very large and very small delays allcurves converge to the finite value defined by (19). In the limit,as , the CRB becomes independent of

VI. CONCLUSIONS

A space-time array manifold model was developed for time-varying communication channels in a multipath propagationenvironment. The existing results for DOA estimation weregeneralized to the more difficult problem of joint angle/delayestimation. The conditional CRB for joint space-time param-eter estimators was derived.

The channel model described in this paper is valid only forlinear modulation schemes (such as PAM, QAM, etc.) whensymbol-spaced sampling is applied at the receiver. However,

the results can be easily extended to nonlinear modulationschemes (such as phase modulation) and fractionally sampleddata models. Although, in the examples, we assumed uniformlinear arrays, the presented methods will work with any arraygeometry, provided that the space-time array manifolduniquely determines and

Interfering users that transmit data in the same frequencyband can be taken into account by assuming that the estimationerror is colored. In this case, the adaptive algorithmsdescribed in [31]–[32] and [36]–[39] can be used to obtainboth an instantaneous channel estimate and an averageinterference-subspace estimate for eachof the users.

APPENDIX ADERIVATION OF THE CRB

The objective is to extend the derivation of the classicalCRB formula (see [10, Appendix E]) to accommodate the extraparameters Without repeating the entire proof, we shall onlyhighlight those steps that must be modified.

The log-likelihood function of the observed process in thiscase can be expressed as

const

where the parameter vector

Re Im Re

Im

includes both angle of arrivals and path delays. The Fisherinformation matrix has the particular form

......

......

......

...


where

Re ImIm Re

ReIm

ReIm

Re

Re

Re

Re

The inverse CRB matrix for and is obtained by forming theSchur complement of the trailing block in the Fisherinformation matrix as

......

......

......

Inverting both sides of the above equation concludes theproof.

APPENDIX BASYMPTOTIC FORMULA FOR THE CRB

The proof of [10, Theorem 4.3] can be carried overmutatismutandis, namely, let Observethat

Re

Re

Re (B.1)

where

The ergodicity condition (15) implies that for large

(B.2)

Substituting (B.1) and (B.2) into (14) concludes the proof.

ACKNOWLEDGMENT

The authors would like to thank M. A. Pollack, V. K. Jones,and S. Diggavi for valuable discussions on the subject.

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Gregory G. Raleigh received the B.S.E.E. degreefrom California State Polytechnic University, SanLuis Obispo, and the M.S.E.E. degree in electri-cal engineering from Stanford University, Stanford,CA. He has completed his thesis research and ex-aminations and expects to receive the Ph.D. degreein electrical engineering from Stanford Universityin the Spring of 1998.

He is Chief Technical Officer and a cofounder ofClarity Wireless, Inc., Belmont, CA. He overseesresearch and development for a new generation of

broadband wireless access modems that employ adaptive digital modulationand coding to enhance performance in severe multipath environments. Hisresearch interests include signal processing algorithms, coding systems, andradio frequency systems for multivariate wireless communication. Beforejoining Clarity, he was Staff Scientist at Watkins Johnson Company, SanJose, CA, responsible for leading research, design, and development for anumber of commercially successful microwave radio and radar products. Hehas several publications in the areas of advanced microwave radio techniquesand adaptive multivariate signal processing for wireless communications. Healso has several patents and patents pending in wireless communications.

Tibor Boros received the Diploma degree (withHonor) from the Technical University of Budapest,Budapest, Hungary in 1991 and the M.S. and Ph.D.degrees in electrical engineering from Stanford Uni-versity, Stanford, CA, in 1993 and 1996, respec-tively.

From 1994 to 1995, he was affiliated with thecolor image processing group of the RICOH Califor-nia Research Center, Menlo Park, where he workedon spatially embedded coding schemes for imagecompression. From 1995 to 1996, he was a Member

of Technical Staff in the Wireless Research Group, Watkins Johnson Company,San Jose, CA, where he developed adaptive antenna array signal processingalgorithms for wireless communications. He currently works as a researchengineer for ArrayComm, Inc., San Jose, CA, where he is involved in thedesign and implementation of smart antenna arrays. His main interests includecomputationally efficient algorithms for signal processing, automatic control,and linear algebra.

Documents

Joint Space-Time Parameter Estimation for Wireless ... Space-Time Parameter Estimation for Wireless Communication ... by Watkins Johnson Company. ... Estimation for Wireless Communication