12 IEEE COMMUNICATIONS LETTERS, VOL. 1, NO. 1, JANUARY 1997

Joint Angle and Delay Estimation (JADE) forMultipath Signals Arriving at an Antenna Array

Michaela C. Vanderveen, Constantinos B. Papadias,Student Member, IEEE,and Arogyaswami Paulraj,Fellow, IEEE

Abstract—We propose a novel subspace approach to esti-mate the angles-of-arrival and delays of multipath signals fromdigitally modulated sources arriving at an antenna array. Ourmethod uses a collection of estimates of a space–time vectorchannel. The Cramer–Rao bound (CRB) and simulations areprovided.

I. INTRODUCTION

I N WIRELESS communications, mobiles emit signals thatarrive at a base station via multiple paths. Estimating

each path’s angle-of-arrival (AOA) and propagation delay isnecessary for several applications, such as mobile localizationfor emergency services. It is in fact a classical radar problem.This work proposes a novel approach (JADE) to estimating theAOA’s and delays of the multipath signals using a collectionof space–time channel estimates (analogous to snapshots insubspace methods for AOA estimation) which have constantparameters of interest but different path fade amplitudes.JADEcan work in cases when the number of paths exceeds thenumber of antennas, unlike the traditionalMUSICandESPRITalgorithms [1], [2].

II. DATA MODEL

We focus on the case for a single user first and show laterhow this approach can be extended to multiple users. Thereceived baseband signal at theth element of an -elementantenna array is given by

(1)

where is the number of multipaths, is the responseof the th antenna to theth path arriving from angle ,is the complex envelope of the path fading, is the pathdelay, is the additive noise, and is the transmittedsignal, given by , where is thesequence of data bits, is the modulation waveform, and

is the symbol period.We collect all the sensor responses tointo an -element

vector and similarly the received

Manuscript received September 12, 1996. The associate editor coordinatingthe review of this letter and approving it for publication was Dr. Y. Bar-Ness.

M. C. Vanderveen is with Scientific Computing Program, Stanford Univer-sity, Stanford, CA 94305-9025 USA (e-mail: vandervn@sccm.stanford.edu).

C. B. Papadias and A. Paulraj are with Information Systems Laboratory,Stanford, CA 94305 USA.

Publisher Item Identifier S 1089-7798(97)01345-8.

signals into a vector such that

(2)

We sample this signal at the symbol rate (i.e., instants).Let be the number of the symbol-spaced samples of thechannel impulse response, , where is thesymbol waveform duration and is the maximum integerpath delay. We obtain

(3)

where the th element of the vector of data isand is an channel matrix capturing the

effects of the array response, delay, symbol waveform, andpath fading, and taking the form:

......

(4)

whereis a -long row vector of samples of . Since weassume the path fadings to be constant within a data burst,we have suppressed their dependence on the sampling instant

. We assume we know the number of multipaths,1 themaximum path delay , the modulation waveform , andthe structure of the array response .

III. JADE

Let vect be a vector of length obtained bytaking the transpose of each row of the matrixand stackingit below the transpose of the previous row. Then

diag (5)

where denotes the Khatri–Rao (columnwise Kronecker)matrix product (that is, theth column of is

, where denotes the Kronecker product). Thematrix is called the space–time matrix, and is

parametrized by the AOA’s and the path delays. The vectoris called space–time response vector to

a path of unit amplitude arriving at anglewith delay . As

1The number of multipaths can be estimated from the data matrix, but thisis beyond the scope of this paper.

1089–7798/97$10.00 1997 IEEE

VANDERVEEN et al.: JADE FOR MULTIPATH SIGNALS 13

varies over the range of angles andvaries over the range ofdelays, traces a multidimensionalspace–time manifold.

In this paper, the radio channel from the mobile to theantenna array is time-slotted, modeled after the GSM standard.The channel from the mobile to the antenna array can thusbe assumed to be constant over each time slot, but it variesfrom one time slot to the next. This variation is due to thechanging complex fadings . However, the AOA’s anddelays are not changing significantly from each time slot tothe next, and thus we take to be constant over a fewtime slots. This assumption is reasonable in practice becausewe consider only a relatively small number of time slots andthus during this very short time the mobile, which is far away,appears to be almost stationary with respect to the base station.

The first step in our approach consists of estimating thechannel impulse response from the user to the antenna array.This can be accomplished by using training bits or blindly.We collect data from consecutive time slots and use it toobtain estimates of . If we let be the time slot index, ourestimates of the true channel take the form:

(6)

where is the estimation noise matrix. Applying the vectoperation yields, with the obvious notation,

(7)

or, if we let and similarly for ,

(8)

The second step in our approach consists of estimating theparameters of interest, namely’s and ’s, and eliminating

the nuisance parameters’s. In other words, given thenoisy estimated matrix and the known structure of thespace–time matrix , we seek the desired parametersaccording to (8). We assume that the space–time manifolddoes not have any ambiguities, therefore, leading to uniqueestimates (this is the case whenever either the array or de-lay manifold are unambiguous). In order for an angle-delaysubspace to exist, we also need to be a tall matrix (i.e.,

). It can also be easily shown that the space–timechannel matrix corresponding to a collection of distinctpaths is full column rank.

IV. A LGORITHMS AND THE CRAMER–RAO BOUND

The second step in our approach consists of estimating themultipath parameters from the estimated channel, using(8). Among the various ways to solve this problem, we focuson two of them.

A. Maximum Likelihood

We will assume that the estimation noise is white andGaussian, a fact which can be readily proved for the case ofnonblind channel estimation. The entries of the complex fadingmatrix can be modeled as unknown deterministic quantities.Then employing deterministic maximum likelihood techniques

Fig. 1. MUSIC-like spectrum.

Fig. 2. Standard deviation of estimates versus CRB.

yields the following minimization problem:

(9)

It is well known that this is a separable optimization prob-lem that reduces to tr , where

, and denotes thecomplex-conjugate transpose. This search can be done usingthe damped Newton method.

B. JADE–MUSIC

The technique described above involves a-dimensionalsearch and may thus be computationally prohibitive. A faster,though suboptimal, approach is based on theMUSICalgorithmand involves only a two-dimensional search. We know thatthe true space–time channel vector is orthogonal tothe noise subspace , whose columns are the eigenvectorsof corresponding to the smallest eigenvalues. Wethus look for peaks in the two-dimensionalMUSIC spectrum

. The peaks should occur close to the truecoordinates (see Fig. 1).

C. Cramer–Rao Bound

The Cramer–Rao bound (CRB) provides a lower bound onthe variance of any unbiased estimator. The bound for AOAestimation (without delay spread) was derived in [3] and isreadily adapted to the present situation. Assuming the pathfadings to be deterministic but unknown, we obtain for themodel in (8) that

real (10)

where is the variance of the estimation noise,diag , and(prime denotes differentiation with respect to the individualparameters and all matrices are evaluated at the true parametervalues).

14 IEEE COMMUNICATIONS LETTERS, VOL. 1, NO. 1, JANUARY 1997

V. SIMULATION RESULTS AND EXTENSIONS

Computer simulations were run to demonstrate the perfor-mance of theJADE–MUSICalgorithm. We assume a singleuser, three multipaths, and a two-element antenna. The classi-cal approaches to this problem, such asMUSIC andESPRIT,will not yield satisfactory results, since the number of antennasis smaller than the number of multipaths. However, theJADEalgorithm can handle this case successfully.

The AOA’s are [-5, 0, 20] relative to the array broadsideand the corresponding path delays are [1.0, 0.7, 2.0]seconds,where is normalized to one. The collected data are corruptedby noise with inverse variance and dB.The modulation waveform is a raised cosine pulse with excessbandwidth 0.35, assumed to be zero outside the interval

. We sample at rate (the purpose of oversampling isto provide improved definition of the delay manifold). Data iscollected over 20 time slots, and at each time slot the channelis estimated via least squares using 27 training bits. The exper-imental variance of the AOA and delay estimates is computedfrom 100 runs. The results are summarized in Fig. 2. The biasof the estimates was on the average 1%–27% or their standarddeviation (STD). Notice the STD is about 5 dB above the CRB.

A typical MUSIC spectrum for noise with inverse variance20 dB is shown in Fig. 1.

When we have more than one user in the same time slot,we can independently estimate the channel matricesusingeach user’s unique (usually orthogonal) training signal. Wecan then proceed as above, with decoupled problems. If notraining signals are available, we can still find the channelfor each user using blind methods, which exploit finite alphabetstructures and oversampling [4].

REFERENCES

[1] R. O. Schmidt, “A signal subspace approach to multiple emitter loca-tion and spectral estimation,” Ph.D. dissertation, Stanford University,Stanford, CA, Nov. 1981.

[2] R. Roy, A. Paulraj, and T. Kailath, “ESPRIT—A subspace rotationapproach to estimation of parameters of cisoids in noise,”IEEE Trans.Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 1340–1342, Oct.1986.

[3] P. Stoica and A. Nehorai, ‘‘MUSIC, maximum likelihood andCramer–Rao bound,’’IEEE Trans. Acoust., Speech, Signal Processing,vol. 37, pp. 720–741, May 1989.

[4] A. van der Veen, S. Talwar, and A. Paulraj, ‘‘Blind identification ofFIR channels carrying multiple finite alphabet signals,’’ inProc. IEEEICASSP, vol. 2, 1995, pp. 1213–1216.