TST-MUSIC for joint DOA-delay estimation - Signal ... - IEEE Xplore

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 4, APRIL 2001

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TST-MUSIC for Joint DOA-Delay Estimation Yung-Yi Wang, Jiunn-Tsair Chen, Member, IEEE, and Wen-Hsien Fang, Member, IEEE

Abstract—In this paper, a multiple signal classification (MUSIC)-based approach known as the time-space-time MUSIC (TST-MUSIC) is proposed to jointly estimate the directions of arrival (DOAs) and the propagation delays of a wireless multiray channel. The MUSIC algorithm for the DOA estimation is referred to as the spatial-MUSIC (S-MUSIC) algorithm. On the other hand, the temporal-MUSIC (T-MUSIC), which estimates the propagation delays, is introduced as well. Making use of the space-time characteristics of the multiray channel, the proposed algorithm—in a tree structure—combines the techniques of temporal filtering and of spatial beamforming with three one-dimensional (1-D) MUSIC algorithms, i.e., one S-MUSIC and two T-MUSIC algorithms. The incoming rays are thus grouped, isolated, and estimated. At the same time, the pairing of the estimated DOA’s and delays is automatically determined. Furthermore, the proposed approach can resolve the incoming rays with very close DOAs or delays, and the number of antennas required by the TST-MUSIC algorithm can be made less than that of the incoming rays. Index Terms—DOA-delay estimation, MUSIC, wireless communication.

multiray

channel,

I. INTRODUCTION

T

HE DOA-delay estimation is a classical problem encountered in radar, sonar, and geophysics. It also finds applications in source localization, accident reporting, cargo tracking, and intelligent transportation [1]. Furthermore, in a multiray wireless communication system, one can obtain a better channel estimate by jointly exploring the ray DOAs and the ray propagation delays, thus significantly improving the system performance [2], [3]. Some algorithms for joint estimation of the DOAs and the multiray propagation delays were suggested recently. For example, Swindlehurst et al. [4]–[6], proposed several computational efficient algorithms for the estimation of the delays of a multiray channel and solved the spatial signatures (or DOAs) as a least square problem. Clark et al. [7] proposed a two-dimensional (2-D) IQML algorithm that could be extended to jointly estimate the channel parameters. All of the algorithms proposed in [4]–[7] take advantage of the Vandermonde structure of the estimated channel pulse response in the frequency domain. However, if two or more rays have close time delays, Manuscript received February 28, 2000; revised December 29, 2000. This work was supported by the National Science Council of the R.O.C. under Contract 89-2213-E-011-106. The associate editor coordinating the review of this paper and approving it for publication was Dr. Lal C. Godara. Y.-Y. Wang is with the Department of Electronic Engineering, St. John’s and St. Mary’s Institute of Technology, Taipei, Taiwan, R.O.C. (e-mail: [email protected]). J.-T. Chen is with the Department of Electrical Engineering, National Tsing-Hua University, Hsinchu, Taiwan, R.O.C. (e-mail: [email protected]. tw). W.-H. Fang is with the Department of Electronic Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. (e-mail: [email protected]). Publisher Item Identifier S 1053-587X(01)02240-1.

the data covariance matrix turns ill-conditioned, and these algorithms may not work properly, even though these rays possess diverse DOAs. In addition, the IQML-based algorithms suffer from the initialization problems. Based on the knowledge of the transmitted signals, Bertaux et al. [8] developed a PML technique that uses the iterative Gauss–Newton procedure to estimate the spatial–temporal parameters of a multipath channel. However, as a consequence of stacking of the observed data matrix into a high-dimensional vector, Bertaux et al.[8] call for enormous computational overhead. Among other alternatives are the subspace-based algorithms. Ogawa et al. [9] presented a channel sounding method using unmodulated carriers. The parameter pairs were then extracted by invoking a 2-D windowed MUSIC algorithm. Vanderveen et al. proposed the JADE-MUSIC [10] algorithm, which exploited the properties of the space-time structure by stacking the received data. After a high-dimensional eigendecomposition on the covariance matrix is performed, the channel parameters can be estimated by a 2-D searching on the DOA-delay plane. The computations required, however, make JADE-MUSIC also unfavorable for real-time implementation. To lower the computational overhead, the SI-JADE and JADE-ESPRIT algorithms [11]–[13] were advocated, which utilized shift invariance property of the estimated channel matrix. By stacking the submatrices of the estimated channel matrix, the SI-JADE transforms the joint estimation problem into a matrix pencil problem. It follows that the rank-reducing numbers of the matrix pencil are the corresponding DOAs/delays. The JADE-ESPRIT algorithm is similar to the SI-JADE, except that the former uses multiple channel estimates, whereas the latter uses only one channel estimate. As a result, the JADE-ESPRIT usually performs much better than the SI-JADE in terms of accuracy. In this paper, we present a low complexity, yet high accuracy, MUSIC-based algorithm [14]—the TST-MUSIC algorithm—which combines the techniques of temporal filtering and of spatial beamforming with three one-dimensional (1-D) MUSICs, i.e., one S-MUSIC and two T-MUSIC algorithms to jointly estimate the DOAs-delays of interest based on the data samples received from an antenna array. The basic idea behind the proposed approach is to group and isolate the signal of each incoming ray using the space-time characteristics of the multiray wireless channel. To achieve this, the T-MUSIC algorithm and the S-MUSIC algorithm are first employed to estimate the group delays1 and the DOAs, which are used for the following processes of temporal filtering and of spatial beamforming, respectively. Thereafter, the other T-MUSIC algorithm is employed to estimate the ray delays. The proposed approach possesses some distinctive features. First, compared with the algorithms mentioned in [4]–[7], the 1Differing from the ray delay, the group delay is defined as the average delay of a group of rays.

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tree-structured TST-MUSIC algorithm not only inherently resolves incoming rays with either very close DOA’s or very close delays but also renders automatic pairing of the estimated DOAs and delays. In addition, due to the employment of the temporal filtering process, the number of antennas required by the TST-MUSIC can be less than that of the incoming rays. Second, in contrast to the JADE-MUSIC algorithm [10], the TST-MUSIC algorithm needs only one-dimensional (1-D) searches and their associated eigendecompositions for smaller-sized covariance matrices. Thus, it calls for substantially lower computational complexity. Third, compared with the SI-JADE and JADE-ESPRIT algorithms [11], [12], the proposed approach performs much better in estimation accuracy, especially in a low signal-to-noise ratio (SNR) environment. This paper is organized as follows. Section II introduces the system model of the fading multiray channels, which assume the propagation rays to be specular rather than dispersed. In Section III, the S-MUSIC algorithm [14] and the T-MUSIC algorithm [5] are reviewed, followed by the proposition of the TST-MUSIC algorithm. The issue of computational complexity is addressed as well. In Section IV, simulation results are presented to verify the performance of the proposed approach. Section V provides a concluding remark to summarize the paper. II. DATA MODEL In this paper, we assume a TDMA wireless system, such as the IS-136 [15] and GSM [16], as our target application system. The radio channel in a wireless communication system is often characterized by a multiray propagation model. In large cells with high base station antenna platforms, the propagation environment is aptly modeled by a few dominant specular rays—typically 2 to 6. In such a case, the baseband signals received at the antenna array during the th burst can be expressed as follows [17]:

(1) where received baseband signals in the th time burst; spatially and temporally white additive Gaussian noises with zero-means and equal variances ; normalized steering vector for all of a signal arriving from direction ; ray amplitude that is a complex Gaussian random process; transmitted complex baseband signal; propagation delay of the th ray; total number of rays present in the system. In a linear time-invariant system, the transmitted signal can be represented as a convolution of the data bits and the pulse-shaping function

where is the symbol period. Therefore, after sampling under , the signal received during the th time burst can be a rate written as

(2)

where is the sampling reference time of the th data burst, and represents the matrix transpose operation. In the superscript consecutive samples in time are considered, where (2), number of antennas; maximum length of the channel pulse response divided by the symbol period ; oversampling factor. The subscripts of the matrices, which are neglected in the rest of this paper to simplify the notation, describe the size of each corresponding matrix. The array re, where sponse matrix denotes the array response vector of the th ray; diag with being the complex fading amplitudes of the th ray during the th burst; with , is the time delay of the th incoming ray. As a where result of the convolution between the data bits and the pulse is a Toeplitz matrix shaping function, the data matrix as its first column with as its first row, where and is the transmitted data bits of the th burst, and is a . The noise matrix of the zero vector of dimension th burst is assumed to be spatially and temporally white Gaussian noise with noise power . Note that in this paper, and are assumed constants among data bursts. In a TDMA system, a known training sequence is usually embedded in each transmitted burst. We denote the data matrix , independent of formed by the training sequence as . Specifically, is regarded as prior information in the proposed approach to estimate the channel parameters. Extracting the training portion from (2), we have (3) , and is the convolution between the training sequence and the time-shifted pulse-shaping function. Similar to the normalfor ization of the array steering vector, we assume and . We define all by adjusting power between as the normalized temporal array vector. The dimension of is , where is the length of the extracted training sequence. In the applications of wireless communication, the fading rays are usually assumed to be mutually uncorrelated, and their fading amplitudes are assumed to be zero-mean complex where

WANG et al.: TST-MUSIC FOR JOINT DOA-DELAY ESTIMATION

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Gaussian distributed [18]. Hence, the covariance matrix of the is fading vector diag

(4)

and (5) represents the statistical average operation, the suwhere denotes the Hermitian operation, and is the perscript average signal power of ray . Since is a known Toeplitz ma. trix, the only stochastic term in (3) is the fading matrix as the We refer to the covariance matrix of the columns of , which is given by spatial covariance matrix

and (9) and are the eigenvectors that span The column vectors of and , respectively, corresponding the signal subspace of and , to the largest eigenvalues. The column spaces of and eigenvectors of spanned by the rest of the and , are the orthogonal complement of the column spaces and , respectively. Both ( , ) and ( , ) are of diagonal matrix pairs with the associated eigenvalues as their diagonal elements. Using the orthogonality property between the signal and the noise subspaces, the S-MUSIC estimates the DOAs by (10)

(6)

Similarly, the T-MUSIC algorithm estimates the ray delays by

is used to simplify the notation. From where instead of (6), after the exclusion of the noise subspace, the spatial covariand the spatial signature matrix share the ance matrix same column space. are the temporal sampling vectors Similarly, the rows of of the received signals. We may express the temporal covariance as matrix

(11)

(7) denotes the complex conjugate operwhere the superscript ation. From (7), after the exclusion of the noise subspace, the and the temporal signature matemporal covariance matrix trix share the same column space. III. PROPOSED ALGORITHM In this section, we first briefly review the S-MUSIC algorithm [14] and the T-MUSIC algorithm [5], respectively, for the DOA estimation and for delay estimation. Next, the TST-MUSIC algorithm is proposed and a three-ray scenario is used as an example to illustrate the proposed algorithm.

After searching and over the range of interest, the spectrum of the S-MUSIC and that of the T-MUSIC are, respectively, defined as (12) and (13) Since the covariance matrix of the received signal is usually unavailable, the implementation of the MUSIC thus employs the sample covariance matrices instead. For example, if data bursts are observed, the sampled spatial covariance matrix and the sampled temporal covariance matrix are estimated, respectively, by [20] (14) and

A. S-MUSIC and T-MUSIC The S-MUSIC and T-MUSIC algorithms use the covariance , respectively, to matrices of the columns and the rows of estimate the ray DOAs and the ray delays. In the T-MUSIC, the temporal samples can be regarded as a kind of temporal antenna array. Compared with the commonly used antenna array for spatial sampling, adding extra antennas in the temporal array is cost-free, and the known pulse-shaping function provides a perfect temporal array manifold in which no calibration is required. Note that the number of the incoming rays is assumed to be known a priori; otherwise, one may estimate by thresholding the magnitude of the eigenvalues of the covariance matrices involved or by using the AIC and the MDL detection methods provided in [19]. Starting from the eigendecomposition of the covariance matrices, (6) and (7) can be, respectively, expressed as (8)

(15) To simplify the notation, we will ignore the burst index superof hereafter. script In the cases where the incoming rays contain only mild contamination and are neither close in ray DOAs nor in ray delays, the S-MUSIC and the T-MUSIC algorithms can, respectively, obtain a fairly precise estimate of the ray DOAs and delays. However, if the ray-gathering scenarios happen either spatially or temporally, the related signature matrix or in (3) tends to be ill conditioned. A hump, instead of several peaks, may occur in the MUSIC spectrum around each ray-gathering area, which thus blurs the estimation results. In a multiray environment, the ray-gathering scenarios may be overcome by using the space-time characteristics of the propagation channels. In the next section, we will describe how the proposed TST-MUSIC

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Fig. 2. Tree structure of the TST-MUSIC in solving the scenario shown in Fig. 1(a). Fig. 1. Evolution of the signal contents for the estimation of the parameters of ray one.

processes the blurred estimates to further separate the close-by rays.

temporal filtering process to separate the rays with delays from the ray with delay . As a result, two group matrices, and , are thus generated as which are denoted as

B. TST-MUSIC The rationale of the TST-MUSIC is to incorporate three 1-D MUSIC (S-MUSIC and T-MUSIC) algorithms with beamforming techniques and the filtering techniques to group, isolate, and then estimate and pair the 2-D parameters of a fading channel. To simplify the algorithm description, we first assume that there are only three rays present in the system. The general procedure of the TST-MUSIC is summarized at the end of this subsection. As shown in Fig. 1(a), three rays are characterized by their temporal–spatial coordinates on the DOA-delay plane. Note that ) but diverse ray 1 and ray 2 possess close time delays ( ), whereas ray 1 and ray 3 are close in the DOAs DOAs ( ) but with far-apart delays ( ). The tree structure ( of the TST-MUSIC algorithm for this scenario is illustrated in Fig. 2. In addition, corresponding to Fig. 2, Fig. 1 shows the evolution of the data contents as the parameters of ray 1 are estimated in the tree structure. The TST-MUSIC treats those temporally close rays, which flatten the T-MUSIC spectrum, as a group. Therefore, ray 1 and ray 2 in Fig. 1(a) are regarded as one group, whereas ray 3 is considered another group. By applying the T-MUSIC to the rows of , the resulting group delays are estimated and denoted by and . Based on the group delay estimates and , we define as the temporal filtering matrices and Note that (or ) is also the complement projection matrix [or ] with [or of ]. In the three-ray scenario shown in Fig. 1(a), we have , which implies that and , where the notation denotes the 2-norm of a vector. With these facts, the TSTMUSIC postmultiplies to , which is referred to as the

(16) and

(17) where “ ” in (16) and (17) means that the residue signals from ray 3 and from ray 1 and 2, respectively, are neglected. Discussions about the magnitude of the neglected residue signal will be given at the end of this section. It is shown in the Appendix that the transformed noise matrices in (16) and (17) are still temporally and spatially white within the projected subspace. Note that the two dominant rays contained in (16) have their . The DOA estimates and can thus be accuDOAs . Similarly, rately obtained by applying the S-MUSIC to is obtained by applying the S-MUSIC to . Note that , but the signal of ray 1 and that of ray 3 are separated into two different signal groups before the S-MUSIC is applied. Therefore, and can also be accurately estimated with the help of the temporal filtering following the first T-MUSIC. In addition, in the S-MUSIC, the estinote that right after estimating are determined and will be used in mated array vectors the spatial beamforming described below. To further divide each group matrix into several single-ray can be defined matrices, the spatial beamforming matrices as

for

, and


for

, respectively. Note that

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nulls the signal from ray as and

Similar to the temporal filtering process, the TST-MUSIC alby and , which is referred to gorithm premultiplies as the spatial beamforming process, to null the corresponding and are ray. It follows that two single-ray matrices formed, respectively, as

We then have and . This and why ray again explains why ray 3 can be neglected in . Similarly, the covariance 1 and ray 2 can be neglected in and are matrices of the rows of

(22) (18) and

(19) Again, the residues of ray 2 and ray 1 are neglected in (18) and (19), respectively. The noise matrices in (18) and (19) are again still temporally and spatially white within the projected and subspace. In (18) and (19), the single-ray structure of implies that the two rays with close ray delays (ray 1 and ray 2) are separated into different subgroups by the spatial beamforming process. As a result, by applying the T-MUSIC and , ray delay estimates and can algorithm to then be accurately estimated, respectively. It also follows that and is automatically achieved. the pairing of Note that in the process of the second T-MUSIC algorithm, the , as shown in (18) and temporal array vector should be . Furthermore, since the group-delay infor(19), rather than mation is known, the searching region shrinks to the vicinity of instead of the whole axis. For the other branch of the , only the single ray is found. Theresignal with respect to and fore, no spatial beamforming is needed, that is, . Next, we investigate the magnitude of the residue signals neglected in (16)–(19). It is obvious that these residue signals are the results of the leakage in the filtering and in the beamforming processes. For example, in (16), the neglected residue signal for is equal to . Similar terms also exist in , , and . By including the residue signals, the cocan be expressed as variance matrices of

. In (22), the modified temporal array vector is for , and the defined as average signal power of the th ray in the th subgroup after beamforming is (23) and . Thus, we have , , and , for the three-ray scenario. Note that the signals from ray 1 and ray 2 are thus isolated. Finally, the covariance matrix of the signal from ray 3 is given by

for

(24)

for . where as shown earlier, signals from ray 3 Because remain isolated since the temporal filtering is applied. The above discussion can be readily extended to more general rays distributed in cases. Suppose that in total, there are temporal groups and that the number of rays contained in the th group is denoted by , where is assumed to be known a priori. In addition, assume that the rays are not close in both of the DOAs and delays. The TST-MUSIC described above can be generalized as follows. C. TST-MUSIC Algorithm

(20)

Step 1) Grouping: Applying the T-MUSIC algorithm to the . received , we obtain the group delays Step 2) Temporal Filtering: The temporal filtering matrices are generated by (25)

where

whereas the output of the th temporal filter is given by (21)

is the average signal power of the th rays in the th group after and . Assume that the varifiltering for ances of these three fading amplitudes are equal before filtering.

(26) for

.

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Step 3) DOA Estimation: Applying the S-MUSIC algo, we estimate the DOAs of the rays in group rithm to each , as given by

for , where is the number of rays in the th group, and . Step 4) Spatial Beamforming: The spatial beamforming is obtained by matrix (27) whereas the signal at the output of the is given by

th spatial beamformer

(28) and . for Step 5) Delay Estimation: Apply the T-MUSIC algorithm again but with different temporal array manifolds to each . If single ray is in the th subgroup and is obtained, pair it with ; . otherwise, pair with Remarks: 1) Since the DOA of each ray is estimated in the group data , the number of antennas required in the TSTmatrices MUSIC is less than that of rays. More specifically, in an antennas uncorrelated propagation environment, are required in the S-MUSIC algorithm to identify rays [21]. As a result, the minimum number of antennas required by the TST-MUSIC algorithm for estimating rays is provided

(29)

rays have distinct delays, we have If all the for all ’s. In such a scenario, only two antennas are needed for the TST-MUSIC algorithm. On the other hand, groups can be resolved by the T-MUSIC since antennas are employed, the TST-MUSIC algorithm, if algorithm can identify a maximum number of rays. 2) The computational complexity of the TST-MUSIC and , includes a) the eigendecomposition of and ; b) the respectively, of orders formulation of the temporal filtering matrices and the spatial beamforming matrices, respectively, of orders and ; and c) the 1-D spatial and temand poral searches, respectively, of orders , where and are the numbers of searches conducted along the DOA axis and the time delay axis. In general, the length of the temporal array is greater ). We than that of the antennas employed, (i.e., thus conclude that the computational complexity of the . On the other TST-MUSIC is hand, the JADE-MUSIC requires eigendecomposition

on covariance matrices of order and the . It is obvious 2-D searches of order that the computational burden of the TST-MUSIC is substantially less than that of the JADE-MUSIC. However, the computational complexity of the SI-JADE and JADE-ESPRIT are much lower than that of the proposed algorithm since no parameter searching is required in them. 3) In general, wireless communication channels might be much more complicated than the three-ray example described earlier. Depending on the antenna array geometry and the signal bandwidth, respectively, the 1-D S-MUSIC and the 1-D T-MUSIC can reach the resolution of the DOAs and of delays only to a certain extent, whereas the TST-MUSIC can resolve extremely close rays either in the DOAs or in delays as long as these rays are not close in both parameters. As for the scenario where both the DOA’s and delays are very close, the proposed algorithm will obtain estimate of only one path. It also has been shown that distinguishing close rays in both DOA’s and in delays does not help improve the data demodulation accuracy at the receivers in a wireless communication system [22]. In other words, to maintain the diversity of the wireless channel, we have to distinguish a) rays with close DOA’s but with far-apart delays or b) rays with very close delays but with far-apart DOAs. The TST-MUSIC is developed with these goals in mind to maintain the channel diversity. . 4) We do not assume perfect knowledge of either or Just as with the standard MUSIC algorithm, the proposed algorithm may overestimate either the number of groups or that of paths within each group. The proposed algorithm will still work fine if we re-estimate the number of groups or that of paths in each group from the peaks of the are smaller MUSIC spectrum. Note that both and than or equal to , which means, for the worst case, if we assume groups and paths in each group, the proposed algorithm is guaranteed to work. IV. SIMULATIONS AND DISCUSSIONS In this section, we conduct several simulations to evaluate the TST-MUSIC algorithm. We assume narrowband signals ) and received that are transmitted through four rays ( ). The anby a three-element uniform linear array ( tennas are of equal gains and are spaced a half wavelength apart corresponding to the carrier frequency. Assuming the GSM system model [16], the GMSK modulated signals are is sampled during 20 data tested. The received signal bursts. In the basic setup, we set the angles of arrival to be , and the propagation delays to be , where s is the symbol , period of the GSM system. The oversampling factor and the training sequence of each burst was truncated from the sixth training bit to the 21st training bit to keep the samples of the training sequence from being corrupted by the data bits. The average fading amplitudes of the four rays are equal and normalized to 0 dB with randomly selected but constant fading


Fig. 3. (a) MUSIC spectrum of the S-MUSIC in the TST-MUSIC. (b) MUSIC spectrum of the T-MUSIC in the TST-MUSIC.

phases. The average power of the additive Gaussian noise is adjusted to achieve the required SNR. Fig. 3(a) demonstrates the S-MUSIC spectra for the group and . Fig. 3(b) shows the T-MUSIC spectra data matrices for the received data matrix and the single-ray data matrices , , , and . All of the group and the single-ray matrices are generated by the temporal filtering process and the spatial beamforming process during the procedure of the TST-MUSIC algorithm. As shown in Fig. 3(a), the TST-MUSIC precisely estimates the DOAs of the two closely spaced rays. Similarly, Fig. 3(b) shows that the rays with close time delays are successfully isolated. Fig. 4 compares the root-mean-square-error (RMSE) of the DOA and the delay estimates of the TST-MUSIC, the JADEESPRIT, and the JADE-MUSIC algorithms with respect to the SNRs from 0 dB to 27 dB. For each specific SNR, 200 Monte Carlo trials are conducted. As shown in Fig. 4, the TST-MUSIC outperforms the JADE-MUSIC (2-D MUSIC) and the JADEESPRIT at low SNR. This is because the JADE-MUSIC and the ( ) data matrix into an JADE-ESPRIT stack each

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Fig. 4. (a) Comparison of the root mean square error of the estimated DOA’s between TST-MUSIC, JADE-ESPRIT, and JADE-MUSIC. (b) Comparison of the root mean square error of the estimated delays between TST-MUSIC, JADE-ESPRIT, and JADE-MUSIC.

snapshot vector, and 20 observation data bursts can provide only 20 snapshot vectors for them. Thus, the sample covariance matrice of the JADE-MUSIC and the JADE-ESPRIT are quite noisy. On the other hand, 20 bursts can offer spatial snapshot vectors for the S-MUSIC and temporal snapshot vectors for the T-MUSIC to estimate the associated sample covariance matrix. However, the RMSE curves of the TST-MUSIC become flat at high SNRs, where the residue signals caused by the temporal filtering and spatial beamforming processes dominate the RMSE of the TSTMUSIC performance. Fig. 5 is plotted to illustrate how the close-by rays affect the parameter estimation in the TST-MUSIC. Fig. 5 shows the RMSE of the estimated delays and the DOAs of the four rays in a noise-free environment with setting to . As shown in Fig. 5, larger results in higher RMSE because worse group delays are provided in the TST-MUSIC.

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Fig. 5. Root mean square errors of the TST-MUSIC estimates.

Fig. 6 illustrates the parameter estimates obtained by three algorithms: the JADE-ESPRIT, the JADE-MUSIC, and the TST-MUSIC. Each point on the DOA-delay [or time of arrival dB. (TOA)] plane represents an independent trial with These figures again manifest the superiority of the TST-MUSIC to the other two algorithms in a highly contaminated environment. Note that in all the simulation scenarios above, we assume time-varying channels, which make the temporal array vectors inaccurate and result in higher RMSE than expected. However, the simulation results prove the resolvability of the proposed TST-MUSIC. V. CONCLUSIONS This paper proposes a novel algorithm—the TSTMUSIC—which combines three 1-D MUSICs along with the temporal filtering techniques and the spatial beamforming techniques to jointly estimate the DOAs and the delays of the multiple rays in a wireless channel. The S-MUSIC and the T-MUSIC algorithms in the TST-MUSIC are used to estimate the DOA’s and the propagation delays, respectively. The TST-MUSIC is biased due to the propagation of the residue signals, which come from the leakage of the spatial beamforming process and of the temporal filtering process, whereas the amount of the bias depends on how accurate the group delays and the DOAs are estimated. Compared with the JADE-ESPRIT, the TST-MUSIC algorithm is poor in computational complexity but produces considerably fewer estimation errors. Furthermore, the TST-MUSIC is far less complex than JADE-MUSIC, using only a few observation bursts under a highly contaminated environment. APPENDIX NOISE PROPERTY IN (16) in (16) is spatially In this Appendix, we prove that and temporally white with unchanged noise power within the , i.e., within the projected subspace, where row space of is a projection matrix as defined in Section III-B.

Fig. 6. (a) Distribution of the JADE-ESPRIT estimates. (b) Distribution of the JADE-MUSIC estimates. (c) Distribution of the TST-MUSIC estimates.

Proof: Let noise vector, and

, where

is a . We then have


The correlation of

with and

729

. is given by

where is the Kronecker delta function. It thus proves that the transformed noise matrix is spatially white. Furthermore, is the th element of the projection matrix , since the noise is temporally white with the same power within the projected subspace. ACKNOWLEDGMENT The authors would like to thank the reviewers for many useful comments and suggestions, which have enhanced the quality and readability of this paper. REFERENCES [1] T. S. Rappaport, J. H. Reed, and B. D. Woerner, “Position location using wireless communications on highways of the future,” IEEE Commun. Mag., Oct. 1996. [2] J.-T. Chen, A. Paulraj, and U. Reddy, “Multi-channel MLSE equalizer for GSM using a parametric channel model,” IEEE Trans. Commun., pp. 53–63, Jan. 1999. [3] J.-T. Chen, J. Kim, and J. Liang, “Multi-channel MLSE equalizer with parametric FIR channel identification,” IEEE Trans. Veh. Technol., vol. 48, pp. 1923–1935, Nov. 1999. [4] A. L. Swindlehurst, “Time delay and spatial signature estimation using known asynchronous signals,” IEEE Trans. Signal Processing, vol. 46, pp. 449–461, Feb. 1998. [5] A. Jakobsson, A. L. Swindlehurst, and P. Stoica, “Subspace-based estimation of time delays and doppler shifts,” IEEE Trans. Signal Processing, vol. 46, pp. 2472–2483, Sept. 1998. [6] A. L. Swindlehurst and J. H. Gunther, “Methods for blind equalization and resolution of overlapping echos of unknown shape,” IEEE Trans. Signal Processing, vol. 47, pp. 1245–1254, May 1999. [7] M. P. Clark and L. L. Scharf, “Two-dimensional modal analysis based on maximum likelihood,” IEEE Trans. Signal Processing, vol. 42, pp. 1443–1452, June 1994. [8] N. Bertaux, P. Larzabal, C. Adnet, and E. Chaumette, “A parameterized maximum likelihood method for multipaths channels estimation,” in Proc. Signal Process. Adv. Wireless Commun., May 1999, pp. 391–394. [9] Y. Ogawa, N. Hamaguchi, K. Ohshima, and K. Itoh, “High-resolution analysis of indoor multipath propagation structure,” IEICE Trans. Commun., vol. E78B, pp. 1450–1457, Nov. 1995. [10] M. C. Vanderveen, C. B. Papadias, and A. Paulraj, “Joint angle and delay estimation (JADE) for multipath signals arriving at an antenna array,” IEEE Commun. Lett., vol. 1, pp. 12–14, Jan. 1997. [11] A. J. van der Veen, M. C. Vanderveen, and A. Paulraj, “Joint angle and delay estimation (JADE) using shift-invariance techniques,” IEEE Signal Processing Lett., pp. 142–145, May 1997. [12] M. C. Vanderveen, A. J. van der Veen, and A. Paulraj, “Estimation of multipath parameters in wireless communications,” IEEE Trans. Signal Processing, vol. 46, pp. 682–690, Mar. 1998. [13] A. J. van der Veen, M. C. Vanderveen, and A. Paulraj, “Joint angle and delay estimation (JADE) using shift-invariance properties,” IEEE Trans. Signal Processing, vol. 46, pp. 405–418, Feb. 1998.

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Yung-Yi Wang was born in Yunlin, Taiwan, R.O.C., in 1964. He received the B.S. degree from the Department of Electronic Engineering, National Taiwan University of Science and Technology (NTUST), Taipei, in 1989, the M.S. degree from the Department of Electrical Engineering, University of Massachusetts, Lowell, in 1993, and the Ph.D. degree from the Department of Electronic Engineering, NTUST, in 2000. In the Fall of 1994, he joined the faculty of St. John’s and St. Mary’s Institute of Technology, Taipei, where he is currently an Associate Professor with the Department of Electronic Engineering. His research interests include statistical signal processing, array signal processing, and wireless communications.

Jiunn-Tsair Chen (M’99) was born in Taiwan, R.O.C., on April 20, 1964. He received the B.S. degree from National Chiao-Tung University, Hsinchu, Taiwan, in 1986, the M.S. degree from National Taiwan University, Taipei, in 1989, and the Ph.D. from Stanford University, Stanford, CA, in 1998, all in electrical engineering. Since August 1999, he has been with National Tsing-Hua University, Hsinchu, as an Assistant Professor. His current research interests are in wireless communications, antenna array signal processing, adaptive digital signal processing, and power amplifier linearization.

Wen-Hsien Fang (S’88–M’91) was born in Taipei, Taiwan, R.O.C., in 1961. He received the B.S. degree from National Taiwan University, Taipei, in 1983 and the M.S.E. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 1988 and 1991, respectively, all in electrical engineering and computer science. In the Fall of 1991, he joined the faculty of National Taiwan University of Science and Technology, Taipei, where he is a Professor with the Department of Electronic Engineering. His research interests include signal processing for wireless communications, adaptive signal processing, fast algorithms and their VLSI implementations, and video coding.