Angle and Delay Estimation of Space–Time Channels ... - IEEE Xplore

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

Angle and Delay Estimation of Space–Time Channels for TD-CDMA Systems José Picheral and Umberto Spagnolini, Senior Member, IEEE

Abstract—In mobile communications the antenna arrays make it possible to estimate the path delay, angle of arrival (or angle), and amplitude for the multipath propagation channel. This paper considers the problem of joint angle and delay estimation (JADE) in a time-division code-division multiple access system. The angle/delay estimation is made from multiple slots by exploiting the angle/delay invariance of the channel regardless of the fast faded amplitude variation (i.e., angle/delay is assumed quasistatic). JADE can be approximated by methods that estimate the frequencies of the two-dimensional complex sinusoids where the faded amplitudes act as sources. The shift invariance method (JADE-ESPRIT) is chosen for its capacity to pair delay and angle for each path and each user. The consistency of JADE-ESPRIT for a large enough number of slots guarantees the feasibility of the method for spatially correlated noise. The performance of the JADE method for channel estimation is evaluated analytically and numerically in propagation environments with increasing complexity. Index Terms—Array processing, code-division multiple access (CDMA), delay estimation, direction of arrival, location methods, multiuser detection, space–time channel estimation, third generation system, time-division code division multiple access (TD-CDMA), time-division multiple access (TDMA).

I. INTRODUCTION

I

N MOBILE communications the propagation channel is characterized by multipaths that are made time varying by mobile terminal movements, each path being described by delay, angle of arrival (or angle), and amplitude. Parametrization of the multipath channel in terms of angle/delay is closely related to the physics of the propagating field. Usually the time variation in angle/delay is slow compared to amplitude variations so that channel angle/delay can be considered quasistatic over long time intervals. In this paper, we exploit the angle/delay invariance to improve channel estimation accuracy in time-slotted systems such as hybrid time-division multiple access (TDMA)–code-division multiple access (CDMA) (e.g., UTRA TDD [1] or TD-SCDMA [2] systems), thus showing that (asymptotically for long time intervals) the resulting

Manuscript received July 4, 2001; revised August 12, 2002, and February 17, 2003; accepted February 18, 2003. The editor coordinating the review of this paper and approving it for publication is K. Wilson. This work was partially supported by SIEMENS Mobile Communications and by Ministero dell’Istruzione dell’Universita’ e della Ricerca (project VICOM). This paper was presented in part at VTC’01 Spring, Rhodes May 6–9, 2001. J. Picheral is with the Ecole Supérieure d’Electricité (SUPELEC), Gif sur Yvette, France (e-mail: [email protected]). U. Spagnolini is with the Dipartimento Elettronica e Informazione, Politecnico di Milano, I-20133 Milano, Italy (e-mail: Umberto.Spagnolini@ elet.polimi.it). Digital Object Identifier 10.1109/TWC.2004.826320

estimation error depends only on the accuracy of the estimation for the fast-varying amplitudes. The estimation of angle and delay has been widely investigated and the field is currently one of continuous innovation. Maximum likelihood estimation (MLE) methods [3] are too expensive to be carried out online. Sequential methods [4], [5] estimate the angles at first and then, after appropriate beamforming, the delays (or the temporal channel). The path resolution of sequential methods depends on the capability to resolve paths in angle and time. Methods for joint angle/delay estimation (JADE) were proposed in [6]–[10]. Recently, JADE was exploited for channel estimation in TDMA systems [11]. In JADE, the path resolution is limited by the resolution in the angle-time domain (i.e., paths can be resolved when they are separated in at least one of the two domains, either angle or time). The advantages of JADE can be exploited by CDMA systems where receivers exploit both angular and temporal diversity in multipath channels. Benefits have been demonstrated for two–dimensional (2-D) RAKE [12] and linear multiuser detection (MUD) [13] receivers. In TD-CDMA systems for each time slot (say th slot), the discrete space-time (S-T) multiuser channel is described by a set (one for each user, ) esof matrixes timated from the training sequences. In this paper, it is shown that for a given user , and assuming the angles/delays constant over slots, the unstructured (or conventional) S-T channel can be used to reestimatrix estimates mate a unique angle/delay set by constraining the angles/delays to be stationary over slots independently on the fading decorrelation. The angle/delay reestimation is the same as estimating the frequencies of 2-D complex sinusoids where the faded amplitudes act as sources; therefore, the 2-D frequency estimation can be solved by any method. Here, we chose ESPRIT [14] for the consistency, the simplicity, and the capacity to pair delay and angle for each path and each user. In this paper, we consider JADE as a parametric method to improve channel estimation. The lower bound on the channel estimation error is derived to quantify the benefits of the structured methods with respect to the unstructured MLE. Since the angle/delay estimation accuracy increases for large , in Section III it is demonstrated that the mean square error (MSE) of the channel estimate depends asymptotically (for ) only on the number of the faded amplitudes that have to be estimated on each slot (as these are varying from slot to slot). In addition, we show that this result remains true for spatially correlated noise even if the angle/delay estimation is not tailored for correlated noise. Numerical analysis for the whole transmission system shows a small degradation of

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PICHERAL AND SPAGNOLINI: ANGLE AND DELAY ESTIMATION OF SPACE–TIME CHANNELS

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Fig. 1. Frame structure of TD-CDMA system.

the parametric estimate with respect to the case of a known channel. To make the numerical analysis more realistic, the modulation and frame structure are those of the UTRA TDD standard [1]. In a simple propagation environment with few paths, performance in terms of bit-error rate (BER) attains (for large ) that of the MUD with known S-T channels. In complex propagation environments, such as the COST 259 channel model [15], the advantage of JADE with respect to the conventional MLE is preserved (gain is of approximately 3–4 dB in the signal-to-noise ratio). The paper is organized as follows. Section II introduces the multipath channel model, and TD-CDMA signals for the purpose of channel estimation. Section III proposes an alternative form of the MLE of the multipath parameters based on a weighted least squares fitting (or reestimation) between a parametric model and the unconstrained MLE. Performance bounds are derived analytically when and a consistent estimator of angles/delays is employed. The JADE-ESPRIT method for the reestimation of angle/delay is discussed in Section IV. The simulations for UTRA TDD standard (Section V) in different propagation environments validate the analytic bounds and demonstrate the advantage of the JADE method for channel estimation. Basic Notations: Lowercase (uppercase) bold denotes vecis the matrix transpose, is the comtors (matrixes), is the Hermitian transposition, is the plex conjugate, is the matrix determinant, Moore–Penrose pseudoinverse, is the norm weighted by a positive definite is the unit matrix, is the Kronecker mamatrix , is the Kronecker trix product (for properties see [16]), and for and elsewhere). delta ( II. MODELS In the TD-CDMA system, a set of users is synchronized to be active within the same time slot but with different spreading codes and transmit periodically with an interslot period according to the frame structure in Fig. 1. Transmitted symbols within each data block are spread to chips/symbol. The training sequences are designed to estimate the chip-spaced channels for all users. Here, we focus the attention on signal and channel models. , for , be the Signal Model: Let users. The block known chip sequence transmitted by the antennas are of base-band signals received by the array of with respect to the low-pass filtered and oversampled by

. The received signals (one for each antenna) chip-rate within the th time slot can be compactly written for the block of chips as (in matrix notation) (1) where the

columns of the matrix contain the samples received at time intervals , and is multiuser channel matrix as each block the is the channel matrix for the th user with temporal . The convolution matrix contains support the oversampled training sequences obtained from (slot independent) matrix where is a Toplitz matrix composed of the th training sequence

.. .

..

.

.. .

(2) The noise and interference is assumed Gaussian. It is temporally uncorrelated but spatially indecorrelated with covariance pendent of the slot (i.e., the angles of the interferers remain the same across slots). To avoid data interference, each training sefor ) or quence can be samples in length ( periodical with cyclic-prefix and period . This latter solution is adopted in some CDMA systems by samples longer and dismaking the training sequence samples (see Fig. 1 and Section V). carding the first Channel Model: Within the th time slot, the th path of the th user can be modeled by the delay , the direction of arrival , and the complex valued slot-varying amplitude (fading) . The overall channel response is a combination of paths. Moving terminals make the channels be time varying. If the velocity is not too great, angles/delays are invariant for time slots (see Section V for a discussion on the validity of this model). The amplitudes are assumed constant within the training period (or within the burst) but vary from slot to slot. Fading variaacross different slots is a stationary process, the tion correlation of the faded amplitudes depending on the Doppler ( is the velocity of the mobile terminal) frequency according, for instance, to compared to the frame length Clarke’s model of 2-D isotropic scattering [17].

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The discrete-time channel matrix for the th user is obtained by oversampling 1: with respect to the chip rate , the channel response at the array of antennas is: for (3) is and depends on the known deVector sampled at rate : layed waveform , denotes the channel length in chip inis the vector of the array response tervals. to the narrowband signal impinging the array from the supported over . For a uniform angle linear array of omnidirectional antennas spaced half a wavewhere length apart it is . By stacking the matrix columns on top of each other (i.e., matrix vectorization: of ), model (3) becomes (4) where U(è , is a

matrix and are the faded amplitudes for the th slot. The columns arrangement of (4) factorizes the problem by and decoupling the stationary terms into matrix . the slot-varying terms into vector Remark: For the analytic derivations the unknowns parameters are arranged into a vector of ordered into amplitude, angle, and delay for each time slot and slots, each path. When considering the ensemble of the parameters are the set of amplitudes , and angle/delay for each path: of size . The set of parameters for the multiuser . system is denoted as III. REESTIMATION OF ANGLE/DELAY be the Let multiuser channel matrix as function of the multipath parame. The MLE of the multiters path parameters for an unknown covariance matrix of the noise can be obtained by minimizing the loss function

(5) where ,

, and (6a) (6b)

The proof of the equivalence of MLE and minimization (5) can be verified by substitutions. The terms of (6a) and (6b) are the unstructured MLE of the (slot-varying) multiuser channel matrix and the covariance matrix of noise, respectively. Miniin (5) shows that the MLE of the multipath mization of parameters can be reduced to the cascade of unstructured MLE and the reestimation of the angles/delays from the unstructured estimate. Incidentally, the unstructured estimate (6a) coincides with the least squares (LS) estimate as carried out conventioncan be simplially in most receiving systems. Notice that . This occurs asymptotically for fied when and . A. Multislot Angle/Delay Estimation Optimization (5) can be asymptotically approximated by the minimization of the loss function (7) where and ; see [18] for proof. The asymptotic equivalence between MLE (5) and the reestimation (7) was first proposed in [6] and has a simple interpretation. It is a weighted minimization of the misfit between the unstructured estimate of the multiuser and the parametric model (3) or (4). S-T channel Weighting depends on the joint correlation properties of the and on the estimate training sequences for all the users that asymptotically of the spatial covariance matrix ) is . Reestimation (7) is still a (for nonlinear optimization over a large number of unknowns and is computationally prohibitive even for moderate values of paths and users. However, some simplifications are necessary: 1) to reduce the problem from multiuser to single user and 2) to exploit efficient (and suboptimum) search methods that approximate the MLE. In the TD-CDMA system, the training sequences for all the users are designed to be mutually uncorrelated (or almost uncorrelated) so that the matrix reduces to a block-diagonal and the minimizacan be decoupled into the minimization of the tion of loss function for each user (8) Here, is the vectorization of the th block of and . This first approximation reduces the problem to a single-user system but still prefor all the users. Alterserves the same covariance matrix natively, a sequential cancellation of the residual correlation of the training sequences can be employed, though the loss of performance is negligible for a small residual correlation [19]. Model (4) allows the reduction of the minimization of (8) into a separable optimization (when fading is modeled as deterministic unknowns). The amplitudes are then estimated as (9)

PICHERAL AND SPAGNOLINI: ANGLE AND DELAY ESTIMATION OF SPACE–TIME CHANNELS

once all the

the

angles/delays

are

known (or estimated), . The estimator from

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a channel matrix with temporally, or spatially, resolved paths the MSE is (Appendix A) and

slots is (12) (10)

. where Optimization (10) is far from being trivial even for small . JADE-ESPRIT is a subspace-based method to replace (9) for angle/delay estimation that performs similarly to MLE for spatially uncorrelated noise and ideal training sequences ). However, the consistency (i.e., guarantees that JADE-ESPRIT is suitable for for angle/delay estimation even for correlated noise. Performance and convergence versus can be quantified by some examples in Section V. observations Notice that the reestimation based on the complex value entries consists of reducing the into real-valued parameters. increases the number of Increasing the number of slots parameters so consistency in the estimate of the fading ampliand ) is not guaranteed. Given the tudes (for equivalence with frequency estimation problem, their estimate . It is shown that the is consistent but inefficient [20] for consistency of the estimator for angles/delays can be exploited ) in terms to evaluate the asymptotic performance (for of MSE of the channel estimate. Optimization (10) would be impractical if not derived as single user. However, once angles/delays have been estimated from slots, amplitudes can be estimated in a multiuser fashion by slightly complicating estimator (9). B. Performance Analysis

Compared to the LS estimate (11), the improvement of the in MSE when angle/delay is reestimation is up to estimated jointly. The result is fairly obvious as the structured estimate is based on the estimation of the propagating parameters; what makes the reestimation not so intuitive is the fact that MSE is lower even though the reestimate is based on (noisy) LS estimates. When the reestimation is carried out over a large number of slots with independent fading, the angles/delays can reach ), provided that any accuracy (as is large enough, while the amplitudes are estimated on a slot-by-slot basis. In this case, the MSE-bound depends on amplitude only, and it is (Appendix A)

(13) Under assumptions 1) and 2), it simplifies into (14) Compared to the slot-by-slot reestimation (12), the MSE for the multislot JADE (14) can reach an improvement of up to 4.5 dB. Here, it is worth mentioning that simply averaging the LS estimate (6a) over slots (i.e., ) causes the MSE (11) to decrease by only for a static channel (or still terminals). The advantages of the averaging (if any) are lost for small fading decorrelation (or slowly moving terminals); see Section V.

The channel estimation based on the reestimation (8) is expected to have a lower MSE than the LS estimate (6a). The purpose here is to quantify the improvement for some simple denote the MSE for the cases. Let ) implies estimate . The asymptotic analysis (for . The MSE can be evaluated in closed form that for: 1) temporally and mutually uncorrelated training sequences ) and 2) spatially uncorrelated (i.e., . noise The MSE of the LS estimate can easily be derived as ; see [19]. Under assumptions 1) and 2), the MSE for the th channel matrix

Spectral methods for delay estimation are based on the estimation of the linear phase variation after Fourier transformation and deconvolution for the known pulse [22]. The S-T channel is known. Therematrix can be similarly handled as pulse samples fore, the discrete Fourier transform (DFT) over of the S-T channel matrix after deconvolution can over the nonzero support of the Fourier transform of be written as a combination of 2-D complex exponentials (for the sake of simplicity, the subscript that indicates a specific user is omitted within the whole section)

(11)

(15)

increases linearly with the overall number of unknowns for the and the oversampling factor , and dechannel matrix creases with training sequence length . The MSE for the reestimate can be evaluated by a sensitivity analysis that first implies an evaluation of the accuracy (see, e.g., [21]) and then the corof the parameter estimate responding MSE for a channel matrix obtained from the estimate of the propagating parameters . For

IV. MULTISLOT JADE

Matrix

(è , is and each of its columns depends on the spatial and temporal response of the path. The spatial term is where and the temporal term is where . Remember that the double Vandermonde is the result of the regular spacing of the structure of sensors in the array and of the DFT samples.

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 3, MAY 2004

The DFT and waveform deconvolution applied now to the LS estimate yield

TABLE I SIMULATION PARAMETERS OF THE UTRA-TDD SYSTEM (BURST TYPE 1)

(16) ). Here, unlike in (15), we account for (for the model mismatch by the term . Model (16) resembles the models for line spectral analysis, the main differences are: are obtained from the LS estimate 1) the observations represent of the multiuser channel and 2) the amplitudes Gaussian sources. The multislot method assumes angle/delay invariance such that the S-T manifold remains unchanged for the . For the estimaset of channel estimates tion of multipath parameters by exploiting the structure (16) the faded amplitudes have to be uncorrelated. for and the covariance matrix has full rank . For identifiability, has to be full column [10]. The rank, which implies at least paths can be identified even if few angles (or de(or lays) are identical and matrix ) is rank deficient (but is still full , column rank). Since matrix should be . For the sample covariance matrix of to be full rank, a necessary fading . Uncorrelation of fading guarantees condition is to be full rank and the amplitudes are uncorrelated from slot to is larger than the coherence time of the fading, slot when . The fading correlation that could lead i.e., to is a constraint that can be relaxed by considering the temporal/spatial smoothing similar to smoothing methods in array processing. The single-slot estimate represents a case limit of correlated amplitudes for JADE; this can be similarly solved by appropriate smoothing as shown in [9]. A. JADE-ESPRIT Optimization (10) re-rewritten for model (16) after DFT and deconvolution can be solved as the frequency estimation of 2-D sinusoids in Gaussian noise. The method proposed in [23] (2-D IQML algorithm) is based on the iterative search but it is not appropriate when as it requires more antennas than paths. The shift invariance method for frequency estimation of 2-D sinusoids (2-D-ESPRIT, [14]) was exploited in [10] to estimate the angles/delays from multiple estimates of the channel response (JADE-ESPRIT). The advantage of JADE-ESPRIT is that it estimates jointly, and pairs automatically, all the angles/delays. The JADE-ESPRIT algorithm is based on the shift-invariance properties of different partitions of the channel according to the double Vandermonde structure matrix . The same properties hold for the partitions of of obtained from the first leading eigenvalues of the basis so that . The estimation of angle/delay is based on the joint exploitation of the shift-invariant structures as described in detail in [10]. However, some remarks are necessary to address the practical issues related to the JADE-ESPRIT implementation.

Spatial/temporal (or angular/frequency) smoothing is mandaeven if is rank deficient. tory for a full rank matrix the This can be easily obtained by extracting from submatrixes that consist of rows and columns and by estimating the covariance matrix from terms. Spatial/temporal smoothing the averaging of is beneficial when fading is highly correlated from slot to slot or when [9]. The number of paths can be determined “offline” by the minimum description length (MDL) criterion [24]. The number of paths that can be realistically estimated depends on the temporal and spatial resolution compared to the signal bandwidth . Simulations and the array beamwidth. In general, it is in Section V show that depends on the space/time diversity of the propagation environment and can be set to a fixed value without any meaningful loss of receiver performance. The additional advantage of having a fixed lies, from the point of view of real-time implementation, in the fact that the number of eigenvectors to be computed remains unchanged. Algorithm complexity is mainly related to the computation with size of the eigenvectors of the square matrix . Such computation can be carried out by exploiting the parallelism in the implementation of the eigenvector decomposition. The computation of the eigenvectors of matrix can be carried out iteratively by minimizing the an norm of the off-diagonal elements with Givens rotations (Jacobi itermethod). In [25], it has been shown heuristically that ations of Brent–Luk “parallel ordering” are enough for converin computation time. Altergence, thus leading to natively, in the multislot approach the eigenvectors of can be updated from the previous slots with a fast (and approximate) subspace tracker like the one proposed by [26]. V. SIMULATIONS The system parameters for numerical analysis are based on the UTRA TDD standard [1], whose essential parameters are listed in Table I. The channel is normalized so that , and the signal to noise ratio is defined for one antenna as where is the energy . Performance for the chip waveform and is evaluated in terms of normalized MSE of the channel : . estimate BER is also evaluated for demodulated sequences after linear decorrelating multiuser detection (MUD) for an array of antennas [27]. The smoothing parameters of JADE-ESPRIT and . Simulations have been carried are out for propagation models with increasing complexity and

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Fig. 2. Performance of LS and JADE versus SNR and varying L: simulations (markers) and bounds (dashed lines). Noise is (a) and (b) spatially uncorrelated or (c) and (d) spatially correlated with background noise 10 log (Q E = ) = 15 dB. RMSE of amplitudes in (b) and (d) are compared with covf ^ g = =N (dashed line) derived in Appendix A (plain and empty markers in (d) are for ^ and ^ , respectively). [J ]

stationary angles/delays: two-path model, 12-path typical urban (TU), and generalized typical urban (GTU). The generalized propagation environments are stochastic models of the S-T channels proposed within the COST 259 working group [15] (see Appendix B for a summary of the parameters). A. Single User and Correlated Noise Performance in terms of MSE is evaluated for a two-path with angles channel in single-user mode , delays (relative separation is approximately four-chip intervals), amplitudes that are independent of each other and from slot to . Noise is either uncorrelated slot, and or spatially correlated with an interferer placed close to that of the path with angle angularly at . In this latter case, the SNR for a specified background is defined as . Fig. 2(a) and (c) shows the MSE of channel noise level versus SNR for varying , and in Fig. 2(b) and (d) the RMSE of the amplitudes after having estimated angles/delays by JADE-ESPRIT. Under uncorrelated noise [Fig. 2(a), (b)] the simulated MSE (markers) attains the bounds (dashed line) in (12) and in (14); simulated evaluated for JADE for is for (remember that for ). JADE for outperforms the LS estimate dB in , and the multislot method by approaches the additional asymptotic gain of for 4.5 dB. Performance for is close to the bound for , thus showing that for the multislot method . Fig. 2(c) is practically on convergence when and (d) shows the MSE of the channel and the RMSE of

amplitudes versus SNR for correlated noise and background dB. Lower bounds on the noise variance of the amplitudes and [dashed lines in Fig. 2(d)] (see Appendix A) and differ because are evaluated for is of the different levels of interference on each path (i.e., angularly closer to ). For large SNR values dB , (markers) approaches the the MSE performance for . Smaller SNR bound derived in Appendix A for values experience a slower convergence to the bound as JADE-ESPRIT is tailored for spatially uncorrelated noise and angle/delay estimate is affected by larger errors (notice that the bound derived in Appendix A holds true for large SNR). The MSE of the channel [Fig. 2(c)] is close to the bound (13) for and dB (dashed lines). (13) Convergence versus to the MSE bound for is shown in Fig. 3 with spatially correlated noise for dB (filled marks) and dB (empty marks), with a background noise set to dB. This is a reasonable value for (pracexample confirms that shows an additional tical) convergence. The MSE for error compared to the MSE bound (dashed lines); this is due to the bias in angle/delay estimation [10] and can be considered as negligible for practical purposes. The same figure shows (for reference) the performance when the amplitudes of the channel are estimated under the wrong assumption of uncorrelated noise in (9) is an identity matrix). In this case, the loss of (i.e., . performance is not tolerable for low B. Multiuser System and TU Channel The interaction between channel estimates and MUD for can be appreciated by considering, for each user, a

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Fig. 3. Convergence of JADE versus L for spatially correlated noise (background noise is 10 log (Q E = ) = 15 dB) when channel is estimated under is in dashed lines. the assumption of spatially correlated (marker ) or uncorrelated (marker ) noise. MSE bound (13) for L

!1

Fig. 4. MSE (left figure) and BER (right figure) versus SNR and varying L for K = 8 users and 12-paths TU channel. Angles of the k th user are independent random variables  ( ; (10 deg) ),  ( 60 deg; 60 deg) is independent of the other users. MSE bound for L is in dashed lines. S-T channels used for MUD are: known channel, LS estimate, JADE for d^ = 12.

N

U 0

12-path TU channel [17] and uncorrelated noise . All the paths have angles normally distributed (standard deviation is 10 deg) around a nominal angle uniformly chosen for each user within the cell coverage [ 60 deg, 60 deg], and fading is uncorrelated from slot to slot. The MSE in Fig. 4

!1

shows that the advantage of the JADE algorithm with is approximately 15.5 dB in with respect to the LS . The estimate and approaches the MSE bound for depends on the fact that some paths cannot be loss for resolved when reestimating from one slot. This example shows

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K = 8). JADE parameters are d^ = 20, fading

Fig. 5. MSE (left figure) and BER (right figure) versus SNR for GTU channel model and TDD-UTRA standard ( is uncorrelated from burst to burst.

that multislot JADE can exploit the information redundancy for all the paths of the channel and gets an improved channel estimate even for rather complex multipath channels. Fig. 4 demonstrates, in terms of BER, the performance for the overall receiver (i.e., the MUD is carried out in the same way but with a different channel estimate) for known channel, LS estimate, ). BER performance and the JADE method ( shows that the multislot estimate attains almost the same when and performance as for the actual channel that single-slot JADE outperforms the LS estimate only when is below 6 dB. C. Multiuser System and GTU Channel Performance in terms of MSE and BER for GTU is shown in Fig. 5 for uncorrelated noise. Since the numbers of paths are generated randomly and spaced closely in angle and/or delay, the MSE bounds in Section III are not meaningful as derived for a fixed number of paths. However, the MSE for simulations shows the advantage of the multislot parametric method ) compared to the conventional LS estimate. (here, This conclusion is supported by the BER performance as the gain of the multislot method is of about 3–4 dB in terms of SNR. Convergence versus the number of slots for varying is shown in Fig. 6 slot-by-slot fading correlation dB . Convergence is reached in slots independently of the fading correlation as far as it is (or km/s according to the Clarke model [17]). As a reference, the same figure shows the performance of the slot averaging of the LS es(dashed lines). Except for still terminals timates

and large numbers of slots, the MSE is uniformly greater than the multislot JADE method. In a complex propagation environment, the multislot method performance when the shows a loss of asymptotic number of paths is below the true and the average number of paths (i.e., underparameterization error). However, for finite (or small) the choice of is a tradeoff between distortion (small ) and noise (large ) in the estimates. This is shown in Fig. 7 where the performance in terms of MSE versus is evaluated for the GTU model with a fixed number of paths (here ), dB and . For simple prop, agation environments ( small), the optimum choice is . For large , while for dense multipaths (large ) it is slightly decreases the performance as the estithe use of mates of the angles/delays and amplitudes are affected by larger errors. Remark: To validate the stationarity of angles/delays by ms as in Table II and let simple arguments, let and km/h. Within ms the terminal moves by less than 10 m. This is negligible in term of delays as it shifts up to 0.03 s (i.e., a fraction below 13% of the chip interval). The angle variation could be considered as negligible if it is, say, under 1 deg. In this case, it is enough that the terminal-base station distance is greater than 550 m (this value can be scaled accordingly for any smaller velocity areas, e.g., urban). In brief, the stationarity of the angle/delay depends on the propagation environment and the cell size. Propagation over large distances (or large cells) is more likely to have stationary angles/delays for large . With respect to large cells, small cells (e.g., micro/pico-cells) have the drawback that the

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Convergence of JADE (solid lines) and averaged LS (dashed lines) versus L for GTU model and varying fading correlation r .

Fig. 7. Sensitivity of multislot JADE to the number of paths d^ for GTU channel with varying d = 5; 10; 20; 30 (SNR = 0 dB, fading is uncorrelated from burst to burst).

angle/delay pattern is more complex and it is stationary for a smaller . VI. CONCLUSION This paper proposes the use of a parametric method based on JADE for the estimation of the space-time channels in

TD-CDMA systems. Given the equivalence with frequency estimation of 2-D complex sinusoids, the shift-invariance method for frequency estimation (JADE-ESPRIT) was adopted as it leads to the pairing of angle/delay and can estimate (for each user) more paths than the number of antennas. In addition, the consistency of JADE-ESPRIT makes it useful to handle spatially correlated noise.

PICHERAL AND SPAGNOLINI: ANGLE AND DELAY ESTIMATION OF SPACE–TIME CHANNELS

The analytical bounds show that the MSE depends on the number of effective parameters to be estimated and closely describe the performance of the method for a large number of slots. In addition, parametric channel estimation is convenient in any propagation environment, even for dense and complex multipaths. In spite of the degrees of freedom in the parameter setting of JADE, it has been shown that reasonable values can be found for its practical use in realistic propagation environments. The reciprocity of the angles/delays in propagation channels allows one to define appropriate strategies for down-link beamforming in order to reduce, if needed, the overall interference at the mobile terminals.

each path

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matrix

is referring to one path. For the

th

, each column of being the derivative with respect to amplitude, time (i.e., ), and angle (i.e., ). Each term in (19) can be evaluated separately and reflects the interaction between the th and the th paths

APPENDIX A ACCURACY AND MSE MSE bound on the channel estimate is evaluated by first deriving the accuracy of the parametric estimate and then by the linearization of the vector with respect to the or for angles/delays. The bound is evaluated either for . The analysis for is based on the assumption that angle/delay is stationary across slots and can be considered as a known parameter since JADE is consistent with (i.e., the MSE depends on the estimate of the amplitudes to be carried out on a slot-by-slot basis). To keep the notation simple, the th user index is dropped within the whole Appendix as performance is evaluated on a user-by-user basis from the so that, loss function (8). The analysis here is for . asymptotically, : Let be the loss Parametric Estimate function rewritten herein without the user index as , and let be the consistent estimate of the multipath parameters (for ) properties of can the th user). The asymptotic (for be established as follows (see [28] for an extensive discussion): around the true value is the Taylor series expansion of (17) , where and denote the gradient and the Hessian of or after respectively. The Hessian can be evaluated for expectation as , where . Since , the asymptotic distribution of is normal and the covariance matrix for the multipath parameter is bounded

(20) In order to make the evaluation of the accuracy more tractable, some simplifications from (20) are necessary. Let the training sequence for each user have ideal correlation properties and let the noise be spatially white . In addition, the paths are temporally or spatially resolved so that in (20) or for and thus the blocks for . The terms reduce to in (20) for

(21) where , is the square is the total of the effective bandwidth, and energy of the pulse shaping filter. The lower bound of the covariance matrix (18) for the th path are the elements of the inverse of block (21)

(18) According to the way the parameters have been grouped (see text) is a block matrix where the th 3 3 block is (22) (19)

The elements of the derivatives matrix need to , angle , be evaluated separately for each of the amplitude in the multipath as , where and delay

Diagonal entries of are the variance of the estimate similarly to the Cramer–Rao bounds derived in [3]. : The MSE of the channel estimate for the reestimate is (23)

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is obtained by assuming that around the MS with delay there is a uniform circular distribution of scatterers such that with . The th path of the th cluster has angle and delay . The angle deviation is Gaussian and delay deviation is exponential with decay rate . The power decreases with the distance according to the path-loss ; the power is lognormal distributed with standard deviation ; the mean power is exponentially decaying with respect to angle and delay

TABLE II ESSENTIAL PARAMETERS OF THE S-T CHANNEL MODEL COST259

It can be evaluated by the first-order approximation as

(27) (24) For temporally or spatially resolved paths with independent fading, the MSE reduces to the summation of the MSE for each path (25) Each term of (25) can be evaluated separately from the elements of matrix in (22), after some algebra it follows:

(26) The summation over angularly or temporally resolved paths gives (12) in the text. : In principle, the derivation of the covarican follow the same steps as those ance matrix for for (see also [21]). However, used to derive the variance of the angle/delay estimate is the for large same as that derived in (22) except for an scaling factor: and . Since the estimator is consistent for any value of , the and for we covariance of angle/delay is can assume that angles/delays are known (or ). This assumption greatly simplifies the derivation of the MSE as it depends on fading amplitudes only. In this is scalar, and for it case, . The is MSE bound can be derived similarly to (25), except that , which leads to (13) in the text. APPENDIX B COST 259 PARAMETERS [15] The paths are grouped into clusters so that . There is always one cluster corresponding to the main path and the additional clusters are Poisson distributed with mean value : . For each cluster the paths are where . For the th cluster the azimuth , and the delay for

Table II shows the essential model parameters. The number is a random variable and the average number of paths reflects the complexity of the propagation environment. For GTU and GRA it is , and for GBU and GHT . REFERENCES [1] M. Haardt, A. Klein, R. Koehn, S. Oestreich, M. Purat, V. Sommer, and T. Ulrich, “The TD-CDMA based UTRA TDD mode,” J. Select. Area Commun., vol. 18, pp. 1375–1385, Aug. 2000. [2] CWTS, China Wireless Telecommunication Standard (CWTS) Working Group 1 (WG1): Physical channels and mapping of transport channels onto physical channels. [3] M. Wax and A. Leshem, “Joint estimation of time delays and directions of arrival of multiple reflections of a known signal,” IEEE Trans. Signal Processing, vol. 45, pp. 2477–2484, Oct. 1997. [4] J. J. Blanz, A. Papathanassiou, M. Haardt, I. Furió, and P. W. Baier, “Smart antennas for combined DOA and joint channel estimation in time-Slotted CDMA mobile radio system with joint-detection,” IEEE Trans. Veh. Technol., vol. 49, pp. 293–306, Mar. 2000. [5] Y. Y. Wang, J. T. Chen, and W. H. Fang, “TST-MUSIC for joint DOAdelay estimation,” IEEE Trans. Signal Processing, vol. 49, pp. 721–729, Apr. 2001. [6] M. Cedervall and A. Paulraj, “Joint channel and space-time parameter estimation,” in 30th Asilomar Conf. Signals, Systems, Comput., vol. 1, Nov. 1996, pp. 375–379. [7] A. Swindlehurst, “Time delay and spatial signature estimation using known asynchronous signals,” IEEE Trans. Signal Processing, vol. 46, pp. 449–462, Feb. 1998. [8] G. G. Raleigh and T. Boros, “Joint space-time parameter estimation for wireless communication channels,” IEEE Trans. Signal Processing, vol. 46, pp. 1333–1343, May 1998. [9] A. Van der Veen, M. C. Vanderveen, and A. Paulraj, “Joint angle and delay estimation using shift-invariance techniques,” IEEE Trans. Signal Processing, vol. 46, pp. 405–418, Feb. 1998. [10] M. C. Vanderveen, A. Van der Veen, and A. Paulraj, “Estimation of multipath parameters in wireless communications,” IEEE Trans. Signal Processing, vol. 46, pp. 682–690, Mar. 1998. [11] J. T. Chen and Y. C. Wang, “Performance analysis of the parametric channel estimators for MLSE equalization in multipath channels with AWGN,” IEEE Trans. Commun., vol. 49, pp. 393–396, Mar. 2001. [12] M. D. Zoltowski and C. Yung-Fang, “Joint angle and delay estimation for reduced dimension space-time RAKE receiver with application to IS-95 CDMA uplink,” in Proc. 5th IEEE Spread Spectrum Techniques Applicat., vol. 2, Sept. 1998, pp. 606–610. [13] J. Picheral and U. Spagnolini, “Shift invariance algorithms for the angle/delay estimation of multipath space-time channel,” in Proc. 53rd IEEE Veh. Technol. Conf.—Spring, vol. 1, 2001, pp. 83–87. [14] M. D. Zoltowski, M. Haardt, and C. P. Mathews, “Closed-form 2-D angle estimation with rectangular arrays in element space or beamspace via unitary ESPRIT,” IEEE Trans. Signal Processing, vol. 44, pp. 316–328, Feb. 1996.

PICHERAL AND SPAGNOLINI: ANGLE AND DELAY ESTIMATION OF SPACE–TIME CHANNELS

[15] COST 259, Mission report—Modeling unification workshop, in COST 259, SWG 2.1 Modeling Unification Workshop, Vienna, Austria, Apr. 1999. [16] A. Graham, Kronecker Products and Matrix Calculus. New York: Wiley, 1981. [17] G. L. Stuber, Principles of Mobile Communication. Norwell, MA: Kluwer, 1996. [18] J. Li, B. Halder, P. Stoica, and M. Viberg, “Computationally efficient angle estimation for signals with known waveforms,” IEEE Trans. Signal Processing, vol. 43, pp. 2154–2163, Sept. 1995. [19] M. Nicoli, “Multiuser reduced-rank receivers for TD-CDMA systems,” Ph.D. dissertation, Politecnico di Milano, Milano, Italy, Dec. 2001. [20] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” IEEE Trans. Acoust., Speech Signal Processing, vol. 37, pp. 720–741, May 1989. [21] G. Zhenghui and E. Gunawan, “Cramer-Rao bound for joint direction of arrival, time delay estimation in DS-CDMA systems,” in Proc. IEEE MILCOM 2000, vol. 2, 2000, pp. 614–618. [22] M. A. Pallas and G. Jourdain, “Active high resolution time delay estimation for large BT signals,” IEEE Trans. Signal Processing, vol. 39, pp. 781–788, Apr. 1991. [23] 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. [24] M. Wax and I. Ziskind, “Detection of the number of coherent signals by the MDL principle,” IEEE Trans. Acoust. Speech Signal Processing, vol. 37, pp. 1190–1196, Aug. 1989. [25] R. P. Brent, F. T. Luk, and C. F. Van Loan, “Computation of the singular value decomposition using mesh connected processors,” J. VLSI Comput. Syst., vol. 1, no. 3, pp. 242–270, 1985. [26] P. Strobach, “Low-rank adaptive filters,” IEEE Trans. Signal Processing, vol. 44, pp. 2932–2947, Dec. 1996. [27] P. Jung and J. Blanz, “Joint detection with coherent receiver antenna diversity in CDMA mobile radio systems,” IEEE Trans. Vehic. Technol., vol. 44, pp. 76–88, Feb. 1995. [28] T. S˝oderstr˝om and P. Stoica, System Identification. Englewood Cliffs, NJ: Prentice-Hall, 1989.

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José Picheral was born in France in 1975. He graduated from the Ecole Supéieure d’Electricité, Paris, France, and from the Politecnico di Milano, Milan, Italy, in 1999. He is currently working toward the Ph.D. degree at the Ecole Superieur d’Electricité, France. From 1998 to 2001, he was a Visiting Scholar at the Electronic and Information Department, Politecnico di Milano. His research interests include the high resolution methods for spectral analysis and their applications for space-time channel estimation in the field of wireless telecommunication and for velocity and polarization estimation of seismic waves.

Umberto Spagnolini (SM’99) received the Dott. Ing. Elettronica (cum laude) degree from Politenico di Milano, Milan, Italy, in 1988. Since 1988, he has been with the Dipartimento di Elettronica e Informazione, Politenico di Milano, where he held the position of Associate Professor of digital signal processing since 1998. His research interests include the area of signal processing, estimation theory, and system identification. The specific areas of interest include channel estimation and array processing for communication systems, parameter estimation and tracking, signal processing and wavefield interpolation with applications to radar (SAR and UWB), geophysics, and remote sensing. Dr. Spagnolini is a member of the SEG and EAGE and serves as an Associate Editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING. He was awarded the AEI Award (1991), Van Weelden Award of EAGE (1991), and the Best Paper Award from EAGE (1998).