DOA Estimation in the Uplink of Multicarrier CDMA Systems
© Antonio A. D'Amico et al. 2008
Received: 15 May 2007
Accepted: 23 October 2007
Published: 20 November 2007
We consider the uplink of a multicarrier code-division multiple-access (MC-CDMA) network and assume that the base station is endowed with a uniform linear array. Transmission takes place over a multipath channel and the goal is the estimation of the directions of arrival (DOAs) of the signal from the active users. In a multiuser scenario, difficulties are primarily due to the large number of parameters involved in the estimation of the DOAs which makes this problem much more challenging than in single-user transmissions. The solution we propose allows estimating the DOAs of different users independently, thereby leading to a significant reduction in the system complexity. In the presence of multipath propagation, however, estimating the DOAs of a given user through maximum-likelihood methods remains a formidable task since it involves a search over a multidimensional domain. Therefore, we look for simpler solutions and discuss two alternative schemes based on the SAGE and ESPRIT algorithms.
Antenna arrays at the base station (BS) can dramatically improve the capacity of a communication system [1–3]. Actually, they can be exploited in various ways. First, to form retrodirective beams that select the desired signals and attenuate the interfering ones. Secondly, antenna arrays make it possible to implement space-time selective transmission in the downlink. Finally, they can provide accurate localization of the user terminals , which is of interest in advanced handover schemes, public safety services, and intelligent transportation systems. In all these applications, accurate estimation of the directions of arrival (DOAs) of the desired signals is required.
DOA estimation has received much attention in the past years and several solutions are available in the technical literature (see [5–9] and the references therein). In particular, the schemes discussed in [5, 6] have good performance but are only devised for single-user applications and cannot be directly used in the uplink of a multiuser system. A tutorial review of subspace-based methods for DOA estimation is provided in . The main drawback of these algorithms is that they can only handle a limited number of users since the overall number of resolvable paths cannot exceed the number of sensors in the antenna array. For this reason their application to a scenario with tens of users and several paths per user (as envisioned in fourth generation wireless systems) seems hardly viable. Schemes for estimating the DOAs in a CDMA multiuser system have been recently proposed in [8, 9]. In particular, the method discussed in  concentrates on a single user's parameters and models the multiple-access interference (MAI) as colored Gaussian noise. This idea is effective as it splits the multiuser DOA estimation problem into a series of simpler tasks in which DOAs of different users are estimated independently instead of jointly. A possible shortcoming of this method is that it requires knowledge of the MAI covariance matrix, which must be estimated in some manner.
In the present paper, we consider the uplink of a multicarrier code-division multiple-access (MC-CDMA) network [10, 11] and propose a method for estimating the DOAs of each active user. Transmission takes place over a multipath time-varying channel in which several paths with possibly different DOAs are present for each user. In a multiuser scenario the main obstacle is the large number of parameters involved in the estimation of the DOAs which makes this problem much more challenging than in single-user transmissions. A practical solution to this problem consists of separating each user from the others before applying conventional DOA estimation schemes. For this purpose, we first estimate the channel response and the data symbols of each active user by resorting to the method discussed in . Once channel estimates and data decisions are obtained, they are exploited to reconstruct the interfering signals, which are then subtracted from the received waveform. This produces an MAI-free signal which is finally used for DOA estimation. In this way the DOAs are estimated independently for each user but, contrarily to , no knowledge of the MAI statistics is required.
In spite of the significant simplification achieved by means of users' separation, estimating the DOAs of a given user through ML methods is still difficult as it involves a numerical search over a multidimensional domain. To reduce the system complexity we investigate two alternative schemes. The first is based on the space-alternating generalized expectation-maximization (SAGE) algorithm , in which the DOAs of a given user are estimated sequentially instead of jointly. This reduces the original multidimensional problem to a sequence of one-dimensional searches. The second scheme exploits the ESPRIT (Estimation of Signal Parameters by Rotational Invariance Techniques) algorithm  and estimates the DOAs in closed form.
The main contribution of this paper is a method for estimating the DOAs of all active users in an MC-CDMA scenario characterized by multiple resolvable paths. As mentioned previously, the major difficulty comes from the need of separating each user from the others before his DOAs can be estimated. Notice that conventional DOA estimation algorithms cannot be employed in such a scenario unless users' separation has been successfully completed, since otherwise the number of sensors in the antenna array should be prohibitively high (on the order of the total number of resolvable paths). To the best of the authors' knowledge, a similar problem has previously been addressed only in . In particular, the solution proposed in  is tailored for the rate space-time block code introduced by Tarokh in  and assumes a static channel with a single DOA for each user. Unfortunately, its extension to a time-varying multipath channel with possibly multiple DOAs for each user does not seem straightforward. A second contribution is a comparison between two popular schemes, namely, the SAGE and ESPRIT algorithms, both in terms of estimation accuracy and system complexity.
The rest of the paper is organized as follows. Section 2 describes the signal model and introduces basic notation. In Section 3 we derive the methods for estimating the DOAs. Simulation results are discussed in Section 4 and some conclusions are offered in Section 5.
2. Signal Model
2.1. MC-CDMA System
We consider the uplink of an MC-CDMA network employing N subcarriers for the transmission of data symbols. The modulated subcarriers are located in the middle of the signal bandwidth and are divided into smaller groups of Q elements . The remaining subcarriers at the edges of the spectrum are not used to limit the out-of-band radiation (virtual carriers). The BS is equipped with P antennas and employs the subcarriers of a given group to communicate with K users that are separated through orthogonal Walsh-Hadamard (WH) codes of length . Without loss of generality, we concentrate on a single group and assume that the Q subcarriers are uniformly spread over the signal bandwidth so as to exploit the channel frequency diversity. We denote the subcarrier indices in the group, with .
The i th symbol of the k th user is spread over Q chips using the code sequence , where and the notation means transpose operation. The resulting vector is then mapped onto Q subcarriers using an OFDM modulator. The channel is assumed static over an OFDM block (slow-fading) and an -point cyclic prefix (longer than the channel impulse response) is inserted to avoid interference between adjacent blocks.
and is the k th user's channel frequency response over the th subcarrier at the p th antenna. Also, is thermal noise, which is modeled as a Gaussian vector with zero mean and covariance matrix (we denote the identity matrix of order Q).
2.2. Channel Model
where is the free-space wavelength and is the DOA of the -path. From (4) we see that measuring is equivalent to measuring since there is a one-to-one relation between these quantities provided that is limited within and . In the following we assume that the path delays and DOAs do not change significantly with time, that is, we set and . Vice versa, the path gains are modeled as independent Gaussian random processes with zero-mean and average power .
where has been set equal to unity without loss of generality.
3. DOA Estimation
It is worth noting that apart from thermal noise, only the contribution of the k th user is present in the right-hand side (RHS) of (9). This amounts to saying that the quantities are MAI-free and, therefore, they can be used to estimate the DOAs of the k th user. In this way, DOA estimation is performed independently for each active user instead of jointly and the complexity of the overall estimation process is significantly reduced.
As mentioned in Section 2.2, measuring is equivalent to measuring the DOA . Without loss of generality, in this section we concentrate on the first user and aim at estimating based on the observation of . Since the ML estimation of is prohibitively complex as it involves a numerical search over a multidimensional domain, in the sequel we discuss two practical DOA estimators based on the SAGE and ESPRIT algorithms. For notational simplicity, we drop the subscript identifier for the first user.
3.1. ML Estimation
where has entries for .
Unfortunately there is no closed form solution to the maximization of (16). The only possible approach is to perform a search over the L-dimensional space spanned by . As the computational load would be too intense, in the next subsection we employ the SAGE algorithm to find an approximate solution of (16).
The ML estimators (15)-(16) have been derived using channel estimates given in (7). In principle, one can directly use the estimates provided by the LMS channel tracker, which are more or less correlated depending on the value of the step-size employed in the tracking algorithm. In contrast, assuming perfect interference cancellation, it is easily recognized that (7) provides uncorrelated channel estimates that facilitate the derivation of the joint ML estimator of and . Since the additional complexity involved by (7) is negligible, we have adopted the latter approach.
3.2. SAGE-Based Estimation
In a variety of ML estimation problems the maximization of the likelihood function is analytically unfeasible as it involves a numerical search over a huge number of parameters. In these cases the SAGE algorithm proves to be effective as it achieves the same final result with a comparatively simpler iterative procedure. Compared with the more familiar EM algorithm , the SAGE has a faster convergence rate. The reason is that the maximizations involved in the EM algorithm are performed with respect to all the unknown parameters simultaneously, which results in a slow process that requires searches over spaces with many dimensions. Vice versa, the maximizations in SAGE are performed varying small groups of parameters at a time. In the following, the SAGE algorithm is applied to our problem without further explanation. The reader is referred to  for details.
Note that only one-dimensional searches are involved in (19).
The following remarks are of interest.
The maximization in the RHS of (19) is pursued through a two-step procedure. The first (coarse search) computes over a grid of values, say , and determines the location of the maximum. In the second step (fine search) the quantities are interpolated and the local maximum nearest to is found.
From (21) it follows that is a periodic function of with period . Thus, the maximum of lies in the interval and, in consequence, the estimator (19) gives correct results provided that π. From (4) it is seen that this condition is easily met using an antenna array with interelement spacing less than half the free-space wavelength.
- (3)In applying the SAGE we have implicitly assumed knowledge of the number L of paths. In practice L is unknown and must be established in some way. One possible way is to choose L large enough so that all the paths with significant energy are considered. Alternatively, an estimate of L can be obtained in the first cycle as follows. Physical reasons and simulation results indicate that in any cycle the multipath components are taken in a decreasing order of strength. On the other hand, if are the estimates of at the first cycle, an indication of the energy of the th path is(22)
Thus, the first cycle may be stopped at that step, say , where drops below a prefixed threshold and may be taken as an estimate of the number of significant paths.
The computational load of the SAGE is assessed as follows. Evaluating for and needs operations at each step. The complexity involved in the computation of in (20) is while operations are required to compute the quantities for . Denoting the number of cycles and bearing in mind that each cycle is made of L steps, it follows that the overall complexity of the SAGE is .
3.3. ESPRIT-Based Estimation
An alternative approach for estimating the DOAs relies on subspace-based methods like the \MUSIC (MUltiple Signal Classication)  or ESPRIT algorithms . In the following we discuss DOA estimation based on ESPRIT. The reason is that this method provides estimates in closed form while a grid-search is needed with MUSIC.
in which J is the exchange matrix with 1's on its antidiagonal and 0's elsewhere.
and denotes the phase angle of in the interval .
The following remarks are of interest.
A necessary condition for the existence of in RHS of (26) is that the number of rows in is greater than or equal to the number of columns. Since has dimension , the above condition implies that , that is, the number of antennas must be greater than the number of multipath components. We also observe that the inverse of in the ML estimator (16) exists provided that is full rank and . Thus, DOA estimation with ESPRIT needs one more antenna compared with the ML estimator. It is worth noting that the minimum number of antennas required by both schemes is independent of the number K of contemporarily active users.
- (2)The number of paths can be estimated using the minimum description length (MDL) criterion . To this purpose, let be the eigenvalues of the correlation matrix in (24) (arranged in a nonincreasing order of magnitude). Then, an estimate of L is computed as(27)where is a trial value of L while and denote the geometric and arithmetic means of respectively, that is,(28)
The complexity of the ESPRIT is assessed as follows. Evaluating in (23) needs operations. Bearing in mind that inverting an matrix requires operations, it follows that the complexity involved in the computation of S in (26) is approximately . Finally, computing the eigenvectors of S needs operations. In summary, the overall complexity of the ESPRIT is 3P. In writing this figure we have ignored the operations required to compute , , and since these matrices are easily obtained from with negligible complexity.
4. Simulation Results
4.1. System Parameters
where is chosen such that the channel energy is normalized to unity, that is, . Each path varies independently of the others within a frame and is generated by filtering a white Gaussian process in a third-order lowpass Butterworth filter. The 3-dB bandwidth of the filter is taken as a measure of the Doppler rate , where v denotes the speed of the mobile terminal and m/s is the speed of light.
A simulation run begins with the generation of the channel responses of each user. Channel acquisition is performed using Walsh-Hadamard training sequences of length while channel tracking is accomplished by exploiting data decisions provided by a parallel interference cancellation (PIC) receiver . Throughout simulations the number of data blocks per frame is set to . Once channel estimates and data decisions are obtained, they are passed to the proposed SAGE- or ESPRIT-based DOA estimators. The SAGE computes the function over a grid of values and it is stopped at the end of the second cycle ( ). Parameter in (11) is fixed to 16, so that . The mobile velocity, the number of users, and the number of antennas are varied throughout simulations so as to assess their impact on the system performance.
4.2. Performance Assessment
The system performance has been assessed in terms of root mean-square-error (RMSE) of the DOA estimates. For simplicity, the number L of paths is assumed perfectly known at the receiver.
We have discussed a method for estimating the DOAs of the active users in the uplink of an MC-CDMA network. Conventional DOA estimation schemes cannot be directly applied in a multiuser scenario due to the large number of parameters involved in the estimation process. Our solution exploits channel estimates and data decisions to isolate the contribution of each user from the received signal. In this way, DOA estimation is performed independently for each user employing either SAGE or ESPRIT algorithms.
Comparisons between the proposed schemes are not simple because of the different weights that may be given to the various performance indicators, that is, estimation accuracy and computational complexity. It is likely that the choice will depend on the specific application. For example, the ESPRIT is simpler and has good accuracy. On the other hand, the SAGE outperforms ESPRIT at low SNR values but has limited resolution. Using more antenna elements can alleviate this problem at the cost of an increased complexity. Computer simulations indicate that both schemes are robust against multiuser interference and channel variations.
This work has been partly presented to the 6th International Workshop on Multi-Carrier Spread Spectrum (MC-SS 2007), Herrsching, Germany, 2007.
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