Efficient Tracking of Eigenspaces and its Application to Eigenbeamforming

Transcription

1 Efficient Tracking of Eigenspaces and its Application to Eigenbeamforming Clemens Michalke, Matthias Stege, Frank Schäfer and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Dresden University of Technology, Dresden, Germany Abstract To achieve high performance for MIMO wireless systems with multiple transmit and receive antennas a detailed understanding of the characteristics of the Eigenspace of the channel is essential. It represents the spatial characteristics of the propagation scenario in general and can be expressed by different covariance matrices of the channel coefficients. In many MIMOalgorithms the knowledge of the Eigenvectors and Eigenvalues is required 1. As typical for the wireless channel, the spatial characteristics continuously change. Therefore, an efficient method for tracking the Eigenspace with moderate computational complexity is required. Tracking of the Eigenspace using incremental Jacobi rotations is presented and compared to other known algorithms for subspace tracking. Eigenbeamforming serves as an example, where such a tracking algorithm can be applied. I. INTRODUCTION MIMO wireless systems with multiple transmit and receive antennas are an important part of discussions on future wireless communication systems, in the 3GPP standards and for use in WLAN systems. In a rich scattering environment, these systems offer large capacity gains, as shown by information theory [1]. The Eigenspace in general represents the spatial characteristics of the propagation scenario. Independent orthogonal subchannels can be defined according to the orthogonal Eigenvectors of the channel that have a capacity proportional to the corresponding Eigenvalues. Eigenbeamforming [2], [3] is one approach that utilizes average knowledge of the Eigenspace at the transmitter to access a major part of the MIMO-capacity gain by using the Eigenvector corresponding to the strongest Eigenvalue. Other approaches use the Eigenspace for an efficient statistical modeling of the spatial statistics of a MIMOchannel [4]. A method to enhance the reliability of MIMOchannel estimates is based on a signal-dependent Karhunen- Loève-Transform (KLT). This approach also uses knowledge of the characteristic Eigenspace to derive an estimate of the signal subspace [5]. All these examples demonstrate the importance of the 1 Throughout the paper we use the terms Eigenvector and Eigenvalue even if the more general singular vector/value are meant. Because in the MIMO case the singular value decomposition of the channel coefficients matrix and the Eigenvalue decomposition of the two covariance matrices derived therefrom yield the same matrices (see II-B). Eigenspace for MIMO-systems. Therefore, an efficient technique that estimates the Eigenvectors and the Eigenvalues of a matrix is needed. The complexity of such algorithms in general limits their implementation in practical systems. Eigenspaces alter over time due to changes of the spatial propagation scenario, caused by movement. Consequently, an Eigenvalue decomposition (EVD) would have to be performed frequently. Hence, a tracking algorithm is desirable that allows tracking the changes of the Eigenbasis of a MIMO-channel with minimum complexity. There are a number of different algorithms for adaptive Eigenspace tracking. Section III-A will give a short summary and demonstrate the demand for a more general approach, that allows for a simultaneous estimation of all Eigenvectors and Eigenvalues. The paper is organized as follows: Section II introduces the signal model and different spatial covariance matrices that represent the spatial channel characteristics. After the introduction of the Eigenspace tracking in Section III, the performance of such tracking schemes is shown for Eigenbeamforming as an example of MIMO-algorithms in Section IV. Conclusions complete the paper. II. FUNDAMENTALS OF MIMO-COMMUNICATION A. The MIMO-Signal Model Flat fading channel characteristics are assumed throughout the paper. Furthermore, a discrete-time representation of the signals is used. For a system with M Rx receive antennas, the received signal vector y(k) has M Rx components and is defined as y(k) = r(k) + n(k) = H(k)s(k) + n(k), (1) where the white Gaussian noise vector process n(k) accounts for thermal noise and interferences: n(k) = [n 1 (k)... n M Rx(k)] T. (2) The components of n(k) are assumed to be zero mean, complex random variables with variance σ 2 n. Thus, this random process is also referred to as temporally and spatial white. The transmitted signal vector s(k) of M Tx complex symbols,

2 where M Tx is the number of transmit antennas, may be represented by s(k) = [s 1 (k)... s M Tx(k)] T. (3) The channel is represented by H(k) C M Rx M Tx, a matrix of complex channel coefficients h i,j (k) characterizing the path between the i-th TX-antenna and the j-th RX-antenna: H(k) = h 1,1 (k)... h M Tx,1(k)..... h 1,M Rx(k)... h M Tx,M Rx(k). (4) The channel is assumed to be passive and normalized, resulting in E { h i,j (k) 2} = 1. Due to local scattering there is some angular spread which results in replicas of the signal that are non-resolvable in time, but arise from different angle of arrival (AOA). A similar angular spread can be defined for the angle of departure (AOD). With a variation of the distribution of AOA and AOD different spatial scenarios can be simulated. A detailed description of the MIMO-channel model can be found in [6]. S c a t t e r e r c l u s t e r Tr a n s m i t t e r v = k m / h R e c e i v e r Fig. 1. The microcell environment with distinct distances between transmitter, receiver and the scatterer cluster. The MIMO microcell propagation scenario depicted in Figure 1 and described in detail in [6] shall be used for simulations of the MIMO-algorithm in Section IV. The Microcell channel scenario demonstrates a scenario with low spatial diversity. Angular spreads at both transmitter and receiver are small. This leads to highly correlated channel coefficients and thus the Microcell scenario can be considered as a typical beamforming scenario with low spatial diversity. B. Characteristics of the MIMO Eigenspace A singular value decomposition (SVD) of the channel matrix H(k) yields: H(k) = U(k)Λ(k)V H (k). (5) The left- and right-side singular vector matrices U(k) C M Rx M Rx and V(k) C M Tx M Tx contain the singular vectors that determine the Eigenspace of the channel. The matrix Λ(k) R M Rx M Tx is a diagonal matrix which consists of the singular values λ m (m = 1... min {M Tx, M Rx }). The singular values can be viewed as attenuations of orthogonal Eigenchannels. The spatial characteristics of a MIMO-channel can be described by its spatial covariance matrices [4]. Correlation between the channel coefficients are in general different at the transmitter and the receiver antenna array. Therefore, two covariance matrices must be considered here. The receivecovariance matrix R Rx (k) C M Rx M Rx depends on the angular spread of the AOA s: R Rx (k) = H(k)H H (k). (6) Similarly the transmit-covariance matrix R Tx C M Tx M Tx depending on the AOD s reads: R Tx (k) = H H (k)h(k). (7) It can be shown that the Eigenvectors of the covariance matrix R Tx (k) are equal to the right-side singular vectors in V(k) of H(k): R Tx (k) = V(k) [Λ Tx (k)] 2 V H (k), (8) whereas U(k) is equal to the Eigenvector matrix of the receiver covariance matrix: R Rx (k) = U(k) [Λ Rx (k)] 2 U H (k). (9) The non-zero Eigenvalues in [Λ Tx (k)] 2 R M Tx M Tx and [Λ Rx (k)] 2 R M Rx M Rx are equal and their square roots yield in the singular values of H(k). In a number of applications the long-term average of the covariance matrices is of importance. One reason to choose a long-term averaging is to minimize the influence of short-term fading effects and noise. The long-term covariance matrices R Tx and R Rx can also be decomposed into their Eigenvector matrices V and Ū: R Tx = E R Tx (k) = V [ ΛTx (k) ]2 VH and (10) R Rx = E R Rx (k) = Ū [ ΛRx (k) ]2 Ū H. (11) Averaging leads in general to differences between the shortand long-term Eigenvector matrices. Hence, the long-term Eigenvalues of the transmitter- and receiver-covariance matrix are not equal. More details on the characteristics of spatial correlation matrices can be found in [7]. The importance of the Eigenspace led to various approaches that utilize the knowledge of Eigenvectors for use in communication algorithms. Since spatial characteristics may change continously, but relatively slowly, some adaptation is needed. In the following an algorithm is presented that allows a continuous tracking of time-variant Eigenvectors and Eigenvalues. III. TRACKING OF EIGENSPACES A. Overview of established Tracking Algorithms In the area of signal processing exist several algorithms using subspace-based methods. Typical examples are high resolution signal parameter estimation algorithms like MUSIC [8] and ESPRIT [9] and data dependent efficient audio and video

3 compression exploiting the Karhunen-Loève transformation [10]. The quality of the algorithms depends on an accurate determination of the Eigenspace, because the signal subspace has to be separated from the noise subspace. The dimension of the two subspaces is determined from the Eigenvalue distribution. For non-time-critical applications usually batch Eigenvalue decomposition or singular value decomposition is used. These two methods are of very high computational complexity and therefore not well-suited for real-time processing. In order to solve this general problem a large number of adaptive algorithms for subspace tracking were developed with respect to different groups of applications they are used for. Anyhow these methods can be divided into three different approaches [11]: classical batch EVD/SVD methods as there are QR algorithm, power iteration and Lanczos method which are modified slightly for adaptive processing [12], [13], rank-one updating algorithms like subspace averaging or reduced power iteration that find only some strong Eigenpairs [14], [15] and methods considering the computation of singular values and vectors or Eigenvalues and Eigenvectors respectively as a problem of constrained or unconstrained optimization [11], [16]. The signal processing application discussed in section IV uses only distinct Eigenspace dimensions. This dimension reduction requires necessarily to know the whole Eigenspace as exactly as possible and also the Eigenvalues corresponding to the Eigenvectors. Additional requirements for the tracking algorithm are low computational complexity and a high level of parallelism for proper DSP implementation. Hence applying algorithms listed in the above mentioned first group is unfavorable. The second group requires computations whenever a new observation is accessible. This is unnecessary, if the Eigenspace is changing slowly as it does in typical MIMO scenarios. B. The PAST algorithm In Section III-C we present an algorithm for subspace tracking based on incremental Jacobi rotations. For comparison the Projection Approximation Subspace Tracking (PAST) [11], an optimization algorithm, is considered. It minimizes the cost function of the deviation of the noisy data vector y (e.g. the observation vector at the receive antenna array) from its projection into a subspace spanned by the orthogonal matrix W: J(W) = E { y WW H y 2}. (12) At the global minimum the columns of W span the signal subspace. Remark, that W defines only an arbitrary basis, a rotated copy of the desired basis of Eigenvectors. The PASTd algorithm [11] which uses a signal subspace dimension of one and repeated deflation of signal energy in the computed direction from y overcomes this problem. Admittedly the estimated vectors are not strictly orthogonal. C. Jacobi Tracking Assuming a conjugate complex symmetric (Hermitian) matrix R 0 (k) C L L (e.g. RTx or R Rx ), of which the Eigenvalues and Eigenvectors shall be determined. One way for the solution of the symmetric Eigenvalue problem is the Jacobi method [17], [18]. The algorithm applies Jacobi rotations to iteratively diagonalize the source matrix R 0 (k). The i-th rotation leads to: R i+1 (k) = J H i (k)r i(k) J i (k). (13) The rotation matrix J i = J i (k, p, q, φ, ϕ) C L L is an identity matrix of same size as R 0 (k) in which entries at the matrix positions (p, p), (p, q), (q, p) and (q, q) are replaced: J i (k, p, q, φ, ϕ) = cos(φ)e jϕ sin(φ)e jϕ p..... sin(φ) cos(φ) q p (14) One rotation causes the element ρ p,q of the matrix R i+1 (k) to be zero. Note, that all matrices R i (k) (i = 0, 1,... ) are Hermitian - so one only needs to concentrate on the upper tridiagonal matrix. Hence, after L (L 1) 2 rotations all offdiagonal elements are zeroed out exactly once. Such a series of L (L 1) 2 rotations is called a sweep. Unfortunately, subsequent rotations destroy already existing zeros in R i (k). Therefore, several sweeps have to be performed until a desired degree of diagonality is achieved. Then the main diagonal contains approximations of the Eigenvalues of R 0 (k). Note, that all matrices R i (k) have the same Eigenvalues but not the same Eigenvectors! The product of all rotation matrices J i (k) yields the approximate Eigenvector matrix U(k) of R 0 (k), q U(k) = J i (k), i (15) with Λ 2 (k) = U H (k) R 0 (k) U(k), (16) where Λ 2 (k) is the diagonal matrix of the real Eigenvalues. Details about the calculation of the rotation parameters cos(φ), sin(φ) and e jϕ can be found in [19]. If the system parameters change only slightly over time some tracking of the Eigenvalues/Eigenvectors could be applied. In case that the Eigenvector matrices of R 0 (k) and R 0 (k + 1) are similar, the both-sided multiplication of R 0 (k + 1) with.

4 the previous Eigenvector matrix U(k) results in a matrix R 1 (k + 1) which is almost diagonal (comp. 16). R 1 (k + 1) = U H (k) R 0 (k + 1) U(k) (17) After premultiplying Jacobi-rotations proceed in the usual way in order to achieve the desired degree of diagonality. The orthonormality of U(k) ensures that R 0 (k + 1) and R 1 (k + 1) possess identical Eigenvalues. So, even if the system parameters change completely from k k + 1 the premultiplication with the previous Eigenvectors introduces no problem in terms of the Eigenvalue decomposition. However, no diagonalization is achieved then. In correspondence with (15) the Eigenbasis at time (k + 1) is obtained after premultiplication and subsequent rotations according to (17) as U(k + 1) = U(k) J i (k + 1). (18) D. Complexity The purpose of the proposed tracking algorithm is to reduce the number of arithmetic operations, which is equivalent to a higher convergence rate compared to the conventional EVD. The original Jacobi EVD algorithm requires approximately 7 sweeps to obtain a degree of diagonality that is sufficient enough for applications in the area of control and communications. However, simulations [19] showed that the application of Jacobi Tracking can save about 4 sweeps by the premultiplication with the previous Eigenvector matrix (17). With the premultiplication step and additional 3 sweeps a similar precision in diagonality is achieved as otherwise within 7 sweeps. The accuracy is not limited, the more sweeps follow the premultiplication the higher becomes the precision. Further details about the complexity of the Jacobi Tracking and the comparison to the original Jacobi EVD algorithm can be found in [19]. IV. APPLICATION OF EIGENSPACE-TRACKING TO EIGENBEAMFORMING One approach to use the advantage of more than one base station transmit antenna is the so called Downlink- Eigenbeamforming [2], [3]. With this method the beamforming gain can be exploited for transmissions independent from the receivers signal processing algorithm and the number of receive antennas 2. At the same time the interference for other users is decreased. The method of Eigenbeamforming is a beamforming approach where the antenna weights are determined from an Eigenvalue decomposition of the channels long-term covariance matrix 2 A similar signal processing approach like the Downlink- Eigenbeamforming is implementable at a mobile station with multiple receive antennas, exploiting a receive beamforming gain. i R Tx (10) instead of an estimation of directions. The Eigenvector v corresponding to the strongest Eigenvalue in [Λ Tx (k)] 2 is used to allocate the transmit data d(k) to the antennas: s(k) = vd(k) (19) Obviously one Eigenvector leads to only one distinct beam. Therefore Downlink-Eigenbeamforming is suited for highly correlated transmit antennas where only one dominant spatial direction is available. This corresponds to the Microcell channel scenario. Long-term and short-term Eigenvectors are nearly equal in this case due to the slow changing and the high antenna correlation. Therefore the need to estimate the Eigenspace at a high rate is reduced. In addition to the savings in computational effort the feedback of Eigenvectors from the receiver to the transmitter is reduced. To overcome the short-term influences of fast fading the Eigenvectors corresponding to the two strongest Eigenvalues are signaled to the transmitter. Short-term beam switching can be done according to the evaluation with the short-term covariance matrix R Tx (k) [20]. uncoded Bit Error Rate Comparison between 4-Tx-EBF with tracking and SIMO 500ms Eigenvector updating rate QPSK modulation Microcell channel model 4x4 MIMO antenna configuration EVD Jacobi SIMO EVD Jacobi PASTd SIMO PASTd SNR (E /N ) at receive antenna [db] S 0 Fig. 2. Raw bit error rate for transmission with the Eigenbeamformer concept using Jacobi tracking, PASTd algorithm and batch Eigenvalue decomposition compared to the BER of a system with only one transmit antenna (SIMO). The ability of tracking the long-term Eigenspace was investigated with simulations and the performance of the Eigenbeamformer with the Jacobi Tracking approach and the PASTd tracking method respectively was compared to that of Eigenbeamforming with batch Eigenvalue decomposition. The averaged bit error rate of uncoded transmission for these approaches can be seen in Figure 2. In highly correlated scenarios like the Microcell environment only one dominant Eigenvalue exists. As mathematical investigations show the one-dimensional signal subspace is very stable [21]. Therefore the Eigenvector undergoes a very slow

5 rotation. For the simulation of Jacobi Tracking only one sweep was computed for the Eigenvector update, which proved to be sufficient enough. Compared to the description of necessary sweeps for a batch Eigenvalue decomposition in Section III-D a computational complexity reduction of approximately 70% is achieved. Using Jacobi Tracking with only one sweep to update the Eigenvector means no markable decrease in the bit error rate compared to that of the Eigenbeamformer with batch Eigenvalue decomposition. In contrast using the PASTd algorithm, which works on the observation vector instead of averaging the covariance matrix, leads to an average loss of 4dB. This amount is certainly influenced by the step size of the optimization method. Finding an appropriate step size considering a time variant channel means additional computational complexity in the end. Remark, that in the beginning of the simulation the Eigenvectors are undefined and therefore are initialized with the identity matrix. This corresponds to a situation where either the tracking procedure starts or where the spatial characteristics of the channel and therefore the Eigenspace has changed completely as a result of slow fading processes and the tracking application has to resume. In general it can be seen that applying the transmit scheme of Eigenbeamforming to an four element antenna array has an advantage over the single antenna transmission by exploiting the beamforming gain. In the simulated Microcell environment the update of the two Eigenvectors at the transmitter every 500 ms seems to be sufficient for a channel velocity of 100 km/h (62 mph) as other simulations showed. This corresponds to a distance of around 14 meters whereas the displacement between transmitter, receiver and scatterer cluster is some hundred meters. V. CONCLUSIONS The Eigenspace of the flat fading MIMO-channel matrix has shown to represent the spatial characteristics of the propagation scenario. Knowledge of the Eigenspace is needed not only for the estimation of the MIMO-channel capacity, but also offers significant information for enhanced communication algorithms. Eigenbeamforming is only one example. Since the Eigenspace of the MIMO-channel continuously changes, adaptive tracking of the Eigenstructure is needed. This would allow for reduced rate feedback of channel information for Eigenbeamforming, since only incremental changes need to be signaled. Conventional subspace tracking algorithms were found to fail for such a task, because they often estimate an arbitrary basis of the Eigenspace by assuming that the Eigenspace is of lower dimension than the overall available degrees of freedom. Further, they often show a slow adaption that results in inferior performance of the communication system until the Eigenvectors are adapted. Otherwise the original EVD/SVDmethods are computationally complex and are not adaptive either. The presented method using premultiplication with the previous computed Eigenvector matrix and subsequent incremental Jacobi-rotations is both, computationally efficient and allows for tracking of smooth changing Eigenspaces. It has been shown, that such an adaptive Eigenspace tracking scheme can be applied to Eigenbeamforming and achieves good results. This tracking algorithm can be applied generally in all algorithms that are based on the knowledge of the time variant Eigenvectors/values. REFERENCES [1] G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications 6, [2] S. A. Jafar, S. Vishwanath, and A. Goldsmith, Channel capacity and beamforming for multiple transmit and receive antennas with covariance feedback, in Proc. ICC 2001, Helsinki, [3] C. Brunner, J. S. Hammerschmidt, and J. A. 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