Efficient Tracking and Feedback of DL-Eigenbeams in WCDMA

Transcription

1 Efficient Tracking and Feedback of DL-Eigenbeams in WCDMA Wolfgang Utschick and Christopher Brunner Institute for Circuit Theory and Signal Processing Technische Universität München Abstract A tracking solution to the downlink eigenbeamforming in WCDMA is presented. To this end, we propose a distributed implementation of the eigenspace/-beam tracking at the UE and BS, respectively. Moreover, the specific nature of the deployed tracking scheme offers a advantageous feedback signalling. The proposed performance measure is given by the ratio of the power for the user of interest and the total TX power of the relevant base station which is required to obtain a certain raw bit error ratio for the user of interest. 1. Introduction The performance of digital mobile radio communication systems is limited by fast fading and interference from co-channel users. Both effects can be reduced by the use of antenna arrays at the base station with the appropriate signal processing, i.e., by diversity combining and interference suppression. On the downlink, the spatial processing 1 is carried out prior to transmission and, therefore, before the signal encounters the channel. This considerably differs from uplink processing with adaptive antennas, where the spatial processing is performed after the channel has affected the signal. Consequently, various closed-loop Tx diversity concepts have been suggested in standardization (3GPP) for,, and > antenna elements, respectively, which solely exploit short-term channel fluctuations [1, ] or both consider short-term and longterm (spatial) channel properties [3,, 5]. The latter has become known as the downlink eigenbeamforming concept, which is based on a principal component analysis (PCA) of the long-term spatial covariance matrices of the radio channel [5, 6]. In this paper, we present a tracking solution to the downlink eigenbeamforming in WCDMA. To this end, we propose a distributed implementation of the eigenspace/-beam tracking at the UE and BS, respectively. Moreover, the specific nature of the deployed tracking scheme offers a advantageous feedback signalling. The proposed performance 1 In the sequel, downlink space-time processing is separated between base station (BS) and the user equipment (UE), resp. mobile terminal, i.e., spatial processing takes place at the BS and a conventional rake receiver performs temporal processing at the UE. measure is given by the ratio of the power for the user of interest and the total TX power of the relevant base station which is required to obtain a certain raw bit error ratio for the user of interest. Basically, the simulation environment compares to that used in [7].. Downlink Eigenbeamforming The general idea behind the eigenbeamformer concept (Figure 1) is a decorrelation of (spatial) diversity branches to achieve a reduction in dimension for subsequent short-term processing and an improved short-term channel estimate at the UE enabled by an increase in diversity and antenna gain and interference suppression [5]. To this end, the eigenvectors or eigenbeams of the long-term spatial covariance matrices with the largest eigenvalues (largest average SNR) are determined and fed back step by step to the base station. This process takes place on the same time scale as the physical terminal movement. Accordingly, required operations at the UE as well as required feedback bits are distributed over a very large number of slots. In addition, a short-term selection between the eigenbeams is carried out at the terminal to account for fast fading and is fed back. Thus, one is able to efficiently address a large number of M antenna elements by having the terminal select one out of a reduced set of d eigenbeams ("

2 user :::K short-term processing long-term processing RRC channel filter D/A M csamp./chip HF modulation orthogonal common pilot sequences w 1 user 1 d ST rake ngers M antennas MS 1 w d user :::K RRC channel filter D/A M csamp./chip HF modulation - long-term downlink feedback (dominant eigenbeams) - short-term downlink feedback (selected eigenbeam) Figure 1: Structure of the downlink eigenbeamformer using long-term and short-term feedback. reduction in dimension) and feed back this information to the BS. The computation of the dominant eigenvectors wi C M comprises the estimation and the PCA of the long-term spatial covariance matrix R: where R = WW H ; (1) W = [w 1 ; : : : ; wm ] C MM () = diag [ 1 ; : : : ; M ] C MM (3) denote the matrices of eigenvectors and eigenvalues, respectively. At first, the estimation of long-term spatial covariance matrices (second order statistics) requires orthogonal pilot sequences transmitted from each BS antenna element. Since the second order statistics of the signals change slowly over time, a forgetting factor is used which, in the example below, is applied to the long-term spatial signal covariance matrix as follows: R R + (1? ) NX n=1 hnh H n C MM ; () where the hn denote the spatial channel estimates of the n = 1; ; : : : ; N dominant temporal channel taps of the current slot and is the applied forgetting factor [6]. Note, that it is sufficient to perform this updating once every frame or even in larger intervals but not necessarily once per slot. 3. Tracking of Eigenspaces/-beams Even when the rate of updating the long-term estimates of channel properties is not demanding, the efficiency of any closed-loop Tx diversity concept depends on two vital items: (i) the amount of required feedback information per time and (ii) computational and numerical effort spent at the mobile terminal. To this end, we deploy a recently proposed subspace tracking technique [8] for tracking the eigenbeams which accomplishes both requirements. The new tracking algorithm resumes the tradition of estimating subspaces by the solution of an unconstrained optimization problems [9, 10]. Resuming the ideas of [10], it has been shown in [8] that the objective function J() with the projection matrix =?E x H P H x =?tr V H RV ; (5) P = W ow H o C MM (6)

3 and V = W o C Md ; (7) atains its global minimum at if and only if V = W o = W d ; (8) where W d is the matrix of the eigenvectors of the d largest eigenvalues of R. Here, the matrices C MM and C dd are unitary rotation matrices. It turns out that the iterative minimization of J() considerably benefits from an alternative parameterization of. To this end, is denoted as the product of elementary rotation matrices, (cf. 9), where k = M? 1; : : : ; 1 l = k + 1; : : : ; M: The k;` are Givens rotation matrices with the characteristic entries at (k; k), (k; `), (`; k), and (`; `), i.e. the defining submatrix, the Givens rotor G k;` R, is equal to + cos( k;`) k;` G =? sin(k;`) + sin(k;`) + cos(k;`) : (10) The rescaling matrix? is given by h i? = diag e j1 ; e j ; : : : ; e j M : (11) Hereby, the parameterization of and thus, regardless from the dimension d, the parameterization of span [W d] which equals the subspace of dominant eigenbeams requires M (M +1) elements. Note, that all are real-valued and only take values from [?; +[ and? ; + 3. Obviously, the tracking of the dominant eigenbeams is closely related to the tracking of the ; (1) constitute the gradient of J. It has been shown that the partial differentials can be obtained In the case of M = antenna elements the parameterization of the unitary rotation matrix reads =? 3; ;3 ; 1; 1;3 1;. 3 For more readability the indices of k;` and k are generally omitted. as a rather straightforward function of P and R (see Appendix). Consequently, for all the update of the matrices, V, and P are equal to ( + ); (13) V V ; (1) P P H ; (15) respectively. In spite of the complex nature of the algorithm, the total update of one iteration cycle requires only 3M (M? 1) real-valued additions and multiplications [11, 1], i.e. considering a - antenna-tx concept it takes 1 real-valued adds and mults to compute one update cycle. Although the colums vectors v of the matrix V constitute the eigenspace of the dominant eigenbeams, the vectors V are not fully decorrelated. Therefore, if perfect decorrelation of channels is a must, a further decorrelation step by means of the unitary rotation matrix is performed by W d V H (16) which directly results from (8). The matrix again can be parameterized and estimated very likely as, however at the lower dimension d < M. Consequently, in the sequel we distinguish between parameters and. Note, that J() is invariant to short-term fluctuations of the estimates of R. Accordingly, the proposed scheme of separating the eigenbeam tracking in rotations of and allows to reduce the forgetting factor without giving up the access to the long-term characteristics of the channel.. Long/Short-Term Feedback The nature of the proposed eigenbeam tracking algorithms ditto offers an alternative concept for the feedback of the eigenbeams from the UE to BS. To this end, instead of directly communicating the eigenvectors via the closed-loop feedback channel, we propose to transmit the increments of the parameters. The general idea behind this feedback concept is founded in a distributed implementation (Figure ) of the eigenspace/-beam tracking. Accordingly, the 3

4 =? M?1;M M?;M?1 M?;M k;` 1;M?1 1;M ; (9) long-term processing w 1 V V ( + ( + ) P H d ST rake ngers MS 1 w d - long-term downlink feedback (dominant eigenbeams) f: : : ; ; : : :g Figure : Distributed implementation of the eigenspace tracking. The decorrelation step (16) is omitted. tracking of the parameters (1) and is iteratively performed at the UE, however the tracking of the eigenspace V (1) and the eigenbeams W d (16), respectively, is accomplished at the BS. Hereby, the size of the feedback signalling is solely determined by the number of parameters according to, i.e. M (M +1) incremental rotation angles 5. The proposed feedback signalling would offer a number of beneficial facets: The numerical complexity of the tracking algorithm is lower or equal than for standard techniques [1]. The size of feedback signalling is independent from the number of dominant eigenbeams d (Table 1). The choice between updating and enables to differentiate between estimation of long-term and short-term properties of the spatially correlated fading channel. In addition to maintain the tracking algorithm at the UE the tracking of the projector matrix (15) is performed at the BS. 5 Note, in order to constrain the feedback effort the parameters and are transmitted from UE to BS in a rotatory modus without degradation of the estimation. Table 1: The number of required bits for M = ; 6; and 8 antenna elements to transmit d = or 3 eigenbeams from the UE to the BS by means of 3 bits per parameter increment. The figures in () equal the number of required bits to feedback the eigenvectors by means of 3 bits per real/imaginary part of a vector element. M d = 1 d = d = 3 30 () bits 30 (8) bits 30 (7) bits 6 63 (36) bits 63 (7) bits 63 (108) bits (5) bits 108 (108) bits 108 (16) bits Although the increase of required parameters grows quadratic with the number of antenna elements instead of the linear increase with standard feedback of eigenvectors, the size of feedback signalling for realistic implementations is even lower (Table 1). Since the proposed subspace tracking algorithm converges very rapidly to the true eigenspace, the initialization of eigenbeams at the BS is not required [8].

5 mean Ec/Ior.5 PCA (0.5) PCA (0.9) PCA (0.995) 3 Tracker (0.5) Tracker (0.9) Tracker (0.995) speed v Figure 3: Performance of downlink eigenbeamforming based on the standard PCA method and the subspace tracking method for the case of three different forgetting factors = 0:5, 0:9, and 0: Simulation Results The signal model of the user of interest used for the following simulations is described by x(t) = NX n=1 w H i h n(t)s(t? n); where N denotes the number of temporal taps, w H i corresponds to the i-th eigenbeam, s(t? n) is the signal, and n denotes the delay of the n-th tap. Moreover, the spatial channel impulse response for the n-th tap, hn(t), is generated according to hn(t) = R 1= n g n (t); where the correlation between antennas for the n-th tap is described by Rn and g n (t) is a normalized Gaussian fading process with Jakes power density spectrum, cf. [13]. For the simulations, we chose M = antenna elements and a frequency flat channel, i.e., N = 1, where the (Hermitian) spatial covariance matrix corresponds to R = 6 1 :7e?j: :1e j1: :e?j3:0 : 1 :7e?j: :1e j1: : : 1 :7e?j: : : : : The performance measure is given by the ratio of the power for the user of interest and the total TX power of the relevant base station which is required to obtain a raw bit error ratio of 10% for the user of interest. Basically, the simulation environment compares to that used in [7]. Figure 3 presents the proposed performance measure for different velocities of the UE 6 : 3, 10, 0, and 10 km/h. The downlink eigenbeamforming is either based on a standard PCA, or the proposed eigenbeam tracking algorithm. In both cases a quantization of 3 bits per real/imaginary feedback quantity has been applied. However, the simulation results do not yet consider the time delay between the UE and the BS due to the constrained feedback capacity in a realistic Tx concept 7. 6 Note that variations in performance as a function of the velocity also depend on the power control. Here, power control is optimized for a target SINR at the rake receiver output which not necessarily maximizes mean raw BER for a low mean transmit power. 7 Therefore the comparison is still biased to the benefit of the standard PCA method which needs approximately 1 & 1/ more feedback effort than the tracking algorithm. 5

6 Obviously, without considering the time delay the lower the forgetting factor the better the overall performance of the Tx concept. This may be interpreted as follows: In the case of # the long-term downlink feedback more and more supports the task of the short-term downlink feedback, but for the price of a worse long-term estimate of the dominant subspace. However, in the case of the eigenbeam tracking ( #) the performance is surprisingly even better than for the PCA method which computes the eigenbeams exactly and each time completely new on the basis of the currently estimated covariance matrix R. Note, that the proposed objective function (5) accounts for the estimation of long-term properties (signal subspace) of the radio channel even when using a low forgetting factor. It can be shown that independently from the forgetting factor minimizing the referred objective function J() estimates the signal subspace almost independently from the short-term fluctuations of the radio channel as if " in a standard PCA. Generally speaking, this is due to the fact that the representation of a linear projector 8, which maps a linear space onto a linear subspace, is of course independent from any rotation 9 of the basis system which lies within the subspace. 6. Conclusion We have presented a tracking solution to the downlink eigenbeamformer concept. In spite of the complex nature of the algorithm it offers a straightforward implementation with low complexity and a number of benefits for the feedback signalling. The reliability of the downlink eigenbeam tracking has been approved by simulation results. At the conference session we will present further results considering a realistic time delay between the UE and the BS. 8 Note, that a linear projector defines the corresponding linear signal subspace uniquely. 9 Note, that the short-term fluctuation of the radio channel may be interpreted as a rotation of the short-term estimates of the eigenvectors of the corresponding long-term covariance matrix. Appendix Given the objective function J(()) =?tr W H () H R()W it has been shown that ; (17) the matrix Do R MM with off-diagonal rotation increments and Do;k;` =?Do;l;kj k6=` ; the vector of rescaling increments do R M with k can be obtained as a function of P and R [8]: Do do = Real G T? G ; (19) = Imag fdiag[g]g ; (0) where G = P R, and the partial differentials of J are taken at k;` = 0 and k = 0, respectively. Note, that only diagonal elements and above of G are relevant. Acknowledgement The authors would like to thank Josef A. Nossek, Institute for Circuit Theory and Signal Processing, Technische Universität München, for his support and the valuable dicussions and inputs on the topic of adaptive array processing. The authors also thank Joachim S. Hammerschmidt, Institute for Integrated Circuits, BRIDGELAB, Technische Universität München who made parts of the channel model and the simulation tools available. References [1] Third Generation Partnership Project (3GPP), 3G TS 5.1, [] M. Raitola, A. Hottinen, and R. Wichman. Transmission diversity in wideband CDMA. In Proc. 9th IEEE Vehicular Technology Conf. Spring (VTC 99 Spring), pages , Houston, Texas, May

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