Uplink and Downlink Beamforming for Fading Channels. Mats Bengtsson and Björn Ottersten

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1 Uplink and Downlink Beamforming for Fading Channels Mats Bengtsson and Björn Ottersten In Proceedings of 2nd IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, pp IR S3 SB 992 ROYAL INSTITUTE OF TECHNOLOGY Department of Signals, Sensors & Systems Signal Processing S STOCKHOLM KUNGL TEKNISKA HÖGSKOLAN Institutionen för Signaler, Sensorer & System Signalbehandling STOCKHOLM

2 UPLINK AND DOWNLINK BEAMFORMING FOR FADING CHANNELS Mats Bengtsson and Björn Ottersten Signal Processing, S3, Royal Institute of Technology Stockholm, SWEDEN Tel: , fax: ABSTRACT We highlight some issues in the design of beamformers for transmission and reception in communications systems with antenna arrays at the base stations. We assume a radio channel characterized by frequency flat Rayleigh fading which is correlated (but not coherent) from antenna element to antenna element. Different design criteria are considered and we show the relationship between downlink and uplink formulations and solutions. The resulting beamformers are typically given by quadratically constrained optimization problems which could be expected to give increased robustness to signal cancellation. Unfortunately, signal cancellation is still present, but diagonal loading can be introduced to minimize the problem.. INTRODUCTION When antenna arrays are used at the base station of a cellular system in environments with multipath propagation, the standard plane wave model used in much of the array processing literature is not applicable. As an alternative, some authors have used a fading channel where the fading process in independent between the different antenna elements [5, 6]. Since the assumption of independently fading subchannels only holds in environments with severe multipath or large separation between the antenna elements, we assume here a channel with correlated fading between the antenna elements. Such models can be used to model e.g. the diffuse scattering caused by reflections close to each mobile, see [, 8, 8]. We assume that the channel is constant within the time frame allocated to one data burst but varies randomly from burst to burst. When the antenna array is used as a receiver, i.e. in uplink mode, the instantaneous channel can be estimated directly from the received data, whereas in the downlink, the transmitting beamformer must be based on information collected in the uplink. Several schemes have been proposed for the transformation from uplink to downlink. In a Time Division Duplex (TDD) system with sufficiently short time slots, the downlink channel is virtually identical to the uplink channel, whereas in a Frequency Division Duplex (FDD) system, the channel fades independently at the two duplex frequencies. However, a statistical model of the downlink channel can be obtained from the collected uplink data using a physical model [5, 8] or model-free techniques [2, 9]. In this paper we assume that the statistics of the downlink channel is given, exactly or in the form of a noisy estimate. The maximum Signal to Interference plus Noise Ratio (SINR) solution has been used for different classes of problems, see for example [7]. For the model with local scattering, the uplink scenario is studied in [3] and the corresponding downlink formulation is given in [8]. Here, we show that the maximum SINR solution can be interpreted as quadratically constrained minimum variance beamforming [7, 2, 3]. Since the phase of the fading channel fluctuates randomly, a naive treatment of the Minimum Mean Square Error (MMSE) problem gives the all-zero solution. Using the assumption of a coherent receiver, we give two different solutions to the problem. For the downlink, it is not obvious how to formulate an MMSE problem and interpret the result. We propose to apply the uplink solution also in the downlink. Simulations have been performed to compare the different beamformers and study the sensitivity to modeling errors. Diagonal loading is successfully used to avoid the problem of signal cancellation. 2.. Uplink 2. DATA MODEL We assume a frequency flat Rayleigh fading channel, where the baseband data at the antenna array is collected in the complex valued Ñ ½ vector Ü u ص given by

3 Ü u ص Ú u ص Ò Øµ () where ص is the baseband signal transmitted at the :th mobile, Ò Øµ is spatially and temporally white Gaussian noise with covariance matrix Ò Ø ½ µò Ø ¾ µ Æ Ø½ Ø ¾ ¾ Ò Á (2) and the array response vector Ú u corresponding to mobile is complex Gaussian with 2.2. Downlink Ú u ¾ Æ ¼ Êu Ú µ (3) If the baseband signal transmitted at the antenna array is Ü d ص, then the signal Ö Øµ received at the :th mobile is given by Ö Øµ Ú d µ Ü d ص Ò Øµ (4) where Ò Øµ is temporally white Gaussian noise with variance Ò ¾ and the array response vector Ú d corresponding to mobile is complex Gaussian with Ú d ¾ Æ ¼ Êd Ú µ (5) In general, Ê u Ú and Ê d Ú can be full rank matrices, even though in many applications several of the eigenvalues are significantly smaller than the noise variance, which means that numerically, the rank can be considered lower than the number of antenna elements. Examples of this kind of models can be found in [, 8]. 3. ALGORITHMS We consider beamformers of the form ½ ص Û Ü u ص for estimation of the signal from mobile number one. Similarly, we use beamformers of the form Ü d ص Û ½ ص for transmission of the signal ½ ص to the first mobile. 3.. Maximal SINR In the uplink, the signal to interference plus noise ratio is given by [3, 7] SINR u Û Ê u Ú ½ Û Û È ¾ Êu Ú ¾ Ò ÁµÛ (6) Define the interference plus noise covariance matrix Ê ÁÆ È ¾ Êu Ú Ò ¾ Á. Then the maximum SINR beamformer is given by the eigenvector corresponding to the maximum eigenvalue of the generalized eigenvalue problem Ê u Ú ½ Û Ê ÁÆ Û (7) It is easy to show that Ê ÁÆ in (7) can be replaced by Ê u Ü Ê u Ú ½ Ê ÁÆ without changing the solution. Thus, the maximum SINR beamformer can alternatively be characterized by (up to a scaling) Ö Ñ Ò Û Ê u Û Ü Û (8) Ê u Ú Û ½ ½ which is a Quadratically Constrained Minimum Variance (QCMV) beamformer, closely related to the linearly constrained minimum variance or Capon beamformer [0, 4]. One tempting implementation of (8) is to estimate Ê u Ú ½ using a training sequence or a blind DOA based method and use this estimate together with the unstructured estimate Ê u Ü ½ Æ Æ Ø ½ Ü u ص Ü u صµ (9) in the calculation of Û. However, as will be illustrated in Section 4, this can easily lead to problems with signal cancellation. Just as for the Capon beamformer, several methods can be used to avoid this problem. One solution is to use a structured estimate also for Ê u Ü, as suggested in [], another is to use diagonal loading, i.e., to replace Ê u Ü by Ê u Ü Á, see e.g. [6]. For the downlink, a design criterion similar to (7) can be derived as the beamformer that gives maximum signal power at the desired mobile while keeping the total power transmitted to all other users below a certain threshold. The details can be found in [8]. Similarly, the criterion (8) can motivated as the beamformer that transmits the minimum total power to all users while keeping the power transmitted to the desired user at a fixed level MMSE A direct application of the data model () (3) on the Minimum Mean Square Error (MMSE) criterion would give the all-zero solution, since Ú u ¼ which results in ½ صÜu ص ¼. However, this is only a problem with the mathematical treatment, since a coherent detector can track the phase of the signal. In the traditional plane wave models, this problem is handled mathematically fixing one element of the array response vector to. This procedure cannot be directly used on the fading channel, so we give two alternative solutions to the problem. Perform an eigenvalue decomposition of Ê u Ú ½, Ê u Ú ½ Ñ (0) ½ ¾ Ñ () 2

4 Then, each realization of Ú½ u can be written as Ú u ½ Ñ (2) where are independent Rayleigh distributed random variables with ¾ and are independent uniformly distributed over ¼ ¾. Now, let Ü Øµ Ü u ص, then the MMSE beamformer for ½ µ given Ü Øµ is Û Ü Ü ½ Ü ½ ص» Êu Ü µ ½ ½ (3) The same Û can be applied to Ü u ص if combined with a coherent detector. Note that normalizing Ü Øµ also by a factor ½ ½ would give a data vector with infinite variance. An even better solution is to first consider the conditional MMSE solution given a specific realization of Ú u ½, Û cond. It is easy to show, using the matrix inversion lemma, that Û cond ¾ ½ Ú u ½ Úu ½ Ê ÁÆ µ ½ ¾ ½ Ú u ½» Ê ½ ÁÆ Úu ½ (4) and since the phase and the amplitude of the beamformer is irrelevant, we select the beamformer as Û Ö Ñ Ü Ú u ÛÛ ½ Û Û cond ¾ ½ Ö Ñ Ü Û Ê ½ ÛÛ ÁÆ Êu Ú ½ Ê ½ ÁÆ Û (5) ½ With this criterion, the optimal beamformer is thus given by the principal eigenvector of Ê ½ ÁÆ Êu Ú ½ Ê ½ ÁÆ. 4. NUMERICAL ILLUSTRATIONS We have studied an interference limited scenario with two mobiles subject to local scattering with rectangular angular distribution and a spread angle of Æ. We assume a uniform linear array with ½ ¾ wavelengths element separation, thus Ê Î is given by [4] ¾ Ô Ðµ Ó ÊÚ µ Ð ¾ е Ò Ò The signal of interest is kept at DOA ½ ½¼ Æ while the 0 db stronger interferer is moved between ¾¼ Æ and Æ. The SNR is 0 db compared to the signal of interest. Data is processed in bursts of Æ ½¼¼ symbols. The channel is constant within each burst but fades independently from burst to burst. The covariance matrices Ê Ú½ and Ê Ü are estimated from the last 0 bursts in order to average over the fading. We define a normalized MSE estimate as È Æ ÅË Ñ Ò Ø ½ ½ ص «½ صµ ¾ «È Æ (6) Ø ½ ¾½ ص The following algorithms were evaluated: Normalized MSE QCMV, true MMSE2, true BF, true QCMV, estim. MMSE, estim Source separation, degrees Figure : Estimated MSE and SINR, using the algorithms without diagonal loading. QCMV The QCMV formulation (8) of the maximum SINR beamformer. MMSE The eigen-decomposition based solution (3) to the MMSE problem. MMSE2 The best matched beamformer solution (5) to the MMSE problem. BF The traditional beamformer without any interference suppression, given by the principal eigenvector of Ê Ú½. Both the true and estimated covariance matrices were used. Figure shows the performance of the original algorithms, whereas in Figure 2, Ê Ü, (Ê Ü ) was replaced by Ê Ü ½¼¼ Ò ¾ Á, (Ê Ü ½¼¼ Ò ¾ Á) in the examples using the estimated covariance, in order to decrease the signal cancellation problems. All plotted results are averaged from 500 data bursts. 5. CONCLUSIONS AND SUMMARY We have illustrated the use of the maximum SINR and MMSE criteria in the design of beamformers for fading channels. The maximum SINR beamformer can be given a minimum variance formulation which is easier to calculate from measurement data. The MMSE formulation causes some problems for the particular data model and we have suggested two different solutions. For a point source scenario, both solutions reduce to the traditional MMSE beamformer. As is shown in Figure, the algorithms suffer from signal cancellation problems. However, when the data covariance matrix is regularized, the suggested algorithms improve significantly, as illustrated in Figure 2, although the 3

5 Normalized MSE QCMV, true MMSE2, true BF, true QCMV, estim. MMSE, estim Source separation, degrees Figure 2: Estimated MSE and SINR, using the algorithms with diagonal loading. theoretical performance drops. The only algorithm that does not react positively to regularization is the second MMSE solution (5) which performs similarly to the traditional beamformer when used with estimated channels (not included in the graphs for clarity). With known channels, MMSE2 performs slightly better than MMSE. The design of downlink beamformers is really a multiobjective optimization problem, but the uplink solutions can be given reasonable motivations also for the downlink problem. 6. REFERENCES [] F. Adachi, M. Feeney, A. Williamson, and J. Parsons, Cross correlation between the envelopes of 900 MHz signals received at a mobile radio base station site, IEE Proc., Pt. F, vol. 33, pp , Oct [2] T. Asté, P. Forster, L. Féty, and S. Mayrargue, Downlink beamforming avoiding DOA estimation for cellular mobile communications, in Proc. IEEE ICASSP 98, vol. 6, pp , 997. [3] M. Bengtsson and B. Ottersten, Signal waveform estimation from array data in angular spread environment, in Proc. ¼ Ø Asilomar Conf. Sig., Syst.,Comput., pp , Nov [4] M. Bengtsson and B. Ottersten, Rooting techniques for estimation of angular spread with an antenna array, in Proceedings of VTC 97, pp , May 997. [5] M. Bengtsson and B. Ottersten, On approximating a spatially scattered source with two point sources, in Proc. NORSIG 98, pp , June 998. [6] H. Cox, R. M. Zeskind, and M. M. Owen, Robust adaptive beamforming, IEEE Trans. ASSP, vol. 35, pp , Oct [7] M. Er and A. Cantoni, An alternative formulation for an optimum beamformer with robustness capability, IEE Proc., Pt. F, vol. 32, pp , Oct [8] R. Ertel, P. Cardieri, K. Sowerby, T. Rappaport, and J. Reed, Overview of spatial channel models for antenna array communication systems, IEEE Personal Communications, vol. 5, pp. 0 22, Feb [9] J. Goldberg and J. R. Fonollosa, Downlink beamforming for cellular mobile communications, in Proceedings of VTC 97, vol. 2, pp , 997. [0] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. Wiley, 980. [] B. Ottersten, R. Roy, and T. Kailath, Signal waveform estimation in sensor array processing, in Proc. ¾ Ö Asilomar Conf. Sig., Syst.,Comput., pp , Nov [2] F. Qian and B. D. V. Veen, Quadratically constrained adaptive beamforming for coherent signals and interference, IEEE Trans. SP, vol. 43, pp , Aug [3] B. D. V. Veen, Minimum variance beamforming with soft response constraints, IEEE Trans. SP, vol. 39, pp , Sept. 99. [4] B. D. V. Veen and K. M. Buckley, Beamforming: A versatile approach to spatial filtering, IEEE ASSP Magazine, pp. 4 24, Apr [5] A. J. Weiss and B. Friedlander, Fading effects on antenna arrays in cellular communications, IEEE Trans. SP, vol. 45, pp. 09 7, May 997. [6] G. W. Wornell and M. D. Trott, Efficient signal processing techniques for exploiting transmit antenna diversity on fading channels, IEEE Trans. SP, vol. 45, pp , Jan [7] J. Yang and A. L. Swindlehurst, Maximum SINR beamforming for correlated sources, in Proc. IEEE ICASSP 95, pp , 995. [8] P. Zetterberg and B. Ottersten, The spectrum efficiency of a base station antenna array for spatially selective transmission, IEEE Trans. VT, vol. 44, pp , Aug

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