Recent Advances on MIMO Processing. Mats Bengtsson, Cristoff Martin, Björn Ottersten, Ben Slimane and Per Zetterberg. June 2002

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1 Recent Advances on MIMO Processing in the SATURN Project Mats Bengtsson, Cristoff Martin, Björn Ottersten, Ben Slimane and Per Zetterberg June 22 In proceedings of IST Mobile & Wireless Telecommunications Summit 22 IR S3 SB 226 ROYAL INSTITUTE OF TECHNOLOGY Department of Signals, Sensors & Systems Signal Processing S-1 44 STOCKHOLM KUNGL TEKNISKA HÖGSKOLAN Institutionen för Signaler, Sensorer & System Signalbehandling 1 44 STOCKHOLM

2 Recent Advances on MIMO Processing in the SATURN Project Mats Bengtsson, Cristoff Martin, Björn Ottersten, Ben Slimane and Per Zetterberg Signal Processing Royal Institute of Technology S-1 44 Stockholm, Sweden Tel: , FAX: mats.bengtsson@s3.kth.se ABSTRACT We highlight some recent results on the theory and algorithms for MIMO systems, obtained at the Royal Institute of Technology (KTH) within the SATURN project. The paper includes a a new approximate expression for the ergodic channel capacity of a MIMO system with correlated fading, algorithms for MIMO beamforming under EIRP (equivalent isotropic radiated power) constraints and a low-complexity algorithm for MIMO channel estimation in OFDM systems. Also, we show how transmit and beamformers can be determined in a decentralized fashion without explicitly estimating the channel response. I. INTRODUCTION The SATURN project explores different aspects on the use of multiple antenna systems. In this presentation, we focus on the application in WLAN systems, in particular HIPERLAN/2 and similar OFDM based indoor communication systems. It is by now well-known that the concepts of MIMO (Multiple Input Multiple Output) processing for wireless communication can provide significantly improved performance, in terms of data rate and quality, both in theory and in practice. Still, there are many aspects to explore further before these promised gains can be implemented fully in a practical implementation. Here, we try to fill some of the gaps. The presentation starts in Section II with a theoretical analysis of the channel capacity of a MIMO system with correlated fading. Second, we look at low-complexity channel estimation techniques that can be implemented with only minor changes to the HIPERLAN/2 standard. Finally, in Section IV we study the problem of joint transmit and receive beamforming under EIRP constraints, where the solution can be found either based on a channel estimate or using a decentralized training algorithm. EIRP constraints are commonly believed to prevent any antenna gain, but on the contrary, we show a significant gain for realistic multipath channels obtained from indoor channel measurements performed by University of Bristol within the SATURN project [1]. This report is a part of the work done within the IST-SATURN project. II. ANALYTIC APPROXIMATIONS OF MIMO CHANNEL CAPACITY Channel capacity defines an upper limit of the rate at which data can be sent at arbitrary low probability of error. For a MIMO channel the expression for capacity is complicated, especially in cases where realistic channel models are used. This section summarizes the results of [2] where an approximation of MIMO channel capacity is derived. The approximation is able to handle a realistic indoor model [3] while it is still simple enough to provide intuition. A. Preliminaries Consider a narrow band flat Gaussian channel between an N T antenna transmitter and an N R antenna receiver. The received data x C NR is commonly modeled as (see e.g. [4]) x = Hs + n, (1) where s C NT contains transmitted data, H (the channel matrix) is an (N R N T ) complex Gaussian matrix while n is zero-mean complex Gaussian noise with E{nn } = I. In this paper we focus on MIMO channels where either N T or N R is large. Receiver and transmitter covariance matrices are defined as R Rx = E{h(k)h (k)}, k = 1,..., N T R Tx = E{(h (k)h (2) (k)) T }, k = 1,..., N R where h(k) and h (k) denote the kth column and row of H respectively. The channel is assumed zero-mean Gaussian distributed with covariance matrix R H = E{vec H vec H } = R Tx R Rx, (3) where denotes the Kronecker product if the data is normalized such that Tr R Tx = N T and Tr R Rx = N R. This model is proposed in e.g. [5] and recently validated from measurements within the project in [3, 6]. In the case where the channel is unknown at the transmitter it is known that, see e.g [4], with uniform power allocation, the expected channel capacity is given by { C = E log det ( I + P HH )} (4) N T

3 Simulated Capacity Asymptotic Approximation Capacity (bits/s/hz) rag replacements Simulated Capacity Asymptotic Approximation Capacity (bits/s/hz) PSfrag replacements N T ρ T = ρ R Figure 1: Mean channel capacity (P N R = 2, ρ T = ρ R =.5) =, 1, 2 db, Figure 2: Mean channel capacity (P N T = 4, N R = 2) =, 1, 2 db, where P denotes the signal to noise ratio at the receiver. In [4] an exact expression of (4) was derived for the case when all elements of H are uncorrelated (corresponding to R Tx = I and R Rx = I). However this result is too complex to give intuition of the capacity and also difficult to extend to the correlated case. In [7] useful bounds on the capacity are derived in the case where the fading is correlated at either the receiver or the transmitter (corresponding to R Tx = I or R Rx = I). In the high SNR case, an approximate expression for the correlated channel channel capacity is derived in [5] based on a limiting distribution of the eigenvalues. That distribution is derived under the assumption that N T = N R is large. Here, we derive a simple approximate expression of the mean channel capacity where correlation is allowed. Also, different number of antennas at both sides can be handled, since the expression is asymptotic in the number of antennas on one side. During the remainder of this Section, we consider a case where N T is large (N T > N R ). However, following the approach in [2] it should be straightforward to derive similar expressions for the case when the number of receive antennas is large. B. Approximate MIMO-Channel Capacity In [2] it was derived that the mean channel capacity can be approximated as N R C i=1 ln ( ( 1 + P λ R ) i P λ R i N T (1 + P λ R i ) ) 2 NT (λ T k) 2 k=1 where λ R i and λ T j denotes the ith and jth eigenvalue of R Tx and R Rx respectively. The approximation above is asymptotic in the number of transmit antennas and can be expected to improve as the number of transmit antennas increases. The approximate expression (5) has been compared with (5) simulations to verify that it is valid also for arrays of realistic sizes. The results of two of those simulations are displayed in Fig. 1 and 2. In order to understand the effects of correlation the transmitter covariance matrix R Tx was chosen according to R Tx = 1 ρ T ρ 2 T ρ NT 1 T ρ T 1 ρ T ρ NT 2. ρ NT 1 T... ρ NT 2 T 1 T (6) which might be a reasonable model for a uniform linear array in a rich scattering environment and is proposed in e.g. [8]. The receiver covariance matrix, R Rx, has a similar structure with ρ R instead of ρ T, ρ T < 1, ρ R < 1. From Fig. 1 the performance of the channel capacity approximations can be evaluated at various number of transmit antennas. Under the conditions present (equal correlation at transmitter/receiver, ρ T = ρ R =.5), the approximation predicts the mean capacity quite well even for a small number of transmit antennas. For high signal to noise ratios, N T must be larger for accurate prediction. In Fig. 2 the sensitivity of the capacity approximations to different correlation factors is investigated. The investigated system has equal correlation factors (ρ R = ρ T ) in both the receiver and transmitter correlation matrices. As expected, the approximations lose accuracy as ρ T grows. The simulations indicate that our approximation (5) is valid also for realistically sized arrays. Also the approximation is simple enough to provide insight to the problem. III. CHANNEL ESTIMATION FOR HIPERLAN/2 WITH TRANSMITTER DIVERSITY Many techniques have been proposed to estimate channel characteristics of OFDM systems with no diversity. These techniques are directly applicable to OFDM systems with receiver diversity when a different FFT demodulator is used after each receive antenna. However, such

4 techniques are not directly applicable to OFDM systems with transmitter diversity. To apply these methods several changes need to be done. For instance, one should assign a different training sequence to every transmit antenna. The next step, which is not straightforward, is to design these training sequences such that the channel state information of the different antennas are separated with minimum estimation errors. Gain db 15 1 Performance gain of ACO with sounding updating based on measured channels Max Median Min A channel estimation technique for HIPERLAN/2 with transmitter diversity was proposed in [9]. The objective was to separate and estimate all channel frequency responses, corresponding to each transmit and receive antenna pair, required for space-time processing in this OFDM based system. To achieve this objective, the proposed technique avoids using a different training sequence for every antenna. It keeps a common training sequence for all antennas and tries to exploit the structure of the OFDM modulation technique instead. The principle of this method is to feed the first antenna with the OFDM block sequence and to feed the second antenna with a cyclic shift of the same block. With this cyclic shifting of the OFDM block at one of the transmitter antenna, the receiver antenna will see a different training sequence for every antenna, the original training sequence for antenna one p 2 = {p, p 1,, p N 1 } with N the number of subcarriers, and a modulated version of the same training sequence for antenna two p 2 = {p }, p 1 e j2πd/n,, p N 1 e j2πd(n 1)/N where D is the cyclic shift applied at the transmitter side. As shown in [9], the obvious choice of the cyclic shifting is D = N/2. The obtained results showed that this technique is able to extract all channel state information for Multiple- Input Multiple-Output (MIMO) HIPERLAN/2 type systems with two transmit antennas without changing the preamble structure of the system. In that, no new pilot sequences and no modification of the used pilot sequence are needed. It was also shown that this channel estimation method works very well and provides accuracy comparable to and in some cases better than that of the single antenna case. The method proposed in this paper works for two transmit antennas and it will be interesting to extend it to larger number of antennas. IV. BEAMFORMING UNDER EIRP LIMITATIONS ISM and Hiperlan-bands have limitations on the maximum allowed equivalent isotropic radiated power (EIRP) of the transmissions. As a result of this, some of the research within the SATURN project has focused on achieving maximum performance under a constraint on EIRP. This section summarizes the results of [1, 11]. The work has assumed a MIMO WLAN-type system with time division duplex (TDD). Even though the theoreti Delay ms Figure 3: Performance of the ACO algorithm using sounding updating as a function of delay based on measured channels. cally predicted boost in link capacity requires using several spatial dimensions for each link, it may often be practically more feasible to use only one spatial channel for each link, considering the total interference level in the system as well as errors in channel estimation and feedback [12, 13]. Therefore, the approach considered so far has been beamforming at both the receiver and transmitter i.e. the transmitter distributes its signal over the transmit antennas using a vector of complex weights and the receiver, reciprocally, combines the signal from its receiving antennas using another complex vector. The beamforming strategies proposed change the antenna pattern in the order of a thousand times per second. The EIRP constraint is then implemented by scaling the beamformer weights such that this constraint is maintained for every single weight vector. For the EIRP constraints we have derived a method called: alternating convex optimization (ACO) that given an estimate of the full MIMO matrix channel attempts to maximize performance (taken as the SNR after combining in the receiving end) subject to an EIRP constraint. The ACO algorithm iteratively optimizes with respect to the receiver and transmitter weights. When calculating the transmitter weights, a convex optimization problem arises. This problem is solved using Se- DuMi [14], an efficient interior point algorithm for convex optimization. For single antenna receivers, the algorithm is guaranteed to find the global optimum, while it may theoretically get stuck in a local optimum for the general MIMO case. The approach can readily be extended to OFDM systems, where the EIRP constraint applies to the total transmitted signal. However, in OFDM systems, the received SNR can be optimized in several ways, either maximizing the total transmitted power summed over all subcarriers or maximizing the power of the worst subcarrier. When combined with interleaving and channel coding across the carriers, it is not obvious which strategy is the best, see also [15]. The matrix channel information can be obtained in a

5 CDF Received signal power averaged over frequency Figure 4: Multicarrier beamforming with EIRP constraints. Strategies from left to right: 1. SISO, optimizing worst carrier, 2. SISO, equal transmit power on all carriers, 3. MISO, individual EIRP constraints on each carrier, 4. MISO, optimizing worst carrier, 5. MISO, optimizing total received SNR. sounding-phase, for example using the algorithm of Section III. However, as we has also shown, it is possible to arrive at the same solutions without sounding using a second approach we call ping-pong updating. An advantage of this approach is that the overheads of sounding can be avoided. This solution has previously been touched at in [16]. In [16] however, the ping-pong approach was merely proposed as a numerical technique and not as system solution that avoids the overheads of feed-back. Figure 3 shows the 1%-percentile of the received SNR as a function of delay between the time of transmission and the time of the channel estimation at the transmitter, based on channel measurements performed by University of Bristol in a indoor open office environment. The results are based on 3*15=45 different base-mobile positions. The plotted gain is the gain in received signal power of a narrowband system with 4 transmit and 2 receive antennas compared to a system with a single transmit and 2 receiver antennas. In the figure, the min, median and maximum diversity gain results over all 45 cases are shown. Results with ping-pong updating are almost identical to those in the figure. As is evident from figure 3, the performance degradation is negligible for delays less than 13ms. This implies that the channel is temporally very stationary although there were occasionally moving people. However, the antenna used during the measurements was mounted on a pole, and this may not be representative of e.g. a notebook application since in this case the movements of the antenna caused by the user may come into play - this needs to be clarified by future studies. On simulated channels delays of.1/fd where fd is the Doppler frequency, are tolerable for both sounding and ping-pong updating. Figure 4 illustrates EIRP constrained beamforming in an OFDM system. A MISO (multiple input single output) setup is considered with a single antenna receiver and a transmitter equipped with 8 antenna elements. For reference a SISO system is included. The cumulative distribution function of the received signal power, averaged over all subcarriers, is plotted for a number of different strategies. Clearly, from Figures 3-4, multiple element transmitter antennas offer a substantial antenna gain even under EIRP constraints, contrary to the common belief that no antenna gain is possible with these constraints. Also, figure 4 illustrates clearly the diversity gain of antenna arrays, since distributing the power based on the worst carrier degrades the total received power significantly in the SISO case whereas the same strategy in MISO and MIMO is close to optimum. V. DECENTRALIZED CHANNEL DIAGONALIZATION According to the information theoretic channel capacity expressions, the optimal link capacity may be realized using the right and left hand singular vectors as linear transformations at the transmitter and receiver, respectively, to transform the channel into a set of independent scalar channels. This diagonalization or spatial multiplexing scheme requires perfect channel knowledge at both the transmitter and the receiver. One approach is to estimate the channel at the receiver, for example using the algorithm in Section III and feed back the information to the transmitter. However, it is possible to obtain the same information using a decentralized scheme without explicitly estimating the full channel. In contrast to the ping-pong principle described above, this is not an iterative approach, it only requires the transmission of one packet of data in each direction. Similar ideas have appeared in [17] but without fully exploiting the duplex character of the communication. The principle is extremely simple; to establish the link between node A and node B, transmit some spatially white data from node A. At node B, calculate the eigenvalue decomposition of the covariance matrix of the received data. Use the eigenvectors as linear transformation and the eigenvalues to determine the spatial waterfilling. The procedure is repeated in the other direction. Figure 5 illustrates the resulting intra-channel interference of the two principal eigenchannels (corresponding to the two largest singular values), stemming from the channel estimation errors. VI. CONCLUSIONS We have presented a mixed collection of results related to the use of multiple antennas for wireless networks and other radio based communication systems. For further conclusions, please refer to the end of each section and [2, 9 11]. REFERENCES [1] P. Karlsson, D. McNamara, and M. Beach, Indoor & campus single & dual multi-sensor characterisation: Measurement data, part 2: Indoor dual multisensor characterisation data, Tech. Rep. D522, part 2, IST SATURN, 2.

6 Signal to interference level, db First spatial channel Second spatial channel Raw SNR at each receiving end, db Figure 5: Performance of decentralized channel diagonalization. 5 transmit and receive antennas. [2] C. Martin and B. Ottersten, Analytic approximations of eigenvalue moments and mean channel capacity for MIMO channels, in Proceedings IEEE International Conference on Acoustics, Speech, and Signal Processing, May 22. [3] K. Yu, M. Bengtsson, B. Ottersten, D. McNamara, P. Karlsson, and M. Beach, Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz measurements, in Proc. GLOBECOM. IEEE, November 21, vol. 1, pp [4] I. E. Telatar, Capacity of multi-antenna gaussian channels, Technical Memorandum, Bell Laboratories (Published in European Transactions on Telecommunications, Vol. 1, No.6, pp , Nov/Dec 1999), [5] C.-N. Chuah, D. N. C. Tse, J. M. Kahn, and R. A. Valenzuela, Capacity scaling in MIMO wireless systems under correlated fading, IEEE Trans. IT, vol. 48, no. 3, pp , March 22. [6] M. Bengtsson, K. Yu, and B. Ottersten, Single and dual multi-sensor channel characterisation analysis and models stochastic models, Tech. Rep. D523, Part 1, IST SATURN, 21. [1] P. Zetterberg, M. Bengtsson, D. McNamara, P. Karlsson, and M. Beach, Downlink beamforming with delayed channel estimates under total power, element power and equivalent isotropic radiated power (EIRP) constraints, in Proceedings IEEE Vehicular Technology Conference, Fall, Oct. 21, vol. I, pp [11] P. Zetterberg, M. Bengtsson, D. McNamara, P. Karlsson, and M. Beach, Performance of multiple-receive multiple-transmit beamforming in WLAN-type systems under power or EIRP constraints with delayed channel estimates, Tech. Rep., Royal Institute of Technology, Feb. 22. [12] R. S. Blum, J. H. Winters, and N. R. Sollenberger, On the capacity of cellular systems with MIMO, in Proc. VTC21 Fall. IEEE, 21, vol. 2, pp [13] O. Gasparini, E. de Marinis, and M. Iarossi, Evaluation of MIMO spatial multiplexing for wireless LAN with channel models from experimental data, in Proc. IST Mobile Communications Summit, Thessaloniki - Greece, June 22. [14] J. F. Sturm, Using SeDuMi 1.x, a MATLAB Toolbox for Optimization over Symmetric Cones, Department of Quantitative Economics, Maastricht University, The Netherlands, nl/sturm/software/sedumi.html, [15] D. P. Palomar, J. M. Cioffi, M. A. Lagunas, and A. Pascual, Convex optimization theory applied to joint beamforming design in multicarrier MIMO channels, in Proc. GLOBECOM, 22, Submitted. [16] J. Andersen, Array gain and capacity for known random channels with multiple element arrays at both ends, IEEE Journal on Selected Areas in Communications, vol. 18, no. 11, pp , November 2. [17] A. S. Y. Poon, D. N. C. Tse, and R. W. Brodersen, An adaptive multi-antenna transceiver for slowly flat fading channels, submitted to IEEE Trans. Commun., March 2. [7] D.-S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, Fading correlation and its effect on the capacity of multielement antenna systems, IEEE Transactions on Communications, vol. 48, no. 3, pp , March 2. [8] S. L. Loyka, Channel capacity of MIMO architecture using the exponential correlation matrix, IEEE Communications Letters, vol. 5, no. 9, pp , September 21. [9] B. Slimane, Channel estimation for HIPER- LAN/2 with transmitter diversity, in Proc. IST Mobile Communications Summit, Barcelona, Spain, September 21.

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