Approaching Eigenmode BLAST Channel Capacity Using V-BLAST with Rate and Power Feedback

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1 Approaching Eigenmode BLAST Channel Capacity Using V-BLAST with Rate and Power Feedback Seong Taek Chung, Angel Lozano, and Howard C. Huang Abstract- Multiple antennas at the transmitter and receiver can achieve enormous capacities by transmitting on the channel s eigenmodes when the channel realization is known at the transmitter. If the transmitter has no knowledge of the channel, a significant fraction of the eigenmode capacity can be achieved in an open-loop mode, but multidimensional coding is required. In this paper, we show how the open-loop capacity can be achieved with conventional single-dimensional coding using Optimum Successive Decoding (OSD) and simple per-antenna rate control. Using power allocation, the capacity can be further increased, although only slightly. I. INTRODUCTION Information theory has shown that the richscattering wireless channel can support enormous theoretical capacities if the multipath propagation is properly exploited using multiple transmit and receive antennas [I]. In order to attain the largest capacity supported by such architectures, it is necessary to signal through the channel s eigenmodes with optimal power and rate allocation across those modes [2]. Such an approach requires instantaneous channel information at the transmitter and, hence, a closed-loop implementation. Furthermore, since it requires a very specialized transmit structure, incorporating it into existing systems may be problematic. Open-loop schemes that eliminate the need for instantaneous channel information at the transmitter have also been proposed [3], [4]. Although their capacity penalty is generally small [5], they require multi-dimensional coding whose detection complexity grows very rapidly with the number of antennas. Alternative signaling approaches wherein every transmit antenna radiates an independently encoded stream of data are much more attractive S. T. Chung is with STAR Labs., Stanford University, CA, 94305, USA. stchungodsl.stanford.edu A. Lozano and H. C. Huang are with Bell Labs., Lucent Technologies, NJ 07733, USA. { aloz,hchuang}@lucent.com from an implementation standpoint for the following three reasons. First, the transmitter can use a simple spatial demultiplexer which can be easily incorporated into existing systems. Second, those transmit structures enable the use of well-known successive detection techniques at the receiver [6]. Third, standard single-dimensional coding schemes can be utilized. Within this class of structures, open-loop schemes such as the Vertical BLAST (V- BLAST) architecture have been proposed [7]. In open-loop fashion, V-BLAST simply allocates equal power and rate to every transmit antenna and, as a result, it becomes limited by the antenna with the smallest capacity as dictated by the channel. As shown in [I], open-loop V-BLAST can attain roughly 50% of the open-loop capacity. In this paper, we introduce an extension of V- BLAST, which incorporates rate and power adaptation. We show that, with per-antenna rate adaptation, V-BLAST can achieve the same capacity available to much more complex structures [3]. Moreover, with per-antenna rate and power adaptation, V-BLAST can achieve even higher capacities and approach the limit attainable with eigenmode signaling. There are many similarities between the V- BLAST and the Gaussian Multiple-Access Channel (GMAC) problems. Every transmit antenna within V-BLAST can be regarded as an individual user in GMAC. It was shown in [9] that, with Optimum Successive Decoding (OSD), the total capacity of the GMAC at any vertex of the capacity region can be achieved. AS will be shown, this result can be directly translated to the V-BLAST context by simply incorporating the notion of per-antenna rate adaptation. The organization of this paper is as follows. The models are introduced in Section 11. In Section 111, we map OSD into V-BLAST and we derive the o p timal set of antenna rates when the antenna powers are given. In Section Iv, we find the antenna power /01/$ IEEE 915

2 allocation that maximizes capacity. We present an exact solution when the number of receive antennas is less than 3, and we suggest a sub-optimal solution for all other cases. Simulations results are shown in Section V and conclusions follow in Section VI. 11. SYSTEM MODEL Let us assume a general architecture with M transmit and N receive antennas and perfect chanriel information at the receiver. Rate and/or power information can be fed back to the transmitter. The M x 1 transmit signal vector is x and the N x 1 received signal vector is y. The N x M channel matrix I$ is composed by independent complex Gaussian random variables. The zero-mean AWGN vector at the receiver, denoted by n, has a covariance matrix equal to the identity matrix scaled by cr2. For simplicity, we assume cr' = 1 and scale the channel appropriately. The variance of each component of H is indicated by g, while the total power available to the transmitter is denoted by PT. We define an average SNR as p = PTg. This model can be expressed mathematically as y=hx+n (1) per-antenna rate and power control. The rate control scheme is based on that in Section 111. However, since the powers can also be adjusted, we expect the resulting capacities to be higher. When the number of receive antennas is less than 3, we find the optimum power and rate allocations. If the number is equal to or greater than 3, we rely on a sub-optimal scheme for the allocations. We describe the optimal and sub-optimal schemes in Sections IV-A and IV- B, respectively. The upper bound capacity of a multiple antenna system is a closed-loop technique with eigenmode signaling (eigenmode scheme). Here the actual channel realizations H are fed back to the transmitter, and the rate and power control scheme is based on [2] PER-ANTENNA RATE CONTROL In this section we present the rate allocation method among transmit branches derived from [9] which achieves the open-loop capacity. Here we assume that transmit power devoted to each antenna, P, is predetermined. The capacity of the ith antenna, Ci, can be expressed in terms of the channel matrix and the transmit power of each branch. Let us define h, as the mth column of H and H(m) (m=l,...,m) as the N x (M-m+l) matrix [h,h,+l.*. ~M-I~M]. Let us also define P(m) as an (M - m + 1) x (M - m + 1) diagonal matrix with (P,, P,+I,..., PM) along the diagonal. E[xxH] = - - PI P2..' 0 0 i i.. (2) 0 0.*. PM PM- According to the OSD (Optimum Successive Decoding) procedure described in (91, the signals radiated from the M transmit antennas are decoded in any agreed-upon arbitrary order. In the remainder, we assume, without loss of generality, that they are decoded according to their index order. It is interesting to note that, unlike in open-loop V-BLAST, the ordering has no impact on the capacity attained by the sum of all M antennas. It does, however, impact the fraction of that capacity that is all* cated through rate adaptation to every individual antenna. It also impacts a total rate when each rate and power is quantized [lo]. The process is parameterized by a set of projection vectors F, (m = 1,..., M), and cancellation vectors Brnl,Brn2,...,B,, (m = l,...,m- l), all of them of dimension N x 1. In decoding the mth transmit antenna signal, interference from the m - 1 already decoded signals is subtracted from 916

3 y by applying the proper cancellation vectors to re-encoded versions of their decoded symbols. An inner product of the result of that cancellation process and the projection vector corresponding to the mth antenna is fed to the mth antenna decoder. The first antenna, in particular, is decoded based on 21, which is obtained as the inner product of F1 and the receive vector Y1 = y expressed as 21 =< F1,Yl >= FIHY1. The decoded bits are re-encoded to produce X 2. The second antenna is similarly decoded based on 22, where Zz is now the inner product of Fz and a vector Yz obtained by subtracting the vector B11X1 from y. Therefore, YZ = y - BllXl and 22 =< F2,& >. In general, the mth antenna is decoded based on Zm =< Fm,Ym >= F,H(y - CGy'B(m-1)jXj). Here we assume that all decoded bits are error-free. The optimal cancellation vectors are given by B(m-llj = hj and the projection vectors admit the expressions Fm = (H(m + l)p(m+ 1)H(m + l)h + l)-'hm [9]. Furthermore, the transmit rate of the mth antenna can be expressed as c, =log2(l + P,h,H. (H(m + I)P(m + 1)H(m + 1lH + IN>-lhm) (m= l,...,m) (3) where h, is a scalar. Under the total power constraint, the optimal power allocation corresponds to assigning the entire power budget to the transmit antenna with the largest Ihml. When N=2, the total capacity can be expressed as Under the total power constraint, the optimal power allocation can be found using a Lagrangian method: and it was proved in [9] that M m=l Cm = log2 det(1n + HE[2zH]HH) (4) which, with equal power per antenna, is precisely the open-loop capacity attainable with vector coding [l]. Hence, the same capacity can achieved with single-dimensional coding, but at the expense of requiring per-antenna rate adaptation. Iv. V-BLAST WITH PER-ANTENNA RATE AND POWER CONTROL In this section, we consider methods to allocate the power P, (m=1,..., M ) under the total power constraint. For any set of powers Pm (m=1,..., M), the optimal rates are those given by (3). m= 1 (7) where J(P1,..., PM) is convex with respect to Pm. Hence, the optimal power assignment policy can be found from -& = 0 (1=1,..., M ) as follows: A. Optimal Scheme for N=l or N=2 When N=l, the total capacity can be expressed as 917

4 Since P, (m=l,..., M) should be positive, all hi + 1 equations (total power constraint and equations (8)) might not be satisfied with positive P, (m=l,..., M). In that case, by the Kuhn-Tucker Theorem, set one of the P, to zero and recalculake the other P, using the M equations. We cannot use 65 = o once we set certain P, to zero. ap, Then, set two of the P, to zero and recalculate the other P, using the M - 1 equations. Repeat the process and pick a transmission power set with the maximum total transmission rate. This algorithm works up to M = 4. Simulation results are shown in Section V. B. A Suboptimal Scheme for N > 2 We were not able to find the optimal power and rete allocations with N > 2. By solving the nth order linear equations, we can get the optimal power solution, but obtaining a closed form even for N = 3 is extremely complicated. However, from the optimal solution for N = 1 and N = 2 we observe the following: The optimal power allocation scheme usually corresponds to selecting 1 or 2 antennas while switching off the remaining ones completely. With suboptimal power allocations (e.g. equalpower allocation), the capacity loss is small. Based on these observations, we suggest a suboptimal power allocation algorithm that works for any combination of M and N. First, divide the total power PT by M and consider PTIM as a power unit. There are M such power units. Then, consider every possible power unit distribution over antennas, calculate the capacity of each one, and pick the one that yields the largest capacity elpenmode (upper lhmht) 4 optimal power altocaton wth rate mmml + equal power allocation Mth rate mntml + sub optimal attocatlon mth rate mnlrol + equal power and rate atlocallon (MMSE V-BLAST) averaw SNR p (db) Fig. 1. Average capacity when M=2 and N=2 Fig. 2. Average capacity when M=4 and N=2 1 V. SIMULATIONS We use the average capacity as a performance measure. The outage capacity also of interest, turns out to follow the same trend as the average capacity. Fig. 1-4 show such average capacity for various combinations of M and N. For each combination, the following cases are depicted: eigenmode capacity, optimal power allocation with per-antenna rate control, equal power allocation with per-antenna rate control, suboptimal power allocation with perantenna rate control, and equal power and equal rate allocation (MMSE V-BLAST). We observe that, as long as rate is controlled under OSD, equal power allocation across antennas works almost as well as the optimal (or subopti- mal) power allocation. Hence, power adaptation becomes largely irrelevant with per-antenna rate control. These results also show that the capacity loss from eigenmode transmission is not significant. (Except in Fig. 2, where the gap between eigenmode capacity and equal-power capacity is not reduced even though we increase the average SNR.) We conclude that equal power allocation combined with per-antenna rate control under OSD is a practical and efficient method to approach the eigenmode capacity. All the schemes proposed in this paper perform better than MMSE V-BLAST. The performance improvement is about 2 bps/hz in most cases. 918

5 channel when eigenmode signaling is utilized. As the number of antennas increases, power adaptation becomes irrelevant when the rates are properly adjusted. Although adaptation at the fading rate is required, the number of parameters to be adjusted with an equal-power policy is only M. On the other hand, the number of parameters required for eigenmode signaling is A4. N. REFERENCES [l] G J Foschini, and M J Gans, On limits of wireless Communications in a Fading Environment when Using Multiple Antennas, Wzreless Personal Communacatzons, No. 6, 1998, pp [2] I. E. Telatar, Capacity of Multi-Antenna Gaussian Channels, AT&T Bell Labs Technical Memorandum BL TM. [3] G. J. Foschini, Layered Space-Time Architecture for Fig 3 Average capacity when M=4 and N=4 Wireless Communications in a Fading Environment When Using multi-element Antennas, Bell Labs Tech. J., Autumn 1996, pp [4] V. Tarokh, N. Seshadri and A. R. Calderbank, Space- Time Codes for High Data Rate Wireless Communications: Performance Criterion and Code Construction, IEEE Trans. On Inform. Theory, Vol 44, March 1998, pp [5] F. R Farrokhi, G. J. Foschini, A. Lozano and R A Valenzuela, Link-Optimal BLAST Processing with Multiple-Access Interference, VTC 2000 Fall, Boston, MA [6] S Verdu, Multiuser Detection, Cambridge University Press,1998. (71 G. D Golden, G. J. Foschini, R. A Valenzuela, and P. W. Wolniansky, Detection Algorithm and Initial Laboratory Results using the V-BLAST Space-Time Communication Architecture, Electronic Letters, Vol. 35, No. 1, Jan. 7, 1999, pp [8] G J Foschini, G D Golden, R. A Valenzuela, and P W. Wolniansky, Simplified Processing for High Spectral Efficiency Wireless Communications employing Multiaverage SNR p (db) Element Arrays, IEEE JSAC., Nov 1999, pp Fig. 4 Average capacity when M=8 and N=8 [9] M. K Varanasi, and T. Guess, Optimum Decision Feedback Multiuser Equalization with Successive Decoding Achieves the Total Capacity of the Gaussian multiple- Access Channel, Aszlomar conference, 1998, pp VI. CONCLUSIONS we have introduced an improved form of Verti- [lo] S T Chung, H. C Howard and A. Lozano, Low Complexity Algorithm for Rate And Power Quantization in BLAST with per-antenna rate and Extended V-BLAST, to appear in VTC 2001 Fall, power adaptation that retains many of the attractive features of V-BLAST. Chief among those are the use of single-dimensional coding along with its easy multiplex transmit structure. We have shown that, with just per-antenna rate adaptation (equal power per antenna), this architecture attains the same capacity that can be achieved by open-loop architectures employing multi-dimensional codes. Furthermore, with per-antenna rate and power adaptation, even higher capacities can be attained. In rich-scattering environments, this scheme also approaches the ultimate capacity supported by the 919