Guaranteed Performance Region in Fading Orthogonal Space-Time Coded Broadcast Channels

Recently, the capacity region of the MIMO broadcast channel (BC) was completely characterized and duality between MIMO multiple access channel (MAC) and MIMO BC with perfect channel state information (CSI) at transmitter and receiver was established. In this work, we propose a MIMO BC approach in which only information about the channel norm is available at the base and hence no dirty paper preceding (DPC) can be applied. However, a certain set of individual performances in terms of MSE or zero-outage rates can be guaranteed at any time by applying an orthogonal space time block code (OSTBC). The guaranteed MSE region without superposition coding is characterized in closed form and the impact of diversity, fading statistics, and number of transmit antennas is analyzed. The guaranteed MSE region with superposition coding is also studied. Finally, the guaranteed sum MSE is briefly discussed.

Abstract-Recently, the capacity region of the MIMO broadcast channel (BC) was completely characterized and duality between MIMO multiple access channel (MAC) and MIMO BC with perfect channel state information (CSI) at transmitter and receiver was established. In this work, we propose a MIMO BC approach in which only information about the channel norm is available at the base and hence no dirty paper precoding (DPC) can be applied. However, a certain set of individual performances in terms of MSE or zero-outage rates can be guaranteed at any time by applying an orthogonal space time block code (OSTBC). The guaranteed MSE region without superposition coding is characterized in closed form and the impact of diversity, fading statistics, and number of transmit antennas is analyzed. The guaranteed MSE region with superposition coding is also studied. Finally, the guaranteed sum MSE is briefly discussed.

I. INTRODUCTION
Wireless multiuser systems are characterized by different performance measures. The choice of the performance measure depends either on the fading characteristics (e.g. fastor slow-fading correspond to ergodic and outage capacity [1]) or on the type of service (e.g. elastic or non-elastic traffic correspond to average and outage or zero-outage capacity. Consider the downlink broadcast channel (BC). In [2] the ergodic BC region was analyzed. Further on, in [3] the zerooutage BC region was studied, where code division with SIC (CD) and without SIC (CDWO) are the full interference cases. The optimality of allowing for full interference is also shown in [4]. With respect to the uplink multiple access channel (MAC), in [5] the ergodic MAC region, and in [6] the delaylimited capacity (DLC) region were characterized. A useful property for the analysis and optimal power allocation is the polymatroid structure of the capacity region [5], [6]. In [7], the capacity region with minimal rate requirements of the fading BC is studied. Recently,in [8], the capacity of fading broadcast channels with rate constraints is analyzed. More recently, the performance under these hard fairness constraints was compared to the performance of the proportional scheduler in [9]. All these results were derived under the assumption of perfect channel state information (CSI) at the base as well as at the mobiles.
We consider the downlink and assume that information about the average channel power instead of perfect CSI is available at the base as well as perfect CSI at the receivers. This is a form of partial CSI which can be achieved by norm-

Aydin Sezgin and Arogyaswami Paulraj
Information Systems Laboratory Stanford University, USA sezgin,apaulraj @ stanford.edu feedback. Then the base applies an orthogonal space-time code (OSTBC) and can either apply superposition coding or dirty paper precoding (DPC).
The disadvantage of the notion of delay-limited capacity or zero-outage capacity is that capacity in general can only be achieved with long codes. In contrast, the mean square error (MSE), 0 < MSE < 1, for the linear multiuser MMSE receiver can be computed for each transmitted symbol. This implies, that the approach using the polymatroidal structure of the capacity region cannot directly be used. In this work, we study the guaranteed MSE region' in a fading BC under longterm sum power constraints. All MSE tupels that lie in the guaranteed MSE region can be achieved for all joint fading states and for each transmitted symbol vector.
We compare the cases where the mobiles are either assumed to perform successive decoding or treat all other user signals as noise. For single-input single-output Rayleigh fading channels, it turns out that the guaranteed MSE point is the tupel (1, 1, ...1). Thus, in order to achieve non-trivial MSE points, spatial diversity has to be exploited. Since full CSI feedback seems impractical, only the channel norm is fed back from the mobiles to base and an OSTBC is applied at the transmitter. The optimality of a full-rate OSTBC has been shown for the MIMO BC without CSI at the base in [10].
The contributions of the paper are the following issues.
. Optimal resource allocation with and without successive decoding to guarantee MSE requirements in all fading states with minimal long-term sum transmit power. Closed form characterization of full interference guaranteed MSE region. . System concept for how to achieve non-unity guaranteed MSE region by utilizing OSTBC and limited channel quality indicator (CQI) feedback. . Analysis of guaranteed sum MSE: impact of spatial correlation, number of transmit antennas and moment constraints on achievable guaranteed MSE region.

II. SYSTEM MODEL AND PRELIMINARIES
The system model in figure 1 consists of K mobile users and one base station. Each user K requests a certain QoS level that has to be fulfilled throughout the transmission in 'This region could also be called delay-limited MSE region.
1-4244-1429-6/07/$25.00 ©2007 IEEE The base station has multiple antennas (nf), denote the channels to the users as hl,..., hK. The base applies an OSTBC [11] as shown in figure 2. The data streams di, ..., dK of dimension 1 x nT of the K users are weighted by a power allocation P1, ---,PK and added before they come into the OSTBC as sl,...,snT.
Each mobile first performs channel matched filtering according to the effective OSTBC channel. Afterwards the received signal at user k is given by at the base station as P, i.e.

Lk=1
The noise power at the receivers is or 2 1 The transmit k p' power to noise power is given by SNR= Pp which is called transmit SNR.
The mobiles feed back their fading coefficient al, ..., aK to the base and we assume these numbers are perfectly known at the base station. The base has perfect information about the channel norm. Further on, in the case with SIC at the mobiles, we assume that the signals X 1, ..., XK are encoded by e.g. superposition coding and the mobiles perform ideal SIC.

III. GUARANTEED PERFORMANCE REGIONS
For non-elastic traffic, like video stream or gaming services, a certain performance measure has to be guaranteed for all channel states. The MSE is a measure which works on a symbol by symbol basis. Therefore, hard delay constraints can be nicely expressed in terms of guaranteed MSE requirements. Since also many other performance measures can be mapped to the MSE, we study the guaranteed MSE region in this section.

A. Guaranteed MSE region without SIC
The individual instantaneous MSE of user k without precoding is given by Pak 2k~1 Pipaps (2) with the instantaneous sum power P5 = K1 pk. The following result describes the guaranteed MSE region without SIC and full collisions. (1) with fading coefficients ak = ak = lhk 2, transmit signal x1 intended for user I and noise nkr. We assume that the fading processes of user k and I for k 1 are independently distributed. Let Pk be the power allocated to user k, i.e. Pk = E[ Xk l2]. Denote the long-term sum transmit power constraint 2There are nT parallel streams for each mobile. However, all streams have the same properties and the same interference. Therefore, we restrict our attention without loss of generality to the first stream.

Ii2±SNR
The hatched area is the guaranteed MSE region. It is lower bounded by the line through ml(O) and m2(0) in (11) and (10). The dashed line in figure (3) corresponds to the feasibility condition in (4). Note that MSE tupels in which one or more components are greater than one are not achievable. Therefore, the guaranteed MSE region is inside the unit box. 2) Impact of diversity andfading statistics: The guaranteed MSE region depends on the expectations E [ 1 1 for 1 < k < K. Therefore, the expectation has been analyzed in [12] with respect to spatial correlation. The results apply also to the multiuser setting.
Corollary 2: The guaranteed MSE region with full collisions shrinks with increasing spatial correlation at the mobile terminals.

This follows from [12, Theorem 111.1] and (3).
For OSTBC using nT antennas, the expectation in (3) For nT approaching infinity the first term on the RHS goes to one. The SNR in (12) converges then to the required SNR of an ideal AWGN BC. However, the rate achievable with OSTBC tends to one half [13]. Note that the rate loss is characterized by [14] as nT2± L + 1 r-C(nT) -12Ln2 (13) One the one hand, increasing diversity has the positive effect on improving the LHS of (12) but also the negative effect by decreasing the code rate. This tradeoff is analyzed for singleuser systems in [15]. Assume rl = r2 = ... = rK = R. From (8) follows Proof: Assume that the channel realization are ordered according to a, > a2 > ... > aK. The cases that two or more realizations have equal power have zero probability. According to (16), the achievable MSE with power allocation Pk are given by SNR > nT 1 1 (14) nTr 1 -K(1 2 -J(T)) J In (14) the first term on the RHS decreases with increasing nT. The second term increases with increasing nT. The analysis of this tradeoff is beyond the scope of this paper, but (14) indicates that increasing the number of transmit antennas may decrease the guaranteed performance region. 4) Moment constraints: Additional moment constraints P. that limit the £-th moment of the transmit power probability distribution E [PI] < Pe lead to the following guaranteed MSE region Tnk)PE ) < Pf (15) Note that for diversity systems the expectation in (15) is only if f + 1 diversity branches, e.g. transmit antennas available [12].
In conjunction with the code rate loss that was studi( the last section, additional moment constraints lead direct a code rate loss and to a smaller zero-outage rate region B. Guaranteed MSE region with superposition coding anG If the user apply successive decoding without error pi gation, the MSE of user k is given by Tnk PkPak 1 + aZkPpk + (kp E Pi lIESk with the interference set Sk containing all users nol subtracted, i.e. Sk(al, ..,aK) {1 < I < K: a > ak} Sort the fading channel realizations by a,1 > a°72 > a7KK. Denote the probability that a certain order 7 o possible K! orders occur by p(7). The set of the K! o: is denoted by P. The function 1(x) is the indicator func i.e. 1(x) = 1 if event x is true or 1(x) = 0 if event x is I  figure 4, the guaranteed MSE region using superposition yet coding and SIC and without SIC are compared for the symmetric fading scenario and two transmit antennas nT = 2. The (17) gain by superposition coding and SIC is visible especially for high SNR. The corresponding zero-outage capacity region is * > convex for superposition coding and SIC whereas it is concave f all without [3]. Irders The third curve in figure 4 shows the guaranteed MSE ,tion, region achieved without SIC and beamforming with perfect false. CSI at the transmitter (BFWO). The optimal beamformers and e de-power allocation is found according to [16,Section 4.3.2]. The fourth curve shows the region achieved with SIC and optimal beamforming (BF) with perfect CSI at the transmitter.
It can be observed that for small SNR the beamforming gain weights more than the nonlinear precoding and BFWO as well as BF outperform CD and CDWO. However, for SNR (18) of 10 dB, there is an intersection between the BFWO and the CD curve. The reason for this behavior is that the system gets interference limited rather than power limited for higher SNR. IV. GUARANTEED SUM MSE In this section, we are interested in only one point of the MSE region, namely the sum MSE point. From a sum MSE point of view, both scenarios with and without superposition coding and SIC have the same optimal power allocation, namely only the best user is supported at each time. The guaranteed sum MSE is then given by MO (22) with 3 to E [maxik ! hk j* The expectation is simplified [7] k= with cumulative distribution function (CDF) Fk(t) of user k and channel coefficient hkl 2. For a short motivation and comparison with other sum performance based measures, the interested reader is referred to section 3.3 in [17].
Let the CDF of user k be given by Fk (t) (1 -Yr(n)) with the incomplete Gamma function -(n, x) L Xtn-lexp(-t)dt. Obviously, decreases with increasing number of users K and increasing number of transmit antennas n.
Note that the guaranteed sum MSE point cannot be identified in the guaranteed MSE region that was derived in section III. Since only the best user is supported at any one time instance, the guaranteed individual MSE is zero for all users. The performance measure in (22) is more system oriented. This difference is highlighted if the number of transmit antennas at the base is increased limn, 1(K, n) = 0 and the guaranteed sum MSE approaches zero.
Further on, the guaranteed sum MSE in (22) is smaller than 1 even for two Rayleigh fading users. The multiuser diversity provided by choosing always the best user provides the necessary degrees of freedom to keep the expectation of the inverse effective channel gain in Q finite.

V. CONCLUSION
The guaranteed MSE region of the orthogonal space-time block coded MIMO BC was characterized in closed form for the case without superposition coding in Theorem 111.1 and implicitly for the case with superposition coding in Theorem 111.2. Interestingly, it is possible to guarantee MSE values smaller than one for K > nT even in the linear precoding setting. The inherent code rate loss of OSTBC causes the guaranteed MSE region not to monotonically grow with increasing the number of transmit antennas. Further on, spatial correlation decreases the guaranteed MSE region in general.