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  1468 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 4, JULY 1998 Uplink Channel Capacity of Space-Division-Multiple-Access Schemes Bruno Suard, Guanghan Xu, Member, IEEE  , Hui Liu, Member, IEEE  , and Thomas Kailath, Life Fellow, IEEE   Abstract— Signal-to-noise ratio (SNR) and signal bandwidthhave been viewed as the dominant factors determining channelcapacity. In wireless communications, the channel capacity can beincreased for a given SNR and a given spectral region, by exploit-ing the spatial diversity provided by the use of multiple antennasand transceivers at a base station. In this paper, we calculatethe channel capacity enhancement of a so-called Space-Division-Multiple-Access (SDMA) system and investigate its dependencewith respect to different decoding schemes, terminal positions,and receiver numbers. Inner and outer capacity boundaries for joint decoding and independent decoding are presented, alongwith physical explanations as to how these boundaries can beachieved. We show that exploitation of the spatial diversity notonly increases the overall achievable rates of both joint andindependent decoding, but also closes the gap between theircorresponding capacity regions, thus bringing the performanceof the low-cost independent decoding scheme close to that of the  optimal  joint decoding. Practical issues of optimum projectionand power control are also briefly addressed.  Index Terms— Antenna array, channel capacity, multiple-access channel, power control, spatial diversity multiple access,wireless communications. I. I NTRODUCTION A prime issue in current wireless systems is the conflictbetween the finite spectrum available and the increasingdemands for wireless services. To mitigate this problem,various multiplexing schemes (e.g., time-division multipleaccess (TDMA) and code-division multiple access (CDMA))have been proposed to increase the channel capacity for a Manuscript received February 2, 1996; revised February 20, 1998. Thiswork was supported in part by the Air Force Office of Scientific Research(AFOSR) under Grant F49620-97-1-0318, the National Science Founda-tion CAREER Program under Grants MIP-9703074 and MIP-9502695, theOffice of Naval Research under Grant N00014-95-1-0638, the Joint Ser-vices Electronics Program under Contract F49620-95-C-0045, Motorola, Inc.,Southwestern Bell Technology Resources, Inc. and TI Raytheon. The work of B. Suard was supported in part by the National Science Foundation underContract DDM 8903385 and by U.S. Army Research Office under ContractsDAAL03-86-K-0171. The work of T. Kailath was supported in part by theJoint Services Program at Stanford University (U.S. Army, U.S. Navy, U.S.Air Force) under Contract DAAL03-88-C-0011 and by grants from GeneralElectric Company and Boeing ARGOSystem Inc. The material in this paperwas presented in part at the 28th Asilomar Conference on Signals, Systems,and Computers, Asilomar, CA 1994.B. Suard is with the Laboratory of Information and Decision Systems,Massachusetts Institute of Technology, Cambridge, MA 02139 USA.G. Xu is with the Telecommunications and Information Systems Engineer-ing, The University of Texas at Austin, Austin, TX 78711 USA.H. Liu is with the Communications and Signal Processing Laboratory,University of Virginia, Charlottesville, VA 22903-2442 USA.T. Kailath is with the Information Systems, Laboratory, Stanford University,Stanford, CA 94305-4055 USA.Publisher Item Identifier S 0018-9448(98)03753-5. given amount of spectrum. Most research efforts are focusedon searching for more efficient ways of using the existingtime–frequency resources [1]–[8]. Recently, it has been pro-posed that the use of multiple receivers (or of an antenna array)at the base station can significantly increase the channel ca-pacity by exploiting the spatial diversity among different users[5], [9]–[11]. Such a system has been referred to as Spatial-Diversity Multiple Access (or SDMA). Significant progresshas been made in incorporating antenna arrays into existingwireless communication systems. However, one fundamentalquestion remains unanswered: “What is (an upper bound on)the channel capacity of an SDMA system?” or “How tightlycan we pack information in space?”.In this paper, we present a study on the channel capacityof an SDMA system from an information-theoretic viewpoint.In particular, we derive the capacity region , i.e., the closureof the class of achievable rate vectors [7], of an antennaarray multiple-access system. Since the capacity region definesthe limit of error-free communications given certain channelcharacteristics, it is used as the ultimate measure of channelcapacity in this field. Our objective is to evaluate the increasein channel capacity made possible by using multiple receivers,and further to examine the system performance under differentwireless scenarios. The formulas we derived are based on anidealized one-cell model, which may seem restrictive, espe-cially given that interference from neighboring cells is criticalto cellular network operations. However, our motivation wasthat simple models can disclose the fundamentals of an SDMAsystem, and provide insights into system behavior.The paper is organized as follows. Section II presents amathematical model for and some basic assumptions on amultireceiver multiple-access Channel (MAC) that will beused throughout this paper. Section III derives the capacityregions for an SDMA system. A heuristic study on the channelcapacities of a simple two-user system is provided in SectionIV. The analysis is then extended to a general-user systemin Section V. In Section VI, several practical issues suchas optimal projection (combining) and capacity sensitivity topower variations are discussed.II. A M ATHEMATICAL M ODEL We consider a one-cell network where mobiles (users)transmit to the base station. We look at the mobile-to-base-station link (uplink) and model it as a multiple-access systemwith Gaussian channels. Further, multiple receivers are used atthe base station to exploit the spatial diversity among multipleterminals. 0018–9448/98$10.00 © 1998 IEEE  SUARD et al. : UPLINK CHANNEL CAPACITY OF SDMA SCHEMES 1469 Fig. 1. A typical array scene.  A. The Antenna Array Without loss of essential generality, we consider only auniform linear array, i.e., sensors uniformly spaced withdisplacement Fig. 1 shows a typical MAC with an arrayof receivers. For tractability in the subsequent analysis, weinvoke the following usual assumptions concerning the signaland noise, as well as the system configuration [6].ã The sources are sufficiently far from the array (far-field sources) that the wavefronts are planar. Also, theincoming signals are narrowband, 1 i.e., the basebandsignal does not vary over the length of the array exceptfor a phase shift.ã The noise components observed at all the receivers haveidentical, independent Gaussian distributions and are spa-tially and temporally white.ã The signals from different users are independent andGaussianly distributed and are temporally white.ã The array response to each source is slowly varying andhence is constant over many information bits.In the absence of noise, the array output vector correspond-ing to a single source can be written aswhere denotes transposition, andis the array response vector.When there is only a direct-path component associated with, two neighboring elements in differ only by a constantphase shift using complex representation, where isthe carrier wavelength and is the direction-of-arrival (DOA)of the incoming signal. Normalizing each element in withrespect to the first element yields(1) 1 This narrowband assumption is reasonable in cellular telephony since thecarry frequency is usually 800 MHz while the bandwidth of the informationtransmission from each terminal is usually less than 2 MHz. The above is termed as the steering vector  ; it is afunction of the DOA and the array configuration. In a direct-path-only environment, the array response vector is identicalto the steering vector (up to a complex scalar). In mostpractical situations, however, contains both the direct-pathand multipath components(2)where denotes the total number of paths associated with, represents the amplitude and phase difference be-tween the th multipath component and the direct path atTo facilitate our derivations in the remainder of this paper, wenormalize so that For a multiuser systemwith additive noise, can be written as(3)(4)where is the number of users, is the noise vector, andis defined as the array manifold  whose columns are thearray response vectors. For simplicity, we normalize the noisepower to unity and define(5)where denotes Hermitian. Consequently,(6)III. C APACITY R EGIONS OF SDMA S YSTEMS The problem under consideration is at what rates we can reliably transmit signals to an array of multiple receivers ina standard Gaussian channel with a given signal-to-noise ratio(SNR). Of course, the system capacity also depends on howthe decoding is performed. In this paper, we only discuss twotypes of decoding methods: joint decoding and independent decoding . Joint decoding means that decoding of all signals isperformed simultaneously, while independent decoding meansthat different signals are decoded independently and in parallel.Though joint decoding is optimal, it requires computa-tional complexity which may be too expensive to implement.Independent decoding, on the other hand, requirescomplexity and thus may be more practical. The fundamentaldifference between joint and independent decoding is thatthe former regards all as signals whereas the latterconsiders , , as interference when decoding  A. Capacity Region for Joint Decoding Let denote a subset of and its complement,i.e., We defineã (respectively, ) to be the matrix whose columnsare array response vectors of the sources in  1470 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 4, JULY 1998 ã (respectively, ) to be the covariance matrixof the sources inThe covariance matrix in (6) can be decomposed asUsing the above notation and recognizing that the systemdepicted in Fig. 1 is a multiple-access channel (MAC) withparallel channels, any achievable rate-tuplecorresponding to the users must satisfy(7)where denotes the mutual information betweenand , given Proof: The first part of the inequality comes from stan-dard information theory (see, e.g., [7], [12]).For the second part, we haveby the assumption of Gaussian noise withby the assumption of Gaussian signals and noise.Here and denote the entropy of a randomvector , and the joint entropy of two random vectors and, respectively [7].The convex hull of the region delimited by (7) is the capac-ity region. The above results can be equivalently expressedas......(8)  B. Capacity Region for Independent Decoding In the case of independent decoding, the MAC is an interfer-ing channel. The decoder for considers background noiseas well as other signal sources as interference. The achievablerate satisfies(9) Proof: Let andãããIV. T WO -U SER S YSTEMS With the above formulas, we can now investigate theperformance of an SDMA system under different scenarios.But first, let us build up some intuition by examining a simpletwo-user system. Results from this section will be extended tothe general case in the next section, where we examine SDMAsystem behavior and highlight some of its salient features.  A. Capacity Region From (8) and (9) we see that the capacity regions are clearlyfunctions of , , and of the users’ SNR’s. Since ,, and are matrices, theirdeterminants can be difficult to evaluate. We can alleviate thisdifficulty using the following well-known lemmas.  Lemma 1: For any matrices and(10)  Lemma 2: Let and be invertible, then(11)  Joint Decoding: By Lemma 1(12)(13)Scalar expressions for achievable rates of a two-user systemfollows immediately(14)where denotes the angle between the vectors andand(15)  SUARD et al. : UPLINK CHANNEL CAPACITY OF SDMA SCHEMES 1471 Fig. 2. Capacity regions for a two-user M  -receiver system.  Independent Decoding: Similarly, from (9) and with thehelp of Lemma 1, one can show that the capacity region forindependent decoding is a rectangle bounded by(16)(17)Typical capacity regions are depicted in Fig. 2. Clearly, boththe regions will depend upon the physical locations, or moreprecisely the array response vectors, of the two users. In thefollowing, we show that the capacity regions expand and varybetween two boundaries as the positions of the users change.  B. Effect of Users’ PositionsOuter Boundary: When the users’ positions are such thatthe array response vectors are orthogonal to one another,and the capacity region for joint de-coding is given by(18)(19)For independent decoding(20)These capacity regions are obviously maximal. Remarkably,the capacity region for independent decoding coincides withthat for joint decoding, which has a well-known rectangularshape [7]. The MAC is orthogonal, which means that by usingSDMA one can separate one signal from the other exactly.  Inner Boundary: When the users’ positions are such thatthe array response vectors are aligned with each other,and the capacity region of joint decoding Fig. 3. Capacity regions with different users’ positions. is given by(21)For independent decoding(22)These capacity regions are obviously the smallest over allpossible However, one does gain a factor of inSNR in joint decoding by using multiple antennas. This isbecause with sensors, the receiver gets replicas of thesignal, which can be added up coherently whereas noise addsup incoherently. On the other hand, there is not much gainin the capacity for independent decoding, especially whenand are low. The variation of capacity regions with respectto different users’ positions are illustrated in Fig. 3.V. -U SER S YSTEMS In this section, some important features of a -user MACare elucidated by theoretical derivations and are also high-lighted by numerical examples.  A. Effect of Users’ PositionsThe Outer Boundary: From the discussions in the previoussection, one may expect that the capacity region to be maxi-mum when the array response vectors are orthogonal to eachother. This expectation does hold, but only for certain limitedcases. In particular, it is only true if the number of users issmaller than the number of receivers. In the following, weshall use the symmetric channel capacity of joint decoding, i.e.,, to quantify the variation of the capacity regions.
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