Sum Capacity and TSC Bounds in Collaborative Multi-Base Wireless Systems

Transcription

1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL X, NO X, DECEMBER Sum Capacity and TSC Bounds in Coaborative Muti-Base Wireess Systems Otiia Popescu, Student Member, IEEE, and Christopher Rose, Member, IEEE Abstract We consider a wireess system with base stations which coaborate and derive bounds on sum capacity and tota squared correation for uniform channes between users and bases The paper aso investigates structura properties which must be satisfied by user transmit covariance matrices at the optima sum capacity/tsc point, and shows that for muti-base systems, maximizing sum capacity and minimizing TSC are in genera not equivaent probems Index Terms muti-base systems, coaboration, MIMO, sum capacity, tota squared correation I INTRODCTION We consider a muti-base wireess system consisting of mutipe users and base stations distributed over a given geographica area The avaiabe spectrum is shared by a users and bases as woud be the case in unicensed bands Furthermore, the stations are aowed to coaborate and we here make no particuar assignment of users to base stations Thus, transmitted signas from a users are observed at a bases, unike the usua ceuar setup which assumes no cooperation among bases, and in which users are observed ony at the bases with which they are associated As such, the ensembe of bases is ogicay a super receiver with mutipe distributed antennas The overabundance of optica fiber depoyed in the past but not yet used (so caed dark fiber ) makes such widespread coaboration pausibe, and as a specific exampe, imagine an abstraction of WiFi/8011 access points which can share information at high speed over a fiber backbone to do joint decoding Such a coaborative scenario has been considered in previous work deaing with systems with mutipe transmitters and receivers [], [3], [10] and provides upper bounds on various measures of interest since one can do no better than to jointy decode We consider two goba criteria, sum capacity and tota squared correation (TSC) which have been extensivey used to characterize the performance of singe-base systems [1], [5], [9], [11], [1] In this paper we derive anaytic bounds on these metrics for muti-base coaborative systems in a genera signa space framework, and investigate structura properties that must be satisfied by user transmit covariance matrices for the specia case where the gains between any Manuscript received September 4, 00; revised February 3, 004 The editor coordinating the review of this paper is Giuseppe Caire This paper was presented in part at the 39 th Aerton Conference on Communication, Contro, and Computing The authors are with the Wireess Information Network Laboratory (WIN- LAB), Rutgers niversity, 73 Brett Rd, Piscataway, NJ Digita Object Identifier given user and any given base are identica over a timebandwidth signa space dimensions used by the system what we ca a uniform channe assumption Such uniform channe modes are appropriate when a singe path between any pair of transmitters and receivers is dominant, as we as for subchannes within a given coherence bandwidth The paper is organized as foows: in section II we introduce our mode for muti-base coaborative systems, and show that the uniform channe assumption impies a specia structure on the received covariance matrix In sections III and IV we present the main resuts of the paper: bounds on sum capacity and TSC impied by the specia structure of the received covariance matrix In section V we extend the resuts to consider carrier phase offsets II SYSTEM DESCRIPTION We consider a system with B base stations and L users distributed over a given area (described schematicay in Figure 1) for which we assume a common signa space representation of dimension N for a users/bases impied by finite bandwidth and finite signaing interva constraints [6] nike the usua ceuar scenario, we assume no particuar assignment of users to bases, and transmissions from a users are received by a bases 1 B /04$000 c 004 IEEE 1 B Fig 1 A muti-base system with L users and B base stations Trianges denote receivers/bases and circes denote transmitters/users 4 3 L B

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL X, NO X, DECEMBER 004 The received signa during an arbitrary symbo interva at base station j is written as r j G j x + w j j 1,,B (1) where x is the N-dimensiona codeword transmitted by user, r j is the N-dimensiona received vector at base station j containing the additive Gaussian noise vector w j G j is the N N gain matrix that characterizes the vector channe between user and base station j For simpicity of notation we assume genera gain matrices now and wi ater speciaize the gain matrices appropriatey for what we ca uniform channes We assume a (tempora) sequence of {x } is transmitted by user during successive symbo intervas and is decoded at the receiver Since we wi assume Gaussian channes, we assume a Gaussian codebook for the x which aows us to worry ony about the covariance matrix X E[x x ] when considering issues of capacity We consider two potentia sources of noise: 1) independent therma noise at the receiver n j with covariance matrix V j E[n j n j ], and ) noise from random emitters (possiby associated with other systems) from µ discrete geographic ocations e n, n 1,,µ, with covariances W n E[e n e n ], n 1,,,µ Thus, we can write µ w j n j + H nj e n () n1 with H nj representing the gain from emitter n to base station j, and the noise covariance matrix at base station j is written as µ W j E[w j wj ] V j + H nj W n H nj (3) n1 Our goa is to derive bounds on goba performance measures for this system, ike information theoretic capacity which characterizes achievabe data rates for reiabe transmission, or tota squared correation which characterizes the tota interference in the system In order to do this we assume a coaborative scenario in which received signas at a bases are coected and used for joint decoding since one can do no better than to jointy decode Assuming coaboration a BNdimensiona received vector is constructed by gathering a received vectors from a bases r 1 r B r with correation matrix R E[rr ] G 1 G B G x + w 1 w B w (4) R() + W (5) where matrix R() represents the user contribution to R and is expressed in terms of its transmit covariance matrix and corresponding gain matrices as R() G X G (6) and W is the covariance matrix of the resuting noise vector w Due to the structure of the noise at each base station the resuting noise vector is written as w 1 w B w n 1 n B n + µ n1 H n1 H nb H n e n (7) and its covariance matrix can be written as V 1 V µ W + H n W n H n (8) Thus we have R VB n1 G X G + W (9) We note that the received signa in equation (4) which characterizes the coaborative scenario corresponds to a base with mutipe antennas and singe-antenna users a type of MIMO system Joint decoding occurs through use of a backbone network as woud be the case for bases distributed ike 8011 access points connected to the Internet Since decoding is coaborative at a what amounts to an aggregate super-receiver, we can use goba performance measures such as information theoretic sum capacity or tota squared correation (TSC) as a measure of system performance Sum capacity characterizes the sum of achievabe rates for reiabe transmission by a users, and under Gaussian signaing and noise assumptions is expressed as [3], [14] Csum 1 og R 1 og W (10) The TSC characterizes the tota interference in the system and is computed at the super-receiver which coects received signas from a bases and forms r in equation (4) as the sum of squared correations between any two user received signas, that is TSC m1 E{[(G x ) (G m x m )] } ( L ) Trace G X G (11) This measure can be regarded as an extension of the TSC used for singe base systems [1], [5], [9], [11] to our coaborative muti-base system By treating the noise vector as coming from a virtua interferer and adding terms corresponding to the squared correations between a users and this additiona interferer (incuding the correation of the noise vector with itsef) we define the generaized TSC as the trace of the

3 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL X, NO X, DECEMBER squared received signa correation matrix ( L ) GTSC Trace G X G + W Trace [ R ] (1) We note that sum capacity is concave in R and GTSC is convex in R [5], [8], [9], [13] We assume that channes between users and bases are uniform, and characterized by identica gains across a signa space dimensions, that is G j g j I N, 1,,L, j 1,,B with I N the identity matrix of order N Such uniform channes coud arise in a variety of settings where a singe path (not necessariy ine of sight) predominates between transmitters and receivers, or for narrowband subchannes within some coherence bandwidth Thus, user gain matrices are expressed as which impies that R() G g 1 I N g B I N (13) g1 X g 1 g B X g B g 1 X gb X (14) and we assume that these gains are stabe for sufficienty ong sequences of x transmissions We then note that the uniform channe assumption impies that R has the foowing specia structure: 1) R is composed of N N sub-bocks ) Each sub-bock, R ij is trace constrained That is, Trace[R ij ] E ij g i g j Trace[X ] + Trace[W ij] P ω ij g i g j P + ω ij E ji (15) with P being the power corresponding to user We wi show that this specia structure impies a particuar form for the R which maximizes equation (10) (or minimizes equation (11)) We wi aso show that maximizing sum capacity and minimizing GTSC are in genera not equivaent probems, as it was the case for singe-base systems III BONDS ON SM CAPACITY Here we identify properties of the R which maximize sum capacity under our uniform gain matrix assumptions Note that we do not attempt to sove the probem of maximizing sum capacity in equation (10), and/or propose an agorithm that does this we refer readers to the paper by Yu et a [14] which soved the probem of maximizing sum capacity for a genera mutiaccess vector channe as a spectra optimization probem max Csum subject to Trace[X ] P, 1,,L X (16) and proposes an iterative agorithm for finding optima user transmit covariance matrices X which maximize sum capacity Rather, we investigate structura properties of matrix R at the maximum sum capacity point Since for stationary noise the noise covariance matrix is fixed, then maximum sum capacity in equation (10) impies that the determinant of the system covariance matrix R is maximized We note that in genera, maximizing R is subject to a trace constraint on R impied by the tota energy in the system However in our case, maximization of R is subject to additiona constraints on traces of sub-bocks of R as specified by equation (15) We aso note that whie in genera, a positive definite K K matrix A with a trace constraint has maximum determinant when it is a scaed identity matrix [4], that is max A A Trace[A] I K (17) K this is not the case with our received covariance matrix R, and uness the off-diagona bocks of R have zero trace, the covariance matrix R cannot be a scaed identity matrix Thus, we must seek the structure of R which maximizes R subject to the imposed trace constraints on individua N N subbocks in equation (15) The foowing mathematica resut, proven in [7], is usefu as we seek to maximize the determinant of our sub-bock traceconstrained matrix R: Theorem 1: Let Q NJ be the cass of symmetric positive definite matrices Q of the foowing form Q 11 Q 1 Q 1J Q Q 1 Q Q J1 Q JJ where the subscript N denotes the size of the square subbocks and J denotes the number of vertica and horizonta sub-bocks Q ij Q ji with trace constraint Trace[Q ij] E ij Then the determinant of Q is maximized when Q ij E ij N I N, 1 i, j J (18) and its maximum vaue is equa to Q 1 N NJ E N (19) where E is a J J symmetric matrix with eements {E ij } If we assume independent white noise/interference then each bock of W is of the form W ij ω ij N I N i, j 1,,B (0) and direct appication of Theorem 1 impies that g i g j X E ij ω ij I N (1) N

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL X, NO X, DECEMBER which is aways satisfied if user covariance matrices X are scaed identity matrices (athough there may be other soutions as we) Thus, for white noise/interference with sub-bock traces ω ij, sum capacity is maximized when a user transmit covariances are X P N I N () with impicit Gaussian codebooks for the {x } The corresponding maximum sum capacity vaue is C max N (og E og Ω ) (3) where E is the B B symmetric matrix with eements E ij in equation (15), and Ω is the B B matrix with eements ω ij Trace[W ij ] For coored noise, the trace constraints are identica to those in equation (1) and appication of Theorem 1 requires that g ik g jk X + W ij E ij N I N 1 i, j B (4) which, due to the symmetry of matrix R, can be regarded as a system of B(B + 1)/ matrix equations with L unknown covariances X, 1,,L In this case one needs to answer the foowing questions: a) does there exist a reaizabe/feasibe set of user transmit covariances {X }, 1,,L which satisfies the system of matrix equations (4), and b) if no such set {X } exists, what is the actua sum capacity maximizing set We note that, when a feasibe soution to the system of matrix equations in (4) exists, then equation (4) provides necessary and sufficient conditions for user transmit covariance matrices at the maximum sum capacity point That is, maximum sum capacity is achieved if and ony if user transmit covariance matrices satisfy the system of matrix equations (4), provided it has a soution, and maximum sum capacity vaue is C max 1 (N og E NB og N og W ) (5) If no set {X } that satisfies the system of matrix equations (4) exists 1, then the vaue in equation (5) is an upper bound Of course, the sum capacity maximizing set of user transmit covariance matrices can aways be obtained numericay through water fiing schemes [14] But more carefu characterization of when the bound can be achieved and providing tighter anaytic bounds when it cannot, woud be usefu and coud be the subject of future work IV BONDS ON GTSC In this section we identify the properties that are satisfied by the received signa covariance matrix R at the minimum GTSC point We provide bounds for GTSC and show that under certain circumstances, minimizing the GTSC in uniform channe muti-base systems is equivaent to maximizing sum capacity 1 most ikey when B(B +1)/ > L; otherwise a system with the number of equations ess than or equa to the number of unknowns has in genera at east one soution The foowing mathematica resut, aso proven in [7], is usefu since the matrix mutipications impied by GTSC and the specia structure of R ead to off-diagona sub-bock products whose Trace[ ] extrema are not obvious Theorem : Let A and B be two square matrices, such that A is positive definite, Trace[A] E A, and Trace[B] E B Then min B Trace[ BA 1 B ] E B E A for B E B E A A (6) In order to derive bounds on GTSC we first note that GTSC Trace [ R ] B B Trace[R ij R ji ] i1 j1 B i1 j1 B Trace [ R ] ijr ij (7) and then using Theorem with A I we obtain that the GTSC is minimized when each of the sub-bocks of R is a scaed identity matrix R ij E ij I, 1 i, j B (8) N and the minimum GTSC is obtained as B B Eij GTSC min N 1 N Trace[ E ] (9) i1 j1 with E the same matrix as in equation (3) When white noise/interference is assumed, each bock of W is an identity matrix as in equation (0), minimum GTSC can be obtained when a user transmit covariance matrices X are scaed identity matrices as in equation () For coored noise, equation (8) impies that the same system of matrix equations as in equation (4) must be satisfied by user transmit covariance matrices in order to minimize the GTSC As before, we note here that there may exist no feasibe set of user transmit covariance matrices that satisfy the conditions in equation (4), in which case equation (9) serves ony as a ower bound for GTSC We aso note here that as ong as a feasibe soution to the system of matrix equations (4) exists, then the same set of user transmit covariance matrices {X } wi both maximize sum capacity in equation (10) and minimize GTSC in equation (1) However, this is not true in genera, as can be seen from the foowing exampe in which a singe user is active in the muti-base system In this case there is no need for the user index, and the correation matrix of the composite received signa is written as where G g 1 I N g B I N R GXG + W (30) g 1 g B g I N g I N (31)

5 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL X, NO X, DECEMBER is the gain matrix that corresponds to the user, and can aso be expressed as a Kronecker product [4] between the B- dimensiona vector containing the uniform gain vaues to a bases g and the identity matrix of order N We can then define 1 g G (3) which is a matrix with orthonorma coumns, that is I N We aso define the orthonorma compement of as Ū and note that [ Ū ] I BN (33) Ū Appying the foowing simiarity transformation to R R [ Ū ] R where Ū [ g X + W 11 W 1 W 11 W W 1 WŪ W 1 Ū W W Ū WŪ ] W 1 W (34) (35) We can now write R using the Schur factorization [4] R W g X + W 11 W 1(W ) 1 W 1 (36) and Trace [ (R ) ] Trace [ ( g X + W 11 )] + (37) Trace[W 1W 1] + Trace[W 1W 1] + Trace [ (W ) ] When maximizing R in equation (36), the eigenvaues of the user covariance matrix X wi be determined by the eigenvaues of matrix W 11 W 1 (W ) 1 W 1 In contrast, when minimizing Trace [ (R ) ] in equation (37), ony the eigenvaues of W 11 matter Therefore, uness W 1 (W ) 1 W 1 0, the optimizing equation (36) and equation (37) in X may ead to different resuts This exampe shows that unike singe base systems for which sum capacity maximization and TSC minimization are equivaent probems [8], [9], for muti-base systems they are in the most genera case not equivaent, and may resut in different soutions However, so ong as R can be reaized with scaed identity sub-bocks, then maximizing sum capacity for muti-base systems wi be equivaent to minimizing the GTSC V INCORPORATING CARRIER PHASE DELAYS So far we have assumed compete synchronization at a receivers between a users, which may be justified in baseband by assuming sufficienty ong signaing intervas reative to the communication bandwidth aotted However, for carriermoduated signas, simpe propagation deay can cause signas moduated on the in-phase rai to appear on the quadrature rai at the receiver, and vice versa We aso note that reative phases can be compensated for a singe user, but that compensation for mutipe users with different deays to the same receivers is not possibe in genera for omnidirectiona transmission In this section we show that the same structura resuts derived for synchronized systems appy to uniform gain systems where propagation deay is considered expicity We start by assuming a set of baseband orthonorma waveforms that span the signa space of interest, which is impied by the aotted bandwidth and duration of signaing interva [6] To be consistent with the signa space representation used in the previous sections we assume N even, and et {φ i (t)}, i 1,,,N/ be the corresponding baseband orthonorma functions set Moduation with both cosπf c t and sin πf c t provides N passband orthonorma basis functions which can be used to represent user transmitted waveform x (t) x (1) φ 1 (t)cosπf c t φ 1 (t)sin πf c t x () x (n 1) φ n (t)cosπf c t φ n (t)sin πf c t x (n) x (N 1) x (N) φ N (t)cos πf c t (t)sin πf c t φ N using the N-dimensiona vector x (1) x () ḷ x x (n 1) x (n) x (N 1) x (N) (38) (39) We aso assume that baseband basis functions are not affected by the deay, that is φ i (t) φ i (t τ) for i 1,,, N/, which is reasonabe for typica propagation deays τ However, the carrier wi be affected and we can write cosπf c (t τ) cosπf c t cosπf c τ + sinπf c t sinπf c τ sin πf c (t τ) sin πf c t cosπf c τ sinπf c τ cosπf c t (40) which impies that for any pair of passband basis functions φ n (t)cos πf c t and φ n (t)sin πf c t we can write φn (t)cosπf c (t τ) φ n (t)sin πf c (t τ) [ cosθ sin θ φn (t)cosπf c t sinθ cosθ φ n (t)cosπf c t ] (41) where θ πf c τ, and is a standard rotation matrix which satisfies O 1 (θ) O( θ) Thus, after

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL X, NO X, DECEMBER propagation deay the signa vector that corresponds to the transmitted signa in equation (39) can be written as x Θ(θ)x (4) which shows that the effect of propagation deay is a pairwise rotation of signa components We now denote the carrier phase rotation that corresponds to the received signa from user at base j as θ j with the corresponding N N rotation matrix Θ j Θ(θ j ) We can then write the covariance matrix of user signa received at base j as Θ j XΘ j which impies that each N N bock R ij in the covariance matrix R can be written as R ij g i g j Θ i X Θ j + W ij (43) The foowing properties of rotation matrices, which can be easiy verified Trace[A] Trace[A] Trace[A] cosθ A R (44) O(θ i ) O(θ j ) O(θ i θ j ) (45) extend to matrix Θ(θ) in equation (4) as Thus Trace[AΘ(θ)] Trace[Θ(θ)A] A R N Trace[A] cosθ N even Trace (46) Θ(θ i ) Θ(θ j ) Θ(θ i θ j ) (47) Θ i X Θ j which impies that Trace[R ij ] E ij Trace X Θ j Θ i Trace[X Θ(θ i θ j )] P cos(θ i θ j ) (48) g i g j P cos(θ i θ j )+Trace[W ij ] (49) This shows that when carrier phase deays are taken into account, the traces of a sub-bocks R ij in R wi continue to be trace constrained as it was when no phase deays were considered, and Trace[R ij ] wi aso depend on the phase deay through θ i θ j πf c (τ j τ i ) (50) The fact that the N N sub-bocks of R which have the expression in equation (43) are trace constrained as shown by equation (49) ensures that the genera necessary and sufficient conditions in equation (4) derived in the previous sections aso hod when carrier phase deays are considered that is R is maximized when g i g j Θ i X Θ j + W ij E ij N I N 1 i, j B (51) And as before, it is possibe that no such set of X exists, in which case our resuts provide an upper/ower bound on sum capacity/tsc VI CONCLSIONS The overa structure of coaborative but geographicay dispersed bases is interesting in ight of the proiferation of consumer wireess systems ike 8011 and the amount of dark fiber avaiabe from past fiber (over)depoyments In this paper we consider an abstraction of such systems as mutipe coaborating base stations and uniform channes between users and bases and derived bounds on sum capacity and TSC via structura properties of the received covariance matrix We aso showed that as compared to singe-base systems where maximizing sum capacity and minimizing TSC are equivaent probems, in muti-base systems TSC and sum capacity optimization can ead to different resuts ACKNOWLEDGMENT We are indebted to our Editor, Giuseppe Caire, and the three anonymous reviewers for their constructive comments on an earier version of the paper REFERENCES [1] P Anigstein and V Anantharam Ensuring Convergence of the MMSE Iteration for Interference Avoidance to the Goba Optimum IEEE Transactions on Information Theory, 49(4): , Apri 003 [] A Godsmith, S Jafar, and G J Foschini Exporing Optima Muticeuar Mutipe Antenna Systems In Proceedings 00 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