Fig. 3. An example of the replica in[0; f ) for (a) f = 240 MHz without an ordering constraint, and (b)f = 417:778 MHz with an ordering con-straint.
[8] is more efficient. (The method in [8] is not compared in Table I as it does not compute minimum sampling frequency without ordering restriction.) Comparing Tables I and II, we can see that the minimum sampling frequency without a constraint can be much smaller than that with a constraint. Fig. 3 shows the replica in[0; fs) with a constraint
(fs;min= 417:78 MHz) and without a constraint (fs;min= 240 MHz)
when the bandpass signals are GSM 900 and GSM1800 as in the first case of Table II.
VI. CONCLUSION
We have proposed a new algorithm for finding the minimum sam-pling frequency for multiband signals. We have derived a new set of conditions for alias-free sampling. These conditions lead to an itera-tive algorithm for finding the minimum sampling frequency. There is no need to consider ordering of the signal bands in the folded spectrum in the implementation of algorithm. The method can be generalized to find alias-free sampling frequency intervals and to find the minimum sampling frequency when the ordering of replicas is constrained.
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Cooperative Interference Management With MISO Beamforming
Rui Zhang and Shuguang Cui
Abstract—In this correspondence, we study the downlink transmission
in a multi-cell system, where multiple base stations (BSs) each with mul-tiple antennas cooperatively design their respective transmit beamforming vectors to optimize the overall system performance. For simplicity, it is as-sumed that all mobile stations (MSs) are equipped with a single antenna each, and there is one active MS in each cell at one time. Accordingly, the system of interests can be modeled by a multiple-input single-output (MISO) Gaussian interference channel (IC), termed as MISO-IC, with in-terference treated as noise. We propose a new method to characterize dif-ferent rate-tuples for active MSs on the Pareto boundary of the achievable rate region for the MISO-IC, by exploring the relationship between the MISO-IC and the cognitive radio (CR) MISO channel. We show that each Pareto-boundary rate-tuple of the MISO-IC can be achieved in a decentral-ized manner when each of the BSs attains its own channel capacity subject to a certain set of interference-power constraints (also known as interfer-ence-temperature constraints in the CR system) at the other MS receivers. Furthermore, we show that this result leads to a new decentralized algo-rithm for implementing the multi-cell cooperative downlink beamforming.
Index Terms—Beamforming, cooperative multi-cell system, interference
channel, multi-antenna, Pareto optimal, rate region.
I. INTRODUCTION
Conventional wireless mobile networks are designed with a cel-lular architecture, where base stations (BSs) from different cells control communications for their associated mobile stations (MSs) independently. The resulting inter-cell interference is treated as
Manuscript received November 26, 2009; accepted June 14, 2010. Date of publication July 08, 2010; date of current version September 15, 2010. The asso-ciate editor coordinating the review of this manuscript and approving it for pub-lication was Prof. Wolfgang Utschick. This work was presented in part at IEEE Wireless Communications and Networking Conference (WCNC), Sydney, Aus-tralia, April 18–21, 2010. This work was supported in part by the National Uni-versity of Singapore under the research Grant 263-000-589-133.
R. Zhang is with the Institute for Infocomm Research, A*STAR, Singapore, and the Department of Electrical and Computer Engineering, National Univer-sity of Singapore, Singapore (e-mail: rzhang@i2r.a-star.edu.sg).
S. Cui is with the Department of Electrical and Computer Engineering, Texas A&M University, TX 77843 USA (e-mail: cui@ece.tamu.edu).
Digital Object Identifier 10.1109/TSP.2010.2056685
additive noise and minimized by applying a predesigned frequency reuse pattern such that the same frequency band is reused only by non-adjacent cells. Due to the rapidly growing demand for high-rate wireless multimedia applications, conventional cellular networks have been pushed towards their throughput limits. Consequently, many beyond-3G wireless technologies such as WiMAX and 3GPP UMTS Long Term Evolution (LTE) have relaxed the constraint on the frequency reuse such that the total frequency band becomes available for reuse by all cells. However, this factor-one frequency reuse pattern renders the overall network performance limited by the inter-cell interference; consequently, more sophisticated interference manage-ment techniques with multi-cell cooperation become crucial. Among others, one effective method to cope with the inter-cell interference in the cellular network is via joint signal processing across different BSs. In this correspondence, we study a particular type of multi-BS cooperation for the downlink transmission, where we are interested in evaluating the benefit in terms of network throughput by coopera-tively optimizing the transmit beamforming vectors for different BSs each with multiple antennas. Notice that the problem setup of our work is different from that for a fully cooperative multi-cell system considered in, e.g., [1]–[6], where a central processing unit is assumed with the global knowledge of all the required downlink channels and user messages to jointly design the transmitted signals for all BSs. In contrast, our work focuses on the decentralized implementation of the multi-cell cooperative downlink beamforming assuming only the local message and neighboring-channel knowledge at each BS, which is more practical than implementing the full baseband-level coordination. It is worth noting that decentralized multi-cell coop-erative downlink beamforming has been studied in [7] to minimize the total power consumption of all BSs to meet with MSs’ individual signal-to-interference-plus-noise ratio (SINR) targets, based on the uplink-downlink beamforming duality. In this work, we provide a different design approach for rate-optimal strategies in decentralized multi-cell cooperative beamforming.
For the purpose of exposition, in this work we consider a simplified scenario, where each MS is equipped with a single antenna, and at any given time there is only one active MS in each cell (over a particular frequency band). Accordingly, we can model the multi-cell cooperative downlink transmission system as a multiple-input single-output (MISO) Gaussian interference channel (IC), termed as MISO-IC. From an information-theoretic viewpoint, the capacity region of the Gaussian IC, which constitutes all the simultaneously achievable rates for all users, is still unknown in general [8], while significant progresses have recently been made on approaching this limit [9]. Capacity-approaching techniques for the Gaussian IC in general require certain signal-level encoding/decoding cooperations among the users, while a more pragmatic approach that leads to subop-timal achievable rates of the users is to allow only single-user encoding and decoding by treating the interference from other users as additive Gaussian noise. In this work, we adopt the latter approach to study the design of cooperative transmit beamforming for the MISO-IC. Partic-ularly, we focus on the design criterion to achieve different rate-tuples for the users on the Pareto boundary of the achievable rate region for the MISO-IC. Due to the coupled signal structure, the achievable rate region for the MISO-IC with interference treated as noise is in general a non-convex set,1 which renders the joint optimization of beamforming vectors to achieve different Pareto-boundary rate-tuples a challenging task. Note that this problem has been studied in [10], where for the special two-user case, it was shown that the optimal transmit beamforming vector to achieve a Pareto-boundary rate-pair for the MISO-IC can be expressed as a linear combination of the zero-forcing (ZF) and maximum-ratio transmission (MRT) beam-formers. The rate maximization for the IC with interference treated as noise has also been studied in the literature via various “pricing” algorithms (see, e.g., [11] and references therein), while in general
1It is noted that the non-convex rate region is obtained without time-sharing
(convex-hull operation) between different achievable rate-tuples. With time-sharing, the achievable rate region will become a convex set.
the price-based approach does not achieve the Pareto-optimal rates for the MISO-IC. In [12], the maximum sum-rate for the Gaussian IC is characterized in terms of degrees of freedom (DoF) over the interference-limited regime.
In this correspondence, we develop a new parametrical characteri-zation of the Pareto boundary for the MISO-IC in terms of the inter-ference-power levels at all receivers caused by different transmitters, also known as the interference temperature (IT) levels in the newly emerging “cognitive radio (CR)” type of applications [13]. We show that each Pareto-boundary rate-tuple can be achieved in a decentral-ized manner when each of the users maximizes its own MISO channel capacity subject to a certain set of IT constraints at the other users’ re-ceivers, which is identical to the CR MISO channel transmit optimiza-tion problem studied in [14] and thus shares the same soluoptimiza-tion struc-ture. We derive new closed-form solutions for the optimal transmit co-variance matrices of all users to achieve an arbitrary rate-tuple on the Pareto boundary of the MISO-IC rate region, from which we see that the optimal transmit covariance matrices should all be rank-one (i.e., beamforming is optimal).2Furthermore, we derive the conditions that are necessary for any particular set of mutual IT constraints across all users to guarantee a Pareto-optimal rate-tuple for the MISO-IC. Based on these conditions, we propose a new decentralized algorithm for im-plementing the multi-cell cooperative downlink beamforming. For this algorithm, all different pairs of BSs independently search for their mu-tually desirable IT constraints (with those for the MSs associated with the other BSs fixed), under which their respective beamforming vectors are optimized to maximize the individual transmit rates. This algorithm improves the rates for the BSs in a pairwise manner until the transmit rates for all BSs converge with their mutual IT levels.
Notation: III and 0 denote the identity matrix and the all-zero
ma-trix, respectively, with appropriate dimensions. For a square matrixSSS; Tr(SSS); jSSSj; SSS01, andSSS1=2denote the trace, determinant, inverse, and
square-root ofSSS, respectively; and SSS 0 means that SSS is positive semi-definite [16].Diag(aaa) denotes a diagonal matrix with the diag-onal elements given byaaa. For a matrix MMM of arbitrary size, MMMH; MMMT, andRank(MMM) denote the Hermitian transpose, transpose, and rank ofMMM , respectively. [1] denotes the statistical expectation. The dis-tribution of a circularly symmetric complex Gaussian (CSCG) random vector with the mean vectorxxx and the covariance matrix 6 is denoted byCN (xxx; 6); and stands for “distributed as”. m2ndenotes the space ofm 2 n complex matrices. kxxxk denotes the Euclidean norm of a complex vector (scalar)xxx. The log( 1 ) function is with base 2 by default.
II. SYSTEMMODEL
We consider the downlink transmission in a cellular network con-sisting ofK cells, each having a multi-antenna BS to transmit inde-pendent messages to one active single-antenna MS. It is assumed that all BSs share the same narrowband spectrum for downlink transmis-sion. Accordingly, the system under consideration can be modeled by aK-user MISO-IC. It is assumed that the BS in the kth cell, k = 1; . . . ; K, is equipped with Mktransmitting antennas,Mk 1. The
discrete-time baseband signal received by the active MS in thekth cell is then given by
yk= hhhHkkxxxk+ j6=k
hhhH
jkxxxj+ zk (1)
wherexxxk 2 M 21denotes the transmitted signal from thekth BS; hhhH
kk 2 12M denotes the direct-link channel for thekth MS, while
hhhH
jk 2 12M denotes the cross-link channel from thejth BS to the
kth MS, j 6= k; and zkdenotes the receiver noise. It is assumed that zk CN (0; k2); 8k, and all zk’s are independent.
2We thank the anonymous reviewer who brought our attention to [15], in
which the authors also showed the optimality of beamforming to achieve the Pareto-boundary rates for the Gaussian MISO-IC with interference treated as noise, via a different proof technique.
We assume independent encoding across different BSs and thusxxxk’s are independent overk. It is further assumed that a Gaussian codebook is used at each BS andxxxk CN (0; SSSk); k = 1; . . . ; K, where SS
Sk = [xxxkxxxHk] denotes the transmit covariance matrix for the kth
BS, withSSSk 2 M 2M andSSSk 0. Notice that the CSCG distri-bution has been assumed for all the transmitted signals.3Furthermore, the interferences at all the receivers caused by different transmitters are treated as Gaussian noises. Thus, for a given set of transmit covariance matrices of all BSs,SSS1; . . . ; SSSK, the achievable rate of thekth MS is expressed as Rk(SSS1; . . . ; SSSK) = log 1 + hhh H kkSSSkhhhkk j6=khhhjkHSSSjhhhjk+ k2 : (2)
Next, we define the achievable rate region for the MISO-IC to be the set of rate-tuples for all MSs that can be simultaneously achievable under a given set of transmit-power constraints for the BSs, denoted by P1; . . . ; PK:
R
fSSS g:Tr(SSS )P ;k=1;...;K
f(r1; . . . ; rK) :
0 rk Rk(SSS1; . . . ; SSSK); k = 1; . . . ; Kg : (3) The upper-right boundary of this region is called the Pareto boundary, since it consists of rate-tuples at which it is impossible to improve a particular user’s rate, without simultaneously decreasing the rate of at least one of the other users. More precisely, the Pareto optimality of a rate-tuple is defined as follows [10].
Definition 2.1: A rate-tuple(r1; . . . ; rK) is Pareto optimal if there
is no other rate-tuple(r01; . . . ; rK0 ) with (r10; . . . ; r0K) (r1; . . . ; rK)
and(r01; . . . ; r0K) 6= (r1; . . . ; rK) (the inequality is component-wise). In this work, we consider the scenario where multiple BSs in the cellular network cooperatively design their transmit covariance ma-trices in order to minimize the effect of the inter-cell interference on the overall network throughput. In particular, we are interested in the design criterion to achieve different Pareto-optimal rate-tuples for the corresponding MISO-IC defined as above.
It is worth noting that in prior works on characterizing the Pareto boundary for the MISO-IC with interference treated as noise (see, e.g., [10] and references therein), it has been assumed (without proof) that the optimal transmit strategy for users to achieve any rate-tuple on the Pareto boundary is beamforming, i.e.,SSSkis a rank-one matrix for all k’s. Under this assumption, we can express SSSkasSSSk= wwwkwwwHk; k =
1; . . . ; K, where wwwk 2 M 21 denotes the beamforming vector for
thekth user. Similarly as in the general case with Rank(SSSk) 1, the
achievable rates and rate region of the MISO-IC with transmit beam-forming (BF) can be defined in terms ofwwwk’s. However, it is not evident whether the BF case bears the same Pareto boundary as the general case withRank(SSSk) 1 for the MISO-IC. In this work, we will show that this is indeed the case (see Section III). Accordingly, we can choose to use the rate and rate-region expressions in terms of eitherSSSk’s orwwwk’s to characterize the Pareto boundary of the MISO-IC, for the rest of this correspondence.
In the following, we review some existing approaches to characterize the Pareto boundary for the MISO-IC with interference treated as noise. For the purpose of illustration, in Fig. 1, we show the achievable rate region for a two-user MISO Gaussian IC with interference treated as
3It is worth noting that in [17] the authors point out that the CSCG distribution
for the transmitted signals is in general non-optimal for the Gaussian IC with interference treated as noise, since it can be shown that the complex Gaussian but not circularly symmetric distribution can achieve larger rates than the symmetric distribution for some particular channel realizations.
Fig. 1. Achievable rate region and Pareto boundary for a two-user MISO Gaussian IC with interference treated as noise.
noise (prior to any time-sharing of achievable rate-pairs), which is ob-served to be non-convex. A commonly adopted method to obtain the Pareto boundary for the MISO-IC is via solving a sequence of weighted sum-rate maximization (WSRMax) problems, each for a given set of user rate weights,k 0; 8k, and given by
Max. fwww g K k=1 klog (1 + k(www1; . . . ; wwwK)) s.t. kwwwkk2 Pk; k = 1; . . . ; K (4)
where k(www1; . . . ; wwwK) is the receiver SINR for the kth user defined
as
k(www1; . . . ; wwwK) = khhh H kkwwwkk2
j6=kkhhhHjkwwwjk2+ k2; k = 1; . . . ; K: (5)
This problem can be shown non-convex, and thus cannot be solved effi-ciently. Moreover, the WSRMax method cannot guarantee the finding of all Pareto-boundary points for the MISO-IC (cf. Fig. 1).
An alternative method to characterize the complete Pareto boundary for the MISO-IC is based on the concept of rate profile [18]. Specifi-cally, any rate-tuple on the Pareto boundary of the rate region can be obtained via solving the following optimization problem with a partic-ular rate-profile vector, = ( 1; . . . ; K):
Max.
R ;fwww g Rsum
s.t. log(1 + k(www1; . . . ; wwwK)) kRsum;
k = 1; . . . ; K kwwwkk2 Pk; k = 1; . . . ; K (6)
with kdenoting the target ratio between thekth user’s achievable rate and the users’ sum-rate,Rsum. Without loss of generality, we assume that k 0; 8k, and Kk=1 k= 1. For a given , let the optimal
so-lution of Problem (6) be denoted byR?sum. Then, it follows thatR?sum1 must be the corresponding Pareto-optimal rate-tuple, which can be ge-ometrically viewed as (cf. Fig. 1) the intersection between a ray in the direction of and the Pareto boundary of the rate region. Thereby, with
different ’s, solving Problem (6) yields the complete Pareto boundary for the rate region, which does not need to be convex.
Next, we show that Problem (6) is solvable via solving a sequence of feasibility problems each for a fixedrsumand given by
Find fwwwkg
s.t. log(1 + k(www1; . . . ; wwwK)) krsum;
k = 1; . . . ; K kwwwkk2 Pk; k = 1; . . . ; K: (7)
If the above problem is feasible for a given sum-rate target,rsum, it follows thatR?sum rsum; otherwise,R?sum< rsum. Thus, by solving Problem (7) with differentrsum’s and applying the simple bisection method [16] overrsum; R?sum can be obtained for Problem (6). Let
k = 2 r 0 1; k = 1; . . . K. Then, for Problem (7), we can
replace the rate constraints by the equivalent SINR constraints: k(www1; . . . ; wwwK) k; k = 1; . . . ; K: (8)
Similarly as shown in [19], the resultant feasibility problem can be transformed into a second-order cone programming (SOCP) problem, which is convex and can be solved efficiently [20].
III. CHARACTERIZINGPARETOBOUNDARY FORMISO-ICVIA INTERFERENCETEMPERATURECONTROL
In this section, instead of investigating centralized approaches, we present a new method to characterize the Pareto boundary for the MISO-IC in a distributed fashion, by exploring its relationship with the CR MISO channel [14]. We start with the general-rank transmit covariance matricesSSSk’s for the MISO-IC. First, we introduce a set of auxiliary variables,0kj; k = 1; . . . ; K; j = 1; . . . ; K; j 6= k,
where0kjis called the interference-power or interference-temperature (IT) constraint from thekth BS to jth MS, j 6= k; 0kj 0. For
notational convenience, let0 be the vector consisting of all K(K 0 1) different 0kj’s, and 0k be the vector consisting of all 2(K 0 1) different 0kj’s and 0jk’s, j = 1; . . . ; K; j 6= k, for any given k 2 f1; . . . ; Kg.
Next, we consider a set of parallel transmit covariance optimization problems, each for one of theK BSs in the MISO-IC expressed as
Max. SSS log 1 + hhhH kkSSSkhhhkk j6=k0jk+ k2 s.t. hhhH kjSSSkhhhkj 0kj; 8j 6= k Tr(SSSk) Pk; SSSk 0 (9) wherek 2 f1; . . . ; Kg. Note that in the above problem for a given k; 0kis fixed. For notational convenience, we denote the optimal value
of this problem asCk(0k). If in the objective function of (9) we set
0jk = hhhHjkSSSjhhhjk; 8j 6= k; Ck(0k) becomes equal to the
max-imum achievable rate of an equivalent MISO CR channel [14], where thekth BS is the so-called “secondary” user transmitter, and all the otherK 0 1 BSs, indexed by j = 1; . . . ; K; j 6= k, are the “primary” user transmitters, each of which has a transmit covariance matrix,SSSj, and its intended “primary” user receiver is protected by the secondary user via the IT constraint:hhhHkjSSSkhhhkj 0kj. In [14], it was proved that the solution for Problem (9) is rank-one, i.e., beamforming is op-timal, and in the special case ofK = 2 (i.e., one single primary user), a closed-form solution for the optimal beamforming vector was derived. In the following proposition, we provide a new closed-form solution for Problem (9) with arbitrary values ofK, from which we can easily infer that beamforming is indeed optimal.
Proposition 3.1: The optimal solution of Problem (9) is rank-one,
i.e.,SSSk = wwwkwwwHk, and ww wk= j6=k kjhhhkjhhhHkj+ kkIII 01 hhhkkppk (10) wherekj; j 6= k, and kk, are non-negative constants (dual variables) corresponding to thekth BS’s IT constraint for the jth MS and its own transmit-power constraint, respectively, which are obtained as the op-timal solutions for the dual variables in the dual problem of Problem (9); andpkis given by pk= ln 21 0 j6=k0jk+ 2 k kAAAkhhhkkk2 + 1 kAAAkhhhkkk2 (11)
whereAAAk ( j6=kkjhhhkjhhhHkj+kkIII)01=2and(x)+ max(0; x).
Proof: Please see Appendix I.
Now, we are ready to present a parametrical characterization of the Pareto boundary for the MISO-IC in terms of0 as follows.
Proposition 3.2: For any rate-tuple(R1; . . . ; RK) on the Pareto
boundary of the MISO-IC rate region defined in (3), which is achiev-able with a set of transmit covariance matrices,SSS1; . . . ; SSSK, there is a corresponding interference-power constraint vector,0 0, with 0kj = hhhHkjSSSkhhhkj; 8j 6= k; j 2 f1; . . . ; Kg, and k 2 f1; . . . ; Kg,
such thatRk = Ck(0k); 8k, and SSSk is the optimal solution of Problem (9) for the givenk.
Proof: Please see Appendix II.
From Proposition 3.2, it follows that the Pareto boundary for the MISO-IC is parameterized in terms of a lower-dimensional manifold over the non-negative real vector0, in comparison with that over the complex transmit covariance matrices,SSSk’s, or with that over the com-plex beamforming vectors,wwwk’s. Furthermore, by combining Propo-sitions 3.1 and 3.2, it follows that beamforming is indeed optimal to achieve any rate-tuple on the MISO-IC Pareto boundary.
Remark 1: It is worth noting that the dimensionality reduction
ap-proach proposed in this work for characterizing the Pareto boundary of the MISO-IC is in spirit similar to that proposed in [10], where it has been shown that the transmit beamforming vectors to achieve any Pareto-boundary rate-tuple of theK-user MISO-IC with interference treated as noise can be expressed in the following forms:
ww wk=
K j=1
kjhhhkj; k = 1; . . . ; K (12)
wherekj’s are complex coefficients. Note that under the assumption of independenthhhkj’s, the above beamforming structure is non-trivial only whenMk > K. For this case, from Remark 2 in Appendix I, it is known that for the optimal beamforming structure given in (10), we havekk > 0. With this and by applying the matrix inversion lemma [21], it can be shown (the detailed proof is omitted here for brevity) that the optimal beamforming structure given by (10) is indeed in ac-cordance with that given by (12). The main difference for these two methods to characterize the Pareto boundary for the MISO-IC lies in their adopted parameters: The method in our work usesK(K 0 1) real0kj’s, while that in [10] usesK(K 0 1) complex kj’s. Note that 0kj corresponds to the IT constraint from thekth user transmitter to
thejth user receiver, whereas there is no practical meaning associated withkj. Consequently, as will be shown next, the proposed method in our work leads to a practical algorithm to implement the multi-cell cooperative downlink beamforming, via iteratively searching for mu-tually desirable IT constraints between different pairs of BSs.
IV. DECENTRALIZEDALGORITHMFORMULTI-CELL COOPERATIVEBEAMFORMING
In this section, we develop a new decentralized algorithm that prac-tically implements the multi-cell cooperative downlink beamforming based on the results derived in the previous section. It is assumed that each BS in the cellular network has the perfect knowledge of the chan-nels from it to all MSs. Furthermore, it is assumed that all BSs operate according to the same protocol described as follows. Initially, a set of prescribed IT constraints in0 are set over the whole network, and the kth BS is informed of its corresponding 0k; k = 1; . . . ; K.
Accord-ingly, each BS sets its own transmit beamforming vector via solving Problem (9) and sets its transmit rate equal to the optimal objective value of Problem (9), which is achievable for its MS since the actual IT levels from the other BSs must be below their prescribed constraints. Then, by assuming that there is an error-free link between each pair of BSs, all different pairs of BSs start to communicate with each other for updating their mutual IT constraints (the details are given later in this section), under which each pair of BSs reset their respective beam-forming vectors via solving Problem (9) such that the achievable rates for their MSs both get improved. Notice that each pair of updating BSs keeps the IT constraints for the MSs associated with the other BSs ex-cluding this pair fixed; and as a result, the transmit rates for all the other MSs are not affected. Therefore, the above algorithm can be im-plemented in a pairwise decentralized manner across the BSs, while it converges when there are no incentives for all different pairs of BSs to further update their mutual IT constraints.
Next, we focus on the key issue on how to update the mutual IT con-straints for each particular pair of BSs to guarantee the rate increase for both of their MSs. To resolve this problem, in the following propo-sition, we first provide the necessary conditions for any given0 0 (component-wise) to correspond to a Pareto-optimal rate-tuple for the MISO-IC, which will lead to a simple rule for updating the mutual IT constraints between different pairs of BSs. Note that from Proposition 3.2, it follows that for any Pareto-optimal rate-tuple of the MISO-IC, there must exist a0 such that the optimal solutions of the problems given in (9) for allk’s are the same as those for the general formu-lation of MISO-IC to achieve this rate-tuple. However, for any given 0 0, it remains unknown whether this value of 0 will correspond to a Pareto-optimal rate-tuple.
Proposition 4.1: For an arbitrarily chosen0 0, if the optimal rate
values of the problems in (9) for allk’s, Ck(0k)’s, are Pareto-optimal on the boundary of the MISO-IC rate region defined in (3), then for any pair of(i; j); i 2 f1; . . . ; Kg; j 2 f1; . . . ; Kg, and i 6= j, it must hold thatjDDDijj = 0, where DDDij is defined as
DDDij = @C (0 ) @0 @C (0 )@0 @C (0 ) @0 @C (0 )@0 : (13)
Proof: Please see Appendix III.
Note thatDDDij’s for all different pairs of(i; j) can be obtained from the (primal and dual) solutions of the problems given in (9) for allk’s with the given 0 (for the details, please refer to Appendix I). More specifically, we have
@Ci(0i)
@0ij = ij (14)
whereijis the solution for the dual problem of Problem (9) withk = i, which corresponds to the jth IT constraint, and from the objective function of Problem (9), @Ci(0i) @0ji = 0hhhH iiSSS?ihhhii ln 2 l6=i0li+ i2 l6=i0li+ 2i+ hhhHiiSSS?ihhhii (15)
whereSSS?i is the optimal solution of Problem (9) withk = i. Simi-larly,(@Cj(0j))=(@0ij) and (@Cj(0j))=(@0ji) can be obtained from
solving Problem (9) via the Lagrange duality method withk = j. From Proposition 4.1, the following observations can be easily ob-tained (the proofs are omitted for brevity):
• for any particular0 that corresponds to a Pareto-optimal rate-tuple, it must hold that0ij 0ij; 8i; j; i 6= j, where 0ij =
(khhhH
ijhhhiik2Pi)=(khhhiik2) corresponds to the case of using
max-imum transmit power with MRT beamforming for theith BS; • for any particular0 that corresponds to a Pareto-optimal
rate-tuple, it must hold thathhhHijSSS?ihhhij = 0ij; 8i; j; i 6= j, i.e., the
IT constraints across all BSs must be tight.
From the above observations, we see that if we are only interested in the values of0 that correspond to Pareto-optimal rate-tuples for the MISO-IC, it is sufficient for us to focus on the subset of0 within the set0 0, in which 0ij 0ijand0ij = hhhHijSSS?ihhhij; 8i; j; i 6= j.
Based on Proposition 4.1, we can develop a simple rule for different pairs of BSs in the cooperative multi-cell system to update their mutual IT constraints for improving both of their transmit rates, while keeping those of the other BSs unchanged. From the Proof of Proposition 4.1 given in Appendix III, it follows that the method for any updating BS pair(i; j) to fulfill the above requirements is via changing 0ij and 0jiaccording to (38). Note that in general, the choice fordddij in (38) to makeDDDijdddij > 0 is not unique. For notational conciseness, let D
D
Dij = a bc d ; it can then be shown that one particular choice for
dddij is
dddij= sign(ad 0 bc) 1 [ ijd 0 b; a 0 ijc]T (16)
wheresign(x) = 1 if x 0 and = 01 otherwise; ij 0 is a
constant that controls the ratio between the rate increments for theith andjth BSs. It can be easily verified that when ij > 1, a larger
rate increment is resulted for theith BS than that for the jth BS, and
vice versa when ij < 1 (provided that the step-size ij in (38) is sufficiently small).
More specifically, the procedure for any BS pair(i; j); i 6= j; i 2 f1; . . . ; Kg, and j 2 f1; . . . ; Kg, to update their mutual IT constraints is given as follows. First, theith BS computes the elements a and b in D
D
Dijaccording to (14) and (15), respectively, with the present value of 0i. Similarly, thejth BS computes c and d with the present value of
0j. Next, theith BS sends a and b to the jth BS, while the jth BS sends c and d to the ith BS. Then, assuming that ijandijare preassigned values known to these two BSs, they can both computedddij according to (16) and update0ijand0jiaccording to (38) in Appendix III. Last, with the updated values00ij and00ji, these two BSs reset their respec-tive beamforming vectors and transmit rates via solving (9) indepen-dently. Note that the above operation requires only local information (scalar) exchanges between different pairs of BSs, and thus can be im-plemented at a very low cost in a cellular system. One version of the decentralized algorithm for cooperative downlink beamforming in a multi-cell system is described in Table I. Since in each iteration of the algorithm the achievable rates for the pair of updating BSs both improve and those for all other BSs are unaffected (non-decreasing), and the maximum achievable rates for all BSs are bounded by finite Pareto-optimal values, the convergence of this algorithm is ensured.
Example 4.1: In Fig. 2 (with the same two-user MISO-IC as for
Fig. 1), we show the Pareto boundary for an example MISO-IC with K = 2; M1 = M2 = 3; P1 = 5; P2 = 1, and 21 = 22 = 1,
which is obtained by the proposed method in this correspondence, i.e., solving the problems given in (9) fork = 1; 2, and a given pair of values012and021with0 012 012and0 021 021, and
TABLE I
ALGORITHM FORCOOPERATIVEDOWNLINKBEAMFORMING
Fig. 2. Achievable rates for the proposed algorithm in a two-user MISO Gaussian IC with interference treated as noise.
then taking a closure operation over the resultant rate-pairs with all dif-ferent values of012and021within their respective ranges. We demon-strate the effectiveness of the proposed decentralized algorithm for im-plementing the multi-cell cooperative downlink beamforming with two initial rate-pairs, indicated by “ZF” and “MRT” in Fig. 2, which are ob-tained when both BSs adopt the ZF and the MRT beamforming vectors, respectively, with their maximum transmit powers. It is observed that the achievable rates for both MSs increase with iterations and finally converge to a Pareto-optimal rate-pair.4Comparing the two cases with 12= 1 and 12= 10, it is observed that a larger value of 12results in larger rate values for the first MS in the converged rate-pairs, which is in accordance with our previous discussion.
V. CONCLUDINGREMARKS
In this correspondence, based on the concept of interference tem-perature (IT) and under a cellular downlink setup, we have developed a new method to characterize the complete Pareto boundary of the
4We have verified with a large number of random channels and a variety of
system parameters that the proposed algorithm always converges to Pareto-op-timal pairs for the two-user MISO-IC with randomly selected initial rate-pairs. However, we could not prove this result in general by, e.g., showing that the conditions given in Proposition 4.1 are not only necessary (as proved in this work) but also sufficient for any given0 to achieve a Pareto-optimal rate-tuple for the MISO-IC.
achievable rate region for theK-user Gaussian MISO-IC with interfer-ence treated as noise. It is shown that the proposed method also leads to a new decentralized algorithm for implementing the downlink beam-forming in a cooperative multi-cell system to achieve maximal rates with a prescribed fairness guarantee.
There are a number of directions along which the developed results in this work can be further investigated. First, it would be interesting to extend the multi-cell cooperative beamforming design based on the principle of IT to the scenario where each BS supports simultaneous transmissions to multiple active MSs each with a single antenna or mul-tiple antennas. Second, it remains yet to be proved whether the neces-sary conditions derived in this work for any particular set of IT con-straints across the BSs to guarantee a Pareto-optimal rate-tuple for the MISO-IC are also sufficient, even for the special two-user case. This proof is essential for the proposed downlink beamforming algorithm to achieve the global convergence (Pareto-optimal rates). Last but not least, it is pertinent to analyze the proposed decentralized algorithm that iteratively updates the mutual IT constraints between different pairs of BSs from a game-theoretical viewpoint.
APPENDIXI PROOF OFPROPOSITION3.1
It can be verified that Problem (9) is convex, and thus it can be solved by the standard Lagrange duality method [16]. Letkj; j 6= k, and
kkbe the non-negative dual variables for Problem (9) associated with
thekth BS’s IT constraint for the jth MS and its own transmit-power constraint, respectively. The Lagrangian function for this problem can be written as L(SSSk; k) = log 1 + hhh H kkSSSkhhhkk j6=k0jk+ 2k 0 j6=k kj(hhhHkjSSSkhhhkj0 0kj)0kk(Tr(SSSk)0Pk) (17)
wherek= [k1; . . . ; kK]. The dual function of Problem (9) is given by
g(k) = max S S
S 0L(SSSk; k): (18)
Accordingly, the dual problem is defined as min
0g(k) (19)
wherek 0 means component-wise non-negative. Since Problem (9) is convex with strictly feasible points [16], the duality gap between its optimal value and that of the dual problem is zero; thus, Problem (9) can be equivalently solved via solving its dual problem. In order to solve the dual problem, we need to obtain the dual functiong(k) for any givenk 0. This can be done by solving the maximization problem
given in (18), which, according to (17), can be explicitly written as (by discarding irrelevant constant terms)
Max. SS S log 1 + hhhH kkSSSkhhhkk j6=k0jk+ 2k 0 Tr(BBBk(k)SSSk) s.t. SSSk 0 (20)
whereBBBk(k) j6=kkjhhhkjhhhHkj+ kkIII and BBBk(k) 0 of
dimensionMk2 Mk. In order for Problem (20) to have a bounded objective value, it is shown as follows thatBBBk(k) should be a
full-rank matrix. Suppose thatBBBk(k) is rank-deficient, such that we could defineSSSk = qkvvvkvvvHk, whereqk 0 and vvvk 2 M 21 satisfying
kvvvkk = 1 and BBBk(k)vvvk = 0. Thereby, the objective function of
Problem (20) reduces to
log 1 + qkkhhhHkkvvvkk2 j6=k0jk+ k2 :
(21) Due to the independence ofhhhkkandhhhkj’s, and thus the independence ofhhhkkandvvvk, it follows thatkhhhHkkvvvkk > 0 with probability one such that (21) goes to infinity by lettingqk! 1. Since the optimal value
of Problem (9) must be bounded, without loss of generality, we only need to consider the subset ofkin the setk 0 to make BBBk(k) full-rank.
Remark 2: Note that from the definition of BBBk(k) and the
Karush–Kuhn–Tucker (KKT) optimality conditions [16] of Problem (9), it follows thatBBBk(k) is full-rank only when either of the fol-lowing two cases occurs:
• kk > 0: in this case, the transmit power constraint for the kth BS is tight for Problem (9);
• kk= 0, but there are at least Mkkj’s,j 6= k, having kj> 0:
in this case, regardless of whether the transmit power constraint for thekth BS is tight, there are at least Mkout of theK 0 1 IT constraints of thekth BS are tight in Problem (9). Note that this case can be true only whenMk K 0 1.
From the above discussions, it is known that(BBBk(k))01 exists. Thus, we can introduce a new variable SSSkfor Problem (20) as
SSSk= (BBBk(k))1=2SSSk(BBBk(k))1=2 (22)
and substituting it into (20) yields Max. SS S log 1 + hhh H kk(BBBk(k))01=2SSSk(BBBk(k))01=2hhhkk j6=k0jk+ k2 0 Tr(SSSk) s.t. SSSk 0: (23)
Without loss of generality, we can express SSSk into its eigenvalue decomposition (EVD) as SSSk = UUUk2kUUUHk,
where UUUk = [uuuk1; . . . ; uuukM ] 2 M 2M is unitary and 2k = Diag([k1; . . . ; kM ]) 0. Substituting the ED of SSSkinto (23) yields Max. U U U ;2 log 1 + M i=1kikhhhHkk(BBBk(k))01=2uuukik2 j6=k0jk+ k2 0 M i=1 ki
s.t. kuuukik = 1; 8i; uuuHkiuuukl= 0; 8l 6= i
ki 0; 8i: (24)
For any givenUUUk, it can be verified that the optimal solution of2kfor Problem (24) is given by ki= ln 21 0 0 + khhh (BBB ( )) uuu k + if i = i? 0 otherwise: (25)
wherei?= arg maxl2f1;...;M gkhhhHkk(BBBk(k))01=2uuuklk. Thus, it fol-lows that for the optimal solution of Problem (23),Rank(SSSk) 1.
Furthermore, leti0denote the index ofi for which ki 0. The ob-jective function of Problem (24) reduces to
log 1 + kikhhhHkk(BBBk(k))01=2uuukik2 j6=k0jk+ k2
0 ki: (26)
Clearly, the above function is maximized with anyki > 0 when uuuki = (BBBk(k))
01=2hhh kk
k(BBBk(k))01=2hhhkkk: (27)
From (25) and (27), it follows that the optimal solution for Problem (23) is SSSk= 1 ln 20 0 + khhh (BBB ( )) k + k(BBBk(k))01=2hhhkkk2 2(BBBk(k))01=2hhhkkhhhHkk(BBBk(k))01=2: (28)
Combining the above solution and (22), it can be shown that the optimal solutionSSSkfor Problem (9) is as given by Proposition 3.1.
With the obtained dual functiong(k) for any given k, the dual problem (19) can be solved by searching overk 0 to minimize
g(k). This can be done via, e.g., the ellipsoid method [22], by utilizing
the subgradient ofg(k) that is obtained from (17) as 0kj0hhhHkjSSS?khhhkj forkj; j 6= k and Pk0 Tr(SSSk?) for kk, whereSSS?k is the optimal solution for Problem (20) with the givenk. Whenkconverges to the optimal solution for the dual problem, the correspondingSSS?kbecomes the optimal solution for Problem (9). Proposition 3.1 thus follows.
APPENDIXII PROOF OFPROPOSITION3.2
Since the given set ofSSS1; . . . ; SSSKachieves the Pareto-optimal rate-tuple(R1; . . . ; RK) for the MISO-IC, from (2) and (3) it follows that
for anyk 2 f1; . . . ; Kg
Rk= log 1 + hhh H kkSSSkhhhkk
j6=khhhHjkSSSjhhhjk+ 2k : (29)
Since0jk= hhhHjkSSSjhhhjk; 8j 6= k, (29) can be rewritten as
Rk= log 1 + hhh H kkSSSkhhhkk j6=k0jk+ 2k :
(30) Note that (30) is the same as the objective function of Problem (9) for the givenk. Furthermore, from (3) it follows that Tr(SSSk) Pk. Using this and the fact that0kj = hhhHkjSSSkhhhkj; 8j 6= k, it follows that SSSk
satisfies the constraints given in Problem (9) for the givenk. Therefore, SS
Skmust be a feasible solution for Problem (9) with the givenk and 0k. Next, we need to prove thatSSSk is indeed the optimal solution of Problem (9) for any givenk, and thus the corresponding achievable rateRkis equal to the optimal value of Problem (9), which isCk(0k). We prove this result by contradiction. Suppose that the optimal solution for Problem (9), denoted bySSS?k, is not equal toSSSkfor a givenk. Thus, we have Rk< log 1 + hhh H kkSSS?khhhkk j6=k0jk+ k2 (31) = log 1 + hhhHkkSSS?khhhkk j6=khhhjkHSSSjhhhjk+ k2 rk: (32) Furthermore, sincehhhHkjSSS?khhhkj 0kj; 8j 6= k, we have for any
j 6= k, Rj = log 1 + hhh H jjSSSjhhhjj i6=jhhhHijSSSihhhij+ 2j (33) = log 1 + hhhHjjSSSjhhhjj i6=j0ij+ 2j (34)
log 1 + hhhHjjSSSjhhhjj i6=j;k0ij+ hhhHkjSSS?khhhkj+ 2j (35) = log 1 + hhhHjjSSSjhhhjj i6=j;khhhHijSSSihhhij+ hhhHkjSSS?khhhkj+ 2j rj: (36)
Thus, for another set of transmit covariance matrices given by SS
S1; . . . ; SSSk01; SSS?k; SSSk+1; . . . ; SSSK, the corresponding achievable
rate-tuple for the MISO-IC,(r1; . . . ; rK), satisfies that rk> Rkand rj Rj; 8j 6= k, which contradicts the fact that (R1; . . . ; RK) is
a Pareto-optimal rate-tuple for the MISO-IC. Hence, the presumption thatSSSk6= SSS?kfor any givenk cannot be true. Thus, we have SSSk= SSS?k andRk= Ck(0k); 8k. Proposition 3.2 thus follows.
APPENDIXIII PROOF OFPROPOSITION4.1
As given in Proposition 4.1, with 0, the corresponding optimal values of the problems in (9) for allk’s, Ck(0k)’s, correspond to a
Pareto-optimal rate-tuple for the MISO-IC, denoted by(R1; . . . ; RK).
LetSSS1; . . . ; SSSK denote the set of optimal solutions for the problems in (9). We thus have Ck(0k) = Rk= log 1 + hhh H kkSSSkhhhkk j6=k0jk+ 2k ; k = 1; . . . ; K: (37) Next, we prove Proposition 4.1 by contradiction. Suppose that there exists a pair of(i; j) with jDDDijj 6= 0, where DDDij is defined in (13). Define a new00over0, where all the elements in 0 remain unchanged except[0ij; 0ji]T being replaced by
[00
ij; 00ji]T = [0ij; 0ji]T+ ij1 dddij (38) whereij > 0 is a small step-size, and dddijis any vector that satisfies DD
Dijdddij > 0 (component-wise), with one possible value for such dddij
is given by (16) in the main text. With00, the optimal solutions for the problems in (9) remain unchanged8k 6= i; j, while for those with k = i and k = j, the optimal solutions are changed to be SSS?
i andSSS?j,
respectively. Accordingly, the new achievable rates in the MISO-IC for anyk 6= i; j are given by
rk= log 1+ hhh H kkSSSkhhhkk l6=k;i;jhhhHlkSSSlhhhlk+ hhhikHSSSi?hhhik+ hhhHjkSSS?jhhhjk+ k2 (39) = log 1 + hhhHkkSSSkhhhkk l6=k;i;j0lk+ hhhHikSSS?ihhhik+ hhhHjkSSS?jhhhjk+ k2 (40) Rk (41)
where (41) is due to (37) and the facts thathhhikHSSS?ihhhik 0ik and hhhH
jkSSS?jhhhjk 0jk. Also, it can be shown that
ri= log 1 + hhh H iiSSS?ihhhii l6=i;jhhhHliSSSlhhhli+ hhhHjiSSSj?hhhji+ 2i (42) = log 1 + hhhHiiSSS?ihhhii l6=i;j0li+ hhhHjiSSS?jhhhji+ i2 (43) Ci(00i) (44)
where (44) is due to the facts thathhhHjiSSS?jhhhji 00jiandSSS?i achieves the
optimal value of Problem (9) withk = i and the given 00i, denoted by
Ci(00i). Similarly, it can be shown that rj Cj(00j). Thus, from (38)
andDDDijdddij> 0, it follows that with sufficiently small ij
ri rj Ci(00i) Cj(00j) (45) = CCi(0i) j(0j) + ijDDDijdddij (46) > RRi j : (47)
Therefore, we have found a new set of achievable rate-tuple for the MISO-IC with00; (r1; . . . ; rK), which has ri > Ri; rj > Rj, and rk Rk; 8k 6= i; j. Clearly, this contradicts the fact that (R1; . . . ; RK) is Pareto-optimal for the MISO-IC. Thus, the
presump-tion that there exists a pair of(i; j) with jDDDijj 6= 0 cannot be true.
Proposition 4.1 thus follows.
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Optimality of Beamforming for MIMO Multiple Access Channels Via Virtual Representation
Hong Wan, Rong-Rong Chen, and Yingbin Liang
Abstract—In this correspondence, we consider the optimality of
beamforming for achieving the ergodic capacity of multiple-input multiple-output (MIMO) multiple access channel (MAC) via virtual repre-sentation (VR) model. We assume that the receiver knows the channel state information (CSI) perfectly but that the transmitter knows only partial CSI, i.e., the channel statistics. For the single-user case, we prove that the capacity-achieving beamforming angle (c.b.a.) is unique, and there exists a signal-to-noise ratio (SNR) threshold below which beamforming is optimal and above which beamforming is strictly suboptimal. For the multi-user case, we show that the c.b.a is not unique and we obtain explicit conditions that determine the beamforming angles for a special class of correlated MAC-VR models. Under mild conditions, we show that a large class of power allocation schemes can achieve the sum-capacity within a constant as the number of users in the system becomes large. The beamforming scheme, in particular, is shown to be asymptotically capacity-achieving only for certain MAC-VR models.
Index Terms—Beamforming, multiple access, input
multiple-output, sum-capacity, power allocation, virtual representation.
I. INTRODUCTION
The multiple-input multiple-output (MIMO) techniques provide powerful means to improve reliability and capacity of wireless channels. Significant amount of work has been done to study op-timal input distributions and the channel capacity of single-user and multi-user MIMO channels (see, e.g., [1]–[7]). Several models have been adopted to capture the spatial correlation between the channel gains corresponding to different transmit-receive antenna pairs. These models include the i.i.d. model [1], the Kronecker model [2], [8]–[10], the virtual representation (VR) model [4], [11], and the unitary-in-dependent-unitary (UIU) model [5]. The i.i.d. model assumes that the channel gains are independent and identically distributed (i.i.d.),
Manuscript received November 26, 2009; accepted June 14, 2010. Date of publication June 28, 2010; date of current version September 15, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xiqi Gao. This work is supported in part by NSF under Grants ECS-0547433 and CCF-1026565. The material in this correspondence has been presented in part at the IEEE International Symposium on Information Theory (ISIT) Toronto, Ontario, Canada, July 6–11, 2008.
H. Wan and R.-R. Chen are with Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, UT 84112 USA (e-mail: (rchen@ece.utah.edu; wan@ece.utah.edu).
Y. Liang is with Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 USA (e-mail: yliang06@syr.edu).
Digital Object Identifier 10.1109/TSP.2010.2055241
and the Kronecker model assumes that the correlation between the channel gains can be written in terms of the product of the transmit correlation and the receive correlation. These two models apply only to wireless environments with rich or locally rich scattering at either the transmitter or the receiver. The VR and UIU models are more general, and both transform the MIMO channel to a domain such that the channel gains can be justified to be approximately independent.
In this correspondence, we adopt the VR model [11], which repre-sents the MIMO channel in a virtual angular domain with each channel gain corresponding to one virtual transmit and receive angle pair. The channel gains in the angular domain can be justified to be approxi-mately independent of each other, although not necessarily identically distributed, because they include different signal paths (corresponding to different transmit and receive angle pairs) with independent random phases.
The single-user MIMO channel based on VR was studied in [4]. In this correspondence, we generalize this study to the MIMO multiple ac-cess channel (MAC) based on VR, denoted by MAC-VR. We first char-acterize the optimal input distribution that achieves the sum-capacity. Then we study the optimality of beamforming, which is a simple scalar coding strategy desirable in practice. We first strengthen the conditions for the optimality of beamforming for the single-user VR model in [4] by proving that there exists a signal-to-noise ratio (SNR) threshold below which beamforming is optimal and above which beamforming is strictly suboptimal. This result was illustrated in [4] only numeri-cally. For the multi-user case, we present an example to show that the capacity-achieving beamforming angle (c.b.a) of a given user may vary with SNR and beamforming angles of other users. This is in contrast to the single-user case in which the c.b.a. is independent of SNR. We also derive explicit conditions to determine possible c.b.a. for certain MAC-VR channels. For systems withK users, we show that as K goes to infinity, the sum-rates achieved by a large class of power allocation schemes are within a constant of the sum-capacity, and they grow in the order ofnrlog K, where nr is the number of receive antennas. Fur-thermore, we obtain conditions under which beamforming is asymp-totically capacity-achieving.
Our study for the single-user case generalizes that in [2], [6] for the Kronecker model, and is different from [12] for the double-scattering model [13]. Our study for the MAC-VR also differs from [7] which assumes perfect channel state information at the transmitter, and from [14], which assumes finite feedback. We also note that the results we derive for the MAC-VR are applicable to the MIMO-MAC Kronecker (MAC-Kr) model in [9]. However, certain results valid for the MAC-Kr may not hold for the MAC-VR as demonstrated in later sections.
II. CHANNELMODEL ANDVIRTUALREPRESENTATION We consider theK-user MIMO MAC, in which K users transmit to one base station (BS) with each user equipped withnt antennas and the BS equipped withnrantennas. The channel between each userk and the BS is assumed to be a frequency-flat, MIMO fading channel. The received signal at the BS is annr-dimensional vectorY 2 Cn and is given by Y = K k=1 pk ntH kXk+ W (1)
whereXk 2 Cn is the input vector of userk that satisfies the power constraintE[XkyXk] nt; ( 1 )y denotes the Hermitian operator, pk represents the effective SNR of user k at each receive antenna,
W 2 Cn is a proper complex Gaussian noise vector that consists
of i.i.d. entries with zero-mean and unit-variance, andHk 2 Cn 2n 1053-587X/$26.00 © 2010 IEEE
