On the Optimality of Beamforming for Multi-User MISO Interference Channels with Single-User Detection
aa r X i v : . [ c s . I T ] A ug On the Optimality of Beamforming for Multi-UserMISO Interference Channels with Single-UserDetection
Xiaohu Shang
Department of Electrical EngineeringPrinceton UniversityEmail: [email protected]
Biao Chen
Department of EECSSyracuse UniversityEmail: [email protected]
H. Vincent Poor
Department of Electrical EngineeringPrinceton UniversityEmail: [email protected]
Abstract —For a multi-user interference channel with multi-antenna transmitters and single-antenna receivers, by restrictingeach receiver to a single-user detector, computing the largestachievable rate region amounts to solving a family of non-convex optimization problems. Recognizing the intrinsic con-nection between the signal power at the intended receiver andthe interference power at the unintended receiver, the originalfamily of non-convex optimization problems is converted into anew family of convex optimization problems. It is shown that,for such interference channels with each receiver implementingsingle-user detection, transmitter beamforming can achieve allboundary points of the achievable rate region.
Index terms — Gaussian interference channel, achievable rateregion, beamforming I. I
NTRODUCTION
The interference channel (IC) models a multi-user com-munication system in which each transmitter communicatesto its intended receiver while generating interference to allunintended receivers. Determination of the capacity region ofan IC remains an open problem. To date, the best achievablerate region was established by Han and Kobayashi in [1],herein termed the HK region, which combines rate splittingat transmitters, joint decoding at receivers, and time sharingamong codebooks. The HK region was later simplified in [2].Recently, [3] showed that the HK region is within 1-bit ofthe capacity region of the Gaussian IC. The results in [4]–[6], whose genie-aided approach is largely motivated by [3],established the sum-rate capacity of the Gaussian IC in noisyinterference: treating interference as noise at both receivers issum rate optimal , i.e., each receiver should simply implementsingle-user detection (SUD). In addition, even if the noisyinterference condition is not satisfied, practical constraintsoften limit the receivers to implementing SUD. For example,the receivers may know only the channels associated with theirown intended links.We assume in the present work that each receiver im-plements SUD, i.e., it treats interference as channel noise.In a preliminary work [7], we showed that beamforming is This research was supported in part by the National Science Foundationunder Grant CNS-06-25637. optimal for the entire SUD rate region for a two-user realmultiple-input single-output (MISO) IC. This result was usedin [8] to characterize the beamforming vectors that achieve theboundary rate points on the SUD rate region. Later, the resultin [7] was also used in [9] to derive the noisy-interferencesum-rate capacity of the symmetric real MISO ICs. We notethat the proof in [7] is applicable only to a two-user real MISOIC.There have been various studies concerning throughputoptimization in a multi-user system under the assumption that each receiver treats interference as channel noise . However,even for the simple scalar Gaussian IC, computing the largestachievable rate region with SUD at each receiver is in generalan open problem [10]. Exhaustive search over the transmitterpowers is typically unavoidable due to the non-convexity of theproblem. The difficulty is much more acute for the MISO ICcase as one needs to exhaust all covariance matrices satisfyingthe power constraints, which renders the computation highlyintractable. In this paper we propose an alternative way toderive optimal signaling for the SUD rate region for MISOICs. Our approach is to convert a family of non-convexoptimization problems for the original formulation to an equiv-alent family of convex optimization problems. What is moresignificant is that, given that each transmitter uses Gaussianinput and each receiver implements SUD, all boundary pointsof the rate region can be achieved by transmitter beamforming.
The rest of the paper is organized as follows. In Section II,we prove that beamforming is optimal for the SUD rate regionof a multi-user MISO IC. Numerical examples are providedin Section III. We conclude in Section IV.Before proceeding, we introduce the following notation. • Bold fonts xxx and X denote vectors and matrices respec-tively. • ( · ) T and ( · ) † denote respectively the transpose and theHermitian (conjugate transpose) of a matrix or a vector. • I is an identity matrix, is an all zero matrix, anddiag ( · · · ) is a diagonal matrix with its diagonal entries. • X (cid:23) means that X is a symmetric positive semi-definite matrix. • tr ( X ) and rank ( X ) denote the trace and the rank of matrix respectively. • ( xxx ) i denotes the i th entry of vector xxx , and X m × n meansthat X is an m × n matrix. • k xxx k is the norm of a vector xxx , i.e., k xxx k = √ xxx † xxx . • E [ · ] denotes the expectation.II. M ULTI - USER
MISO IC
WITH SINGLE - USER DETECTOR
We define the received signal for user i of an m -user MISOIC as Y i = m X j =1 hhh Tji xxx j + N i , i = 1 , · · · , m, (1)where xxx i is the transmitted signal vector of user i with dimen-sion t i ; Y i is the scalar received signal; the N i is unit varianceGaussian noise; and hhh ji is the t i × channel vector from the j th transmitter to the i th receiver. The power constraint atthe transmitter is tr ( S i ) ≤ P i , where S i = E (cid:2) xxx i xxx Ti (cid:3) . Weassume that receiver i knows only channel hhh ii , and decodesits own signal by treating the interference from all other usersas noise. The boundary points of the achievable rate regionfor this channel is characterized by the following family ofoptimization problems: max m X i =1 µ i R i subject to R i = 12 log hhh Tii S i hhh ii P mj =1 ,j = i hhh Tji S j hhh ji ! tr ( S i ) ≤ P i , S i (cid:23) , i = 1 , · · · m, (2)where ≤ µ i < ∞ .Apparently problem (2) is a non-convex optimization prob-lem. For each choice of µ = [ µ , · · · , µ m ] , all possible [ S , · · · , S m ] must be exhausted. To obtain the entire SUDrate region, one has to go through this exhaustive search forall the µ vectors.Following the same problem reformulation procedure usedin [7], to characterize the achievable rate region of m -userMISO IC, it is equivalent to solve the following family ofconvex optimization problems: max hhh Tii S i hhh ii subject to hhh Tij S i hhh ij ≤ z ij , tr ( S i ) ≤ P i , S i (cid:23) i, j = 1 , . . . , m, i = j, (3)where z ij is a preselected constant denoting the interferencepower at the j th receiver caused by the i th transmitter. Theproblem reformulation is summarized in the following lemma. Lemma 1:
For any vector µ with non-negative components,the optimal solution S ∗ i for problem (2) is also an optimalsolution for problem (3) with z ij = z ∗ ij = hhh Tij S ∗ i hhh ij . Proof:
Problem (2) is equivalent to the following optimiza-tion problem for the same µ i : max m X i =1 µ i hhh Tii S i hhh ii P mj =1 ,j = i z ∗ ji ! subject to hhh Tij S i hhh ij ≤ z ∗ ij tr ( S i ) ≤ P i , S i (cid:23) ,i, j = 1 , · · · , m, i = j. (4)The equivalence is due to the following. First, the maximumof problem (2) is no smaller than that of problem (4), sinceproblem (4) has extra constraint hhh Tij S i hhh ij ≤ z ∗ ij . (Thisconstraint is active, since the rates associated with S ∗ ii are onthe boundary of the SUD rate region.) On the other hand, themaximum of problem (2) is no greater than that of problem(4), since the S ∗ i ’s are also feasible for problem (4). Therefore,problems (2) and (4) are equivalent. We now recognize thatproblem (4) is equivalent to problem (3) by setting z ij = z ∗ ij .We remark that the optimization problem (4) can not besolved directly as the constraint parameters z ∗ ji all depend onthe unknown optimal covariances. That is, unless the optimal S ∗ i of problem (2) is obtained, the equivalent optimizationproblem (4) cannot be parameterized. Although we do notgive an explicit solution of problem (2) for a given µ vector,Lemma 1 provides the following essential fact which is enoughto obtain the entire SUD rate region defined by problem (2): [ all µ i ,i =1 , ··· ,m { S ∗ i ( µ i ) } ⊆ [ all z ji ,i,j =1 , ··· m,i = j { S ∗ i ( z ji ) } , (5)where the left-hand side denotes the collection of the optimalsolutions of problem (2) by exhausting µ , and the right-handside denotes the collection of the optimal solutions of problem(3) by exhausting z ij . Since the SUD rate region is determinedby the left-hand side of (5), Lemma 1 successfully converts afamily of non-convex optimization problems (2) into a familyof convex optimization problems (4).Based on Lemma 1, we obtain the following theorem. Theorem 1:
For an m -user MISO IC, the boundary pointsof the SUD rate region can be achieved by restricting eachtransmitter to implementing beamforming.Theorem 1 can be readily extended to complex channels.Before proving Theorem 1, we first introduce the followinglemma. Lemma 2:
Let xxx and yyy be two vectors with dimensions t and t respectively, and K (cid:23) be a ( t + t ) × ( t + t ) matrix with tr ( K ) ≤ P . If K = (cid:20) K K T K K (cid:21) (6)and K (cid:23) is a preselected t × t matrix, then (cid:20) xxxyyy (cid:21) T K (cid:20) xxxyyy (cid:21) ≤ (cid:16)p xxx T K xxx + k yyy k p P − tr ( K ) (cid:17) , (7)and the equality can be achieved by choosing K = K ∗ :1) When xxx T K xxx = 0 and k yyy k 6 = 0 , we have K ∗ = K p P − tr ( K ) k yyy k p xxx T K xxx K xxxyyy T p P − tr ( K ) k yyy k p xxx T K xxx yyyxxx T K P − tr ( K ) k yyy k yyyyyy T (8)) When xxx T K xxx = 0 and k yyy k 6 = 0 , we have K ∗ = K p P − tr ( K ) k yyy k K T yyy T p P − tr ( K ) k yyy k yyy T K P − tr ( K ) k yyy k yyyyyy T , (9)where = (cid:20) ( t − × (cid:21) , K = (cid:20) Λ
00 0 (cid:21) Q , with K = Q T (cid:20) Λ 00 0 (cid:21) Q being the eigenvalue decomposition and Λ being a strictlypositive diagonal matrix.3) When k yyy k = 0 , we have K ∗ = (cid:20) K
00 0 (cid:21) . (10)For all three cases, we haverank ( K ∗ ) ≤ max { rank ( K ) , } . (11)The proof is omitted due to the space limitation.Lemma 2 is useful for the following optimization problems: max (cid:20) xxxyyy (cid:21) T K (cid:20) xxxyyy (cid:21) subject to h i ( K ) = 0 , i = 1 , · · · , n,g j ( K ) ≤ , j = 1 , · · · , m, tr ( K ) ≤ P, K (cid:23) , (12)where h i ( · ) and g j ( · ) are fixed functions. By Lemma 2, wecan convert the above problem into max (cid:16)p xxx T K xxx + k yyy k p P − tr ( K ) (cid:17) subject to h i ( K ) = 0 , i = 1 , · · · , n,g j ( K ) ≤ , j = 1 , · · · , m, tr ( K ) ≤ P, K (cid:23) . (13)Problems (12) and (13) have the same maximum. Once theoptimal K for problem (13) is obtained, we can constructthe optimal K for problem (12) by (8), (9) and (10). We notethat the choices of (9) and (10) are not unique. One can choose K different from that of (8) and (9) and still achieve the samemaximum of problem (12).With Lemma 2, we prove Theorem 1 as follows. Proof:
By symmetry, it suffices to show that for the m thuser, the optimal covariance matrix S ∗ m for the followingoptimization problem satisfies rank ( S ∗ m ) ≤ : max hhh Tmm S m hhh mm subject to hhh Tmj S m hhh mj ≤ z mj , j = 1 , · · · , m − , tr ( S m ) ≤ P m , S m (cid:23) , (14)where all the hhh mj ’s are t m × vectors.We first show that problem (14) can be written as max (cid:20)q hhh T ˜ S hhh + p k hhh mm k − k hhh k · q P − tr (˜ S ) (cid:21) subject to hhh Tj ˜ S hhh j ≤ z mj , j = 1 , · · · , m − tr (cid:16) ˜ S (cid:17) ≤ P m , ˜ S (cid:23) , (15)where hhh and all the hhh j ’s, j = 1 , · · · , m − , are ¯ m × vectors, ˜ S is an ¯ m × ¯ m matrix, and ¯ m is defined as ¯ m = min { t m , m − } . (16)Obviously, when ¯ m = t m ≤ m − , problem (14) is exactlyproblem (15) by choosing hhh = hhh mm , ˜ S = S and hhh j = hhh mj .We need only to show the equivalence of problems (14) and(15) when ¯ m = m − < t m .Let the singular value decomposition (SVD) of hhh m be hhh m = U (cid:20) k hhh m k ( t m − × (cid:21) , and define S (1) m = U T S m U (17) h (1) mj = U T h mj , j = 1 , · · · , m. (18)Substituting (17) and (18) into (14), we obtain max hhh (1) Tmm S (1) m hhh (1) mm subject to hhh (1) Tmj S (1) m hhh (1) mj ≤ z mj , j = 2 , · · · , m − , (cid:20) k hhh m k ( t m − × (cid:21) T S (1) m (cid:20) k hhh m k ( t m − × (cid:21) ≤ z m , tr (cid:16) S (1) m (cid:17) ≤ P m , S (1) m (cid:23) . (19)Consider h (1) m and let h (1) m = (cid:16) h (1) m (cid:17) (cid:16) h (1) m (cid:17) , ··· ,t m = (cid:20)
00 U (cid:21) (cid:16) h (1) m (cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) h (1) m (cid:17) , ··· ,t m (cid:13)(cid:13)(cid:13)(cid:13) ( t m − × , (20)where (cid:16) h (1) m (cid:17) , ··· ,t m is a vector consisting of the last t m − elements of h (1) m . The SVD of (cid:16) h (1) m (cid:17) , ··· ,t m is (cid:16) h (1) m (cid:17) , ··· ,t m = U (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) h (1) m (cid:17) , ··· ,t m (cid:13)(cid:13)(cid:13)(cid:13) ( t m − × , (21)where U T U = I ( t m − × ( t m − . Therefore (cid:20)
00 U (cid:21) T (cid:20)
00 U (cid:21) = I t m × t m . (22)Define S (2) m = (cid:20)
00 U (cid:21) T S (1) m (cid:20)
00 U (cid:21) (23) h (2) mj = (cid:20)
00 U (cid:21) T h (1) mj , j = 1 , · · · m. (24)On Substituting (20), (23) and (24) into (19), we have max hhh (2) Tmm S (2) m hhh (2) mm subject to hhh (2) Tmj S (2) m hhh (2) mj ≤ z mj , j = 3 , · · · , m − , k hhh m k ( t m − × (cid:21) T S (2) m (cid:20) k hhh m k ( t m − × (cid:21) ≤ z m , (cid:16) h (1) m (cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) h (1) m (cid:17) , ··· ,t m (cid:13)(cid:13)(cid:13)(cid:13) ( t m − × T S (2) m (cid:16) h (1) m (cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) h (1) m (cid:17) , ··· ,t m (cid:13)(cid:13)(cid:13)(cid:13) ( t m − × ≤ z m , tr (cid:16) S (2) m (cid:17) ≤ P m , S (2) m (cid:23) . (25)We note that the above transformation does not change theform of the existing constraint (see the third lines of problems(19) and (25)). Now we continue the above procedure up to h m,m − . In the j th transformation, we keep the first j − elements of h ( j − mj and apply the SVD to the remaining ( t m − j + 1) elements, and update the optimization problem. Weformulate the j th iteration, j = 2 , · · · , m − , as follows: hhh ( j − mj = (cid:16) hhh ( j − mj (cid:17) , ··· ,j − (cid:16) hhh ( j − mj (cid:17) j, ··· ,t m = (cid:20) I ( j − × ( j −
00 U j (cid:21) (cid:16) hhh ( j − mj (cid:17) , ··· ,j − (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) hhh ( j − mj (cid:17) j, ··· ,t m (cid:13)(cid:13)(cid:13)(cid:13) ( t m − j ) × ,hhh ( j ) mk = (cid:20) I ( j − × ( j −
00 U j (cid:21) T hhh ( j − mk , k = 1 , · · · , m, S ( j ) m = (cid:20) I ( j − × ( j −
00 U j (cid:21) T S ( j − m (cid:20) I ( j − × ( j −
00 U j (cid:21) , where (cid:16) hhh ( j − mj (cid:17) j, ··· ,t m denotes the j th to the t m th elementsof hhh ( j − mj , and its SVD is (cid:16) hhh ( j − mj (cid:17) j, ··· ,t m = U j (cid:13)(cid:13)(cid:13)(cid:13)(cid:16) hhh ( j − mj (cid:17) j, ··· ,t m (cid:13)(cid:13)(cid:13)(cid:13) , (26)where U Tj U j = I ( t m − j +1) × ( t m − j +1) .Finally, we convert problem (14) into the following form: max (cid:20) hhh b hhh (cid:21) T ˜ S (cid:20) hhh b hhh (cid:21) subject to (cid:20) hhh j (cid:21) T ˜ S (cid:20) hhh j (cid:21) ≤ z mj , j = 1 , · · · m − , tr (cid:16) ˜ S (cid:17) ≤ P m , ˜ S (cid:23) , (27)where hhh and hhh j are ( m − × vectors and b hhh is a ( t m − m +1) × vector. Furthermore, k hhh mm k = k hhh k + (cid:13)(cid:13)(cid:13)b hhh (cid:13)(cid:13)(cid:13) . Let ˜ S = (cid:20) ˜ S ˜ S T ˜ S ˜ S (cid:21) , (28) where ˜ S is an ( m − × ( m − matrix. The quadraticconstraints in problem (27) are (cid:20) hhh j (cid:21) T ˜ S (cid:20) hhh j (cid:21) = hhh Tj ˜ S hhh j ≤ z mj , j = 1 , · · · m − . (29)Therefore, the quadratic constraints in problem (27) are relatedonly to ˜ S . By Lemma 2, problem (27) is equivalent toproblem (15).We summarize that we have shown the equivalence ofproblems (14) and (15) with all the vectors in (15) being ¯ m × and ˜ S being ¯ m × ¯ m .By Lemma 2, we can reconstruct ˜ S in a way such thatrank (cid:16) ˜ S (cid:17) ≤ max { rank (cid:16) ˜ S (cid:17) , } . Let ˜ S ∗ be optimal forproblem (27). To prove Theorem 1, it is equivalent to proverank (cid:16) ˜ S ∗ (cid:17) ≤ . (30)Furthermore, it suffices to prove that the rank of the optimalcovariance matrix for the following optimization problem isno greater than : max h T ˜ S hhh subject to hhh Tj ˜ S hhh j ≤ z mj , j = 1 , · · · , m − tr (cid:16) ˜ S (cid:17) ≤ ¯ P , ˜ S (cid:23) , (31)where ¯ P = tr (cid:16) ˜ S ∗ (cid:17) ≤ P m . (32)The equivalence is due to the fact that the optimal ˜ S forproblem (15) is also optimal for problem (31) and vice versa.Since ˜ S (cid:23) , we can define ˜ S = B T B , (33)where B is an ¯ m × ¯ m matrix. Then we can rewrite problem(31) as max k B hhh k subject to k B hhh j k ≤ z mj , j = 1 , · · · , m − , tr (cid:0) B T B (cid:1) ≤ ¯ P . (34)The Lagrangian of problem (34) is L = −k B hhh k + m − X j =1 λ j (cid:0) k B hhh j k − z mj (cid:1) + λ m (cid:2) tr (cid:0) B T B (cid:1) − ¯ P (cid:3) . (35)Let ∂L∂ B = B ( C + λ m I ) = , (36)where C = − hhhhhh T + m − X j =1 λ j hhh j hhh Tj = H ∗ diag [ − , λ , · · · , λ m − ] ∗ H T (37)where H = [ hhh, hhh , · · · , hhh m − ] is an ¯ m × m matrix, and C isan ¯ m × ¯ m matrix.We then introduce the following lemma which is an exten-sion of Sylvester’s Law of Inertia. emma 3: [11, Theorem 7] Let H be an m × n matrixand A be an n × n Hermitian matrix. Denote π ( · ) and υ ( · ) respectively as the numbers of positive and negativeeigenvalues of a matrix. Then we have π (cid:0) HAH † (cid:1) ≤ π ( A ) , υ (cid:0) HAH † (cid:1) ≤ υ ( A ) . By Lemma 3 and the Karush-Kuhn-Tucker (KKT) conditionsthat require λ i ≥ , i = 1 , · · · , m , we have π ( C ) ≤ m − , υ ( C ) ≤ . (38)Since C is an ¯ m × ¯ m matrix, we can write the eigenvaluedecomposition of C as C = Q T diag ( η , · · · , η ¯ m ) Q (39)where Q T Q = I , and η i ’s are the eigenvalues of C inascending order. From (38), η ≤ and η j ≥ , j = 2 , · · · , ¯ m .Since the optimal B ∗ satisfies tr (cid:0) B ∗ T B ∗ (cid:1) = ¯ P , from theKKT conditions we have λ m > . Thus, we haverank ( C + λ m I )= rank ( diag [ λ m + η , λ m + η , · · · , λ m + η ¯ m ]) ≥ ¯ m − . (40)Since B is an ¯ m × ¯ m matrix, from (36), we conclude that theoptimal B ∗ for problem (34) satisfiesrank ( B ∗ ) ≤ . (41)Thereforerank ( S ∗ m ) = rank (cid:16) ˜ S ∗ (cid:17) = rank (cid:0) B ∗ T B ∗ (cid:1) ≤ . (42) Remark:
Theorem 1 proves the sufficiency of transmitterbeamforming for achieveing the SUD rate region. However, itdoes not mean that the SUD rate region can only be achievedby beamforming. This depends on how we construct ˜ S afterwe obtain ˜ S . In the proof we choose ˜ S as (8), (9) and (10) inLemma 2. However, the choices of (9) and (10) are not unique.Another observation is that only (8) and (9) use full power.Equation (10) corresponds to the case that hhh mm is linearlydependent of hhh mj , j = 2 , · · · , m , so that b hhh = in (27). Thisagrees with the result for two-user scalar Gaussian IC in whichthe maximum SUD sum rate sometimes is achieved when oneuser is silent [12, Theorem 6].III. N UMERICAL EXAMPLES
Using Theorem 1, we obtain in Fig. 1 the SUD rate regionof a three-user MISO IC with the power constraint P = 1 , P = 1 . and P = 2 . The channels are H − . . . . . . . . . . − . . − . , H . − . . . . − . − . − . . . . . . , H . . . − . − . . − . . . . . − , where H = [ hhh , hhh , hhh ] , H = [ hhh , hhh , hhh ] and H =[ hhh , hhh , hhh ] . The solid curves are the rate regions for oneuser being inactive or at the maximum rate. That is, they areprojections of the 3-D rate region onto a 2-D plane with onerate fixed at a constant value. IV. C ONCLUSION
We have considered multi-user MISO ICs where eachreceiver is limited to single-user detection. By exploiting therelation between the signal power at the intended receiverand the interference power at the unintended receiver, wehave converted the original family of non-convex optimizationproblems into an equivalent family of convex optimizationproblems. Transmitter beamforming is shown to be sufficientto achieve all boundary points of the SUD rate region. R in nat/Hz/s R R Fig. 1. The SUD rate regions of a three-user MISO IC. R EFERENCES[1] T. S. Han and K. Kobayashi, “A new achievable rate region for theinterference channel,”
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