Cooperative Feedback for MIMO Interference Channels
CCooperative Feedback for MIMO Interference Channels
Kaibin Huang
School of EEE, Yonsei UniversitySeoul, KoreaEmail: [email protected]
Rui Zhang
Institute for Infocomm Research, A*STAR, SingaporeECE Department, National University of SingaporeEmail: [email protected]
Abstract —Multi-antenna precoding effectively mitigates theinterference in wireless networks. However, the precoding ef-ficiency can be significantly degraded by the overhead due tothe required feedback of channel state information (CSI). Thispaper addresses such an issue by proposing a systematic methodof designing precoders for the two-user multiple-input-multiple-output (MIMO) interference channels based on finite-rate CSIfeedback from receivers to their interferers, called cooperativefeedback . Specifically, each precoder is decomposed into innerand outer precoders for nulling interference and improving thedata link array gain, respectively. The inner precoders are furtherdesigned to suppress residual interference resulting from finite-rate cooperative feedback. To regulate residual interference dueto precoder quantization, additional scalar cooperative feedbacksignals are designed to control transmitters’ power using differentcriteria including applying interference margins, maximizing sumthroughput, and minimizing outage probability. Simulation showsthat such additional feedback effectively alleviates performancedegradation due to quantized precoder feedback. I. INTRODUCTION
In multi-antenna wireless networks, precoding can effec-tively mitigate interference between coexisting links. Thispaper presents a new approach of efficiently implementing pre-coding in the two-user multiple-input-multiple-output (MIMO)interference channels by exchanging finite-rate channel stateinformation (CSI). Specifically, precoders are designed tosuppress interference to the interfered receivers based ontheir quantized CSI feedback, and the residual interferenceis regulated by additional feedback of power control signals.Recently, progresses have been made in analyzing thecapacity of multi-antenna interference channels. In particular,interference alignment techniques have been proposed forachieving the channel capacity for high signal-to-noise ratios(SNRs) [1]. Such techniques, however, have limited practi-cality due to their complexity, requirement of perfect globalCSI and their sub-optimality for finite SNRs. This promptsthe development of linear precoding algorithms for practicaldecentralized wireless networks [2]–[5]. For time-divisionmultiplexing (TDD) multiple-input-single-output (MISO) in-terference channels, it is proposed in [2], [5] that forward-link beamformers can be adapted distributively based onreverse-link signal-to-interference-pluse-noise ratios (SINRs).Targeting the two-user MIMO interference channels, lineartransceivers are designed in [3] under the constraint of onedata stream per user and using different criteria includingzero-forcing and minimum mean square error. In [4], therate region for MISO interference channels is analyzed basedon the cognitive radio principle, yielding a message passing algorithm for enabling distributive beamforming. The aboveprior work does not address the issue of finite-rate feedbackthough it is widely used in the practice to enable precoding.Neglecting feedback CSI errors in precoder designs leads toover optimistic network performance.For MIMO precoding systems, the substantiality of CSIfeedback overhead has motivated extensive research on CSIquantization algorithms, forming a field called limited feed-back [6]. Recent limited feedback research has focused onMIMO downlink systems, where multiuser CSI feedbacksupports space division multiple access (SDMA) [7]. It hasbeen found that the number of feedback bits per user has toincrease with the transmit SNR so as to bound the throughputloss caused by feedback quantization [8]. Furthermore, sucha loss can be reduced by exploiting multiuser diversity [9],[10]. Designing limited feedback algorithms for interferencechannels is more challenging due to the decentralized networkarchitecture and the growth of feedback CSI. Cooperative feed-back algorithms are proposed in [11] for a two-user cognitiveradio network, where the secondary transmitter adjusts itsbeamformer to suppress interference to the primary receiverthat cooperates by feedback to the secondary transmitter. Thisdesign is tailored for a MISO cognitive radio network and thusunsuitable for general MIMO interference channels, whichmotivates the current work.We consider two coexisting MIMO links where all nodesemploy equal numbers of antennas and linear precoding isenabled by quantized cooperative feedback. Channels areassumed to have i.i.d. Rayleigh fading. A systematic method isproposed for jointly designing linear precoders and equalizersunder an orthogonality constraint, which decouples the linksin the case of perfect feedback. To be specific, precoders andequalizers are decomposed into inner and outer components,where the former are designed to suppress residual interferencecaused by feedback errors and the latter to enhance link arraygain. Second, additional scalar cooperative feedback, called interference power control (IPC) feedback, is proposed forcontrolling transmitters’ power so as to regulate the residualinterference. Specifically, the IPC feedback algorithms aredesigned using different criteria including fixed interferencemargin, maximum sum throughput, and minimum outageprobability.
Notation:
The superscript † represents matrix Hermitiantranspose. The operator [ X ] k gives the k th column of a matrix X . Let (cid:22) , ≺ , (cid:23) and (cid:31) represent element-wise inequalitiesbetween two real vectors. a r X i v : . [ c s . I T ] S e p c o o p e r a t i v e f e e d b a c k R1T2 R2 local feedback
Fig. 1. MIMO interference channels with local and cooperative feedback.
II. S
YSTEM M ODEL
We consider two interfering wireless links as illustratedin Fig. 1, where the two pairs of transceivers are denotedas ( T , R ) and ( T , R ) . Each transmitter/receiver has L antennas employed for suppressing the interference as wellas supporting spatial multiplexing. These functions requireCSI feedback from receivers to their interferers and intendedtransmitters, called cooperative feedback and local feedback ,respectively. We assume perfect CSI estimation and localfeedback, allowing the current design to focus on suppressinginterference caused by cooperative feedback quantization. Allchannels are assumed to follow independent blocking fading.The channel coefficients are samples of i.i.d. CN (0 , randomvariables. Let H mn denote a L × L i.i.d. CN (0 , matrixrepresenting fading of the channel from T n to R m . Then theinterference channels are modeled as { ν H mn } and the datachannels as { H mm } where m, n ∈ { , } and m (cid:54) = n . Thefactor ν < quantifies the difference in transmission distancebetween the data and interference links.Each link supports M ≤ L spatial data streams by linearprecoding and equalization. To regulate residual interferencecaused by precoder feedback errors, the total transmissionpower of each transmitter is constrained by cooperative IPCfeedback. For simplicity, the scalar IPC feedback is assumedperfect since it requires much less overhead than the precoderfeedback. Each transmitter uses identical transmission powerfor all spatial streams, represented by P n for T n with n = 1 , and the maximum P max . Assume that all additive white noisesamples are i.i.d. CN (0 , random variables. Let G m and F m denote the linear equalizer used by R m and the linearprecoder applied at T m , respectively. Thus the receive signal-to-interference-plus-noise ratio (SINR) at R m for the (cid:96) thstream can be written as SINR [ (cid:96) ] m := P m | [ G m ] † (cid:96) H mm [ F m ] (cid:96) | P n ν (cid:107) [ G m ] † (cid:96) H mn F n (cid:107) , m (cid:54) = n. (1)Two performance metrics, ergodic throughput and outageprobability, are considered. The total ergodic throughput of both links, called sum throughput , is defined as ¯ C := (cid:88) m =1 M (cid:88) (cid:96) =1 E (cid:104) log (cid:16) SINR [ (cid:96) ] m (cid:17)(cid:105) (2)where SINR [ (cid:96) ] m is given in (1). Next, consider the scenariowhere the coding rates for all data streams are fixed at log (1 + θ ) where θ is the receive SINR threshold for correctdecoding. We define an outage event as one that the SINR ofat least one data stream is smaller than θ . It follows that theoutage probability is given by P out := Pr (cid:18) min m =1 , min ≤ (cid:96) ≤ M SINR [ (cid:96) ] m ≤ θ (cid:19) . (3)III. P RECODING WITH L IMITED F EEDBACK
A. Precoder Design
A pair of precoder and equalizer ( G m , F n ) with m (cid:54) = n arejointly designed under the following orthogonality constraint G † m H mn F n = , m, n ∈ { , } , m (cid:54) = n. (4)The constraint aims at decoupling the links and requires that L ≥ M . A key step of the proposed design is to decomposethe precoder F n into an inner precoder F i n and an outerprecoder F o n . Specifically, F n = F i n F o n where F i n and F o n are L × M and M × M matrices, respectively, where the size of F i n is minimized to reduce feedback overhead. Similarly, wedecompose the equalizer G m as G m = G i m G o m where G i m is a L × N inner equalizer and G o m a N × M outer equalizer ,where N is a design parameter under the constraints N ≥ M and N ≤ L − M . The inner precoder/equalizer pair ( G i m , F i n ) is designed to enforce the constraint in (4) while the outerpair ( G o m , F o n ) enhances the link array gain as discussed inthe sequel. It follows that ( G i m ) † H mn F i n = , m (cid:54) = n. (5)Under this constraint, ( G i m , F i n ) are designed by decomposing H mn using the singular value decomposition (SVD) as H mn = V mn (cid:113) λ [1] mn . . . (cid:113) λ [ L ] mn U † mn , m (cid:54) = n (6)where the unitary matrices V mn and U mn consist of the leftand right singular vectors of H mn as columns, respectively,and (cid:110) λ [ (cid:96) ] mn (cid:111) denote the eigenvalues of H mn H † mn followingthe descending order. Let A and B be two subsets of theindices { , , · · · , L } with |A| = N , |B| = M , and A∩B = ∅ .The constraint in (5) can be satisfied by choosing G i m = { [ V mn ] k | k ∈ A} and F i n = { [ U mn ] k | k ∈ B} . (7)Given ( G i m , F i m ) , the outer pair ( G o m , F o m ) are jointlydesigned based on the SVD of the N × M effective channels o mm := G i m H mm F i m : H o mm = V mm (cid:113) λ [1] mm . . . (cid:113) λ [ M ] mm · · · U † mm . (8)Note that the elements of H o mm are i.i.d. CN (0 , randomvariables and their distributions are independent of ( G i m , F i m ) since H mm is isotropic. To transmit data through the eigen-modes of H mm , G o m and F o m are chosen as G o m = { [ V mm ] , [ V mm ] , · · · , [ V mm ] M } and F o m = U mm . With perfect CSI feedback, the above precoder and equalizerjoint design converts each data link into M parallel spatialchannels which are free of interference.Note that increasing N enhances the array gain of bothlinks. Specifically, the expectations of the SNRs increase with N . Thus, N should take its maximum ( L − M ) . However,maximizing N need not be optimal for the link performancein the case of quantized feedback as discussed in the sequel. B. Quantized Precoder Feedback
In this section, we choose the index sets A and B in (7) withthe objective of suppressing the residual interference causedby precoder feedback errors.Recall that the precoding at T n is enabled by quantizedcooperative feedback of F i n from R m with m (cid:54) = n . Let ˆ F i n denote the quantized version of F i n and define the resultantquantization error (cid:15) n as (cid:15) n := 1 − (cid:107) ( F i n ) † ˆ F i n (cid:107) F M , n = 1 , (9)where ≤ (cid:15) n ≤ . The error (cid:15) n is zero in the case ofperfect cooperative feedback, namely F i n = ˆ F i n . A nonzeroerror results in violation of the orthogonality constraint in (5) G i m H mn ˆ F i n (cid:54) = 0 , m (cid:54) = n. (10)The resultant residual interference from T n to the (cid:96) th datastream of R m has the power I [ (cid:96) ] mn := P n ν (cid:107) [ G o m ] † (cid:96) ( G i m ) † H mn ˆ F i n F o n (cid:107) , m (cid:54) = n. (11)Next, we choose A and B in (7) by minimizing an upperbound on the residual interference power as follows. Based on(7), (6) can be rewritten as H mn = (cid:2) B m G i m (cid:3) Σ mn (cid:2) F i n C n (cid:3) † (12)where Σ mn := Π diag (cid:18)(cid:113) λ [1] mn , (cid:113) λ [1] mn , · · · , (cid:113) λ [ L ] mn (cid:19) Π † with Π being an arbitrary permutation matrix that rearrangesthe order of the singular values along the matrix diagonal. Thecolumns of the matrices B m and C n comprise the ( L − N ) left and ( L − M ) right singular vectors of H mn , respectively,which are determined by Π . Let the set D mn contain the last N elements along the diagonal of Σ mn . Lemma 1.
The interference power I [ (cid:96) ] mn in (11) can be upperbounded as I [ (cid:96) ] mn ≤ M νP n (cid:15) n max α ∈D mn α , m (cid:54) = n. (13)Readers can refer to the full paper [12] for the proofs ofthe above lemma as well as analytical results in the sequel.Minimizing the upper bound in (13) gives that D mn consistsof the N smallest singular values of H mn . Equivalently, Π isan identity matrix and thus G i m and F i n are given as G i m = { [ V mn ] L − N +1 , [ V mn ] L − N +1 , · · · , [ V mn ] L } F i n = { [ U mn ] , [ U mn ] , · · · , [ U mn ] M } . (14)Then (13) can be simplified as I [ (cid:96) ] mn ≤ M νP n λ [ L − N +1] mn (cid:15) n , ∀ ≤ (cid:96) ≤ M, m (cid:54) = n. (15)Note that the above upper bound on I [ (cid:96) ] mn is independent ofthe stream index (cid:96) . On one hand, the upper bound reduceswith decreasing N . On the other hand, as mentioned earlier,larger N increases link array gain. These opposite effects of N on link performance make it an important parameter for pre-coder optimization. Finding the optimal N is mathematicallyintractable but a numerical search is straightforward.IV. I NTERFERENCE P OWER C ONTROL F EEDBACK
A. Fixed Interference Margin
The receiver R m sends the IPC signal, denoted as η n , to theinterferer T n for controlling its transmission power as P n = min( η n , P max ) , n = 1 , . (16)The scalar η n is designed to prevent the per-stream interfer-ence power at R m from exceeding a fixed margin τ with τ > ,namely I [ (cid:96) ] mn ≤ τ for all ≤ (cid:96) ≤ M . A sufficient conditionfor satisfying such constraints is to upper bound the right handside of (15) by τ . It follows that η n := τM νλ [ L − N +1] mn (cid:15) n , m (cid:54) = n. (17)Given τ , a lower bound A IM on the sum throughput ¯ C , calledthe achievable throughput , is obtained from (2) as A IM = (cid:88) m =1 M (cid:88) (cid:96) =1 log (cid:32) η m , P max ) λ [ (cid:96) ] mm τ (cid:33) (18)where η m is given (17).It is infeasible to derive the optimal value of τ for eithermaximizing A IM in (18) or minimizing P out in (3). However,for P max being either large or small, simple insight intochoosing τ can be derived as follows. The residual interferencepower decreases continuously with reducing P max . Intuitively, τ should be kept small for small P max . For large P max , thechoice of τ is less intuitive since large τ lifts the constraintson the transmission power but causes stronger interference andvice versa. We show below that large τ is preferred for large P max . Let ´ λ k denote the eigenvalue of the Wishart matrix HH † with H being an i.i.d. N × M CN (0 , matrix. Define ˇ λ k similarly but with H being a L × L matrix. emma 2. For large P max , the achievable throughput is A IM = 2 M (cid:88) (cid:96) =1 E (cid:34) log (cid:32) τ ´ λ (cid:96) (1 + τ ) M ν ˇ λ L − N +1 (cid:15) (cid:33)(cid:35) + o (1) . It can be observed from the above result that the first orderterm of A IM attains its maximum for τ → ∞ . However, thisterm is finite even for asymptotically large P max and τ , whichis the inherent effect of residual interference. Lemma 3.
For large P max , the outage probability is upperbounded as P out ≤ (cid:32) τ τ × ´ λ (cid:96) M ν ˇ λ L − N +1 (cid:15) < θ (cid:33) + o (1) . Similar remarks on Lemma 2 apply to Lemma 3.
B. Sum Throughput Criterion
In this section, an iterative IPC algorithm is designed forincreasing the sum throughput ¯ C in (2). Since ¯ C is a non-convex function of transmission power, directly maximizing ¯ C does not yield a simple IPC algorithm. Thus, we resort tomaximizing a lower bound A ST (achievable throughput) on ¯ C instead, obtained from (2) and (15) as A ST = E [ A ] with A := M (cid:88) (cid:96) =1 (cid:34) log (cid:32) P λ [ (cid:96) ]11 P M νλ [ L − N +1]12 (cid:15) (cid:33) +log (cid:32) P λ [ (cid:96) ]22 P M νλ [ L − N +1]21 (cid:15) (cid:33)(cid:35) . (19)Thus, the optimal transmission power pair is given as ( P (cid:63) , P (cid:63) ) = max P ,P ∈ [0 ,P max ] A ( P , P ) . (20)The objective function A remains non-convex and its max-imum has no known closed-form. However, inspired by themessage passing algorithm in [4], a sub-optimal search for ( P (cid:63) , P (cid:63) ) can be derived using the fact that ∂A ( P (cid:63) , P (cid:63) ) ∂P m = 0 ∀ m = 1 , . To this end, the slopes of A are obtained using (19) as ∂A ( P , P ) ∂P m = µ m + ψ m − ρ m (21)where µ m := log e (cid:88) M(cid:96) =1 λ [ (cid:96) ] mm M νλ [ L − N +1] mn (cid:15) n P n + λ [ (cid:96) ] mm P m ψ m := log e (cid:88) M(cid:96) =1 M νλ [ L − N +1] nm (cid:15) m M νλ [ L − N +1] nm (cid:15) m P m + λ [ (cid:96) ] nn P n ρ m := log eM νλ [ L − N +1] nm (cid:15) m M νλ [ L − N +1] nm (cid:15) m P m . Note that based on available CSI, µ m has to be computed at R m and ( ψ m , ρ m ) at R n with n (cid:54) = m . Therefore, based on(21), an iterative IPC feedback algorithm can be designed tohave the following procedure. Algorithm :
1) The transmitters T and T arbitrarily select the initialvalues for P and P , respectively.2) The transmitters broadcast their choices of transmissionpower to the receivers.3) Given ( P , P ) , the receiver R computes ( µ , ψ , ρ ) and sends µ and ( ψ − ρ ) to T and T , respectively.Likewise, R computes ( µ , ψ , ρ ) and feeds back µ and ( ψ − ρ ) to T and T , respectively.4) The transmitters T and T update P and P , respec-tively, using (21) and the following equation P m ( k +1) = min (cid:40)(cid:20) P m ( k ) + ∂A ( P , P ) ∂P m ∆ γ (cid:21) + , P max (cid:41) where k is the iteration index and ∆ γ a step size.5) Repeat Steps − till the maximum number ofiterations is performed or the changes on ( P , P ) aresufficiently small. C. Outage Probability Criterion
As the problem of minimizing P out in (3) by power controlis analytically intractable, the IPC algorithm is designed byminimizing an upper bound on P out . Using (15), the SINR in(1) is lower bounded by (cid:93) SINR [ (cid:96) ] m where (cid:93) SINR [ (cid:96) ] m := P m λ [ (cid:96) ] mm P n M νλ [ L − N +1] mn (cid:15) n , m (cid:54) = n. (22)Therefore, P out ≤ Pr (cid:18) min m =1 , (cid:93) SINR [ M ] m < θ (cid:19) . (23)Minimizing the above upper bound on P out is similar inthe mathematical structure to the classic problem of optimalpower control for single-antenna interference channels [13].The optimal transmission power for minimizing the right handside of (23) solves the following optimization problem ( P (cid:63) , P (cid:63) ) = arg min P ,P ∈ [0 ,P max ] ( P , P ) s.t. min m =1 , (cid:93) SINR [ M ] m ( P , P ) > θ (24)where the first minimization implies that ( P (cid:63) , P (cid:63) ) (cid:22) ( P , P ) for all ( P , P ) that satisfies the constraint in (24) as well as (0 , (cid:22) ( P , P ) (cid:22) ( P max , P max ) , called feasible power pairs.Using (22) and (24), the constraint in (24) can be written as P m ≥ a m + b mn P n , m (cid:54) = n (25)where a m := θλ [ M ] mm and b mn := Mνλ [ L − N +1] mn (cid:15) n θλ [ M ] mm . The minimumpower ( ´ P , ´ P ) that satisfies the constraints in (25) is ´ P m := a m + b mn a n − b mn b nm , m = 1 , . (26)This expression gives optimal power control as stated below. Proposition 1. If ( P (cid:63) , P (cid:63) ) in (24) exists, ( P (cid:63) , P (cid:63) ) = ( ´ P , ´ P ) with ( ´ P , ´ P ) given in (26) . A c h i e v ab l e S u m T h r oughpu t ( b i t/ s / H z ) Max ThroughputMax Throughput (1 Iteration)Fixed Interference MarginIncreasing Interference MarginPerfect CSI Feedback
Fig. 2. Comparison of achievable sum throughput between different IPCfeedback algorithms for the coupling factor ν = 0 . . Based on Proposition 1, the corresponding IPC feedbackprocedure is obtained as follows.
Algorithm :
1) The receiver R computes b and transmits b to T .Similarly, R computes b and feeds back b to T .2) The receiver R computes a and communicates a to T . Likewise, R computes a and feeds back a to T .3) The transmitters T and T compute ´ P and ´ P , respec-tively and set their transmission power equal to ( ´ P , ´ P ) if they are feasible. Otherwise, arbitrary transmissionpower is used.V. S IMULATION R ESULTS
In the simulation, codebooks for quantizing F i and F i are randomly generated and have equal sizes. The simulationparameters are set as L = 6 , M = 2 , N = 3 , and B = 6 .The interference margin is either fixed at τ = 2 or increasedas τ = 0 . P max .Fig. 2 compares the achievable throughput of differentIPC feedback algorithms. Significant coupling ( ν = 0 . )between links is observed to decrease achievable throughputdramatically with respect to perfect CSI feedback. For large P max , the IPC feedback Algorithm designed for maximizingthe achievable throughput is observed to provide substantialthroughput gain over those based on interference margins.Furthermore, increasing τ with growing P max gives higherthroughput than fixed τ , which is consistent with Lemma 2.Fig. 3 compares the outage probabilities and average trans-mit SNRs of different IPC feedback algorithms. With respectto the IPC feedback using increasing τ or with perfect feed-back, Algorithm dramatically decreases average transmissionpower. Moreover, Algorithm yields lower outage probabilitythan the two algorithms using τ . Finally, fixing τ causes P out to saturate as P max increases.R EFERENCES[1] V. R. Cadambe and S. A. Jafar, “Interference alignment and the degreesof freedom for the k user interference channel,”
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