Wireless MIMO Switching with Network Coding

1 W ireless MIMO Switching with Netw ork Coding Fangg ang W ang and Soung C . Liew Institute of Network Coding The Chinese Uni versity of Hong K ong Abstract In a generic switching problem, a switching pattern consists of a one-to-o ne mapping f rom a set of inputs to a set of outpu ts (i.e., a permutation ). W e propo se and inv estigate a wireless switching framework in wh ich a multi-anten na relay is responsib le for switch ing traffic among a set of N stations. W e refer to su ch a relay as a MIMO switch. W ith b eamformin g and linear d etection, th e MIMO switch controls which station s are conn ected to w hich other stations. Each beamf orming matrix realizes a p ermutation p attern amo ng th e stations. W e refer to the corre sponding p ermutation matrix as a switch matrix. By sch eduling a set of different switch matr ices, full connectivity amon g th e stations can be established . In this pap er , we fo cus on “fair switching” in wh ich equal amounts of traf fic are to be deli vered for all N ( N − 1) o rdered p airs of stations. In particular, we investi gate how the system throug hput can be max imized. In genera l, for large N the n umber of possible switch matr ices (i.e., permutatio ns) is huge, m aking th e sch eduling problem com binatorially ch allenging. W e show that for the cases of N = 4 and 5 , only a subset of N − 1 switch m atrices n eed to be conside red in the sche duling problem to achieve good through put. W e co njecture that this will be the case for large N as well. This conjectur e, if valid, implies th at for practical purp oses, fair-switching scheduling is n ot an intrac table prob lem. W e also investigate MIMO switchin g with phy sical-layer n etwork coding in this pap er . W e find that it can improve thro ughpu t ap preciably . Index T erms MIMO switching, relay , derangem ent, fairness, phy sical-layer n etwork c oding. I . I N T R O D U C T I O N Relaying in wireless networks plays a key role in various communi cation applications [1]. The use of relays can extend covera ge as well as improve ener gy effic iency [2]. In this p aper , we s tudy a set-up in which multipl e stations communicate with each other via a mul ti-antenna relay . W ith beamforming, the Corresponding author *** 2 relay controls which station s are connected to which other st ations. Each beamformi ng matrix realizes a permutation among the statio ns. By schedulin g a s et of di f ferent switch matrices, full connectivity amo ng the stations can be established. Prior work that inv estigated the set-up of multip le s tations exchanging data via a relay includ es [2], [3], [4], and [5]. Ref. [2] studi ed “pairwise data exchange”, in w hich st ations form pairs, and two stati ons in a pair exchange data with each other only . Specifically for pairwise data exchange, if station i transmits to station j , then station j t ransmits t o s tation i as well. In [2], MIMO relays wit h d if ferent forward strategies were considered. The bound s on the symmetric capacity were presented. Ref. [3] also studied pairwise data exchange, but the relay adopts t he decode-and-forward strategy only . The div ersity-mu ltiplexing tradeoffs under reciprocal and non-reciprocal chann els were analyzed. Both [2] and [3] studied the case in which a station com municates with one other s tation only . In a general setti ng, a s tation could hav e data for m ore than o ne station. In this paper , we focus on a uni form traffic setti ng in which the amounts of t raf fic from station i to station j are the same for all i, j ∈ { 1 , · · · , N } , i 6 = j . W e refer to m eeting such a uniform traf fic requirement as “fair switchi ng”. Fair switchi ng is realized by scheduli ng a set o f sw itch mat rices. T o the best of our knowledge, the framew ork of fair switchin g has not been considered in the existing literature. Refs. [4] and [5] in vestigated the case of full data exchange, in which all stati ons want to broadcast their data to all other statio ns 1 . Data transmissi ons in [4] and [5] can be summ arized as follows: in t he first slot, all stations transm it to t he relay sim ultaneously; the first slot is followed by m ultiple slo ts for downlink trans missions; i n each downlink s lot, the relay mul tiplies the signal receiv ed in the first time slot by a different beamforming matrix, such that at the end of all downlink slots, all stations receiv e the broadcast data from all ot her stations . By cont rast, the frame work inv estigated in this paper is more general in that it can accommodate the pure unicast case, t he mixed unicast-mul ticast case, as well as t he pure broadcast case as in [4] and [5]. In particular , a stati on i can ha ve M i data streams, and each stati on j 6 = i i s a targe t receiver of one of t he M i streams. In our framework, the MIMO relay serves as a general switch that swit ches traffic amo ng th e s tations. W e focus on the use of beamformi ng at the relay and l inear detection to realize di f ferent con nectivity patterns among the st ations. Each beamforming matrix realizes a permut ation conn ecti vity among the stations. By schedul ing a set of switch m atrices, the M IMO switching system can realize any g eneral 1 Note that full data excha nge is also discussed in [2]. But they consider a single-antenna relay . 3 transmissio n pattern (uni cast, mul ticast, broadcast, or a mi xture of th em) among the stati ons. Before delving i nto t echnical details, we provide a si mple example to illus trate the scenario of int erest to us here. Consider a network with three statio ns, 1 , 2 , and 3 . The traffic flows among them are shown in Fig. 1: stat ion 1 wants to transm it “ a ” to both s tations 2 and 3 ; s tation 2 wants to t ransmit “ b ” and “ c ” to s tations 1 and 3 , respectively; s tation 3 wants to transmit “ d ” and “ e ” to stations 1 and 2 , respectiv ely . Pairwise data exchange as i n [2] and [3] is not effec tive in this case because when the number of stations is odd, one station will always be left out when formi ng pairs. That is, when the number of s tations is odd, t he connectivity pattern realized by a swit ch/permutation matrix do es n ot correspond t o pairwise communication . Full data exchange is not appropriate either , since in our example, stati on 2 (as well as station 3 ) transm its differ ent dat a t o the other t wo st ations. Under our framework, the traffic flows among stations can be met as shown in Fig. 2. In the first slo t, station 1 t ransmits “ a ” to s tation 3 ; station 2 transmits “ b ” to station 1 ; s tation 3 transmits “ e ” to station 2 . In t he second slot, station 1 transm its “ a ” to st ation 2 ; stati on 2 transmi ts “ c ” to station 3 ; statio n 3 transm its “ d ” to station 1 . In Section III.C, we will present the details on how to realize the switch matrices. T o limit the scop e, this paper focuses on the use of amplify-and-forward relaying and zero forcing (ZF) in establish ing the permutations am ong stati ons. Howe ver , we do generalize t he ZF method to one that e xploi ts physical-layer network coding [6], [7], [8], [9] for performance i mprovement. The rest of the paper is organized as follows: Section II d escribes the framework of wireless MIMO switching and introduces th e ZF relaying method for es tablishing permutat ion among stations. A fair switching schem e i s proposed in Section III. In Section IV , we generalize th e ZF meth od to o ne that exploits network coding . In Section V , we propose two enhanced schemes of MIMO switching. Section VI presents and discusses our si mulation resul ts. Section VII conclu des this paper . I I . S Y S T E M D E S C R I P T I O N A. System Model Consider N s tations, S 1 , · · · , S N , each wit h one antenna. The stati ons commu nicate via a relay R with N antennas and there is no direct link between any two s tations as shown i n Fig. 3. Each tim e slo t is divided into two sub slots. The first subslot is for u plink transmi ssions from the stations t o the relay; the second subslo t is for d ownlink transmission s from the relay to the statio ns. For simplicit y , we assume the two subslo ts are of equal duration. Each time slot realizes a switchi ng permut ation, as described below . 4 Consider one time slot . Let x = { x 1 , · · · , x N } T be t he vector representing the si gnals transmitt ed by the stations. W e assume that all powers (includi ng noi se powers) are norm alized with respect to the transmit power of a stat ion. Furthermore, all stations use the sam e transmit power . Thu s, E { x 2 i } = 1 , ∀ i . W e also assume that E { x i } = 0 , ∀ i , and that there i s n o cooperati ve coding among the stations so that E { x i x j } = 0 , ∀ i 6 = j . Let y = { y 1 , · · · , y N } T be the recei ved signals at the relay , and u = { u 1 , · · · , u N } T be the noise vector with i.i.d. noise sam ples foll o wing t he com plex Gauss ian distribution, i.e., u n ∼ N c (0 , σ 2 r ) . Then y = H u x + u , (1) where H u is t he uplin k channel gain matrix. The relay mult iplies y by a beamform ing matrix G before relaying t he s ignals. W e impose a power const raint on the sig nals transmitt ed by the relay so that E {k Gy k 2 } = p. (2) Combining (1) and (2), w e have E [ X H H H u G H GH u X + u H G H Gu ] = p. (3) This giv es T r [ H H u G H GH u ] + T r [ G H G ] σ 2 r = p. (4) Let H d be the downlink channel m atrix. Th en, the receive d signals at the stations in vector form are r = H d Gy + w = H d GH u x + H d Gu + w , (5) where w is the noise vector at the recei ver , with the i .i.d. noise samples following the complex Gaussi an distribution, i.e., w n ∼ N c (0 , σ 2 ) . B. MIMO Switching Suppose t hat the purposes of G are to realize a particular permutation witho ut d iagonal elem ents represented b y the permutation m atrix P , and to provide signal amplifications for the signals comin g from the s tations. That is, H d GH u = AP , (6) 5 where A = diag { a 1 , · · · , a N } is an amplification diagonal matrix. Thus, r =             a 1 x i 1 . . . a j x i j . . . a N x i N             + AP H − 1 u u + w , (7) where S i j is the s tation transm itting to S j under the permutation P (i .e., in row j of P , element i j is one, and al l other elem ents are zero). Define ˆ r = A − 1 r , i.e., station S j divides its received signal by a j . W e can rewrite (7) as ˆ r =             x i 1 . . . x i j . . . x i N             + P H − 1 u u + A − 1 w , (8) Suppose t hat we require the recei ved si gnal-to-noise rati o (SNR) of each station to be the same. Let h ( − 1) u, ( i,j ) be element ( i, j ) in H − 1 u . Then σ 2 r X k | h ( − 1) u, ( i j ,k ) | 2 + σ 2 | a j | 2 = σ 2 e , ∀ j. (9) In other words, | a j | = v u u t σ 2 σ 2 e − σ 2 r P k | h ( − 1) u, ( i j ,k ) | 2 , ∀ j, (10) where σ 2 e is t he effecti ve no ise power of each station under un it si gnal power (i.e., the noise-to-signal ratio). Substitutin g (6) int o (4), we have X i,j | h ( − 1) d, ( i,j ) | 2 | a j | 2 + σ 2 r X i,k | X j h ( − 1) d, ( i,j ) a j h ( − 1) u, ( i j ,k ) | 2 = p, (11) where h ( − 1) d, ( i,j ) is element ( i, j ) in H − 1 d . If we restrict a j ∀ j t o be real values, then plugging (10) into (11) 6 giv es X i,j | h ( − 1) d, ( i,j ) | 2 σ 2 σ 2 e − σ 2 r P k | h ( − 1) u, ( i j ,k ) | 2 + σ 2 r X i,k | X j h ( − 1) d, ( i,j ) h ( − 1) u, ( i j ,k ) | σ | r σ 2 e − σ 2 r P k | h ( − 1) u, ( i j ,k ) | 2 | 2 = p, (12) Problem Definition 1 : Giv en H u , H d , p, σ 2 , σ 2 r , and a desi red permu tation P , solve for G , σ 2 e . In the following, we will dis cuss wheth er the problem is solvable, and if solvable how to solve the problem. For simpli city , we assum e a j ∀ j are restricted t o be non -negati ve real va lues. For non-negative real a j , σ 2 e ∈ (max i,j,k { σ 2 r P k | h ( − 1) u, ( i j ,k ) | 2 } , + ∞ ) according to (9). The l eft hand s ide (LHS) of (12) h as the following l imits: lim σ 2 e → + ∞ LHS = 0 , lim σ 2 e → max i,j,k { σ 2 r P k | h ( − 1) u, ( i j ,k ) | 2 } + LHS = + ∞ . Thus, there exists a σ 2 e that satisfies (12) for a give n power p . In the SNR regime of i nterest, t he first item of the LHS is generally dom inant because it is the power of desired signals. In this case, the LHS o f (12) decreases monotoni cally as σ 2 e increases, and t here exists a uniqu e σ 2 e for wh ich (12) is satisfied. Given this σ 2 e , a unique | a j | can be found from (10). Therefore, in the SNR regime of interest, “Problem 1” i s alwa ys s olvable and has a un ique soluti on when a j ∀ j are non-negative real, and H u , H d are in vertible. Numerical Method 1 : There are N equations in (10) and one equation in (11). These equations can be used to solve a j for j = 1 , · · · , N and σ 2 e . After that (6) can be us ed to find G from A . I I I . F A I R S W I T C H I N G As has been described in the previous sectio n, in each tim e slot, the statio ns transmi t to one another according to the s witch matrix. In this secti on, we s tudy a specific s cenario in wh ich each station h as an equal amount of t raf fic to be sent to e very other statio n. The data from station i to st ation j could be different for differe nt j , so t his is not restri cted to the mu lticast o r broadcast settin g. W e refer to this setting as ”fair s witching”. T o achiev e fair switchi ng, mult iple transmi ssions using a su ccession of diffe rent swi tch matrices will b e n eeded. W e next discuss t he set of swit ch matrices we need to const ruct ”fair switchin g”. 7 A. Derangement A derangement i s a permutat ion in which i i s not mapped to its elf [10]. While t he number of di stinct permutations with N stat ions is N ! , the number of derangements is given by a recursive form ula d N = N · d N − 1 + ( − 1) N , ! N , (13) where d 1 = 0 , and ! N is subfactorial calcul ated b y ! N = ( N − 1)[!( N − 1)+!( N − 2)] , (14) where !0 = 1 , !1 = 0 . For example, d 4 = 9 althoug h the num ber of permutati ons is 4! = 24 . The nine derangements are listed as foll ows 2 : P 1 = P 2 = P 3 =          0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0          ,          0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0          ,          0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0          , P 4 = P 5 = P 6 =          0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0          ,          0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0          ,          0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0          , P 7 = P 8 = P 9 =          0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0          ,          0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0          ,          0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0          . 2 Note that for pairwise data e xchange [2], [3], pairs of stations want to se nd data to each other . This correspond s to a sy mmetric derangement in which element ( i, j ) = 1 implies element ( j, i ) = 1 , such as P 1 , P 5 and P 9 . 8 It can be shown that lim N →∞ d N N ! = e − 1 and the limit is approached quite quickly . Thus, the number of derangements is i n general very l ar ge for l ar ge N . Performing optim ization over this l ar ge com binatorial set of derangements in o ur problem is a formi dable task. For example, in our fair s witching probl em, we want to maxim ize the system throughput by scheduling over a subs et of derangements. It would be ni ce if for our problem, the optimal solut ion is not very sensi tiv e to the particular selection of derangements . In Part B, we will formalize the concept of “condensed derangement sets”. B. Condensed Derangement Set Definition 1 : A set o f N − 1 derangements, D 1 , D 2 , · · · , D N − 1 , is said to be a condensed derangement set if N − 1 X n =1 D n = J − I , (15) where J is a matrix with all “1” elements, and I is the identi ty m atrix. W i th the help of a computer program, we obt ain all the four cond ensed derangement sets for N = 4 , i.e., Q 1 = { P 1 , P 5 , P 9 } , Q 2 = { P 1 , P 6 , P 8 } , Q 3 = { P 2 , P 4 , P 9 } , and Q 4 = { P 3 , P 5 , P 7 } . Furthermore, there are d 5 = 44 derangement s for N = 5 and the num ber of condensed derangement sets is 56 . In fair switching, we want to s witch an equal amount o f t raf fic from any stati on i to any station j , i 6 = j . Thi s can be achieved by scheduling the d erangements in the con densed derangement set i n a weighted round-robin manner (as detailed i n “ Approach t o Problem 2” below). Given a cond ensed set, the schedulin g to achie ve fair swi tching i s rather sim ple. Howe ver , unless proven otherwise, di f ferent condensed sets may potentially yi eld soluti ons of different performance. A nd the number of condensed derangement sets could b e hu ge for large N . W e define a problem as follows. Problem Definition 2 : Suppose that we want t o send equal amou nts of traf fic from S i to S j ∀ i 6 = j . Which condens ed d erangement sets s hould be used t o schedule transmis sions? Does it matter? Approach to Pr oblem 2 : The derangement s in a condensed d erangement set are th e building bl ocks for scheduling. For example, in a comp lete rou nd transmis sions, w e may schedule derangement D n for k n time slots. Then the length of the com plete round transm issions wi ll be P N − 1 n =1 k n . Consider t he case of N = 4 . There are four condensed derangement sets. The qu estion is which condensed derangement set w ill result in the highest th roughput. T o answer th is question, we can approach the problem as fol lows. 9 Let Q m = { D m 1 , D m 2 , · · · , D m N − 1 } be a particular condensed d erangement set. For each D m n , we use “Numerical Meth od 1” above to compute the correspondi ng σ 2 e , denoted by σ 2 e,n,m . The Shannon rate i s then r n,m = log(1 + 1 σ 2 e,n,m ) . (16) Because of the u niform t raf fic ass umption, w e require k n,m r n,m = c, ∀ n ∈ [1 , · · · , N − 1] , (17) for som e c . That is, c is t he amount of traffic deliv ered from one station to another statio n in one round of transmissio ns. The ef fective throug hput per station (i.e., the amount of traffic from a st ation to all oth er stations) is T m = ( N − 1) c P N − 1 n =1 k n,m = N − 1 P N − 1 n =1 1 /r n,m . (18) Numerically , we could first solve for r n,m ∀ n . Then, we apply (18) to find the throughput. The question we want to answer is whether T m for different Q m are signi ficantly dif ferent. For the case of N = 4 and 5 , we will show some simulation results indicating that the throughputs of differe nt Q m are rather close, and therefore it does no t matt er mu ch which Q m we use. C. Generalization In mu lti-way relay networks, as mentio ned in the introdu ction, most prior works focus on t wo patterns of transm issions. The first is pairwise u nicast, in whi ch stations form pairs, and the two s tations of a pair only com municate wi th each ot her [2], [3]. The second is the full data exchange, in which each station needs to broadcast to all t he other stations [4], [5]. In practice, howev er , the actual transmi ssion patterns could be d if ferent from th ese two patterns. F or example, in a video conference s ession, a subset of stations within th e network forms a mul ticast group, and the transm ission pattern is so mewher e between the two extremes above. More generally , in the same network, there coul d be the co-existence of broadcast sessio ns, multicast sessions, pairwise unicast sessi ons, and uni directional u nicast sessions. The MIMO sw itching frame work here is flexible and encompasses th is generality . A s cheme for the full data exchange (broadcast) traf fic pattern is given in [11]. It turns out that we could expand the idea to cater t o the general t raf fic p attern containing a mi xture of unicast , multicast , and broadcast as well . For easy e xplanation, ou r previous 10 discussion in Part B has an implicit assum ption (focus) that each station i wants to send different data to each other station j 6 = i . If we examine the scheme carefully , this assu mption is not necessary . W e note that under the scheme, a station wil l ha ve chances to transm it to all other stations. In particul ar , a station i will h a ve chances t o transm it data to two diff erent stations j and k in two different derangement s. If so desired, station i could transmi t the same data to stations j and k in the two derangements. This observation in turn im plies that t he general traffic pattern can be realized. For illustration, let us examine how the traf fic pattern of Fig. 1 can be realized. T his example is a pattern consisting of the co-existence of unicast and broadcast. As has been described the data transmission can be realized by scheduling a con densed derangement s et, which i s: D 1 =      0 1 0 0 0 1 1 0 0      , D 2 =      0 0 1 1 0 0 0 1 0      . The transmitted data of station 1 , 2 and 3 are respectively [ a, b, e ] T for D 1 and [ a, c, d ] T for D 2 . I V . M I M O S W I T C H I N G W I T H N E T WO R K C O D I N G The MIMO switch so far makes us e of ZF detection, in which data are s witched according to the derangement matrix. In other words, the elements of “1” denote the s witching desi red, while the elements of “0” are the interferences that have been forced to zero. A non-zero di agonal element corresponds to self transmission, m eaning the relay forwards the self information back to the transm itter . In general, there is no need to force a diagonal element to zero because the self informat ion is known by t he transmitter , and can be subtracted out by t he transmitter . This is the basic idea behind network coding. Thus, t he zero-forcing requirement for the transmission pattern matri x can be relaxed for the diagonal. W ith t his relaxation, t here is an extra degree of freedom in our desig n to improve the sy stem performance. In thi s case we rewrite (6) as H d GH u = A ( P + B ) , (19) where B is a diagonal matrix B = di ag { b 1 , · · · , b N } . For a symmetric derangement such as P 1 , the correspond ing network-coded switch matrix has the 11 following p attern: P s =          b 1 0 0 1 0 b 2 1 0 0 1 b 3 0 1 0 0 b 4          . This particular s witching pattern corresponds to pairwise d ata exchange, in which stations 1 and 4 are a pair and stations 2 and 3 are the other pair . Each pairwis e t ransmission is actual ly a two-way relay channel. Hence, the network codin g-based MIM O swi tching constructs multi ple parallel two-way relay transmissio ns. Howe ver , for an asymmetric derangement such as P 2 , the correspond ing swi tch matri x is P a =          b 1 0 0 1 0 b 2 1 0 1 0 b 3 0 0 1 0 b 4          . This is not a traditional physical-layer network coding setting [8] because the data exchange is not pairwise. For both symm etric and asym metric switch matrices, we shall refer to t he correspondi ng matrices wi th non-zero diagonal as network-coded switch matri x, and th e ass ociated MIM O switching setup as MIMO switching with n etwork codi ng. Note that the concept here generalizes th e concept of physical-layer network coding for two-way relay channel i n [8]. Extending t he previous ZF metho d, ZF with network coding yi elds t he following signal after detection. ˜ r =             x i 1 . . . x i j . . . x i N             + A − 1 B x + ( P + B ) H − 1 u u + A − 1 w . (20) The self-information term A − 1 B x can be cancelled at the station s. Potentially , with network coding, we could decrease the n oise power by designin g B appropriately . The ef fectiv e noise power can be written by σ 2 r X k | h ( − 1) u, ( i j ,k ) | 2 + σ 2 r | b j | 2 X k | h ( − 1) u, ( j,k ) | 2 + σ 2 | a j | 2 = σ 2 e , ∀ j, (21) 12 that i s | a j | = | σ | r σ 2 e − σ 2 r P k | h ( − 1) u, ( i j ,k ) | 2 − σ 2 r | b j | 2 P k | h ( − 1) u, ( j,k ) | 2 , ∀ j. (22) The relay transmissi on power can be rewritten as T r { [ H − 1 d A ( P + B )] H H − 1 d A ( P + B ) } + σ 2 r · Tr { [ H − 1 d A ( P + B ) H − 1 u ] H H − 1 d A ( P + B ) H − 1 u } = p. (23) Problem Definition 3 : Giv en H u , H d , p, σ 2 , σ 2 r , and a desired permutati on P . Solve for G , B , σ 2 e to optimize the effecti ve through put of the network. Numerical Method 3 : W e give a naiv e method here. Assum e B = b I and b is a real scalar for simplicit y . W e set a search range for b , and for each trial of b , there are N equations in (22) and one equation i n (23). Use the N + 1 equ ations to solve a j for j = 1 , · · · , N and σ 2 e . Then find the mi nimum σ 2 e in among all tri als within t he s earch range of b , and the correspond ing A and B as t he solut ion. V . D I S C U S S I O N O N C O M P L E X A A N D B A and B are restricted to be real so far . In this section we consider complex A and B . Recall that with a give n effecti ve n oise power σ 2 e we can calculate th e p o wer consumptio n of t he relay . For compl ex A and B , with the extra degree of freedom of their phases, the relay power consumption can b e reduced. In other words, g iv en the transm it power constraint of the relay w e can o ptimize the ph ases of A and B to reduce σ 2 e . Howe ver , the problem is noncon vex thu s int ractable. W e p ropose two schemes to exploit the phases of A and B . The first schem e is tar geted for the pairwise transmissio n pattern without network coding. The amplitudes o f A are calculated b y Numerical Method 1. Th en for any pair of stations i n the overall pairwise pattern, their elements in A are set to hav e counter phases (i.e., out of phase by π ). Th is setting could be proved to be optimal for two-way relay chann el. T o li mit the scope of t he current paper , th is new result of ours will be publish ed s hortly els e where. For lar ger num bers of s tations, this setti ng can stil l improve the through put performance, as will be shown in Section VI. The s econd scheme is for the general case (pairwise or non-pairwis e transmis sion pattern, with or w ithout network coding). The ampli tudes of A and B are calculated by Num erical Method 1 and 3 respecti vely . Then we di vide the interval of [0 , 2 π ) equally i nto M bins wi th t he values of 0 , 2 π M , · · · , 2( M − 1) π M respectiv ely and randomly pick among them to set the phase values for the variables of A and B . W e perform L trials of t hese random p hase ass ignments. For each trial we could solve for 13 σ 2 e by Numerical M ethod 3, and t hen com pare the L s olutions of σ 2 e and s elect the trial which has the minimum σ 2 e . Thus, t he phase assi gnment of t his trial i s used to set th e phase values of A and B . In Section VI, we will show th at lar ge gains can be achieve d wit h only a small numb er of bins and trials. V I . S I M U L A T I O N In this section, we ev aluate t he t hroughputs of diffe rent s chemes. W e assume that the uplin k channel H u and downlink channel H d are reciprocal, i.e., H d = H T u , and they b oth follo w the complex Gaussian distribution N c ( 0 , I ) . Thus, elements of the channels are i .i.d. and fol low the complex Gaussian distribution N c (0 , 1) . W e assu me the relay has the same transmit power as all the statio ns, i.e., p = 1 . First, we answer the question raised in Probl em Definition 2. W e analyze th e scenarios wh ere N = 4 and N = 5 , and use the random-phase scheme described in Section V to calculate A . The four different condensed derangement set s of N = 4 , Q 1 , Q 2 , Q 3 and Q 4 , are considered for fair s witching. For each channel realization , we ev aluate the t hroughput per station , T m , as g iv en b y (18). W e simu lated a t otal of 10000 channel realizations and com puted the expected throu ghput aver aged over the channel realizations. W e find that the four con densed derangement sets yield essentially the s ame average throu ghput (within 1% i n the medi um and high SNR regimes and within 3% in the low SNR regime). Fig. 4 plots t he throughput for o ne of the con densed derangement set. For N = 5 there are 56 dif ferent condensed derangement sets. As wi th t he N = 4 case, all the sets have roughly the s ame avera ge throughput (within 1%). Fig. 4 p lots t he result of one set. W e conjecture that different condens ed d erangement sets achiev e roughly the same av erage t hroughput for N larger than 5 as well. A concrete proof remains an op en problem. The ram ification of this result , if valid, is as foll o ws. For large N , the n umber of condensed derangement set is hu ge, and choosing the optim al set is a com plex combinatori al problem. Howe ver , if their relative performances do not d if fer m uch, choosing any one of t hem in our engi neering design will do, si gnificantly s implifying the prob lem. W e next ev aluate MIMO switching with and without network codi ng in single slot, i .e., for o ne derangement. First, we cons ider real variables. A scheme is proposed in th e literature [11], which in vestigates a similar problem as ours. It si mply uses a po sitive scalar weight to control the relay power consumption ins tead o f our d iagonal A . As shown in Fig. 5, it has al most the same av erage t hroughput as our MIM O switching scheme wi th real A and witho ut network coding. Henceforth we regard the MIMO switchi ng schem e with real A and without network codin g as a benchmark and call it “th e basic scheme”. Despite the same avera ge throughput performance, the basic scheme h as an advantage over the 14 scalar scheme in [11] in that the basic scheme guarantees fairness. That is, in our basic schem e, each station has exactly the same throughput, while the stations in the s cheme i n [11] could hav e varying throughputs. The scalar scheme i n [11] focuses on opt imizing the sum rate of all s tations; the individual rates of the s tations may vary widely with onl y one degree of freedom g iv en by the scalar . Fig. 5 als o shows that when both A and B are real, our MIMO swi tching wi th network codi ng has 0.5dB gain compared to the basic s cheme. Next, we consider compl ex A wit hout network coding. The random-phase scheme p roposed i n Section V with 10 and 1 00 trials are shown i n F ig. 5, a nd the t wo curves ac hieve 0.6dB and 0.7dB gains r espectively over the basic scheme. Thu s, the scheme with complex A can po tentially achiev e even larger gain than the network coding scheme with real A and B . The other propo sed scheme in Section V is for pairwise transmissio n pattern. W e set the two corresponding A ’ s elements in one pair to have counter phases. Its a verage throughput is in between those of the random-phase schemes with 10 and 100 trials, and achie ves 0 .65dB gain over the bas ic scheme. The advantage of the count er -phase scheme is th at it has low com plexity comparable to the basic scheme. Howe ver , for non-pairwis e transm ission, as s hown in Fig. 6 the count er -phase schem e cannot achie ve any gain over the b asic scheme, whi le random-phase scheme stil l can. Lastly , we consider complex A and B for M IMO sw itching with network coding. W e ev aluate the random-phase scheme with 10 and 100 trials, and th ey can achie ve 1.5dB and 2.2dB gain over t he basic scheme. Therefore, wi th network coding the throughput p erformance can be further improved compared with the other M IMO swi tching s chemes. T o su m up t his section, three g eneral result s are stated as follows: General Result 1 : In t he framew ork of MIMO fair swit ching with 4 or 5 stations, any condensed derangement set can be used to s chedule a fair switching because different cond ensed derangement sets achie ve roughly the same av erage through put. W e conjecture that this will b e the case when N , th e number of stat ions, is large as well. If this conjecture hol ds, then the issue of condensed s et selection will go aw ay , and the complexity of t he optimizatio n problem will be greatly reduced. This is especially so when N is large because the nu mber of different condensed derangement sets grow exponentially with N . General Result 2 : Network codi ng can be appl ied in MIMO switching to significantly improve ave rage throughput performance. It is worth menti oning that n etwork coding helps not onl y for the tradi tional pairwise swit ching pattern but als o for the non-pairwise pattern. The non-pairwise pattern has not been 15 treated in t he existi ng literature p rior to t his paper . General Result 3 : For pairwise swi tch wit hout network coding, the counter-phase scheme outperforms the basi c s cheme with real A . Furthermo re, the counter-phase scheme has low complexity because it s computation does no t optimize over the phases of A . V I I . C O N C L U S I O N S W e hav e proposed a framework for wireless MIMO swi tching to facilitate comm unications among multiple wireless statio ns. In our framework, a mul ti-antenna relay con trols wh ich stations are connected to whi ch other stati ons with beamform ing. Specifically each beamforming m atrix at the m ulti-antenna relay realizes a switching permut ation among the s tations, represented by a sw itch matrix. By scheduli ng a s et o f swi tch matrices, full conn ecti vity among th e st ations can be establis hed. A salient feature of our MIMO swit ching framew ork is th at it can cater to general traffi c p atterns consisting of a mi xture of unicast traf fic, multicast traffi c, and broadcast traffic flows among the station s. There are many nuances and impl ementation v ariations arising ou t of our MIMO switching framework. In th is paper , we h a ve only ev aluated th e performance of some o f the variations. W e first studied the “fair switching” setting in which each stati on wants t o send equal amounts of traffi c to all other stations. In this setting, we aim to deliver the s ame amount of data from each station i to each station j 6 = i by scheduling a set of switch matrices. In general, m any sets of s witch matrices coul d be used for such scheduling . The problem of finding the set t hat yield s opti mal th roughput is a very challenging problem combinatorially . Fortunately , for number of station s N = 4 or 5 , our simulati on results indicate t hat different s ets of switch m atrices achiev e roughly t he same throug hput, essentially rendering the selection of the optimal set a non-is sue. W e conjecture this will b e the case for larger N as well. If this conjecture hol ds, then the com plexity of the opt imization problem can be decreased sign ificantly as far as engineering design is concerned. W e next m ove d to the s tudy of single swi tch matrix and in vestigated the performances of different realizations for t he same switch matrix (i.e., realizations using different beamforming m atrices). Our general conclusion is that the use of physical-layer network coding can improve the throughput performance appreciably . In additi on, we di scove r an interesting resu lt for pairwise switch matrices. The computation cost of the b eamforming matrix could be hig h in general. Howe ver , when the switch matrix is p airwise and network codi ng is not used, the comp utation of the beamforming matrix can be m uch simplified wi th a “counter- phase” approach. 16 There are many future d irections go ing forward. For example, the beamforming matrices used in our simulatio n studies could be further optimized. In addition , the setting in which there are unequal amoun ts of traffic b etween stations wil l be int eresting to explore. Also, this paper has on ly considered s witch matrices those realize full p ermutations in which all stations participate in transmiss ion and reception in each sl ot; i t will be interestin g to explore switch m atrices that realize connectivities am ong stations that are not a full permutation. Finally , fut ure work coul d also explore the case w here the numb er of antenna at the relay i s not exactly N . R E F E R E N C E S [1] T . Cover and A. Gamal, “Capacity theorems f or t he rel ay channel, ” Information Theory , IE EE Tr ansactions on , vol. 25, no. 5, pp. 572–58 4, 1979. [2] D. Gunduz, A. Y ener , A. Goldsmith, and H. Poor, “The multi-way relay channel, ” in Information Theory , 2009. ISIT 2009. IEEE International Symposium on , 2009, pp. 339–343. [3] Y . Mohasseb , H . Ghozlan, G. Kramer, and H. El Gamal, “The MIMO wireless switch: Relaying can increase the multiplexing gain, ” in Information Theory , 2009. I SIT 2009. IEEE I nternational Symposium on , 2009. [4] T . C ui, T . Ho, and J. Kliewer , “Space-time communication protocols for n-way relay networks, ” in IEE E Global T elecommunications Confer ence , 2008. , 2008. [5] F . Gao, T . Cui, B . Jiang, and X. 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Klein, “Non-regen erativ e multi-way relaying wi th l inear beamforming, ” in Pr oc. 20th IEEE International Symposium on P ersonal, Indoor and Mobile Radio Communications Symposium , 2009. 17 1 2 3 a a 2 1 3 b c 3 1 2 d e Fig. 1. Traf fic demand of a three stations example. 1 2 3 3 1 2 MIMO switch MIMO switch 1 2 3 2 3 1 a a b b e e a a c c d d Slot 1 Slot 2 Fig. 2. A t ransmission established by t wo slots of unicast conn ectivity realizes the tr af fic demand in Fig. 1. Relay 1 2 3 N N Subslot 1 Subslot 2 Fig. 3. Wireless MIMO swit ching. 18 −10 −5 0 5 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 SNR(dB) Ave. Throughput Per Station 4 users 5 users Fig. 4. A verage throughput per station under MIMO fair switching. Only the results of one condensed derangement set i s presented because the results of other derangement sets are within 3% of the results sho wn here. 19 −10 −5 0 5 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 SNR(dB) Throughput cplx A rand 10 cplx A rand 100 cplx A counter−phase cplx A&B rand 100 (w/ NetCod) cplx A&B rand 10 (w/ NetCod) real A&B (w/ NetCod) real A scalar a real A cplx A cplx A & B Fig. 5. Throughput comparison of different MIMO switching schemes for pairwise switching pattern. 20 −10 −5 0 5 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 SNR(dB) Throughput real A cplx A counter−phase cplx A rand 10 cplx A&B rand 10 (w/ NetCod) Fig. 6. Throughput comparison of different MIMO switching schemes for non-pairwise switching pattern.