aa r X i v : . [ c s . I T ] A p r Wireless MIMO Switching
Fanggang Wang and Soung Chang Liew
Abstract
In a generic switching problem, a switching pattern consists of a one-to-one mapping from a set of inputs to aset of outputs (i.e., a permutation). We propose and investigate a wireless switching framework in which a multi-antenna relay is responsible for switching traffic among a set of N stations. We refer to such a relay as a MIMOswitch. With beamforming and linear detection, the MIMO switch controls which stations are connected to whichstations. Each beamforming matrix realizes a permutation pattern among the stations. We refer to the correspondingpermutation matrix as a switch matrix. By scheduling a set of different switch matrices, full connectivity amongthe stations can be established. In this paper, we focus on “fair switching” in which equal amounts of traffic are tobe delivered for all N ( N − ordered pairs of stations. In particular, we investigate how the system throughput canbe maximized. In general, for large N the number of possible switch matrices (i.e., permutations) is huge, makingthe scheduling problem combinatorially challenging. We show that for N = 4 and , only a subset of N − switchmatrices need to be considered in the scheduling problem to achieve good throughput. We conjecture that thiswill be the case for large N as well. This conjecture, if valid, implies that for practical purposes, fair-switchingscheduling is not an intractable problem. Index Terms
MIMO switching, relay, derangement, fairness, physical-layer network coding.
I. I
NTRODUCTION
Relaying in wireless networks plays a key role in various communication applications [1]. The useof relays can extend coverage as well as improve energy efficiency [2]. In this paper, we study a set-up in which N stations communicate with each other via a multi-antenna relay. With beamforming, therelay controls which stations are connected to which other stations. Each beamforming matrix realizes apermutation among the stations represented by a switch matrix. By scheduling a set of different switchmatrices, full connectivity among the stations can be established. Corresponding author: Fanggang Wang, [email protected]
Prior work that investigated N stations exchanging data via a relay includes [2], [3], [4], and [5]. Ref.[2] studied “pairwise data exchange”, in which stations form pairs, and two stations in a pair exchange datawith each other only. Specifically for pairwise data exchange, if station i transmits to station j , then station j transmits to station i as well. In [2], MIMO relays with different forwarding strategies were considered.Ref. [3] also studied pairwise data exchange, but the relay adopts the decode-and-forward strategy only.The diversity-multiplexing tradeoffs under reciprocal and non-reciprocal channels were analyzed. Both [2]and [3] studied the case in which a station communicates with one other station only. In a general setting,a station could have data for more than one station. In this paper, we focus on a uniform traffic settingin which the amounts of traffic from station i to station j are the same for all i, j ∈ { , · · · , N } , i = j .“Fair switching” is used to meet the uniform traffic requirement. Specifically, fair switching is realizedby scheduling a set of switch matrices. To the best of our knowledge, the framework of fair switchinghas not been considered in the existing literature.Refs. [4] and [5] investigated the case of full data exchange, in which all stations want to broadcasttheir data to all other stations . Data transmissions in [4] and [5] can be summarized as follows: in thefirst slot, all stations transmit to the relay simultaneously; the first slot is followed by multiple slots fordownlink transmissions; in each downlink slot, the relay multiplies the signal received in the first timeslot by a different beamforming matrix, such that at the end of all downlink slots, all stations receivethe broadcast data from all other stations. By contrast, the framework investigated in this paper is moregeneral in that it can accommodate the pure unicast case, the mixed unicast-multicast case, as well as thepure broadcast case as in [4] and [5]. In particular, a station i can have M i data streams, and each station j = i is a target receiver of one of the M i streams.In our framework, the MIMO relay serves as a general switch that switches traffic among the stations.We use beamforming at the relay and linear detection to realize different connectivity patterns amongthe stations. Each beamforming matrix realizes a permutation among the stations represented by a switchmatrix. By scheduling a set of switch matrices, the MIMO switching system can realize any generaltransmission pattern (unicast, multicast, broadcast, or a mixture of them) among the stations.Before delving into technical details, we provide a simple example to illustrate the scenario of interestto us here. Consider a network with three stations, , , and . The traffic flows among them are shownin Fig. 1: station wants to transmit “ a ” to both stations and ; station wants to transmit “ b ” and “ c ” Note that full data exchange is also discussed in [2]. But they consider a single-antenna relay. to stations and , respectively; station wants to transmit “ d ” and “ e ” to stations and , respectively.Pairwise data exchange as in [2] and [3] is not effective in this case because when the number of stationsis odd, one station will always be left out when forming pairs. That is, when the number of stations isodd, the connectivity pattern realized by a switch/permutation matrix does not correspond to pairwisecommunication. Full data exchange is not appropriate either, since in our example, station (as well asstation ) transmits different data to the other two stations. Under our framework, the traffic flows amongstations can be met as shown in Fig. 2. In the first slot, station transmits “ a ” to station ; station transmits “ b ” to station ; station transmits “ e ” to station . In the second slot, station transmits “ a ”to station ; station transmits “ c ” to station ; station transmits “ d ” to station . In Section III.C, wewill present the details on how to realize the switch matrices. To limit the scope, this paper focuses on theuse of amplify-and-forward relaying and zero forcing (ZF) to establish the permutations among stations.The rest of the paper is organized as follows: Section II describes the framework of wireless MIMOswitching and introduces the ZF relaying method for establishing permutations among stations. The fairswitching framework is presented in Section III. Section IV discusses our simulation results. Section Vconcludes this paper. II. S YSTEM D ESCRIPTION
A. System Model
Consider N stations, S , · · · , S N , each with one antenna, as shown in Fig. 3. The stations communicatevia a relay R with N antennas and there is no direct link between any two stations. Each time slot isdivided into two subslots. The first subslot is for uplink transmissions from the stations to the relay; thesecond subslot is for downlink transmissions from the relay to the stations. We assume the two subslotsare of equal duration. Each time slot realizes a switching permutation, as described below.Consider one time slot. Let x = { x , · · · , x N } T be the vector representing the signals transmitted by thestations. We assume all stations use the same transmit power, normalized to one. Thus, E { x i } = 1 , ∀ i .We also assume that E { x i } = 0 , ∀ i , and that there is no cooperative coding among the stations sothat E { x i x j } = 0 , ∀ i = j . Let y = { y , · · · , y N } T be the received signals at the relay, and u = { u , · · · , u N } T be the noise vector with i.i.d. noise samples following the complex Gaussian distribution,i.e., u n ∼ N c (0 , σ r ) . Then y = H u x + u , (1) where H u is the uplink channel gain matrix. The relay multiplies y by a beamforming matrix G beforerelaying the signals. We impose a power constraint on the signals transmitted by the relay so that E {k Gy k } ≤ p. (2)Combining (1) and (2), we have Tr [ H Hu G H GH u ] + Tr [ G H G ] σ r ≤ p. (3)Let H d be the downlink channel matrix. Then, the received signals at the stations in vector form are r = H d Gy + w = H d GH u x + H d Gu + w , (4)where w is the noise vector at the receiver, with the i.i.d. noise samples following the complex Gaussiandistribution, i.e., w n ∼ N c (0 , σ ) . B. MIMO Switching
Suppose that the purposes of G are to realize a particular permutation represented by the permutationmatrix P , and to amplify the signals coming from the stations. That is, H d GH u = AP , (5)where A = diag { a , · · · , a N } is an “amplification” diagonal matrix. Define ˆ r = A − r , i.e., station S j divides its received signal by a j . We can rewrite (4) as ˆ r = (cid:2) x i , · · · , x i j , · · · , x i N (cid:3) T + P H − u u + A − w , (6)where S i j is the station transmitting to S j under the permutation P (i.e., in row j of P , element i j isone, and all other elements are zero). Suppose that we require the received signal-to-noise ratio (SNR) ofeach station to be the same. Let h ( − u, ( i,j ) be element ( i, j ) in H − u . Then σ r X k | h ( − u, ( i j ,k ) | + σ | a j | = σ e , ∀ j. (7)Note that σ e is the effective noise power for each station under unit signal power. Substituting (5) into (3), we have q , X i,j | h ( − d, ( i,j ) | | a j | + σ r X i,k | X j h ( − d, ( i,j ) a j h ( − u, ( i j ,k ) | ≤ p, (8)where h ( − d, ( i,j ) is element ( i, j ) in H − d . Let a j = | a j | e iθ j , then combining (7) and (8) gives q = X i,j | h ( − d, ( i,j ) | σ σ e − σ r P k | h ( − u, ( i j ,k ) | + σ r X i,k | X j h ( − d, ( i,j ) h ( − u, ( i j ,k ) σe iθ j r σ e − σ r P k | h ( − u, ( i j ,k ) | | ≤ p, (9) Problem Definition 1 : Given H u , H d , p, σ , σ r , and a desired permutation P , solve for minimum σ e and the corresponding G . Random-phase Algorithm : For a given set of θ j , j = 1 , · · · , N , according to (9), lim σ e → + ∞ q = 0 , and lim σ e → max i,j,u { σ r P k | h ( − u, ( ij,k ) | } + q = + ∞ . Furthermore, q is a continuous function of σ e . Thus, there exists a σ e such that q = p . Denote such a σ e by σ e ( θ , · · · , θ N ) . The problem consists of finding σ ∗ e = arg min θ , ··· ,θ N σ e ( θ , · · · , θ N ) . (10)We note that σ e is a complicated nonlinear function of θ j . A time-consuming exhaustive search can beused to find the solution to (10). We use a random-phase algorithm to reduce the complexity. We dividethe interval of [0 , π ) equally into M bins with the values of , πM , · · · , M − πM respectively and randomlypick among them to set the the value of θ j for each and every j = 1 , · · · , N . After that, we compute thecorresponding σ e ( θ , · · · , θ N ) by solving (9) with the inequality set to equality. We perform L trials ofthese random phase assignments to obtain L different values of σ e . We choose the smallest among themas our estimate for σ ∗ e . Substituting the estimated σ ∗ e into (7) yields | a j | for all j ; hence G . Note that thesolution found is a feasible solution and is in general larger than the actual optimal σ ∗ e . In Section IV,we will show that large gains can be achieved with only small M and L . Moreover, increasing M and L further yields very little improvement, suggesting that the estimated σ ∗ e with small M and L is nearoptimal. III. F AIR S WITCHING
As has been described in the previous section, in each time slot, the stations transmit to one anotheraccording to a switch matrix. In this section, we study the fair switching scenario in which each station has an equal amount of traffic for every other station. The data from station i to station j could be differentfor different j , so this is not restricted to the multicast or broadcast setting. To achieve fair switching,multiple transmissions using a succession of different switch matrices over different time slots will beneeded. We next discuss the set of switch matrices. A. Derangement
We assume a station does not transmit traffic to itself. A derangement is a permutation in which i isnot mapped to itself [6]. While the number of distinct permutations with N stations is N ! , the numberof derangements is given by the recursive formula d N = N · d N − + ( − N , (11)where d = 0 . For example, d = 9 although the number of permutations is
4! = 24 . It can be shownthat lim N →∞ d N N ! = e − and the limit is approached quite quickly. Thus, the number of derangements is ingeneral very large for large N . Performing optimization over this large combinatorial set of derangementsin our problem is a formidable task. For example, in our fair switching problem, we want to maximizethe system throughput by scheduling over a subset of derangements. It would be nice if for our problem,the optimal solution is not very sensitive to the particular selection of derangements. In Part B, we willformalize the concept of “condensed derangement sets”. B. Condensed Derangement Set
Definition 1 : A set of N − derangements, D , D , · · · , D N − , is said to be a condensed derangementset if N − X n =1 D n = J − I , (12)where J is a matrix with all “1” elements, and I is the identity matrix.The four condensed derangement sets for N = 4 are Q = { P , P , P } , Q = { P , P , P } , Q = { P , P , P } , and Q = { P , P , P } , where P n are listed in TABLE I. There are d = 44 derangements for N = 5 and the number of condensed derangement sets is .In fair switching, we want to switch an equal amount of traffic from any station i to any station j , i = j . This can be achieved by scheduling the derangements in the condensed derangement set in aweighted round-robin manner (as detailed in “Approach to Problem 2” below). Given a condensed set, the scheduling to achieve fair switching is rather simple. However, different condensed sets could potentiallyyield solutions of different performance. And the number of condensed derangement sets is huge for large N . We define a problem as follows. Problem Definition 2 : Suppose that we want to send equal amounts of traffic from S i to S j ∀ i = j .Which condensed derangement sets should be used to schedule transmissions? Does it matter? Approach to Problem 2 : The derangements in a condensed derangement set are the building blocks forscheduling. For example, in a complete round of transmissions, we may schedule derangement D n for k n time slots. Then the length of the complete round transmissions will be P N − n =1 k n .Consider the case of N = 4 . There are four condensed derangement sets. The question is whichcondensed derangement set will result in the highest throughput. We could approach the problem asfollows.Let Q m = { D m , D m , · · · , D mN − } be a particular condensed derangement set. For each D mn , we userandom-phase algorithm above to compute the corresponding σ e , denoted by σ e,n,m . The Shannon rate isthen r n,m = log(1 + 1 σ e,n,m ) . (13)Because of the uniform traffic assumption, we require k n,m r n,m = c , ∀ n ∈ [1 , · · · , N − , for some c . Thatis, c is the amount of traffic delivered from one station to another station in one round of transmissions.The effective throughput per station (i.e., the amount of traffic from a station to all other stations) is T m = ( N − c P N − n =1 k n,m = N − P N − n =1 /r n,m . (14)Numerically, we could first solve for r n,m ∀ n . Then, we apply (14) to find the throughput.The question we want to answer is whether T m for different Q m are significantly different. For thecase of N = 4 and , we will show simulation results indicating that the throughputs of different Q m arerather close, and therefore it does not matter which Q m we use. C. Generalization
As mentioned in the introduction, most prior works for multi-way relay networks focus on two patternsof transmissions. The first is pairwise unicast, in which stations form pairs, and the two stations of a paironly communicate with each other [2], [3]. The second is the full data exchange, in which each station needs to broadcast to all the other stations [4], [5]. In practice, however, the actual transmission patternscould be different from these two patterns. For example, for video conferencing, a subset of stationswithin the network forms a multicast group, and the transmission pattern is somewhere between the twoextremes above.More generally, in the same network, there could be the co-existence of broadcast sessions, multicastsessions, pairwise unicast sessions, and unidirectional unicast sessions. The MIMO switching frameworkhere is flexible and encompasses this generality. For easy explanation, our previous discussion in Part Bhas an implicit assumption (focus) that each station i wants to send different data to different stations j = i . If we examine the scheme carefully, this assumption is not necessary. In the scheme, a station willhave chances to transmit to all other stations. In particular, a station i will have chances to transmit datato two different stations j and k in two different derangements. If so desired, station i could transmit thesame data to stations j and k in the two derangements. This observation implies that the general trafficpattern can be realized.For illustration, let us examine how the traffic pattern of Fig. 1 can be realized. This example is a patternconsisting of the co-existence of unicast and broadcast. As has been described, the data transmission canbe realized by scheduling a condensed derangement set, which is D = [ e , e , e ] and D = [ e , e , e ] ,and e n contains in the n th position and zeros elsewhere. The transmitted data of station 1, 2 and 3 arerespectively [ a, b, e ] T for D and [ a, c, d ] T for D .IV. S IMULATION
In this section, we evaluate the throughputs achieved by different condensed sets. We assume that theuplink channel H u and downlink channel H d are reciprocal, i.e., H d = H Tu , and they both follow thecomplex Gaussian distribution N c ( , I ) . We assume the relay has the same transmit power as all thestations, i.e., p = 1 .We will answer the question raised in Problem Definition 2. We analyze the scenarios where N = 4 and N = 5 . The four different condensed derangement sets of N = 4 , Q , Q , Q and Q , are consideredfor fair switching. For each channel realization, we evaluate the throughput per station T m as defined by(14). We simulated a total of 10000 channel realizations and computed E { T m } averaged over the channelrealizations. Recall that for random-phase algorithm, there are two associated parameters: number of trials L and number of bins M (see Section II). We find that for a fixed L and a fixed M , the four condensedderangement sets yield essentially the same average throughput (within 1% in the medium and high SNR regimes and within 2% in the low SNR regime). Fig. 4 plots the throughput for one of the condensedderangement set for different L and M . For N = 5 there are different condensed derangement sets.As with the N = 4 case, all the sets have roughly the same average throughput (within 1%). Fig. 5 alsoplots the results of one set. We also note that increasing L and M beyond and respectively yieldslittle throughput gain. This implies that our heuristic yields near optimal result when L = 10 and M = 8 .We conjecture that different condensed derangement sets achieve roughly the same average throughputfor N larger than as well. A concrete proof remains an open problem. The ramification of this result,if valid, is as follows. For large N , the number of condensed derangement set is huge, and choosing theoptimal set is a complex combinatorial problem. However, if their relative performances do not differmuch, choosing any one of them in our engineering design will do, significantly simplifying the problem.A scheme proposed in [7] investigates a similar problem as ours. It simply uses a positive scalar weightto control the relay power consumption instead of our diagonal A . As a comparison, we also plot thethroughputs of the scalar scheme in [7]. Our scheme with diagonal A outperforms the scalar schemeby dB and . dB in Fig. 4 and Fig. 5. Besides the advantage in throughput, our scheme has anotheradvantage over the scalar scheme in that our scheme guarantees fairness. That is, in our basic scheme,each station has exactly the same throughput, while the stations in the scheme in [7] could have varyingthroughputs. The scalar scheme in [7] focuses on optimizing the sum rate of all stations; the individualrates of the stations may vary widely with only one degree of freedom given by the scalar.To sum up this section, we state the following general result: General Result : In our framework of MIMO fair switching with or stations, any condensedderangement set can be used because different condensed derangement sets achieve roughly the sameaverage throughput. We conjecture that this will be the case when the number of stations is large as well.If this conjecture holds, then the issue of condensed set selection will go away, and the complexity of theoptimization problem will be greatly reduced. This conjecture remains to be proven.V. C ONCLUSIONS
We have proposed a framework for wireless MIMO switching to facilitate communications amongmultiple wireless stations. A salient feature of our MIMO switching framework is that it can cater togeneral traffic patterns consisting of a mixture of unicast traffic, multicast traffic, and broadcast trafficflows among the stations. There are many nuances and implementation variations arising out of our MIMO switching framework.In this paper, we have only studied the “fair switching” setting in which each station wants to send equalamounts of traffic to all other stations. In this setting, we aim to deliver the same amount of data fromeach station i to each station j = i by scheduling a set of switch matrices. In general, many sets ofswitch matrices could be used for such scheduling. The problem of finding the set that yields optimalthroughput is a very challenging problem combinatorially. Fortunately, for number of stations N = 4 or ,our simulation results indicate that different sets of switch matrices achieve roughly the same throughput,essentially rendering the selection of the optimal set a non-issue. We conjecture this will be the casefor larger N as well. If this conjecture holds, then the complexity of the optimization problem can bedecreased significantly as far as engineering design is concerned.There are many future directions going forward. For example, the beamforming matrices used in oursimulation studies could be further optimized. Physical-layer network coding could be considered toimprove throughput performance [8]. In addition, the setting in which there are unequal amounts of trafficbetween stations will be interesting to explore. Also, this paper has only considered switch matrices thatrealize full permutations in which all stations participate in transmission and reception in each slot; itwill be interesting to explore switch matrices that realize connectivities among stations that are not fullpermutations. Finally, future work could also explore the case where the number of antenna at the relayis not exactly N . R EFERENCES [1] T. Cover and A. Gamal, “Capacity theorems for the relay channel,”
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ACM Mobicom , 2006. TABLE ID
ERANGEMENTS OF N = 4 . P = P = P = , , , P = P = P = , , , P = P = P = , , . aa bc de Fig. 1. Traffic demand of a three stations example.
123 312MIMOswitch MIMOswitch123 231 a ab be e a ac cd d
Slot 1 Slot 2
Fig. 2. A transmission established by two slots of unicast connectivity realizes the traffic demand in Fig. 1.
Relay 123 NN Subslot 1 Subslot 2
Fig. 3. Wireless MIMO switching. A v e r age T h r oughpu t P e r U s e r P e r S t a t i on scalr adiag A 10 trials & 8 binsdiag A 50 trials & 8 binsdiag A 50 trials & 16 bins Fig. 4. Average throughput per station under MIMO fair switching when N = 4 . In each case only the result of one condensed derangementset is presented because the results of other derangement sets are within 2% of the results shown here. A v e r age T h r oughpu t P e r U s e r P e r S t a t i on scalr adiag A 10 trials & 8 binsdiag A 50 trials & 8 binsdiag A 50 trials & 16 bins Fig. 5. Average throughput per station under MIMO fair switching when N = 5= 5