MIMO Capacity of Wireless Mesh Networks

Transcription

1 MIMO Capacity of Wirele Meh Network 3 Sebatian Max, Bernhard Walke Communication Network (ComNet) Reearch Group Faculty 6, RWTH Aachen Univerity, 74 Aachen, Germany {mx walke}@comnet.rwth-aachen.de Abtract A Wirele Meh Network (WMN) erve to extend the wirele coverage of an Internet gateway by mean of Meh Station (MSTA) that tranparently forward data between Station (STA) and the gateway. Thi concept reduce deployment cot by exchanging the multiple gateway, required to cover a larger area with wirele Internet acce, by a wirele backbone. Unfortunately, thi alo reduce capacity, owing to multiple tranmiion of the ame data packet on it multi-hop route. Hence, different mechanim to increae the capacity of WMN are invetigated. Multiple Input/Multiple Output (MIMO) i a technique that i able to increae the capacity of a ingle link in the ame bandwidth and tranmiion power: Both the tranmitter and the receiver i configured with multiple antenna. If multiple tream are tranmitted in a rich cattering environment, thee tream can be eparated and decoded by the receiver uccefully. However, it i unclear how thi ingle-link capacity increae tranlate into a ytem capacity increae in a WMN. In thi paper, we will combine a realitic MIMO model with a capacity calculation framework to how the combined effect of the two technologie. The reult how that although not the full link capacity increae of MIMO can be exploited, epecially WMN benefit from the MIMO gain. Keyword Wirele Network, Capacity, Multiple- Input/Multiple-Output (MIMO), Meh I. INTRODUCTION In the lat year, two parallel reearch area have produced fundamental innovation for wirele data network: Firt, the exploitation of multipath propagation by multiple tranmit and receive antenna to increae the link capacity uing the ame bandwidth. Second, the upcoming of Mobile Ad-Hoc Network (MANET) where data i forwarded by intermediate node on dynamic, elf-configured path to extend the range of a ingle wirele link. In both area the innovation have uccefully found their way into tandard and product: Multi-antenna technology, alo known a Multiple Input/Multiple Output (MIMO), i for example a crucial part of the latet amendment of IEEE 82.11, n [2], to reach the maximum gro throughput of 6 Mb / (uing, among other technique, 4 tranmit and receive antenna). And while MANET are not deployed themelve, the reult from the reearch of wirele path election protocol are now tandardied and implemented in Wirele Meh Network (WMN), e. g., in [3]: In contrat to MANET, data forwarding i retricted to pecial Meh Station (MSTA), implementing the meh facility. Thi facility enable forwarding of frame between MSTA o that for example the limited radio coverage of a Internetconnected MSTA (named meh gate according to [3]) i extended without new wire. Throughout the paper, it i aumed that MSTA alo provide the Acce Point (AP) facility for aociation and management of mobile Station (STA). From their viewpoint, the coverage extenion via relaying i completely tranparent. The two reearch area are parallel two each other becaue they are applied to different layer of the OSI/ISO protocol tack: MIMO i a technology applied motly by the Phyical Layer (PHY), plu ome intelligence in the Medium Acce Control (MAC) required for the advanced Rate Adaptation (RA) that now incorporate, next to the election of the Modulation- and Coding Scheme (MCS), the number of tream to be tranmitted. In contrat, the ability to forward data tranparently for the application over multiple hop i at the heart of the Network Layer (NL), with probably ome improvement in the MAC to meaure the wirele link quality or to chedule multi-hop tranmiion. Hence, due to the characteritic of the OSI/ISO protocol tack model, it i traightforward to combine the advance of the PHY to thoe of the NL; a a matter of fact, the two technologie hould be tranparent to each other. While thi tatement i true from a qualitative perpective, it quantitative implication are unknown: In theory, MIMO provide a link capacity improvement which cale linearly with the number of tranmit/receive antenna. Of coure, thi capacity increae would be advantageou epecially for the wirele backbone between the MSTA where the aggregated data of the mobile STA i tranported. However, it i not clear how much of the link capacity increae of MIMO remain to the ytem capacity of the WMN. The cope of thi paper i to etimate the improvement of MIMO in a WMN. It i tructured a follow: After reviewing the related work on capacity etimation of wirele network in Section II, Section III detail the applied ytem model. At firt, thi ytem model i

2 4 applied to a ingle Baic Service Set (BSS), coniting of multiple STA and a ingle AP, in Section IV to etimate the upper bound BSS capacity for a WMN. Then, Section V introduce the capacity calculation method for WMN. Thi calculation method i applied in Section VI to evaluate the effect of different MIMO configuration. Finally, the paper conclude with Section VII. II. RELATED WORK Capacity calculation of wirele communication network i a popular reearch topic. Two major trend have evolved: 1) To determine the capacity bound of a random network with certain propertie, where the capacity i conidered to be a random variable and aymptotic propertie are calculated, and 2) To compute the capacity of a given, arbitrary network uing graph-theory baed algorithm. Beide thee trend, work ha been publihed that conider a given network for calculating it capacity from the hortet poible chedule by mean of Linear Programming. 1) Analytical Bound: In their eminal paper Gupta and Kumar [4] explored the limitation of multi-hop radio network with random ource-detination traffic relationhip by computing the achievable throughput for a random network obtained under optimal condition to be Θ (W/ n), where n i the number of node and W the radio bandwidth. Gupta and Kumar conclude that effort hould be targeted to mall network, where node communicate with near neighbour only. Several reearcher have conidered to extend thi baic model, e. g., by incorporating different network tructure [] or mobility of node [6]. Due to the ame approach choen, thee paper have in common that they derive aymptotic caling law to decribe the capacity bound for the conidered random network. Application of thee reult to any real WMN intance with a given topology appear not to be poible. 2) Graph-baed capacity calculation: The mentioned diadvantage i avoided when concentrating on the calculation of capacity bound in a given network intance. In [7] and [8] thi i done by tranlating the propertie of the wirele medium (hadowing, interference, receive probability) into two graph: The connectivity graph G = (V G, E G ) and the conflict graph C = (V C = E G, E C ). While in G each vertex repreent a node and an edge repreent a link between two node, C repreent link that cannot tranmit imultaneouly. The capacity of the network i computed then uing method from graph-theory. Since computation of the capacity bound of a given wirele network i NP-complete even under imple aumption [9], approximation algorithm mut be ued. For real-world wirele network where link adaptation i tate of the art, the problem i even aggravated: A node may chooe among alternate MCS to be ued for a tranmiion. If a high-rate, but interference uceptible MCS i choen, the link hould be operated under low interference only, in contrat to a more robut, low-rate MCS that may function well under high interference. Hence, it i impoible to generate the conflict graph C without aigning to each link a ingle MCS in advance. Therefore, mot paper on capacity calculation retrict the link model to one MCS only, thereby ignoring an important characteritic of current wirele tandard. 3) Linear Programming-baed capacity calculation: Algorithm publihed for calculating the ytem capacity of a given network taking different MCS into account all ue a model imilar to the one introduced in [1]: Alternate aignment of MCS are compared by computing the et of achievable data rate combination between all ource-detination pair in the network. Baic Rate are introduced a a key element, decribing a et of link active at a given time. The challenge i to find the chedule of all feaible baic rate that minimie the chedule duration. A capacity bound can be derived from the duration of the hortet chedule and the amount of carried traffic. Conitent with [9], the number of baic rate and thu the algorithm runtime complexity grow exponentially, rendering it uele for network with more than 3 node. Reference [11] prove the computation of the hortet chedule to be NP-complete and extend the work of [1] by a column-baed approach to olve the Mixed- Integer Programming (MIP)-formulation of the optimiation problem. Although thi method make ue of modern branch-and-price method to olve the MIP, large network with 4 and more node cannot be olved exactly. Intead it i propoed to top the branching proce uing a heuritic. 4) Previou Publication by the Author: The concept of linear programming-baed capacity calculation i picked up by the author in [12], where the heuritic Selective Growth (SG) and Early Cut (EC) to control the number of network tate are introduced and applied to WMN firt. A more detailed analyi and the additional heuritic Selective Growth/Delete (SG/Del) i provided by [13]. Thee extenion to the linear programming method are crucial to compute the capacity of large-cale cenario with 1 and more node. The improved calculation method have been applied by the author to calculate the capacity of WMN under different condition: [14] conider hybrid wirele/wired meh network, [] ue Ultra Wideband

3 (UWB) a tranmiion technology and [1] how the effect of tranmit power control on the capacity. Throughout the publication, the capacity calculation method ha proven to be a veratile tool to etimate the effect of a PHY technology to the ytem capacity. III. SYSTEM MODEL The ytem model i epecially concerned with the characteritic of a wirele network, i. e., the the wirele channel and the performance capability of the PHY to tranmit information uing the wirele channel. In compliance with the topic of the paper. According to the topic of the paper, pecial treadment i given to model MIMO tranmiion. A. Wirele Channel Model The wirele channel determine the received ignal trength of a tranmiion from node N i to N j, poitioned at p i and p j, repectively. Typically, a wirele channel model i of the form where P (N i, N j ) [dbm] = P i + g i + g j (1) pl(p i, p j ) (p i, p j ) (2) P i i the tranmiion power of node N i ; g i and g j are optional tranmit and receive antenna gain; pl(p i, p j ) i the pathlo function that model the attenuation of the radio wave due to the ditance between N i and N j ; (p i, p j ) i a hadowing fading component having log-normal ditribution. The ytem performance highly depend on the characteritic of thi model, i. e., the parameteriation of pl(p i, p j ) and (p i, p j ). Therefore, it i crucial that the election i baed on extenive real-world meaurement campaign. Furthermore, the uage of a tandardied model allow direct comparion with reult from the literature that ue the ame aumption. Baed on thee conideration and the typical application cenario of a WMN, we elect the Urban Micro (UMi) channel model decribed in [16], which i deigned to evaluate radio interface technologie in the IMT-Advanced proce. Thi model provide pathlo function for Line Of Sight (LOS), Non Line Of Sight (NLOS) and Outdoor-to-Indoor (OtoI) link a well a a decription of a random proce with correlated lognormal ditribution for the hadowing fading. B. Phyical Layer Model The Phyical Layer (PHY) model decide under which condition a packet tranmiion i ucceful, i. e., the packet i decoded error-free at the receiver. In our model, the ucce probability depend on two factor: 1) How much noie from the background and other active tranmiion interfere with the ignal and 2) which MCS i elected at the tranmitter. Throughout the paper we will aume IEEE 82.11n- 9 [2], a the phyical layer tandard. Adaptation of the methodology to other wirele tranmiion technologie baed on Orthogonal Frequency Diviion Multiplexing (OFDM) i poible by adapting the calculation to different MCS; however, thi i not in the cope of thi paper. A in [2], the MCS doe not only comprie the modulation and the channel coding, but in addition the number of patial tream n. Hence, the MCS comprie all information that determine the number of data bit per OFDM ymbol. 1) SINR: Signal degradation at the receiver i caued by two factor: Firt, the thermal- and receiver noie; econd, interference from other active tranmiion. The power of thermal noie (dbm) i given by N th = log 1 ( f ), where f i the bandwidth in Hz ; the receiver noie N rx i an additional degradation caued by component in the RF ignal chain and aumed to be db. Let now denote N j the current receiver, trying to decode a ignal from N j, tarting at time t and ending at t 1. Furthermore, let I = {k : k i, k j} be the et of active tranmitter of all other overlapping tranmiion, having tart time t k, and end time t k,1. Then the interference I i,j,i at N j for the tranmiion from N i i computed in mw a I i j,i = min (t 1, t k,1) max (t, t k,) P (N k, N j ). t 1 t k I i j (3) Thi calculation average each interfering ignal over the tranmiion time; hence, the effect of trong but hort interference peak are underetimated. Thi implification i tolerable when uing the capacity calculation method decribed below, a interference will be aligned optimally. The final quality of the received ignal i meaured by the Signal to plu Noie Ratio (SINR): SINR i j = P (N i, N j ) [mw] I i j,i [mw] + (N th + N rx ) [mw]. (4) 2) Packet Error Rate: The reulting Packet Error Rate (PER) of a received frame, which correpond to the probability of a data burt with faulty Cyclic Redundancy Check (CRC), depend on three parameter: the MCS which i elected by the tranmitter, the frame ize, and the SINR meaured at the receiver.

4 6 Baed on the SINR and the modulation cheme, the pre-decoder Bit Error Rate (BER) can be derived analytically for the modulation cheme defined in IEEE a hown in [17]. IEEE pecifie two channel coding cheme, namely Binary Convolutional Code (BCC) and Low- Denity Parity-check Code (LDPC). In thi paper, we retrict ourelve to the BCC cheme. Hence, the reult from [18] can be applied, allowing for etimating an upper bound for the PER dependent on the SINR and the packet length. 3) Link Throughput: For an error-free link, the link gro throughput uing MCS m i given by the number of data bit per ymbol, n m DBP S, divided by the duration of one ymbol: C. Modelling MIMO Link T m = n m DBP S/t ymbol () Coare claification ditinguihe two type of MIMO technique (both part of IEEE 81.11n-9) baed on the propagation channel propertie, i. e., on the tructure of the patial correlation matrix at the antenna array. In the cae of high correlation of the tranmitted ignal beamforming can be applied, wherea in the cae of low correlation diverity and multiplexing approache apply [19]. The focu of thi work i MIMO method in the later ene, namely Spatial Multiplexing (MUX) and Spatial Diverity (DIV) cheme. In MUX cheme, n > 1 tream are tranmitted imultaneouly, each one uing one dedicated antenna of the tranmitter. In a rich cattering environment the ignal of the combined tream take different path with none or low correlation. Hence, different ignal arrive at the multiple receive antenna which can be proceed to gain the different tream. Obviouly, the number of data tream i limited by the number of tranmit antenna, n tx. Furthermore, the receiver mut contain at leat a many receive antenna, n rx, a tream. Conequently, a MUX cheme increae the data rate at mot by min(n tx, n rx ). DIV cheme, in contrat, exploit the diverity of the multiple reception of the ame ignal: The receiver with multiple antenna ha multiple copie of the tranmitted ignal, each ditorted by a different channel function. Thu, appropriate ignal proceing algorithm can increae the SINR of the ignal by combining the different tream. In the cheme combining MUX and DIV, more than one tranmit antenna i active, but the receiver, a in DIV cheme, ha more antenna than the number of patial tream tranmitted. To decribe a link with n tx tranmit and n rx receive antenna, the common notion n tx n rx will be ued. If not mentioned otherwie, we will aume that either n = n tx or that the tranmitter deactivate n tx n antenna to tranmit n < n tx tream. A detailed introduction to the hitory, benefit and problem of MIMO ytem can be found in []. 1) Signal Model: A n tx n rx MIMO ytem i repreented by Equation 6 where it i aumed that the total tranmit power i equally divided over the n tx tranmit antenna: ES y = H + n; (6) n tx C ntx 1 i the tranmitted ignal vector whoe jth component repreent the ignal tranmitted by the jth antenna. Similarly, the received ignal and received noie are repreented by n rx 1 vector, y and n, repectively, where y j and n i repreent the ignal and noie received at the ith antenna. E S denote the average ignal energy during the tranmiion. Finally, H C nrx ntx i the matrix repreenting the n rx n tx channel between the n tx tranmit and n rx receive antenna. If n = n tx = n rx and H ha full rank, i. e., H 1 exit, the Zero-Forcing (ZF)-receiver can extract a follow: ( ) 1 ES ŝ = H y. (7) n tx Thi equation can be generalied for n tx n rx by uing the Moore-Penroe peudo-invere matrix H = H / (H H) intead of H 1, where H i the conjugate tranpoe of H. Under ideal circumtance, one may increae the data rate of the ytem by merely adding tranmit and receiver antenna. Under realitic condition, there i nonneglectable correlation between the tranmit and receive antenna: In the extreme cae, the channel H i equal to ( ), which reemble a completely correlated channel. In thi cae, the matrix i ingular and cannot be inverted by the receiver; hence, the reception fail, independently of the SINR. In practice, the MIMO channel doe not fall completely in either of the theoretical cae decribed. The antenna correlation and the matrix rank are influenced by many different parameter, a the antenna pacing, antenna height, the preence and poition of local and remote catterer, the degree of LOS and more. Uing a widely accepted channel model, the MIMO channel with correlated antenna can be decribed by the matrix product H = R rx 1/2 H R tx 1/2, (8) where H repreent the i. i. d. block fading complex Gauian channel according to [21] and R rx and R tx are the long-term table normalied receive and tranmit correlation matrice.

5 Link Type Angle Spread (rad) AP STA LOS NLOS OtoI TABLE I: Parameter for the UMi angle of arrival/departure pread Under the aumption of a uniform linear array at both the tranmitter and the receiver with identical unipolaried antenna element and the antenna pacing T and R, repectively, the correlation matrice are given by [22]: where R rxi,j = ρ ((j i) R, θ R, σ R ) (9) R txi,j = ρ ((j i) T, θ T, σ T ), (1) ρ (, θ, σ θ ) define the fading correlation between two antenna element having ditance, θ T and θ R denote the mean Angle of Departure (AoD) at the tranmit array and the mean Angle of Arrival (AoA) at the receive array, repectively, and σ T and σ R i the mean AoD pread and mean AoA pread, repectively. A Gauian angular ditribution i ued in [16], implying that θ N(, σ). With thi aumption it i hown in [23] that ρ (, θ, σ) e j2π co(θ) e 1/2(2π in(θ)σ)2. (11) Eentially, thi model reult in a correlation function which i Gauian with pread inverely proportional to the product of antenna pacing and angle pread. Conequently, large antenna pacing and/or large angle pread lead to a mall correlation and vice vera. Support of thi model i given by [24], which find by imulation that correlation reache a maximum with both antenna array inline, i. e., θ T = θ R = While the mean AoA and AoD can be derived from the receiver and tranmitter poition, repectively, the pread depend on the environment. The UMi model, ued for the pathlo and hadowing, alo define value for thee, given in Table I. The model differentiate between node cloe to the ground (STA), where many cloe catterer and therefore a large angle pread can be expected, and higherelevation node with le catterer and a maller angle. 2) Pot-proceing per-tream SINR: To integrate the impact of the MIMO channel model into the PER calculation from Section III-B2, we extend the model from [19] with the help of the reult from [2], [26] to incorporate a correlated channel. For thi, we reconider Equation 7 including the peudo-invere H : The ZF-receiver multiplie the received ignal y with the matrix ntx G ZF = H, (12) E S The error vector e of the proceed ymbol tream i given by ntx E S H n, reulting in a noie power on the k th data tream a [E(ee )] kk = n txn E S [ H H ] kk, (13) where [X] kk denote the (k, k) th element of the matrix X. Hence, the pot-proceing SINR on the k th tream i SINR pot,k = E S [E( )] kk n tx N [H H (14) ] kk = E S 1 1 N n tx [H H] 1. () k,k A viible in the equation, pot-proceing SINR on each tream i a combination of three factor: 1) The pre-proceing SINR. 2) A reduction by n tx, becaue the tranmitter ha to plit it tranmiion energy among the n tx tream. 3) A pot-proceing MIMO lo of [H H] 1 k,k. [2] prove that the pot-proceing MIMO gain on each tream follow a Chi-quared ditribution with 2 (n rx n tx + 1) degree of freedom. From thi fact, it i derived that the mean gain on each tream without tranmit- and receive correlation i 1 log 1 (n rx n tx + 1) db. Furthermore, [2] how that tranmit correlation caue a degradation in effective SINR that can be decribed by ( [R ] ) 1 K T = 1 log 1 tx (16) k,k on the kth tream. [26] calculate the impact of the receive correlation a ( ) 1/(nrx n tx+1) K R = 1 log 1 tr ntx 1(λ(R rx )) ( nrx, n tx 1) det(rrx ) (17) with tr l () the l th elementary ymmetric function defined a tr l (X) = l λ x,αi (18) {α} i=1 for a poitive-definite X C n n, where the um i over all ordered equence α = {α 1,..., α l } {1,..., n} and λ x,i denote the i th eigenvalue of X. 7

6 8 λ() the diagonal matrix containing the eigenvalue of the matrix argument. To viualie the impact of the antenna correlation on the pot-proceing SINR, the exemplary cenario in Figure 1a i ued: A receiving node i poitioned in a half-circle around a tranmitting node; the orientation of the node remain contant, i. e., with an angle α rx = to the x-axi. With different poition, the AoD and AoA varie and o do the correlation matrice. Auming a pre-proceing SINR of 3 db, antenna pacing of. wave-length and angle pread a given in Table I (LOS), the mean pot-proceing SINR i given by Figure 1b. A expected, a 1 1 configuration i independent of the receiver poition. All other configuration reult in a pot-proceing SINR decreae o that the 3 dbm i not reached any more; the upper bound i given by a ytem without correlation, i. e., K T = K R =. Correlation i high if the antenna face each other or if they are parallel. 3) MIMO Link Throughput: With the help of the preented MIMO model it i now poible to compute, for given node poition and pre-proceing SINR the pot-proceing SINR per tream and thu the per-tream BER and PER. A in Section III-B3, a relation SINR v. gro throughput can be derived if AoD and AoA are given. Figure 2 omit for more clarity the different MCS but how only the encloing hull that can be achieved with a given MIMO antenna configuration and two node that face each other with parallel antenna orientation. The graph for the 1 1 cae, tarting at db and levelling off at 2 db/6 Mb / preent the baic cae that would alo be poible uing the legacy IEEE (plu the new 64-QAM /6 MCS). Adding more antenna allow to receive the ignal at lower SINR level (uing DIV) and increaing the throughput (uing MUX), although the full throughput gain can only be reached at very high SINR, i. e., above 33 db for the 4 4 cae. IV. SINGLE BSS OPERATION In thi chapter it i aumed that only one AP exit that ue the carrier frequency f c. Thu, no interference from other AP or STA that do not belong to the AP BSS exit, the SINR i implified to the Signal to Noie Ratio (SNR). For wirele Internet acce, thi theoretical cae only exit if (a) the coverage area of the AP i a large a the ervice area and (b) the carrier frequency i licened to the provider. While thee condition repreent only a theoretical cae, performance metric of a ingle BSS are important for the ubequent multi-bss evaluation, becaue a ingle BSS i the trivial upper bound for the capacity: any deployment of multiple AP will increae the interference and/or the number of orthogonal channel and thu the ued bandwidth. The BSS capacity i given by the maximum throughput that can be achieved in a BSS under the aumption that all STA alway have data to tranmit to the AP and vice vera. The capacity depend on the link capacitie of the STA in the BSS and thu on their poition. A thi differ from cenario to cenario, the evaluation aume that STA are poitioned with a uniform random ditribution over the BSS area. Thu, the capacity of the BSS become a random variable C with Probability Denity Function (PDF) p C. We evaluate thi capacity uing a Monte-Carlo approach: In one cenario, multiple STA are dropped randomly; their capacity i calculated uing the equation for the wirele channel model, the pot-proceing SNR and the throughput from Figure 2. Thi i repeated for multiple cenario which differ by the placement of STA and the tochatic hadowing fading. Figure 3a how the Cumulative Ditribution Function (CDF) of BSS capacity that i generated uing the Monte-Carlo method for a 1x1 to 4x4 antenna configuration. The maximum ditance of a STA to the AP i choen uch that the area covered equal to the mean BSS area ued in the following multi-ap evaluation, namely.49 km 2. In the 1x1 cae, the expected capacity i 37. Mb /, with a probability of roughly 4% that the highet MCS with 6 Mb / i reached. With every antenna added, one more tream can be tranmitted under optimal SINR condition; the maximum capacity increae accordingly up to 26 Mb /. However, the probability that the required SINR can be reached decreae a more tream reult in a lower pot-proceing SINR. In the end, the probability of 26 Mb / i only 22%. Conequently, the expected capacity cale not linearly with the number of antenna, but only by a factor of 1.71, 2.26 and 2.78 for the 2x2, 3x3 and 4x4-cae, repectively. Figure 3b and 3c how the effect of the tranmit and receive antenna correlation on the capacity: If both the tranmit and receive angle pread i π, antenna correlation i minimal; the only factor reducing the pot- SINR i the tranmit power reduction to keep the total emitted power. Hence, only the minimum MIMO lo of 1 log 1 ((n rx n tx + 1) /n tx ) db i incorporated. Accordingly, a viible in Figure 3b, the probability to reach the maximum capacity increae to 4% for all antenna configuration. Thi cale the expected capacity by 1.8, 2.69 and 3.3 for the multi-antenna cae in comparion to the 1x1 cae. A further capacity increae would be poible by uing highly enitive MCS like 26-Quadrature Amplitude Modulation (QAM) (not part of IEEE 82.11n) in the high-sinr region. Figure 3c how the expected capacity if the angle

7 9 3 2 Receover (nrx antenna)... SINR [db] 1 β... Tranmitter (ntx antenna) x1 2x2 3x3 4x Angle β between tranmitter and receiver [rad] (a) Exemplary cenario with varying SINR lo due to antenna correlation. (b) Mean pot-proceing SINR per tream with varying MIMO configuration. Fig. 1: Effect of antenna correlation on the pot-proceing SINR. Link Throughput [] x1 antenna 2x2 antenna 3x3 antenna 4x4 antenna SINR [db] Fig. 2: Link throughput v. SINR; aumed PSDU length i 1 B and maximum PER.1. pread i lower than in the UMi cenario: For demontration, the angle pread from the Suburban Macro (SMa) cenario i taken (.1 rad for the AP,.27 rad for STA). A a reult, the CDF of the multi-antenna converge to the CDF of the 1x1-cae. Conequently, the expected capacity increae only by 1., 1.81 and 2.1: Introducing complex MIMO tranceiver in thi cenario doe not reult in ignificant capacity gain. V. MULTI BSS OPERATION The major difference between a ingle BSS and a WMN i the addition of interference from concurrent tranmiion. In a ingle BSS, a concurrent tranmiion can only be initiated by two STA or a AP and a STA. In the firt cae the AP receive two tranmiion at the ame time; hence, at mot one ignal can be decoded if tranmitted uing a robut MCS. In the econd cae, the AP i buy tranmitting, there i only the chance that the downlink tranmiion to the STA can be received if a robut MCS i ued by the AP. Both cae aume that the interference by the other STA i low, o that at leat one tranmiion will fail and the other require a long time due to the robut MCS. Hence, to optimie capacity, concurrent tranmiion are avoided in a ingle BSS. In a WMN thi concluion i not valid, becaue link exit that are eparated from each other uch that a concurrent tranmiion (uing more robut MCS if neceary) hould be preferred to a equential operation. Thi i already demontrated by the imple network hown in Figure 4, compriing four node and two link. It i aumed that the two link, named 1 and 2, have a maximum throughput of 3 / 22 Mb /, if no interference i preent. If both link tranmit concurrently, the maximum throughput i lowered to / Mb /. If both tranmitter have 1 Mb of data to tranmit, it would take 1/3 + 1/22.79 until the data ha reached the receiver if both link are active equentially. Uing an optimied mix of concurrent and equential tranmiion, both link tranmit concurrently firt; after link 1 ha completed the tranmiion, 1-/ Mb are left to be tranmitted at link 2, which continue at 22 Mb /. In total, the data reache the receiver after , (19) 22 thu the required duration i hortened by 22%. Hence, the trategy deciding which link are active at what time intance i important to determine the achievable capacity of the network. In thi example the calculation of the optimal ditribution between equential and concurrent tranmiion i imple becaue only two concurrent link exit. Every link that i added to

8 6 1 2 Probability throughput < x x1 MIMO (E(C) =37.6).2 2x2 MIMO (E(C) =64.8, 1.71 gain) 3x3 MIMO (E(C) =8.6, 2.26 gain) 4x4 MIMO (E(C) =14.6, 2.78 gain) Link Capacity [] 2 Fig. 4: Example network to demontrate the effect of concurrent tranmiion. the network potentially double the number of poible combination of active link, making the capacity calculation hard for any larger network. Link 2 [] 1 L (a) IMT-A UMi BSS A. Capacity Limit Probability throughput < x x1 MIMO (E(C) =37.6).2 2x2 MIMO (E(C) =69.1, 1.8 gain) 3x3 MIMO (E(C) =11.17, 2.69 gain) 4x4 MIMO (E(C) =132.71, 3.3 gain) Link Capacity [] (b) IMT-A UMi BSS without MIMO antenna correlation The ytem model conider the retriction and opportunitie a node i contrained by and able to exploit, repectively. To find the capacity of the WMN, we apply an optimal cheduler that i able to plan concurrent tranmiion optimally. Time i aumed to be divided into fixed cheduling interval of duration I. During one interval, a node i generate a load of l ij directed to node j. Thi i expreed by the traffic requirement T which define ource, detination and the load. For example, the traffic requirement T = (ource) (rate) (detination) N 2 1 Mb N 1 N 6 1 Mb N 1 N 1 1 Mb N 7 ). Probability throughput < x x1 MIMO (E(C) =37.6).2 2x2 MIMO (E(C) =8.1, 1. gain) 3x3 MIMO (E(C) =67.83, 1.81 gain) 4x4 MIMO (E(C) =7.4, 2.1 gain) Link Capacity [] (c) IMT-A UMi BSS with high MIMO antenna correlation Fig. 3: CDF of the capacity in a BSS. pecifie in three row the load from N 2 to N 1, N 6 to N 1 and N 1 to N 7, with 1 Mb each. The tak of the cheduler i to generate the equence of tranmiion uch that thi load i tranported. ( Thi equence i repreented a the chedule (S1, δ 1 ) ; (S 2, δ 2 ) ;... ; ( )) S S, δ S of network tate Si and duration δ i, with δ i 1 and S the et that contain all network tate. Each network tate repreent a poible combination of active link, given by tranmitter, receiver, rate, ource and detination of the packet flow. An example tate would be S = (ource) (tx) (rate) (rx) (detination) (N 2 ) N 2 4 Mb / N 3 ( N 1 ) (N 6 ) N 12 Mb / N 4 ( N 1 ) (N 1 ) N 1 24 Mb / N 7 ( N 7 ). Thi example pecifie in three row three imultaneou tranmiion, one from node N 2 to node N 3 at 4 Mb / with data originated at node N 2 and addreed to

9 2 2 2 International Journal on Advance in Internet Technology, vol 3 no 1 & 2, year 1, Link 1 [] Link 1 [] Link 1 [] node N 1 ; another from N to N 4 at 12 Mb / originated from node N 6 and addreed to N 1, etc. A network tate i feaible if each tranmiion contained i poible according to the ytem model. A feaible chedule mut contain feaible network tate only; furthermore, it mut fulfil the offered traffic requirement uch that if S i i active for δ i, 1 i S, the requirement from T are met. The um i=1... S δ i give the duration of the complete chedule. If thi duration i larger than the duration of the cheduling interval I, more traffic i generated than what can be tranported by the chedule. A chedule i called optimal if no other feaible chedule exit that ha a maller duration; let δi denote the correponding optimal duration for network tate S i. Then the minimum reource utiliation to carry the traffic given in T uing the network tate S i RU(S, T ) = i=1... S δ i. () A defined in [1], the capacity region C(S) of a WMN with network tate S i the et of all load etting T for which a feaible chedule exit: C(S) = {T : RU (S, T ) I}. (21) The convex hull of C(S), i. e., the et of all T where RU (S, T ) = I, decribe the capacity limit of the WMN under any poible partitioning of reource among the (ource-detination)-pair in the network. Therefore, the dimenion of the capacity region i the number of STA, n ST A, in the WMN, a each can have a different load. To reduce the number of dimenion, we compute only the uniform ytem capacity C u (S), defined a the point of the capacity where all n ST A STA have the ame load l: C u (S) = n ST A l RU (S, T ). (22) The calculation of the optimal chedule i performed C u (S) in two tep: In tep one, the et S of all feaible network tate i computed. The econd tep convert T and each network tate into a matrix uch that the optimiation problem of finding the optimal chedule become an intance of Linear Programming (LP): minimie f(δ) = uch that i=1... S i=1... S δ i δ i i = T 3 δ i 1 i = 1... S. (23) The complexity of both part of the algorithm, namely the creation of the network tate and the olving of the LP intance, depend on the number of network tate S to be conidered. A hown in [1], thi number i 3 3 (a) S 1 22 L 22 (b) S 2 L 22 L (c) S 3 Fig. : The network tate of the network in Figure 4. expected to increae exponentially with the number of node, which limit the applicability to network having le than 3 node. [13] propoe heuritic to optimie the generation of the network tate uch that the upper bound capacity i cloely approximated. Thereby, up to node become feaible. B. Example The example network in Figure 4 i ued to illutrate network tate and the reulting capacity region. The network ha two link and three network tate, depicted in Figure : link 1 active, link 2 active, or both link active. In thi network, the computation of the capacity region C(S) given in Figure 6 i imple owing to the mall number of network tate: The interection with the x- and y-axi are given by chedule where only S 1 repectively S 2 i active. The throughput of a chedule where only S 3 i active cannot be achieved by any linear combination of S 1 and S 2 ; hence, the point (/) i part of the hull. The remaining hull i a linear combination of either S 1 with S 3 if link 1 need to tranmit more than Mb (and link 2 le than Mb), or S 2 with S 3 in the oppoite cae. Any combination of S 1 and S 2 would reult in lower throughput for one of the link. The uniform ytem capacity C u (S) can be found by retricting T to the point where the load of link 1 i equal to link 2. The capacity limit under thi condition i given by T = (44/27, 44/27), reulting in C u (S) 32.6 Mb /. VI. EVALUATION For the evaluation, we apply the WMN cenario creator from [27]. It i ued to generate 1 cenario of 1 km 2 each with different hadowing condition; then, each cenario i covered with around MSTA with AP functionality o that wirele coverage and connectivity of the WMN i enured. 61

10 2 62 Link 2 [] Link 1 [] Fig. 6: Capacity region of the network in Figure 4. Expected BSS Capacity [] x1 2x2 4x2 4x4 In the econd tep, either all, 6, 4 or 2 of the MSTA equipped with meh gate function, i. e., connected to the wired backbone; thi reult in approximately, 2., or 1 MSTA per meh gate, repectively. Then, cot for every link in the WMN are calculated by the maximum tranmiion rate on the link; thi allow for creating a 22 routing matrix uing the Dijktra all-pair hortet path matrix. Traffic i generated in each cenario by 1 STA, poitioned randomly and aociated to the cloet (in term of pathlo) AP. Each STA requet downlink and uplink traffic from/to the Internet, divided a 9 to 1. Downlink traffic origin at the meh gate cloet to the STA (in term of path cot), uplink traffic i detined to thi meh gate. By combining the randomly generated offered traffic and the routing matrix, a traffic requirement for each link in the network can be calculated, compoing the load matrix T 1. Then, the capacity calculation procedure a decribed above i applied in each cenario, reulting in a cenariopecific value for C u. Dividing thi capacity by the number of MSTA reult in the mean BSS capacity of the cenario; thi value allow for comparion to the value found in the ingle BSS cae, Section IV. Finally, the cenario-pecific mean BSS capacity i averaged over the 1 different cenario, reulting in an etimation for the expected BSS capacity in a WMN. Beide thi expected BSS capacity, we calculate and plot the confidence interval for a 9% confidence level of the mean capacity etimator. Similar to Figure 3, three different etting for the MIMO model are conidered: Firt, the default UMi value a given in Table I. Then, for comparion, the antenna correlation i minimied by etting the angle pread of the MSTA and STA to π. Finally, the 1 Optionally, T can only contain the end-to-end load and the optimal route through the WMN are found automatically during the chedule minimiation. However, a STA might be aociated to multiple AP and ditribute it traffic over multiple route, then. A routing i not the cope of thi work, the route are pre-calculated a decribed. Expected BSS Capacity [] Expected BSS Capacity [] MSTA per Meh Gate (a) UMi MIMO etting. 1x1 2x2 4x2 4x MSTA Per Meh Gate (b) No MIMO antenna correlation. 2x2 4x2 4x4 1x MSTA per Meh Gate (c) High MIMO antenna correlation. Fig. 7: Expected BSS capacity of the WMN. In axb, a i for the number of antenna of MSTA, b for STA

11 63 angle pread value from the IMT-A SMa are ued to demontrate a cenario etting with high correlation and low MIMO performance. The baeline antenna configuration i a traditional Single Input/Single Output (SISO) equipment: All device MSTA and STA only have a ingle antenna. Of coure, the BSS capacity for the baeline configuration i independent of the MIMO model parameter etting, a they only impact the performance in the multiantenna cae. Neverthele, the reult from the baeline configuration are given in all figure for comparion. The other MIMO configuration aume either 2 or 4 antenna at all device; Additionally, a 4x2 cae i given where all MSTA are configured with 4 antenna, wherea the STA have 2 antenna only. Figure 7 preent the expected BSS capacity for the different MIMO configuration and model parameter etting. Clearly, the interference from the neighbouring BSS reduce the BSS capacity ignificantly below the expected capacity of a ingle BSS given in Figure 3. In cae of a network without MSTA, i. e., a traditional multi-ap deployment, the BSS capacity decreae by a factor of four. Interetingly, thi decreae i independent of the antenna configuration and the MIMO model etting. Conequently, the capacity increae of MIMO in the ingle BSS i completely tranlated into a capacity increae in the multi BSS network, i. e., the ame nonlinear increae a found in Section IV i alo preent. For example, uing the UMi MIMO model parameter, the expected capacity increae compared to the 1x1 cae by a factor of 1.6 and 2.4 for the 2x2 and 4x4 configuration, repectively. A expected, the introduction of MSTA reduce BSS capacity. The more MSTA per meh gateway are deployed, the higher the mean number of hop in the WMN, which cannot be countered completely by an increae of concurrent tranmiion. However, the theoretical link capacity gain of MIMO i better approached by the higher the number of MSTA. For the MSTA per meh gate deployment and the UMi MIMO model etting, the expected capacity increae factor are 1.9 (2x2) and 2.9 (4x4) when compared to the 1x1 cae. Without antenna correlation, thee factor become 2. and 3., repectively. The reaon for a higher MIMO gain in the WMN deployment i found in the capacity ditribution of link in the backbone of the WMN: According to the node placement algorithm from [27], good poition for MSTA are preferred, leading to a high SINR between adjacent MSTA in the WMN. Hence, a 1x1 link i improved more than what can be expected from the average improvement a calculated in Section IV. Thi improvement of few link i viible in the final reult becaue the capacity of the WMN backbone limit the capacity of the whole WMN; hence, an improvement of few, but important link lead to an improvement of the complete network capacity. VII. CONCLUSION Both WMN and MIMO are intereting reearch field on their own. In thi paper, we how that the combination of both gain valuable inight: WMN benefit ignificantly from the capacity increae of MIMO. The reult, baed on the capacity calculation framework, repreent upper bound capacitie. It i not clear what remain of thi capacity if a real MAC protocol, uing an imperfect (ditributed) cheduler, i applied: Introducing MIMO to WMN increae the chance of the rate adaptation algorithm to apply different MCS; conequently, more error can be made by the cheduler, reulting in uncoordinated concurrent tranmiion. Conequently, ditributed cheduling of MIMOenhanced WMN appear to be a promiing reearch area. ACKNOWLEDGEMENTS The work leading to thee reult ha been partly funded by the European Community Seventh Framework Program FP7/7-13 under grant agreement no alo referred a OMEGA project. REFERENCES [1] S. Max and T. 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