Bounds on the Maximum Number of Concurrent Links in MIMO Ad Hoc Networks with QoS Constraints

1 Bounds on the Maximum Number of Concurrent Links in MI MO Ad Hoc Netw orks with QoS Constraints Pengkai Zhao, and Babak Daneshrad W ireless Integrated System s Research (WISR) Group, Electrical Engineering Department, Univ ersity of California, Los Angeles, CA 90095 USA (e-mail: pengkai@ee.ucla.edu; babak@ee.ucla.edu) Abstract Multiple-In put Multiple-Outp ut (MIMO) based Medium Access Contr ol (MA C) protoco ls have received a good deal of attention as resear chers look to enhanc e o verall performance of Ad Ho c networks by le veraging multi antenna en abled nodes [1]–[5]. T o date such MAC pr otocols have been ev aluated throug h co mparative simulation based studies th at report on the num ber of concu rrent link s the proto col can supp ort. However , a bou nd on th e maxim um numb er of concurr ent links (MNCL) that a MIMO based MAC protocol should stri ve to achieve ha s hitherto been unav ailable. In this pap er we present a theoretical formulatio n for calcula ting the bou nd on the MNCL in a Mo bile Ad Hoc Network (MANET ) where the n odes have multiple antenn a capab ility , while guarantee ing a minim um Qua lity o f Service (QoS). In an attemp t to make o ur findings as practical an d realistic as possible, th e study inco rporates models for the following P HY layer and channel depend ent elements: (a) path loss and f ast fading effects, in order to acc urately mo del adjac ent link interfer ence; (b) a Min imum Mean Squ ared Err or (MMSE) based detector in the receiver which provides a balance between c ompletely n ulling of neighbor ing interferen ce and h ardware complexity . In c alculating the bou nd on the MNCL ou r work also delivers the optimal power control solution for t he n etwork as well as the optimal link selection. The results ar e readily app licable to M IMO systems using Rece i ve Div ersity , Space T ime Block Codin g (STBC), and T ransmit Beamform ing and show tha t with a 4 element an tenna system, as much as 3 × improvement in the total number of concur rent links c an b e ach iev ed relative to a SISO ba sed network. T he re sults also show dim inishing improvemen t as the numbe r of antennas is increased beyond 4 , and the max imum allow able transmit power is increased beyond 10 dBm (f or the simulated parameters). June 10, 2018 DRAFT 2 Index T erms MIMO, MANET , MAC , MMSE, Concur rent Links, Receive Di versity , STBC, Beam formin g. I . I N T RO D U C T I O N W ireless Ad Hoc net working has emer g ed as an i mportant aspect of next generation communi- cation systems. For con ventional Single-Input Singl e-Output (SISO) s ystem, i nterference among nodes drastically limit s the number of concurrent (simultaneous) links in Ad Hoc networks. Multi antenna, mul ti-input-mu lti-output (M IMO), based wireless communication s has the abilit y to spatially n ull interference and in so doi ng increase the number of concurrent links within a Mobile Ad Ho c Networks (MANET), t hus increasing overall network throu ghput. In fact so me work found in t he literature [1]–[6] look to M IMO capable MAC protocols as a means of increasing t he network efficienc y and i ts sum-throughput . The maximum numb er o f concurrent links is a metric used in the literature [1], [7], [8] to ev aluate th e capacity of a network. Examp les of MA Cs that suppo rt concurrent links in a netw ork where all nodes h a ve multi ple antennas at their disposal can be found i n [1]–[5], [7]. For con venience t hese MA C protocols wil l be referred to as concurrent-based MA Cs in this paper . The Null-Hoc [2] and SP A CE-MA C [3] proto cols look t o enable con current links by us ing the Gram-Schmidt Orthonormalizatio n, s o as to create orthogonal channels amo ng diffe rent l inks. Th e prot ocol in [4] us es adaptive interference cancellation both at the transmitter and at the receiver , as well as a simpl e power cont rol p rotocol for each lin k. M ultiple links are assumed to access the channel sequentially and work sim ultaneously . The MIMA-MA C protocol [5] uses space division mul tiple access techniques to guarantee the concurrency of diffe rent comm unicating l inks in t he network. Although these con current-based MA C p rotocols have proved to outp erform the con ventional SISO b ased M A Cs s uch as th e IEEE 802.11 DCF [9], a natural question to ask i s that how close they actually come t o the t heoretical bound (lim it) of concurrency in Ad Hoc networks. Furthermore, since MIM O systems enable a variety of approaches in ut ilizing multipl e antennas in the physical layer [10], i t is also concerned that how this bo und of con currency is affected by the choice of MIMO algorithms and associated physical layer techniques. In this work we identify the th eoretical Maxi mum Num ber of Concurrent Links (MNCL) in the network by cons idering June 10, 2018 DRAFT 3 the foll owing PHY layer and channel dependent elements: (a) path loss and fast fading effects; (b) different MIMO t ransmit/receive algori thms; (c) a Minimu m M ean Squared Error (MMSE) based d etector in the recei ver; (d) opti mal power control and opt imal li nk selection . Th e derived MNCL acts as a performance benchmark for concurrent-based MA C protocols, and is also used as a metric for comp aring di f ferent MIMO t echniques and parameters. Our study is based on the assumpt ion that each transmit/recei ve pair r equires the same Constant Bit Rate (CBR). In this way , the MNCL that can be had in a MIMO capable network is identified subject to a minim um Qu ality of Service (QoS) constraint . In our case the QoS constraint is the received Signal to Interference plus Noise Ratio (SINR) which is directly related to Bit Error Rate (BER) and Pack et Error Rate (PER). The proposed framew ork is exec uted via an iterativ e process for power allocation and a Backtracking-based search strategy for link selection. Results of differe nt MIMO t ransmit algorithm s such as Receiv e Dive rsity , Space- T ime Block Coding (STBC) and Transmit Beamformi ng o n the MNCL bound are studied. Usin g this bound we present design-rele va nt insight regarding the impact of t he nu mber of pairs, the number of antennas, and the m aximum allow able t ransmit power per pair on the MNCL. The reminder of this paper is or ganized as follows. The proposed system m odel i s described in Section II. Th e definit ion of concurrent lin ks is provided i n Section III. Optimal power allocation is in vestigated in Section IV . In Section V , we propose a backtracking-based strategy to find th e optimal link selectio n. Num erical results are shown i n Section VI, and we conclude in Section VII. I I . S Y S T E M M O D E L A. Network Mod el W e focus on a small network (or sub-network), where each node is within the transmissio n range of any other node subject to path los s and Rayleigh flat fading (Fig. 1). Assum e K simultaneous ly communi cating pairs in the network. Each pair is com posed of on e transmit n ode and one receive node, which are all randoml y d istributed in th e network. No des in different pairs are un ique and ind ependent. They are all equi pped with M recei ve antennas, sharing the sam e frequency band and requiring the same Constant Bit Rate (CBR). T he num ber of transmit antennas v aries depending on the MIMO technique being cons idered. For si mplicity , June 10, 2018 DRAFT 4 each communi cating pair is labeled as a T ransceiver P air . For the k th transceiver pair , the transmit n ode and recei ve node are named Tx Node k and Rx Node k , respectively . The propagati on b etween nodes i s characterized by path los s and Rayleigh fading [1 1]–[13]. For path loss, we use the simplified model in [13], which i s: L P ( d )( dB ) = L P ( d 0 ) + 10 α log 10 d d 0 (1) In thi s m odel, α is th e path lo ss exponent, d 0 is the reference di stance, and d is the distance between nodes obtained from no de topology information . Both α and d 0 are parameters in our study and can be set by the us er . W e use ρ k j to denote t he power l oss ratio from Tx Node j to Rx Node k , and ρ k j = 10 − L P ( d ) 10 . W e ass ume a flat fading en vironment , which is modeled by a Rayleigh distribution. H k j ( m ) denotes the Rayleigh fading channel from t he m th antenna of Tx Node j to Rx Node k . It i s an M × 1 vec tor and consists of ind ependent ident ically distributed (i.i.d.) complex Gaussian random v ariables. The background whi te noise i s a circularly complex Gauss ian vector wi th cov ariance matrix σ 2 N I M . I M is an M × M u nitary m atrix, and σ 2 N is g iv en by: σ 2 N (dBm) = η n + 10 log 10 ( W ) + F n (2) W (Hz) is the bandwi dth of the syst em, while η n (dBm/Hz) and F n (dB) are t he power spectral density of the thermal nois e and t he noi se figure o f the receiv er , respectiv ely . They are assu med to b e identical for all the nodes. B. T ransceiver Mod el Three di f ferent MIMO techniques are considered in this paper , t hey are: (a) 1 × M Receiv e Div ersity; (b) 2 × M Space-T ime Block Coding (STBC); and (c) M × M T ransmit Beamforming . In t his paper , we assume that data sym bols in the baseband have the same mod ulation type, regardless of single or multipl e antennas. For fair compariso n, t he num ber of t ransmit antennas in these MIMO t echniques are selected so that their spectral effi ciency is the same as that of a Single-Input Single-Output (SISO) system. June 10, 2018 DRAFT 5 1) 1 × M Receive Diversity: In the 1 × M scenario [14], only one transmit antenna is employed. W ithou t loss of generality , t he 1 st transmit antenna is used. Considering t he k th transceiver pair , the recei ved signal at Rx Node k is giv en b y: Y k = q P k ρ k k H k k (1) x k + R ( S D ) k + N k (3) R ( S D ) k = K X i =1 ,i 6 = k q P i ρ k i H k i (1) x i (4) In these equatio ns, Y , H , R and N are all M × 1 vectors. R denotes the inter -pair interference. x k is the transmit ted symbols for the k th pair , which has zero mean and unit variance. P k is the allocated power for x k . N k is a whi te Gaussian noise vector with cov ariance σ 2 N I M . 2) 2 × M STBC: In the 2 × M Space-T ime Block Coding (STBC ) scenario we us e an Alamouti Code [15] with two transmit antenn as. W e use x k ,m,n to denot e th e transmit ted symbol at m th antenna and n th tim e sl ot of the k th pair . Using the Alamouti Code, we hav e x k , 1 , 2 = − x ∗ k , 2 , 1 and x k , 2 , 2 = x ∗ k , 1 , 1 , (1 ≤ m ≤ 2 , 1 ≤ n ≤ 2 ) . H ere x k , 1 , 1 and x k , 2 , 1 are independent symbols with zero mean and u nit variance, and ( · ) ∗ is the complex conjugati on. P k (1) and P k (2) are allocated p ower for x k , 1 , 1 and x k , 2 , 1 , respective ly . Assu me the receiv ed vectors corresponding to these two t ime slots are Y k , 1 and Y k , 2 , as well as the noise vectors N k , 1 and N k , 2 . Define new vectors by Y k = h Y k , 1 ; Y ∗ k , 2 ; i , H k i, 1 = [ H k i (1); H k i (2) ∗ ; ] , H k i, 2 = [ H k i (2); − H k i (1) ∗ ; ] and N k = [ N k , 1 ; N ∗ k , 2 ; ] . Consequently , the received s ignal at Rx N ode k can be denoted as: Y k = q P k (1) ρ k k H k k , 1 x k , 1 , 1 + q P k (2) ρ k k H k k , 2 x k , 2 , 1 + R ( S T B C ) k + N k (5) R ( S T B C ) k = K X i =1 ,i 6 = k 2 X l =1 q P i ( l ) ρ k i H k i,l x i,l, 1 (6) 3) M × M T ransmit Beamformi ng: The Beamformi ng method out lined in [16], [17] is also employed in this paper . M transmit antennas are adopted, and an M × 1 weight vec tor u k is applied to th ese transm it antennas. Let H k k = [ H k k (1) , H k k (2) , . . . , H k k ( M )] , then according to [16], [17], u k is the eigen vector corresponding to the lar gest eig en value of the matri x H H k k H k k , where ( · ) H denotes conjugate transpos e. W e assu me that u H k u k = 1 and x k has zero m ean and unit var iance. The transmi t power for x k is P k . Then the receiv ed signal under this method is represented as: Y k = q P k ρ k k H k k u k x k + R ( B eam ) k + N k (7) R ( B eam ) k = K X i =1 ,i 6 = k q P i ρ k i H k i u i x i (8) June 10, 2018 DRAFT 6 C. MMSE S olution For all three scenarios listed in the p re vious section we w ill use the M inimum M ean Squared Error (MMSE) solu tion at the recei ver [18], [19], [21] to arri ve at the recei ve MIMO antenna weights. Here we only present the MMSE so lution for the 1 × M Recei ve Div ersity case. Howe ver , the MM SE solution can also b e derive d for the 2 × M STBC and M × M T ransmit Beamforming s cenarios by s light modification (the associated result s are gi ven in Appendix I). For the 1 × M Recei ve Diversity scenario, the estim ate of the transm itted stream x k at the recei ver is give n by (9) where the vector w k is the MMSE solutio n. ˆ x k = w H k Y k (9) The correspondi ng SINR at t he receiver is: Γ k = P k ρ k k w H k H k k (1) H H k k (1) w k w H k Φ ( S D ) ( k ) w k (10) Φ ( S D ) ( k ) = X 1 ≤ i ≤ K,i 6 = k P i ρ k i H k i (1) H H k i (1) + σ 2 N I M (11) W ith the constrain t that w H k H k k (1) = 1 , optim al l inear vector w k in the MMSE soluti on is giv en in [18], [19], [21] w k = Φ − 1 ( S D ) ( k ) H k k (1) H H k k (1)Φ − 1 ( S D ) ( k ) H k k (1) (12) The SINR wit h M MSE solution ˆ Γ k is: ˆ Γ k = P k ρ k k H H k k (1)Φ − 1 ( S D ) ( k ) H k k (1) (13) Finally , we ass ume a QoS requirement at the recei ver . Consider an SINR threshold γ T , for the k th t ranscei ver pair , they can be correctly receive d if and only if the recei ved SINR i s not lower than γ T . The threshold γ T represents the QoS requirement and is bein g used here i n the same manner as t he Physical M odel i n [20]. June 10, 2018 DRAFT 7 I I I . T H E D E FI N I T I O N O F C O N C U R R E N T L I N K S W e assum e t hat all Tx and Rx Nodes are m obile, and their l ocations are randomly dis tributed and varying wit h time. In the following d iscussion, a specific scenario means one s pecific realization of node locations in the re gion, and the channel responses between them. For a specific scenario, given the allocated power in each p air , the SINR for each pair can b e ev aluated using the results of Section II. Based o n these SINR results, we first provide the fol lowing definition. Definition 1 (Link): Consider a sp ecific scenario with K transceiver pairs. The transmit p ower for e very pair is constrained by P T . The k th pair (1 ≤ k ≤ K ) is called a li nk if and only if i t satisfies t he following conditions: For 1 × M Receive Diversity and M × M T ransmit Beamforming: ˆ Γ k ≥ γ T and P k ≤ P T (14) For Space T ime Block Coding: ˆ Γ k ( l ) ≥ γ T , 1 ≤ l ≤ 2 and 2 X l =1 P k ( l ) ≤ P T (15) Assume a sp ecific scenario with K transceiver pairs, l abeled from 1 to K and d enoted by a set U = { 1 , 2 , . . . , K } . A pair set U P ⊆ U is a feasible pai r set if there exists a power allocation for all transmitters in U P such that Definition 1 holds for all transceiver pairs in U P (i.e., the QoS constraint is satisfied at all receive rs subject to the maximum P T constraint). Then all the pairs in U P are nam ed Concurr ent Links . Let us denot e the number of pairs i n U P as: | U P | . Then the MNCL is calculated by t he following optimizatio n prob lem: max | U P | s.t. U P ⊆ U and U P is feasible (16) For notational simp licity , the result of the above optimization problem is denoted as N max ( K , M ) . Note that s e veral di f ferent feasible s ets U P may produce the same N max ( K , M ) . W e then define A verag e MNCL as t he expectation of N max ( K , M ) , avera ged over random locations and random channel responses. The A v erage MNCL will be denoted as C ( K , M ) in this paper . June 10, 2018 DRAFT 8 The optim ization in equation (16 ) can be divided into two steps: 1) to examine whether a pair vector U P is feasible; and 2) to search for the feasible pair set U P with the MNCL. In this paper , t he first step is sol ved by an optimal power allocation process presented in Section IV , while the second st ep, referred to as op timal li nk s election, is so lved in Section V . I V . O P T I M A L P OW E R A L L O C A T I O N Optimal power allocation is an important factor when deriving the bound on the MNCL. In our formulation it is us ed to decide whether a given pair set U P is feasible per Definit ion 1 . In this section, the algorithm for o ptimal power allocatio n is presented from con ventional po wer control techniques [21]–[23]. W ithout loss of generality , only the 1 × M Receiv e Dive rsity scenario is considered in t his s ection. Howe ver , the proposed algorithm can also be applied to the 2 × M STBC and M × M T ransmit Beamforming scenarios after slight modification . A. Iterative P ower Contr ol Consider K transceiv er pairs in the network using 1 × M Receive Diversity . The allocated power for each pair is stacked into a vector P , which is defined as the power vector and giv en by: P = [ P 1 , P 2 , . . . , P K ] (17) Since we o nly focus on pairs i n U P , we ha ve the following constraint: for k / ∈ U P , P k = 0 , 1 ≤ k ≤ K (18) Let w k be the M MSE solution in equ ation (12). W e define the following iteration equation: For k ∈ U P P n +1 k = γ T C ( S D ) k { w k , P n } + σ 2 N w H k w k ρ k k (19) C ( S D ) k { w k , P n } = K X i =1 ,i 6 = k P n i ρ k i G { w k , H k i (1) } (20) G { w k , H k i (1) } = w H k H k i (1) H H k i (1) w k (21) June 10, 2018 DRAFT 9 For 1 ≤ k ≤ K and k / ∈ U P P n +1 k = 0 (22) where n d enotes the n th iteration. For simplicit y , the above iteration is denoted as P n +1 = m ( P n ) (23) Here P n is t he power vector for t he n th iteration. W e define the fixed point of mappi ng as the power vector b P satisfyi ng b P = m ( b P ) . The following theorem holds for the iterative equati on (23 ), which is used t o verify the existence o f optimal p ower allocation i n this paper . Theor em 1: Given K trans cei ver pairs and a sp ecific pair set U P . If U P is feasible, then a unique fixed p oint of mappi ng, b P , exists t hat s atisfies b P = m ( b P ) and b P k ≤ P T , ∀ k ∈ U P . Furthermore, corresponding to the unique power vector b P is a uniq ue receive weigh t vector b w k giv en by the MMSE solutio n. Pr oof : Please refer to Appendix II. Note th at the con verse proposi tion of this th eorem holds obviously . That is, i f there exists b P = m ( b P ) and b P k ≤ P T , ∀ k ∈ U P , then U P is feasib le. B. Decision Criteria Theorem 1 su ggests that a feasible pair s et can be identified if and only i f a fixed poi nt of mapping for the transmit power l e vels exists and the power constraint s are met. Moreover , [21] suggests that the po wer control algorit hm in (23) is guaranteed t o con ver ge to a fixed point of mapping e ven when starti ng from an arbitrary initial power vector P 0 . This suggests a rather straight forward approach for determining the feasibility o f a given pair set U P as captured b y the following three criteria. Criterion 1: Assu me that the iteratio n process st arts from initial condit ion P 0 = 0 . In each iteration, i f the po wer in any transceiver pair excee ds the power const raint P T , then U P is not feasible. The proof of Criterion 1 is referred to Theorem 2 in [21]. Criterion 2: Assu me t hat the iteration process starts from arbit rary condition P 0 . In each iteration, if P n is feasib le, then t hese pairs can be supported sim ultaneously . June 10, 2018 DRAFT 10 Here, sayin g P n is feasible means that b y using P n , for 1 ≤ k ≤ K , we can have ˆ Γ k ≥ γ T and P n k ≤ P T . This criterion i s straightforward and it i s help ful i n reducing the iterations in the decision process. Criterion 3: After I max number of iterations, if the conditi ons in Criterion 2 hav e not yet been satisfied for pairs i n U P , then we declare that U P is n ot feasible. Criterion 3 guarantees that the decision can be made within a finite iteratio n number I max . Using this criterion, some feasible pair set s may be missed. Howev er , proper selection of I max can driv e the probability of m issing a feasible pair set to be arbit rarily clos e to zero. Using the above criteria we now outline a sequential procedure by whi ch w e can automatically determine if a s et U P is feasible. The steps are as follows: Iterative Determinat ion of F easibility (IDF) 1) Given pair set U P . Initialize n = 0 and P 0 = 0 ; 2) Iterate by P n +1 = m ( P n ) . Us ing the updated power vector , P n +1 , calculate the SINR for each pair under the MMSE s olution; 3) If U P is not feasible by Criterion 1, th en go t o step 6; else, go to step 4; 4) If U P is feasible by Criterion 2, then go t o st ep 6; els e, go t o step 5 ; 5) If U P is not feasible by Criterion 3, th en go t o step 6; else, n = n + 1 , g o to step 2. 6) The feasibil ity of U P is determined, stop. V . F I N D I N G U P W I T H M A X I M U M N U M B E R O F C O N C U R R E N T L I N K S In t he previous section, we deriv ed the method to determine if a pair set is feasible. In this section we describe how to find t he feasible pair set with the MNCL. Naturall y , a brute force search can be employed for this probl em. Howe ver , in order to reduce th e search space, we first characterize the property o f the feasible pair set . Consider a t otal of K transceiv er pairs, labeled 1 to K . Define U = { 1 , 2 , . . . , K } and U P is a giv en p air set. Theor em 2: For t wo pair sets U P , 2 ⊆ U P , 1 , if U P , 1 is feasib le, then U P , 2 is also feasible. Pr oof : Consider the 1 × M Recei ve Diversity scenario. Let ( U P , 1 − U P , 2 ) denote the pairs in U P , 1 but not in U P , 2 . If U P , 1 is feasible with the associated power vector P , then for the pai rs in U P , 2 , keep the p ower P k and li near vector w k unchanged. Meanwhile, shu t down the pairs in ( U P , 1 − U P , 2 ) . Here, shutting down means setting the corresponding t ransmit power to zero. As June 10, 2018 DRAFT 11 a result, t here are less interference in Φ ( S D ) ( k ) of equation (10), and conditi ons in Definition 1 are satisfied for the pai rs in U P , 2 . Thus U P , 2 is feasible and the above theorem ho lds. The p roof is extendable for STBC and Beamforming methods as well. Theorem 2 s hows that the backtracking formulatio n in [8], [24] can be adopted to solve the problem of finding the U P with the MNCL. W e start t his form ulation with an empty pair set. At e very le vel, each feasible subset is expanded by including one more pair , constructin g new s ubsets to be validated by the IDF iterations presented in Section IV (namely , forward search). If one subset becomes unfeasible, the algorithm backtracks by removing th e trail ing pair from the subset, and then proceeds by expanding the subs et with alt ernativ e pairs (namely , backward search). Specifica lly , we use a depth-first search strategy to ex ecute this procedure. An example with 4 pairs are illust rated in Fig. 2, where all feasible pair sets are outlined. Here the i nitial feasibl e sub sets are {{ 1 } , { 2 } , { 3 } , { 4 }} , while th e final feasibl e sub sets are {{ 1 , 2 , 3 } , { 1 , 3 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } , { 4 } } , and the MNCL is 3. Let U P ,sear ch denote the pair set candidate during the search, and | U P ,sear ch | is the num ber of elements in U P ,sear ch . M eanwhile, use dir ection = 1 to represent the forward search, and dir ection = 0 the backward s earch. The pseudo code of the proposed alg orithm is describ ed in the following. Back trac king-based Optimal Link Selection (BOLS) 1) Given K p airs (labeled from 1 to K ) in the network and M receive antennas per pair . Initialize by U P ,sear ch = { 1 } , N max ( K , M ) = 0 . 2) If | U p,sear ch | > N max ( K , M ) , t hen: a) Us e the IDF process in Section IV to examine whether U P ,sear ch is feasible. If U P ,sear ch is feasible, then set N max ( K , M ) = | U P ,sear ch | and d ir ection = 1 ; otherwise, set dir ection = 0 . 3) If | U p,sear ch | ≤ N max ( K , M ) , t hen set d ir ection = 1 . 4) Update the pai r set candidate U P ,sear ch : a) If U P ,sear ch = { K } , then all the search have been done, return N max ( K , M ) and sto p. b) If U P ,sear ch 6 = { K } , t hen U P ,sear ch = Pa irSet Gen [ U P ,sear ch , dir ection ] and go to step 2 . In step 3, we o nly examine the pair set U P ,sear ch in which | U P ,sear ch | is larger th an the current June 10, 2018 DRAFT 12 value of N max ( K , M ) . In o ther words, if | U P ,sear ch | ≤ N max ( K , M ) , instead of employing the IDF p rocedure , we set dir ection = 1 and go to step 4 directly . Next, the functio n PairSet Gen is d escribed as follows: Function PairSet Gen 1) Input U P ,sear ch and dir ection . Find k max = max k ∈ U P ,sear ch k , which represnts the pair with the maximum index in U P ,sear ch . 2) If k max < K , then: a) If dir ection = 1 , add new element in U P ,sear ch by U P ,sear ch = U P ,sear ch ∪ { k max + 1 } . Return U P ,sear ch and stop. b) If d ir ection = 0 , update the element k max in U P ,sear ch by k max = k max + 1 . Return U P ,sear ch and stop. 3) If k max = K and | U P ,sear ch | > 1 , then; a) Delet e k max from U P ,sear ch by U del P ,sear ch = U P ,sear ch −{ k max } . Find k max = max k ∈ U del P ,sear ch k , update t he element ¯ k max in U del P ,sear ch by ¯ k max = ¯ k max + 1 . b) Set U P ,sear ch = U del P ,sear ch . Return U P ,sear ch and st op. 4) If k max = K and | U P ,sear ch | = 1 , then all the search ha ve been done. Return U P ,sear ch and stop. Finally , a veraging N max ( K , M ) among random locations and random channel responses, we obtain the C ( K , M ) . V I . S I M U L A T I O N R E S U LT S The si mulation setup random ly distri butes the nodes, in accordance to a uniform di stribution, within a disk of radius 1 00 meters. Num erical results are ave raged over 100 0 Monte Carlo simulatio ns of independent realizations of node topol ogy and channel response. W e assume QPSK modulation in the baseband, and the desired SINR threshold, γ T , is 10dB. This corresponds to an uncoded BER of less than 1e-3 (see T able 6.1 in [14]). The remaining parameters are listed in T able I. A. Comparison of MIMO T echniques In this section we ev aluate the performance of the proposed algorithms, and use the MNCL as a metric to compare d iff erent M IMO met hods. The three M IMO meth ods outlin ed in Section II are June 10, 2018 DRAFT 13 compared, namely , 1 × M Receiv e Dive rsity , 2 × M STBC , and M × M Transmit Beamforming, all wi th M MSE decoding. W e assume up to 15 transceiv er pairs, and vary the number of recei ve antennas from 1 to 4. Results for the three different MIMO methods are reported in Fig. 3 and Fig. 4. W e observe that T ransmit Beamforming has the best p erformance due to the fac t that it explores Channel State Information (CSI) at t he transmi tter . On t he other hand, STBC has a worse performance than Recei ve Diversity . Note that STBC is con ventionall y desi gned t o cope with the white Gaus sian noise [15], but in this study we use MIMO sys tem to combat the colored in terference signals (equation (11)), and our resul ts indicate that Receiv e Diversity outperforms STBC in thi s case. Finally , performance of conv entional SISO s ystem corresponds to that of Receiv e Diversity with M = 1 . W e observe that compared to a SISO system, as m uch as 3 × improv ement in MNCL can be achiev ed by using M = 4 receive antennas (compare th e highli ghted v alues for K = 5 , 10 , 15 in Fig . 3 with those in Fig. 4). Next, we st udy the con vergence of the estimate for th e MNCL. W e assume a total of 12 transceiv er pairs in the network, with the number of recei ve antennas fixed 4. As previously mentioned, we create a realization by dist ributing the pairs in a 100m radi us disk and rando mly generating the channel respons es among t hem. In Fig. 5, we pl ot the av erage n umber o f MNCL versus the number of realizations. W e find that the simulation results become con ver gent after around 500 realizations. Note that we us ed 1000 independent realizations in these simulations, which is large enough to yield a precise esti mate for t he performance of the p roposed algorit hm. Finally , we compare the proposed backtracking strategy (Section V) with the brute force search. W e ha ve verified that, for 1 ≤ K ≤ 15 , the proposed backtracking method has exactly the sam e MNCL as the brute force search (the results are omi tted due to space limit). Here we focus on t he search com plexity of these two search methods , that is, th e number of times the IDF it erations were called. The av erage result ov er 1000 Monte Carlo simulatio ns i s shown in Fig. 6. Note that the associated complexity for the b rute force method is 2 K , where K is t he number of pairs in the network. Compared with the brute force search, we observe a sign ificant reduction in complexity when the backtracking scheme is used. Thi s verifies the effic acy of our proposed scheme. June 10, 2018 DRAFT 14 B. Impact of Simul ation P arameter s Using the formulatio n dev eloped in this work, we can in vestigate the impact of different parameters, such as the numb er of transcei ver pairs, t he number of recei ve antenn as, and the maximum allowa ble transm it power on the M NCLs. W e first look at the num ber of pairs i n th e network. Results in Fig. 3 and Fig. 4 have already s hown that t he MNCL increases substantially with t he number of pairs. Actually , we can use a two-stage linear equation t o approxim ate the result of the MNCL. The equation is presented in (24), and the approximation result for T ransmit Beamforming i s demonstrated in Fig. 7 and T able II. For 1 ≤ K ≤ 15 ( a 1 , b 1 , a 2 , b 2 are parameters t o be fitted): C ( K , M ) =      a 1 + b 1 K, if 1 ≤ K ≤ M a 2 + b 2 K, if M + 1 ≤ K ≤ 15 (24) In t he above equation, when 1 ≤ K ≤ M , the improvement is mainly from the diversity gain; while when M + 1 ≤ K ≤ 15 , the improvement is from mul ti-user d iv ersity (as the number of pairs wanting to com municate increases the likelihood of choosi ng a subset of these pairs t hat exhibit good interference properties increases). Th e resul ts in Fig. 3 and Fig. 4 show that the MNCL is larger t han M when th e number of pairs K is increased beyond M . This observation highlight s so me extra gains in MNCL to b e exploited by the MA C protocol. Next, we focus on the im pact o f the number of receiv e antennas. W e assume 12 transceiver pairs, and the n umber of recei ve ant ennas M va ries from 1 to 8. Simu lation results are shown in Fig . 8. W e see that in this sim ulation, th e MNCL is i mproved dramati cally wi th t he number of receiv e antennas. Ag ain, we find that wi th 4 receive antennas, the im provement is around 3 times relative to the SISO system . Furthermo re, we obs erve that saturation starts to s et in when the nu mber of receiv e antennas is greater than 5. W ith 8 receiv e antennas the gain relative to a SISO system i s 4 × . Lastly , we explore the imp act of the maximum allowa ble transmit po wer P T . Naturally , h igher transmit power is useful t o combat path loss and increase transmission range. Howev er , here we analyze the impact of P T under the assumption that all the nodes are distri buted in a fixed disk with radius 1 00m. This assumpti on is reasonabl e which corresponds to a geographically constrained area wi th m any potenti al trans mission pairs (e.g., classroo m, conference room). W e assume a total of 1 0 pairs in the network, wit h 4 recei ve antennas per pair , and consider diff erent June 10, 2018 DRAFT 15 power constraints from -20dBm to 50dBm. The results are depicted in Fig. 9. W e o bserve that for each M IMO m ethod, initially , the MNCL increases with higher power constraint . Howe ver , when P T is beyond 10dBm, the improv ement becomes diminis hing. The re ason is that co-channel interference among pairs h as domi nated th e network, and increasing P T can not mi tigate these interfering s ignals. V I I . C O N C L U S I O N In this p aper , we ha ve id entified the bound on the m aximum n umber of concurrent links (MNCL) i n MIMO Ad Hoc networks subject t o a min imum QoS (SINR in our case) for each link. This number is derived by considering the practical fac tors of wireless channel, transm it power allo cation, l ink selection and MIMO transceiv er algorith ms in a realisti c MIMO system . W e em ploy an iterative algorithm to examine the existence of an optim al power allocation that guarantees QoS for all pairs in a given set. Based o n this iterativ e algo rithm, we proposed a backtracking s earch algorithm to sel ect the optimal s ubset of pairs, const ituting the M NCL. Extensive si mulations were conducted to verify the efficac y of the propo sed algorithms and e va luate the impact from differ ent parameters. The result s show a 3 × imp rove ment in MNCL with a 4 element antenna system relative to a SISO syst em. For the parameters sim ulated, diminish ing imp rovement is observed when the number of antennas is increased beyond 5, and when the m aximum transmit power is increased beyond 10dBm. A P P E N D I X A In this appendix, we s how the MMSE sol ution for STBC and Tra nsmit Beamforming methods. For STBC, define that the MMSE sol utions for x k , 1 , 1 and x k , 2 , 1 are w k , 1 and w k , 2 , respective ly . Then we can ha ve: w k , 1 = Φ − 1 ( S T B C, 1) ( k ) H k k , 1 H H k k , 1 Φ − 1 ( S T B C, 1) ( k ) H k k , 1 (25) w k , 2 = Φ − 1 ( S T B C, 2) ( k ) H k k , 2 H H k k , 2 Φ − 1 ( S T B C, 2) ( k ) H k k , 2 (26) Φ ( S T B C, 1) ( k ) = K X i =1 ,i 6 = k 2 X l =1 P i ( l ) ρ k i H k i,l H H k i,l + P k (2) ρ k k H k k , 2 H H k k , 2 + σ 2 N I M (27) June 10, 2018 DRAFT 16 Φ ( S T B C, 2) ( k ) = K X i =1 ,i 6 = k 2 X l =1 P i ( l ) ρ k i H k i,l H H k i,l + P k (1) ρ k k H k k , 1 H H k k , 1 + σ 2 N I M (28) For T ransmit Beamformi ng, t he MM SE solution is: w k = Φ − 1 ( B F ) ( k ) H k k u k ( H k k u k ) H Φ − 1 ( B F ) ( k ) H k k u k (29) Φ ( B F ) ( k ) = K X i =1 ,i 6 = k P i ρ k i ( H k i u k )( H k i u k ) H + σ 2 N I M (30) A P P E N D I X B P R O O F O F T H E O R E M 1 Note that we ha ve assumed i n (18) that { P k = 0 , k / ∈ U P } . The it eration equation (23) can be proved to be a s tandard function [23] for { P k , k ∈ U P } via sim ilar manner in [21]. Now assume that wi th p owe r vector P , U P is feasible. Set t he initial p ower vector P 0 = P , and run the iteration process by P n +1 = m ( P n ) . According t o Lemma 1 in [23], P n will con vergence to the fixed point of m apping, which is b P = m ( b P ) . W ith Monotoni city property in standard function, for 1 ≤ k ≤ K , we have b P k ≤ P 0 k ≤ P T . Final ly , it has been proved i n [21] that power and weight vectors correspond ing to the fixed poi nt of mapping are all unique. Thus the theorem fol lows. R E F E R E N C E S [1] K. S undaresan, R. Siv akumar , M. A. Ingram, and T .-Y . Chang, “ A fair medium access control protocol for ad-hoc networks with M IMO links, ” Pr oc. IE EE IN FOCOM’04 , v ol. 4, pp. 2559– 2570, Mar . 7–11 , 2004 . [2] B. Hamdaoui and P . Ramanathan, “ A cross-layer admission control framew ork for wireless ad-hoc networks using multiple antennas, ” W ir eless Communications, IEE E Tr ansac tions on , vol. 6, no. 11, pp. 4014–402 4, Nove mber 2007. 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June 10, 2018 DRAFT 18 T ABLE I P A R A M E T E R S I N T H E S I M U L AT I O N S Parameter V a lue Path Loss d 0 = 1 m L P ( d 0 ) = 46 dB Exponent factor α = 3 Noise Power Spectral Density η n = − 174 dBm/Hz Noise F igure F n = 4 dB Bandwidth W = 1 MHz Maximum Transmit Power P T = 20 dBm SINR t hreshold γ T = 10 dB T ABLE II P A R A M E T E R S I N T H E A P P R O X I M AT I O N R E S U LT S O F A V E R AG E M N C L W I T H T R A N S M I T B E A M F O R M I N G M 1 2 3 4 a 1 0 0 . 0040 − 0 . 0003 0 . 0030 b 1 0 . 9680 0 . 9950 0 . 9990 0 . 9956 a 2 0 . 9961 1 . 9587 2 . 6045 3 . 5086 b 2 0 . 1084 0 . 2121 0 . 3013 0 . 3171 5[1RGH  7[1RGH  7[1RGH  5[1RGH  5[1RGH  7[1RGH  'HVLUHG &RPPXQLFDWLRQ ,QWHUIHUHQFH Fig. 1. Illustration of the considered network. Solid lines are the desired communication, while dashed lines are the i nterference. June 10, 2018 DRAFT 19 ^` ^` ^` ^` ^` ^` ^` ^` ^` ^` ^` Fig. 2. T ree structure for backtracking search. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 Total Number of Transceiver Pairs, K Average MNCL Receive Diversity STBC Transmit Beamforming M =1 M =3 K =5 MNCL=1.544 K =10 MNCL=2.107 K =15 MNCL=2.529 M =2 Fig. 3. A v erage MNCL under different MIMO techniques. M is from 1 to 3. June 10, 2018 DRAFT 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 Total Number of Transceiver Pairs, K Average MNCL Receive Diversity STBC Transmit Beamforming M =4 K =5 MNCL=4.868 K =15 MNCL=8.199 K =10 MNCL=6.741 Fig. 4. A v erage MNCL under different MIMO techniques. M is set as 4. 0 100 200 300 400 500 600 700 800 900 1000 4 4.5 5 5.5 6 6.5 7 7.5 8 The Number of Independent Realizations Average MNCL Transmit Beamforming Receive Diversity STBC Fig. 5. Estimate results for t he MNCL versus the number of independen t reali zations. Assume total 12 transceiv er pairs and M = 4 . June 10, 2018 DRAFT 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 10 0 10 1 10 2 10 3 10 4 10 5 Total Number of Transceiver Pairs, K Average Times of Calling IDF Procedure Receive Diversity STBC Transmit Beamforming Brute Force Fig. 6. A v erage number of times the IDF procedure was called versus the number of transceiv er pairs for M = 4 . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 Total Number of Transceiver Pairs, K Average MNCL Simulation Result, M =1 Simulation Result, M =2 Simulation Result, M =3 Simulation Result, M =4 Fit Result M =1 M =2 M =3 M =4 Fig. 7. Simulation and approximation results for average MNCL with T ransmit Beamforming. Assume 12 pairs equipped with 4 receiv e antennas. June 10, 2018 DRAFT 22 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 9 10 The Number of Receive Antennas, M Average MNCL Receive Diversity STBC Transmit Beamforming M =1 MNCL=2.3017 M =4 MNCL=7.3167 M =8 MNCL=9.63037 Fig. 8. Results under different numbers of receiv e antennas. Assume total 12 pairs. Up to 8 receiv e antennas are simulated. −20 −15 −10 −5 0 5 10 15 20 25 30 35 40 45 50 0 1 2 3 4 5 6 7 8 9 Maximum Allowable Transmit Power Per Pair (dBm) Average MNCL Receive Diversity STBC Transmit Beamforming Fig. 9. A verage MNCL versus differen t maximum allow able transmit power per pair . Assume a total of 10 pairs, M is fixed as 4. June 10, 2018 DRAFT