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Cross-layer Diversity and Scheduling Optimization for Interference-limited MIMO Ad hoc Networks Tamer ElBatt Advanced Technology Center Lockheed Martin Space Systems Company Sunnyvale, CA 94089, USA {[email protected]}

Abstract— In this paper we study the problem of reliable transmission in interference-limited MIMO networks. This is motivated by a fundamental tradeoff between scheduling full diversity non-interfering links vs. scheduling interfering links using lower diversity gain in conjunction with nulling. First, we formulate a distributed cross-layer optimization problem that jointly decides the scheduling, diversity gain and nulling in order to minimize the probability of error of individual links subject to signal-tointerference-and-noise-ratio (SINR) constraints. Second, we characterize the optimal solution for 2 links. It reveals simple decision rules that constitute the basis for solving the problem for arbitrary number of links using Scheduling Space-time coded Links (SSL) algorithm. Numerical results exhibit significant improvement over scheduling non-interfering links with full diversity gain.

I. I NTRODUCTION MIMO research has received considerable attention in the point-to-point literature [3] as opposed to multi-user settings. The problem of exploiting spatial multiplexing (SM) and diversity schemes in network settings has received attention only recently, e.g. [6], [7], [8], [9], [10], [12]. [6] focuses on a similar problem, yet, focusing on rate as opposed to communication reliability under focus in this paper. It focuses on SM and explores the gains of stream control and partial interference suppression. In [7], the authors focus on handling the non-negligible encoding and decoding delays caused by Lucent’s V-BLAST system [15]. Thus, it introduces mechanisms for reducing the MAC overhead (e.g. RTS/CTS) as well as parallel stop-and-wait ARQ scheme to remedy the per packet ACK. SPACE-MAC, proposed in [8], enables denser spatial reuse patterns with the aid of transmitter and receiver beam forming. However, it does not explore the role of diversity and its tradeoffs with nulling, in the MAC. [10] explores the role of spatial diversity schemes (e.g. spacetime coding (STC)) to combat fading and achieve robustness in RTS/CTS-based MIMO ad hoc networks. However, it does not consider the optimal MIMO stream allocation or analyze the diversity-scheduling trade-off. Layered space-time multi-user detection and its role in PHY-MAC cross-layer design are analyzed in [9]. In [12], SM with antenna subset selection for data packet transmission is proposed. Optimally allocating the MIMO streams in network settings has not received sufficient attention in the literature. This problem is motivated by a trade-off between diversity and scheduling inherent to MIMO networks. We analyze this trade-off which reveals SINR-based decision rules that constitute the foundation for developing cross-layer MIMO-MAC protocols for robust communications. This paper focuses primarily on the resource allocation aspect of the MAC problem, as opposed to

protocol aspects (e.g. handshaking, overhead, etc.). Thus, we focus on the optimization problem formulation, complexity, and theoretically-founded sub-optimal distributed algorithms with significant performance gains. Our contribution in this paper is two-fold: i) Formulate the diversity-scheduling cross-layer resource allocation problem subject to SINR constraints and characterize the optimal solution for 2 links and ii) Develop distributed decision rules for Scheduling Space-time coded Links (SSL). We present numerical results for plausible scenarios that not only capture the tradeoff at hand but also confirm considerable performance gains for arbitrary number of links. The paper is organized as follows: In section II, we review the underlying model and assumptions and formulate the crosslayer optimization problem. Next, we develop SINR-based decision rules that characterize the optimal solution for 2 links and constitute the basis for SSL in section III. In section IV, we show performance results for a number of interference scenarios. Finally, conclusions are drawn in section V. II. D IVERSITY AND S CHEDULING C ROSS - LAYER O PTIMIZATION A. MIMO Background We briefly review the distinct role of different gains of a MIMO link with M transmit and N receive antennas [1]. Array Gain: can be made available at the transmitter or the receiver and results in an increase in the average SNR due to coherently combining signals from different antennas, even in the absence of multi-path fading. Since it requires channel state information (CSI), array gain can be attained at receivers where the CSI is typically available. It makes the average SNR at the output of the receiver combiner N times greater than the average SNR at any antenna element. Diversity Gain: Diversity, at the transmitter [2] or receiver, is a powerful technique to exploit fading in wireless channels. Diversity techniques rely on transmitting the signal over multiple independently fading paths, in time, frequency or space. Diversity gain refers to the reduction in the SNR variance at the output of the combiner, relative to the variance of SNR prior to combining. At the transmitter side, the diversity gain is attained through transmitting correlated data, carefully constructed on independent signal paths created between the transmitter and the receiver. It can be achieved via either beamforming, if CSI is available, or Space-time coding, if CSI is not available. The

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maximum diversity gain, that is asymptotically achievable, is M.N if the MIMO channel is full rank and the transmitted signal is suitably constructed. Spatial Multiplexing Gain: Spatial multiplexing exploits fading to increase the link capacity for no additional power or bandwidth expenditure. The spatial multiplexing gain (SMG) is attained via transmitting independent signals on parallel spatial data pipes on the same frequency channel. The maximum SMG, that is asymptotically achievable, is min(M, N ) if the MIMO channel is full rank and a scheme that attains full SMG (e.g. VBLAST) is employed. Notice the linear increase with the number of antennas that is in contrast to the logarithmic increase in capacity if the multiple antennas capture only the array and diversity gains. Interestingly enough, a fundamental trade-off between diversity and multiplexing has been characterized in [5] for point-to-point links. Interference Reduction: when multiple antennas are used, the spatial signatures of the desired user and interferers can be exploited to reduce interference. However, this requires knowledge of the desired user’s CSI, and possibly the CSI of the interferer depending on the interference reduction scheme. If CSI is available, transmitter beamforming achieves interference reduction via minimizing the interference energy sent to neighbors other than the intended receiver. If CSI is available only at the receiver, as assumed in this paper, the receiver can null signals from neighboring interferers. Interference reduction is of particular interest in this paper due to its key role in complimenting diversity techniques to optimize reliable MIMO communications in interference-limited ad hoc networks. In this paper, we focus on the diversity gain, array gain and interference reduction and their inherent trade-offs in network settings. This is primarily attributed to our focus on minimizing the probability of error as opposed to maximizing rate.

(iid) frequency non-selective Rayleigh fading and using a STC that attains full diversity gain is,

B. Assumptions

where Peij is approximated by (2) as given below,

We focus on the interference channel with K single-hop MIMO links with stationary nodes. Each node has M transmit and N receive antennas. We assume that the channel state information (CSI) is known only at the receiver, not at the transmitter. Hence, we focus on open-loop, as opposed to closed-loop MIMO systems, due to its practical relevance. All nodes share a single frequency channel and time is divided into slots. We assume fixed power (P ) and modulation for all nodes. Thus, we focus on a single PHY knob, namely the diversity gain (DG) of space-time coding (STC). In order to support STC in conjunction with interference nulling, the receiver structure is assumed to combine both space-time decoding along with interference nulling algorithms. We assume an independent and identically distributed (iid) frequency non-selective Rayleigh fading MIMO channel. However, the results can be extended to correlated MIMO channels via periodically estimating their ranks. The path loss follows exponential decay with distance from the transmitter, with a path loss exponent α. Receiver thermal noise is modeled as additive white Gaussian noise (AWGN) with power N0 dBm. It has been shown in [4] that the pairwise error probability of a point-to-point space-time coded link with M transmit and N receive antennas, under independent and identically distributed

Pe ≤ (

ηEs −M.N ) 4N0

(1)

where η is the coding gain given by the geometric mean of nonEs zero eigenvalues of the error matrix and N is the average sym0 bol signal to noise ratio. We use the additive Gaussian approximation for interference and account only for the diversity gain as it dominates performance, especially for space-time block codes. Accordingly, the link Pe in multi-user settings can be approximated as a function of the average SINR, Pe ≈ (

SIN R −M.N ) 4

(2)

C. Objective Function In this section, we define a distributed communication reliability objective function, denoted ζi for link i. Given K links and K slots, where each link is required to transmit at least once over the K slots, we define Peij as the error probability of link i given that it transmits in slot j. The next step towards defining ζi is to extend the above per-link per-slot metric to a reliability metric for link i over the K slots. In essence, ζi is defined to capture the improvement in error probability due to the multiple transmission attempts available to each link over different slots. Hence, we assume independence of error events from slot to slot and define the objective function ζi as the probability that link i is in error given that it transmits in any of the K slots, ζi =

K Y

Peij

(3)

j=1

Peij = (

SIN Rij −Yij .Zij ) 4

(4)

and Yij is the number of transmit antennas used by the spacetime coder of link i in slot j, 1 ≤ Yij ≤ M , Zij is the number of receive antennas used by the space-time decoder of link i in slot j, 1 ≤ Zij ≤ N , and SIN Rij is the average SINR at the output of the combiner of link i in slot j and is given by, SIN Rij = N0 +

Pij Gii Gr PK−[ ZNij −2] k=1,k6=i

(5) Pkj Gik

Notice that Gvu is the path loss gain between the transmitter of link u and receiver of link v. Gr = N is the receiver array gain attributed to the coherent combining which exploits the CSI available at the receiver. Finally, the interference term in the denominator assumes interferers are spatially separated from the signal of interest to perfectly null ( ZNij − 2) interferers. This is a reasonable assumption in light of state-of-the-art beamforming algorithms [14]. For instance, the optimal beam former with L antennas can null up to (L − 2) interferers using its (L − 1) degrees of freedom (DoF) where a single DoF is reserved to detect the signal of interest.

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D. Problem Formulation In this section, we formulate a distributed optimization problem that strikes a balance between scheduling full diversity noninterfering links vs. simultaneously activating interfering links with lower diversity gains and exploiting nulling. The problem solved at the receiver of link i over the K slots can be formulated as a constrained optimization problem that minimizes the link probability of error ζi subject to SINR constraints, P1 : s.t.

min ζi Y , Z, P

SIN Rij ≥ β Pij = {0, P } 1 ≤ Yij ≤ M 1 ≤ Zij ≤ N Yij , Zij Integer

(6) ∀j ∀j

∀j ∀j ∀j

where the optimization variables are Y = [Yij ], vector of number of transmit antennas used by STC for link i in slot j, Z = [Zij ], vector of number of receive antennas used by the space-time decoder for link i in slot j, and P = [Pij ], vector of binary variables representing the link-slot assignment for link i in slot j, such that Pij = P when link i is activated in slot j, otherwise Pij = 0. β is a minimum requirement on the SINR that is essential for successful reception. E. Complexity Solving P1 is challenged by: i) The non-linearity of ζi and ii) The discrete optimization variables Yij , Zij , Pij . Accordingly, P1 is characterized as a non-linear integer programming problem which is quite challenging. The second challenge can be resolved via relaxing the optimization variables to be real and using iterative techniques like branch and bound. Nevertheless, we show next that the continuous variable version of the problem is still non-convex. Theorem 1: The objective function ζi is non-convex. Proof Outline: ζi exhibits complex structure as the product of individual SIN Rij raised to the power of (−Yij .Zij ). Motivated by the structure of ζi , one way to examine convexity is to investigate whether P 1 can be approximated to a convex geometric programming problem of posynomial objective function and constraints [13]. We show that ζi cannot even be approximated to a posynomial function due to the role of the 1 STC diversity gain (Yij .Zij ) as the exponent of SIN R . Furthermore, if we use (log ζi ) as the objective function instead, it turns out to be the sum of non-convex terms of the form SIN Rij (−Yij .Zij log ). 4 In conclusion, solving the non-convex integer programming problem P1 in general settings is challenging. Hence, we propose in the rest of the paper a novel solution that is sub-optimal for K > 2, yet, exhibits considerable performance gains and is founded on SINR-based decision rules that characterize the optimal solution for K = 2 links. III. S CHEDULING S PACE - TIME CODED L INKS (SSL) A. Optimal Diversity and Scheduling for K = 2 Links The distributed decision rules derived in this section are executed at the receiver of link i. They stem from the optimality of

P1 and the SINR constraints. For K = 2 links, the scheduling policy space collapses to two simple policies, namely Policy A: 1 link per slot (i.e. TDMA) and Policy B: 2 links per slot. Furthermore, STC with subsets of small M and N can be solved using search which gives rise to minimum and maximum possible diversity gains (Y.Z). Due to the lack of CSI at the transmitter and the desire to fully exploit transmit diversity provided by the M antennas, we set the variable Y = M for the 2 links over the 2 slots and fix it at this value. On the other hand, we characterize the minimum and maximum values of Z as dictated by the optimality condition and SINR constraint respectively. A.1 Minimum Z In this section, we address the following question which determines minimum Zi : When does policy B outperform policy A with respect to ζi ? This can be written formally as, ζiB < ζiA

(7)

where ζiB denotes link i error probability, as given in (3), in the presence of interference from the other link and ζiA denotes link i error probability in the absence of interference. Under policy A, all antennas are configured for maximum reliability using STC with full diversity gain = M .N (e.g. space-time block codes (STBC)[11]) and no nulling is needed. It is straightforward to verify that the error probability of link i under policy A, denoted ²i , is ζiA = ²i = ( SN4Ri )−M.N since each link transmits only in 1 slot. Next, we quantify ζiB where any link i is activated in any slot j among the 2 slots. Due to the presence of interference in each slot, a subset of receiver streams Z is dedicated to spacetime decoding, whereas the remaining streams are dedicated to nulling the other interferer. Based on (3), ζiB = Pei1 .Pei2

(8)

The interplay of diversity and nulling and their effect on SINR under policy B (as opposed to SNR under policy A) and the associated lower DG (Y.Z) yields the outcome of which policy constitutes the optimal. The above formula can be simplified if we factor in the fact that interference under policy B, where the 2 links are activated in each slot, does not vary from slot to slot. Accordingly, Pei1 = Pei2 = Pei and, hence, we drop the slot index j and (7) can be re-written as, √ ∀i (9) Pei < ²i Notice that different values of Zi may or may not satisfy the above condition. This is primarily attributed to the effect of Zi on the LHS since ²i is independent of Zi . As Zi increases from 1 to N (i.e. higher DG), the LHS decreases due to the role of the negative exponent in (4). The objective then is to find the minimum value of Zi , denoted M INi that satisfies (9). A.2 Maximum Z In this section, we shift our attention to the reception success condition which is governed by the SINR constraints and dictates the maximum value of Zi , SINRi ≥ β

∀i

(10)

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The RHS of (10) is independent of Zi whereas the LHS varies with Zi . As Zi decreases from N to 1 (i.e. more antennas dedicated to nulling at the receiver), SINR increases. The objective is to find the maximum value of Zi that satisfies (10), which is denoted M AXi . The interplay between M AXi and M INi yields SINR-based decision rules that we illustrate for general K links in the next section. If M INi ≤ M AXi , then slot sharing with Zi = M AXi is the optimal Diversity-Scheduling policy. Otherwise, activating only link i in slot i with Zi = N is the optimal.

Start: s = 0 Ks links probe

Link i scheduled before? Y N

1/K

for K links. Along the lines of previous section, this condition yields M INi for the optimization variable Zi . In addition, the SINR constraint yields M AXi which, along with M INi , gives rise to the following distributed decision rules: Distributed Decision Rules executed at receiver i over K slots: if M INi ≤ M AXi : Scheduling: Activate K links in each of the K slots MIMO: STC with Yi = M , Zi = M AXi Nulling with ZNi antennas • else: Scheduling: Activate 1 link in each slot (TDMA) MIMO: STC with Yi = M , Zi = N

•

B.2 SSL Algorithm Based on the decision rules, we introduce the scheduling space-time coded links (SSL) algorithm that achieves significant reliability gains for each link over TDMA with full diversity. Fig. 1 shows the flowchart of SSL, executed at the receiver of link i, to schedule K links over minimal number of slots, denoted s. The variable s denotes also the number of iterations

N Y

Rx i examine Decision Rules: MINi ” MAXi?

B.1 Distributed SSL Decision Rules

P e i < ²i

Exit: Ko= K links scheduled over s slots

Y

B. The General Problem: K > 2 Links Based on M AX and M IN of Z which characterize the optimal solution for 2 links in the previous section, we develop decision rules for arbitrary number of links. However, they yield a sub-optimal solution for K > 2 due to comparing only extreme policies (1 link/slot vs. K links/slot) and leaving many other possible schedules unexamined. Along the lines of section III.A, QK the objective function for each link is given by ζi = j=1 Peij as in (3). This enables each receiver to autonomously find a solution, i.e. number of antennas (Zi ) and link activation (Pi ), for its link and feed it back to its transmitter for execution. The unique feature of our approach is that we compare two extreme scheduling policies only to reach a solution: Policy A: 1 link per slot and Policy B: K links per slot. This greatly simplifies the problem as it eliminates the combinatorial complexity of scheduling. Furthermore, it opens room for the distributed scheduling space-time coded links (SSL) algorithm which iteratively packs links into successive slots depending on the levels of interference they experience. Based on the fact that link i experiences similar interference in all slots under policy B, (9) can be generalized to,

N

Rx i sense probe signals in sth interval?

Rx i feedback to Tx i: - Activate in sth slot - Zi = MAXi

Rx i feedback to Tx i: Do not activate in sth slot

No probes for Tx i in s+1, s+2,… probing intervals

Tx i re-probe in s+1 probing interval

s=s+1

Y

Rx i sense energy in sth slot?

N

Schedule Ks links, 1 per slot, over Ks slots (TDMA) s=s+Ks

Fig. 1. SSL flowchart at the receiver of link i to schedule K links over minimal number of slots, s

until SSL finds a solution starting from iteration s = 0. As indicated, SSL commences with K single-hop links which get partitioned over successive iterations to subsets denoted Ks < K where K0 = K. Accordingly, the time axis over which SSL operates is partitioned to s slots. Each time slot is preceded by a short probing interval where the Ks links examined in the sth iteration probe the wireless medium with small probing packets in order for receivers to individually compute M AXi and M INi values of Zi used by the space-time decoder. Hence, the sth iteration accommodates sth probing interval and sth slot. As shown in Fig. 1, SSL involves four conditional statements where the first and last are responsible for exit conditions whereas the third condition examines the decision rules. Given Ks links that probe the wireless medium in iteration s, receiver i computes the M AX and M IN of Zi and compares them to decide whether it can be activated in the next slot or not. If M INi ≤ M AXi , then link i can survive the (Ks − 1) interferers, i.e. achieve SIN R ≥ β with Zi = M AXi and ZNi streams for nulling. Hence, it can be activated in the sth slot and does not need to probe again. These receiver-based decisions are fed back to the transmitter for execution in future slots and iterations. If M INi > M AXi , then no solution exists for link i at current interference levels and, hence, it should not transmit in slot s and should re-probe again in the next probing interval. Next, consider an arbitrary iteration with Ks probing links, the third condition partitions Ks into two subsets: i) Feasible Subset: which can share slot s and do not need to re-probe the medium and ii) Infeasible Subset: which do not activate in slot

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IV. P ERFORMANCE R ESULTS A. Optimal Performance for K = 2 Links In this section, we present numerical results obtained using Matlab. We consider 2 links each with M = N = 4 antennas. We present results in this section for symmetric links where the transmitter-receiver separation is 250m. In addition, the distances between each receiver and the other transmitter (inter-

ferer) are equal and denoted D. D is varied across different runs, from 200m to 1000m, in order to model varying levels of interference. The symmetry in this scenario gives rise to equal interference at both receivers and, hence, same solution for each link. The transmit power per node is fixed at P = 20 mw. The minimum SINR requirement β is set to 5 dB. The path loss exponent is α = 4 and N0 is set to -90 dBm. 0

10

−5

10 Link probability of error (ζ)

s and need to re-probe in the next iteration. Notice that SSL goes through another iteration iff the Infeasible Subset is nonempty. In essence, if all links are feasible in iteration s, then no transmitters will re-probe the medium in iteration (s + 1). This constitutes our first observation and defines the exit condition in the first conditional statement, i.e. if any receiver i does not sense a probing signal in the sth probing interval, it exits with its own scheduling and diversity-nulling stream allocation solution along with the information that all K links are now scheduled over s slots. The second key observation stems from the other extreme, i.e. What if the Ks links probing in iteration s are all infeasible? This implies that those links cannot share the same slot and, hence, there is no need for more iterations under same interference conditions since it will not change the infeasibility result. Novel techniques for partitioning infeasible links lie out of the scope of this paper. Instead, SSL simply falls back to TDMA (i.e. 1 link per slot) for the infeasible Ks links, as suggested by the decision rules. This defines our exit condition in the last conditional statement, i.e. if any receiver i does not sense any transmission in the sth slot, it exits with its own scheduling and MIMO solution along with the s slots needed for the K links, where the last Ks slots are scheduled in a TDMA fashion. Finally, the second conditional statement indicates that once receiver i finds a solution it does not need to re-probe or re-solve the problem any more, it just needs to keep track of the evolution of the algorithm for other links, via incrementing s and examining the exit conditions. This is essential for all links to proceed synchronously over the algorithm and exit with consistent results, irrespective of which iteration yields their individual solutions. SSL is distributed since each link takes its link activation and MIMO stream allocation decisions independent of other links. The only communication needed is the feedback from each receiver to its respective transmitter. The knowledge of number of contending links K can be obtained through higher layers, e.g. topology control and routing mechanisms. It should also be noted that if K is finite, the links are guaranteed to find a solution in a finite number of iterations. For the two extreme scenarios, namely K links are feasible and K links are infeasible, a solution is found in a single iteration. Under typical scenarios, interference decreases from iteration to another as we partition the links until iteration s has: i) Ks = 1 which is trivial, ii) Ks > 1 and feasible which activates the links in slot s and exits and iii) Ks > 1 and infeasible which yields a TDMA solution for the Ks links and exits. It can be shown that the worst case number of iterations for SSL is d K 2 e which yields only two feasible links in each iteration, since 2 is the minimum number of links that can share a slot.

−10

10

−15

10

−20

10

−25

10

−5

Policy1: 1 link/slot Policy2: 2 links/slot, Y.Z=16 Policy3: 2 links/slot, Y.Z=8 Policy4: 2 links/slot, Y.Z=4 Policy5: 2 links/slot, Y.Z=2 Policy6: 2 links/slot, Y.Z=1 0

5

10

Avg. SINR (dB)

Fig. 2. Optimality Regions for Two Symmetric 4x4 Links

Fig. 2 plots ζ (which is the same for both links) and associated optimality regions for six MIMO-Scheduling policies. The optimization objective ζ is plotted against the average SINR in dB where SINR is computed with all interferers to capture varying D, irrespective of whether an interferer is nulled under a specific policy or not. Policy 1 represents TDMA with Z = 4 (policy A) whereas policies 2 through 6 represent slot sharing with different values of Z (policy B). Notice that policy 1 performance does not vary with SINR since transmissions are interference-free. Policy 2 and 3 performance varies with SINR due to the impact of interference on the SINR and, hence, on the link error probability. Finally, Policies 4, 5 and 6 performance does not vary with SINR, despite the fact that these are slot sharing policies. This is due to their low diversity gains (Y.Z = 4, 2, 1 respectively) which leave sufficient receiver antenna degrees of freedom to completely null the other interferer. The figure reveals different regions of optimality for different policies. For the rightmost region where SIN R ≥ 5.2 dB (i.e. D>450m), policy 2 achieves minimum ζ due to slot sharing while using the 4x4 MIMO for STC since interference is negligible. For 4.2 ≤ SIN R ≤ 5.2 (i.e. 350m < D < 450m), policy 2 fails to maintain the SINR constraint, due to interference buildup and, hence, ζ goes sharply to one. On the other hand, the less aggressive policy 3 assumes the optimal role for this region due to dedicating antennas to nulling interference. Finally, as interference dominates for D < 350m (SIN R ≤ 4.2), none of the slot sharing policies achieves the optimal and, interestingly,

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naive TDMA with Z = 4 achieves minimum ζ. B. Performance of K > 2 Links In this section, we present a sample scenario that demonstrates the performance gains of SSL. A randomly generated scenario consists of 10 links each with 4x4 MIMO, as shown in Fig. 3, and simulation parameters similar to the previous section. The average transmitter-receiver distance is 250m and the average interferer-receiver distance is approximately 870m. Due to interference, SSL goes through 3 iterations until scheduling the links is complete.

over consecutive slots, depending on the intense of interference. Finally, we conclude that MIMO link scheduling founded on the optimal solution for 2 links constitutes an important step towards developing MAC protocols that efficiently exploit MIMO for reliable communications in network scenarios. V. C ONCLUSIONS We studied the problem of reliable transmission in interference-limited MIMO networks. We formulated a distributed cross-layer optimization problem that jointly decides the scheduling, diversity gain and nulling in order to minimize the probability of error of individual links subject to SINR constraints. Next, we characterize the optimal solution for 2 links and utilize the distributed decision rules as the basis for solving arbitrary number of links using the SSL algorithm. Numerical results exhibit profound probability of error improvement (quadratic for some links) over scheduling non-interfering links with full diversity gain. This work can be extended along the following directions: i) Develop MIMO-MAC protocols for reliable communications based on the developed decision rules and ii) Incorporate fairness into the problem formulation and SSL algorithm. R EFERENCES

Fig. 3. Interference Scenario with ten random 4x4 MIMO Links

In iteration s = 0, the 10 links are grouped into two subsets: i) 6 feasible links (i.e. can share a slot) and ii) Remaining 4 infeasible links (i.e. cannot share a slot). The feasible set of links can thus share the first data slot using: Z = 4, 4, 4, 4, 1, 1. In iteration s = 1, SSL attempts to solve the infeasible subset of K1 = 4 links under reduced interference conditions. This process is repeated until the 10 links are assigned to successive slots that are not only maximally packed but also guarantee successful communication in the presence of interference (i.e. SIN R ≥ β). Reduced interference in this iteration enables 2 out of the 4 links to share the next slot with Z = 4, 4. In the last iteration, SSL is executed for the remaining K2 = 2 links. These two links cannot share a slot primarily since their average interferer-receiver distance (207m) is less than their average transmitter-receiver distance (246m). Hence, they are scheduled over 2 non-interfering slots. In summary, the proposed SSL algorithm partitions the 10 links over 4 subsets scheduled over 4 consecutive slots where reliability is maximal for each link and SINR is guaranteed to be at least β. The previous example exhibits profound improvement (quadratic for some links) in the link error probability (ζi ) over TDMA with full diversity gain. This is due to SSL which utilizes SINR-based distributed decision rules to maximally pack links

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