Spatial Multiplexing Gains for Realistic Sized Ad Hoc Networks with Directional Antenna Arrays

Hindawi Publishing Corporation EURASIP Journal on Wireless Communications and Networking Volume 2007, Article ID 98490, 12 pages doi:10.1155/2007/98490 Research Article Spatial Multiplexing Gains for Realistic Sized Ad Hoc Networks with Directional Antenna Arrays Eugene Perevalov,1 Danny Safi,2 Lang Lin,2 and Rick S. Blum2 1 Department of Industrial and Systems Engineering of Lehigh University, Bethlehem, PA 18015, USA 2 Department of Electrical and Computer Engineering of Lehigh University, Bethlehem, PA 18015, USA Received 7 January 2007; Revised 27 April 2007; Accepted 16 August 2007 Recommended by Wolfgang Gerstacker We concentrate on an ad hoc network model with nodes on integer lattice points over a 2D plane. We examine the limits of ad hoc network performance for systems with antenna arrays capable of allowing both spatial multiplexing and directional processing. Two cases are considered. In the first case, we consider “perfect” directional antenna arrays, in other words, each node can form beams of infinitesimally narrow beamwidth. In this case, the throughput capacity of an ad hoc network is independent of the network size. In the second case, we consider a more practical system where each node can form a fixed number of beams of finite beamwidth. Our results show that the spatial multiplexing gains depend on the system size, antenna beamwidth, and number of antenna beams. Furthermore, we show that spatial multiplexing gains offsetting the interference-related performance degradation can be achieved in ad hoc networks with thousands of nodes. Copyright © 2007 Eugene Perevalov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION The application of multiple antennas at both the transmitter and receiver sides of a wireless system for the purpose of spatial multiplexing (simply put, spatial multiplexing in this context means making use of multiple paths distinct in physical space to deliver information from a source to the corresponding destination) [1, 2] has been shown to have the potential of achieving extraordinary bit rates. As a result, this topic has received significant study recently [3–10]. The issues of MAC/routing protocol design for ad hoc networks utilizing multiple antennas were also studied in [11– 17]. It should be noted that antenna arrays can implement directional processing and beamforming in addition to spatial multiplexing. When these approaches are suitably combined, good network performance is achieved. However, the majority of research has focused on point-to-point communications. Here we study spatial multiplexing at the network level. Further, we assume the antenna arrays used for the spatial multiplexing will also be used for beamforming. We will study the uniform throughput capacity, or simply uniform throughput, which we define as the minimum long-term average rate at which every node in the network can transmit to its corresponding destination. The throughput in wireless ad hoc networks is inherently limited by in- terference since the nodes have to use the common wireless channel in order to transmit different information. The use of multiple directed beams and spatial multiplexing cannot completely eliminate the interference but, as we will see, can greatly alleviate it, even if a small number of beams at every node is used. Previous results [18] have shown poor performance for large ad hoc networks without spatial multiplexing. In this paper, we will show that spatial multiplexing provides large gains in throughput for small networks, and that while these gains shrink for larger networks, there are still spatial multiplexing gains in networks with thousands of nodes. In this paper, we consider a network consisting of n nodes located on a square grid with periodic boundary conditions. We begin by examining a simpler case of infinitely narrow beamwidth where every beam is just a zero-width ray with the origin at the transmitting node. We find that in this case, the uniform throughput is upper bounded by Wg/2, where W is the rate of point-to-point transmission along a single beam, and g is the number of beams each transmitter can form. Furthermore, we show that, under reasonable assumptions, the uniform throughput of Wg/2 can be achieved regardless of the network size (and the distance between sources and destinations). Next, we consider the case of a finite angular beamwidth D where the beams are infinite 2 EURASIP Journal on Wireless Communications and Networking D D D D Figure 1: A node with 4 transmitting beams of angular width D. metric (“Manhattan distance”) in units of lattice space, unless noted otherwise. We assume that each transceiver node is equipped with an antenna array that can produce g antenna beams, each with angular width D (see Figure 1) such that gD ≤ 2π. Node-to-node transmissions on the torus are allowed only in the “shorter” direction, that is, the largest horizontal and vertical transmitting distance allowed by the model is  m/2. The latter requirement is used to imitate a real system with boundaries while disregarding boundary effects where they can lead to unwanted complications. A transmission from node i to node j along a beam bli , l = 1, 2, . . . , g, is assumed to be successful if 2 (1) node j lies inside the beam bli , two-dimensional cones with vertices at the transmitter. We show that in this case the uniform throughput is bounded from above by a quantity, that is proportional to Wg, and  √  for larger network sizes, proportional to l / n, where l is the average of the longest g hops possible from a given node without interference. Moreover, a fixed fraction of this upper bound can be shown to be achievable. The result is that, although the degradation of performance due to interference is still present for the finite beamwidth case, the spatial multiplexing allows one to “postpone” the throughput from falling below W (which is what the throughput would be for just a single source-destination pair) until fairly large network sizes (thousands of nodes for beamwidth of about 10 degrees and no more than 10 beams) which makes practically large network sizes entirely feasible. The directional antenna assumptions used in this paper are consistent with accepted results [19] that imply that antenna arrays (smart antennas) can be used to form beams in n different directions if at least n antennas are available in an array. Further, by proper spacing of the antennas and by employing more antennas, these beams can be made more narrow. Therefore, the number of antennas limits the number of directional beams that each node may employ. The rest of the paper is organized as follows. In Section 2, we formulate the model used in the paper. Section 3 is devoted to evaluating the uniform throughput for both cases of infi (...truncated)