SNR Penalty from the Path-loss Disparity in Virtual Multiple-Input-Single-Output (VMISO) Link Haejoon Jung and Mary Ann Ingram School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250, USA Email: {hjung35, mai}@gatech.edu Abstract—Cooperative transmission (CT), in which spatially separated wireless nodes collaborate to form a virtual antenna array or virtual multiple-input-multiple-output (VMISO) link, is an effective technique to mitigate multi-path fading by spatial diversity. In this paper, we study the disparities in path losses between the randomly placed relay nodes in a transmit cluster and the destination nodes. Many authors assume that the elements in a virtual antenna array are co-located, even though they are spread out. In this paper, we show that a signal-to- noise-ratio (SNR) penalty of up to 3dB should be included when making this assumption. By the high-SNR approximation of the outage rates, we show that the performance degradation caused by the path-loss disparity can be characterized equivalently by log-normal shadowing. Moreover, we derive the upper and lower bounds of the real outage probabilities in closed forms based on the log-normal shadowing model, by which we can estimate the SNR penalty of the co-located assumption. I. I NTRODUCTION For small wireless nodes with limited power, where col- located antennas (a real antenna array) cannot be deployed, cooperative transmission (CT) is an alternative way to achieve spatial diversity in fading channel [1], [2]. CT provides an SNR advantage through array and diversity gains by creat- ing a virtual multiple-input-single-output (VMISO) link that connects a transmitting cluster (multiple nodes) with a single receiver node. Based on the SNR advantage of CT, various higher layer protocols have been proposed, in which the VMISO links provide gains at higher layers such as throughput improvement, energy saving, energy balancing, and range extension [3]–[6]. In such protocols, two abstraction models of the VMISO link are widely used to deal with random topologies in the network-scale analysis and simulations. First, when the node degree is high enough, the continuum approximation in [7] and [6] is used, in which the number nodes goes to infinity while the transmit power of each node becomes extremely small. However, this continuum model cannot be applied to the low node degree situation. The other abstraction model, intended for the low node degree case, assumes physically separated cooperating nodes in a cluster are simplified to be a single node with a multiple-antenna array as in [3]–[5]. In this co- located approximation model, the disparate path losses caused by the different distances between the transmitting nodes in a cluster to the receiver node are ignored. The authors in [8] realized through simulation of some specific topologies that there can be a significant error (i.e., SNR penalty) incurred for making the co-located assumption. Beaulieu et al. [2] derived a closed-form expression for the outage probability at the destination for the case of the decode- and-forward (DF) relays, where the location of the relays are assumed known. [2] has an intermediate result, where the number of relays that successfully decode is assumed known, while the final result in [2] allows for the opportunistic case, where the number of relays that successfully decode is not known a priori. However, this final expression in [2] is long, complicated, and numerically sensitive [1]. Moreover, the SNR penalty for the co-located assumption is not considered in [2], nor are random locations of nodes taken into account. In this paper, we allow the node locations to be random. For the known number of successfully decoding relays, we show that the random locations of nodes produces a random received power, averaged over multi-path fading, with a log- normal distribution, as in shadowing. We also derive a lower bound based on assuming error-free source-relay links. We also treat the opportunistic case and derive an upper bound for the case when the number of relays that successfully decode is not known a priori by assuming the first-hop errors are independent and identically distributed (i.i.d.). These bounds provide the best and worst case SNR penalties for the co-located assumption. To our knowledge, this is the first study that models the SNR loss due to the random path-loss disparities in the VMISO links. II. SYSTEM MODEL We consider a VMISO communication consisting of two phases as shown in Fig. 1, where the source, which is indicated by the left black dot at the center of the dotted lined circle, first transmits a packet to the destination (the right black dot) in Phase 1. After that, the multiple relays (the white-filled circles) around the source decode and then forward (DF) using orthogonal channels to the destination in Phase 2 [2]. This cluster architecture is referred to as the centralized cluster [9], where each cluster has a cluster head that recruits its cooperating relays and triggers the group transmission. A. Network Topology We consider a static VMISO network as shown in Fig. 2, where the source node is located at the origin with a distance d 0 to the destination. Also, there are N number of cooperating 978-1-4673-3122-7/13/$31.00 ©2013 IEEE IEEE ICC 2013 - Wireless Communications Symposium 3943