Transmission capacity for dual-hop relaying in wireless ad hoc networks
© Lee et al; licensee Springer. 2012
Received: 1 May 2011
Accepted: 22 February 2012
Published: 22 February 2012
To account for randomly distributed nodes in a wireless ad hoc network, the transmission capacity is defined as the number of successful transmissions taking place in the network per unit area under an outage constraint. In this paper, we analyze the transmission capacity for dual-hop relaying in a wireless ad hoc network in the presence of both cochannel interference and thermal noise, where interferers are spatially distributed following a Poisson distribution. Specifically, we first present the exact outage probability for amplify-and-forward and decode-and-forward protocols in a Poisson field of interferers without neglecting noise at all nodes. We then derive the transmission capacity of such networks, which determines the maximum allowable density of transmitting nodes for each relay strategy at a specified outage probability and data rate. Numerical results demonstrate that the dual-hop relaying is still beneficial in terms of the transmission capacity in wireless ad hoc Poisson networks.
Cooperative relay communication has recently drawn considerable attention to increase network coverage and reliability [1, 2]. For a decode-and-forward (DF) strategy, a relay node decodes the received signal and transmits it after re-encoding, whereas the relay simply retransmits a scaled version of the received signal in amplify-and-forward (AF) mode [3, 4]. Most of the previous work has focused on noise-limited fading environments, such as Rayleigh fading with additive white Gaussian noise (AWGN). While such work made great strides toward understanding the potential of relay networks, it dealt mainly with ideal configurations with no interference. However, network interference is inevitable in practical wireless networks due to spectral reuse. Motivated by this, relay networks have been recently studied in the presence of cochannel interference [5–7]. In Krikidis et al. , the asymptotic performance was analyzed under the interference scenario only at relays. The fixed-gain AF relaying in an interference network was investigated in Zhong et al. , where interference at the relay and thermal noise at the destination were ignored for analytical tractability. The exact outage probability for DF relaying was further derived in Si et al.  accounting for multiple interferers and noise at both the relay and the destination. In all these work, the locations of network nodes are deterministic--without spatial randomness.
To treat the capacity of a decentralized ad hoc wireless network, the transport capacity, defined as the product of the end-to-end sum throughput and distance, has been introduced due to the difficulty in determining the capacity region of a large ad hoc network . More recently, using the stochastic geometry framework, the transmission capacity has been proposed as the maximum density of active transmitting nodes per unit area to satisfy an outage constraint at a given data rate when interferers are randomly scattered and uncoordinated . For a variety of scenarios, this notion of transmission capacity has been used successfully to characterize the physical layer on the ad hoc network [10–14]. A new metric akin to the transmission capacity, called the random access transport capacity, has been also developed for the end-to-end throughput in multi-hop transmission over some distance .
In this paper, we consider dual-hop relaying with DF and AF protocols in a wireless ad hoc network in the presence of both interference and noise. Each interfering node in the network independently transmits data and is randomly distributed in a Poisson law over a plane. The motivation behind imposing Poisson interference and noise is threefold: (i) many previous works neglected either interference or noise in analysis of a relay network; (ii) previous work on relay networks with interference considered only a limited number of interferers (iii) randomly distributed nodes(i.e., interferers) allowing a Poisson distribution are suitable for realistic communication model. To the best of our knowledge, relay network to qualify interference using stochastic geometry has not been addressed before. Hence, we analyze exact outage probability of a dual-hop relay with both noise and interference in a wireless ad hoc network. Unlike previous work in a relay network, when the node locations are distributed as a Poisson point process (PPP), we need to analyze relay networks using a metric for decentralized wireless network, termed transmission capacity. Furthermore, as transmission capacity has considered single-hop transmission without noise, we focus on the transmission capacity of dual-hop relay with noise and specific relaying protocols. It is worth of finding maximum successful transmitting nodes per unit area to satisfy outage probability and data rate from transmission capacity of dual-hop relay. Although both noise and interference with Poisson distribution are considered, it is noted that dual-hop relay is still beneficial in terms of transmission capacity.
The remainder of the paper is organized as follows. Section 2 presents the system model based on location of nodes, channel models, and distribution of interference. We derive outage probability analysis of DF and AF strategies in Section 3. Based on the outage probability of dual-hop relay, transmission capacity is derived in Section 4. Section 5 compares simulation results with analytical results. Finally, Section 6 concludes the paper.
2 System model
We consider a wireless ad hoc network consisting of a source, a relay, and a destination with no direct source-to-destination link. All nodes have a single antenna operating in a common frequency band and are in half-duplex mode.
2.1 Node locations
The sets of nodes in a relay network are denoted as for sources, for relays, and for destinations. Now, allow the set of nodes to be divided into two different pairs for each hop transmission: (Sx n , Rx n ) and (Rx n , Dx n ), where the distance between two nodes in a pair is d SR and d RD , respectively. In a dual-hop relay network, sources and relays are also interferers in each hop. In particular, all nodes independently transmit and distribute their locations following a Poisson distribution. Let the location of Sx n be S n , Rx n be R n , and Dx n be D n . The source set generating interference at the first time slot is modeled as a homogeneous PPP on the plane of intensity λ1. Since R n is randomly located at a fixed distance d SR from S n , the set also follows a homogeneous PPP on a two-dimensional plane with intensity λ2. As destination node D n is placed at d RD away from the relay node R n , is also a homogeneous PPP with intensity λ3.
In this paper, we consider reference source Sx0, reference relay Rx0, and reference destination Dx0 in a decentralized wireless network that transmits desirable data in the Φ S , Υ R , and Ψ D . According to each hop transmission, reference nodes can be bounded as (Sx0, Rx0) and (Rx0, Dx0). For the (Sx0, Rx0) pair in the first hop, the relay node Rx0 is located at the origin and source node Sx0 is placed at d SR meters away from the relay node Rx0. For the (Rx0, Dx0) pair in the second hop, we place the relay node Rx0d RD meters away from the destination node Dx0, where Dx0 is located at the origin.
2.2 Channel model
where F12 captures the small-scale fading which obeys a Rayleigh fading model and l(∥x1 -x2∥) = ∥x1-x2∥-αcharacterizes the effect of large-scale path loss following the power law with path loss exponent α. We assume that channel responses for all nodes are independent and quasi-static.
2.3 Distribution of aggregate interference
where h i is a instantaneous channel response of interferer i, Φ is a homogenous PPP with intensity λ, and P i is interference power.
where λ is intensity of interferers, c d is the volume of d-dimensional unit ball, r is the radius of finite area located interferers, and δ ≜ d/α. Also, Γ(x) is the Gamma function and γ(s,x) is the lower incomplete gamma function .
3 Outage performance analysis
3.1 Decode-and-forward relaying
where λ1 is intensity of source nodes and r1 denotes the radius of interference area.
Proof See Appendix A
where d RD is the distance between relay and destination and r2is radius of finite area with intensity of interferers λ2.
where R is end-to-end spectral efficiency.
where β is target SINR β = 22R- 1
3.2 Amplify-and-forward relaying
where and .
The random variable X and Y were defined in DF relaying, and the PDF and CDF of them are the same as in Theorem 1 and Corollary 1.
Proof See Appendix B
4 Transmission capacity analysis
In this section, we compute transmission capacity C(ϵ) based on outage probability of a dual-hop relay. Transmission capacity was defined as the maximum density of the transmitting node to satisfy outage probability and data rate on medium access control (MAC) layer performance. That is, it is efficient to find the maximum available transmitting source and relay nodes to satisfy a given outage probability and data rate (i.e., Quality of Service (QoS)) from the transmission capacity of a dual-hop relay. As transmission capacity in Weber et al.  considers single-hop transmission without noise, we focus on the transmission capacity of a dual-hop relay with noise and specific relaying protocols. We assume that the sets of Φ S , ϒ R , and Ψ D are homogeneous PPPs on the two-dimensional plane with the same intensity λ due to the random translation invariance property of PPP .
4.1 Direct transmission
where ϵ is outage probability for target SINR β = 2 R - 1 with spectral efficiency R, q-1(ϵ) is the spatial intensity of attempted transmission associated with outage probability ϵ and it is always greater than 0. The transmission capacity C(ϵ) is thinned by the probability of success 1 - ϵ.
where r is radius of finite area with the intensity of each interferer λ, and d SD is the distance between source and destination.
4.2 Dual-hop relaying
Since we are concerned with the outage probability of the relay network after two-hop transmission, the transmission capacity of dual-hop relay is identical to that of single hop except for the target SINR β.
where ϵ is outage probability of dual-hop relaying for target SINR β.
5 Numerical and simulation results
In this section, we present some numerical examples of the outage probability and transmission capacity for a dual-hop relaying with both interference and noise in a Poisson network. We consider Rayleigh fading channel and path loss exponent α = 4 to illustrate our analytic and simulated results.
5.1 Outage probability
5.2 Transmission capacity
This paper considers a dual-hop relaying in the presence of both noise and interference simultaneously, allowing a Poisson interference model. The outage probability of DF and AF strategies have been derived, especially we verified that the end-to-end SINR of AF relaying has Macdonald r.v's. The analytic and simulated results showed that outage probability of dual-hop relay had an error floor in high density of transmitting nodes and performance was greatly influenced by distance and interference power. Furthermore, we consider a metric of decentralized wireless network, called transmission capacity, for dual-hop relay with AF and DF strategies. We redefine the transmission capacity, because spatial intensity of attempted transmission associated with outage probability q-1(ϵ) can be negative number due to thermal noise. Hence, we note that the existing region (q-1(ϵ) > 0) of transmission capacity of dual-hop relay is growing with increasing transmission power. Our results reveal that transmission capacity of a dual-hop relay considering both noise and interference has better performance than direct transmission in a wireless ad hoc network.
A Proof of Theorem 1
Finally, we can obtain CDF and PDF of X in Theorem 1.
B Proof of Theorem 2
Using y - z = u, we can obtain outage probability of AF relaying in Theorem 2.
This research was supported by the KCC (Korea Communications Commission), Korea, under the "Development of highly efficient transmission technology for the next generation terrestrial 3D HDTV" support program supervised by the KCA (Korea Communications Agency) (KCA-2011-10912-02002).
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