Outage Performance of TwoHop OFDM Systems with Spatially Random DecodeandForward Relays
Abstract
In this paper, we analyze the outage performance of different multicarrier relay selection schemes for twohop orthogonal frequencydivision multiplexing (OFDM) systems in a Poisson field of relays. In particular, special emphasis is placed on decodeandforward (DF) relay systems, equipped with bulk and persubcarrier selection schemes, respectively. The exact expressions for outage probability are derived in integrals for general cases. In addition, asymptotic expressions for outage probability in the high signaltonoise ratio (SNR) region in the finite circle relay distribution region are determined in closed forms for both relay selection schemes. Also, the outage probabilities for free space in the infinite relay distribution region are derived in closed forms. Meanwhile, a series of important properties related to cooperative systems in random networks are investigated, including diversity, outage probability ratio of two selection schemes and optimization of the number of subcarriers in terms of system throughput. All analysis is numerically verified by simulations. Finally, a framework for analyzing the outage performance of OFDM systems with spatially random relays is constructed, which can be easily modified to analyze other similar cases with different forwarding protocols, location distributions and/or channel conditions.
I Introduction
Cooperative communications have become an important topic for research and industry in recent years [1, 2, 3, 4]. It is well known that relayassisted cooperative communications are capable of providing extra diversity and thus a better system performance in terms of energy efficiency, outage performance and network coverage extension [5, 6, 7]. In particular, multicarrier relay systems are of high importance, because it fits a number of applications in practice [8]. A number of representative and useful multicarrier relay systems have been proposed and analyzed. For example, a blockbased orthogonal frequencydivision multiplexing (OFDM) decodeandforward (DF) relay system has been analyzed in [9]. Bulk and persubcarrier relay selection schemes for OFDM systems have been proposed and compared in [10] and [11], respectively. However, all of the above achievements regarding cooperative OFDM systems do not consider the location distribution of relays. The conventional network model employed in these previous OFDMrelated works assumes the locations of all nodes to be deterministic and stationary, which form a stationary network topology. In practice, however, the dynamic nature of communication nodes is common and should be considered in order to provide a more general and meaningful analysis [12]. Hence, a more realistic way to model a communication network is to assume the location of a node to be a random variable. To perform the analysis effectively, Poisson point processes (PPPs) have been used to analyze the location distribution of communication nodes for a large number of applications in wireless communications [13, 14, 15, 16]. Pioneering work related to cooperative transmission in Poisson distributed networks was published in [17], in which an upper bound on outage probability is derived. Then, generalized analyses of DF and AF cooperative systems with spatially random relays distributed within a finite region have been given in [18, 19] and [20, 21], respectively. The system with relays distributed over an infinite space is analyzed in [22, 23, 24]. Opportunistic relaying with different combining techniques can be found in [25, 26, 27].
However, to the best of the authors’ knowledge, the link between OFDM systems and randomly distributed networks is lacking. This motivates us to construct a framework for analyzing the outage performance of twohop OFDM systems with spatially random relays. In this paper, we analyze the outage performance of the twohop OFDM system with spatially random DF relays and investigate a series of important properties of cooperative systems in random networks related to the outage performance. Specifically, the contributions of this paper are summarized infra:

The exact expressions for outage probability for bulk and persubcarrier selections are derived in integral forms for general cases. Meanwhile, the asymptotic expressions for finiteregionbased outage probability in the high signaltonoise (SNR) region for bulk and persubcarrier selections are determined in closed forms. Furthermore, the exact expressions for infiniteregionbased outage probability are determined in closed forms for the case of free space.

It is proved that the cooperative diversity gain in Poisson random networks can either be zero, one or infinite, which is termed the ternary property.

An approximate relation between the outage probability ratio of two selection schemes and the relay node density is determined, which can be used to evaluate the performance advantage of persubcarrier selection over bulk selection in sparse networks.

The relation among system throughput, the number of subcarriers and relay node density is investigated and a concave problem is formulated and proved to be capable of producing the optimal number of subcarriers, so that the system throughput can be maximized. Meanwhile, a special optimization case with reliability requirement is discussed and an approximation of the cutoff relay node density above which the formulated problem is solvable, is also derived.
All analysis is numerically verified by simulations. The results provided in this paper can be easily modified to analyze other similar cases with different forwarding protocols, location distributions and/or channel conditions.
The rest of this paper is organized as follows. The system model is detailed in Section II. We subsequently analyze the outage performance and discuss a series of related system properties in Section III and Section IV. After that, the analysis is numerically verified by simulations in Section V. Finally, Section VI concludes the paper.
Ii System Model
Iia System configurations and channel model
In this paper, we consider a network with a single source located at the origin denoted by and a destination node located at in a twodimensional polar coordinate system. The locations of source and destination are deterministic and stationary. Then, we assume the relays are homogeneously Poisson distributed over a twodimensional region with a constant density , which form a homogeneous PPP denoted as . In particular, a finite circle distribution region centered at the source node with a radius and an infinite distribution region are considered and employed to analyze the system performance in this paper, which are denoted by and , respectively. Besides, for a typical OFDM system, we assume the number of subcarriers is , which is deterministic. The set of all subcarriers is denoted as . Furthermore, it is assumed that the channel state information (CSI) can be perfectly estimated without any delay and overhead by the source node, so that relay selection can be effectively performed. We further suppose that the entire network operates in a halfduplex protocol and there is not a direct transmission link between source and destination due to deep fading, so that two orthogonal phases are required for one complete transmission from source to destination. In particular, the source broadcasts the signal to all relays at the first phase and relays decode and forward the received signal to the destination^{1}^{1}1We choose DF forwarding protocol in this paper due to its low CSI estimation complexity and satisfactory outage performance [28].. For the noise, it is assumed to be independent and identically distributed (i.i.d.) at all nodes with noise power .
Meanwhile, two signal degradation mechanisms encountered in transmission are considered, which are signal attenuation and multipath fading. Assuming randomly distributed relays are organized in the set , for the th relay located at , , if equal power allocation scheme is applied over all communication nodes with transmit power , the received instantaneous SNR on the th subcarrier is
(1) 
where is the path loss exponent; is the th hop channel gain on the th subcarrier due to multipath fading and is modeled as an i.i.d. exponentially distributed random variable with unit mean. Therefore, for , and , the probability density function (PDF) and cumulative distribution function (CDF) of are given by
(2) 
For the second phase, because of the DF forwarding protocol, the received instantaneous SNR at the destination is
(3) 
where is the distance between the th relay and destination; however, it should be noted that because we assume the locations of all relays are unchanged during a complete transmission process, is a dependent random variable on and , which can be expressed by the law of cosines as
(4) 
Finally, the equivalent endtoend SNR in DF relaying network can be regarded as^{2}^{2}2An outage in DF relaying networks depends on the minimum channel coefficient among the sourcerelay and the relaydestination links. Hence, we can employ the minimum singlehop channel SNR as the equivalent endtoend SNR here [29].
(5) 
IiB Relay selection schemes and outage probability
Two selection schemes are considered in the paper. First, only one relay among is selected by the selection criterion below:
(6) 
This selection scheme is termed bulk selection, because all subcarriers will be forwarded by the only one selected relay in bulk. Obviously, this selection scheme is easy to implement for OFDM systems, since only one relay is involved in the entire transmission process. However, its outage performance is obviously not optimal for each individual subcarrier. To obtain the optimal outage performance, we can apply another selection scheme termed persubcarrier selection, in which multiple relays are selected in a persubcarrier manner, so that all subcarriers can be forwarded by their optimal relays. The persubcarrier selection criterion is given as follows^{3}^{3}3Note that, here it is allowed that for . In other words, the relay is capable of forwarding two or more subcarriers simultaneously. Because we do not consider a transmit power limit in all nodes, the power imbalance problem among relays is out of the scope of this paper.
(7) 
For clarity, these two selection schemes are illustrated in Fig. 1.
To consider the outage performance of multiple subcarriers as an entity, let the superscript standing for different relay selection schemes and define the a posteriori outage probability after selection as^{4}^{4}4 is possible when the area of relay distribution region and this special scenario should be regarded as outage as well [18].
(8) 
where denotes the probability of the random event enclosed; is a predefined and fixed target SNR threshold ; is the index of the selected relay forwarding the th subcarrier. In this paper, we will take the outage probability defined above as a metric to evaluate the outage performance.
Iii Outage Performance Analysis
Iiia Bulk selection
By (2) and (5), the CDF of the endtoend SNR can be derived as
(9) 
and by (6) the a posteriori outage probability after performing bulk selection can be determined by^{5}^{5}5The average over includes the case of when is finite; The area integral given in the second line of the equation can be converted to a double integral by adopting a certain coordinate system (Cartesian, polar or biangular) and thus numerically calculated.
(10) 
where denotes the average of the enclosed.
IiiA1 Finite relay distribution region
Due to the symmetry of the finite circle region , we can derive the outage probability in the finite circle region by (10) and obtain:
(11) 
where and .
Although there is no closedform expression of (11) because of the double integral in , we can employ a power series expansion at and obtain the asymptotic expressions of for an arbitrary :
IiiA2 Infinite relay distribution region
Due to the symmetry of the infinite region , we can obtain
(15) 
To the best of the authors’ knowledge, there does not exist a closed form of (15) for an arbitrary . However, for a special case when (free space), we can obtain
(16) 
IiiB Persubcarrier selection
Similarly as the case of bulk selection, by (7) and the binomial theorem, the a posteriori outage probability after performing persubcarrier selection can be determined by
(17) 
IiiB1 Finite relay distribution region
We can obtain the outage probability of persubcarrier selection scheme over finite relay distribution region by
(18) 
where .
Meanwhile, by power series expansion on (18) at , we also obtain the asymptotic expression of as
(19) 
from which we can observe that the outage floor given by
(20) 
is exactly the same as the case of bulk selection. As expected, the outage event at high SNR within a finite circle distribution region is dominated by the case where there is no relay candidate for selection (i.e. ). As a result, it is irrelevant to the selection schemes and/or forwarding protocols and only dependent on the relay node density and the radius . In general, the outage floor for an arbitrary finite distribution region can be derived by
(21) 
IiiB2 Infinite relay distribution region
Similarly, substituting into (17) yields
(22) 
Again, there is no closed form of (22) for an arbitrary . When , we can obtain
(23) 
Iv Discussion of Important System Properties
Iva Cooperative diversity analysis
For deterministic networks in which the number of relays and/or their locations are stationary, diversity gain is an allimportant metric to measure the cooperative performance advantage [3]. However, we show as follows that this metric is not appropriate anymore for cooperative systems in random networks where relays are Poisson distributed. Subsequently we prove the ternary property of diversity gain in Poisson random networks.
For cooperative systems in finite Poisson random networks, the binary property of diversity gain has been proved and it states that the diversity gain can only be either one or zero depending on whether there is a direct transmission link between source and destination or not [18]. Following the discussion of outage floor in Section III, this binary property related to diversity gain is straightforward. Because without a direct transmission link, an increasing power would not affect the outage performance at high SNR and this leads to a zerodiversityorder system.
However, when considering an infinite distribution region, there does not exist such an outage floor, because if the region is infinite, as long as the relay node density is positive, there are always an infinite number of relays distributed over region . As a result, with a sufficiently large transmit power (considering the asymptotic region), the twohop transmission can always be successful, because there must exist a ‘satisfactory’ twohop link among an infinite number of relays. This indicates the diversity gain of cooperative systems in infinite random networks is infinity. Mathematically, for an arbitrary and bounded , we can prove that^{6}^{6}6See Appendix A.
(24) 
Therefore, unlike the situations in deterministic networks, it would be impossible to derive a linear relation among , and in random networks. Instead, the diversity gain can only be either zero, one or infinity. We term this ternary property of diversity gain in Poisson random networks. In the meantime, (24) also indicates that any high but bounded path loss attenuation does not counter the constructive effects of infinite distribution region on asymptotic outage performance.
IvB Comparison of outage performances between bulk and persubcarrier selections
By (10) and (17), we can quantify the outage performance advantage of persubcarrier selection over bulk selection via the outage probability ratio given by
(25) 
where .
Considering the case of sparse networks with small , the outage probability ratio can be approximated by^{7}^{7}7See Appendix B.
(26) 
Eq. (26) indicates that the outage performance advantage brought by persubcarrier selection over bulk selection will become negligible with a decreasing , because the number of relays for selection is small and employing multiple relays to perform persubcarrier selection becomes less likely. Meanwhile, if we would like to ensure the performance advantage by letting the outage probability ratio (i.e. the outage probability of persubcarrrier selection is times lower than that of bulk selection), an approximation of the required minimum density can be derived as
(27) 
which would provide a guidance for whether bulk or persubcarrier selection should be employed or a switching criterion for a dynamic switching mechanism between these two selection schemes when the relay node density is known^{8}^{8}8Although persubcarrier selection scheme has the optimal outage performance, it is not always preferable if the performance advantage is not significant, because the selection and synchronization processes among multiple relays and destination node are complicated and thus will require more overheads [30]..
IvC Optimization of the number of subcarriers for bulk selection
With the development of index modulation and frequency resource allocation, OFDM systems with a varying number of active subcarriers become common in practice [31, 32]. Therefore, it is meaningful to investigate the effects of adopting different numbers of subcarriers on the system throughput. For simplicity, we assume that equal numbers of bits are carried by each subcarrier. Hence, the average system throughput can be characterized by the average number of successfully decoded subcarriers per transmission at the destination [33], which can be determined by
(28) 
According to (8), a large will lead to a large , since all signals transmitted on these subcarriers need to be successfully decoded at the destination, or an outage will occur otherwise. Consequently, for a given , there exists an optimal number of subcarriers denoted by , by which the system throughput can be maximized. Also, because relay node density is varying in practice due to the dynamic on/off switching mechanisms and node mobility [34, 35], it is intuitive that we can dynamically adjust the number of subcarriers to offset the adverse effects of relay node density , so that the system throughput can always be maximized. To be clear, the relation among , and is plotted in Fig. 2 for the case of bulk selection in free space.
Now, let us focus on the method to determine . First, we relax the integer to a positive real number ^{9}^{9}9Currently, this method is only applicable for bulk selection, since can be regarded as a product factor in and thus can be relaxed to . On the other hand, because is the upper limit of the summation in , this relaxing relation is not feasible, and thus the following optimization is not possible for persubcarrier selection. However, it is possible to propose a suboptimal solution for persubcarrier selection in which the is replaced by a certain approximation with as a product factor., so that some optimization tools can be applied. Then, it can be proved that is a concave function of ^{10}^{10}10See Appendix C. Therefore, the optimal solution to can be obtained via a concave problem formulated by
(29) 
which can be solved efficiently using standard optimization techniques (e.g. CVX in MATLAB). Then, is determined by
(30) 
where and denote ceiling and floor functions, respectively.
However, the optimization of given above would not be always applicable, since the chosen would lead to an inappropriate outage probability beyond a threshold , which should be maintained for a prescribed quality of service (QoS). Unlike networks for realtime streaming media in which throughput is the key [36], this reliability requirement is in particular crucial for some special networks, e.g. the Internet of things (IoT) [37] and military wireless sensor networks (WSNs) [38]. In this context, is dependent on and . Therefore, with this constraint on outage probability , the concave optimization problem formulated in (29) becomes
(31) 
Then will be determined by
(32) 
On the other hand, because must be ensured in order to provide transmission service, it is impossible to always maintain a given outage probability by reducing when keeps decreasing. Consequently, there is a cutoff relay node density below which the outage probability cannot be maintained by reducing . By substituting into (10), the cutoff density can be numerically evaluated by
(33) 
By (16), the cutoff relay node density for bulk selection in free space can be approximated by
(34) 
V Numerical Results
Va Outage performance of bulk and persubcarrier selections
First of all, we need to verify the correctness of the analytical results of outage probabilities. To do so, we normalize , and set , and ^{11}^{11}11The lengths given here are relative and thus dimensionless, since other parameters have been normalized.. Then, we simulate the relation between the outage probability and the transmit power with spatially random relays distributed over a finite circle region. The simulation results for bulk and persubcarrier selections are presented in Fig. 3 and Fig. 4, respectively. By the results shown in these two figures, our analysis of the outage performance corresponding to finite circle region has been verified. Meanwhile, we can also see that the variation of the number of subcarriers has a significant impact on the outage performance for bulk selection, while it is relatively trivial for persubcarrier selection. This is because bulk selection can only employ one relay to satisfy transmission requirements of all subcarriers, which is more ‘sensitive’ than the case of persubcarrier selection which is capable of employing multiple relays. Both selection schemes are affected by path loss exponent in a large but similar scale, which aligns with our expectation.
VB Relation between relay node density and multirelay performance advantage
In order to investigate the effects of the relay node density on outage probability ratio , we define the connection probability as , which eases the illustration of some crucial properties of spatially random networks in logarithmic plots when is small^{12}^{12}12Because when is small, the outage probability will approach to one and the logarithmic scale is a nonlinear scale in terms of the power of ten, which cannot show the details around one.. Then, we set and and plot the relation between connection probability and in Fig. 5. From Fig. 5, it is obvious that when is small (i.e. in sparse networks), the multirelay performance advantage brought by persubcarrier selection over bulk selection will become negligible, because the number of relays for selection is small and employing multiple relays to perform persubcarrier selection becomes less likely. Also, in order to show the properties for small in logarithmic plots, we specify and plot the relation between and to verify the proposed approximation (27) in Fig. 6^{13}^{13}13As the same reason that we define connection probability, when is small, will approach to one and the details cannot be shown by the logarithmic scale based on the power of ten. In Fig. 6, the exact is produced by numerically inversely solving (25) for , while the approximate is produced by calculating , as given in (27).. From this figure, it is verified that the proposed approximation on is tight for small and can thereby provide an efficient metric for evaluating the performance advantage brought by persubcarrier selection over bulk selection in sparse networks.
VC Relation between the number of subcarriers and system throughput
To verity the solutions to the optimal number of subcarriers produced by the concave problem formulated in (29), we set , and and plot the simulation results in Fig. 7. The jagged nature of reflects , as the number of subcarriers can only take discrete values in practice. Meanwhile, from this figure, the solutions produced by the concave problem exactly match the numerical results, which verifies the accuracy of the proposed solution. It also aligns with our expectation that with an increasing , both and the maximum will increase, since the signal propagation environment has been improved with more ‘appropriate’ relays. Also, a higher path loss exponent will significantly degrade the system throughput, which can be observed by comparing the results shown for and . The degradation due to high path loss exponent is obvious especially when is large.
Besides, to verify the constrained solution to the optimal number of subcarriers for a given outage probability , the simulation results are presented in Fig. 8. Here, we specify when is lower than the cutoff density for simplicity and frequency resource saving purposes. Again, both numerical and analytical results match each other and these results verify the feasibility of the proposed solution for the scenario with an outage constraint. Also, in comparison with the results illustrated in Fig. 7, the constraint on outage probability will result in a lower optimal number of subcarriers. Furthermore, by comparing the cases with different , it can be found that the optimal number of subcarriers reduces with the constraint on outage.
Vi Conclusion
In this paper, we analyzed the outage performance of twohop OFDM systems adopting bulk and persubcarrier selection schemes, respectively. Also, a series of important properties related to cooperative systems in random networks were investigated, including diversity, outage probability ratio of two selection schemes and the optimization of the number of subcarriers. All analysis has been verified by simulations and some key properties of cooperative OFDM systems over finite and infinite random networks have been revealed and discussed. Moreover, by (10) and (17), an analytical framework for OFDM systems over random networks has been constructed in this paper, which can be easily modified to analyze other similar cases with different forwarding protocols, location distributions and/or channel conditions.
Appendix A Proof of Diversity Gain in Infinite Poisson Random Networks
To determine the diversity gain in infinite Poisson random networks for bulk selection, we first have the relation
(35) 
where holds because of the trigonometric relation among , and expressed in (4); is given by
(36) 
where is the lower incomplete gamma function and is the exponential integral function.
Appendix B Proof of the Approximation of for Small
First, we propose a lemma below and this is the prerequisite for applying power series expansion on for small :
Lemma 1
is positive and bounded for and .
Proof:
Because and is a CDF and thus satisfies , there exists a relation of the integrand in :
(38) 
For , we can employ mathematical induction as below to prove :
For , and the statement is true. Then for , assuming holds, we can have . Because , we can further obtain
(39) 
Therefore, the statement for is true and we thus prove the statement for . Due to (38) and (basic assumption of multicarrier systems), we thereby prove for and . Meanwhile, because will decrease exponentially with an increasing transmission distance, the area integral of the integrand over the region is positive and bounded. As a result, the lemma is proved.
By Lemma 1, is proved to be expandable for small [39]. we can employ a power series expansion on for small and obtain
(40) 
We can truncate by its second order term and substitute into (25), which yields
(41) 
Furthermore, for and , by the binomial theorem, we can derive
(42) 
For the first order term, we can use the additivity property of integrals and swap the order of summation and integral by
(43) 
Appendix C Proof of the Concavity of in Terms of
According to the expressions of outage probability for bulk and persubcarrier selections given in (10) and (17), it is obvious that is a monotonically deceasing function of . Therefore, , and , we have
(45) 
and
(46) 
As a result, it can be derived that
(47) 
This proves the concavity of in terms of , according to the definition of a concave function [40].
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