Uplink achievable rate in underlay random access OFDMbased cognitive radio networks
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Abstract
This paper investigates the uplink achievable rate of secondary users (SUs) in underlay orthogonal frequency division multiplexing based cognitive radio networks, where the SUs randomly access the subcarriers of the primary network. In practice, the primary base stations (PBSs), such as cellular base stations, may not be placed close to each other to mitigate the interferences among them. In this regard, we model the spatial distribution of the PBSs as a βGinibre point process which captures the repulsive placement of the PBSs. It is assumed that in order to alleviate the interferences at the PBSs from the SUs, each SU controls its transmit power based on the average interference level at the closest PBS induced by the SU. We first analytically identify the characteristics of the transmit powers at the SUs. Then, tight approximations of the uplink achievable rate of the secondary network are provided in two different scenarios that assume either a decentralized or centralized allocation of the SUs’ subcarriers, respectively. The accuracy of our analytical results is validated by simulation results.
Keywords
Orthogonal frequency division multiplexing Cognitive radio Random access Underlay network Stochastic geometryAbbreviations
 AWGN
Additive white Gaussian noise
 CDF
Cumulative distribution function
 CRN
Cognitive radio networks
 CSA
Centralized subcarrier allocation
 DSA
Decentralized subcarrier allocation
 GPP
Ginibre point process
 OFDM
Orthogonal frequency division multiplexing
 PBS
Primary base station
Probability density function
 PGFL
Probability generating functional
 PN
Primary network
 PPP
Poisson point process
 PU
Primary user
 SBS
Secondary base station
 SINR
Signaltointerferenceplusnoise ratio
 SN
Secondary network
 SU
Secondary user
1 Introduction
Orthogonal frequency division multiplexing (OFDM)based cognitive radios have been recognized as a promising solution to overcome the radio frequency spectrum scarcity [1]. In cognitive radio networks (CRNs), secondary users (SUs) access the spectrum in general via three different approaches, i.e., interweave, overlay, and underlay, respectively. Among the access models, due to their simplicity, underlay CRNs, where SUs transmit their data without spectrum sensing as long as the interference level at the primary network (PN) remains acceptable, have been widely explored [1, 2, 3].
During the past decade, many researchers have studied underlay OFDMbased CRNs where the locations of nodes in the overlapping networks are deterministic and known [4, 5, 6, 7, 8]. The work [4] investigated the average SU achievable rate when random subcarrier allocation schemes are adopted at the secondary network (SN), and the number of allocated subcarriers for users is fixed. The number of subcarrier collisions between the PN and the SN was analyzed in [5] assuming fixed and random numbers of subcarriers at each user. The authors in [6] proposed a joint subcarrier pairing and power allocation method to maximize the SU’s throughput in relayaided OFDMbased CRNs. The problem of determining the power and subcarrier allocation levels that maximize the average achievable throughput of multiuser OFDMbased CRNs under the constraint of an allowable interference threshold at the PN was examined in [9] and [10].
Assessing the underlay CRN performance by taking into account the spatial distributions of CRN nodes as point processes has attracted much attention recently [11]. In [12], the outage probability of an underlay CRN was established under the assumption that the locations of primary base stations (PBSs) and secondary base stations (SBSs) are modeled as two independent Poisson point processes (PPPs). Also, [13, 14] studied the coverage probability of the uplink transmission of an underlay CRN in which the distributions of PBSs, SBSs, primary users (PUs), and SUs follow independent PPPs. However, to the best of our knowledge, the performance of an underlay OFDMbased CRN, which depends implicitly on the employed subcarrier allocation method, has not been addressed yet.
Due to its mathematical tractability, the PPP whose points are independent has been widely adopted to model various types of wireless networks where nodes are placed in an unplanned fashion [15]. However, when an operator designs a network, transmitters in the network may not be placed in close proximity to each other. This measure is to reduce the interferences among the transmitters and to extend the coverage region, and it incurs a repulsive feature in the networks [16, 17]. The βGinibre point process (GPP) [18] is a repulsive point process which can reflect the repulsive nature and presents the additional benefit of including the PPP as a particular distribution [19]. For this reason, the βGPP has been applied to analyze the coverage probabilities of singletier and multitier downlink cellular networks in [20] and [21], respectively. Despite the fact that PBSs in CRNs may exhibit a repulsive behavior, the performance of an underlay OFDMbased CRN which considers the repulsion has not been investigated yet.
2 Method
This paper examines the performance of an underlay OFDMbased CRN that assumes multiple PBSs, SBSs, PUs, and SUs, and a power control method in the SN to mitigate the interferences at PBSs from the SUs. We study the SU uplink achievable rate under two scenarios: with and without a centralized allocation of subcarriers at the SN, respectively, and which affect differently the collisions among the SUs subcarriers. The PNs can stand for various networks with a repulsive feature such as cellular networks or wireless sensor networks, and hence the locations of the PBSs are modeled as a βGPP. On the other hand, since the SBSs in a SN can represent individual users or devices deployed by multiple different operators/users, the repulsion among the positions of the SBSs is negligible, and thus the spatial distribution of the SBSs is assumed to follow a PPP.

First, we present the expressions of the cumulative distribution function (CDF) and probability density function (PDF) of transmit powers at SUs when the positions of the PBSs are modeled as a βGPP.

Then, we derive approximations of the average of the SU uplink achievable rate for the cases where the SUs subcarriers are allocated in a decentralized or centralized manner at the SN.

Lastly, we explore the impact of subcarrier collisions among the SUs and repulsion among the locations of PBSs on the SU achievable rate.
The organization of this paper is as follows: Section 3 describes the system model of an underlay OFDMbased CRN. The characteristics of the transmit powers at SUs and the SU uplink achievable rate are investigated in Section 4. In Section 5, numerical simulation results are provided to validate the tightness of the derived analytical results. Lastly, conclusions are made in Section 6.
List of notations
Notation  Definition 

Φ _{ P}  βGPP with repulsion parameter β and intensity λ_{P} which represents the locations of PBSs 
Φ _{ S}  PPP with intensity λ_{S} which represents the locations of SBSs 
\(\Phi _{P}^{(u)}(\Phi _{S}^{(u)})\)  PPP with intensity \(\lambda _{P}^{(u)} (\lambda _{S}^{(u)})\) which represents the locations of PUs (SUs) 
N  Number of available subcarriers 
N_{P}(N_{S})  Number of subcarriers for each PU (SU) 
τ  Predefined interference threshold at PBSs 
α_{P}(α_{S})  Path loss exponent for channels between users and PBSs (SBSs) 
P _{ S, x}  Transmit power at the SU at x 
P _{ P}  Transmit power at the PUs 
h_{x,i}(g_{x,i})  Power of smallscale fading channel between the tagged SBS and the SU (PU) at x for the ith subcarrier 
σ ^{2}  Power of additive white Gaussian noise (AWGN) 
γ _{ i}  Signaltointerferenceplusnoise ratio (SINR) for the ith subcarrier 
C  Instantaneous SU achievable rate 
\(\hat {C}\)  Average SU achievable rate 
3 System model
We consider an underlay OFDMbased CRN consisting of a PN and a SN. We will focus on assessing the uplink performance of the SN when each SU (or PU) is associated with the SBS (or PBS) providing the highest average channel gain, i.e., the nearest SBS (or PBS), and transmits its data to the associated SBS (or PBS).
To capture the repulsive behavior among the locations of PBSs, the spatial distribution of the PBSs is modeled as a βGPP Φ_{P} with repulsion parameter β and intensity λ_{P}. Here, the parameter β∈(0,1] determines the degree of repulsion among the positions of the PBSs. More specifically, the PBSs are more evenly distributed when β increases and Φ_{P} converges to a homogeneous PPP with intensity λ_{P} when β→0. The locations of SBSs are assumed to follow a homogeneous PPP Φ_{S} with intensity λ_{S}. Also, we model the distributions of PUs and SUs as a homogeneous PPP \(\Phi _{P}^{(u)}\) with intensity \(\lambda _{P}^{(u)}\) and a homogeneous PPP \(\Phi _{S}^{(u)}\) with intensity \(\lambda _{S}^{(u)}\), respectively. It is presumed that Φ_{P}, Φ_{S}, \(\Phi _{P}^{(u)}\), and \(\Phi _{S}^{(u)}\) are mutually independent.
We assume that there are N available subcarriers, and the number of allocated subcarriers for each PU and each SU is N_{P} and N_{S}, respectively^{1}. In the PN, to alleviate the interferences among the PUs, each PBS allocates each subcarrier to at most one PU which is associated to the PBS so that the PUs communicating with the same PBS do not collide. More specifically, if the number of PUs in the Voronoi cell of a PBS is M_{P}, as each PU requests N_{P} subcarriers, the number of subcarriers requested by the connected PUs is equal to M_{P}N_{P}. When M_{P}N_{P}≤N, the allocations of PUs subcarriers are conducted by maintaining orthogonality among them. On the other hand, if M_{P}N_{P}>N, each of N subcarriers is assigned to a single PU that is chosen in a random fashion.
Two types of subcarrier allocation methods for the SN are considered, i.e., decentralized subcarrier allocation (DSA) and centralized subcarrier allocation (CSA). When the DSA is applied, each SU randomly selects N_{S} subcarriers independently with respect to other SUs and PUs. On the other hand, when the CSA is adopted, each SBS sequentially and randomly assigns orthogonal sets of subcarriers to its associated SUs. More precisely, if the number of SUs in the Voronoi cell of a SBS is equal to M_{S} and M_{S}N_{S}≤N, each SU connected to the SBS utilizes N_{S} subcarriers which are orthogonal with other subcarriers allocated for other SUs in the Voronoi cell. Otherwise, if M_{S}N_{S}>N, each of N subcarriers is allocated to a single SU that is selected in a random fashion [4].
where P_{max} is the maximum transmit power allowed at SUs.
Denote now the set of indices of all subcarriers as \({\mathcal N} = \{ 1, \dots, N \}\). We consider a typical SU and represent the set of indices of the subcarriers selected by the typical SU by \({\mathcal F} = \{ f_{1}, \dots, f_{N_{S}} \}\), where \({\mathcal F} \subset {\mathcal N}\). Since the homogeneous PPP is stationary [22], without loss of generality, we assume that the typical SU is located at the origin o. We term the SBS which is associated with the typical SU as tagged SBS.
where σ^{2} indicates the power of additive white Gaussian noise (AWGN), P_{P} denotes the transmit power at PUs and α_{S} is the path loss exponent for channels between users and SBSs. We define \(\Phi _{S,i}^{(u)}\) and \(\Phi _{P,i}^{(u)}\) as the point processes representing the distributions of the SUs and the PUs employing the subcarrier f_{i}, respectively. Also, h_{x,i} (or g_{x,i}) stands for the power of smallscale fading channel between the tagged SBS and the SU (or PU) at x for the subcarrier f_{i}, which is independent with respect to Φ_{P}, Φ_{S}, \(\Phi _{P}^{(u)}\), and \(\Phi _{S}^{(u)}\). It is assumed that the smallscale fading channels are Rayleigh distributed, and hence {h_{x,i}} and {g_{x,i}} are independent exponential random variables with unit mean.
Remarkably, even when the CSA is adopted in the SN, there exist interferences from other SUs which are connected to other SBSs, i.e., I_{S,i} in (4). In addition, since both the CSA and the DSA do not consider the subcarrier allocation in the PN, signals from PUs may collide with the signal transmitted at the typical SU, and this collision is quantified by I_{P,i} in (5).
In the following section, we will derive approximations of the average of C in (6) and confirm the tightness of the approximations in Section 5.
4 Performance analysis
4.1 Characteristics of transmit powers at SUs
From [20, Prop. 1], for the βGPP \(\Phi _{P} = \{ {\mathbf {y}}_{k} \}_{k \in {\mathbb N}}\), the set \(\left \{ \ {\mathbf {y}}_{k} \^{2} \right \}_{k \in {\mathbb N}}\) follows the same distribution as the set Ξ_{P} constructed from a sequence \(B_{P,i} \sim {\mathcal G}\left (i,\beta /(\pi \lambda _{P}) \right)\) of independent random variables by deleting each B_{P,i} independently and with probability 1−β, where \({\mathcal G}(a,b)\) denotes a gamma random variable with shape parameter a and scale parameter b. As the βGPP is stationary [18], the characteristics of P_{S,x} are invariant to the location x. Thus, for brevity of presentation, we omit the subscript x. We introduce the CDF and PDF of transmit power P_{S} in (1) in the following lemma.
Lemma 1
where \(a_{k}(x) \triangleq 1  \frac {\beta }{\Gamma (k)} \gamma \left (k, \pi \lambda _{P} x^{2/\alpha _{P}}/\left (\beta \tau ^{2/\alpha _{P}}\right) \right)\).
Proof
See See Appendix A. □
where \(\varphi = \prod _{k=1}^{\infty }\! \left (1 \,\, \frac {\beta }{\Gamma (k)} \gamma \left (k, \frac {\pi \lambda _{P} P_{{\text {max}}}^{2/\alpha _{P}}}{ \beta \tau ^{2/\alpha _{P}}} \right) \! \right)\) and \({\mathbbm 1}(\cdot)\) is the indicator function.
Each PU may adjust its transmit power based on the distance between the PU and its associated PBS. In this case, the CDF and PDF of transmit powers at PUs can be derived in the same manner as in Lemma 1. Then, the corresponding SU uplink achievable rate can be identified by utilizing the obtained PDF of the transmit powers at the PUs. The goal of this paper is to investigate the impacts of the power control scheme at SUs and the subcarrier allocation methods at the SN on the SU uplink achievable rate. Therefore, we have not considered the power control at the PUs in this paper, and more sophisticated performance analysis for the networks with power controls at both SUs and PUs is left for future work.
4.2 Analysis
where (a) follows from the fact that the CDF of an exponential random variable X with mean ρ is \({\mathbb P}(X< x) = 1  \exp (x / \rho)\), f_{r}(r)=2πλ_{S}r exp(−πλ_{S}r^{2}) is the PDF of the distance from the typical SU and the tagged SBS [22], \(f_{P_{S}}(p)\) is the PDF of the transmit powers at SUs in (8) or (10), and \({\mathcal L}_{X}(t) \triangleq {\mathbb E}\left [ \exp (tX)\right ]\) denotes the Laplace transform of a random variable X.
However, since the points in \(\Phi _{P,i}^{(u)}\) (or \(\Phi _{S,i}^{(u)}\)) are relevant to the Voronoi tessellation induced by Φ_{P} (or Φ_{S}), the points are correlated, and hence it is intractable to identify the exact distributions of \(\Phi _{S,i}^{(u)}\) and \(\Phi _{P,i}^{(u)}\). To circumvent this difficulty, we approximate \(\Phi _{P,i}^{(u)}\) and \(\Phi _{S,i}^{(u)}\) as homogeneous PPPs with intensities \(\tilde {\lambda }_{P}^{(u)}\) and \(\tilde {\lambda }_{\mathrm {S,CSA}}^{(u)} \left (\text {or}\ \tilde {\lambda }_{\mathrm {S,CSA}}^{(u)}\right)\), respectively.
where (b) follows from the independence between Φ_{P,i} and g_{x,i}, (c) comes from the fact that the Laplace transform of an exponential random variable X with mean ρ is \({\mathcal L}_{X}(t) = 1/(1+\rho t)\), and (d) is due to the PPP probability generating functional (PGFL) [22].
We would like to mention that, in the previous works in [23, 24, 25], each transmitter adjusts its transmit power based on the distance between the transmitter and its intended receiver. Since both transmit power and desired signal term are determined by the distance between a transmitter and its receiver, it is tractable to characterize the performance of the networks in [23, 24, 25]. On the other hand, in underlay cognitive radio networks, transmit power at a SU is computed based on the distance between the SU and its nearest PBS, and the desired signal term is relevant to the distance between the SU and its closest SBS. Therefore, analyzing the performance of underlay cognitive radio networks is more challenging than those of the networks in [23, 24, 25].
In the following lemma, we derive an approximation of \({\mathcal L}_{I_{S,i}}(t)\) in the case with the CSA where the interfering SUs are located outside of the Voronoi cell of the tagged SBS.
Lemma 2
where \(\eta \triangleq \int _{0}^{\infty } q^{2/\alpha _{S}} f_{P_{S}}(q) d q\) and \(f_{P_{S}}(q)\) is the PDF of transmit power P_{S} in (8) or (10).
Proof
See See Appendix B. □
Finally, when the CSA (or DSA) is adopted, we can evaluate \(\hat {C}\) in (11) by plugging (12), (14), (15) (or (16)) into (11). From the derived analytical expressions, we can expect that \(\hat {C}\) is an increasing function of the number of available subcarriers N since an increase in N leads to decreases in the intensities of interfering users, i.e., \(\tilde {\lambda }_{\mathrm {S,CSA}}^{(u)}\), \(\tilde {\lambda }_{\mathrm {S,CSA}}^{(u)}\) and \(\tilde {\lambda }_{P}^{(u)}\). When N→∞, the Laplace transforms \({\mathcal L}_{I_{P,i}}\) and \({\mathcal L}_{I_{S,i}}\) in (12) become one since \(\tilde {\lambda }_{P}^{(u)}\), \(\tilde {\lambda }_{\mathrm {S,CSA}}^{(u)}\), and \(\tilde {\lambda }_{\mathrm {S,CSA}}^{(u)}\) converge to zero. In this case, if N_{S}/N→0, an increase of N_{S} leads to an enhanced total average achievable rate \(\hat {C}\) as the number of summations in (11) gets larger as N_{S} grows. Also, from the definitions of \(\tilde {\lambda }_{\mathrm {S,CSA}}^{(u)}\), \(\tilde {\lambda }_{\mathrm {S,CSA}}^{(u)}\), and \(\tilde {\lambda }_{P}^{(u)}\), we can infer that \({\mathbb P} \left (\gamma _{i} \geq u \right)\) in (12) decays as N_{S} and N_{P} get larger.
5 Simulation results
6 Conclusions
This paper studied an underlay OFDMbased CRN where the locations of PBSs are modeled as a βGPP. The scenario where the SUs control their transmit powers to alleviate the inferences at the PBSs is considered. Also, two types of subcarrier allocation techniques are considered depending on whether the subcarriers for the SN are allocated in a centralized manner or not. First, we have identified the characteristics of the transmit powers at SUs. Then, we have derived approximations of the total average SU achievable rate and verified the tightness of the approximations via numerical simulations. In addition, from simulations, we have observed that the transmit power at SUs and the SU achievable rate become smaller as the degree of repulsion among the PBS increases, and that the SU achievable rate is enhanced when the centralized subcarrier allocation method is applied at the SN.
7 Appendix A Proof of Lemma 1
where (e) follows from the fundamental property of the βGPP and (f) comes from the fact that the CDF of a gamma random variable \(X \sim {\mathcal G}(a,b)\) is \({\mathbb P}(X < x) = \gamma (a,x/b)/\Gamma (a)\).
where \(a_{k}(x) \triangleq 1  \frac {\beta }{\Gamma (k)} \gamma \left (k, \pi \lambda _{P} x^{2/\alpha _{P}}/(\beta \tau ^{2/\alpha _{P}}) \right)\).
8 Appendix B Proof of Lemma 2
Our approximation (18) only takes into account the interfering signals whose powers are larger than the power of the desired signal, and thus the approximation can be interpreted as a lowerbound of I_{S,i}.
where (g), (h), and (i) follow from the approximation in (18), the stationarity of the homogeneous PPP and the assumption that the transmit powers {P_{S,x}} are independent, respectively. Also, (j) and (k) are due respectively to the PGFL of the homogeneous PPP and the variable change \(\phantom {\dot {i}\!}y= x(tq)^{1/\alpha _{S}}\), where \(\eta \triangleq \int _{0}^{\infty } q^{2/\alpha _{S}} f_{P_{S}}(q) d q\).
Footnotes
 1.
In this paper, nonrandom N_{P} and N_{S} are considered for analytical tractability. An extension to networks with randomly varying N_{P} and N_{S} is outside the scope of this paper and represents an interesting problem.
Notes
Acknowledgements
This work was supported by NSF EARS Award No. 1547447.
Funding
This work was supported by NSF EARS Award No. 1547447.
Authors’ contributions
ES and KQ proposed the conception of subcarrier allocation methods of the study. HBK and SE contributed to the analysis and simulation. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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