Spectrum Sharing Between Cellular and Mobile Ad Hoc Networks: Transmission-Capacity Trade-Off
aa r X i v : . [ c s . I T ] A ug Spectrum Sharing Between Cellular and Mobile Ad HocNetworks: Transmission-Capacity Trade-Off
Kaibin Huang, Vincent K. N. Lau, Yan Chen
Abstract
Spectrum sharing between wireless networks improves the efficiency of spectrum usage, and therebyalleviates spectrum scarcity due to growing demands for wireless broadband access. To improve theusual underutilization of the cellular uplink spectrum, this paper studies spectrum sharing between acellular uplink and a mobile ad hoc networks. These networks access either all frequency sub-channelsor their disjoint sub-sets, called spectrum underlay and spectrum overlay , respectively. Given thesespectrum sharing methods, the capacity trade-off between the coexisting networks is analyzed basedon the transmission capacity of a network with Poisson distributed transmitters. This metric is defined asthe maximum density of transmitters subject to an outage constraint for a given signal-to-interference ratio(SIR). Using tools from stochastic geometry, the transmission-capacity trade-off between the coexistingnetworks is analyzed, where both spectrum overlay and underlay as well as successive interferencecancelation (SIC) are considered. In particular, for small target outage probability, the transmissioncapacities of the coexisting networks are proved to satisfy a linear equation, whose coefficients depend onthe spectrum sharing method and whether SIC is applied. This linear equation shows that spectrum overlayis more efficient than spectrum underlay. Furthermore, this result also provides insight into the effectsof different network parameters on transmission capacities, including link diversity gains, transmissiondistances, and the base station density. In particular, SIC is shown to increase transmission capacitiesof both coexisting networks by a linear factor, which depends on the interference-power threshold forqualifying canceled interferers.
Index Terms
Spatial reuse; wireless networks; Poisson processes; spectrum sharing; interference cancellation
I. I
NTRODUCTION
Despite spectrum scarcity, most licensed spectrum are underutilized according to Federal Communi-cations Commission [1]. In particular, in existing cellular systems based on frequency division duplex(FDD) such as FDD UMTS [2], equal bandwidths are allocated for uplink and downlink transmissions,even though the data traffic for downlink is much heavier than that for uplink [3], [4]. Spectrum sharingbetween wireless networks improves spectrum utilization, and will be a key solution for broadband accessin next-generation wireless networks [5]. This motivates the study in this paper on sharing uplink spectrum
K. Huang, V. K. N. Lau, and Y. Chen are with Department of Electronic and Computer Engineering, Hong Kong Universityof Science and Technology, Hong Kong. Email: [email protected], [email protected], [email protected]. Y. Chen is also affiliatedwith Institute of Information & Communication Engineering, Zhejiang University, Hangzhou, 310027, P.R. China.ctober 25, 2018 coexistingnetworks . A basic question is then how is the trade-off between the capacities of these networks.We provide answers to this question in terms of the transmission capacities of the coexisting networksconsisting of Poisson distributed transmitters. By extending the definition in [6], this metric is definedas the maximum weighted sum of the transmitter densities of the coexisting networks so that all linkswill satisfy an outage probability constraint for a target signal-to-interference ratio (SIR), where theweights depend on the spectrum-sharing method. We derive the transmission-capacity trade-off betweenthe networks for different spectrum-sharing methods. Such results are useful for controlling the sizes ofthe coexisting networks for optimizing uplink spectrum usage.
A. Related Work and Motivation
A spectrum band can be either licensed or unlicensed , where a license gives a network the exclusiveright of spectrum usage. Depending on whether holding a licence, a wireless network is referred to as the primary (e.g. cellular networks) or secondary network (e.g. MANETs). Accessing a licensed band, thetransmitters in a secondary network, called secondary transmitters , must not cause significant interferenceto the receivers in the primary network, called primary receivers . One simple method of sharing a licensedband is to spread the signal energy radiated by each secondary transmitter over the whole band usingspread spectrum techniques [7], suppressing the power spectrum density of the resultant interference tothe primary receivers. This method is called spectrum underlay [1], [5], [8], [9].Another method for sharing licensed spectrum is called spectrum overlay , where secondary transmittersaccess frequency sub-channels unused by nearby primary receivers. Recent research on spectrum overlayhas been focusing on designing cognitive-radio algorithms for secondary transmitters to opportunisticallyaccess the spectrum by exploiting the spatial and temporal traffic dynamic of the primary network [5],[8], [10]. Such algorithms require secondary transmitters to continuously detect and track transmissionopportunities by spectrum sensing, and decide on transmission based on sensing results [11]–[13]. Suchalgorithms are vulnerable to sensing errors, and most important require complicated computation at thesecondary transmitters, which usually have limited computational power. For this reason, we considerthe case where base stations in the cellular (primary) network coordinates spectrum overlay. Thereby adhoc (secondary) transmitters use a simple random access protocol rather than complicated cognitive-radioalgorithms.In unlicensed spectrum such as the industrial, scientific and medical (ISM) bands, all networks haveequal priorities for spectrum access. The networks using unlicensed bands include wireless local areanetworks (WLANs) [14] and wireless personal area networks (WPANs) [15]. Due to mutual interference,the coexistence of networks in the unlicensed bands degrades the networks’ performance as shown byanalysis [16], [17], simulation [18], [19], and measurement [20], [21]. Sharing of unlicensed bands ctober 25, 2018 of two coexisting multi-hop ad hoc networks are shown to follow the optimum scaling laws for anasymptotically large number of network nodes. In [24], [25], the network-capacity trade-off betweencoexisting networks is not analyzed.The transmission capacity is used as the performance metric in this paper [6]. Recently, this metric hasbeen employed for analyzing different types of MANETs with Poisson distributed transmitters and anALOHA-like medium-access-control layer, including spatial diversity [27], opportunistic transmissions[27], distributed scheduling [28], bandwidth partitioning [29], successive interference cancellation (SIC)[30], and spatial interference cancelation [31] in MANETs. B. Contributions and Organization
Our main contributions are summarized as follows. The paper targets a cellular uplink network anda MANET sharing the uplink spectrum using either spectrum overlay or underlay, where uplink users,base stations, and ad hoc transmitters all follow Poisson distributions but with different densities. Eachtransmitter modulates signals using frequency-hopping spread spectrum over the frequency sub-channelsassigned to the corresponding network [7]. First, considering an interference-limited environment, boundson the SIR outage probabilities are derived for spectrum overlay and underlay with and without usingSIC at receivers [30], [32]. Second, for small target outage probability, the transmission-capacities of thecoexisting networks are showed to satisfy a linear equation, whose coefficients depend on the overlaymethod and whether SIC is used. Define the capacity region as the set of feasible combinations oftransmission capacities. Third, for small target outage probability, the capacity region for spectrumunderlay is shown to be no larger than that for spectrum overlay. The former can be enlarged to be identicalto the latter by choosing the transmission-power ratio between the two networks as derived. Finally, wecharacterize the effects of different parameters on transmission capacities of the coexisting networks. Inparticular, depending on whether using spectrum overlay or underlay, the transmission capacity of one orboth networks grows linearly with the increasing base station density, linearly with the increasing spatialdiversity gains raised to a fractional power, inversely with the decreasing distance between an ad hoctransmitter and its intended receiver. Moreover, SIC increases both transmission capacities by a linearfactor that is a function of the interference-power threshold for qualifying canceled interferers.Simulation results are also presented. As observed from these results, the derived bounds on outageprobabilities are tight for different spectrum sharing methods with and without SIC. In particular, the This metric introduced in [26] refers to end-to-end throughput per unit distance of a multi-hope wireless network.ctober 25, 2018
ETWORK M ODEL
A. Network Architecture
The spectrum-sharing cellular and ad hoc networks, referred to simply as coexisting networks , areillustrated in Fig. 1. Following [6], [27], [33], the transmitters in the MANET are modeled as a Poissonpoint process (PPP) on the two-dimensional plane, denoted as ˜Π with the density ˜ λ . Each transmitterin the MANET is associated with a receiver located at a fixed distance denoted as ˜ d . The transmissionpower of transmitters is assumed fixed and denoted as ˜ ρ .For the cellular network, the base stations and uplink users are modeled as two independent homoge-neous PPPs denoted as Ω and Π , respectively. Their corresponding densities are represented by λ b and λ .Let B n , U m , D n,m denote the two-dimensional coordinates of the n th base station, the m th uplink user,and their distance, respectively. Thus, D n,m = | B n − U m | . To enhance the long-term link reliability,each uplink user transmits to the nearest base station. Consequently, the cellular network forms a
Poissontessellation of the two-dimensional plane and each cell is known as a
Voronoi cell [34]. The uplink usersin the cell served by the m th base station, denoted as V m , is given as [34] V m = { U ∈ Π || U − B m | < | U − B | ∀ B ∈ Ω \{ B m } } . (1)Based on their distances from the serving base station, the users in each cell are separated into inner-cell and cell-edge users as follows. Consider the largest disk centered at B m and contained inside the m thVoronoi cell, and represent this disk using D m . Specifically [35] D m = (cid:26) Z ∈ R (cid:12)(cid:12)(cid:12)(cid:12) | Z − B m | ≤ | B − B m | ∀ B ∈ Ω \{ B m } (cid:27) . (2) Consideration of the randomness in ˜ d does not provide little insight. It is straightforward to extend the results in this paperto include the randomness in ˜ d . The operator | X | gives the Euclidean distance between X and the origin if X is two-dimensional coordinates, or the cardinalityof X if X is a set.ctober 25, 2018 m th cell are separated depending onwhether they lie inside or outside the disk. In other words, the sets of inner-cell and cell-edge users are { U | U ∈ V m ∩ D m } and { U | U ∈ V m ∩ D cm } , respectively, where D cm = R / D m . Typically, directlinks between cell-edge users and their serving base stations are severely attenuated by pass loss. As aresult, direct transmissions from these users to base stations are potentially difficult due to the requiredlarge transmission power. Furthermore, such direct transmissions cause strong interference to nearby usersand ad hoc receivers. For these reasons, the uplink transmissions of cell-edge users are assumed to beassisted by relay stations near cell edges [36]. For simplicity, it is assumed that by relay transmission theSIR outage probabilities of the cell-edge users are no larger than those of inner-cell users. Thereby it issufficient to consider only inner-cell users in the analysis. B. Channel and Modulation
The uplink spectrum is divided into M frequency-flat sub-channels by using orthogonal frequencydivision multiplexing (OFDM) [37]. Each of the coexisting networks uses a subset or the full set ofsub-channels, depending on the spectrum sharing methods discussed in Section II-C. In each network,a transmitter modulates signals using frequency-hopping spread spectrum, where signals hope randomlyover all sub-channels assigned to the network [6], [7].Consider the link between a typical user and the serving base station, denoted as U and B , respectively.A typical sub-channel accessed by U consists of path loss and a fading factor denoted by W such that thesignal power received by B is ρW D − α , where ρ is the transmission power and D = | U − B | . Similarly,the interference power from an interferer X to B is P X G X R − αX , where P X ∈ { ρ, ˜ ρ } , R X = | X − B | ,and G X is the fading factor.Similar channel models are used for the ad hoc network. Specifically, the received signal power for atypical receiver, denoted as T , is f W ˜ d − α where f W is the fading factor; the interference power from aninterferer X to T is P X G X R − αX where ˜ R X = | X − T | and G X is the fading factor mentioned earlier. C. Spectrum Sharing Methods
For spectrum overlay, the M sub-channels are divided into two disjoint subsets and assigned to twocoexisting networks. Let K and ˜ K denote the numbers of sub-channels used by the cellular and ad hocnetworks respectively, where K + ˜ K = M . Spectrum overlay requires initialization, where the cellularnetwork communicates to the MANET the indices of the available sub-channels and the allowable nodedensity. One initialization method is to use base stations to broadcast control signals to ad hoc nodes. We assume that different cells use the identical sets of sub-channels. Without this assumption, the users and the ad hocnodes accessing one particular sub-channel are non-homogeneous
PPPs. The analysis of this case is complicated and delegatedto future work.ctober 25, 2018 thinning the PPP of ad hoc transmitters [38]. Moreover, K and ˜ K can be adapted to thetime-varying uplink traffic load, increasing spectrum-usage efficiency at the cost of additional initializationoverhead. Next, for spectrum underlay, both coexisting networks use all M sub-channels. Compared withspectrum overlay, spectrum underlay has less initialization overhead as the cellular network need notinform the ad hoc network the indices of available sub-channels.The transmission capacities of the coexisting networks can be increased by employing SIC at eachbase station and ad hoc receiver for reducing interference. The SIC model is modified from that in [30]for making tractable analysis of fading and network coexistence not considered in [30]. For effectiveSIC, the SIC model in [30] requires the interference power from each targeted interferer to be larger thanthe signal power, and furthermore the average number of canceled interferers is upper bounded. In thispaper, by combining these two SIC constraints, the interference power of each targeted interferer mustexceed a threshold equal to the received signal power multiplied by a factor larger than one, denoted as κ . Increasing κ decreases the average number of canceled interferers and vice versa. Finally, perfect SICis assumed. D. Transmission Capacity
Network transmission capacities of the coexisting networks are defined in terms of outage probabilities[6]. As in [27], the networks are assumed to be interference limited and thus noise is neglected forsimplicity. Consequently, the reliability of received data packets is measured by the SIR. Let
SIR and g SIR represent the SIRs at the typical user U and ad hoc receiver T , respectively. The correct decodingof received data packets requires the SIRs to exceed a threshold θ ≥ , identical for all receivers inthe networks. In other words, the rate of information sent from a transmitter to a receiver is no lessthan log (1 + θ ) assuming Gaussian signaling. To support this information rate with high probability, theoutage probability that SIR and g SIR are below θ must be no larger than a given threshold < ǫ ≪ , i.e. P out ( λ ) := Pr( SIR < θ ) ≤ ǫ, ˜ P out (˜ λ ) := Pr( g SIR < θ ) ≤ ǫ (3)where P out and ˜ P out denote the SIR outage probabilities for the cellular and the ad hoc networks,respectively. The transmission capacities of the cellular and the ad hoc networks, denoted as C and ˜ C respectively, are defined as [6] C ( ǫ ) = (1 − ǫ ) λ ǫ , ˜ C ( ǫ ) = (1 − ǫ )˜ λ ǫ (4)where λ ǫ and ˜ λ ǫ satisfy P out ( λ ǫ ) = ǫ and ˜ P out (˜ λ ǫ ) = ǫ .III. O UTAGE P ROBABILITIES
In this section, the outage probabilities for the coexisting networks are derived for spectrum overlayand underlay with and without SIC. ctober 25, 2018 A. Existing Analytical Approach
The analysis in the subsequent sections adopts an existing approach for analyzing the outage probabilitygiven a Poisson filed of interferers [6], [27], [30], [31], [31]. Based on the network model in Section II-B,the aggregate interference power at a receiver in the networks is known as a power-law shot noise process[39]. Analyzing outage probabilities require deriving the complementary cumulative density function(CCDF) of such a process, which, unfortunately, has no closed-form expression [27], [39]. For thisreason, the existing approach resolves to deriving bounds on the CCDF as summarized below in thecontext of the coexisting networks using spectrum overlay.Without loss of generality, assume that the typical user U accesses the m th sub-channel. Let Π m represent the process of users using this sub-channel. By the Marking Theorem [40], Π m can be shownto be a homogeneous PPP with the density λ/K . Furthermore, the interferer process Π m \{ U } is also ahomogeneous PPP with the same density λ/K according to Slivnyak’s Theorem [38]. Define the processof strong interferers for U conditioned on the link realization { W = w, D = d } as Σ S ( w, d ) = { X ∈ Π m \{ U } | R − αX G X > wd − α θ − } , where each interferer alone guaranteers the outage for U . Moreover,the process of weaker interferers is defined as Σ cS ( w, d ) = (Π m \{ U } ) / Σ cS ( w, d ) . Define the interferencepower of the weak interferers as I cS ( w, d ) := P X ∈ Σ cS ( w,d ) ρG X R − αX . Thus, P out can be written as P out = E (cid:2) Pr (Σ S ( W, D ) = ∅ |
W, D ) Pr (cid:0) I cS ( W, D ) > W D − α θ − | W, D (cid:1)(cid:3) +Pr (Σ S ( W, D ) = ∅ ) . (5)Considering only the strong interferers leads to a lower bound on P out , denoted as P l out P l out := E [Pr (Σ S ( W, D ) = ∅ )] = 1 − E h e − E [ | Σ S ( W,D ) | ] i . Let P l out ( w, d ) represent P l out conditioned on { W = w, D = d } . The upper bound on P out , denoted as P u out , is obtained by bounding the term Pr (cid:0) I cS ( W, D ) > W D − α θ − (cid:1) in (5) using Chebyshev’s inequality Pr (cid:0) I cS ( W, D ) > W D − α θ − (cid:1) ≤ var ( I cS ( W, D )) (cid:0) W D − α θ − − E (cid:2) I cS ( W, D ) (cid:3)(cid:1) , W D − α θ − E (cid:2) I cS ( W, D ) (cid:3) > . (6)Using [27, Theorem 2] obtained following the above approach, the bounds on P out and ˜ P out forspectrum overlay are given in the following lemma. Lemma 1 (Spectrum Overlay):
For the coexisting networks based on spectrum overlay, the bounds onSIR outage probabilities are given as follows.1)
Cellular network : E (cid:20) P l out (cid:18) W, D, λK (cid:19)(cid:21) ≤ P out ( K, λ ) ≤ E (cid:20) P u out (cid:18) W, D, λK (cid:19)(cid:21) (7) Note that the processes Σ S ( w, d ) and Σ cS ( w, d ) are independent as a property of the PPP.ctober 25, 2018 P l out ( w, d, λ ) = 1 − exp (cid:16) − ζλw − δ d (cid:17) (8) P u out ( w, d, λ ) = 1 − ξ ( w, d, λ ) exp (cid:16) − ζλw − δ d (cid:17) (9) ξ ( w, d, λ ) = − δ − δ ζd w − δ λ (cid:16) − δ − δ ζd w − δ λ (cid:17) + , δ − δ ζd w − δ λ < , otherwise (10)and ζ := πθ δ E [ G δ ] .2) MANET : E " P l out f W , ˜ d, ˜ λ e K ! ≤ ˜ P out ( ˜ K, ˜ λ ) ≤ E " P u out f W , ˜ d, ˜ λ e K ! (11)where P l out ( · , · , · ) and P u out ( · , · , · ) are given in (8) and (9), respectively. B. Outage Probabilities: Spectrum Underlay
For spectrum underlay, the SIRs for the coexisting networks can be written as(Cellular)
SIR = ρW D − α ρ P X ∈ Π m \{ U } G X R − αX + ˜ ρ P X ∈ ˜Π m G X R − αX (12)(MANET) g SIR = ˜ ρ f W ˜ d − α ρ P X ∈ Π m G X R − αX + ˜ ρ P X ∈ ˜Π m \{ T } G X R − αX . (13)Using (12), the bounds on P out for the cellular network are derived as follows. The parallel derivationfor the MANET is similar and thus omitted for brevity. For the cellular network, all interferers for U (including ad hoc transmitters and other users) can be grouped into a homogeneous marked PPP [40]defined below, where a mark P X ∈ { ρ, ˜ ρ } is transmission power Υ = n ( X, P X ) (cid:12)(cid:12)(cid:12) X ∈ Π m ∪ ˜Π m \{ U } , P X ∈ { ρ, ˜ ρ } o . (14)The distribution of Υ is given in the following lemma. Lemma 2:
The point process Υ is a homogeneous marked PPP with the density ( λ + ˜ λ ) /M , wherethe marks are i.i.d and have the following distribution function P T = P, w.p. λλ + ˜ λ ˜ P , w.p. λλ + ˜ λ . (15) Proof:
See Appendix A. (cid:3)
Using this lemma, the bounds on P out are derived and given in the following proposition. Proposition 1: [Spectrum Underlay] For the coexisting networks based on spectrum underlay, theoutage probabilities are bounded as follows. ctober 25, 2018
Cellular network : E " P l out W, D, λ + η − δ ˜ λM ! ≤ P out ( λ, ˜ λ ) ≤ E " P u out W, D, λ + η − δ ˜ λM ! (16)2) MANET : E " P l out f W , ˜ d, η δ λ + ˜ λM ! ≤ ˜ P out ( λ, ˜ λ ) ≤ E " P u out f W , ˜ d, η δ λ + ˜ λM ! (17)where η := ρ/ ˜ ρ , and P l out ( · , · , · ) and P u out ( · , · , · ) are defined in Lemma 1. Proof:
See Appendix B. (cid:3)
Proposition 1 shows that the outage probability for each network depends on the transmitter densities ofboth networks. This coupling is due to spectrum underlay and the resultant mutual interference betweenthe coexisting networks. As shown in Section IV, such coupling may result in smaller transmissioncapacities for spectrum underlay than those for spectrum overlay. Moreover, Proposition 1 also showsthat the outage probabilities for spectrum underlay depend on the transmission power ratio η . The effectof η is also characterized in Section IV.Finally, the probability density function (PDF) of D for an inner-cell user is given in the followinglemma, which is required for computing the bounds on P out for different overlay methods. Recall theassumption that the outage probabilities of relay-assisted cell-edge users are no smaller than those ofinner-cell users (cf. Section II-A). Thus, the PDF of D for cell-edge users are unnecessary for ouranalysis. Lemma 3:
The probability density function (PDF) of D for an inner cell user is given as f D ( t ) = − πλ b t Ei ( − πλ b t ) (18)where the exponential integral Ei ( x ) = R x −∞ t − e t dt . Proof:
See Appendix C. (cid:3)
It can be observed from (18) that the key parameter of the PDF of D is the density of base station λ b .Intuitively, increasing the density of base stations reduces the cell sizes and thus D and vice versa. C. Outage Probabilities: Spectrum Sharing with SIC
The SIRs for the coexisting networks employing SIC are obtained as follows. With SIC, the conditionalinterferer processes for the typical user U and ad hoc receiver T , denoted respectively as Σ( w, d ) and e Σ( f W ) , are defined as Σ( w, d ) := (cid:8) X ∈ Π m \{ U } (cid:12)(cid:12) G X R − αX ≤ κwd − α (cid:9) , spectrum overlay { X ∈ Π m ∪ ˜Π m \{ U } (cid:12)(cid:12) P X G X R − αX ≤ κρwd − α } , spectrum underlay e Σ( f W ) := { X ∈ ˜Π m \{ T } | G X R − αX ≤ κ f W ˜ d − α } , spectrum overlay { X ∈ Π m ∪ ˜Π m \{ T } | P X G X R − αX ≤ κ ˜ ρ f W ˜ d − α } , spectrum underlay ctober 25, 2018 κ determines the power threshold for qualifying interferers for SIC (cf. Section II-C).Using the above definitions, the SIRs for the cellular and the ad hoc networks, denoted respectively as SIR and g SIR , can be written as(Spectrum overlay)
SIR ( w, d ) = wd − α P X ∈ Σ (3) G X R − αX , g SIR ( f W ) = f W ˜ d − α P X ∈ e Σ (3) G X R − αX (19)(Spectrum underlay) SIR ( w, d ) = ρwd − α P X ∈ Σ P X G X R − αX , g SIR ( f W ) = ˜ ρ f W ˜ d − α P X ∈ e Σ P X G X R − αX (20)where the distribution of P X is given in Lemma 2.The outage probabilities of the SIRs in (19) and (20) are given in the following proposition. Proposition 2:
For spectrum sharing with SIC, the bounds on outage probabilities P out and ˜ P out canbe modified from their counterparts for the case of no SIC as given in Lemma 1 and Proposition 1 byreplacing the functions ˜ P l out and ˜ P u out with ˆ P l out and ˆ P u out correspondingly, which are given as ˆ P l out ( w, d, λ ) = 1 − exp (cid:16) − χζλd w − δ (cid:17) (21) ˆ P u out ( w, d, λ ) = 1 − ξ ( w, d, λ ) exp (cid:16) − χζλw − δ d (cid:17) (22)where χ := (cid:0) − θ − δ κ − δ (cid:1) and the function ξ ( w, d, λ ) is given in Lemma 1. Proof:
See Appendix D. (cid:3)
Note that (21) and (22) differ from respectively (8) and (9) only by the factor χ . The factor χ < represents the SIC advantage of reducing outage probabilities with respect to the case of no SIC ( χ = 1 ).Moreover, decreasing the SIC factor κ reduces χ and thus outage probabilities. Nevertheless, κ beingtoo small may invalidate the assumption of perfect SIC. Specifically, small κ implies small SIR for theprocess of decoding interference prior to its cancelation and potentially results in significant residualinterference after SIC [32].IV. N ETWORK C APACITY T RADE -O FF : A SYMPTOTIC A NALYSIS
Using the results obtained in the preceding section, the trade-off between the transmission capacitiesof the coexisting networks, namely C and ˜ C as defined in (4), is characterized in the following theoremfor small target outage probability ǫ → . Theorem 1:
For ǫ → , transmission capacities of the coexisting networks satisfy ˜ µ ˜ C + µC = Mϕ ǫ + O (cid:0) ǫ (cid:1) (23)where the weights µ and ˜ µ are given as ˜ µ o = ζ E [ f W − δ ] ˜ d , µ o = ζ E [ W − δ ](8 πλ b ) − , spectrum overlay ˜ µ u = ˜ µ o ∨ ( η − δ µ o ) , µ u = ( η δ ˜ µ o ) ∨ µ o , spectrum underlay (24) The subscripts o and u identify spectrum overlay and underlay, respectivelyctober 25, 2018 ϕ depends on if SIC is used ϕ = 1 , no SIC − θ − δ κ − δ ≤ ϕ ≤ − δ − θ − δ κ − δ , SIC . (25) Proof:
See Appendix D. (cid:3)
Theorem 1 shows that the trade-off between C and ˜ C follows a linear equation. Specifically, the slopeat which ˜ C increases with decreasing C is − µ/ ˜ µ , which depends on different network parameters asobserved from (24). The results in Theorem 1 are interpreted using several corollaries in the sequel.To facilitate discussion, define an outage limited network as one whose transmission capacity is achievedwith the outage constraint being active. For instance, the cellular network is outage limited if C = (1 − ǫ ) λ ǫ with P out ( λ ǫ ) = ǫ . For spectrum overlay, both the coexisting networks are outage limited. Nevertheless,for spectrum underlay, it is likely that only one of the two networks is outage limited as explained shortly.As implied by the proof for Theorem 1, for spectrum underlay, both networks are outage limited onlyif µ u = ˜ µ u , where µ u and ˜ µ u are given in (24). Otherwise, µ u > ˜ µ u correspond to only the cellularnetwork being outage limited; µ u < ˜ µ u indicates that only the MANET is outage limited.Spectrum overlay is shown to be more efficient than spectrum underlay as follows. Define the capacityregion of the coexisting networks as the region enclosed by the capacity trade-off curve in (23) andthe positive axes of the C - ˜ C coordinates. This region contains all feasible combinations of transmissioncapacities of coexisting networks. Thus, the size of the capacity region measures the efficiency of theoverlaid network. The capacity regions for spectrum overlay and underlay are compared in the followingcorollary. Corollary 1:
For ǫ → , the capacity region for spectrum underlay is no larger than that for spectrumoverlay. They are identical if and only if the transmission-power ratio is chosen as η = (cid:18) µ o ˜ µ o (cid:19) δ (26)where µ o and ˜ µ o are given in (24). Proof:
See Appendix F. (cid:3)
Corollary 1 shows that spectrum overlay is potentially more efficient than spectrum underlay due to net-work coupling for the latter. Specifically, the possibility that a network is not outage limited compromisesthe efficiency of spectrum underlay, which, however, can be compensated by setting η as given in (26).This optimal value of η ensures both networks are outage limited for the case of spectrum underlay.The next corollary specifies the effects of several parameters on transmission capacities of the coexistingnetworks. Corollary 2:
For ǫ → , transmission capacities vary with network parameters as follows.1) Spectrum overlay : C increases linearly with the base station density λ b ; ˜ C increases inversely with the ad hoc transmitter-receiver distance ˜ d . ctober 25, 2018 Spectrum underlay : If the cellular network is outage limited, both C and ˜ C increase linearly with the base station density λ b . Otherwise, both C and ˜ C increase inversely with the ad hoctransmitter-receiver distance ˜ d .3) For both spectrum sharing methods, C and ˜ C increase linearly with ǫ and the number of sub-channels M , and inversely with ϕ related to SIC.Finally, we analyze the transmission-capacity gains due to spatial diversity gains contributed by multi-antennas [37]. To obtain concrete results, the fading factors W and f W are assumed to follow the chi-squared distributions with the degrees of freedom L and ˜ L respectively, which are the diversity gains .These fading distributions can result from using spatial diversity techniques such as beamforming overmulti-antenna i.i.d. Rayleigh fading channels [37], [41]. Thus E [ W − δ ] = Γ( L − δ )Γ( L ) , E [ f W − δ ] = Γ( ˜ L − δ )Γ( ˜ L ) . (27)The following corollary is obtained by combining Theorem 1, (27) and the following Kershaw’s Inequal-ities [42] (cid:16) x + s (cid:17) − s < Γ( x + s )Γ( x + 1) < " x −
12 + (cid:18) s + 14 (cid:19) − s , x ≥ , < s < . (28) Corollary 3 (Spatial Diversity Gain):
Consider the diversity gains per link of L and ˜ L for the coex-isting cellular and ad hoc networks, respectively.1) Spectrum overlay : The spatial diversity gains multiply C by a factor between ( L − δ and L δ ,and ˜ C by a factor between ( ˜ L − δ and ˜ L δ .2) Spectrum underlay : The spatial diversity gains multiply both C and ˜ C by a factor between ( L − δ and L δ if the cellular network is outage limited, or otherwise between ( ˜ L − δ and ˜ L δ .Note that similar results are obtained in [43] for a standing-alone MANET by using a more complicatedmethod than the current one based on Kershaw’s Inequalities.V. S IMULATION AND N UMERICAL R ESULTS
In this section, the tightness of the bounds on outage probabilities derived in Section III is evaluatedusing simulation. Moreover, the asymptotic transmission capacity trade-off curves obtained in Theorem 1are compared with the non-asymptotic ones generated by simulation. The simulation procedure summa-rized below is similar to that in [44]. The typical base station (or the ad hoc receiver) of the coexistingnetwork lies at the centers of two overlapping disks, which contain interfering transmitters (either adhoc nodes, users or both) and base stations respectively. Both the transmitters and the base stationsfollow the Poisson distribution with the mean equal to . The disk radiuses are adjusted to provide thedesired densities of transmitters or base stations. For simulations, the distance between the typical ad hoc ctober 25, 2018 d = 5 m, the required SIR θ = 3 or . dB, the path-loss exponent α = 4 , thebase station density λ b = 10 − , the SIC factor κ = 2 dB, and the transmission-power ratio η = 5 dB.Fig. 2 compares the bounds on outage probabilities in Section III and the simulated values. As observedfrom Fig. 2, for all cases, the outage probabilities converge to their lower bounds as the transmitterdensities decrease; the upper and lower bounds differ by approximately constant multiplicative factors.Fig. 2 also shows that SIC reduces outage probabilities by a factor of about . approximately equal to χ in Proposition 2. Moreover, SIC loosens the bounds on outage probabilities for relatively large transmitterdensities since SIC reduces the number of strong interferers to each receiver. Finally, outage probabilitiesbecome proportional to transmitter densities as they decrease.Fig. 3 compares the asymptotic transmission-capacity trade-off curves in Theorem 1 and those generatedby simulations for the target outage probability ǫ = 10 − . In Fig. 3(b) for the case of SIC, the bounds onthe asymptotic trade-off curves correspond to those on ϕ as given Theorem 1. By comparing Fig. 3(a)and Fig. 3(b), the capacity regions for spectrum overlay are larger than those for spectrum underlay. Forthe case of no SIC, the asymptotic results closely match their simulated counterparts. When SIC is used,the capacity trade-off curves generated by simulation are close to the corresponding asymptotic upperbounds. In particular, for spectrum overlay with SIC, the simulation results are practically identical totheir asymptotic upper bounds. In summary, the asymptotic results derived in Section IV are useful forcharacterizing the transmission capacities of the coexisting networks in the non-asymptotic regime.VI. C ONCLUSION
In this paper, the transmission-capacity trade-off between the coexisting cellular and ad hoc networksis analyzed for different spectrum sharing methods. To this end, bounds on outage probabilities for bothnetworks are derived for spectrum overlay and underlay with and without SIC. For small target outageprobability, the transmission capacities of the coexisting networks are shown to satisfy a linear equations,whose coefficients are derived for the cases considered above. These results provide a theoretical basisfor adapting the node density of the ad hoc network to the dynamic of the traffic in cellular uplink underthe outage constraint for both networks. The trade-off relationship suggests that transmission capacitiesof coexisting networks can be increased by adjusting various parameters such as decreasing the distancesbetween intended ad hoc transmitters and receivers, increasing the base station density and link diversitygains, or by employing SIC. In particular, SIC increases the transmission capacities by a linear factorthat depends on the interference power threshold for qualifying canceled interferers. Simulation resultsshow that the derived bounds on outage probabilities are tight and the asymptotic liner capacity trade-offis valid even in the non-asymptotic regime.This paper opens several issues for future work on spectrum sharing between networks including theimpact of cognitive radio, the capacity trade-off between competing networks, and the extension to more ctober 25, 2018
PPENDIX
A. Proof for Lemma 2
By using the superposition property of Poisson processes, the combined PPP Π m ∪ ˜Π m is also ahomogeneous PPP with the density λ +˜ λM . Consider a typical point X ∈ Π m ∪ ˜Π m . Let B ( A, r ) denote adisk centered at a point A ∈ R and with a radius r , thus B ( A, r ) = { X ∈ R || X − A | ≤ r } . Moreover,the area of B ( A, r ) is denoted as A ( B ( A, r )) . Thus the probability for the event that X belongs to Π m ,or equivalently P X = ρ , is Pr( X ∈ Π m ) = lim r → − exp (cid:0) λM A ( B ( X, r )) (cid:1) − exp (cid:16) λ +˜ λM A ( B ( X, r )) (cid:17) = lim r → λ exp( λπr /M )( λ + ˜ λ ) exp(( λ + ˜ λ ) πr /M ) = λλ + ˜ λ . Similarly,
Pr( X ∈ ˜Π m ) = ˜ λλ +˜ λ . This completes the proof. B. Proof for Proposition 1
The marked point process in (14) is modified to include the fading factor G X as an additional markas follows ˆΥ := n ( X, P X , G X ) (cid:12)(cid:12)(cid:12) X ∈ Π m ∪ ˜Π m \{ U } , P X ∈ { ρ, ˜ ρ } , G X ∈ R + o . (29)Following the approach discussed in Section III-A, ˆΥ is divided into a strong-interferer sub-processconditioned on ( W = w, D = d ) , denoted as ˆΥ S ( w, d ) and given as Υ S ( w, d ) = (cid:8) ( X, P X , G X ) (cid:12)(cid:12) ( X, P X , G X ) ∈ Υ ′ , P X | X | − α G X > ρwd − α θ − (cid:9) (30)and the weak-interferer process defined as ˆΥ cS ( w, d ) = ˆΥ / Υ S ( w, d ) . Thus, the sum interference powerfrom weak interferers can be written as I cS ( w, d ) = P ( X,P X ,G X ) ∈ ˆΥ cS ( w,d ) P X | X | − α G X . To apply the ana-lytical procedure in Section III-A, it is sufficient to obtain E [ | ˆΥ S ( w, d ) | ] , E [ I cS ( w, d )] and var [ I cS ( w, d )] .Using the Marking Theorem [40] and Lemma 2, E [ | Υ S ( w, d ) | ] = 2 π ( λ + ˜ λ ) M " Pr ( P X = ρ ) Z ∞ Z ( w − d α θg ) α rf G ( g ) drdg +Pr ( P X = ˜ ρ ) Z ∞ Z ( η − w − d α θg ) α rf G ( g ) drdg = ζw − δ d ( λ + η − δ ˜ λ ) M (31) ctober 25, 2018 ζ is defined in Lemma 1. Next, E [ I cS ( w, d )] and var [ I cS ( w, d )] are derived using Campbell’sTheorem [40] and Lemma 2 as follows E [ I cS ( w, d )] = 2 π ( λ + ˜ λ ) M " Pr ( P X = ρ ) Z ∞ Z ∞ ( w − d α θg ) α ( ρr − α g ) rf G ( g ) drdg +Pr ( P X = ˜ ρ ) Z ∞ Z ∞ ( η − w − d α θg ) α (˜ ρr − α g ) rf G ( g ) drdg = ρδ − δ λ + η − δ ˜ λM ! ζ ( w − d α ) δ − θ − , (32) var [ I cS ( w, d )] = 2 π ( λ + ˜ λ ) M " Pr ( P X = ρ ) Z ∞ Z ∞ ( w − d α θg ) α ( ρr − α g ) rf G ( g ) drdg +Pr ( P X = ˜ ρ ) Z ∞ Z ∞ ( η − w − d α θg ) α (˜ ρr − α g ) rf G ( g ) drdg = ρ δ − δ λ + η − δ ˜ λM ! ζ ( w − d α ) δ − θ − . (33)Combining (31), (32), (33) and the analytical approach in Section III-A gives the desired results. C. Proof for Lemma 3
Let Z denote the largest disk centered at a typical base station B and contained inside the correspondingVoronoi cell. Conditioned on Z = z , the CDF of D of a typical inner-cell user is Pr( D ≤ t | Z = z ) = , t ≥ zt z , otherwise. (34)As a property of the random tessellation, the event ( Z ≤ z ) has the same probability as that where thereis at least one other base station lying with in the distance of z from B [35]. Mathematically Pr( Z ≤ z ) = 1 − e − πλ b z . (35)From (34) and (35) Pr( D ≤ t ) = Z ∞ Pr( D ≤ t | Z ) f Z ( z ) dz = Pr( Z ≤ t ) + Z ∞ t t z × πλ b ze − πλ b z dz = 8 πλ b te − πλ b t + 4 πλ b t Z ∞ πλ b t z − e − z dz. (36)Differentiating the above equation gives the desired result. D. Proof for Proposition 1
Only the bounds on P out are proved. The proof for those on ˜ P out is similar and thus omitted. ctober 25, 2018
1) Spectrum Overlay:
The interferers that are canceled at B using SIC form a process defined as Σ C ( w, d ) := { X ∈ Π m \{ U } | G X D − αX ≥ κwd − α θ − } . Define the process of strong interferersafter SIC as Σ S ( w, d ) := { X ∈ Π m \{ U } | θ − wd − α ≤ G T D − αT ≤ κwd − α } . Note that κwd − α >θ − wd − α since κ > and θ > . Thus, the process of weak interferers can be defined as Σ cS ( w, d ) :=(Π m \{ U } ) / [Σ S ( w, d ) ∪ Σ C ( w, d )] , which is observed to be identical to the counterpart for the caseof no SIC. Since Σ cS ( w, d ) ∩ Σ S ( w, d ) = ∅ , Σ cS ( w, d ) and Σ S ( w, d ) are independent processes. Fromthe discussion in Section III-A, the exponential terms in (8) and (9) depends only on Σ S ( w, d ) , and thefunction ξ ( w, d, λ/K ) only on Σ cS ( w, d ) . Since Σ cS ( w, d ) is invariant to SIC, and Σ cS ( w, d ) and Σ S ( w, d ) are independent, the bounds on P out in Lemma 1 can be extended to the case of SIC by replacingthe exponential term in (8) and (9) with exp( − E [ | Σ S ( w, d ) | ]) , where E [ | Σ S ( w, d ) | ] is obtained usingCampbell’s Theorem E [ | Σ S ( w, d ) | ] = 2 πλ Z ∞ Z ( θw − d α g ) α ( κ − w − d α g ) α rf G ( g ) drdg = χζw − δ d λK (37)and χ is defined in the statement of the proposition.
2) Spectrum Underlay:
With SIC, the strong and weak interferer process for U are defined as b Σ S ( w, d ) := { X ∈ Π m \{ U } | θ − wd − α < P X G X D − αX ≤ κwd − α } and b Σ cS ( w, d ) := { X ∈ Π m \{ U } | P X G X D − αX ≤ θ − wd − α } , respectively, where the distribution of P X is given in Lemma 2. Based onthe same arguments in the preceding section, the bounds on P out in 16 can be extended to the case ofSIC by replacing their exponential terms with exp (cid:16) − E h | b Σ S ( w, d ) | i(cid:17) , where E h | b Σ S ( w, d ) | i is obtainedusing Campbell’s Theorem as follows E h | b Σ S ( w, d ) | i = 2 π ( λ + ˜ λ ) M " Pr ( P X = ρ ) Z ∞ Z ( w − d α θg ) α ( κ − w − d α g ) α rf G ( g ) drdg +Pr ( P X = ˜ ρ ) Z ∞ Z ( η − w − d α θg ) α ( κ − w − d α g ) α rf G ( g ) drdg = χζw − δ d ( λ + η − δ ˜ λ ) M .
E. Proof for Theorem 11) Spectrum Overlay:
The convergence ǫ → implies λ → and ˜ λ → . Using the series represen-tation of the PDF of a power shot-noise process [39], the asymptotes of the outage probabilities followfrom [27, Theorem 2] P out = λζ E h W − δ i E (cid:2) D (cid:3) + O (cid:0) λ (cid:1) , ˜ P out = ˜ λζ E hf W − δ i ˜ d + O (cid:16) ˜ λ (cid:17) . (38)By using (34) and (35), the term E (cid:2) D (cid:3) in (38) is obtained as follows E (cid:2) D (cid:3) = E (cid:20)Z z t f D ( t | Z ) dt (cid:21) = E (cid:20) Z (cid:21) = Z ∞ z × πλ b ze − πλ b z dz = 18 πλ b . (39) ctober 25, 2018
2) Spectrum Underlay:
By using the series expression of the PDF of the power shot noise [39] aswell as Proposition 1, P out ( λ, ˜ λ ) = λ + η − δ ˜ λM ζ E [ W − δ ] E [ D ] + O (max( λ , ˜ λ )) (40) ˜ P out ( λ, ˜ λ ) = η δ λ + ˜ λM ζ E [ f W − δ ] ˜ d + + O (max( λ , ˜ λ )) . (41)For ǫ → , the transmission capacities C and ˜ C satisfy the constraints P out ( C/M, ˜ C/M ) ≤ ǫ and ˜ P out ( C/M, ˜ C/M ) ≤ ǫ . By combining these constraints, (40) and (41) C + η − δ ˜ CM ζ max (cid:16) E [ W − δ ] E [ D ] , η δ E [ f W − δ ] ˜ d (cid:17) = ǫ + O ( ǫ ) . (42)The desired result follows from the above equation.
3) Spectrum Sharing with SIC:
Consider spectrum overlay with SIC. By canceling the strongestinterferers using SIC, the PDF “upper-tail” of the power shot noise process is trimmed and its seriesexpansion is difficult to find [39]. Nevertheless, the asymptotic transmission capacities can be characterizedby expanding the bounds on P out in Proposition 2. Specifically P l out ( λ/K ) = λK ˆ ζ E [ W − δ ] E [ D ] + O ( λ ) P u out ( λ/K ) = 1 − E (cid:20)(cid:18) − δ − δ ζW − δ D λK + O ( λ ) (cid:19) (cid:18) λK ˆ ζW − δ D + O ( λ ) (cid:19)(cid:21) = (cid:18) − δ − θ − δ κ − δ (cid:19) ζ E [ W − δ ] E [ D ] λK + O ( λ ) . (43)Thus P out ( λ/K ) = χζ E [ W − δ ] E [ D ] λK (44)where (cid:0) − θ − δ κ − δ (cid:1) ≤ χ ≤ (cid:16) − δ − θ − δ κ − δ (cid:17) . Similarly ˜ P out (˜ λ/ ˜ K ) = χζ E [ f W − δ ] ˜ d ˜ λ ˜ K . (45)The desired results for spectrum overlay with SIC are obtained by combining (4), (44), and (45). Theresults for spectrum underlay with SIC are derived following a similar procedure.
F. Proof for Corollary 1
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