Spectrum_Sharing Between Cellular and Mobile Ad Hoc

8/4/2019 Spectrum_Sharing Between Cellular and Mobile Ad Hoc

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1256 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 7, SEPTEMBER 2009

Spectrum Sharing between Cellular and Mobile AdHoc Networks: Transmission-Capacity Trade-Off

Kaibin Huang, Vincent K. N. Lau, and Yan Chen

AbstractSpectrum sharing between wireless networks im-proves the efficiency of spectrum usage, and thereby alleviatesspectrum scarcity due to growing demands for wireless broad-band access. To improve the usual underutilization of the cellularuplink spectrum, this paper addresses spectrum sharing betweena cellular uplink and a mobile ad hoc networks. These networksaccess either all frequency subchannels or their disjoint subsets,called spectrum underlay and spectrum overlay, respectively. Giventhese spectrum sharing methods, the capacity trade-off betweenthe coexisting networks is analyzed based on the transmissioncapacity of a network with Poisson distributed transmitters. Thismetric is defined as the 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 coexisting networks is analyzed,where both spectrum overlay and underlay as well as successiveinterference cancelation (SIC) are considered. In particular, forsmall target outage probability, the transmission capacities ofthe coexisting networks are proved to satisfy a linear equation,whose coefficients depend on the spectrum sharing method andwhether SIC is applied. This linear equation shows that spectrumoverlay is more efficient than spectrum underlay. Furthermore,this result also provides insight into the effects of networkparameters on transmission capacities, including link diversitygains, transmission distances, and the base station density. Inparticular, SIC is shown to increase the transmission capacities ofboth coexisting networks by a linear factor, which depends on theinterference-power threshold for qualifying canceled interferers.

Index TermsSpatial reuse; wireless networks; Poisson pro-cesses; spectrum sharing; interference cancelation

I. INTRODUCTION

DESPITE spectrum scarcity, most licensed spectrums areunderutilized according to the Federal CommunicationsCommission [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 down-

link is much heavier than that for uplink [3], [4]. Spectrumsharing between wireless networks improves spectrum utiliza-

tion, and will be a key solution for broadband access in next-

generation wireless networks [5]. This motivates the study in

this paper on sharing the uplink spectrum between a cellular

network and a mobile ad hoc network (MANET), called the

Manuscript received 30 August 2008; revised 31 January 2009.K. Huang is with the School of Electrical and Electronic Engineering,

Yonsei University, 262 Seongsanno, Seodaemun-gu Seoul 120-749, Korea (e-mail: [email protected]).

V. K. N. Lau is with the Department of Electronic and Computer Engi-neering, Hong Kong University of Science and Technology, Clear Water Bay,Hong Kong (e-mail: [email protected]).

Y. Chen is with Huawei Technology, Pudong, Shanghai, China (e-mail:[email protected]).

Digital Object Identifier 10.1109/JSAC.2009.090921.

coexisting networks. A basic question is what is the trade-off

between the capacities of these networks?

We provide answers to this question using the metric of

transmission capacity [6], [7]. By extending the definition

in [6], the transmission capacities of the coexisting networks

are defined as their maximum transmitter densities under an

outage probability constraint for a target signal-to-interference

ratio (SIR). We derive the transmission-capacity trade-off

between the networks for different spectrum-sharing methods.

Such results are useful for controlling the sizes of the coex-

isting networks for optimizing uplink spectrum usage.

A. Related Works and Motivation

A spectrum band can be either licensedor unlicensed, where

a licence gives a network the exclusive right 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, the secondary transmitters must not cause significant

interference to the primary receivers. One simple method

of sharing a licensed band is to spread the signal energy

radiated by each secondary transmitter over the whole band

using spread spectrum techniques [8], suppressing the power

spectrum density of the resultant interference to the primary

receivers. This method is called spectrum underlay [1], [5],

[9], [10].

Another method for sharing a licensed spectrum is called

spectrum overlay, where secondary transmitters access fre-

quency subchannels unused by nearby primary receivers.

Recent research on spectrum overlay has been focusing on

designing cognitive-radio algorithms for secondary transmit-

ters to opportunistically access the spectrum [5], [9]. Such

algorithms require secondary transmitters to continuously de-

tect and track transmission opportunities by spectrum sensing,

and decide on transmission based on sensing results [11], [12].

These operations are vulnerable to sensing errors, and most

important require complicated computation at the secondary

transmitters. For this reason, we consider the case where base

stations in the cellular (primary) network coordinates spectrum

sharing. Thereby ad hoc (secondary) transmitters use a simple

random access protocol rather than complicated cognitive-

radio algorithms.

In unlicensed spectrums such as the industrial, scientific

and medical (ISM) bands, all networks have equal priorities

for spectrum access. The networks using unlicensed bands

include wireless local area networks (WLANs) and wireless

personal area networks (WPANs). Due to mutual interference,

the coexistence of networks in unlicensed bands degrades the

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HUANG et al.: SPECTRUM SHARING BETWEEN CELLULAR AND MOBILE AD HOC NETWORKS: TRANSMISSION-CAPACITY TRADEOFF 1257

networks performance as shown by analysis [13], [14], sim-

ulation [15], [16], and measurement [17]. Sharing unlicensed

bands between competing networks is also studied using game

theory [18].

There exist few theoretical results on the network capacity

trade-off between the coexisting networks despite this being

a fundamental issue. In [19], the transmission capacities of

a two-tier network are analyzed, which comprises a cellularnetwork and a network of femtocell hot-spots. In [20], the

transport capacities1 of two coexisting multi-hop MANETs are

shown to follow the optimum scaling laws for asymptotically

large network sizes. In [19], [20], the network-capacity trade-

off between coexisting networks is not analyzed.

The metric of transmission capacity has been applied to de-

signing MANETs with Poisson distributed transmitters and an

ALOHA-like medium-access-control layer, addressing issues

including spatial diversity [22], opportunistic transmission [7],

bandwidth partitioning [23], successive interference cance-

lation (SIC) [24], and spatial interference cancelation [25].

Besides the transmission capacity, a new performance metriccalled capacity region is used in this paper, which is tailored

for spectrum sharing. The capacity region of the coexisting

networks is defined as the set of feasible combinations of

transmitter densities under the SIR outage constraints. By

definition, transmission capacities of the coexisting networks

give the boundary points of the capacity region. Both the

transmission capacities and capacity region of the coexisting

networks are analyzed in this paper.

B. Contributions and Organization

This paper targets a cellular uplink network and a 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. Each transmitter modulates signals using frequency-

hopping spread spectrum over the frequency subchannels

assigned to the corresponding network [8]. This modulation

decreases the density of transmitters accessing the same sub-

channel. As a result, the inter-link interference reduces and

the transmission capacities increase.

Our main contributions are summarized as follows. First,

considering an interference-limited environment, bounds on

the SIR outage probabilities are derived for spectrum over-

lay and underlay with and without using SIC at receivers[24], [26]. Second, for small target outage probability, the

transmission-capacities of the coexisting networks are showed

to satisfy a linear equation, whose coefficients depend on the

spectrum-sharing method and whether SIC is used. Given the

above linear capacity trade-off, the capacity region of the co-

existing networks is a triangular. Third, for small target outage

probability, the capacity region for spectrum underlay is shown

to be no larger than that for spectrum overlay. The former can

be enlarged to be identical to the latter by choosing the derived

transmission-power ratio between the two networks. Finally,

we characterize the effects of different parameters on network

transmission capacities. In particular, for spectrum overlay,

1This metric introduced in [21] refers to end-to-end throughput per unitdistance of a multi-hop wireless network.

the transmission capacity C of the cellular network increaseslinearly with the base station density b, and the capacity Cof the MANET grows inversely with the decreasing distance dbetween a pair of communicating ad hoc nodes. For spectrum

underlay, C and C both increase linearly with b or otherwiseinversely with decreasing d. Moreover, C and C both growsub-linearly with spatial diversity gains. Furthermore, SIC

increases both C and C by a linear factor that is a functionof the interference-power threshold for qualifying canceled

interferers.

To distinguish our contributions from exiting results, several

new challenges addressed in the current work are emphasized.

First, the current analysis of the network outage probabilities

requires characterizing the distribution of sum interference

power from two classes of interferers from separate networks,

which differ in properties such as transmission power and

densities. This issue does not exist in prior works focusing

on stand-alone networks. Second, due to spectrum sharing, the

transmission capacities of the coexisting networks must satisfy

simultaneously two SIR outage constraints rather than onewhen a stand-alone network is considered. Consequently, the

current transmission-capacity analysis must account for new

issues including heterogeneous network architectures and sizes

as well as resource allocation cross networks. These issues

complicate the relation between the transmission capacities of

the coexisting networks. Whether a simple relation exists is

unknown from prior works. Finally, we generalize the study

of SIC in [24] for a MANET with path-loss to the coexisting

networks with both fading and path-loss.

The remainder of this paper is organized as follows. The

mathematical models are described in Section II. The back-

ground is provided in Section III. In Section IV, the bounds onoutage probabilities are derived for different spectrum sharing

methods. For small target outage probability, the transmission-

capacity trade-off is analyzed in Section V. Numerical results

are presented in Section VI, followed by concluding remarks

in Section VII.

II . MATHEMATICAL MODELS

A. Network Models

The coexisting networks are illustrated in Fig. 1. Their

models are described as follows.

Following [6], [7], [27], we make the following assumption.

Assumption 1. The transmitters in the MANET form a Pois-

son point process (PPP) on the two-dimensional plane, which

is denoted as with the density .

This Poisson assumption as well as the similar one in Assump-

tion 2 are made for mathematical tractability. However, they

are supported by experiments for networks with mobility. In

particular, network models based on similar assumptions havebeen employed by cellular service providers to successfully

predict network traffic load and blocking probabilities [28].

Next, each transmitter in the MANET is associated with a



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Base station

Uplink user

Ad hoc node

Fig. 1. The coexisting cellular and ad hoc networks

receiver located at afi

xed distanced.

2

The transmission powerof transmitters is assumed fixed and denoted as P.The cellular network is modeled based on the following

assumption

Assumption 2. The base stations and uplink mobiles form

two independent stationary PPPs denoted as and , re-spectively.

Their corresponding densities are represented by b and .Let Bn, Um, Dn,m denote the two-dimensional coordinatesof the nth base station, the mth uplink mobile, and theirdistance, respectively. Thus, Dn,m = |Bn Um|.3 To en-

hance the long-term link reliability, each mobile transmits tothe nearest base station. Consequently, the cellular network

forms a Poisson tessellation of the two-dimensional plane

and each cell is known as a Voronoi cell [29]. The mo-

biles served by the mth base station form the set Vm :={U ||U Bm| < |U B| B \{Bm} }.

A typical point of a PPP is defined as a point selected

using the procedure where every point of the PPP has the

same probability of being selected [29]. The typical points

of the PPPs , and are referred to as the typical basestation B0, the typical mobile U0, and the typical ad hoctransmitter T0, respectively. Moreover, the intended receiver

for T0 is called the typical ad hoc receiver and denoted as R0.Finally, represent the random distance between B0 and U0 asD := |B0 U0|.

B. Channel and Modulation

The uplink spectrum is divided into M frequency-flat sub-channels by using orthogonal frequency division multiplexing

(OFDM) [30]. Each of the coexisting networks uses a subset

or the full set of subchannels, depending on the spectrum

2Consideration of the randomness in d provides little new insight. It isstraightforward to extend the results in this paper to include the randomness

ind

. Specifi

cally, ifd

is random, Proposition 1-4 still apply except that theexpectation operations in Proposition 1-2 must account for the randomnessof d. Furthermore, d in Theorem 1 should be replaced with E[d].

3The operator |X| gives the Euclidean distance between X and the originifX is two-dimensional coordinates, or the cardinality of X ifX is a set.

sharing methods discussed in Section II-C. In each network, a

transmitter modulates signals using frequency-hopping spread

spectrum, where signals hop randomly over all subchannels

assigned to the affiliated network [6], [8].

Consider the link between U0 and B0. The typical subchan-nel accessed by U0 and B0 comprises path loss and a fadingfactor denoted by W such that the signal power received by B0

is P W D

, where P is the transmission power. Similarly, theinterference power from an interferer X to B0 is PXGXR

X ,

where PX {P, P}, and RX = |X B0|, and GX is thefading factor.

Similar channel models are used for the MANET. Specifi-

cally, the received signal power for T0 is PWd where Wis the fading factor; the interference power from an interferer

X to T0 is PXGXRX where RX = |X T0|.

C. Spectrum Sharing Methods

1) Spectrum Overlay: For spectrum overlay, the M sub-channels are divided into two disjoint subsets and assigned to

two coexisting networks. Let A and A denote the index setsof the subchannels assigned to the cellular network and the

MANET, respectively. Their cardinalities are represented by

K := |A| and K := |A|, where K+ K = M.Spectrum overlay requires initialization where base stations

broadcast to ad hoc nodes the indices of the available sub-

channels and the allowable transmitting node density. This

density can be controlled by distributed adjustments of nodes

transmission probability [7]. In practice, K and K can beadapted to the time-varying uplink traffic load to increase the

spectrum-usage efficiency.

2) Spectrum Underlay: For spectrum underlay, both coex-

isting networks use all M subchannels. Therefore the trans-mitters accessing the same subchannel comprises both mobiles

and ad hoc nodes. Consequently, spectrum overlaid networks

are coupled rather than decoupled as for the case of spectrum

overlay.

D. Successive Interference Cancelation

SIC decodes multiuser signals and subtracts them from the

aggregate received signal sequentially [30]. This technique is

effective for mitigating interference in the MANET given the

disparity in received power of multiuser signals due to random

node positions as well as path loss and fading [24]. The SICdecoding of a users signal treats the signals of undecoded

users as noise. Therefore, for users with equal data rates, it is

optimal to decode multiuser signals in the descending order

of their power [31]. The above decoding order implies that

canceled interferers signals are stronger than the desired one.

This motivates the current simplified SIC model based on the

following assumption.

Assumption 3. The interference power of each interferer

targeted for cancelation must exceed a threshold equal to the

desired signal power multiplied by a factor > 1.

This assumption is made for mathematical tractability. Moreaccurate SIC models must account for cancelation errors and

the decoding order [26], [30], [32]. Increasing decreases theaverage number of canceled interferers and vice versa. This



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model is very similar to that in [24] where only path-loss is

considered and interferers within a fixed distance are canceled.

III. BACKGROUND

In this section, the shot noise process is introduced for

modeling interference in the coexisting networks. Existing

results on the shot-noise distribution are discussed, which are

useful for analysis in the sequel. Finally, the transmission

capacity and capacity region are defined.

A. Shot Noise Process

A general shot noise process refers to a functional resulting

from feeding a memoryless and linear filter with impulses

derived from a stationary PPP [33]. Such processes have been

widely applied in modeling spatial interference in wireless

networks (see e.g. [7], [34]). Consider a wireless network

where transmitters form a stationary PPP with the density and narrow-band channels comprise path-loss and fading. For

a receiver A0 located at the origin, the aggregate interferencepower I is a shot noise process given as

I =

X\{F0}P QX |X|

(1)

where F0 denotes the transmitter for A0 and {QX} arei.i.d. fading factors. It is important to note that the pro-

cess \{F0} is identically distributed as accordingly toSlyvnyaks theorem [29]. Analyzing the outage probability

for the transmission from F0 to A0 requires deriving thecomplementary cumulative density function (CCDF) of I.Unfortunately, this function has no closed-form expression [7],

[33].

The CCDF of I can be bounded using the approach in[6], [7]. This approach separates the interferers in \{F0}into strong and weak interferers. Define the process of strong

interferers as S = {X \{F0} | P QX |X| > t}, where

each interferer alone guaranteers the outage I > t. It followsthat the process of weaker interferers is cS := (\{F0})\S.Using these definitions

Pr(I > t) = Pr(S = ) P r (IcS > t) + Pr(S = ) (2)

where IcS :=

XcS

P QX |X| represents the interferencepower from weak interferers. From (2), a lower bound on the

outage probability is given as

Pr(I > t) > Pr(S = ) = 1 eS (3)

where S := E[|S |] is the average number of strong interfer-ers. The upper bound on Pr(I > t) is obtained by boundingthe term Pr (IcS > t) in (2) using Chebyshevs inequality

Pr (IcS > t) var(IcS)

(t E [IcS ])2 , t > E [I

cS ] (4)

where the right-hand-side of the first inequality is Chebychevs

upper bound. Closed-form expressions for S , var(IcS) and

E [IcS ] can be obtained using Campbells theorem [7], [29].

By substituting the resultant expressions into (3) and (4), thebounds on Pr(I > t) are obtained in [7, Theorem 2] andshown in the following lemma.

Lemma 1. The CCDF of the shot noise process in (1) is

bounded as

Pout

(t, ) < P r(I > t) < Puout

(t, )

where Pout

(a, b) = 1 exp

ab

, Puout

(a, b) = 1 (a, b)exp

ab

, and

(a, b) =

1 2 ab

1 1 ab

2+

,

1 ab < 1

0, otherwise

(5)

with := E[Q].

B. Transmission Capacity

We define two related performance metrics for the coexist-

ing networks. As in [7], the networks are assumed to be in-

terference limited and noise is neglected for simplicity. Hence

the SIR measures the reliability of received data packets. LetSIR and SIR represent the SIRs at U0 and R0, respectively.The correct decoding of received packets requires the SIRs

to exceed a threshold . To satisfy this constraint with highprobability, the following outage constraints are applied for

given 0 < 1

Pout(, ) := Pr(SIR(, ) < )

Pout(, ) := Pr(SIR(, ) < ) (6)where the functions Pout and Pout map the SIRs to the outageprobabilities. The transmission capacities C of the cellularnetwork and C of the MANET are defined as the maximum

transmitter densities under the outage constraints in (6). Thisdefinition differs slightly from that in [6], [7] by a linear factor

(1 ), which has a negligible effect on the analysis.Next, we define the capacity region R as the set comprising

all combinations (, ) that satisfy the outage constraints in(6)

R =

(, ) R2 | Pout(, ) , Pout(, )

. (7)

As mentioned earlier, transmission capacities specify the

boundary points of R.

IV. NETWORK OUTAGE PROBABILITIESIn this section, the outage probabilities for the coexisting

networks are derived for spectrum overlay and underlay with

and without SIC.

For the notation used in this section, the superscripts (a)-(d) identify four cases: (a) spectrum overlay, (b) spectrumunderlay, (c) spectrum overlay with SIC and (d) spectrumunderlay with SIC.

A. Outage Probabilities: Spectrum Overlay

For spectrum overlay, the SIRs for the coexisting networks

are obtained as follows. Let

(a)

denote the process ofmobile transmitters accessing the th subchannel. Similarly,(a)m represents the process of ad hoc nodes transmitting overthe mth subchannel. Since spectrum overlay decouples the



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networks, mobile receivers are interfered with only by unin-

tended mobile transmitters, and ad hoc receivers by unintended

ad hoc transmitters. Without loss of generality, consider the

time slot where the receivers B0 and R0 decode signalsfrom the th and the mth subchannels, respectively, where A and m A. The interference power for B0 can bewritten as I

(a) := PX(a) GXR

X and that for R0 as

I(a)m := PXe(a)m GXRX . It follows that the SIRs aregiven as

Celluar : SIR(a) = P W D

/I(a)

MANET : SIR(a)m = PWd/I(a)m . (8)Given frequency-hopping spread spectrum, multiple sub-

channels assigned to each network contributes a processing

gain for reducing the density of transmitters accessing the

same subchannel. Specifically, the densities of (a) and

(a)mare shown to be inversely proportional to K and K, respec-tively, as follows. Define the mark MX of a mobile transmitterX as the index of the subchannel X accesses in the currenttime slot. It follows from the marking theorem [35] that

(a)

is a stationary PPP with the density Pr(MX = ) = /K,

which is a thinned process generated by (a). Similarly, (a)mis a stationary PPP with the density /K.

Using the above results and (8), the bounds on the out-

age probabilities for spectrum overlay are obtained using

Lemma 1. Note that the distribution ofSIR(a) is independent

of . Thus the outage probability for the cellular network is

P(a)out

= Pr

SIR

(a) <

= E PrI() > W D1 | W, D . (9)Recognize that I

(a) is a shot noise process (see Section III-A).

Thus from Lemma 1 and (9), P(a)out

is bounded as shown in

the following proposition. Similarly, we obtain the bounds

on the outage probability P(a)out

for the MANET as given in

Proposition 1.

Proposition 1. For spectrum overlay, the outage probabilities

are bounded as

Cellular : E PloutW D

,

K P(a)out

(K, )

E

Puout

W D,

K

MANET : E

Plout

Wd, K

P(a)out

(K, ) E

Puout

Wd, K

where Pout

(, ) and Puout

(, ) are given in Lemma 1.

Recall from Section II-B that W and

W follow the same

distributions as the fading factors of data links in the cel-lular network and the MANET, respectively; G is identicallydistributed as the fading factors of the interference channels

in the coexisting networks.

Finally, the CDF ofD is obtained. Note that the event D tis equivalent to that there exists at least one base station within

a distance of t from U0. Lemma 2 follows.

Lemma 2. The CDF of D is

P r(D t) = 1 ebt2

. (10)

From (10), the CDF of D depends on the density of basestation b. Intuitively, increasing b reduces the cell sizes andthus D and vice versa.

B. Outage Probabilities: Spectrum Underlay

The SIRs for spectrum underlay are obtained as follows.

Define the transmitter processes (b) and

(b)m similarly as(a) and

(a)m in the preceding section. Again, using the mark-ing theorem,

(b) and

(b)m are shown to be stationary PPPswith the densities /M and /M, respectively. For spectrumunderlay, B0 accessing the th subchannel is interfered with

by two processes of transmitters, namely (b) \{U0} and

(b) .

Thus the interference power for B0 is given as

I(b) := P

X

(b) \{U0}

GXRX +

P Xe

(b)

GXRX . (11)

Similarly, we can write the interference power for R0 access-ing the mth subchannel as

I(b)m := P

X(b)m

GXRX + P

Xe

(b)\{T0}

GXRX . (12)

Given the above definitions, the SIRs for spectrum underlay

are readily written as

Celluar : SIR(b) = P W D/I(b)

MANET : SIR(b)m = PWd/I(b)m (13)where , m {1, 2, , M}.

We consider the combined process := (b)

(b) . Apoint in is either a mobile or ad hoc transmitter. As a result,its transmission power is a random variable with the support

set {P, P}. To analyze the shot noise process generated by ,we derive the distributions of and the transmission powerof the points in . These results are summarized below.

Lemma 3. is a stationary PPP with the density (+)/M.

Given X , the transmission power PX of X has the following distribution function

PX =

P, w.p.

+

P , w.p.

+ .

(14)

Proof: See Appendix A.

In terms of , (11) and (12) can be simplified as

I(b) :=

X\{U0}

PXGXRX

I(b)m := Xm\{T0} PXGXRX .

(15)

Using Lemma 3 and (15) and following the approach in

Section III-A, the bounds on the outage probabilities are

obtained and shown below.



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Proposition 2. For spectrum underlay, the outage probabili-

ties are bounded as follows.

Cellular : E

Plout

W D,

+

M

Pout(, )

E

Puout

W D, +

M MANET : E

Plout

Wd, + M

Pout(, )

E

Puout

Wd, + M

where the power ratio := /, and Pl

out(, ) and Pu

out(, )

are defined in Lemma 1.

Proof: See Appendix B.

Proposition 2 shows that the outage probability for each

network depends on the transmitter densities of both net-

works. This coupling is due to spectrum underlay and theresultant mutual interference between the coexisting networks.

As shown in Section V, such coupling may result in smaller

transmission capacities for spectrum underlay than those for

spectrum overlay. Moreover, Proposition 2 also shows that

the outage probabilities for spectrum underlay depend on the

transmission power ratio . The effect of is characterized inSection V.

C. Outage Probabilities: Spectrum Overlay with SIC

As in the preceding sections, consider B0 and R0 accessing

the th and the mth subchannels, respectively. SIC effectivelyremove the strongest interferers for B0 and R0. Based onthe SIC model in Section II-D, the interferer process for B0conditioned on the link power W D is

(c) (W D

) :=

X (a) \{U0} | GXR

X W D

(16)

where (a) is defined for spectrum overlay in Section IV-A.

Similarly, the conditional interferer process for R0 is given by(c)m (Wd) := X (a) \{T0} | GXRX Wd(17)

where (a) follows from Section IV-A. Thus the conditionalinterference power for B0 and R0 can be written asI(c) (W D

) := P

X(c)(WD)

GXRX

I(c)m (Wd) := P

Xe(c)m (fWd)

GXRX .

(18)

It follows that SIRs for spectrum overlay with SIC are

Celluar : SIR(c) = P W D

/I(c) (W D

)

MANET : SIR(c)m = PWd/I(c)m (Wd). (19)The above SIR derivation shows that spectrum overlay with

and without SIC are closely related. This relation is specified

in the following proposition.

Proposition 3. For spectrum overlay with SIC, the bounds on

the outage probabilities P(c)out

and P(c)out

can be modified from

those in Proposition 1 by replacing the functions Plout

and

Puout

with Plout

and Puout

correspondingly, which are defined as

Plout

(a, b) := 1 exp

ab

(20)

Puout

(a, b) := 1 (a, b)exp

ab

(21)

where := 1 .

Proof: See Appendix C.Note that (20) and (21) differ from their counterparts in

Lemma 1 only by the factor . The factor < 1 represents theSIC advantage of reducing outage probabilities with respect

to the case of no SIC ( = 1). Moreover, decreasing the SICfactor reduces and thus outage probabilities. Nevertheless, being too small may invalidate the assumption of perfectSIC. Specifically, small implies small SIR for the processof decoding interference prior to its cancelation and potentially

results in significant residual interference after SIC [26].

D. Outage Probabilities: Spectrum Underlay with SICThe outage probabilities for spectrum underlay with SIC can

be obtained from those for spectrum underlay in Section IV-B.

The procedure is similar to that in the preceding section.

Specifically, the conditional interferer processes for B0 andR0 are defined similarly as in (16) and (17)

(d) (W D

) :=

X \{U0} | PXGXRX PWD

(d)m (Wd) := X m\{T0} | PXGXRX PWdwhere is the combined PPP defined Section IV-B, and PXis distributed as in Lemma 3. Thus the SIRs can be written as

Celluar : SIR(d) = P W D/I(d) (W D

)

MANET : SIR(d)m = PWd/I(d)m (Wd). (22)where the conditional interference power

I(d) (a) := P

X

(c)(a)

GXRX

I(d)m (a) := P

Xe(c)m (a)

GXRX .

(23)

The similar results as in Proposition 3 are obtained for

spectrum underlay with SIC and shown in the following

proposition. Its proof is similar to that for Proposition 3 and

thus omitted.

Proposition 4. For spectrum underlay with SIC, the bounds

on the outage probabilities P(d)out

and P(d)out

can be modified

from those in Proposition 2 by replacing the functions Plout

and Puout

with Plout

and Puout

defined in Proposition 3.

The remarks on Proposition 3 are also valid for Proposition 4.

V. NETWORK CAPACITY TRADEOFF: ASYMPTOTIC

ANALYSIS

Using the results obtained in the preceding section, the

trade-off between the transmission capacities C and C of thecoexisting networks is characterized in the following theorem

for small target outage probability 0.



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Theorem 1. For 0, transmission capacities of thecoexisting networks satisfy

C+ C =M

+ O

2

(24)

where the weights and are given as 4

o = E[W ]d2, o = 2E[W](b)1u = o (o), u = (o) o, (25)and depends on if SIC is used

= 1, no SIC

1 22 , SIC.

(26)

Proof: See Appendix D.

It can be observed that for spectrum overlay the numbers

of subchannels assigned to the coexisting networks, namely

K and K, do not appear in Theorem 1. The reason is that Kand K are merged into M based on the equality M = K+ K(see Appendix D). Theorem 1 shows that the trade-off between

C and C follows a linear equation. Specifically, the slope atwhich C increases with decreasing C is /, which dependson different network parameters as observed from (25). 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 achieved with the outage

constraint being active. For instance, the cellular network is

outage limited if Pout(C) = . For spectrum overlay, boththe 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 networksare outage limited only ifu = u, where u and u are givenin (25). Otherwise, u > u correspond to only the cellularnetwork being outage limited; u < u indicates that only theMANET is outage limited.

Spectrum overlay is shown to be more efficient than spec-

trum underlay as follows. By definitions, the capacity region

in (7) is the region enclosed by the capacity trade-off curve

in (24) and the positive axes of the C-C coordinates. Thisregion contains all feasible combinations of the densities

of the coexisting networks. Thus, the size of the capacity

region measures the spectrum-sharing efficiency. The capacity

regions for spectrum overlay and underlay are compared inthe following corollary.

Corollary 1. For 0, the capacity region for spectrumunderlay is no larger than that for spectrum overlay. They are

identical if and only if the transmission-power ratio is

= (o/o)1 (27)

where o and o are given in (25).

Proof: See Appendix E.

Corollary 1 shows that spectrum overlay is potentially more

efficient than spectrum underlay due to network coupling

for the latter. Specifically, the possibility that a network is

4The subscripts o and u identify spectrum overlay and underlay, respec-tively

not outage limited compromises the efficiency of spectrum

underlay, which, however, can be compensated by setting as given in (27). This optimal value of ensures that bothnetworks are outage limited for the case of spectrum underlay.

The next corollary specifies the effects of several parameters

on transmission capacities of the coexisting networks.

Corollary 2. For 0, transmission capacities vary withnetwork parameters as follows.

1) Spectrum overlay: C increases linearly with the basestation density b; C increases inversely with the ad hoctransmitter-receiver distance d.

2) Spectrum underlay: If the cellular network is outage

limited, both C and C increase linearly with the basestation density b. Otherwise, both C and C increaseinversely with the ad hoc transmitter-receiver distance

d.3) For both spectrum sharing methods, C and C increase

linearly with and the number of subchannels M, andinversely with related to SIC.

Finally, we analyze the transmission-capacity gains due to

spatial diversity gains contributed by multi-antennas [36]. To

obtain concrete results, the fading factors W and W are as-sumed to follow the chi-squared distributions with the complex

degrees of freedom L and L, respectively, which are called thediversity gains [30]. These fading distributions can result from

using spatial diversity techniques such as beamforming over

multi-antenna i.i.d. Rayleigh fading channels [37]. Thus

E[W] =(L )

(L), E[

W] =

(L )

(L). (28)

The following corollary is obtained by combining Theorem 1,(28) and the following Kershaws inequalities [38]

x +s

2

1s 1W D since > 1 and > 1.Thus, the process of weak interferers can be defined as

cS(W D) :=

(c) \S(W D

), which is observed to beidentical to the counterpart for spectrum overlay without SIC

considered in Section B. Since cS(W D) S(W D) =, cS(W D

) and S(W D) are independent processes

based on a basic property of Poisson processes [35]. From

the discussion in Section III-A, the exponential terms in (3)

and (4) depends only on S(W D), and the function (, )

only on cS(W D). Since cS(w, d) is invariant to SIC, and

cS(w, d) and S(w, d) are independent, the bounds on Pout inLemma 1 can be extended to the case of SIC by replacing the

exponential terms in (3) and (4) with exp(E [|S(w, d)|]),where E [|S(w, d)|] is obtained using Campbells theorem

E [|S

(w, d)|] =2

0 (w1dg)

1

(1w1dg) 1rf

G(g)drdg

=wd2

K

and is defined in the statement of the proposition.

D. Proof for Theorem 1

1) Spectrum Overlay: The convergence 0 implies 0 and 0. Using the series representation of the PDF of apower shot-noise process [33], the asymptotes of the outage

probabilities follow from [7, Theorem 2]

Pout = K

E WE D2 + O 2Pout =

KE

W d2 + O2 . (32)

By using Lemma 2, the term E

D2

in (32) is obtained as

E

D2

=

0

2bt3ebt

2

dt

=

0

btebtdt =

(2)

b=

2

b. (33)

Combining (6), (32), and (33) gives the desired asymptotic

capacity trade-off function for spectrum overlay.

2) Spectrum Underlay: By using the series expression of

the PDF of the power shot noise [33] as well as Proposition 1,

Pout(, )= +

ME[W ]E[D2] + O(max(2, 2))(34)

Pout(, )= +

ME[W ]d2 + O(max(2, 2)). (35)

For 0, the transmission capacities C and C satisfy theconstraints Pout(C/M, C/M) and Pout(C/M, C/M) .By combining these constraints, (34) and (35)

C+ C

Mmax

E[W]E[D2], E[W ]d2 = + O(2).

The desired result follows from the above equation.

3) Spectrum Sharing with SIC: Consider spectrum overlay

with SIC. By canceling the strongest interferers using SIC, the

PDF upper-tail of the power shot noise process is trimmed

and its series expansion is difficult to find [33]. Nevertheless,

the asymptotic transmission capacities can be characterized by

expanding the bounds on Pout in Proposition 3. Specifically

P

l

out(/K) =

KE

[W

]E

[D

2

] + O(

2

)

Puout

(/K) = 1 E

1

2 WD2

K+ O(2)

KWD2 + O(2)

=

2

2

1

E[W ]E[D2]

K+ O(2).

Thus

Pout(/K) = E[W]E[D2]

K(36)

where 1 22 . SimilarlyPout(/K) = E[W ]d2

K. (37)

The desired results for spectrum overlay with SIC are obtained

by combining (6), (36), and (37). The results for spectrum

underlay with SIC are derived following a similar procedure.

E. Proof for Corollary 1

First, the capacity region for spectrum underlay is proved to

be no larger than for spectrum overlay. It is sufficient to provethat u o and u o, which follow from (25). Next,substituting (27) into (25) results in u = o and u = o.This proves the second claim in the theorem statement.



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E[IcS(a)] =2( + )

M

Pr (PX = P)0

( ga)1

(P rg)rfG(g)drdg + Pr

PX = P 0

( ga)1

(P rg)rfG(g)drdg

=

P

1

+

M

(a1)11, (30)

var[IcS(a)] =2( + )

M

Pr (PX = P)0

( ga)1

(P rg)2rfG(g)drdg + Pr

PX = P 0

( ga)1

(P rg)2rfG(g)drdg

=

P2

2

+

M

(a1)22. (31)

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Kaibin Huang (S05M08) received the B.Eng.(first-class hons.) and the M.Eng. from the NationalUniversity of Singapore in 1998 and 2000, respec-tively, and the Ph.D. degree from The University ofTexas at Austin (UT Austin) in 2008, all in electricalengineering.

Since Mar. 2009, he has been an assistant pro-fessor in the School of Electrical and ElectronicEngineering at Yonsei University, Seoul, Korea.From Jun. 2008 to Feb. 2009, he was a PostdoctoralResearch Fellow in the Department of Electrical and

Computer Engineering at the Hong Kong University of Science and Technol-ogy. From Nov. 1999 to Jul. 2004, he was an Associate Scientist at the Institutefor Infocomm Research in Singapore. His research interests focus on limitedfeedback techniques for wireless networks. Dr. Huang received the MotorolaPartnerships in Research Grant, the University Continuing Fellowship at UTAustin, and the Best Student Paper award at IEEE GLOBECOM 2006.

Vincent K. N. Lau obtained B.Eng (Distinction 1stHons) from the University of Hong Kong (1989-1992) and Ph.D. from Cambridge University (1995-1997). He was with HK Telecom (PCCW) as systemengineer from 1992-1995 and Bell Labs - LucentTechnologies as member of technical staff from1997-2003. He then joined the Department of ECE,Hong Kong University of Science and Technol-ogy (HKUST) as Associate Professor. His currentresearch interests include the robust and delay-sensitive cross-layer scheduling of MIMO/OFDM

wireless systems with imperfect channel state information, cooperative andcognitive communications, dynamic spectrum access as well as stochasticapproximation and Markov Decision Process.

Yan Chen received her B.Sc degree from ChuKochen Honored College, Zhejiang University,Hangzhou, China, in 2004. She is expected to re-ceive her Ph.D degree in information and com-munication engineering from Zhejiang Universityin 2009. Since Jan 2008, She has been a visitingresearcher in the group of Prof. Vincent Lau inHKUST. Her current research interests lie in com-bined information theory and queueing theory inwireless communications, with particular emphasison exploiting communication opportunities via co-

operation and cognition.

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