05659492

8102019 05659492

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1

Adaptive Modulation in Spectrum-Sharing Channels

Under Delay Quality of Service ConstraintsLeila Musavian Member IEEE Sonia Aiumlssa Senior Member IEEE and Sangarapillai Lambotharan Senior

Member IEEE

AbstractmdashWe propose a variable-rate variable-power M-levelQuadrature Amplitude Modulation (MQAM) scheme employedunder delay quality-of-service (QoS) constraints over spectrum-sharing channels An underlay cognitive radio system withone primary user and a secondary user with constraints oninterference leakage imposed by the primary receiver is con-sidered The transmission parameters of the secondary user areset optimally such that the successful communications for theprimary user in terms of a minimum rate to be supported issatisfied irrespective of co-existence with the secondary userWe obtain interference constraints that when satisfied by the

secondary users is sufficient for satisfying the service outagerequirement of the primary user We further study the per-formance of the secondary userrsquos link employing an adaptiveMQAM scheme when on top of the above-mentioned interferenceconstraint the secondary user is also required to satisfy astatistical delay QoS constraint Considering two modulationschemes namely continuous MQAM and discrete MQAM withrestricted constellations we obtain the effective capacity of thesecondary userrsquos link and derive the optimum power allocationthat maximizes the effective capacity of the secondary user

Index TermsmdashSpectrum sharing Adaptive modulation DelayQoS constraint Effective capacity Service-outage constraint

Fading channels

I INTRODUCTION

Adaptive resource allocation is considered a powerful tool

for enhancing spectrum efficiency in current and future-

generation wireless networks In particular adaptive power

and rate allocation is known to improve the performance of

wireless fading channels [1] Several approaches and studies

have taken place to investigate the capacity gains that can be

achieved by these techniques For instance it has been shown

in [2] that adaptive power and rate allocation based on M-

level Quadrature Amplitude Modulation (MQAM) achieves a

Copyright (c) 2010 IEEE Personal use of this material is permitted

However permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubsminuspermissionsieeeorg

Manuscript received October 01 2009 revised June 02 2010 and Septem-ber 30 2010 accepted October 12 2010 The associate editor coordinatingthe review of this paper and approving it for publication was Y Ma Thiswork has been supported by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada under research grant RGPIN22907-2005 andby the Engineering and Physical Sciences Research Council (EPSRC) UKunder grant EPG0204421 Part of this work is published in the proceedingsof ICC 2009

Leila Musavian and Sonia Aiumlssa are with the INRS-EMT University of Quebec Montreal QC Canada Emailmusavian aissaemtinrsca

Sangarapillai Lambotharan is with the Advanced Signal ProcessingGroup Loughborough University Loughborough Leicestershire UK Emailslambotharanlboroacuk

Digital Object Identifier

20dB gain in spectral efficiency as compared to a nonadap-

tive scheme The spectral efficiency can be enhanced further

through dynamic spectrum allocation Enforced by regulatory

bodies the spectrum allocation has traditionally followed poli-

cies where non-overlapping parts of the spectrum are allocated

to specific applications and users Nevertheless while we

witness a huge surge for new wireless applications recent

spectrum allocation chart suggests that not much spectrum

is left for new applications and for the growing number of

wireless users [3] Fortunately recent spectrum measurementshave also shown that significant parts of the spectrum are

inefficiently utilized [3] paving the way for feasible sharing of

the spectrum using the so-called cognitive radio (CR) concept

One of the major challenges for next-generation wireless

systems in general and CR systems in particular is to support

quality-of-service (QoS) requirements for different applica-

tions Indeed providing QoS measures for secondary users

is even more challenging due to the secondary type of access

to the radio spectrum One of the critical QoS requirements is

the delay requirement for real-time or delay-sensitive applica-

tions Generally two different kinds of delay constraints are

considered in communications systems namely deterministic

and statistical Imposing deterministic delay requirements thatis the delay should be less than a threshold at all times is

very challenging or even impossible in fading channels due to

the variations in the capacity as a function of the channel gain

and the availability of channel state information (CSI) at the

transmitter andor receiver [4] On the other hand statistical

delay QoS constraints where delay is required to be lower

than a specific threshold only for a certain percentage of time

are considered more pragmatic in various applications [5]

Studying the performance of wireless communication sys-

tems using the current physical-layer channel models is very

complicated since these models can not be translated to the

complex link-layer requirements such as delay bound QoS

requirements [4] Recently the concept of effective capacitywhich is a link-layer channel model and aims to model the

wireless channel in terms of functions that can be mapped into

link-layer performance metrics has been introduced in [4]

Effective capacity is the dual of effective bandwidth [5] and

can be interpreted as the maximum constant arrival-rate that

can be supported by the channel given that the delay QoS

requirement of the system is satisfied [4] In this regard an

optimum power and rate allocation strategy that maximizes the

effective capacity in fading channels has been obtained in [6]

It is worth mentioning that providing the QoS constraint in

cognitive radio channels is a further complicated task due to

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2

the secondary type of access to the spectrum for secondary

users This issue is very challenging and has attracted many

researchers eg [7]ndash[9]

In assessing the performance of spectrum-sharing chan-

nels in fading environments we refer to the earlier work

of Gastpar who presented capacity investigations of additive

white Gaussian noise (AWGN) spectrum-sharing channels

under interference power constraint rather than transmit power

constraint [10] Later ergodic and outage capacity metrics of

a point-to-point system with constraints on the received-power

at the primaryrsquos receiver in fading environment were derived

in [11] and [12] The underlay spectrum-sharing approach is

considered in [13] wherein optimum power allocation strate-

gies are proposed such that the interference to the primary

user is minimized while a set of minimum signal-to-noise

ratio (SNR) targets is provided at the secondary receivers

A delay QoS-driven power and rate allocation scheme under

spectrum-sharing constraint was proposed in [14] wherein

the effective capacity of a point-to-point channel in Rayleigh

fading environment was determined

In this paper we consider spectrum sharing systems forwhich the transmission of the secondary user is subject to

constraints on the interference-power inflicted on the primary

receiver In general we assume that there are guidelines and

limitations set by the regulatory bodies on the maximum

interference power in terms of peak or average values in-

flicted on the primary users In addition there are certain

service outage constraints for the primary users that should be

satisfied irrespective of the existence of the secondary users

in the network Specifically we wish to limit the transmission

parameters of the secondary transmitter such that the primary

user is supported with a minimum-rate for a certain percent-

age of time Translating this limitation into an interference-

power constraint either on peak or average interference limitswe obtain the maximum throughput of the secondary userrsquos

channel under delay QoS constraint by obtaining the effective

capacity of the channel We determine the maximum arrival-

rate that can be supported by the secondary userrsquos link subject

to satisfying a statistical delay QoS constraint by obtaining

the effective capacity of the channel under adaptive MQAM

with interference-power constraint We further obtain closed-

form expressions for the effective capacity and its power

allocation in Nakagami-m block-fading channels The service

outage constraint considered in this work is different from

our previous work on effective capacity of cognitive radio

channel In addition in this paper we assume that secondary

users implement MQAM which has not been studied in ourprevious publications

The subsequent sections are organized as follows In Section

II we provide the channel and system models The inter-

ference power is studied in Section III wherein the primary

userrsquos service outage constraint is translated into an average

or peak interference power constraint The effective capacity

of the secondary user channel under average interference

power constraint is provided in closed-form in Section IV The

effective capacity of the channel under peak interference power

constraint is studied in Section V Numerical results are given

in Section VI followed by conclusions in Section VII

I I SYSTEM M ODEL

The transmission parameters of the secondary user are

chosen such that the service outage requirement of the primary

user is satisfied The effect of the transmission of the primary

user on the secondary receiver is assumed as AWGN In the

secondary user communication system the upper layer packets

are organized into frames with duration T f The secondary

transmitter employs adaptive MQAM with continuous or dis-crete constellations Discrete-time block-fading channels are

assumed for both the secondary and primary usersrsquo links

The channel gain between the transmitter and receiver of

the secondary user and the AWGN are denoted by hs[t] and

zs[t] respectively where t denotes the time index We define

the channel gain between the secondaryrsquos transmitter and the

primaryrsquos receiver by hsp[t] We assume hs[t] and hsp[t] are

statistically independent and identically distributed (iid) and

also independent from the noise The channel envelopes are

distributed according to Nakagami-m fading Channel gains

are stationary and ergodic random processes

The secondary transmitter is provided with knowledge of

hs[t] and hsp[t] Information about the latter can be obtainedfrom a band manager that intervenes between the primary and

secondary users [15] or can be directly fed back from the

primaryrsquos receiver to the secondary user as proposed in [16]

[17] where the protocols allow the primary and secondary

users to collaborate and exchange CSI The effect of imper-

fection in the knowledge of the channel gains between the

secondary transmitter and primary receiver at the secondary

transmitter on the ergodic capacity of the secondary userrsquos link

has been investigated in [18] for Rayleigh fading channels The

secondary user knows only the statistical information of the

link between the transmitter and receiver of the primary user

hp[t] The instantaneous channel knowledge of hp[t] is known

to the primary userrsquos transmitter

We consider a statistical delay constraint according to

Pr D(t) ge Dmax le P outdelay (1)

where D(t) indicates the delay experienced by a packet at

time instant t and Dmax is the maximum delay that can be

tolerated for 1 minus P outdelay percentage of time We further assume

that the transmission technique of the secondary user must

satisfy a statistical delay QoS constraint It is shown that

the probability for the queue length of the transmit buffer

exceeding a certain threshold x decays exponentially fast as

a function of x [5] [6] We now define θ as the delay QoS

exponent given by

θ = minus limxrarrinfin

ln(Pr q (infin) gt x)

x (2)

where q (n) denotes the transmit buffer length at time n and

Pr a gt b denotes the probability that the inequality a gt bholds true Considering a data source with constant data rate r

the QoS exponent θ is related to the delay violation probability

according to

supt

Pr D(t) ge Dmax asymp γ (r)eminusθDmax (3)

where γ (r) = P rQ(t) ge 0 is the probability of a non-

8102019 05659492




3

empty buffer Therefore the maximum constant arrival rate for

providing the delay constraint (1) can be obtained from

P outdelay = γ (r)eminusθDmax (4)

Note that θ rarr 0 corresponds to a system with no delay

constraint while θ rarr infin implies a strict delay constraint

Considering θ as the delay QoS exponent we obtain the

secondary userrsquos maximum supported arrival-rate given that

the QoS constraint is satisfied An interested reader is referred

to [4] for more details Note that effective capacity relates to

the asymptotic case for the delay and is defined for large value

of Dmax However it has been shown in [4] that this model

also provides a good estimate for small values of Dmax

III INTERFERENCE-POWER C ONSTRAINT

We recall that the transmission power of the secondary

user is limited such that the primary user is guaranteed

with a minimum-rate Rmin for a certain percentage of time

(1 minus P outp ) We formulate the interference constraint starting

with the following outage probability

Pr

Rp le Rmin

le P outp (5)

where Rp indicates the rate of the primary user link The

transmission power of the primary user is assumed to be

constrained by an average level P p ie1

E hp P p(hp) le P p (6)

where E hp defines the expectation over the probability density

function (PDF) of hp and P p(hp) is the input transmit power

of the primary user as a function of hp

We consider two different transmission strategies for the

primary user constant transmit power (cons) and optimum

power and rate allocation (opra) [1] For opra scheme thetransmission power at the primary transmitter for each time

instant is chosen using the CSI of hp available at the primary

transmitter Note that the primary user chooses its transmit

power without taking into consideration the existence of the

secondary user in the network The secondary transmitter on

the other hand should limit its transmit power such that the

communication process of the primary user is not harmed in

the sense defined in (5)

A Constant Transmit Power Scheme (cons)

For this scheme we assume that the primary user transmits

with fixed power P p at all time As such one can show that

ln

10486161 +

P php

P s(θ hs hsp)hsp + N 0B

1048617 le Rp (7)

where P s(θ hs hsp) denotes the transmit power of the sec-

ondary user as a function of θ hs and hsp The noise power

spectral density and received signal bandwidth are denoted by

N 0 and B respectively Hereafter for the ease of notations

we use P s to denote the transmit power of the secondary user

Lemma1 When the following average interference con-

straint is satisfied by the secondary user who operates in

1Hereafter we omit the time index t whenever it is clear from the context

a Rayleigh fading environment so is the service outage

constraint of the primary user

E h P shsp le I avg (8)

with k1 = eRminminus1P p

and I avg = minusln(1minusP outp )

k1minus N 0B which is

referred to as average interference-limit

Proof We start by formulating the service outage constraint

of the primary user according to

P outp ge Pr852091

0 le hp le k1 (P shsp + N 0B)852093

(9)

holds then the interference power constraint (5) will be

satisfied Now the condition (9) can be expanded as

P outp ge

991787 infin0

991787 infin0

991787 k1(P shsp+N 0B)

0

f hp(hp)dhp I 0

times f hsp(hsp)f hs(hs)dhspdhs

(10)

where f x(x) indicates the PDF of the random variable x The

PDF of hp is given by [19]

f hp(hp) = m

mpp h

mpminus1p

Γ(mp) eminusmphp (11)

where mp indicates the Nakagami-m parameter for the channel

gain between the transmitter and receiver of the primary user

and Γ(middot) =int infin0

wzminus1eminuswdw is the Gamma function [20]

The integral function in I 0 can be extended using a change of

variable x = mphp according to

I 0 = 1

Γ(mp)

991787 f cons(P shsp)0

xmpminus1eminusxdx (12)

where f cons(P shsp) = mpk1 (P shsp + N 0B) For integer mp

the solution for the integration operation (12) can be found as

I 0 =

852008minuseminusx

mpminus1sumk=0

xk

k

852009f cons(P shsp)0

(13)

For the special case of Rayleigh fading ie mp = 1 the

interference power constraint in (10) simplifies to

P outp ge E hshsp

8520911 minus eminusf cons(P shsp)

852093 (14)

where E hshsp defines the expectation over the joint PDF of hs

and hsp Hereafter we refer to E hshsp as E h The inequality

in (14) can be further simplified by using Jensenrsquos inequality

according to (8) This conclude the proof

It is worth noting that when I avg le 0 no feasible power

allocation satisfying (8) exists hence the capacity lower

bound is zero In the following we assume I avg gt 0

For the cases with integer mp gt 1 the interference power

simplifies to

P outp ge E hshsp


mpminus1sumk=0

(f cons(P shsp))k

k

1048701

(15)

Satisfying constraint (15) guarantees that the achievable-rate

of the primary user is bigger than Rmin for at least (1 minus P outp )

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4

percentage of time Analyzing the throughput of the secondary

user link under the constraint in (15) is however complicated

Hence we proceed by simplifying the inequality in order to

obtain a simple peak interference power constraint which will

be a sufficient condition to satisfy (5)

We proceed by simplifying (15) assuming that the amount of

the interference-power P shsp should satisfy the inequality (15)

at all times Hence we have P shsp le I peak where I peak is

the solution for x from

eminusf cons(x)

mpminus1sumk=0

(f cons(x))k

k = 1 minus P outp (16)

For non-integer mp the interference constraint can be found

P outp ge E hshsp γ (mp f cons(P shsp)) (17)

where γ (s t) =int t0

xsminus1eminusxdx is the lower incomplete

Gamma function [20] A peak constraint can be obtained when

the above inequality is satisfied at all the time

To summarize we translated the service-outage constraint

of the primary user into a peak or average interference power

constraint

B Optimum Power and Rate Allocation Scheme (opra)

Recall that the transmission parameters of the primary user

is chosen without any consideration to the presence of the

secondary user in the frequency band

Lemma2 Satisfying average interference power constraint

E h P shsp le I avg with

I avg = minusln(

1 minus P outp

)+ k2

k1minus N 0B (18)

where k1 =

eRminminus1

micro and

k2 =

N 0B

micro by a secondary user

who operates in a Rayleigh fading environment is sufficient to

satisfy the service outage constraint of the primary user who

employs opra technique at its transmitter

ProofThe primary transmitter employs adaptive power and

rate allocation technique (opra) [1] Hence the power alloca-

tion of the primary user can be found as

P p(hp) =

852059micro minus

N 0B

hp

852061+ (19)

where [x]+ indicates maxx 0 The cutoff threshold micro is

found so that the power constraint (6) is satisfied with equality

The left-hand-side of the constraint in (5) can now be

expanded according to

P outP gePr

983163hp lt

N 0B

micro

983165+ Pr

983163Rp le Rmin hp ge

N 0B

micro

983165

Using the inequality

Pr

983163ln

10486161 +

P p(hp)hp

P shsp + N 0B

1048617 le Rmin

983165 ge Pr Rp le Rmin

(20)

where P p(hp) is given in (19) Hence satisfying the inequality

P outp ge Pr852091

0 le hp le (k1 (P shsp + N 0B) + k2)852093

(21)

is sufficient for (20) to be satisfied By following similar

approach as in Subsection A we can obtain the interference-

limit in Lemma 2 This concludes the proof

In order to obtain the peak power constraint we first

set f opra(P shsp) = mp (k1 (P shsp + N 0B) + k2) Now peak

power constraints for integer mp and non-integer mp can be

found by replacing f cons(P shsp) with f opra(P shsp) in (15)

and (17) respectively and assuming that the inequalities are

satisfied at all time

IV AVERAGE I NTERFERENCE-P OWER C ONSTRAINT

A Continuous MQAM

In this section we consider the case when the service-outage

constraint of the primary user is translated into an average

interference-power constraint The secondary transmitter em-

ploys MQAM with no restriction on the constellation size

The constellation set is chosen adaptively while satisfying the

interference-power constraint

Defining M (θ hs hsp) as the number of points in the signal

constellation ie modulation order one can obtain an upper

bound on the required bit-error-rate (BER) of the systemaccording to [2]

BER le 02eminus15P shs

N 0B(M (θhshsp)minus1) (22)

which leads to

M (θ hs hsp) = 1 + KP shs (23)

where K = minus15N 0B ln(5BER) We further define R[t] t =

1 2 as the stochastic service rate which is assumed to

be stationary and ergodic We can now obtain the service rate

of the MQAM scheme according to

R[t] = T f B ln (1 + KP s[t]hs[t])We now introduce the concept of effective capacity and

obtain the effective capacity of the secondary user link when

employing adaptive MQAM under the interference-power con-

straint (8) Effective capacity was originally defined in [4] as

the dual concept of effective bandwidth Assuming that the

function

Λ(minusθ) = limN rarrinfin

1

N ln983080

E 852091

eminusθsum

N t=1 R[t]

852093983081 (24)

exists the effective capacity is outlined as [4]

E c(θ) = minusΛ(minusθ)

θ = minus lim

N rarrinfin

1

N θ ln 983080E 852091eminusθ

sumN t=1 R[t]

852093983081

It is worth noting that the effective capacity quantifies the

maximum arrival-rate that can be supported by the channel

under the constraint of QoS exponent θ interpreted as the

delay constraint Moreover in block-fading channels where

the sequence R[t] t = 1 2 is uncorrelated the effective

capacity can be simplified to

E c(θ) = minus1

θ ln983080

E 852091

eminusθR[t]852093983081

(25)

Having introduced the formulation for effective capacity we

now obtain the optimum power and rate allocation that maxi-

mizes the effective capacity of the channel This maximization

8102019 05659492




5

problem can be formulated as

E optc (θ) =maxP sge0

1048699minus

1

θ ln

852008E hshsp

983163eminusθT f B ln

(1+KP shs

)9831658520091048701st E hshsp P shsp le I avg (26)

where E optc (θ) indicates the maximum of the effective capac-

ity Using a similar approach as in [14] the solution for the

maximization problem in (26) can be obtained as

P s =

983131 β

11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khs

983133+

(27)

where α = θT f B β = γ 0α [x]+ denotes max0 x and

γ 0 = 1λ0

λ0 being the Lagrangian multiplier chosen to satisfy

the interference-power constraint in (8) with equality The

power allocation policy can be expressed as

P s =

β 11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khsif hsp le Kβhs

0 otherwise

(28)

In order to obtain a solution for γ 0 = βα

we need to evaluate

the integration in

I avg =

991787 infin0

991787 Kβhs

0

1048616β

11+α

1048616 hsp

Khs

1048617 α1+α

minus hsp

Khs

1048617times f hsp(hsp)f hs(hs)dhspdhs

(29)

Noting that (29) depends on the channel gains only through

ratio values we define a new random variable v = hsphs

Using

the fact that the distribution of the ratio between two Gamma

distributed random variables with parameters α1 and α2 is a

beta prime distribution with parameters α1 and α2 [11] [21]

we can determine the distribution of the random variable v as

f v(v) = ρms

β (ms msp)

vmspminus1

(v + ρ)ms+msp

(30)

where ρ = ms

mspand β (ms msp) =

Γ(ms)Γ(msp)Γ(ms+msp)

with Γ(z) =int infin0

tzminus1eminustdt defining the Gamma function [20] We now

obtain the solution for γ 0 by evaluating the integration in (29)

as follows

I avg = ρms

Kβ (ms msp)

991787 Kβ

0

983080(Kβ )

11+α v

α1+α minus v

983081times

vmspminus1

(v + ρ)ms+msp

dv

(31)

= ρms(Kβ )msp+1

Kβ (ms msp)(Kβ + ρ)ms+msp(32)

times

983131991787 10

(1 minus x)mspminus1+ α1+α

10486161 minus

Kβ

Kβ + ρx

1048617minus(ms+msp)

dx J 0

minus

991787 10

(1 minus x)msp

10486161 minus

Kβ

Kβ + ρx


dx J 1

983133

where x = 1 minus vKβ

A closed-form expression for the first

integral in (32) J 0 can be obtained using [14] according to

J 0 =Γ983080

msp + α1+α

983081Γ983080

msp + 1+2α1+α

983081times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

(33)

where 2F 1(a b c z) denotes the Gaussrsquos hypergeometric func-tion [20] A closed-from expression for J 1 can also be ob-

tained by following a similar approach Now by inserting (33)

into (32) and using the equality Γ(1+ z) = zΓ(z) we obtain

a closed-form expression for (32) according to

I avg = ρms(Kβ )msp+1

Kβ (ms msp)(Kβ + ρ)ms+msp

9831311048616msp +

α

1 + α

1048617minus1

times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

minus 1

msp + 12F 1

1048616ms + msp 1 msp + 2

Kβ

Kβ + ρ

1048617 983133

from which γ 0 can be obtained We now derive a closed-

from expression for the effective capacity of the channel by

evaluating the integration in (26) as follows

E optc (θ) = minus1

θ ln

852008E v

104869910486161 +

1048667(Kβ )

11+α v

minus11+α minus 1

1048669+1048617minusα1048701852009

= minus1

θ ln

1048616ρms(Kβ )

minusα1+α

β (ms msp)

991787 Kβ

0

vmspminus 11+α

(v + ρ)ms+msp

dv

+ ρms

β (ms msp)

991787 infinKβ

vmspminus1

(v + ρ)ms+msp

dv

1048617 (34)

By using ρms

β(msm

sp) int infin0 vmspminus1

(v+ρ)

ms+msp dv = 1 we get (37)

B Restricted MQAM

We now consider the case when the number of signal

points in the MQAM is not continuous but restricted to a set

M n n = 0N where M n = 2n The spectral efficiency

related to each constellation is given by n(bitssHz) As such

the service rate can be found according to rn = T f Bn [6]

with rn denoting the service rate of the n_th mode At

each time the secondary transmitter chooses an appropriate

constellation size based on its own channel gain hs the

channel gain between its transmitter and the primary receiver

hsp and the delay QoS exponent θ In addition the secondary

transmitter should determine the transmission power that satis-fies the BER requirement of the system the interference-power

restriction (8) and the delay QoS constraint

As stated earlier the effective capacity of the channel in the

continuous constellation case depends on the channel gains hs

and hsp only through the ratio of these two parameters Using

this fact we partition the entire range for the random variable

w hshsp

into N non-overlapping intervals and denote the set

pertaining to the boundaries of these intervals as W n n =0N + 1 with W 0 = 0 and W N +1 = infin We associate

the constellation M n to the n-th boundary which refers to the

case when W n le w lt W n+1 The constellation employed in

8102019 05659492




6


θ ln

8520081+

(Kβ )mspρms

β (ms msp)(ρ + Kβ )ms+msp

983080msp + α

1+α

9830812F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

ρ + Kβ

1048617

minus (Kβ )mspρms

β (ms msp)msp(ρ + Kβ )ms+msp 2F 1

1048616ms + msp 1 msp + 1

Kβ

ρ + Kβ

1048617852009 (37)

the 0-th interval is M 0 = 0 meaning that the transmission is

cut off when v lt W 1 or equivalently when the secondary

userrsquos channel gain is weak compared to hsp

We now need to find the boundary points and the trans-

mission power for each interval that maximizes the effective

capacity of the secondary user while satisfying the interference

power constraint and the BER requirement of the system For

this purpose we first obtain the optimal boundary points by

inserting the power allocation (27) into (23) yielding

M (θ hs hsp) = 1048616wKα

λ0 1048617 11+α

(38)

Using (38) as a guideline we obtain the boundary points as

W n = M 1+αn

λlowast0Kα

(39)

where λlowast0 should be found such that the interference-power

constraint in (8) is satisfied with equality Once the boundary

points and their associated constellations are found we need

to obtain the transmission power level at each boundary A

fixed BER means that the received SNR is fixed As such the

power allocation can be obtained using (23) according to

P s =

M n minus 1

KhsW n le w lt W n+1 n = 1N

0 0 le w lt W 1

(40)

The parameter λlowast0 can be obtained by inserting (40) into the

interference-power constraint (8) and replacing the inequality

with equality thus yielding

I avg =N minus1sumn=1

991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0

Kα

M n minus 1

K times

1

w f w(w)dw

+

991787 infinM

α+1N

λlowast0

Kα

M N minus 1

K times

1

w f w(w)dw

(41)

where

f w(w) = ρminusmsp

β (msp ms)

wmsminus1

(w + 1ρ)m

sp+m

s

(42)

Finally the effective capacity in this case can be found as

E disc (θ) = minus1

θ ln

1048616N minus1sumn=1

991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0Kα

M minusαn f w(w)dw

+

991787 infinM

α+1N

λlowast0Kα

M minusαN f w(w)dw

1048617

(43)

V PEA K I NTERFERENCE-P OWER C ONSTRAINT

Here we consider the case when the service-outage con-

straint of the primary user is translated into peak interference-

power constraint and obtain the maximum arrival rate for the

secondary user under delay QoS constraint

A Continuous MQAM

In this case the power of the secondary user can be found

as P s = I peak

hsp Therefore the service rate is given by

R[t] = T f B ln

10486161 + I peakK

hs

hsp

1048617 which leads to the effective

capacity

E c(θ) = minus 1θ ln 852008E hshsp 104869910486161 + I peakK hshsp

1048617minusα1048701852009 (44)

A closed-from expression for the effective capacity can

be obtained according to (45) see Appendix A where

F 1(a β β prime γ x y) is the appell hypergeometric function of

the first kind defined in [22] as

F 1(a β β prime γ x y) =

infinsumm=0

infinsumn=0

(a)m+n(β )m(β prime)nmn(γ )m+n

xmyn

with (x)n = x(x+1) (x+nminus1) indicating the Pochhammer

symbol [20]

B Restricted MQAM

Here we study the effective capacity of the secondary

userrsquos link under peak interference power constraint when

the secondary transmitter emblements discrete MQAM We

partition the entire range for the random variable W into

N + 1 non-overlapping regions In order to satisfy the peak

interference power constraint the secondary userrsquos transmit

power should be limited to I peak

hsp Now using (23) we get

M (θ hs hsp) = wI peak where can be used as a guideline to

obtain boundary points according to W n = M nI peak

Therefore

the effective capacity can be obtained according to


θ ln


991787 M n+1Ipeak

M nIpeak

M minusαn f w(w)dw

+

991787 infinM N Ipeak

M minusαN f w(w)dw

1048617

(46)

VI NUMERICAL R ESULTS

In this section we numerically evaluate the effective capac-

ity of the secondary userrsquos link in Nakagami-m block fading

under peak or average interference-power constraints when the

8102019 05659492




7

E c(θ) =

minus1

θ ln

1048616F 1

1048616msp + ααms + msp ms + msp + α 1 minus

1

KI peak 1 minus

1

ρ

1048617 ρminusmsp(KI peak)minusαΓ(ms)Γ(msp + α)

β (msp ms)Γ(ms + msp + α)

1048617

for 05 le KI peak and 05 le ρ

minus1

θ ln

1048616F 1 (ms α ms + msp ms + msp + α 1 minus KI peak 1 minus ρ)

ρmsΓ(ms)Γ(msp + α)


1048617

for K I peak le 2 and ρ le 2

(45)

01 02 03 04 05 06 07 08 09 1 11

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I a v g

( w a t t s )

Pout

p =1

Pout

p =2

Pout

p =3

Iavg

gt0

Iavg

gt0

Iavg

gt0

Iavg

gt0Iavg

gt0

Iavg

gt0

Fig 1 Average Interference-limit versus Rmin for various outageprobabilities for cons (solid lines) and opra (dashed lines)

06 08 1 12 14 16 18 2 22

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I p e a k

( w a t t s )

mP=4

mP=3

mP=2

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0Ipeak

gt0

Fig 2 Peak Interference-limit versus Rmin for various outageprobabilities for cons (solid lines) and opra (dashed lines)

secondary transmitter employs MQAM adaptive modulationscheme Hereafter we assume T f B = 1

We start by examining the effect of different transmis-

sion techniques namely opra and cons adopted by the pri-

mary user on the interference constraints obtained in this

paper Fig 1 depicts the average interference-limit versus

the minimum-rate required by the primary user with P p =15dBW The solid and dashed lines represent opra and cons

techniques respectively The arrows indicate the regions for

which I avg ge 0 holds true The figure shows that after certain

thresholds for Rmin the interference-limit decreases rapidly as

the minimum rate Rmin increases or as the outage probability

minus5 minus4 minus3 minus2 minus1 001

02

03

04

05

06

07

Interference limit (dBW)

N o

r m a l i z e d E f f e c t i v e C a p a c i t y ( n a t s s H z )

ms=m

sp=1

ms=1 m

sp=15

ms=1 m

sp=2

Fig 3 Normalized effective capacity of the secondary link versusinterference-limit average (solid lines) or peak (dashed lines)

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

045

θ (1nats)

N o r m a l i z e d E f f e c t i v e C a p a c i t y ( n a t s s H z )

ms=m

sp=1

ms=2 m

sp=1

ms=3 m

sp=1

ms=1 m

sp=2

ms=1 m

sp=3

Fig 4 Normalized effective capacity of the secondary user versusQoS exponent for various Nakagami parameters ms and msp

decreases The figure also reveals that the interference-powerconstraint obtained when the primary user employs cons

techniques is much tighter than those with opra case

Fig 2 on the other hand shows the results for the peak

interference power limit I peak obtained in Section III for

Nakagami fading parameters mp = 1 The plots depict the

peak interference-limit values versus the required minimum-

rate for the primary user with P p = 15dBW for different Nak-

agami fading parameters mp The figure shows that when mp

increases the peak interference-limit increases significantly

We continue by examining the effective capacity of the

secondary userrsquos when the secondary transmitter employs

8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d

E f f e c t i v e C a p a c i t y ( n a t s a H z )

Rayleigh BER=10(minus3)


ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)

Fig 5 Normalized effective capacity of the secondary userrsquos link versus QoS exponent θ for various Nakagami parameters ms and

msp and BER requirements


03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case

Continuous MQAMDiscrete MQAM

Fig 6 Normalized effective capacity of the secondary userrsquos link

versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2

continuous MQAM for different Nakagami fading parame-

ters Fig 3 depicts the normalized effective capacity versus

average (solid lines) and peak (dashed lines) interference-

limit values with θ = 01(1nats) and BER = 10minus3 This

figure includes the plots for the expectation equations of the

effective capacity ie (34) and (44) and their corresponding

closed-from expressions ie (37) and (45) The plots from

the expectation equations are shown by different markers with

no lines The closed-from expressions are shown with lines

steady and dashed lines with no markers As the figure shows

the closed-from expressions and the expectation equationsmatch perfectly We further observe that when the Nakagami

parameter of the interference link msp increases the effective

capacity decreases The figure also reveals that the capacity

under average interference constraint is considerably higher

than that under peak interference power constraint

On the other hand in Fig 4 we keep the fading parameter

of one of the links either hs or hsp fixed and change the

parameter on the other link The figure includes plots for

the effective capacity versus θ with I avg = minus5dBW and

BER = 10minus3 The figure reveals that the changes in the

fading parameter of the secondary userrsquos link have negligible

1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z

e d E f f e c t i v e C a p a c i t y ( n a t s s H z )

mp=2

mp=3


versus P outp with P p = 15dBW Rmin = 01natssHz θ = 01

BER=10minus3 and Nakagami parameters ms = msp = 1

01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)

N o r m a l i z e d E f f e c t i v e C a p a c i t y

( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW

Fig 8 Normalized effective capacity of the secondary userrsquos link versus Rmin under opra technique with mp = 3 θ = 01

BER=10minus3 and ms = msp = 1

effects on the effective capacity as long as the fading parameter

pertaining to hsp is fixed On the other hand increasing the

Nakagami parameter of hsp degrades the effective capacity of

the secondary userrsquos link significantly

Plots for the normalized effective capacity versus the delay

QoS exponent θ under average interference-power constraint

at I avg = minus5dBW are provided in Fig 5 We observe that

the capacity increases as θ decreases however the gain in the

effective capacity decreases for lower values of θ

Fig 6 depicts the effect of different modulation techniques

on the effective capacity of the secondary userrsquos link The

figure includes plots for three different cases namely con-tinuous MQAM discrete MQAM and the case when there

is no restriction on the coding employed by the secondary

transmitter referred to as the optimum case In this figure θhas been set to θ = 01 (1nats) BER=10minus3 and N = 5

The figure shows that the capacity with discrete MQAM is

smaller than that with continuous MQAM The loss in the

capacity however is small when compared to the one between

the optimum case and continuous MQAM

We further examine the effect of the service-outage prob-

ability of the primary user P outp on the achievable effective

capacity of the secondary userrsquos link in Fig 7 and Fig 8 In

8102019 05659492




9

particular Fig 7 depicts the plots for the effective capacity

of the secondary user versus P outp for various Nakagami

parameters for the primary userrsquos link mp under opra (solid

lines) and cons (dashed lines) schemes with P p = 15dBW

Rmin = 01natssHz θ = 01 BER=10minus3 and ms = msp =1 The figure reveals that under the same fading parameters

and service-outage constraints the effective capacity of the

secondary user link is higher when primary user employs cons

scheme compared to opra technique

Fig 8 includes the plots for the effective capacity versus

the minimum-rate required by the primary user for various

primary service-outage probabilities under opra transmission

technique with θ = 01 BER=10minus3 and Nakagami parameters

mp = 3 and ms = msp = 1 The solid and dashed lines

refer to P p = 15dBW and P p = 12dBW respectively The

figure shows that the capacity decreases significantly when

the minimum-rate required by the primary user increases

VII CONCLUSIONS

We considered spectrum-sharing channels in Nakagami-

m fading environments and studied the effects of adaptive

MQAM modulation on the capacity gain of the secondary

userrsquos channel under delay QoS constraints We assumed

that the spectrum band occupied by a primary user may be

accessed and utilized by a secondary user as long as the

latter adheres to interference limitations set by the primary

user Specifically the successful communication process of the

primary user requires a minimum-rate to be supported by its

channel for a certain percentage of time We obtained average

or peak interference-power constraints as a sufficient condition

for satisfying the service-outage requirement of the primary

user Under average or peak interference-power constraint we

obtained the effective capacity of the secondary userrsquos channelfor two different modulation schemes namely continuous

MQAM and discrete MQAM with limited constellations For

these schemes we determined the optimal power and rate

allocation strategies that maximize the effective capacity Also

we obtained closed-form expressions for the capacity and

the corresponding power allocation policy under Nakagami-

m block-fading for continuous MQAM Considering the Nak-

agami parameter m as a measure of fading severity it has been

observed that the effective capacity of the secondary user is

more sensitive to the fading severity of the interference link

between secondary transmitter and primary receiver compared

to the one between the secondary transmitter and receiver of

the secondary user

APPENDIX A

The integration in the effective capacity formula in (45) can

be expanded as follows

E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0

(1 + KI peakw)minusα wmsminus1983080

w + 1ρ

983081ms+mspdw

I

852009

where w = 1v

and I can be simplified by using the change

of variable x = 11+w

according to

I = (KI peak)minusα

991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα

times (1 minus x)msminus1

10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)

Then using the following expression [20]

Γ(a)Γ(γ minus a)

Γ(γ ) F 1(a β β prime γ x y) =

991787 10

taminus1

times (1 minus t)γ minusaminus1(1 minus tx)minusβ(1 minus ty)minusβprime

dt

(48)

for Re(a) gt 0 Re(γ minus a) gt 0 |x| lt 1 and |y| lt 1

and inserting (48) into (47) when setting a = msp +α β = α

β prime = ms + msp γ = ms + msp + α x = 1 minus 1KI peak

and

y = 1 minus 1ρ

we get

I =(KI peak)minusαΓ(ms)Γ(msp + α)

Γ(ms + msp + α) F 1

983080msp + α α

ms + msp ms + msp + α 1 minus 1KI peak

1 minus 1ρ

983081

(49)

Note that the condition |x| lt 1 and |y| lt 1 imply that

KI peak gt 05 and ρ gt 05 respectively

We now obtain an alternative solution for the closed-from

expression of the effective capacity when the above-mentioned

inequalities on K I peak and ρ do not hold We first apply the

change of variable x = w1+w

on I

I = ρms+msp

991787 10

xmspminus1(1 minus x)ms+αminus1 (50)

times (1 minus (1 minus KI peak) x)minusα

(1 minus (1 minus ρ) x)minus(ms+msp) dx

Now by setting a = msp β = α β prime = ms + msp γ =ms + msp + α x = 1 minus KI peak y = 1 minus ρ and inserting (48)

into (50) we get

I = ρms+mspΓ(ms)Γ(msp + α)

Γ(ms + msp + α) (51)

times F 1 (ms α ms + msp ms + msp + α 1 minus KI peak 1 minus ρ)

where the conditions |x| le 1 and |y| le 1 imply that KI peak lt2 and ρ lt 2 and as such (51) is correct when 0 le KI peak lt 2and 0 le ρ lt 2 This concludes the proof for (45)

REFERENCES

[1] A J Goldsmith and P P Varaiya ldquoCapacity of fading channels withchannel side informationrdquo IEEE Trans Inf Theory vol 43 no 6 pp1986ndash1992 Nov 1997

[2] A J Goldsmith and S-G Chua ldquovariable-rate variable-power MQAMfor fading channelsrdquo IEEE Trans Commun vol 45 no 10 pp 1218ndash1230 Oct 1997

[3] T A Weiss and F K Jondral ldquoSpectrum pooling An innovative strategyfor the enhancement of spectrum efficiencyrdquo IEEE Commun Magvol 42 no 3 pp S8ndashS14 Mar 2004

[4] D Wu and R Negi ldquoEffective capacity A wireless link model forsupport of quality of servicerdquo IEEE Trans wireless Commun vol 2no 4 pp 630ndash643 July 2003

[5] C-S Chang ldquoStability queue length and delay of deterministic andstochastic queueing networksrdquo IEEE Trans Automatic Control vol 39no 5 pp 913ndash931 May 1994

8102019 05659492


Copyright (c) 2010 IEEE Personal use is permitted For any other purposes Permission must be obtained from the IEEE by emailing pubs-permissionsieee org


10

[6] J Tang and X Zhang ldquoQuality-of-service driven power and rateadaptation over wireless linksrdquo IEEE Trans Wireless Commun vol 6no 8 pp 3058ndash3068 Aug 2007

[7] W Chen K B Letaief and Z Cao ldquoA joint coding and schedulingmethod for delay optimal cognitive multiple accessrdquo in Proc IEEE ConfCommun (ICC 2008) China Beijing May 2008

[8] M Rashid M Hossain E Hossain and V Bhargava ldquoOpportunisticspectrum scheduling for multiuser cognitive radio a queueing analysisrdquo

IEEE Trans Wireless Commun vol 8 no 10 pp 5259 ndash 5269 Nov2009

[9] L Gao and S Cui ldquoPower and rate control for delay-constrainedcognitive radios via dynamic programmingrdquo IEEE Trans Veh Technolvol 58 no 9 pp 4819 ndash 4827 Nov 2009

[10] M Gastpar ldquoOn capacity under received-signal constraintsrdquo in Proc42nd Annual Allerton Conf on Commun Control and Comp Monti-cello USA Sept 29-Oct 1 2004

[11] A Ghasemi and E S Sousa ldquoCapacity of fading channels underspectrum-sharing constraintsrdquo in Proc IEEE Int Conf Commun (ICC)Istanbul Turkey June 2006 pp 4373ndash4378

[12] L Musavian and S Aiumlssa ldquoCapacity and power allocation for spectrum-sharing communications in fading channelsrdquo IEEE Trans WirelessCommun vol 8 no 1 pp 148ndash156 Jan 2009

[13] K T Phan S A Vorobyov N D Sidiropoulos and C TellamburaldquoSpectrum sharing in wireless networks via QoS-aware secondary mul-ticast beamformingrdquo IEEE Trans Signal Proc vol 57 no 6 pp 2323ndash2335 June 2009

[14] L Musavian and S Aiumlssa ldquoQuality of service based power allocation

in spectrum-sharing channelsrdquo in Proc IEEE Global Commun Conf(Globecom 2008) Neworleans LA USA 30 Nov-4 Dec 2008

[15] J M Peha ldquoApproaches to spectrum sharingrdquo IEEE Commun Magvol 43 no 2 pp 10ndash12 Feb 2005

[16] A Jovicic and P Viswanath ldquoCognitive radio An information-theoreticperspectiverdquo submitted to IEEE Trans Inf Theory May 2006 availableonline at httparxivorgPS_cachecspdf06040604107v2pdf

[17] L Zhang Y-C Liang and Y Xin ldquoJoint beamforming and powerallocation for multiple access channels in cognitive radio networksrdquo

IEEE J Sel Areas Commun vol 26 no 1 pp 38ndash51 Jan 2008[18] L Musavian and S Aiumlssa ldquoOutage-constrained capacity of spectrum-

sharing channels in fading environmentsrdquo IET Commun vol 2 no 6pp 724ndash732 July 2008

[19] M K Simon and M-S Alouini Digital Communication over FadingChannels A Unified Approach to Performance Analysis John Wileyand Sons Inc 2000

[20] M Abramowitz and I A Stegun Handbook of mathematical functions

New York Dover 1965[21] E W Weisstein (From mathWorldndasha wolfram

web resource) Gamma distribution [Online] AvailablehttpmathworldwolframcomGammaDistributionhtml

[22] W N Bailey ldquoOn the reducibility of appellrsquos functionrdquo Quart J Math(Oxford) no 5 pp 291ndash292 1934

Leila Musavian (Srsquo05-Mrsquo07) obtained her PhDdegree in Electrical and Electronics Engineeringfrom Kingrsquos College London London UK in 2006and her BSc degree in Electrical Engineering fromSharif University of Technology Tehran Iran in1999 she was a post doctoral fellow at National In-stitute of Scientific Research-Energy Materials andTelecommunications (INRS-EMT) University of Quebec Montreal Canada Afterwards she joinedthe Advanced Signal Processing Group Electronicsand Electrical Engineering Loughborough Univer-

sity Leicestershire UK as a research associate She has been TPC memberof ICC2008 ICC2009 Globecom2008 WCNC2009 and Globecom2009

Dr Musavianrsquos research interests include radio resource management fornext generation wireless networks Cognitive radios performance analysis of MIMO systems and cross-layer design

Sonia Aiumlssa (Srsquo93-Mrsquo00-SMrsquo03) received her PhDdegree in Electrical and Computer Engineeringfrom McGill University Montreal QC Canada in1998 Since then she has been with the NationalInstitute of Scientific Research-Energy Materialsand Telecommunications (INRS-EMT) Universityof Quebec Montreal Canada where she is currentlya Full Professor

From 1996 to 1997 she was a Researcher withthe Department of Electronics and Communications

of Kyoto University Kyoto Japan and with theWireless Systems Laboratories of NTT Kanagawa Japan From 1998 to 2000she was a Research Associate at INRS-EMT Montreal From 2000 to 2002while she was an Assistant Professor she was a Principal Investigator inthe major program of personal and mobile communications of the CanadianInstitute for Telecommunications Research (CITR) leading research in radioresource management for code division multiple access systems From 2004to 2007 she was an Adjunct Professor with Concordia University MontrealIn 2006 she was Visiting Invited Professor with the Graduate School of Informatics Kyoto University Japan Her research interests lie in the area of wireless and mobile communications and include radio resource managementcross-layer design and optimization design and analysis of multiple antenna(MIMO) systems and performance evaluation with a focus on Cellular AdHoc and Cognitive Radio networks

Dr Aiumlssa was the Founding Chair of the Montreal Chapter IEEE Womenin Engineering Society in 2004-2007 a Technical Program Cochair for theWireless Communications Symposium (WCS) of the 2006 IEEE International

Conference on Communications (ICC 2006) and PHYMAC Program Chairfor the 2007 IEEE Wireless Communications and Networking Conference(WCNC 2007) She was also the Technical Program Leading Chair for theWCS of the IEEE ICC 2009 and is currently serving as Cochair for the WCSof the IEEE ICC 2011 She has served as a Guest Editor of the EURASIP

journal on Wireless Communications and Networking in 2006 and as Asso-ciate Editor of the IEE E W IRELESS COMMUNICATIONS MAGAZINE in 2006-2010 She is currently an Editor of the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS the IEEE TRANSACTIONS ON C OMMUNICATIONS andthe IEEE COMMUNICATIONS M AGAZINE and Associate Editor of the WileySecurity and Communication Networks Journal Awards and distinctions toher credit include the Quebec Government FQRNT Strategic Fellowship forProfessors-Researchers in 2001-2006 the INRS-EMT Performance Awardin 2004 for outstanding achievements in research teaching and service theIEEE Communications Society Certificate of Appreciation in 2006 2009 and2010 and the Technical Community Service Award from the FQRNT Centerfor Advanced Systems and Technologies in Communications (SYTACom)

in 2007 She is also co-recipient of Best Paper Awards from IEEE ISCC2009 WPMC 2010 and IEEE WCNC 2010 and recipient of NSERC (NaturalSciences and Engineering Research Council of Canada) Discovery AcceleratorSupplement Award

Dr Sangarapillai Lambotharan holds a Reader-ship in Communications within the Advanced SignalProcessing Group Department of Electronic andElectrical Engineering Loughborough UniversityUK He received a PhD degree in Signal Process-ing from Imperial College UK in 1997 where

he remained until 1999 as a postdoctoral researchassociate working on an EPSRC funded project onmobile communications He was a visiting scientistat the Engineering and Theory Centre of CornellUniversity USA in 1996 Between 1999 and 2002

he was with the Motorola Applied Research Group UK and investigatedvarious projects including physical link layer modelling and performance char-acterization of GPRS EGPRS and UTRAN He has been with Kingrsquos CollegeLondon UK and Cardiff University UK as a lecturer and senior lecturerrespectively from 2002 to 2007 His current research interests include spatialdiversity techniques wireless relay networks and cognitive radios He servesas an associate editor for EURASIP Journal on Wireless Communications andNetworking

8102019 05659492




2

the secondary type of access to the spectrum for secondary

users This issue is very challenging and has attracted many

researchers eg [7]ndash[9]

In assessing the performance of spectrum-sharing chan-

nels in fading environments we refer to the earlier work

of Gastpar who presented capacity investigations of additive

white Gaussian noise (AWGN) spectrum-sharing channels

under interference power constraint rather than transmit power

constraint [10] Later ergodic and outage capacity metrics of

a point-to-point system with constraints on the received-power

at the primaryrsquos receiver in fading environment were derived

in [11] and [12] The underlay spectrum-sharing approach is

considered in [13] wherein optimum power allocation strate-

gies are proposed such that the interference to the primary

user is minimized while a set of minimum signal-to-noise

ratio (SNR) targets is provided at the secondary receivers

A delay QoS-driven power and rate allocation scheme under

spectrum-sharing constraint was proposed in [14] wherein

the effective capacity of a point-to-point channel in Rayleigh

fading environment was determined

In this paper we consider spectrum sharing systems forwhich the transmission of the secondary user is subject to

constraints on the interference-power inflicted on the primary

receiver In general we assume that there are guidelines and

limitations set by the regulatory bodies on the maximum

interference power in terms of peak or average values in-

flicted on the primary users In addition there are certain

service outage constraints for the primary users that should be

satisfied irrespective of the existence of the secondary users

in the network Specifically we wish to limit the transmission

parameters of the secondary transmitter such that the primary

user is supported with a minimum-rate for a certain percent-

age of time Translating this limitation into an interference-

power constraint either on peak or average interference limitswe obtain the maximum throughput of the secondary userrsquos

channel under delay QoS constraint by obtaining the effective

capacity of the channel We determine the maximum arrival-

rate that can be supported by the secondary userrsquos link subject

to satisfying a statistical delay QoS constraint by obtaining

the effective capacity of the channel under adaptive MQAM

with interference-power constraint We further obtain closed-

form expressions for the effective capacity and its power

allocation in Nakagami-m block-fading channels The service

outage constraint considered in this work is different from

our previous work on effective capacity of cognitive radio

channel In addition in this paper we assume that secondary

users implement MQAM which has not been studied in ourprevious publications

The subsequent sections are organized as follows In Section

II we provide the channel and system models The inter-

ference power is studied in Section III wherein the primary

userrsquos service outage constraint is translated into an average

or peak interference power constraint The effective capacity

of the secondary user channel under average interference

power constraint is provided in closed-form in Section IV The

effective capacity of the channel under peak interference power

constraint is studied in Section V Numerical results are given

in Section VI followed by conclusions in Section VII

I I SYSTEM M ODEL

The transmission parameters of the secondary user are

chosen such that the service outage requirement of the primary

user is satisfied The effect of the transmission of the primary

user on the secondary receiver is assumed as AWGN In the

secondary user communication system the upper layer packets

are organized into frames with duration T f The secondary

transmitter employs adaptive MQAM with continuous or dis-crete constellations Discrete-time block-fading channels are

assumed for both the secondary and primary usersrsquo links

The channel gain between the transmitter and receiver of

the secondary user and the AWGN are denoted by hs[t] and

zs[t] respectively where t denotes the time index We define

the channel gain between the secondaryrsquos transmitter and the

primaryrsquos receiver by hsp[t] We assume hs[t] and hsp[t] are

statistically independent and identically distributed (iid) and

also independent from the noise The channel envelopes are

distributed according to Nakagami-m fading Channel gains

are stationary and ergodic random processes

The secondary transmitter is provided with knowledge of

hs[t] and hsp[t] Information about the latter can be obtainedfrom a band manager that intervenes between the primary and

secondary users [15] or can be directly fed back from the

primaryrsquos receiver to the secondary user as proposed in [16]

[17] where the protocols allow the primary and secondary

users to collaborate and exchange CSI The effect of imper-

fection in the knowledge of the channel gains between the

secondary transmitter and primary receiver at the secondary

transmitter on the ergodic capacity of the secondary userrsquos link

has been investigated in [18] for Rayleigh fading channels The

secondary user knows only the statistical information of the

link between the transmitter and receiver of the primary user

hp[t] The instantaneous channel knowledge of hp[t] is known

to the primary userrsquos transmitter

We consider a statistical delay constraint according to

Pr D(t) ge Dmax le P outdelay (1)

where D(t) indicates the delay experienced by a packet at

time instant t and Dmax is the maximum delay that can be

tolerated for 1 minus P outdelay percentage of time We further assume

that the transmission technique of the secondary user must

satisfy a statistical delay QoS constraint It is shown that

the probability for the queue length of the transmit buffer

exceeding a certain threshold x decays exponentially fast as

a function of x [5] [6] We now define θ as the delay QoS

exponent given by

θ = minus limxrarrinfin

ln(Pr q (infin) gt x)

x (2)

where q (n) denotes the transmit buffer length at time n and

Pr a gt b denotes the probability that the inequality a gt bholds true Considering a data source with constant data rate r

the QoS exponent θ is related to the delay violation probability

according to

supt

Pr D(t) ge Dmax asymp γ (r)eminusθDmax (3)

where γ (r) = P rQ(t) ge 0 is the probability of a non-

8102019 05659492




3



















Pr

Rp le Rmin

le P outp (5)





















ln

10486161 +

P php


1048617 le Rp (7)


















P outp ge Pr852091


(9)



P outp ge

991787 infin0

991787 infin0

991787 k1(P shsp+N 0B)

0

f hp(hp)dhp I 0


(10)



f hp(hp) = m

mpp h

mpminus1p








I 0 = 1

Γ(mp)





I 0 =

852008minuseminusx

mpminus1sumk=0

xk

k


(13)



P outp ge E hshsp


852093 (14)









simplifies to

P outp ge E hshsp


mpminus1sumk=0

(f cons(P shsp))k

k

1048701

(15)



8102019 05659492




4










eminusf cons(x)

mpminus1sumk=0

(f cons(x))k










constraint







I avg = minusln(

1 minus P outp

)+ k2

k1minus N 0B (18)

where k1 =

eRminminus1

micro and

k2 =

N 0B








P p(hp) =

852059micro minus

N 0B

hp

852061+ (19)





P outP gePr

983163hp lt

N 0B

micro

983165+ Pr


N 0B

micro

983165


Pr

983163ln

10486161 +

P p(hp)hp

P shsp + N 0B

1048617 le Rmin


(20)


P outp ge Pr852091


(21)











A Continuous MQAM












which leads to











function


1

N ln983080

E 852091

eminusθsum

N t=1 R[t]

852093983081 (24)



θ = minus lim

N rarrinfin

1


sumN t=1 R[t]

852093983081







E c(θ) = minus1

θ ln983080

E 852091

eminusθR[t]852093983081

(25)




8102019 05659492




5



1048699minus

1

θ ln

852008E hshsp


(1+KP shs





P s =

983131 β

11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khs

983133+

(27)


γ 0 = 1λ0




P s =

β 11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khsif hsp le Kβhs

0 otherwise

(28)


we need to evaluate

the integration in

I avg =

991787 infin0

991787 Kβhs

0

1048616β

11+α

1048616 hsp

Khs

1048617 α1+α

minus hsp

Khs


(29)



Using





f v(v) = ρms

β (ms msp)

vmspminus1

(v + ρ)ms+msp

(30)

where ρ = ms






as follows

I avg = ρms

Kβ (ms msp)

991787 Kβ

0

983080(Kβ )

11+α v

α1+α minus v

983081times

vmspminus1

(v + ρ)ms+msp

dv

(31)

= ρms(Kβ )msp+1


times

983131991787 10


10486161 minus

Kβ

Kβ + ρx


dx J 0

minus

991787 10

(1 minus x)msp

10486161 minus

Kβ

Kβ + ρx


dx J 1

983133




J 0 =Γ983080

msp + α1+α

983081Γ983080

msp + 1+2α1+α

983081times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

(33)







9831311048616msp +

α

1 + α

1048617minus1

times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

minus 1

msp + 12F 1

1048616ms + msp 1 msp + 2

Kβ

Kβ + ρ

1048617 983133





θ ln

852008E v

104869910486161 +

1048667(Kβ )

11+α v

minus11+α minus 1

1048669+1048617minusα1048701852009

= minus1

θ ln

1048616ρms(Kβ )

minusα1+α

β (ms msp)

991787 Kβ

0

vmspminus 11+α

(v + ρ)ms+msp

dv

+ ρms

β (ms msp)

991787 infinKβ

vmspminus1

(v + ρ)ms+msp

dv

1048617 (34)

By using ρms

β(msm


(v+ρ)


B Restricted MQAM

















w hshsp





8102019 05659492




6


θ ln

8520081+

(Kβ )mspρms


983080msp + α

1+α

9830812F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

ρ + Kβ

1048617

minus (Kβ )mspρms


1048616ms + msp 1 msp + 1

Kβ

ρ + Kβ

1048617852009 (37)











λ0 1048617 11+α

(38)


W n = M 1+αn

λlowast0Kα

(39)







P s =

M n minus 1


0 0 le w lt W 1

(40)





991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0

Kα

M n minus 1

K times

1

w f w(w)dw

+

991787 infinM

α+1N

λlowast0

Kα

M N minus 1

K times

1

w f w(w)dw

(41)

where

f w(w) = ρminusmsp

β (msp ms)

wmsminus1

(w + 1ρ)m

sp+m

s

(42)



θ ln


991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0Kα

M minusαn f w(w)dw

+

991787 infinM

α+1N

λlowast0Kα

M minusαN f w(w)dw

1048617

(43)






A Continuous MQAM


as P s = I peak


R[t] = T f B ln

10486161 + I peakK

hs

hsp


capacity


1048617minusα1048701852009 (44)






infinsumm=0

infinsumn=0


xmyn


symbol [20]

B Restricted MQAM











Therefore



θ ln


991787 M n+1Ipeak

M nIpeak

M minusαn f w(w)dw

+


M minusαN f w(w)dw

1048617

(46)





8102019 05659492




7

E c(θ) =

minus1

θ ln

1048616F 1


1

KI peak 1 minus

1

ρ



1048617


minus1

θ ln




1048617


(45)

01 02 03 04 05 06 07 08 09 1 11

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I a v g

( w a t t s )

Pout

p =1

Pout

p =2

Pout

p =3

Iavg

gt0

Iavg

gt0

Iavg

gt0

Iavg

gt0Iavg

gt0

Iavg

gt0


06 08 1 12 14 16 18 2 22

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I p e a k

( w a t t s )

mP=4

mP=3

mP=2

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0Ipeak

gt0













02

03

04

05

06

07


N o


ms=m

sp=1

ms=1 m

sp=15

ms=1 m

sp=2


10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

045

θ (1nats)


ms=m

sp=1

ms=2 m

sp=1

ms=3 m

sp=1

ms=1 m

sp=2

ms=1 m

sp=3













8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d




ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)




03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case



versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2






















1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z


mp=2

mp=3




01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)


( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW
























8102019 05659492




9

















VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






8102019 05659492




10




































8102019 05659492




3



















Pr

Rp le Rmin

le P outp (5)





















ln

10486161 +

P php


1048617 le Rp (7)


















P outp ge Pr852091


(9)



P outp ge

991787 infin0

991787 infin0

991787 k1(P shsp+N 0B)

0

f hp(hp)dhp I 0


(10)



f hp(hp) = m

mpp h

mpminus1p








I 0 = 1

Γ(mp)





I 0 =

852008minuseminusx

mpminus1sumk=0

xk

k


(13)



P outp ge E hshsp


852093 (14)









simplifies to

P outp ge E hshsp


mpminus1sumk=0

(f cons(P shsp))k

k

1048701

(15)



8102019 05659492




4










eminusf cons(x)

mpminus1sumk=0

(f cons(x))k










constraint







I avg = minusln(

1 minus P outp

)+ k2

k1minus N 0B (18)

where k1 =

eRminminus1

micro and

k2 =

N 0B








P p(hp) =

852059micro minus

N 0B

hp

852061+ (19)





P outP gePr

983163hp lt

N 0B

micro

983165+ Pr


N 0B

micro

983165


Pr

983163ln

10486161 +

P p(hp)hp

P shsp + N 0B

1048617 le Rmin


(20)


P outp ge Pr852091


(21)











A Continuous MQAM












which leads to











function


1

N ln983080

E 852091

eminusθsum

N t=1 R[t]

852093983081 (24)



θ = minus lim

N rarrinfin

1


sumN t=1 R[t]

852093983081







E c(θ) = minus1

θ ln983080

E 852091

eminusθR[t]852093983081

(25)




8102019 05659492




5



1048699minus

1

θ ln

852008E hshsp


(1+KP shs





P s =

983131 β

11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khs

983133+

(27)


γ 0 = 1λ0




P s =

β 11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khsif hsp le Kβhs

0 otherwise

(28)


we need to evaluate

the integration in

I avg =

991787 infin0

991787 Kβhs

0

1048616β

11+α

1048616 hsp

Khs

1048617 α1+α

minus hsp

Khs


(29)



Using





f v(v) = ρms

β (ms msp)

vmspminus1

(v + ρ)ms+msp

(30)

where ρ = ms






as follows

I avg = ρms

Kβ (ms msp)

991787 Kβ

0

983080(Kβ )

11+α v

α1+α minus v

983081times

vmspminus1

(v + ρ)ms+msp

dv

(31)

= ρms(Kβ )msp+1


times

983131991787 10


10486161 minus

Kβ

Kβ + ρx


dx J 0

minus

991787 10

(1 minus x)msp

10486161 minus

Kβ

Kβ + ρx


dx J 1

983133




J 0 =Γ983080

msp + α1+α

983081Γ983080

msp + 1+2α1+α

983081times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

(33)







9831311048616msp +

α

1 + α

1048617minus1

times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

minus 1

msp + 12F 1

1048616ms + msp 1 msp + 2

Kβ

Kβ + ρ

1048617 983133





θ ln

852008E v

104869910486161 +

1048667(Kβ )

11+α v

minus11+α minus 1

1048669+1048617minusα1048701852009

= minus1

θ ln

1048616ρms(Kβ )

minusα1+α

β (ms msp)

991787 Kβ

0

vmspminus 11+α

(v + ρ)ms+msp

dv

+ ρms

β (ms msp)

991787 infinKβ

vmspminus1

(v + ρ)ms+msp

dv

1048617 (34)

By using ρms

β(msm


(v+ρ)


B Restricted MQAM

















w hshsp





8102019 05659492




6


θ ln

8520081+

(Kβ )mspρms


983080msp + α

1+α

9830812F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

ρ + Kβ

1048617

minus (Kβ )mspρms


1048616ms + msp 1 msp + 1

Kβ

ρ + Kβ

1048617852009 (37)











λ0 1048617 11+α

(38)


W n = M 1+αn

λlowast0Kα

(39)







P s =

M n minus 1


0 0 le w lt W 1

(40)





991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0

Kα

M n minus 1

K times

1

w f w(w)dw

+

991787 infinM

α+1N

λlowast0

Kα

M N minus 1

K times

1

w f w(w)dw

(41)

where

f w(w) = ρminusmsp

β (msp ms)

wmsminus1

(w + 1ρ)m

sp+m

s

(42)



θ ln


991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0Kα

M minusαn f w(w)dw

+

991787 infinM

α+1N

λlowast0Kα

M minusαN f w(w)dw

1048617

(43)






A Continuous MQAM


as P s = I peak


R[t] = T f B ln

10486161 + I peakK

hs

hsp


capacity


1048617minusα1048701852009 (44)






infinsumm=0

infinsumn=0


xmyn


symbol [20]

B Restricted MQAM











Therefore



θ ln


991787 M n+1Ipeak

M nIpeak

M minusαn f w(w)dw

+


M minusαN f w(w)dw

1048617

(46)





8102019 05659492




7

E c(θ) =

minus1

θ ln

1048616F 1


1

KI peak 1 minus

1

ρ



1048617


minus1

θ ln




1048617


(45)

01 02 03 04 05 06 07 08 09 1 11

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I a v g

( w a t t s )

Pout

p =1

Pout

p =2

Pout

p =3

Iavg

gt0

Iavg

gt0

Iavg

gt0

Iavg

gt0Iavg

gt0

Iavg

gt0


06 08 1 12 14 16 18 2 22

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I p e a k

( w a t t s )

mP=4

mP=3

mP=2

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0Ipeak

gt0













02

03

04

05

06

07


N o


ms=m

sp=1

ms=1 m

sp=15

ms=1 m

sp=2


10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

045

θ (1nats)


ms=m

sp=1

ms=2 m

sp=1

ms=3 m

sp=1

ms=1 m

sp=2

ms=1 m

sp=3













8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d




ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)




03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case



versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2






















1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z


mp=2

mp=3




01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)


( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW
























8102019 05659492




9

















VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






8102019 05659492




10




































8102019 05659492




4










eminusf cons(x)

mpminus1sumk=0

(f cons(x))k










constraint







I avg = minusln(

1 minus P outp

)+ k2

k1minus N 0B (18)

where k1 =

eRminminus1

micro and

k2 =

N 0B








P p(hp) =

852059micro minus

N 0B

hp

852061+ (19)





P outP gePr

983163hp lt

N 0B

micro

983165+ Pr


N 0B

micro

983165


Pr

983163ln

10486161 +

P p(hp)hp

P shsp + N 0B

1048617 le Rmin


(20)


P outp ge Pr852091


(21)











A Continuous MQAM












which leads to











function


1

N ln983080

E 852091

eminusθsum

N t=1 R[t]

852093983081 (24)



θ = minus lim

N rarrinfin

1


sumN t=1 R[t]

852093983081







E c(θ) = minus1

θ ln983080

E 852091

eminusθR[t]852093983081

(25)




8102019 05659492




5



1048699minus

1

θ ln

852008E hshsp


(1+KP shs





P s =

983131 β

11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khs

983133+

(27)


γ 0 = 1λ0




P s =

β 11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khsif hsp le Kβhs

0 otherwise

(28)


we need to evaluate

the integration in

I avg =

991787 infin0

991787 Kβhs

0

1048616β

11+α

1048616 hsp

Khs

1048617 α1+α

minus hsp

Khs


(29)



Using





f v(v) = ρms

β (ms msp)

vmspminus1

(v + ρ)ms+msp

(30)

where ρ = ms






as follows

I avg = ρms

Kβ (ms msp)

991787 Kβ

0

983080(Kβ )

11+α v

α1+α minus v

983081times

vmspminus1

(v + ρ)ms+msp

dv

(31)

= ρms(Kβ )msp+1


times

983131991787 10


10486161 minus

Kβ

Kβ + ρx


dx J 0

minus

991787 10

(1 minus x)msp

10486161 minus

Kβ

Kβ + ρx


dx J 1

983133




J 0 =Γ983080

msp + α1+α

983081Γ983080

msp + 1+2α1+α

983081times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

(33)







9831311048616msp +

α

1 + α

1048617minus1

times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

minus 1

msp + 12F 1

1048616ms + msp 1 msp + 2

Kβ

Kβ + ρ

1048617 983133





θ ln

852008E v

104869910486161 +

1048667(Kβ )

11+α v

minus11+α minus 1

1048669+1048617minusα1048701852009

= minus1

θ ln

1048616ρms(Kβ )

minusα1+α

β (ms msp)

991787 Kβ

0

vmspminus 11+α

(v + ρ)ms+msp

dv

+ ρms

β (ms msp)

991787 infinKβ

vmspminus1

(v + ρ)ms+msp

dv

1048617 (34)

By using ρms

β(msm


(v+ρ)


B Restricted MQAM

















w hshsp





8102019 05659492




6


θ ln

8520081+

(Kβ )mspρms


983080msp + α

1+α

9830812F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

ρ + Kβ

1048617

minus (Kβ )mspρms


1048616ms + msp 1 msp + 1

Kβ

ρ + Kβ

1048617852009 (37)











λ0 1048617 11+α

(38)


W n = M 1+αn

λlowast0Kα

(39)







P s =

M n minus 1


0 0 le w lt W 1

(40)





991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0

Kα

M n minus 1

K times

1

w f w(w)dw

+

991787 infinM

α+1N

λlowast0

Kα

M N minus 1

K times

1

w f w(w)dw

(41)

where

f w(w) = ρminusmsp

β (msp ms)

wmsminus1

(w + 1ρ)m

sp+m

s

(42)



θ ln


991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0Kα

M minusαn f w(w)dw

+

991787 infinM

α+1N

λlowast0Kα

M minusαN f w(w)dw

1048617

(43)






A Continuous MQAM


as P s = I peak


R[t] = T f B ln

10486161 + I peakK

hs

hsp


capacity


1048617minusα1048701852009 (44)






infinsumm=0

infinsumn=0


xmyn


symbol [20]

B Restricted MQAM











Therefore



θ ln


991787 M n+1Ipeak

M nIpeak

M minusαn f w(w)dw

+


M minusαN f w(w)dw

1048617

(46)





8102019 05659492




7

E c(θ) =

minus1

θ ln

1048616F 1


1

KI peak 1 minus

1

ρ



1048617


minus1

θ ln




1048617


(45)

01 02 03 04 05 06 07 08 09 1 11

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I a v g

( w a t t s )

Pout

p =1

Pout

p =2

Pout

p =3

Iavg

gt0

Iavg

gt0

Iavg

gt0

Iavg

gt0Iavg

gt0

Iavg

gt0


06 08 1 12 14 16 18 2 22

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I p e a k

( w a t t s )

mP=4

mP=3

mP=2

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0Ipeak

gt0













02

03

04

05

06

07


N o


ms=m

sp=1

ms=1 m

sp=15

ms=1 m

sp=2


10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

045

θ (1nats)


ms=m

sp=1

ms=2 m

sp=1

ms=3 m

sp=1

ms=1 m

sp=2

ms=1 m

sp=3













8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d




ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)




03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case



versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2






















1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z


mp=2

mp=3




01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)


( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW
























8102019 05659492




9

















VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






8102019 05659492




10




































8102019 05659492




5



1048699minus

1

θ ln

852008E hshsp


(1+KP shs





P s =

983131 β

11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khs

983133+

(27)


γ 0 = 1λ0




P s =

β 11+α

h1

1+αsp (Khs)

α1+α

minus 1

Khsif hsp le Kβhs

0 otherwise

(28)


we need to evaluate

the integration in

I avg =

991787 infin0

991787 Kβhs

0

1048616β

11+α

1048616 hsp

Khs

1048617 α1+α

minus hsp

Khs


(29)



Using





f v(v) = ρms

β (ms msp)

vmspminus1

(v + ρ)ms+msp

(30)

where ρ = ms






as follows

I avg = ρms

Kβ (ms msp)

991787 Kβ

0

983080(Kβ )

11+α v

α1+α minus v

983081times

vmspminus1

(v + ρ)ms+msp

dv

(31)

= ρms(Kβ )msp+1


times

983131991787 10


10486161 minus

Kβ

Kβ + ρx


dx J 0

minus

991787 10

(1 minus x)msp

10486161 minus

Kβ

Kβ + ρx


dx J 1

983133




J 0 =Γ983080

msp + α1+α

983081Γ983080

msp + 1+2α1+α

983081times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

(33)







9831311048616msp +

α

1 + α

1048617minus1

times 2F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

Kβ + ρ

1048617

minus 1

msp + 12F 1

1048616ms + msp 1 msp + 2

Kβ

Kβ + ρ

1048617 983133





θ ln

852008E v

104869910486161 +

1048667(Kβ )

11+α v

minus11+α minus 1

1048669+1048617minusα1048701852009

= minus1

θ ln

1048616ρms(Kβ )

minusα1+α

β (ms msp)

991787 Kβ

0

vmspminus 11+α

(v + ρ)ms+msp

dv

+ ρms

β (ms msp)

991787 infinKβ

vmspminus1

(v + ρ)ms+msp

dv

1048617 (34)

By using ρms

β(msm


(v+ρ)


B Restricted MQAM

















w hshsp





8102019 05659492




6


θ ln

8520081+

(Kβ )mspρms


983080msp + α

1+α

9830812F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

ρ + Kβ

1048617

minus (Kβ )mspρms


1048616ms + msp 1 msp + 1

Kβ

ρ + Kβ

1048617852009 (37)











λ0 1048617 11+α

(38)


W n = M 1+αn

λlowast0Kα

(39)







P s =

M n minus 1


0 0 le w lt W 1

(40)





991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0

Kα

M n minus 1

K times

1

w f w(w)dw

+

991787 infinM

α+1N

λlowast0

Kα

M N minus 1

K times

1

w f w(w)dw

(41)

where

f w(w) = ρminusmsp

β (msp ms)

wmsminus1

(w + 1ρ)m

sp+m

s

(42)



θ ln


991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0Kα

M minusαn f w(w)dw

+

991787 infinM

α+1N

λlowast0Kα

M minusαN f w(w)dw

1048617

(43)






A Continuous MQAM


as P s = I peak


R[t] = T f B ln

10486161 + I peakK

hs

hsp


capacity


1048617minusα1048701852009 (44)






infinsumm=0

infinsumn=0


xmyn


symbol [20]

B Restricted MQAM











Therefore



θ ln


991787 M n+1Ipeak

M nIpeak

M minusαn f w(w)dw

+


M minusαN f w(w)dw

1048617

(46)





8102019 05659492




7

E c(θ) =

minus1

θ ln

1048616F 1


1

KI peak 1 minus

1

ρ



1048617


minus1

θ ln




1048617


(45)

01 02 03 04 05 06 07 08 09 1 11

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I a v g

( w a t t s )

Pout

p =1

Pout

p =2

Pout

p =3

Iavg

gt0

Iavg

gt0

Iavg

gt0

Iavg

gt0Iavg

gt0

Iavg

gt0


06 08 1 12 14 16 18 2 22

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I p e a k

( w a t t s )

mP=4

mP=3

mP=2

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0Ipeak

gt0













02

03

04

05

06

07


N o


ms=m

sp=1

ms=1 m

sp=15

ms=1 m

sp=2


10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

045

θ (1nats)


ms=m

sp=1

ms=2 m

sp=1

ms=3 m

sp=1

ms=1 m

sp=2

ms=1 m

sp=3













8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d




ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)




03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case



versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2






















1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z


mp=2

mp=3




01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)


( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW
























8102019 05659492




9

















VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






8102019 05659492




10




































8102019 05659492




6


θ ln

8520081+

(Kβ )mspρms


983080msp + α

1+α

9830812F 1

1048616ms + msp 1 msp +

1 + 2α

1 + α

Kβ

ρ + Kβ

1048617

minus (Kβ )mspρms


1048616ms + msp 1 msp + 1

Kβ

ρ + Kβ

1048617852009 (37)











λ0 1048617 11+α

(38)


W n = M 1+αn

λlowast0Kα

(39)







P s =

M n minus 1


0 0 le w lt W 1

(40)





991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0

Kα

M n minus 1

K times

1

w f w(w)dw

+

991787 infinM

α+1N

λlowast0

Kα

M N minus 1

K times

1

w f w(w)dw

(41)

where

f w(w) = ρminusmsp

β (msp ms)

wmsminus1

(w + 1ρ)m

sp+m

s

(42)



θ ln


991787 M α+1n+1

λlowast0Kα

M α+1n

λlowast0Kα

M minusαn f w(w)dw

+

991787 infinM

α+1N

λlowast0Kα

M minusαN f w(w)dw

1048617

(43)






A Continuous MQAM


as P s = I peak


R[t] = T f B ln

10486161 + I peakK

hs

hsp


capacity


1048617minusα1048701852009 (44)






infinsumm=0

infinsumn=0


xmyn


symbol [20]

B Restricted MQAM











Therefore



θ ln


991787 M n+1Ipeak

M nIpeak

M minusαn f w(w)dw

+


M minusαN f w(w)dw

1048617

(46)





8102019 05659492




7

E c(θ) =

minus1

θ ln

1048616F 1


1

KI peak 1 minus

1

ρ



1048617


minus1

θ ln




1048617


(45)

01 02 03 04 05 06 07 08 09 1 11

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I a v g

( w a t t s )

Pout

p =1

Pout

p =2

Pout

p =3

Iavg

gt0

Iavg

gt0

Iavg

gt0

Iavg

gt0Iavg

gt0

Iavg

gt0


06 08 1 12 14 16 18 2 22

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I p e a k

( w a t t s )

mP=4

mP=3

mP=2

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0Ipeak

gt0













02

03

04

05

06

07


N o


ms=m

sp=1

ms=1 m

sp=15

ms=1 m

sp=2


10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

045

θ (1nats)


ms=m

sp=1

ms=2 m

sp=1

ms=3 m

sp=1

ms=1 m

sp=2

ms=1 m

sp=3













8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d




ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)




03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case



versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2






















1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z


mp=2

mp=3




01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)


( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW
























8102019 05659492




9

















VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






8102019 05659492




10




































8102019 05659492




7

E c(θ) =

minus1

θ ln

1048616F 1


1

KI peak 1 minus

1

ρ



1048617


minus1

θ ln




1048617


(45)

01 02 03 04 05 06 07 08 09 1 11

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I a v g

( w a t t s )

Pout

p =1

Pout

p =2

Pout

p =3

Iavg

gt0

Iavg

gt0

Iavg

gt0

Iavg

gt0Iavg

gt0

Iavg

gt0


06 08 1 12 14 16 18 2 22

10minus4

10minus3

10minus2

10minus1

100

101

102

Rmin

(natssHz)

I p e a k

( w a t t s )

mP=4

mP=3

mP=2

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0

Ipeak

gt0Ipeak

gt0













02

03

04

05

06

07


N o


ms=m

sp=1

ms=1 m

sp=15

ms=1 m

sp=2


10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

045

θ (1nats)


ms=m

sp=1

ms=2 m

sp=1

ms=3 m

sp=1

ms=1 m

sp=2

ms=1 m

sp=3













8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d




ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)




03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case



versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2






















1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z


mp=2

mp=3




01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)


( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW
























8102019 05659492




9

















VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






8102019 05659492




10




































8102019 05659492




8

10minus3

10minus2

10minus1

100

101

005

01

015

02

025

03

035

04

θ (1nats)

N o r m a l i z e d




ms=m

sp=2 BER=10

(minus3)

ms=m

sp=2 BER=10

(minus5)




03

04

05

06

07

08

09

1

Iavg

(dBW)


Optimum case



versus I avg with

θ = 01 BER=

10minus3

and m

s = m

sp = 2






















1 12 14 16 18 2 22 24 26 28 316

18

2

22

24

26

28

3

Pout

p ()

N o r m a l i z


mp=2

mp=3




01 02 03 04 05 06 07 08 09 10

05

1

15

2

25

3

Rmin

(natssHz)


( n a t s s H z )

Pp

out=1

Pp

out=2

Pp

out=3

PP=15dBW

Pp=12dBW
























8102019 05659492




9

















VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






8102019 05659492




10




































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VII CONCLUSIONS


























the secondary user

APPENDIX A



E c(θ) = minus1

θ ln

852008 ρminusmsp

β (msp ms)

times

991787 infin0


w + 1ρ

983081ms+mspdw

I

852009

where w = 1v



according to


991787 10

xα+mspminus1

10486161 minus

10486161 minus

1

KI peak

1048617x

1048617minusα


10486161 minus

10486161 minus

1

ρ

1048617x


dx (47)


Γ(a)Γ(γ minus a)


991787 10

taminus1


dt

(48)




and

y = 1 minus 1ρ

we get



983080msp + α α


1 minus 1ρ

983081

(49)







on I

I = ρms+msp

991787 10





into (50) we get


Γ(ms + msp + α) (51)



REFERENCES






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