Asymptotic Scheduling Gains in Point-to-Multipoint Cognitive Networks

0.77 MB·English

by  Nadia Jamal

1
Asymptotic Scheduling Gains in Point-to-Multipoint
Cognitive Networks
Nadia Jamal, Student Member, IEEE, Hamidreza Ebrahimzadeh Saffar, Student Member, IEEE,
and Patrick Mitran, Member, IEEE
Abstract—We considercollocated primary andsecondary net- Thus far, three different approaches namely, interweave,
works that have simultaneous access to the same frequency overlay, and underlay have been proposed to enable spectrum
bands. Particularly, we examine three different levels at which
sharing of primary and secondary systems and consequently
primaryandsecondarynetworksmaycoexist:pureinterference,
increase the spectral efficiency of licensed bands [4].
asymmetric co-existence, and symmetric co-existence. At the
asymmetric co-existence level, the secondary network selectively In the interweave approach, the secondary system has ad-
deactivates its users based on knowledge of the interference vanced capabilities to dynamically monitor the licensed spec-
0
and channel gains, whereas at the symmetric level, the primary trum, detect frequency gaps (whitespaces), and opportunis-
1 network also schedules its users in the same way.
tically communicate over these gaps. Much of the research
0 Our aim is to derive optimal sum-rates (i.e., throughputs)
2 of both networks at each co-existence level as the number of on sequential detection of whitespaces [5], [6] is motivated
usersgrowsasymptoticallyandevaluatehowthesum-ratesscale by the interweave approach. Clearly, a technical challenge
n
with network size. In order to find the asymptotic throughput in this approach is developing accurate sensing methods in
a
results, we derive a key lemma on extreme order statistics and order to detect the presence of the primary’s signal. Among
J
a proposition on the sum of lower order statistics. methods proposed thus far, matched filter detection, energy
9 Asabaselinecomparison,wecalculatethesum-ratesforchan-
detection, cyclostationary detection, and wavelet detection are
1 nel sharing via time-division (TD). We compare the asymptotic
secondary sum-rate in TD with that under simultaneous trans- mostcommon[7],[8].Incaseswithmultiplesecondaryusers,

] mission,whileensuringtheprimarynetworkmaintainsthesame cooperative detection of whitespaces helps improve perfor-
T throughput in both cases. The results indicate that simultaneous mance significantly. The interested reader is referred to [7]
I transmission at both asymmetric and symmetric co-existence and [9]–[11] for an overview of cooperative spectrum sensing
s. levels can outperform TD. Furthermore, this enhancement is in interweave cognitive systems. After detecting whitespaces,
c achievable when uplink activation or deactivation of users is
[ basedonlyontheinterferencegainstotheoppositenetworkand thesecondarysystemmustapplyflexiblemodulationschemes
not on a network’s own channel gains. such as orthogonal frequency division multiplexing (OFDM)
1 to communicate over the licensed frequency gaps [12].
v IndexTerms—CognitiveRadio,Interference,ScalingLaw,Co-
The overlay approach is based on the assumption that the
5 existence.
secondary system has knowledge of the primary system’s
6
messages or possibly its codebooks. Thus, it can dedicate a
3
3 I. INTRODUCTION part of its power to relay the primary’s message and use the
. Fixed spectrum allocation policy serves as a traditional remaining power for its own communication. This power split
1
0 way to allow co-existence of various wireless applications canbeadjustedsothatthedegradationintheprimary’sperfor-
0 in the same spectrum. Using this policy, significant portions mance caused by the interference from the secondary system
1 of the limited frequency spectrum have been assigned to iscompensatedforbytheprimary’sperformanceenhancement
: licensed(primary)systemsonalongtermbasis,whileleaving duetotheassistancefromthesecondarysystem[4].Therefore,
v
only some tight bands available for unlicensed (secondary) theprimarysystem’sratecanremainunchanged.Ontheother
i
X applications [1]. hand, in order to mitigate the interference at the secondary
r As the popularity of wireless services increase and innova- receiver and provide positive secondary throughput, the sec-
a tive systems appear, the demand for higher bandwidths and ondary transmitter can apply dirty paper coding (DPC) [13],
data rates grows dramatically. On the other hand, measure- based on the knowledge of the primary’s messages.
mentsconductedbygovernmentagencies,suchastheFederal The simplest overlay network is a two-user interference
Communications Commission (FCC), indicate that licensed channel, where one user has knowledge of the other user’s
bandsareunderutilizedbytheirlicenseesbothtemporallyand transmission. These systems and their capacity results are
geographically [2]. Motivated by this observation, there has further analyzed in [14]–[18].
been recent interest in opportunistic access of the secondary In the underlay approach, primary and secondary systems
system to the licensed bands as first envisioned by Mitola [3]. are allowed to simultaneously communicate over the same
The basic idea of opportunistic spectrum access is to allow frequency bands. In this case, unlike the overlay approach,
secondary users to communicate over licensed spectrum in none of the primary and secondary systems have knowledge
a controlled fashion so that the performance of the primary of the other system’s messages or codebooks. Thus, no DPC
system does not severely degrade. A user with the mentioned is applied and interference is treated as noise. Due to the
capabilities is broadly referred to as a cognitive radio (CR). constraint on the interference power at the primary receiver,
thecapacityofthesecondarysystemintheunderlayapproach
The authors are with the Department of Electrical and Computer En- canbederivedunderreceivedpowerconstraintsattheprimary
gineering at the University of Waterloo, Ontario, Canada (nadia, hamid, receiver rather than transmitted power constraints at the sec-
[email protected]).
ondary transmitter [19], [20].
These results were presented in part at the 2009 IEEE International
SymposiumonInformationTheory(ISIT09). In point-to-point AWGN channels, due to the non-fading
2
characteristic of the channel, the capacity under the received the CDF of the power gains corresponding to the well-known
power constraint and that under the transmitted power con- fadingdistributionssuchasRayleigh,Rician,andNakagami-m
straint are identical. On the other hand, in fading environ- have both of the mentioned properties. Therefore, the results
ments, the secondary system can take advantage of the fading presented in this paper can be applied to these three fading
characteristic of the channel by transmitting at higher power distributions.
levelswheneverthechannelbetweenthesecondarytransmitter Unlike[22]whichconsidersactivatingusersinuplinkmode
and the primary receiver is in a deep fade. In [20], a specific only based on interference knowledge, in this paper, we also
system model is examined and the capacity of the secondary consider scheduling users jointly based on interference to the
system is derived under received power constraints at the other network and channel gains to one’s own network.
primary receiver. Interestingly, it is shown that an increase The results obtained in this paper indicate that, significant
in the variance of the fading distribution gives rise to a higher asymptoticperformanceimprovementscanbegainedoverTD
capacityandinsomecases,thiscapacityresultissignificantly by selectively deactivating users in uplink mode. Specifically,
more than that under a transmitted power constraint at the someconclusionsthatwedrawarethatatsuitableco-existence
secondary transmitter. levels
Inthispaper,weconsidercollocatedprimaryandsecondary • If both networks operate in uplink mode and gains are
point-to-multipoint networks that simultaneously operate in Rayleigh or Rician distributed, in some specific cases,
the same frequency bands in a fading environment. We treat for 1/2 < f ≤ 1 underlay operation results in a better
the secondary network’s interference to the primary system asymptotic sum-rate for the secondary network over TD
as noise and do not allow more advanced techniques such and in some other cases, for 0 < f < 1/2, underlay
as DPC. Since each network may operate in either of uplink operation outperforms TD.
or downlink modes, there are four possible uplink-downlink • If the primary network is in downlink and the secondary
scenarios to be considered. network is in uplink mode, in case of Rayleigh or Rician
When a network operates in downlink mode, we assume fading, for 1/2 < f ≤ 1, underlay is asymptotically
that its base station transmits to the user with the highest preferred over TD. When the gains are Nakagami-m
channel gain. For the uplink transmission, we consider three distributed, then for m < f ≤ 1, simultaneous
1+m
different levels of co-existence based on utilization of the transmission outperforms TD.
channel side information (CSI), i.e., the networks can apply • If the primary network is in uplink and the secondary is
the information on the channel and interference power gains in downlink mode, in case of Rayleigh or Rician fading,
to selectively activate or deactivate their users such that they for 0 < f ≤ 1/2, underlay is asymptotically preferred
can achieve a higher sum-rate and impose less interference to over TD. When the gains have Nakagami-m distribution,
theoppositenetwork’sreceiver.Thelevelsofco-existenceare then for 0 < f ≤ 1 , simultaneous transmission
1+m
further discussed in Section II. outperforms TD.
In [21], co-existence of a large collection of networks is • If both networks are in downlink mode, simultaneous
consideredbutwithouttheco-existencefeaturesconsideredin transmission asymptotically outperforms TD for any 0<
this work. f ≤ 1. In other words, both networks may operate
Ourmaininterestisstudyingthetradeoffsbetweenthesum- simultaneously in the same band at no loss. This result
rate of each network and the number of users. Consequently, isachievedwhenthechannelandinterferencegainshave
we measure the asymptotic sum-rate of the secondary net- an arbitrary CDF with the exponential tail property.
work for each co-existence level while ensuring the primary Interestingly, the above enhanced sum-rate results are
network’s sum-rate is not reduced by more than a specified achievable when uplink scheduling and deactivation of users
primary protection factor 0 < f ≤ 1. As a base reference, is based only on the interference gains to the opposite net-
we consider sum-rate results of channel sharing via time- work. Moreover, in uplink mode, user scheduling based on
division(TD)wheretheprimarynetworkemploysthechannel interference to the other network as well as gains to one’s
for a fraction f of time, leaving a fraction 1 − f for the own network does not improve the results. Therefore by only
secondarynetwork’stransmission.Then,ineachoftheuplink- allowing activation of users that generate interference less
downlink scenarios, we compare the asymptotic secondary than a certain threshold, the secondary network can achieve a
sum-rate under simultaneous transmission to that in TD for significant throughput while the primary network is protected
the same value of f. Sum-rates of primary and secondary to the desired protection factor. It can be easily verified that
networks in TD are similar to those in channel sharing by an increase in the variance of the Nakagami-m distribution
means of orthogonal subchannels and those in opportunistic (decrease in m) expands the range of f for which better sum-
channel sharing. In the first case, f refers to the fraction of rates are achieved. In other words, the richer the scattering
frequency bands dedicated to the primary’s transmission, and, environment, the better the asymptotic sum-rates over TD.
in the second, it is the fraction of licensed frequency bands in The rest of this paper is organized as follows. In the
which the primary is active. next section, we introduce our system model, notation, and
As an extension of the previous results in [22], in this definitions. We also present a key lemma on the maximum
work, we assume channel and interference power gains to of order statistics and a proposition on the sum of lower
have any arbitrary CDF, possessing two specific properties. orderstatistics.InsectionsIIItoVI,sum-ratesofprimaryand
The first property corresponds to the order of the CDF in the secondary networks are derived in each of the four uplink-
very low gain regime while the second one corresponds to its downlink scenarios. We compare the secondary sum-rates
behavior in the very high gain regime. These properties are in simultaneous transmission to that in TD in section VII.
presented in detail in section II. As we shall illustrate later, Discussions about the value of the threshold are presented in
3
As stated earlier, each of the primary and secondary net-
works can operate in either uplink (i.e., multiple access
channel)ordownlink(i.e.,broadcastchannel)modes,resulting
in four possible scenarios for their simultaneous activity.
In both networks, each transmitter is assumed to employ a
Primary Base Station Secondary Base Station
complex Gaussian codebook with power P, and each receiver
is subject to circularly symmetric additive white Gaussian
noise with power N.
Throughout the paper, we denote by R (n) the sum-rate
Σ
p of a single network with n users either in uplink or downlink
G s
i Gp s Gs p Gj mode. When both primary and secondary networks operate
i j in the same band via TD, Rp (n,k) denotes the sum-
Σ,TD
rate of the primary network, while Rs (n,k) denotes the
i j Σ,TD
sum-rate of the secondary network, where n and k are the
number of primary and secondary users, respectively. In case
of simultaneous transmission, Rp (n,k) and Rs (n,k)
Σ,Sim Σ,Sim
represent primary and secondary sum-rates, respectively.
Fig.1. Channelpowergainbetweenprimary’si’th(resp.secondary’sj’th)
user and primary (resp. secondary) base station. Interference power gain
between primary’s i’th (resp. secondary’s j’th) user and secondary (resp. B. Definitions
primary)receiver.
Definition 1: Let X ,X ,... be a sequence of random
1 2
variables whose mean may not converge. We say that the
section VIII and simulation results on the accuracy of the key sequenceXnconcentratesasthenon-randomsequencean >0
lemma and the proposition are provided in section IX. We if
conclude this work in section X and discuss some directions
Pr[|X −a |≥(cid:15)a ]→0, (1)
for further research. The proof of the key lemma and the n n n
proposition are relegated to the appendices. asn→∞forall(cid:15)>0,i.e.,X /a convergesinprobability
n n
to 1. We denote this fact by X ∼˙ a .
n n
WealsoabusenotationandwriteX ∼˙0whenX converges
II. PRELIMINARIES n n
in probability to zero.
A. System Model and Notations
We consider two collocated point-to-multipoint networks Remark 2: It can be verified that given positive sequences
that share the same frequency band. We assume that the an and bn, if Xn ∼˙ an and limn→∞an/bn = 1, then
network with priority access to the band (referred to as the Xn ∼˙ bn. We then write Xn ∼˙ an ∼˙ bn. Also, if Xn ∼˙ an
primary network) comprises a base station and n users. The andlimn→∞an =0,thenXn∼˙ 0.WethenwriteXn∼˙ an∼˙ 0.
channel power gains between each primary user and the
primary base station are denoted by Gp, i=1,...,n. Remark 3: Let Xn and Yn be two sequences of random
i
Likewise, the second network (referred to as the secondary variables where Xn ∼˙ xn and Yn ∼˙ yn for strictly positive
network) consists of a base station and k users. Also, we sequences xn and yn. Then
edaecnhoteofbytheGsjs,ecjo=nd1a,ry..u.s,ekr,sthaendchtahnenseelcpoonwdearrygabianssebsettawtieoenn. Xn ∼˙ xn· (2)
Y y
Throughout the paper, we choose k =nα for some α>0. n n
poSssiimblueltiannteerofuesrentrcaensgmaiinsssiboentwgeievnesthreisperimtoar(yna+nd1s)e(ckon+da1ry) Ipfrotbhaebilliimtyitt,oltihmant→lim∞itx,ni./ey.,n exists, then XYnn converges in
networks. However, at each instant, only one single user or
base station will act as a receiver1. We thus denote by Gsj→p plim Xn = lim xn, (3)
the interference power gain between secondary user j and n→∞ Yn n→∞ yn
the primary receiver. Similarly, the interference power gain
where “plim” refers to limit in probability.
between primary user i and the secondary receiver is denoted
Throughout the paper, the sequences of random variables
by Gp→s, with the special case that i = 0 (resp. j = 0)
i represent the sum-rates of the networks. Sum-rates of a single
denotes the interference from the primary (resp. secondary)
network with n users in uplink or downlink modes are two
base station.
well-known examples.
We further assume that the interference power gains and
Example 4: (Single Network Uplink) The sum-rate of a
channel power gains are i.i.d. with CDF F (g).
G Gaussian multiple access channel with n users is known to
Fig. 1 shows primary and secondary networks, channel be2 [23]
power gains, and interference power gains. For simplicity, we
illustrate only one user in each network and its corresponding R (n)=log(cid:18)1+ P (cid:80)ni=1Gi(cid:19). (4)
channel and interference power gains. Σ N
1Inuplink,onlyabasestationisreceiving,whileindownlink,transmitting 2Throughout the paper, log is logarithm base-2 while ln is the natural
tothereceiverwiththestrongestgainmaximizessum-rate. logarithm.
4
TABLEI
Since the channel power gains are i.i.d. with unit mean, then
ASYMPTOTICPRIMARYANDSECONDARYSUM-RATESINTD.
(cid:18) (cid:19)
Pn Primary Secondary Rp (n,nα) Rs (n,nα)
RΣ(n)∼˙ log 1+ N ∼˙ logn. (5) up up Σ,fTDlogn α(Σ1,T−Df)logn
up down flogn (1−f)loglogn
Example 5: (Single Network Downlink) Consider a broad- down up floglogn α(1−f)logn
cast channel with n potential receivers. It is well-known that down down floglogn (1−f)loglogn
transmitting to the receiver with the strongest gain optimizes
the system throughput [24]. Hence, the sum-rate is
(cid:18) P maxn G (cid:19)
R (n)=log 1+ i=1 i · (6) Proposition 12: Let X ,...,X be i.i.d. random vari-
Σ N 1 n
ables with unit mean, CDF F (x) and parameters λ and γ as
X
Definition 6: Let X be a random variable with prob- x → 0. Furthermore, let f : N → N be such that f(n) → ∞
ability distribution function (pdf) f (x). We say X has an andf(n)/n→0asn→∞.Thenthesumofthef(n)lowest
X
exponential tail with parameter c if: order statistics
lnf (x) f(n)
lim X =−c, (7) (cid:88)
x→∞ x Sn = Xi:n (11)
where c is a positive real number. i=1
concentrates as
Definition 7: Let X be a random variable with CDF n(f(n))1+γ1
FX(x). We say X has parameters λ>0 and γ >0 as x→0 S ∼˙ n · (12)
if and only if there exist positive numbers δ and M such that n λγ1(1+ 1)
γ
|FX(x)−λxγ|≤M(cid:12)(cid:12)xγ+1(cid:12)(cid:12) (8)
Proofs of Lemma 11 and Proposition 12 are provided in
for every |x|<δ. We express this fact by,
Appendix B.
F (x)=λxγ +O(xγ+1)· (9) Now recall Example 5 and assume the channel gains have
X
an exponential tail with parameter c>0, then by Lemma 11,
the network’s sum-rate in downlink mode concentrates as
Now,wepresentsomeexamplesofthepropertiesmentioned
(cid:32) (cid:33)
in definitions 6 and 7 where X is a random variable with P · lnn
R (n)=log 1+ c ∼˙ loglogn· (13)
E[X2]=2. Σ N
Example 8: If X is Rayleigh distributed, then the random
variable G = X2/2 will have an exponential tail with
parameter c=1 and it also has parameters λ=1 and γ =1 D. Baseline Comparison
as g →0. As a baseline, we consider channel sharing via TD and we
Example 9: If X is Rician distributed with parameter K define the primary protection factor f as the limit
(K-factor) as the ratio of the LOS component’s power to the
Rp (n,k)
power of the multipath fading component, then G = X2/2 f = plim Σ,TD , (14)
will have an exponential tail with parameter c = K +1 and n→∞ RΣ(n)
parameters λ=(1+K)e−K and γ =1 as g →0.
i.e., it represents the fraction of time 0 < f ≤ 1 that the
Example 10: If X is Nakagami distributed, with parameter primary network uses the channel. If we take k = nα (α >
m (Nakagami-m distributed), then G = X2/2 will have an
0) and let the channel and interference power gains have an
exponential tail with c = m and parameters λ = mΓ(mm−)1 and exponential tail with parameter c>0, then using Lemma 11,
γ =m as g →0. weobtainasymptoticsum-rateslistedintableIforthechannel
Proofs of the three examples are provided in Appendix A. sharing via TD. We refer to the limit
Rs (n,nα)
C. Proposition and Lemma plim Σ,TD (15)
R (nα)
GivenasetofrandomvariablesX ,X ,...,X ,wedenote n→∞ Σ
1 2 n
its order statistics by X ,...,X , i.e., X ≤ X ≤ as the secondary throughput factor in TD which is easily
1:n n:n 1:n 2:n
···≤X . verified to be equal to 1−f.
n:n
In the following, we have a lemma on the asymptotic Unsurprisingly,iftheprimarynetworkisnottobeimpacted
behavior of the maximum of n i.i.d. random variables and a at all (i.e., f =1), then the secondary system is silent.
proposition on the asymptotic sum of lower order statistics,
both playing key roles in the sequel.
E. CSI and Co-existence
In downlink mode, CSI refers to the knowledge of a
Lemma 11: Let X ,X ,...,X be i.i.d. random vari-
1 2 n
network’s own channel power gains which is available to
ables with pdf f (x) and an exponential tail with parameter
X
c>0. Define X =maxn X . Then, the network’s base station. When the network operates in
n:n i=1 i downlinkmodewithsuchCSI,thehighestsum-rateisobtained
lnn by transmitting to the user with the greatest channel power
X ∼˙ · (10)
n:n c gain(thestrongestuser).Knowledgeoftheinterferingchannel
5
gains between the base station and the opposite network’s In what follows, we examine each of the uplink-downlink
receiver has no influence on which receiver is scheduled. scenarios and find the asymptotic sum-rates at each co-
Inuplinkmode,CSIreferstotheknowledgeofanetwork’s existence level.
own channel power gains as well as the knowledge of the
interference power gains between the users in the network
and the other network’s receiver. This information is available III. PRIMARYANDSECONDARYUPLINK
to the network’s scheduler, which applies the CSI to activate
In this section, we derive the asymptotic primary and
or deactivate users of the network so that the network obtains
secondary sum-rates under simultaneous transmission when
the highest sum-rate and also the interference imposed on the
both networks operate in uplink mode. We examine the three
other network is minimized.
different co-existence levels, and, at each level, we find the
For uplink transmission, three different levels of co-
asymptotic result for the maximum secondary throughput
existence are considered based on utilization of the CSI.
factor provided that the primary protection factor be at least
Pure Interference: At this simplest level of co-existence,
f.
the primary and secondary networks are considered as two
independent entities. None of the networks employ the CSI to
schedule their users. Therefore, all the primary and secondary
A. Pure Interference
users are active and interfere with each other.
We first consider the pure interference case. Having n and
Asymmetric co-existence: At the asymmetric co-existence nα (α > 0) active primary and secondary users respectively,
level,onlythesecondarynetworkschedulesitsusers.Inother the sum-rates of the primary and secondary networks are,
words, only the secondary network’s scheduler selectively
activates or deactivates secondary users. The primary network (cid:32) P (cid:80)n Gp (cid:33)
Rp (n,nα)=log 1+ i=1 i , (18)
however, operates as usual. Σ,Sim N +P (cid:80)nα Gs→p
j=1 j
Symmetric co-existence: The third level is referred to as and
symmetricco-existencelevel.Inthiscase,theprimarynetwork
also applies its CSI and selectively schedules its users. (cid:32) P (cid:80)nα Gs (cid:33)
Rs (n,nα)=log 1+ j=1 j , (19)
Σ,Sim N +P (cid:80)n Gp→s
i=1 i
F. Scheduling Strategies
respectively.Forprimaryandsecondaryuplinkscenario,Gs→p
Inuplinktransmission,theprimaryandsecondaryuserscan j
is the interference power gain from secondary user j to the
be activated or deactivated based on two different scheduling
primary base station and Gp→s is the interference power gain
strategieswhicharereferredtoasjointoptimization,andleast i
from primary user i to the secondary base station.
interference strategies. These strategies are performed by the
For any 0 < f ≤ 1, (17) should be satisfied in order to
scheduler in each network.
protect the primary network. Since
Using the joint optimization strategy, the scheduler of a
networkcanoptimallydeactivatethenetwork’susersbasedon (cid:18) (cid:19)
Pn
the network’s own channel power gains and the interference Rp (n,nα)∼˙ log 1+ ∼˙ logn1−α, (20)
Σ,Sim N +Pnα
power gains between its users and the opposite network’s
receiver (CSI). The joint optimization scheduling can be
for 1−α>0, and
performed in either of the networks such that the highest
(cid:18) (cid:19)
secondary throughput factor is achieved while the primary Pn
R (n)∼˙ log 1+ ∼˙ logn, (21)
network is protected. Σ N
However, when using the least interference scheduling
method, activation or deactivation of users is based only on the primary protection constraint, (17), cannot be satisfied for
theirinterferencepowergainstotheoppositenetworkandnot any value α > 0 when f = 1. For 0 < f < 1, the primary
onthenetworks’ownchannelpowergains.Inotherwords,the network is protected when 1−α≥f which results in
scheduler only considers the interference power gains rather
than the network’s own channel power gains. Rs (n,nα)∼˙ log(cid:18)1+ Pnα (cid:19) ∼˙ 0· (22)
Forthepureinterferenceaswellaseachschedulingstrategy Σ,Sim N +Pn
in the symmetric or asymmetric co-existence levels, we are
Therefore,
interested in maximizing the secondary throughput factor
defined as (cid:20) Rs (n,nα)(cid:21)
Rs (n,nα) max plim Σ,Sim =0. (23)
plim Σ,Sim , (16) 0<α≤1−f n→∞ RΣ(nα)
R (nα)
n→∞ Σ
for any 0<f <1.
subject to the constraint,
Thisnegativeresultisnotsurprising,becausethesecondary
Rp (n,nα) network has no method to limit its interference to the primary
plim Σ,Sim ≥f (17) network, except by decreasing the number of active users.
R (n)
n→∞ Σ Thus, at the pure interference level, no better secondary
for some primary protection factor 0<f ≤1. sum-rate can be achieved over TD.
6
B. Co-existence Proof of Theorem 13: In the first part of the proof,
Using CSI, the secondary and primary networks can suf- we show that the above result is achievable when scheduling
ficiently mitigate the interference to the opposite network by users is based only on the least interference power gains.
deactivating some of their users. Therefore, a positive sec- Letting FG(g) be the CDF of the channel and interference
ondary throughput factor may be achieved while the primary power gains, then FG(g)=λgγ +O(gγ+1) as g →0. Using
network is protected. Proposition 12, for any α>0 we have
First, we examine the joint optimization strategy. In this
case, if the primary network activates nβ, 0 ≤ β ≤ 1, users (cid:88)nα¯ Gs→p∼˙ nα¯(1+γ1)−αγ , (30)
oouftnoαf,n0 a≤ndα¯th≤e sαe,cothnednarythenestwumor-krataecstivoaftethsenpα¯r,imusaerrysaonudt j=1 j:nα λγ1(1+ γ1)
secondary networks can be written as follows, and
RΣp,Sim(n,nα)=log1+ N +PP(cid:80)(cid:80)n(cid:96)=βn(cid:96)¯=1α¯1GGpi(cid:96)sj(cid:96)→¯ p, (24) (cid:88)in=β1Gpi:→ns∼˙ nλβγ1((11++γ1)−γ1γ1)· (31)
and Thus, sum-rate of the primary network concentrates as
 P (cid:80)nα¯ Gs   
RΣs,Sim(n,nα)=log1+ N +P (cid:80)(cid:96)¯=n1β Gj(cid:96)¯p→s· (25) Rp (n,nα)∼˙ log1+ Pnβ  (32)
(cid:96)=1 i(cid:96) Σ,Sim  N +P · nα¯(1+γ1)−αγ 
1
The joint optimization is over α¯, β, and the set of vectors λγ(1+1)
γ
(j1,j2,...,jnα¯)and(i1,i2,...,inβ),suchthattheprimaryis (cid:32) (cid:20) 1 α(cid:21)+(cid:33)+
protected and the secondary throughput factor is maximized. ∼˙ β− (1+ )α¯− logn. (33)
On the other hand, following the least interference strategy, γ γ
the primary network activates nβ users, 0 ≤ β ≤ 1, out
Likewise, the sum-rate of the secondary network concentrates
of n that result in the least interference power gains to
as
the secondary receiver and similarly, the secondary network
activates nα¯ users out of nα, 0≤α¯ ≤α, which generate the  
least interference power gains to the primary receiver. Using Rs (n,nα)∼˙ log1+ Pnα¯  (34)
thisstrategy,thesum-ratesofprimaryandsecondarynetworks Σ,Sim  N +P · nβ(1+γ1)−γ1 
can be written as 1
λγ(1+1)
  γ
RΣp,Sim(n,nα)=log1+ N +PP(cid:80)(cid:80)n(cid:96)=βn1α¯GGpi(cid:96)s→p, (26) ∼˙ (cid:32)α¯−(cid:20)(1+ γ1)β− γ1(cid:21)+(cid:33)+logn. (35)
j=1 j:nα
and We now consider two co-existence levels:
(cid:32) P (cid:80)nα¯ Gs (cid:33) • Case β = 1 (Asymmetric Co-existence): In this case,
RΣs,Sim(n,nα)=log 1+ N +P (cid:80)(cid:96)=n1β Gj(cid:96)p→s . (27) we have to maximize [α¯−1]+ subject to the primary
i=1 i:n protection constraint
Theorem 13: Let interference and channel power gains
(cid:20) 1 α(cid:21)+
be i.i.d. random variables with unit mean and an exponential 1− (1+ )α¯− ≥f, (36)
tail with parameter c > 0 and parameters λ > 0 and γ > 0. γ γ
Given the protection of the primary network and using the
for every 0<f ≤1, as well as 0≤α¯ ≤α.
joint optimization strategy, an upper bound on the maximum
Solving the optimization problem when α > 1, we find
achievable secondary throughput factor at the asymmetric co- α¯ = γ−fγ+α as an optimal value. In the case that
(cid:104) (cid:105)+ 1+γ
existencelevelis α−1−fγ forevery0<f ≤1andα>0. α ≤ 1, the secondary sum-rate trivially concentrates as
α(1+γ)
In case of symmetric co-existence, for 0<f ≤ 1 we have 0. Therefore, the maximum secondary throughput factor
1+γ is
the upper bound
(cid:40) (cid:104) γ −( γ )f + 1 (cid:105)+, α≥ 1 −f max(cid:20)plim RΣs,Sim(n,nα)(cid:21)=(cid:20)α−1−fγ(cid:21)+ (37)
α(γ+1)2 γ+1 α 1+γ 1+γ α¯ n→∞ RΣ(nα) α(1+γ)
1, 0<α< 1 −f,
1+γ for any 0<f ≤1 and α>0.
(28)
• Case 0 ≤ β < 1 (Symmetric Co-existence): In this case,
and for 1 <f ≤1, we have the upper bound (cid:20) (cid:104) (cid:105)+(cid:21)+
1+γ we must maximize α¯− (1+ 1)β− 1 subject to
γ γ
(cid:20) 1 1 f 1 (cid:21)+
the primary protection constraint
−(1+ ) + , (29)
αγ γ α 1+γ
(cid:20) 1 α(cid:21)+
forα>0,wherex+ :=max(0,x).Furthermore,thesebounds β− (1+ )α¯− ≥f, (38)
γ γ
are achievable by using the least interference scheduling
strategy. for every 0<f ≤1, as well as 0≤α¯ ≤α.
7
For 0<f ≤ 1 , the optimal solution is to select and
1+γ
 
(cid:40)
ββ == 11++11γγ,,αα¯¯ ==αγ(,β1−+fγ)+α, 0α<≥α1+1<γ −1+1fγ −f· RΣs,UB,Sim(n,nα)∼˙ log1+ N +PPn·α¯n·β(l1n+cnγ1α)n−γ1 
(39) 1
λγ(1+1)
γ
 
For 1 <f ≤1,theoptimalsolutionistoselectβ =f
and 1α¯+γ= α when α > (1+γ)2f − 1+γ. In the case ∼˙ log1+ Pnα¯ . (47)
that α≤ (11++γγ)2f−1+γ the seγcondary suγm-rate trivially  N +P · nβ(11+γ1)−γ1 
γ γ λγ(1+1)
concentrates as 0. γ
Thus, for 0 < f ≤ 1 the maximum secondary But (46) and (47) are identical to (32) and (34) respectively,
1+γ
throughput factor is thus by finding
(cid:40) (cid:104) (cid:105)+ (cid:20) Rs (n,nα)(cid:21)
γ −( γ )f + 1 , α≥ 1 −f max plim Σ,UB,Sim , (48)
α(γ+1)2 γ+1,1 α 1+γ α< 1+1γ −f, α¯,β n→∞ RΣ(nα)
1+γ
(40) subject to
Rp (n,nα)
and for 1+1γ <f ≤1, the maximum secondary through- plim Σ,UB,Sim ≥f, (49)
put factor is n→∞ RΣ(n)
(cid:20) 1 1 f 1 (cid:21)+ for 0 < f ≤ 1, we obtain the upper bound for the maximum
−(1+ ) + , (41) secondary throughput factor to be equal to the maximum
αγ γ α 1+γ
secondary throughput factor when user scheduling is based
when α>0. on the least interference strategy. Therefore, by using the
joint optimization strategy, the secondary network can not
The final step of the proof is to show that using joint
achieveabettersum-ratethanthatachievedbyusingtheleast
optimization, we achieve the same results. Scheduling users
interference strategy.
based on joint optimization, one can find upper bounds for
primary and secondary sum-rates as follows
The results obtained in this section indicate that, at asym-
Rp (n,nα)≤Rp (n,nα), (42) metric and symmetric co-existence levels, it is possible for
Σ,Sim Σ,UB,Sim
the secondary network to provide positive asymptotic sum-
and rate while the primary is still protected by factor f, whereas
at the pure interference level, the secondary is asymptotically
RΣs,Sim(n,nα)≤RΣs,UB,Sim(n,nα), (43) unable to deliver a positive sum-rate.
where
(cid:32) (cid:33) IV. PRIMARYDOWNLINKANDSECONDARYUPLINK
PnβGp
Rp (n,nα)=log 1+ n:n , (44) As the primary network is in downlink mode, there is no
Σ,UB,Sim N +P (cid:80)nj=α¯1Gjs:→nαp symmetric co-existence for this scenario to be considered.
and
(cid:32) (cid:33) A. Pure Interference
Pnα¯Gs
Rs (n,nα)=log 1+ nα:nα · (45) In this case, all nα (α>0) secondary users are active and
Σ,UB,Sim N +P (cid:80)nβ Gp→s
i=1 i:n the primary base station transmits to the primary user with
the highest channel power gain. Therefore, the sum-rates of
These upper bounds are obtained by assuming all activated
primary and secondary networks are
users have the maximum channel power gains to their own
networksandtheleastinterferencepowergainstotheopposite (cid:32) P maxn Gp (cid:33)
network. Rp (n,nα)=log 1+ i=1 i , (50)
Σ,Sim N +P (cid:80)nα Gs→p
Since the channel power gains have exponential tail with j=1 j
parameter c>0, we have
and
 
(cid:32) P (cid:80)nα Gs (cid:33)
Rp (n,nα)∼˙ log1+ Pnβ · lncn  RΣs,Sim(n,nα)=log 1+ N +Pj=G1p→js , (51)
Σ,UB,Sim  N +P · nα¯(1+γ1)n−αγ  0
1
λγ(1+1) respectively.
γ
 
Let the channel power gains have an exponential tail with
 Pnβ  parameter c>0. Then, by Lemma 11
∼˙ log1+ , (46)
 N +P · nα¯(11+γ1)−αγ  mnaxGp∼˙ lnn· (52)
λγ(1+γ1) i=1 i c
8
Thus, sum-rates of the primary and secondary networks concentrate
(cid:32) (cid:33) as
P · lnn
Rp (n,nα)∼˙ log 1+ c ∼˙ 0, (53)  
Σ,Sim N +Pnα
Rp (n,nα)∼˙ log1+ P · lncn . (58)
for any α>0. Σ,Sim  N +P · nα¯(1+γ1)−αγ 
Due to the excessive interference from the secondary net- 1
λγ(1+1)
worktotheprimaryreceiver,theprimaryprotectionconstraint, γ
(17),cannotbesatisfiedforanyvalueofα>0andtherefore and
the networks can not coexist at this level. (cid:18) Pnα¯ (cid:19)
Rs (n,nα)∼˙ log 1+ ∼˙ α¯logn. (59)
Σ,Sim N +PGp→s
0
B. Asymmetric co-existence
respectively.
Using the joint optimization scheduling strategy in the
To obtain the highest secondary throughput factor, we must
secondary network with nα¯ (0 ≤ α¯ ≤ α) secondary active
maximize α¯ subject to the primary protection constraint (17).
users and letting the primary base station transmit to the
In (58), when α¯ >α/(1+γ), the sum-rate of the primary
strongestprimaryuser,sum-ratesoftheprimaryandsecondary
network concentrates as 0. On the other hand, in the case that
networks can be written as
α¯ ≤ α/(1+γ), (17) will be satisfied for every 0 < f ≤ 1.
(cid:32) (cid:33)
P maxn Gp Thus, α¯ =α/(1+γ) is the optimal value and the maximum
Rp (n,nα)=log 1+ i=1 i , (54)
Σ,Sim N +P (cid:80)nα¯ Gs→p secondary throughput factor is then
(cid:96)¯=1 j(cid:96)¯
(cid:20) Rs (n,nα)(cid:21) 1
and max plim Σ,Sim = , (60)
 P (cid:80)nα¯ Gs  α¯ n→∞ RΣ(nα) 1+γ
RΣs,Sim(n,nα)=log1+ N +(cid:96)P¯=G1p→j(cid:96)¯s, (55) forany0<f ≤1andα>0.Hence,thefirstpartofTheorem
0 14 is proven.
In what follows, we show that applying the joint optimiza-
respectively. To solve the above joint optimization problem,
tion strategy, the sum-rate results will be the same in the best
we should find α¯ and the vector (j ,j ,...,j ) such that
1 2 nα¯ case.
theprimarynetworkisprotectedandthesecondarythroughput
If users are scheduled based on the joint optimization
factor is maximized.
strategy, we will have
Using the least interference strategy and letting the sec-
ondary network activate only nα¯ users that generate the least (cid:32) P maxn Gp (cid:33)
interference power gains to the primary receiver, sum-rates of Rp (n,nα)=log 1+ i=1 i , (61)
the primary and secondary networks are as follows Σ,UB,Sim N +P (cid:80)nα¯ Gs→p
j=1 j:nα
(cid:32) (cid:33)
P maxn Gp and
Rp (n,nα)=log 1+ i=1 i , (56)
Σ,Sim N +P (cid:80)nα¯ Gs→p (cid:18) Pnα¯Gs (cid:19)
j=1 j:nα Rs (n,nα)=log 1+ nα:nα , (62)
Σ,UB,Sim N +PGp→s
and 0
 P (cid:80)nα¯ Gs  as the upper bounds for the primary and secondary sum-rates
RΣs,Sim(n,nα)=log1+ N +(cid:96)P¯=G1p→j(cid:96)¯s. (57) rthesepesecctiovnedlya.rTyhaecsteivueppuesrerbsouhnadvseatrheedehriigvheedstbychaassnunmelinpgotwheart
0
gains to their own base station and they also have the least
Given a scheduling strategy, we are interested in finding interference power gains to the primary receiver.
the maximum secondary throughput factor while ensuring the Using Lemma 11 and Proposition 12, we have
primary is protected by a protection factor at least equal to f.
 
Theorem 14: Let the interference and channel power Rp (n,nα)∼˙ log1+ P · lncn ,
gains be i.i.d. random variables with unit mean and an ex- Σ,UB,Sim  N +P · nα¯(1+γ1)n−αγ 
ponential tail with parameter c > 0 and parameters λ > 0 1
λγ(1+1)
and γ > 0. At the asymmetric co-existence level, provided γ (63)
theprotectionoftheprimarynetworkandusingthejointopti-
mizationschedulingstrategy,anupperboundonthemaximum and
secondary throughput factor is 1+1γ for any 0 < f ≤ 1 and (cid:32) Pnα¯ · lnnα (cid:33)
α>0. Furthermore, this bound can be achieved by using the Rs (n,nα)∼˙ log 1+ c
Σ,UB,Sim N +PGp→s
least interference scheduling strategy. 0
∼˙ α¯logn· (64)
Proof of Theorem 14: First, we examine sum-rate
results when user scheduling is based only on the least Since (63) and (64) are identical to (58) and (59) respec-
interference power gains. Since the channel and interference tively,bymaximizing(48)overα¯ subjectto(49),wefindthat
powergainshaveanexponentialtailwithparameterc>0and the maximum secondary throughput factor is upper bounded
parameters λ > 0 and γ, by Lemma 11 and Proposition 12, by that achieved in the least interference strategy.
9
V. PRIMARYUPLINKANDSECONDARYDOWNLINK Using the joint optimization strategy, we find the upper
boundsfortheprimaryandsecondarysum-ratestoconcentrate
As the secondary network is in downlink mode, there is no
as
asymmetric co-existence for this scenario to be considered.
(cid:32) (cid:33)
Pnβ · lnn
Rp (n,nα)∼˙ log 1+ c
A. Pure Interference Σ,UB,Sim N +PGs→p
0
In this case, following the same line of development as in ∼˙ βlogn, (71)
the first two scenarios, we find
(cid:18) (cid:19) and
Pn
Rp (n,nα)∼˙ log 1+ ∼˙ logn, (65)
Σ,Sim N +PGs→p  
0
and Rs (n,nα)∼˙ log1+ P · lncnα , (72)
(cid:18) Pn(cid:19) Σ,UB,Sim  N +Pnβ(1+γ1)n−γ1 
RΣ(n)∼˙ log 1+ N ∼˙ logn. (66) λγ1(1+γ1)
respectively. Again, by solving the optimization problem (48)
Thus, (17) is satisfied for any 0<f ≤1.
subject to (49), one can show that no better result is achieved
By Lemma 11,
by applying the joint optimization strategy.
(cid:32) P · lnnα(cid:33)
Rs (n,nα)∼˙ log 1+ c ∼˙ 0, (67)
Σ,Sim N +Pn
VI. PRIMARYANDSECONDARYDOWNLINK
for any α>0 and thus
As the primary and secondary networks are both in down-
(cid:20) Rs (n,nα)(cid:21) link mode, there is no co-existence level to be considered.
max plim Σ,Sim =0, (68) In this scenario, both primary and secondary base stations
α>0 n→∞ RΣ(nα) transmittotheirownuserwiththehighestchannelpowergain.
for any 0 < f ≤ 1. Hence, the maximum secondary Thus,thesum-ratesoftheprimaryandsecondarynetworksare
throughputfactorisequaltozerointhepureinterferencecase.
(cid:18) P maxn Gp(cid:19)
Rp (n,nα)=log 1+ i=1 i , (73)
Σ,Sim N +PGs→p
B. Symmetric co-existence 0
Theorem 15: Let the interference and channel power and
gains be i.i.d. random variables with unit mean and an expo- (cid:32) P maxnα Gs(cid:33)
nential tail with parameter c > 0 and parameters λ > 0 and Rs (n,nα)=log 1+ j=1 j , (74)
γ > 0. For any α > 0, given the protection of the primary Σ,Sim N +PGp→s
0
network and using the joint optimization strategy, an upper
bound on the maximum secondary throughput factor is 1 for respectively. Using Lemma 11
any 0 < f ≤ 1 and 0 for any 1 < f ≤ 1. Furthermore,
1+γ 1+γ (cid:32) (cid:33)
this result can be achieved by using the least interference P · lnn
Rp (n,nα)∼˙ log 1+ c ∼˙ loglogn,
scheduling strategy. Σ,Sim N +PGs→p
0
(75)
Proof of Theorem 15: When scheduling users is based
on the least interferences, the primary sum-rate concentrates
and
as
(cid:18) Pnβ (cid:19) (cid:32) P · lnnα (cid:33)
RΣp,Sim(n,nα)∼˙ log 1+ N +PGs→p ∼˙ βlogn. (69) RΣs,Sim(n,nα)∼˙ log 1+ N +PGcp→s ∼˙ loglogn.
0 0
(76)
for 0<β ≤1.
Using Lemma 11 and Proposition 12, we find that
Therefore,
 
RΣs,Sim(n,nα)∼˙ log1+ N +PP··nlnβc(n1α+γ1)−γ1 · (70) np→lim∞RΣp,RSimΣ((nn,)nα) =1≥f, (77)
1
λγ(1+1) for any 0<f ≤1 and
γ
By choosing β = f, the primary network is protected the
Rs (n,nα)
the secondary throughput factor is, plim Σ,Sim =1. (78)
R (nα)
n→∞ Σ
Rs (n,nα) (cid:26) 1 0<f ≤ 1
plim ΣR,Sim(nα) = 0 1 <f ≤1+1γ, As a result, without any restriction on the value of α>0, for
n→∞ Σ 1+γ anygiven0<f ≤1,theprimarynetworkisalwaysprotected,
for any α>0. while the secondary achieves a positive throughput factor.
10
VII. COMPARISONTOTD
1
The results obtained in the first three uplink-downlink
Time Division
scenariosarevalidincaseofanyarbitraryCDF,FG(g),which 0.9 Asymmetric Co-existence
satisfies the two mentioned properties, whereas in the last Symmetric Co-existence
0.8
scenario, only the exponential tail property is required. As
shown in Appendix A, channel gains with either Rayleigh, rotc 0.7
a
Rician or Nakagami-m distributions have both properties for f tu 0.6
some parameters c>0, λ>0 and γ >0. ph
g
u
In this section, we compare the maximum secondary orh 0.5
throughput factor achievable under simultaneous transmission t y
ra 0.4
to that achievable in TD. As it was shown earlier, the asymp- dn
o
totic secondary throughput factor in TD is equal to 1−f in ce 0.3
S
all of the uplink-downlink scenarios.
0.2
0.1
A. Primary and Secondary Uplink:
In this scenario, generally with asymmetric co-existence 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
when α > 1+γ, for 1+γα < f ≤ 1 simultaneous Primary protection factor f
α−γ+γα
transmission outperforms TD.
In case of symmetric co-existence, when γ ≥1 for
 γα+γ21−1 <f ≤1, α>1+γ Fpriigm.a2r.y pArsoytemcptitoonticfarecstourltfforfomraαxim=um4seacnodndwarhyenthrbooutghhpnuettwfaocrtkosrvareersuins
γα−1−γ (79)
 0<f < γα−γ+γ1, 0<α< 1 , uaspylimnkmemtroicd)e.cRo-aeyxliesitgehncoeriRsicbieatnterdisthtrainbutTioDnwishaesnsu1m/e2d.<Syfmm≤etri1c ((rreesspp..
γα+α−γ 1+γ
5/7<f ≤1).
the secondary network achieves a greater sum-rate compared
to TD.
As shown in Appendix A, we obtain γ = 1 when channel
1
and interference gains have Rayleigh or Rician distributions.
Time Division
Therefore, in case of Rayleigh or Rician fading, when α>2 0.9 Asymmetric Co-existence
symmetricco-existenceoutperformsTDfor1/2<f ≤1,and Symmetric Co-existence
0.8
when 0 < α < 1/2 symmetric co-existence outperforms TD
for 0<f <1/2. Furthermore, with asymmetric co-existence, rotc 0.7
a
secondarysum-rateishigherthanthatinTDfor(1+α)/(2α− f tu 0.6
p
1)<f ≤1 when α>2. h
g
u
The tradeoff between the secondary throughput factor and orh 0.5
theprimaryprotectionfactorf isillustratedinFig.2forγ =1 t y
ra 0.4
and α=4. Surprisingly, when f =1, for both symmetric and dn
o
asymmetriclevels,thesecondarynetworkcanprovidepositive ce 0.3
S
sum-rates.
0.2
Fig. 3 shows the secondary throughput factor versus f
under simultaneous transmission when channel and interfer- 0.1
ence gains have Nakagami-m distribution with m = 3/2 and
0
α = 4. At the asymmetric co-existence level, for 14/17 < 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Primary protection factor f
f ≤ 1, simultaneous transmission is better than TD and at
the symmetric co-existence level, simultaneous transmission
outperforms TD for 26/35<f ≤1.
Fig. 3. Coefficients of sum-rate scalings for secondary versus primary
protection factor f for α = 4, when channel and interference gains have
B. Primary Downlink and Secondary Uplink: Nakagami-mdistributionwithm=3/2andprimaryandsecondarynetworks
both operate in uplink mode. Symmetric (resp. asymmetric) co-existence is
In this scenario, simultaneous transmission asymptotically
betterthanTDwhen26/35<f ≤1(resp.14/17<f ≤1).
outperforms TD for γ < f ≤ 1. Particularly, in case of
1+γ
RayleighorRiciandistributionswithγ =1,for1/2<f ≤1,
simultaneous transmission asymptotically outperforms TD.
C. Primary Uplink and Secondary Downlink:
When the channel and interference gains have the Nakagami-
m distribution with γ = m, for m < f ≤ 1, simultaneous In simultaneous transmission, for 0 < f ≤ 1 , the
1+m 1+γ
transmission results in a higher sum-rate for the secondary maximum secondary throughput factor is 1 which is greater
network compared to TD. Decreasing m results in expan- than that in TD. In case of Rayleigh and Rician distributions,
sion of the interval of f for which a better performance is for0<f ≤1/2,simultaneoustransmissionresultsinahigher
achievable. Since m is reversely proportional to the variance secondary throughput factor than TD. In case of Nakagami-m
of the Nakagami-m distribution, an increase in the variance distribution, for 0 < f ≤ 1 , we obtain better results over
1+m
improves the range of f for which simultaneous transmission TD. Also, smaller m increases the interval of f for which a
outperforms TD. higher sum-rate is achieved.