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Optimal Hybrid Spectrum Sensing under Control
Channel Usage Constraint
Nilanjan Biswas, Student Member, IEEE Goutam Das, Member, IEEE and Priyadip Ray, Member, IEEE
Abstract—Cooperative spectrum sensing significantly improves
the detection reliability of a cognitive radio network. In coopera-
tive spectrum sensing, the cognitive radio nodes send their hard
decisions/detections to the fusion center via a control channel.
Traditional control channels are bandwidth constrained, and a
large number of hard decisions to the fusion center may saturate
the control channel, thereby degrading the performance of the
network. In this paper, we consider a cooperative spectrum
sensing scheme, where each cognitive radio node performs a
fixed sample sensing test and sends a hard decision to the fusion
center via a common control channel. The fusion center collects
the hard decisions from all the cognitive radio nodes to form an
observation vector, and performs a sequential probability ratio
test to make the final decision. We term this sensing scheme as
hybrid spectrum sensing. An optimization problem is formulated
for the hybrid sensing strategy in order to maximize the cogni-
tive radio network’s throughput while considering interference
(interference on primary user by cognitive radio network) and
control channel’s bit rate constraint. The decision thresholds for
the cognitive radio nodes and the fusion center are considered as
the optimization parameters. Though the optimization problem
is non-convex, we provide an efficient algorithm for obtaining the
global optimal solution. Extensive simulation results are provided
to demonstrate the efficacy of our proposed approach.
Index Terms—Cognitive radio networks, spectrum sensing,
distributed detection, sequential test, throughput optimization
I. INTRODUCTION
The concept of cognitive radio (CR) has emerged to al-
leviate the spectrum scarcity problem [1]. In recent time, it
can be found in [2], [3], that the spectrum can be reused
using CR technology from cellular or televisions (TV) bands.
In CR, unlicensed users, often referred to as secondary users
(SUs), sense and opportunistically access the radio spectrum
while ensuring that the interference to the primary user (PU) is
below some acceptable threshold [4]. Interference to the PU is
primarily caused when the SU fails to detect the PU’s activity
in a licensed band. Hence, accurate and efficient spectrum
sensing is one of the fundamental problems in CR networks.
For different kind of spectrum sensing techniques and their
advantages/disadvantages one may refer to [5]. Cooperative
spectrum sensing (CSS) is often preferred over single SU
sensing as it improves the spectrum sensing performance for
CR networks [6], [7]. In CSS, each SU senses a common band
of interest and sends it’s local sensing information to a central
entity which is often denoted as the fusion center (FC), using
Nilanjan Biswas and Goutam Das are with the G.S. Sanyal School
of Telecommunications, Indian Institute of Technology (IIT) Kharagpur,
West Bengal 721302, India, Email: gdas@gssst.iitkgp.ernet.in. Priyadip Ray
was with the G.S. Sanyal School of Telecommunications, Indian Insti-
tute of Technology (IIT) Kharagpur, West Bengal 721302, India, Email:
priyadipr@gmail.com.
a predefined control channel. Using an appropriate fusion rule,
the FC makes the global decision regarding the PU’s activity
on a licensed band [8].
Sensing duration is an important parameter for CR net-
works’ as the performance metrics (e.g., energy efficiency,
detection error probability, and throughput) of CR networks’
strongly depends on it. To reduce the sensing duration (com-
pared to traditional fixed sample spectrum sensing (FSS)),
while maintaining an acceptable detection performance, se-
quential detection based spectrum sensing has been proposed
in [9]. Sequential probability ratio test (SPRT) is optimal
among all the sequential tests [9], [10], as it requires the
smallest number of average samples to arrive at a final decision
having identical constraints on detection error probabilities. In
SPRT, two different thresholds are used and decision is made
as soon as the test statistic exceeds the upper threshold or falls
below the lower threshold. In CSS, primarily four different
sensing strategies exist in the literature:
•FSS at both CR nodes and the FC [11]–[19]
•SPRT at both CR nodes and the FC [20]–[24]
•SPRT at CR nodes and FSS at the FC [25]
•FSS at CR nodes and SPRT at the FC [26]–[31].
The first sensing strategy may not be an efficient one in terms
of sensing duration. If CR nodes perform SPRT, then CR
nodes’ decision time instants become random. Therefore, the
event that more than a single CR node send their decisions to
the FC, may take place with a finite probability. If multiple
CR nodes send their decisions at the same time, then their
signals get superimposed on the channel. The FC will be able
to evaluate the likelihood ratio properly when two decision
thresholds (i.e., upper and lower thresholds of the SPRT) of
a CR node become identical [23]. However, this imposes a
separate constraint on the optimal decision thresholds’ design
problem. This constraint may be relaxed if CR nodes send
their decisions at separate time instants. Therefore, the control
channel’s access protocol may be designed using one of the
existing random channel access mechanisms. Carrier sense
multiple access with collision avoidance (CSMA-CA) is the
most popular random access protocol for wireless medium.
However, usage of CSMA-CA increases the control channel
overhead as well as the energy consumption of CR nodes [32].
It can be observed that in third and fourth sensing strategies,
two different kinds of spectrum sensing techniques, i.e., FSS
and SPRT, are used. For this reason, we can term third and
fourth sensing strategies as hybrid sensing strategy (HSS) of
first kind and HSS of second kind respectively. It is to be noted
that second sensing strategy (i.e., SPRT at both CR nodes
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at http://dx.doi.org/10.1109/TSP.2018.2838575
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and the FC), HSS of first kind, and HSS of second kind may
help in reducing the sensing time. However, the throughput
analysis for second sensing strategy and HSS of first kind are
beyond the scope of this paper. We consider HSS of second
kind in this paper. The sensing information which is sent to
the FC may be in the form of raw log likelihood ratio (LLR)
[28], [29] or soft decision (multi-bits) [26] or hard decision
(single bit) [27], [30]. LLR fusion may help in improving the
detection performance compared to the soft decision fusion
and hard decision fusion at a cost of significantly high control
channel overhead. The control channel overhead reduces for
soft decision fusion. However, both LLR transmission and soft
decision transmission causes more energy consumption at CR
nodes. Moreover, the evaluation of optimal decision thresholds
at CR nodes become complex. In literature, we find two
different approaches for HSS of second kind sensing strategy;
the FC may perform SPRT after either collecting a single
CR node’s sensing information [26]–[28], [31] or collecting a
vector of sensing information (i.e., sensing information from
all CR nodes at a time instant) [29], [30]. We analyse both of
these approaches for hard decision fusion in our paper.
Beside sensing duration, CR networks’ performance metrics
have strong dependence on the detection probability as well as
the false-alarm probability. This necessitates the design of the
decision thresholds’ for both CR nodes and the FC. In liter-
ature, we find [11]–[14], [17]–[19], [25], where authors have
considered FSS at CR nodes and the FC and designed decision
thresholds in various contexts. Authors of [25] have considered
AND/OR fusion rule at the FC and have designed CR nodes’
decision thresholds to maximize energy efficiency. In the
context of detection error probability minimization, the authors
of [17], [19], have designed the decision threshold for the FC
only; whereas, in [18], authors have optimized the decision
thresholds of CR nodes’ and the FC’s in a separate manner.
Decision threshold design for throughput maximization has
been considered in [11]–[14]. It is to be noted that in [11]–
[14], authors have considered joint optimization of decision
thresholds, i.e., the thresholds of CR nodes and the FC. In
[33], authors have considered HSS of second kind, where the
FC performs SPRT after collecting a single CR node’s LLR.
The authors have evaluated the FC’s decision thresholds as
CR nodes’ decision threshold design is not relevant in [33].
In another related paper [34], the authors have evaluated CR
nodes’ hard decision thresholds by maximizing the Kullback-
Leibler (KL) divergence of the quantizer’s output, where CR
nodes report their hard decisions to the FC over separate
control channels. However, from literature, it is observed that
the joint optimization of decision thresholds for the HSS of
second kind has not been considered earlier. Further, it is to be
noted that none of [11]–[14], [17]–[19], [25], [33], [34] have
considered control channel’s bit rate constraint in their work.
The assumption of any available control channel bandwidth is
not a realistic one in the context of CR networks.
In this paper, we consider a throughput optimization prob-
lem for a CR network, which try to access a licensed channel
opportunistically, i.e., only when the PU is sensed as idle.
We consider HSS of second kind, where the FC performs
SPRT after collecting an observation vector consisting of the
hard decisions from all CR nodes. In our paper, we term
this sensing strategy by hybrid spectrum sensing with batch
collection (HSS-BC) for the ease of discussion. Please note
that in [30], authors considered the HSS-BC in the context of
sensor networks. However, in the context of CR networks, the
HSS-BC has not been explored earlier. We aim to design the
optimal decision thresholds to maximize the CR network’s
throughput, taking control channel’s bit rate constraint into
account. The thresholds at the FC are set by two parameters
defined by αand β, which are also the desired false-alarm
and missed-detection probabilities at the FC respectively. We
formulate a joint optimization problem, i.e., considering α,
β, and CR nodes’ false-alarm probability, i.e., Pfa 1, as
optimization parameters, in order to maximize the secondary
network’s throughput. The optimization problem is observed
to be non-convex with respect to α,β, and Pfa. We show
that for a predefined value of Pfa , the feasible set forms
a convex set; whereas, the objective function remains non-
convex. However, we develop an efficient way for solving the
optimization problem to get the optimal result.
We summarize our contributions in this paper as following:
•Joint optimization, (i.e., considering α,β, and Pf a as op-
timization parameters) is performed in order to maximize
the secondary network’s throughput.
•We impose control channel’s bit rate constraint and show
it’s effect on the optimal throughput of the CR network.
•We show that the optimization problem is in general non-
convex and provide an efficient mechanism to obtain the
global optima.
•We compare our proposed strategy with a sub-optimal
strategy (i.e., optimizing jointly over αand βonly, and
choosing a Pfa which maximizes the KL-divergence
of the quantizer output) in terms of throughput. The
optimal solution becomes equivalent to this sub-optimal
solution, under certain conditions. However, in general
the proposed algorithm substantially outperforms the sub-
optimal strategy.
•We initially assume hard decisions and identical decision
thresholds at CR nodes. Later we relax these assumptions
and discuss about the optimization problems for soft
decisions and non-identical decision thresholds at CR
nodes in Section V-C and V-B respectively.
The remainder of the paper is organized as follows: Sec-
tion II presents the system model of HSS-BC. In Section III,
we evaluate average control channel usage for both hypothe-
ses, detection error probabilities, and throughput for the sec-
ondary network. Section IV gives the detailed analysis of
solving the proposed optimization problem. We relax some
conditions in Section V, and discuss the corresponding opti-
mization problems. Section VI shows simulation results and
shows the efficacy of our proposed algorithm. We conclude in
Section VII.
II. SYSTEM MODEL: HYBRID SCHEME
In Fig. 1, we provide the block diagram of our system
model, where CRi,i= 1, ..., M , denotes cooperating cognitive
1As CR nodes perform energy detection, choosing the false-alarm proba-
bilities also corresponds to the decision thresholds for CR nodes.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at http://dx.doi.org/10.1109/TSP.2018.2838575
Copyright (c) 2018 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
3
radio nodes. We assume that each CR node senses a particular
licensed band to collect observations; in Fig. 1, we denote
the licensed band by radio frequency (RF) environment. To
avoid asynchronous arrival of local decisions at the FC, we
consider FSS at cognitive radios instead of SPRT. Energy
detection is the widely accepted spectrum sensing technique,
as it does not require any knowledge about the PU signal
and the associated system complexity for the detector is also
low [5]. For this reason, we choose energy detection at CR
nodes. Each CR node collects Nobservations from the radio
environment and performs energy detection to take a hard
decision (i.e., 1 or 0) about the presence or absence of the PU.
After completing the local sensing procedure, MCR nodes
send their hard decisions to the FC. The FC performs SPRT
using the received local decision vector (LDV) (consisting of
MCR nodes’ hard decisions) and either takes a global decision
or asks CR nodes to conduct another local sensing procedure,
such that, the FC receives another LDV. In Fig. 1, we present
a scenario where the FC receives sth LDV, which is denoted by
us= [us
1, us
2, .., us
M],s= 1,2, .., S, where Sdenotes the index
of time sample when the SPRT terminates (Sis random for
SPRT); us
i∈ {0,1}2, denotes the decision of the ith CR node,
i= 1,2, ..., M . It is to be noted that, to take the decision us
i,
ith CR node collects observation vector xs
i= [xs
i1, ..., xs
iN ]from
the RF environment. We define the presence and absence of
the PU by hypotheses H1and H0respectively and write the
received signal at the ith CR node as:
H1:xs
ij =hs
i.ps
j+ws
ij
H0:xs
ij =ws
ij ,(1)
where j= 1, ...., N ,hs
iis the channel coefficient between the
PU and the ith CR node. ps
jand ws
ij are the PU’s signal during
jth sampling interval and the additive white Gaussian noise
(AWGN) of known variance respectively at the ith CR node.
RF
Environment
1
...
Fusion Center
.....
...
2
.....
...
th th
1st
1
2
M
1
2
M
1
2
M
u
u
u
u
u
u
u
u
u
u uu
xx x2M
M
CRCRCR1 2 ....................
1 M
1
1
1
0
Final Decision u
s
s
s
s
s
s
s
s
s
s
S
S
S
S
LDV LDV LDV
Figure 1: Hybrid Sensing Architecture
We make the following assumptions in our work:
•During a sensing batch, channel coefficient hs
iremains
constant. However, the channel coefficient changes for
different sensing batches [20], [29]. hs
iis zero-mean, unit-
variance complex Gaussian random variable [35]. How-
ever, if we relax the channel’s fast changing scenario and
consider slow fading, then also the optimization problem
does not change, which we discuss in Section V-A.
•The PU’s activity remain constant (i.e., either idle or
active) throughout the frame [35], [36].
2In Section V-C, we discuss about soft decision fusion at the FC.
•Primary signal ps
jis independent and identically dis-
tributed (i.i.d.) random process, which has zero mean and
known σ2
pvariance [35].
•Noise samples wn
ij are i.i.d and follows circular sym-
metric complex Gaussian distribution with zero mean and
known variance σ2
w.
•PU signal’s samples and noise samples are mutually
independent.
•Given the hypotheses, observations at CR nodes are
conditionally independent and identically distributed.
•We assume a noiseless or perfect common control chan-
nel [33] over which CR nodes send their decisions in a
TDMA fashion [14]. The sequence of decisions reporting
is also predefined.
We follow the frame structure of a CR node as shown
in Fig. 2, where a fixed frame duration of Tis considered.
Each CR node collects Nsamples simultaneously from the
environment and sends their hard decisions to the FC. A
different frame structure may be adopted where a CR node
may perform spectrum sensing when it is not transmitting
hard decision, which may help in reducing the effective
sensing time. However, for simultaneous sensing and decision
transmission, CR nodes need extra memory to store the sensing
observations. Ts= 1/fs, is the sensing interval, where fsis the
sensing frequency of CR nodes. The reporting time for each
CR node is considered to be Tr= 1/Rb, where Rbis the bit
rate over the control channel. Each sensing batch requires the
time duration (NTs+MTr).
Figure 2: Frame Structure
A. Local Sensing Procedure
Test statistic for energy detection at the ith CR node for
taking the decision us
iis:
ys
i=1
N
N
X
j=1 |xs
ij |2
H1
R
H0
ǫi,(2)
where ǫiis the detection threshold for ith CR node. It is to be
noted that under the assumption of fast fading (i.e., channel
coefficient changes for different sensing batches), traditional
energy detection technique as given in [35], is optimal. Each
CR node observes a common phenomenon and makes a local
decision us
ias per following procedure
us
i=1ys
i≥ǫi
0ys
i< ǫi.(3)
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at http://dx.doi.org/10.1109/TSP.2018.2838575
Copyright (c) 2018 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
4
Note that, under the assumption of sufficiently large value of
N, the false-alarm and detection probabilities of the ith sensor
may be written as [35]:
Pi
fa = Pr{ys
i≥ǫi;H0}=Qǫi
σ2
w−1√N(4a)
Pi
d= Pr{ys
i≥ǫi;H1}=Qǫi
σ2
w−γi−1rN
2γi+ 1 ,
(4b)
where Q(x) = 1
√2πR∞
xexp −t2
2dt and γi=γ=E[|hs
i|2]σ2
p
σ2
w
3
is the received SNR at each CR node; E[z]denotes the
expectation of the random variable z.
In [37], [38], authors have shown that if the sensor nodes
receive i.i.d. observations in a collaborative sensor network,
then choosing identical sensors’ threshold is asymptotically
optimal. In this paper, we assume identical CR nodes’ decision
rules and hence identical false-alarm and detection probabil-
ities as denoted by: Pi
fa =Pf a and Pi
d=Pdrespectively. In
Section V-B, we relax the identical threshold condition and
discuss about the corresponding optimization problem.
B. Fusion Strategy
The FC performs SPRT after updating the likelihood ratio
with received LDV from the CRs. Updated likelihood ratio is
compared with two thresholds µ1and µ0to take the final deci-
sion u0. These thresholds are fixed by two predetermined pa-
rameters, i.e., αand β, where αand βare targeted false-alarm
and missed-detection probabilities at the FC. The expressions
for µ1and µ0may be approximated as: µ1= (1 −β)/α and
µ0=β/(1 −α)[9]. As the observations from CR nodes are
conditionally statistically independent over space and time, the
FC’s likelihood ratio after sth LDV reception may be written
as:
Ls=
s
Y
k=1
M
Y
i=1
Pr(uk
i|H1)
Pr(uk
i|H0).(5)
Decision strategy at the FC is expressed as [9]:
If Ls
≥µ1,Decide H1
≤µ0,Decide H0
∈(µ1, µ0),Continue. (6)
After following the simplification steps as given in Ap-
pendix A, the test procedure can be written as:
If Ts
H≥A−sB, Decide H1
If Ts
H≤C−sB, Decide H0
Otherwise collect another LDV, (7)
where Ts
H=Ps
k=1 mk, is the test statistic after the FC receives
the sth LDV; ms∈ {0,1, .., M }denotes the number of CR
nodes out of M, which decides in favor of the hypothesis
H1in the sth LDV. A,B, and Ccan be derived as:
A=ln(µ1)
ln( Pd(1−Pfa )
Pfa (1−Pd));B=
Mln( 1−Pd
1−Pfa )
ln( Pd(1−Pfa )
Pfa (1−Pd));C=ln(µ0)
ln( Pd(1−Pfa )
Pfa (1−Pd)).
From the nature of the test at the FC as given in (7), it may
be observed that the test terminates at random time instant,
such that, the required number of LDVs become also random,
which can be represented as:
S=inf{s|Ts
H≥A−sB or Ts
H≤C−sB}.(8)
3As we have assumed unit variance for the channel coefficient hs
i, the
average SNR at CR nodes become identical.
III. CONTROL CHANNEL’S BIT RATE, ERROR
PROBABILITIES,AND SECONDARY NETWORK’S
THROUGHPUT
In our system model, the FC collects the LDVs, each of
which is formed with the hard decisions of MCR nodes.
From (8), it is observed that, the required number of such
vectors to terminate the SPRT is random. We evaluate the
average number of LDVs using WALD’s approximation [9],
where negligible overshoot at the termination time has been
assumed. Following [9], the lower bounds of the required
expected number of LDVs under hypotheses H1and H0can be
written as functions of α,β, and Pfa , which are respectively:
SW ALD
H1(α, β, Pf a )≥1
D(p1||p0)βln β
1−α+ (1 −β) ln 1−β
α
(9a)
SW ALD
H0(α, β, Pf a )≥1
−D(p0||p1)(1 −α) ln β
1−α+αln 1−β
α,
(9b)
where p1= Pr(us|H1)and p0= Pr(us|H0)are probability
mass functions (pmfs) of the sth LDV under hypotheses H1
and H0respectively. D(p0||p1)and D(p1||p0)are the KL-
divergences 4from p1to p0and from p0to p1respectively.
As the decisions from different CR nodes are independent and
identically distributed, we can write:
D(p0||p1) =
M
X
i=1
D(pi
0||pi
1)(10a)
D(p1||p0) =
M
X
i=1
D(pi
1||pi
0),(10b)
where pi
0= Pr(us
i|H0)and pi
1= Pr(us
i|H1), such that, ∀i:
D(pi
0||pi
1) = X
us
i∈{0,1}
Pr(us
i|H0) log2Pr(us
i|H0)
Pr(us
i|H1)
=Pfa log2Pfa
Pd+ (1 −Pfa ) log21−Pf a
1−Pd
(11a)
D(pi
1||pi
0) = X
us
i∈{0,1}
Pr(us
i|H1) log2Pr(us
i|H1)
Pr(us
i|H0)
=Pdlog2Pd
Pfa + (1 −Pd) log21−Pd
1−Pfa .
(11b)
We consider that SW ALD
H0(α, β, Pf a )and SW ALD
H1(α, β, Pf a )
are equal to their corresponding lower bounds. From Fig. 2, it
can be observed that the control channel is used for Mtimes
during each local sensing batch. Without loss of generality, we
consider that single bit is transmitted over the control channel
every time and define the average number of bits per second,
which are sent over the control channel under hypotheses H0
and H1respectively as:
SW ALD
H0(α, β, Pf a )·M
T=M
T
f1(α, β)
g1(Pfa )bits/sec. (12a)
SW ALD
H1(α, β, Pf a )·M
T=M
T
f2(α, β)
g2(Pfa )bits/sec., (12b)
4For a discrete random variable xwhose two probability distributions
are p(x)and q(x), the KL-divergence from q(x)to p(x)is defined
as:D((p(x)||q(x)) = Px∈Xp(x) ln p(x)
q(x), where Xis the set over
which p(x)and q(x)are defined.
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5
where
f1(α, β) = (1 −α) ln 1−α
β+αln α
1−β
f2(α, β) = βln β
1−α+ (1 −β) ln 1−β
α
g1(Pfa ) = D(p0||p1), g2(Pf a ) = D(p1||p0).
The false-alarm and missed-detection probabilities at the FC
may be approximated as [9]:
PW ALD
F A ≈αand PW ALD
MD ≈β. (13)
The total sensing time depends on the required number of
LDVs to terminate the test at the FC. Total time spent for
receiving Snumber of LDVs and taking decision by the FC is
S·(NTs+MTr). As the required number of LDVs is random,
total sensing duration also becomes random. Under hypothesis
Ht,t∈ {0,1}, the average sensing time becomes:
τHt=SW ALD
Ht(α, β, Pf a )· {N Ts+M Tr}.(14)
We use the average sensing times under both hypotheses as
given in (14) and write the secondary network’s throughput as
function of α,β, and Pfa :
R(α, β, Pf a )
=RH0(α, β, Pf a ) + RH1(α, β , Pfa)
=T−τH0
TP(H0)(1 −PF C
F A )r0+T−τH1
TP(H1)PF C
MD r1
=P(H0)r0(1 −α)−f3(α, β)
g1(Pfa )+P(H1)r1β−f4(α, β )
g2(Pfa ),
(15)
where RHt(α, β, Pf a )denotes the throughput under
hypothesis Ht,t∈ {0,1}.r0=E[log2(1 + |g|2.σ2
cr
σ2
w)] and
r1=E[log2(1 + |g|2.σ2
cr
σ2
w+σ2
I
)] are the normalized average rates
received at the SU-Rx under hypotheses H0and H1
respectively, where gdenotes the channel coefficient between
the SU-Tx. and SU-Rx; the expectations are taken over the
random variable g.σ2
Iand σ2
cr are the received interference
power at the SU-Rx and the transmission power of the SU-Tx
respectively. We further define f3(α, β)and f4(α, β)as:
f3(α, β) = N Ts+M Tr
T(1 −α)f1(α, β)(16a)
f4(α, β) = N Ts+M Tr
Tβf2(α, β ).(16b)
We next formulate an optimization problem, where we max-
imize R(α, β, Pf a )under the constraints on control channel’s
bit rate and interference created on the PU by the CR network.
IV. THROUGHPUT OPTIMIZATION FOR HYBRID SENSING
STRATEGY
We maximize the throughput as shown in (15) with respect
to α,β,and Pfa . The optimization problem is presented as:
P1 : maximize
(α,β,Pf a )R(α, β, Pf a )(17a)
subject to: α≥0(17b)
β≥0(17c)
α+β≤1(17d)
β≤ζ(17e)
M
T
f1(α, β)
g1(Pfa )≤Ψbits/sec. (17f)
M
T
f2(α, β)
g2(Pfa )≤Ψbits/sec., (17g)
where (17b) and (17c) represents the non-negativity of false-
alarm and missed-detection probabilities at the FC respec-
tively, whereas, the constraint shown in (17d) represents the
fact that the detection probability remains greater or equals
to the false-alarm probability [39]. (17e) is for interference
constraint (interference created on the PU by the CR network),
where the maximum tolerable interference probability is de-
noted by ζ5. (17f) and (17g) are for the control channel’s
instantaneous bit-rate (as given in (12a) and (12b)) constraints
for the CR network under hypotheses H0and H1respectively,
where the maximum allowed bit-rate is Ψbits/sec.
In general, lower missed-detection probability at the FC
(i.e., β) is considered to reduce the interference on the PU
caused by the secondary network [14], [35]. If the PU trans-
mits at a high power, then the received interference power at
the SU-Rx. becomes high, which makes the normalized aver-
age rate under hypothesis H1, i.e., r1, very small. Under the
assumptions of lower βand r1, we can neglect RH1(α, β, Pf a)
[14], [40] and rewrite the optimization problem P1as:
P2 : maximize
(α,β,Pf a )RH0(α, β, Pf a )(18)
subject to: (17b) −(17g).
As the optimization problem P2is dependent on three vari-
ables, convexity check of the optimization problem is not
trivial. However, if we can show that the objective function or
any of the constraints is non-convex, then we can conclude that
the optimization problem P2does not belong to the convex
optimization family. In that direction, we choose the constraint
as given in (17g). We can rewrite the constraint as:
M
Tf2(α, β)−Ψ·g2(Pf a )≤0.(19)
In Appendix C, we show that f2(α, β)is a convex function
of αand β. So, we can conclude that the constraint given
in (17g) will be convex if and only if g2(Pfa )is concave.
However, we check that g2(Pf a )is not a concave function of
Pfa (proof is given in Appendix D). Hence, we conclude that
the optimization problem P2is not convex.
We next provide a summary of the steps for solving the
optimization problem P2for obtaining the global optima:
•Step1: In Proposition IV.1, we show that, for a fixed value
of Pfa , the optimal feasible set (i.e., αand βvalues) of
the optimization problem P2is convex; moreover, the
contour lines of (17f) and (17g), follow a monotonically
decreasing pattern with respect to the βvariation. We give
a pictorial overview about the feasible set and from there
we show that the optimal value for αchanges with the
value of Pfa . This is the motivation behind our further
analysis considering fixed value of Pfa .
•Step2: In Proposition IV.2, we show that for fixed value
of Pfa , the objective function as given in (18), becomes
monotonically increasing and concave function of β
(when αis fixed) and a concave function of α(when
βis fixed).
5If we consider the average received power at the PU from the SU-
Tx. to be P′
swatt., then we can write the average interference power
at the PU as: P(H1)·β·P′
swatt. We denote the maximum tolerable in-
terference power at the PU as Pint watt, such that, the constraint be-
comes: P(H1)·β·P′
s≤Pint, which can be rewritten as: β≤ζ, where
ζ=Pint/{P(H1)P′
s}.
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•Step3: In Proposition IV.3, we show that irrespec-
tive of any values of αand Pfa, maximum value for
RH0(α, β, Pf a )is achieved for β=ζ. This helps us in
omitting the optimization parameter βfrom the optimiza-
tion problem P2.
•Step4: In Proposition IV.4, after considering β=ζ, we
show that the value of Pfa which maximizes each CR
node’s KL-divergence from pi
1to pi
0, i.e., D(pi
0||pi
1), in
(10a), becomes optimal if the constraint as given in
(17f) becomes active for the corresponding Pfa . As for a
fixed value of Pfa the optimization problem P2becomes
convex, optimal value for αcan be easily evaluated while
considering the corresponding Pf a (which maximizes
D(pi
0||pi
1))and β=ζin P2. After a detailed analysis,
we find this particular case happens under lower values
of N,Ψ,and SNR at CR nodes.
•Step5: One-dimensional search over Pfa is performed for
finding out the optimal Pf a and αvalues if the constraint
as given in (17f) does not become active for β=ζand
maximum value of D(pi
0||pi
1).
Proposition IV.1. For a fixed value of Pf a (= Pf a), the
feasible set constructed by the constraints as given in (17b)-
(17g), becomes convex. Moreover, if we consider equality
constraints in (17f) and (17g), then the corresponding contour
lines follow a monotonically decreasing pattern with respect
to the βvariation in α, β-plane.
Proof. Initially, we prove the first statement of this propo-
sition. It may be observed that the constraints as given in
(17b)-(17e) are nominally convex as they are linear functions.
Whereas, in Appendix C, we show that f1(α, β)and f2(α, β )
are convex functions of αand β, which means that for a fixed
value of Pfa , constraints given in (17f) and (17g), are convex
with respect to αand β. So, we can conclude that the feasible
set becomes convex after considering fixed value for Pfa.
Now, we move to the proof of the second statement of this
proposition. For this, we consider equality constraints in (17f)
and (17g) and evaluate the first order derivatives (i.e., ∂α/∂β)
respectively:
∂α
∂β =−d1
d2(20a)
∂α
∂β =−d2
d1
,(20b)
where d1=1−α−β
β(1−β)and d2= ln h(1−α)(1−β)
αβ i.
It may be observed from (20a) and (20b) that, d1and d2
are always positive for any set of feasible values of αand
β. Therefore, the derivatives in (20a) and (20b) are always
negative. Hence, we can conclude that the contour lines
monotonically decrease with β.
For given values of T, M, Ψ, and Pf a =Pfa , we define two
different αvalues from (17f) and (17g) for β=ζconsidering
equality in those corresponding equations as:
α1(Pfa , ζ ) = α|f1(α, ζ) = Ψ·T·g1(Pf a )
M(21a)
α2(Pfa , ζ ) = α|f2(α, ζ) = Ψ·T·g2(Pf a )
M.(21b)
Figure 3: Feasible Region of αand β
In Fig. 3, we plot the contour lines for (17d)-(17g) (considering
equality sign) in α, β-plane based on following observations:
•From the nature of the functions f1(α, β)and f2(α, β )
as given in (12a) and (12b), it may be observed that for
β= 0, we get: α1(Pfa ,0) and α2(Pf a ,0) ≥0.
•In a similar fashion, we can say that
α1(Pfa , β ′) = α2(Pfa, β ′′) = 0, for β′, β′′ ≤1.
•Contour-3 and 4, as shown in Fig. 3, monotonically
decrease with respect to β(Proposition IV.1). Moreover,
the contour lines should be also convex as both f1(α, β)
and f2(α, β)are convex(Appendix C).
The blue shaded region in Fig. 3, represents the feasible
region of αand β. It is to be noted that Fig. 3 is shown
considering a fixed value of Pfa . The cut points (i.e., with α
and βaxis, and β=ζline) of the curves 3 and 4, and hence
the feasible region (of αand β) change depending upon the
chosen value of Pfa .
Now, we analyse the objective function as given in (18),
and frame our findings in the following proposition.
Proposition IV.2. For fixed value of α,RH0(α, β, P f a )is a
monotonically increasing and concave function of β, whereas,
for fixed value of β,RH0(α, β, Pf a )becomes a concave func-
tion of α.
Proof. Proof is given in Appendix E.
From the fact that, RH0(α, β, Pf a )is a monotonically in-
creasing function of βfor fixed α(Proposition IV.2), we
present the following proposition.
Proposition IV.3. Maximum value for RH0(α, β , Pfa)is
achieved at the boundary value of β, i.e., β=ζ.
Proof. We prove this proposition by contradiction. Therefore,
we assume that RH0(α, β , P fa )becomes maximum for α=α∗
and β=β∗(where, 0≤β∗< ζ), such that, there exist a
δ(δ > 0) for which β′=β∗+δ(hence β′> β∗), is always in
the feasible set. As α∗and β∗are the corresponding optimal
values for αand βrespectively, we can say that:
RH0(α∗, β∗,Pf a )≥RH0(α∗, β
′
, P f a ).(22)
It may be observed that for β′≥β∗, (22) contradicts the fact
that RH0(α, β, Pf a )is a monotonically increasing function of
βfor fixed α(proof is given in Proposition IV.2). Hence, we
conclude that our initial assumption is false. We can fix αand
increase βto be in the feasible region, which means that the
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7
optimal feasible βshould always lie on the contour-1 (i.e., for
β=ζ) or contour-2 (i.e., for α+β= 1) as shown in Fig. 3.
For the condition, i.e., α+β= 1, after relevant substitution
in (15), we get:
RH0(α, β, Pf a ) = P(H0)·r0·(1 −α).(23)
It is observed that (23) is a linear function of α, which
increases while we reduce the value of α. As it is evident from
Fig. 3, that the lowest feasible value of αis received for β=ζ,
we can conclude that the optimal value for RH0(α, β, Pf a )is
received for β=ζirrespective of any values of α. This proof
holds for any value of Pfa as (23) does not depend on Pf a.
In our optimization problem P2, we substitute βby ζand
rewrite the optimization problem as:
P3 : maximize
(α,Pfa )RH0(α, ζ , Pfa )(24a)
subject to (17b)
α≤1−ζ(24b)
M
T
f1(α, ζ)
g1(Pfa )≤Ψbits/sec. (24c)
M
T
f2(α, ζ)
g2(Pfa )≤Ψbits/sec.. (24d)
We investigate the cut points (i.e., corresponding αvalues)
of contour-3 and contour-4 (as given in Fig. 3) with the
β=ζline. It is observed that depending upon the value of
Pfa ,α1(Pf a , ζ )(given in (12a)) may be greater or lesser than
α2(Pfa , ζ )(given in (12b)).
To understand the effect of the parameter, i.e., Pfa , on the
optimization problem P3, we analyse the objective function
(given in (24a)) and the constraints (given in (24c) and (24d)).
For our further analysis, we define the following αvalues:
αmax =α|f1(α, ζ) = Ψ·T·g1(Pmax
fa )
M(25a)
α
′
=α|f2(α, ζ) = Ψ·T·g2(Pmax
fa )
M,(25b)
where Pmax
fa is defined as:
Pmax
fa = arg max
Pfa
g1(Pfa ).
From (10a), it can be observed that Pmax
fa maximizes in-
dividual CR node’s KL-divergence from pi
1to pi
0, i.e.,
D(pi
0||pi
1),i= 1,2, .., M , as given in (10a).
As the instantaneous number of bits per second is a positive
quantity, from (12a), we can say that f1(α, β)should be a
positive quantity for any values of αand βas KL-divergence,
i.e., g1(Pfa )can never be a negative quantity. This implies
that f3(α, ζ)as given in (16a), is also a positive quantity.
Therefore, from (15), we can say that RH0(α, ζ , Pfa)attains
higher value when we increase the value of g1(Pfa). For these
reasons, initially we check whether the constraints are satisfied
for αmax or not and present Proposition IV.4.
Proposition IV.4. For αmax ≥α′, maximum throughput, i.e.,
RH0(eα, ζ, P ∗
fa )(eαstands for optimal αvalue), is received for
the P∗
fa =Pmax
fa , which maximizes the each CR node’s KL-
divergence, i.e., D(pi
0||pi
1), in (10a).
Proof. From (25a) and (25b), for Pfa =Pmax
fa , we get
α1(Pfa , ζ ) = αmax and α2(Pfa , ζ) = α′. As we have consid-
ered αmax ≥α′in this proposition, from Fig. 3, we can
conclude that αmax lies in the feasible region.
We prove this proposition by contradiction theorem. For
that, we assume that α∗and P∗
fa (P∗
fa 6=Pmax
fa ) are optimal
values for αand Pfa respectively, and define the sets of αas
given in (26a) and (26b), on the top of next page.
As per our initial assumption, we get α∗∈S. Now, we try
to prove that α∗∈S′. In that direction, we define the following
level sets for f1(α∗, β)and f1(αmax, β )as respectively:
γ∗=Ψ·T·g1(P∗
fa )
M(27a)
γmax =Ψ·T·g1(Pmax
fa )
M.(27b)
As g1(Pmax
fa )> g1(P∗
fa ), we can write:
f1(α∗, ζ)(a)
<Ψ·T·g1(Pmax
fa )
M
(b)
=f1(αmax, ζ ),(28)
where (a): as γmax > γ∗, with the help of Theorem B.2 as
given in Appendix B, we write this; (b): from (25a).
As f1(α, ζ)is a monotonically decreasing function of α,
with the help of (28), we can write:
αmax < α∗.(29)
From the relation as given in (29) and our initial consideration,
i.e., αmax ≥α′, we get:
α′< α∗.(30)
With the help of (30) and the monotonically decreasing
property of f2(α, ζ), we can write that:
f2(α∗, ζ)< f2(α′, ζ )
⇒f2(α∗, ζ)<Ψ·T·g2(Pmax
fa )
M.(31)
From the inequality relations as given in (28) and (31), it
may be clearly observed that: α∗∈S′. So, we can say that
(α∗, P max
fa )is a possible solution point of the optimization
problem P3, as it satisfies all the constraints of the correspond-
ing problem. As per our initial assumption, we can write:
R(α∗, ζ, P ∗
fa )> R(α∗, ζ, P max
fa ).(32)
It can be observed that (32) contradicts the fact that Pmax
fa
yields the maximum throughput. So, we can conclude that
our initial assumption, i.e., P∗
fa is optimal Pfa value, does
not hold anymore. As we have considered eαas the optimal α
value corresponding to Pmax
fa , we can say that (eα, P max
fa )is the
optimal solution point in this case, which may be evaluated
after solving the following optimization problem:
P4 : maximize
αRH0(α, ζ , P max
fa )(33a)
subject to: (17b) and (24b)
M
T
f1(α, ζ)
g1(Pmax
fa )≤Ψbits/sec.. (33b)
As the constraint given in (24d), becomes inactive for the
condition αmax ≥α′, we neglect that in the optimization
problem P4. The optimal value of α, i.e., eα, becomes:
eα= max αuncons (Pmax
fa ), αmax,(34)
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S=α|α≥0, α ≤1−ζ , f1(α, ζ)≤Ψ·T·g1(P∗
fa )
M, f2(α, ζ)≤Ψ·T·g2(P∗
fa )
M(26a)
S′=α|α≥0, α ≤1−ζ , f1(α, ζ)≤Ψ·T·g1(Pmax
fa )
M, f2(α, ζ)≤Ψ·T·g2(Pmax
fa )
M.(26b)
where αuncons(Pmax
fa )is the inflection point of the concave
objective function as given in (33a):
αuncons(Pmax
fa ) = α|∂
∂α RH0(α, ζ , P max
fa )= 0.(35)
Proposition IV.4 represents the fact that we can write the
optimization problem P3in the form P4only when both
of the tests (i.e., under hypotheses H0and H1) terminates
for Pfa =Pmax
fa and α=αmax. This also implies that for
αmax ≥α′, optimal quantization threshold for taking decision
at CR nodes are evaluated by maximizing CR nodes’ KL-
divergence, i.e., D(pi
0||pi
1).
However, the relation, i.e., αmax ≥α′, may not hold every
time. For given value of T, based on the values of Pmax
fa and Ψ,
we may get αmax < α′. Under the condition, i.e., αmax < α′,
we need to perform an one dimensional search over Pf a. In
the following subsection, we provide some reasons behind the
search operation’s optimality.
A. Conditions under which search operation is optimal
As f2(α, ζ)is a monotonically decreasing and convex func-
tion of α(Appendix C), for αmax < α′we get:
f2(αmax, ζ )> f2(α′, ζ)
(c)
⇒f2(αmax, ζ )
g2(Pmax
fa )>Ψ·T
M,(36)
where (c): from (25b).
It may be observed that if we choose Pfa =Pmax
fa and
α=αmax, then after substituting (25a) in (36), we get:
f2(αmax, ζ )
g2(Pmax
fa )>f1(αmax, ζ )
g1(Pmax
fa )
(d)
⇒SW ALD
H1(αmax, ζ , P max
fa )> SW ALD
H0(αmax, ζ , P max
fa ),(37)
where (d): from (12a) and (12b).
From (37), we can observe that for αmax < α′, the required
number of LDVs for terminating the test is more for hypothesis
H1compared to hypothesis H0. It is to be noted that as f2(α, ζ)
is a monotonically decreasing and convex function of α(proof
is given in Appendix F), we can say lower number of LDVs
for hypothesis H0compared to H1leads to the condition
αmax < α′. It can be observed that (37) always holds for:
g2(Pmax
fa )< g1(Pmax
fa )and f2(αmax, ζ )> f1(αmax, ζ).(38)
We analyse the functions g1(Pf a ),g2(Pf a ),f1(α, ζ ),and
f2(α, ζ)and give the possible reasons for which (38) holds.
•g2(Pmax
fa )< g1(Pmax
fa ): It means, the KL-divergence
from p1to p0is greater than the KL-divergence from
p0to p1for Pfa =Pmax
fa , where p0and p1are two pmfs
and have been defined in (10a) and (10b).
Both g1(Pfa )and g2(Pf a )depends on Pf a and Pd,
which in turn depends on the SNR at CR nodes and num-
ber of samples collected for spectrum sensing, i.e., N. We
analyse the functions g1(Pfa )and g2(Pf a )pictorially for
different values of SNR and N.
From Fig. 4, it may be observed that under
higher values for SNR and N, the condition, i.e.,
g2(Pmax
fa )< g1(Pmax
fa ), becomes more prominent. So, we
can conclude that the search operation becomes optimal
when CR nodes’ detection performance improves.
•f2(αmax, ζ )> f1(αmax, ζ):f1(α, ζ )and f2(α, ζ)are
the average LLR values at the FC for hypotheses H0and
H1respectively. So, f2(αmax , ζ)> f1(αmax , ζ)indicates
that in order to get the false-alarm probability αmax
and missed-detection probability ζ, required average LLR
value is more for the hypothesis H1compared to H0. In
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
0.5
1
1.5
2
2.5
3
3.5
4
α
f1(α,ζ)
f2(α,ζ)
αcut=ζ
Figure 5: f1(α, ζ)and f2(α, ζ )vs α(ζ= 0.1)
Fig. 5, we plot f1(α, ζ)and f2(α, ζ )for different values
of α, where
αcut ={α|f1(α, ζ) = f2(α, ζ ), α ∈(0,1−ζ)}.(39)
As both f1(α, ζ)and f2(α, ζ )are convex functions of α
(proof is given in Appendix F), for α∈(0,1−ζ), we
get one crossover point between f1(α, ζ)and f2(α, ζ ).
From (12a) and (12b), we get the crossover point as
αcut =ζ. Fig. 5 shows that we may get the condition,
i.e., f2(αmax, ζ )> f1(αmax, ζ ), for αmax < ζ. From the
nature of f1(α, ζ)as shown in Fig. 5, it may be observed
that for given values of Tand Mif we increase the values
for either Ψ,g1(Pmax
fa )or both, then the chance of getting
the condition, i.e., f2(αmax, ζ )> f1(αmax, ζ ), is higher.
One dimensional search over Pfa is performed in order
to get the optimal solution for αmax < α′. During the search
operation, we solve the following optimization problem after
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9
0 0.2 0.4 0.6 0.8 1
0
0.02
0.04
0.06
0.08
0.1
0.12
Pfa
g1(Pfa)
g2(Pfa)
(a) SNR=-15 dB, N= 50
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
Pfa
g1(Pfa)
g2(Pfa)
(b) SNR=-5 dB, N= 50
0 0.2 0.4 0.6 0.8 1
0
10
20
30
40
50
60
70
Pfa
g1(Pfa)
g2(Pfa)
(c) SNR=0 dB, N= 50
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Pfa
g1(Pfa)
g2(Pfa)
(d) SNR=-8 dB, N= 10
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
Pfa
g1(Pfa)
g2(Pfa)
(e) SNR=-8 dB, N= 100
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
Pfa
g1(Pfa)
g2(pfa)
(f) SNR=-8 dB, N= 200
Figure 4: g1(Pfa )and g2(Pf a )vs. Pf a
considering Pfa =Pf a, where Pf a ∈(0,1).
P5 : maximize
αRH0(α, ζ , P fa )(40a)
subject to: (17b) and (24b)
M
T
f1(α, ζ)
g1(Pfa )≤Ψbits/sec. (40b)
M
T
f2(α, ζ)
g2(Pfa )≤Ψbits/sec.. (40c)
It is to be noted that for a given value ofPfa , the optimization
problem P5belongs to convex optimization family (as the
objective function and the constraints are convex over α). From
Fig. 3, it is evident that for given Pfa, the optimal value of
α, i.e., eα(Pf a ), may be evaluated as:
eα(Pfa ) = max α1(Pf a , ζ ), α2(Pfa, ζ ), αuncons.(Pf a ),(41)
where α1(Pfa , ζ ), α2(Pfa , ζ),and αuncons.(Pf a )may be evalu-
ated from (21a), (21b), and (35) respectively. From this search
operation, we choose the corresponding values of Pf a and
eα(Pfa ), which gives maximum value for RH0(α, ζ, Pf a ).
V. OPTIMIZATION PROBLEMS FOR SLOW FADING, CR
NODES’NON-IDENTICAL DECISION THRESHOLDS,AND
SOFT DECISION FUSION
In Section IV, we have assumed fast fading for the channel
between the RF environment and each CR node. Moreover,
CR nodes get i.i.d. observations and take hard decisions. In
this section, we relax some assumptions and discuss about the
corresponding optimization problems.
A. Optimization problem for slow fading scenario
In Section IV, we have solved the optimization problem
considering fast fading for the sensing channel, i.e., the
channel between the RF environment and a CR node. In this
subsection, we discuss about the scenario when the channel
changes slowly over time, such that, the observations at a
CR node over different sensing batches as shown in Fig. 2,
may no longer remain uncorrelated. We assume that CR nodes
possess knowledge about the correlation model. However, due
to the control channel’s bandwidth constraint the FC may
not have any knowledge about the correlation. CR nodes
may incorporate the channel correlation model in order to
perform the sensing optimally. As CR nodes are performing
energy detection, it will be more relevant to consider the
fading amplitude rather than the phase. Typically, the fading
amplitude is considered to be Rayleigh distributed [41]. In
literature, we find a good amount of papers (e.g. [42], [43]),
which have modelled the fade amplitudes at successive epochs
of interest by finite state Markov chain (FSMC).
CR nodes’ detection and false-alarm probabilities will
change from (3), after considering FSMC in modelling the
channel’s correlation, which also changes the values for KL-
divergences, i.e., D(p0||p1)and D(p1||p0), as given in (10a) and
(10b). After considering the correlation model, the detection
performance of CR nodes will improve, which will increase
the values of D(p0||p1)and D(p1||p0). As the FC does not
incorporate the correlation model in the SPRT process as given
in (6), the forms for the required number of LDVs as given
in (9a) and (9b), do not change. Moreover, the false-alarm
and detection probabilities (i.e., αand (1 −β)respectively)
also remain identical to the fast fading scenario. Hence, it can
be concluded that the corresponding form for the throughput
optimization problem for the slow fading scenario will remain
identical to the fast fading scenario as given in P2. Therefore,
our proposed approach for getting the optimal throughput is
applicable for the slow fading scenario.
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10
0.5 1 1.5
x 104
3.25
3.3
3.35
3.4
3.45
ψ (bits/sec.)
RH0(α,β,Pfa) (bits/sec./Hz.)
Optimal
Sub−optimal
SNR = −6 dB
SNR = −7 dB
(a) Varying Ψ(N= 25 and ζ= 0.1)
−10 −9.5 −9 −8.5 −8 −7.5 −7 −6.5 −6
2.95
3
3.05
3.1
3.15
3.2
3.25
3.3
3.35
3.4
3.45
SNR (in dB)
RH0(α,β,Pfa) (bits/sec./Hz.)
Optimal
Sub−optimal
ψ = 5000 bits/sec.
ψ = 8000 bits/sec.
(b) Varying SNR (N= 65 and ζ= 0.1)
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
3.32
3.34
3.36
3.38
3.4
3.42
3.44
ζ
RH0(α,β,Pfa) (bits/sec./Hz.)
Optimal
Sub−optimal
ψ = 8000 bits/sec.
ψ = 10000 bits/sec.
(c) Varying ζ(N= 65 and SNR = -6dB)
Figure 6: Average throughput for varying Ψ, SNR at CR nodes, and ζ
B. Optimization problem for non-identical decision thresholds
at CR nodes
In this subsection, we consider the scenario when CR
nodes’ decision thresholds are non-identical, which makes CR
nodes’ false-alarm probabilities non-identical. We formulate
the corresponding optimization problem as follows:
P6 : maximize
(α,β,Pfa )RH0(α, β , Pfa)(42)
subject to: α≥0(43)
β≥0(44)
α+β≤1(45)
β≤ζ(46)
M
T
f1(α, β)
g1(Pfa)≤Ψbits/sec. (47)
M
T
f2(α, β)
g2(Pfa)≤Ψbits/sec., (48)
where Pfa = [P1
fa , .., P M
fa ], is the corresponding vector consist-
ing the false-alarm probabilities of MCR nodes. g1(Pfa)and
g2(Pfa)can be evaluated from (10a) and (10b), which are
evaluated with the help of (11a) and (11b) respectively after
putting Pfa =Pi
fa and Pd=Pi
d.RH0(α, β, Pfa )is evaluated by
considering g1(Pfa ) = g1(Pfa )in (15).
It can be observed that Proposition IV.1,IV.2, and IV.3,
are valid for the optimization problem P6. Hence, we can
conclude that the optimal value for βis ζfor P6. Similarly,
Proposition IV.4, also holds for the optimization problem P6.
We can evaluate the matrix Pmax
fa = [P1,max
fa , .., P 1,max
fa ](where
Pi,max
fa = arg maxPi
fa D(pi
0||pi
1)), by evaluating the following
equation:
Pmax
fa = arg max
Pfa
g1(Pfa)
(e)
= arg max
P1
fa
D(p1
0||p1
1) + .. + arg max
PM
fa
D(pM
0||pM
1),(49)
where (e): as KL-divergence is always positive. Pmax
fa becomes
optimal for the condition αmax ≥α′satisfies, where αmax and
α′can be evaluated from (25a) and (25b) respectively by
considering Pmax
fa =Pmax
fa . However, Mdimensional search
operation needs to be performed over the vector Pfa for the
condition αmax < α′.
C. Optimization problem for soft decision combining
In this subsection, we discuss about the optimization prob-
lem related to soft decision combining at the FC. CR nodes
represent their decisions by using multiple bits rather than
a single bit. Therefore, the number of decision thresholds
become more than one in case of soft decision combining.
If the observations at CR nodes become i.i.d., then it is fair
to assume that CR nodes follow identical decision thresholds,
such that, the optimization problem becomes:
P7 : maximize
(α,β,t)RH0(α, β , t)(50)
subject to: α≥0(51)
β≥0(52)
α+β≤1(53)
β≤ζ(54)
log2K·M
T
f1(α, β)
g1(t)≤Ψbits/sec. (55)
log2K·M
T
f2(α, β)
g2(t)≤Ψbits/sec., (56)
where Kis the number of different decisions, L={l1, .., lK}
represents the set of different decisions of CR nodes, and
t= [t0, ..tK]represents different decision thresholds. More-
over,
g1(t) =
M
X
i=1 X
us
i∈L
Pr(us
i|H0) log2Pr(us
i|H0)
Pr(us
i|H1)
g2(t) =
M
X
i=1 X
us
i∈L
Pr(us
i|H1) log2Pr(us
i|H1)
Pr(us
i|H0)
Pr(us
i=lk|Hj) = Ztk
tk−1
f(Ys
i|Hj)dY s
i,
where f(Ys
i|Hj)is the density function of the received ob-
servation ys
i(as defined in (2)) at the ith CR node during
taking sth decision. RH0(α, β, t)is evaluated by considering
g1(Pfa ) = g1(t)in (15). Please note that in order to represent
Knumber of decisions, the required number of bits will
be log2K. Therefore, we have multiplied that term in (55)
and (56) to represent the number of control channel bits for
hypothesis H0and H1respectively.
The optimization problem P7possess the same form like
P1. Therefore, Proposition IV.1,IV.2, and IV.3, are valid for
P7. Hence, the optimal value for βbecomes ζfor P7. We
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11
can solve αmax and α′from (25a) and (25b) respectively by
considering g1(Pmax
fa ) = g1(tmax)and g2(Pmax
fa ) = g2(tmax),
where:
tmax = arg max
tg1(t).(57)
From Proposition IV.4, we can say that if αmax ≥α′holds,
then tmax becomes the optimal decision thresholds vector for
CR nodes. Please note that it is not trivial to mathematically
solve the value for tmax. We may use search operation over
the vector tin order to solve (57). The search operation is
also required for αmax < α′.
VI. RESULTS AND DISCUSSION
In this section, we present the simulation results
for the HSS-BC. The channel between the SU-
Tx. and the SU-Rx. is assumed to follow complex
normal distribution, i.e., g∼ CN(0,1). We consider
P(H0) = 0.6, fs= 6 MHz., T= 1 msec., Rb= 250 Kbps, and
σ2
w= 1,for evaluating the secondary network’s throughput
under hypothesis H0, i.e., RH0(α, β, Pf a ).M= 5 is considered
in Fig. 6(a), 6(b), 6(c), and 8(a).
A. Effects of different parameters
In this subsection, we compare our proposed optimal
method with the sub-optimal one, where we consider fixed
Pfa , i.e., Pf a =Pmax
fa , and solve the optimization problem
for αand βonly. From Proposition IV.3, we can say that
the optimal value of RH0(α, β, P max
fa )is received for β=ζ,
which makes the sub-optimal optimization problem equivalent
to P5for Pfa =Pmax
fa . The comparison is done in terms
of the achieved throughput of the secondary network under
hypothesis H0, i.e., RH0(α, β, Pf a ).
1) Effects of Ψ:In Fig. 6(a), we plot RH0(α, β , Pfa)for
two different values of SNR at CR nodes while varying the
value for Ψ. It is observed that in both cases, i.e., for SNR
= -6 dB and -7 dB, the optimal method performs better than
the sub-optimal method. As we increase the value for Ψ, the
optimal value for αreduces which increases the throughput in
turn. Achieved throughput for SNR = -6 dB is more compared
to SNR = -7 dB under a fixed value of Ψ, which is due to the
better detection performance at CR nodes. Better detection
performance at CR nodes helps in reducing the required
average number of LDVs at the FC as well as improves the
FC’s detection performance, such that, we get higher values
for RH0(α, β, Pf a )in high SNR regime.
2) Effects of SNR at CR nodes: In Fig. 6(b), we plot
RH0(α, β, Pf a )while varying the SNR and number of observa-
tions collected at CR nodes respectively. We have considered
Ψ=5000 and 8000 bits/sec. It may be observed that the optimal
method outperforms the sub-optimal one in both cases. The
improvement under higher values for SNR, and Ψmay be
explained from the previous discussed analogy (i.e., better
detection performance at CR nodes and the FC).
3) Effects of ζ:Fig. 6(c) represents the plot of
RH0(α, β, Pf a )for different values of ζ. It is pretty intuitive
that as we increase the interference tolerance level at the PU,
the transmission opportunity for the CR network increases,
which leads to higher throughput. Fig. 6(c) reflects this fact.
20 40 60 80 100
2.95
3
3.05
3.1
3.15
3.2
3.25
3.3
3.35
3.4
3.45
N
RH0(α,β,Pfa) (bits/sec/Hz.)
Optimal
Sub−optimal
ψ = 6000
bits/sec.
ψ = 10000
bits/sec.
(a) Varying N
2 4 6 8 10
3.2
3.25
3.3
3.35
M
RH0(α,β,Pfa) (bits/sec/Hz.)
Optimal
Sub−optimal
ψ = 15000
bits/sec.
ψ = 20000
bits/sec.
(b) Varying M
Figure 7: Average throughput for varying Nand M
4) Effects of Nand M:In Fig. 7(a) and Fig. 7(b), we
plot the average throughput for different values of Nand
Mrespectively. We consider M= 5, ζ = 0.1,S N R =−7dB,
and Ψ = 6000 and 10000 bits/sec. in Fig. 7(a), whereas, in
Fig. 7(b), we consider N= 20, ζ = 0.04,SN R =−7dB, and
Ψ = 15000 and 20000 bits/sec. In both Fig. 7(a) and Fig. 7(b),
it is observed that the optimal solution outperforms the sub-
optimal one. Moreover, we also observe that the throughput
increases while the values of Nand Mare increased. The
improvement in throughput over Ncan be explained from our
analysis as given in Section IV-A. The optimal value of α, i.e.,
eα, reduces for increasing the values of Nand M, which may
be explained from (25a), (25b), and (35). As the false-alarm
probability at the FC, i.e., α, reduces, the throughput gets
better for higher values of Nand M. However, it is to be noted
that g1(Pfa ), g2(Pf a )→ ∞ for high value of N, which makes
the sensing time under hypothesis H0, i.e., τH0→0(τH0can
be evaluated from (14)). For this reason, the tradeoff between
the sensing duration and the throughput is not captured in
Fig. 7(a). This happens due to the consideration of the lower
limit of the required number of LDVs (given in (9a) and (9a)).
From Fig. 6(a), 6(b), 6(c), 7(a), and 7(b), it is evident that
optimal method becomes advantageous while varying Ψ,SNR,
ζ,N, and M. From the plots, it is evident that the search oper-
ation over Pfa becomes optimal for higher values of Ψ, SNR,
and N, which supports our analysis as given in Section IV-A.
However, it is observed that the optimal and sub-optimal
methods converge at a point while increasing the values for
Ψ, SNR, and N. The values of αmax,α′,α1(Pf a , ζ ),and
α2(Pfa , ζ )decreases significantly for higher values of Ψ, SNR,
and N(high SNR and Nleads to high g1(Pfa )and g2(Pf a )).
Therefore, from (34) and (41), we get: eα=αuncons (Pmax
fa )and
eα(Pfa ) = αuncons (Pfa ),∀Pf a ∈(0,1). As the inflection point
of the concave objective function becomes the solution for
both optimal and sub-optimal optimization problems, from our
previous discussion we can conclude that Pmax
fa becomes the
optimal value for Pfa in high Ψand SNR regime, which makes
the throughput same for optimal and sub-optimal methods.
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−10 −9 −8 −7 −6 −5
3.15
3.2
3.25
3.3
3.35
3.4
3.45
3.5
SNR (in dB)
Throughput under hypothesis H0 (bits/sec./Hz.)
HSS−BC
HSS−NNC
ψ=15000
bits/sec.
ψ=7000
bits/sec.
(a)
−10 −9 −8 −7 −6 −5
2.9
3
3.1
3.2
3.3
3.4
3.5
SNR (in dB)
Throughput under hypothesis H0 (bits/sec./Hz.)
ψ=15000 bits/sec.
ψ=15000 bits/sec.
ψ=7000 bits/sec.
ψ=7000 bits/sec.
NSS
HSS−BC
(b)
Figure 8: Comparison for varying SNR (ζ= 0.1and N= 65):
(a)with HSS-NNC and (b)with NSS
B. Comparison with node by node sequential sensing strategy
In this subsection, we compare the optimal HSS-BC with
the sequential sensing strategy as proposed in [26]–[28], where
the FC performs SPRT each time after collecting a CR
node’s sensing observation. We term this sensing process by
hybrid spectrum sensing with node by node collection (HSS-
NNC). In Appendix G, we evaluate the average number of
observations required to terminate the test at the FC and
frame the corresponding equivalent throughput optimization
problem, i.e., P6, for the HSS-NNC.
In Fig. 8(a), we plot the optimal throughput for both HSS-
BC and HSS-NNC strategies for M= 5 and two different
values for Ψ = 7000 and 15000 bits/sec. It is observed that
for identical decision thresholds at CR nodes the optimal
throughput becomes identical for both HSS-BC and HSS-NNC
strategies. The difference between the HSS-BC and the HSS-
NNC may have not been captured as the optimization problems
have been framed with the help of Wald’s approximation.
C. Comparison with non-sequential sensing strategy
In this subsection, we compare the HSS-BC with traditional
non-sequential sensing (NSS) strategy as given in [11], where
CR nodes perform energy detection and the FC takes global
decision by using optimal k−out-of−Mfusion rule. It is to
be noted that authors of [11] have evaluated optimal sensing
duration at CR nodes and the decision thresholds at CR
nodes and the FC. To make the comparison fair, we evaluate
the number of CR nodes from the control channel bit rate
constraint. As total Mbits are communicated over the control
channel to take the final decision at the FC, we can evaluate
the number of CR nodes from M/T = Ψ.
In Fig. 8(b), we compare the HSS-BC method with the
non-sequential sensing method in terms of throughput for
M= 5 and two different values of Ψ = 7000 and 15000
bits/sec. It is observed that the average throughput for the
HSS-BC increases with the value of Ψ, whereas, for the NSS,
this property does not hold. As the number of communicating
CR nodes increase, the transmission duration reduces, which
reduces the throughput for the NSS. However, at lower SNR
regime, higher number of CR nodes helps in improving the
detection performance. Therefore, we can see that at lower
SNR, Ψ = 15000 bits/sec. gives better performance for the
NSS compared to Ψ = 7000 bits/sec. It is observed that for
Ψ = 15000, the HSS-BC outperforms the NSS, which does not
hold for Ψ = 7000. Please note that we have not considered
the sensing duration in optimization for HSS-BC like NSS.
D. Comparison with two-bits decision combining
We compare the hard decision combining with two-bits
decision combining. Each CR node sends two-bits rather a
single bit to the FC. There will be four different decisions for
two-bits decision, which we can write as L={00,01,10,11}.
We consider same signal model as has been considered in Sec-
tion II-A. Therefore, we can say that the accumulated energy
at the ith CR node, i.e., ys
i, as given in (2), follows normal dis-
tribution. Mathematically, we can write ys
i∼ N (σ2
w, σ4
w/N)
under hypothesis H0and ys
i∼ N ((γ+ 1)σ2
w,(2γ+ 1)σ4
w/N)
under hypothesis H1.
−10 −9 −8 −7 −6 −5
3.1
3.15
3.2
3.25
3.3
3.35
3.4
3.45
3.5
SNR (in dB)
Throughput under hypothesis H0 (bits/sec/Hz.)
Two−bits combining
Hard combining
Ψ = 15000 bits/sec.
Ψ = 10000 bits/sec.
Figure 9: Comparison for varying SNR with two-bits decision
fusion (ζ= 0.1, N = 65)
In Fig. 9, we have given a plot comparing the hard decision
fusion and two-bits decision fusion at the FC for M= 5 and
Ψ = 15000,10000 bits/sec.. It is observed that for Ψ = 10000,
hard decision strategy gives higher throughput than the two-
bits decision strategy in the low SNR regime. However, two-
bits decision performs better when the SNR is increased.
For Ψ = 15000, it is observed that the region over which
the hard decision strategy performs better than the two-bits
decision strategy becomes narrower compared to Ψ = 10000.
In case of two-bits decision, the detection performance at CR
nodes become better than the hard decision. However, the
better detection performance increases the accumulated KL-
divergence at the FC, which in turn makes the search space
over αnarrower. 6This may be a possible reason behind
the hard decision strategy’s better performance compared to
two-bits decision strategy. Please note that we have performed
search operation while evaluating the optimal throughput for
the two-bits decision strategy; as we make the search grids
finer, the result must improve.
6In Appendix F, we have proved that both f1(α, ζ)and f2(α, ζ)are
monotonically decreasing and convex functions of α. If the KL divergence,
i.e, g1(t)and g2(t)increases, then from (55) and (56), it is evident that the
search region for αwill reduce.
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13
VII. CONCLUSION
In this paper, we maximized the throughput of a cooperative
CR network, where CR nodes perform energy detection and
take hard decisions. The FC performs SPRT after collecting
all CR nodes’ decisions to take the global decision. We
considered CR nodes’ false-alarm probability (i.e., Pf a) and
the FC’s decision threshold parameters (i.e., αand β) as
the optimization parameters, which has not been attempted
earlier. We have compared our optimization set-up with a sub-
optimal one, which depends on αand βonly; whereas, the
Pfa has been fixed at Pfa =Pmax
fa (Pmax
fa maximizes the KL
divergence, i.e., D(pi
0||pi
1)). It was proved that based on CR
nodes’ detection performance and control channel’s bit rate
constraint’s value we get some conditions under which the
optimal and the sub-optimal methods become equivalent. One
dimensional search over Pfa is required to find out the optimal
solution for the rest of the conditions.
If the sensing channels at CR nodes experience slow fading,
then there will be temporal correlation between the decisions
of a CR node. It will be challenging to design the SPRT
with the knowledge of a correlation model and then solving
the corresponding throughput optimization problem. It would
be also interesting to design the throughput optimal decision
thresholds at CR nodes and the FC, when both of CR nodes
and the FC perform SPRT as has been considered in [23]. The
optimization problem becomes even more complicated when
the control channel is accessed with the help of CSMA-CA.
APPENDIX A
FUSION RULE SIMPLIFICATION STEPS
Following [44], we can write:
Ls=
s
Y
k=1
M
Y
i=1
Pr(uk
i|H1)
Pr(uk
i|H0)
=
s
Y
k=1 "Y
S0
Pr(uk
i= 0|H1)
Pr(uk
i= 0|H0)Y
S1
Pr(uk
i= 1|H1)
Pr(uk
i= 1|H0)#
=
s
Y
k=1 "Y
S0
Pd
Pfa Y
S1
1−Pd
1−Pfa #,(58)
where Sncorresponds to the set of local decisions which are
equal to n= 0,1, such that, S0∪ S1={1,2, .., M }.
After taking logarithm, we can write:
ln{Ls}=
s
X
k=1
M
X
i=1 uk
iln Pd
Pfa + (1 −uk
i) ln 1−Pd
1−Pfa
= ln Pd(1 −Pf a )
Pfa (1 −Pd)s
X
k=1
M
X
i=1
uk
i+sM ln 1−Pd
1−Pfa .
(59)
In (6), we take logarithm in both side and use (59) to get the
fusion rule as given in (7).
APPENDIX B
THEOREMS USED
Theorem B.1. Let y⊆R2be a convex open set and let
f:y→Rbe a function with continuous partial derivatives
of first and second order. Now, if the hessian matrix of fat
the point x∈ybecomes:
H(x)=a b
b c
(where, a, b, and care functions of x), then the conditions for
convexity or concavity are respectively
1) a, c ≥0and ac −b2≥0, then fis strictly convex.
2) a, c ≤0and ac −b2≥0, then fis strictly concave.
Theorem B.2. For a function f:Rn→R, where fis a convex,
we can say that
{x|f(x)≤γ
′
, x ∈Rn} ⊆ {x|f(x)≤γ , x ∈Rn}for all γ
′
≤γ.
APPENDIX C
PROOF:f1(α, β )AND f2(α, β )ARE CONVEX FUNCTIONS OF
αAND β.
1) Convexity proof of f1(α, β ):Elements of hessian matrix
of f1(α, β)are:
∂2{f1(α, β)}
∂α2=1
α(1 −α)(60a)
∂2{f1(α, β)}
∂α∂β =∂2{f1(α, β)}
∂β∂α =1
β(1 −β)(60b)
∂2{f1(α, β)}
∂β2=(1 −β)2+α(2β−1)
β2(1 −β)2.(60c)
It may be observed that for the feasible values of αand
β,∂2{f1(α,β)}
∂α2>0, and we can also show ∂2{f1(α,β)}
∂β2>0as
follows:(1 −β)2+α(2β−1)
β2(1 −β)2=1
β2−α(1 −2β)
β2(1 −β)2
>1
β2−α(1 −2β)
β2−2β3
=1−α
β2>0.(61)
Determinant of the hessian matrix of f1(α, β)is:
Hf1(α,β)=(1 −β−α)2
αβ2(1 −α) (1 −β)2(62)
which is also positive for feasible values of αand β. So, from
Theorem B.1 as given in Appendix B, we can conclude that
f1(α, β)is a convex function of αand β.
2) Convexity proof of f2(α, β ):Elements of hessian matrix
of f2(α, β)are:
∂2{f2(α, β)}
∂α2=(1 −α)2+β(2α−1)
α2(1 −α)2(63a)
∂2{f2(α, β)}
∂α∂β =∂2{f2(α, β)}
∂β∂α =1
α(1 −α)(63b)
∂2{f2(α, β)}
∂β2=1
β(1 −β).(63c)
It may be observed that for the feasible values of
β,∂2{f2(α,β)}
∂β2>0, moreover, we can show that ∂2{f2(α,β)}
∂α2>0
as follows:
(1 −α)2+β(2α−1)
α2(1 −α)2=1
α2−β(1 −2α)
α2(1 −α)2
>1
α2−β(1 −2α)
α2−2α3
=1−β
α2>0.(64)
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14
Determinant of the hessian matrix of f2(α, β)is:
Hf2(α,β)=(1 −β−α)2
βα2(1 −β) (1 −α)2(65)
which is also positive for feasible values of αand β. So, from
Theorem B.1 as given in Appendix B, we can conclude that
f2(α, β)is a convex function of αand β.
APPENDIX D
PROOF:g2(Pf a )IS NOT CONCAVE FUNCTION OF Pf a
From (10b), we observe that g2(Pfa )is the addition of M
identical terms, i.e., D(pi
1||pi
0),i= 1,2, .., M . So, if we can
prove that any of those terms is non-convex, then we can
conclude that g2(Pfa )is non-convex. From (4a) and (4b), we
write Pd=Q(c1Q−1(Pfa )−c2), where c1= 1/p(2γ+ 1) and
c2=γ·p(N/(2γ+ 1)).
We evaluate the second order derivative of D(pi
1||pi
0)with
respect to Pfa as given in (66), on the top of next page, where,
˙
Pd=c1exp "Q−1(Pfa )2−c1Q−1(Pf a )−c22
2#
¨
Pd=√2πc1exp "2Q−1(Pf a )2−c1Q−1(Pf a)−c22
2#
c2
1Q−1(Pfa )−c1c2−Q−1(Pf a ).
It is to be noted that, g2(Pfa )will follow the concavity property
if and only if the second order derivative of D(pi
1||pi
0)with
respect to Pfa as given in (66) becomes always negative
for any values of Pfa . In Fig. 10, we plot ∂2{D(pi
1||pi
0)}
∂P 2
fa for
different values of Pfa considering SNR=-7 dB and N= 30,
where we show that: ∂2{D(pi
1||pi
0)}
∂P 2
fa 0,∀Pfa . It indicates that
g2(Pfa )is not a concave function of Pf a .
0.4 0.45 0.5 0.55 0.6 0.65
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
0.02
Pfa
∂2[D(p1
i||p0
i)]/∂ Pfa
2
Figure 10: ∂2{D(pi
1||pi
0)}
∂P 2
fa vs. Pfa
APPENDIX E
PROOF OF PROPOSITION IV.2
At first, we consider α=α, and evaluate the first and
second order derivatives of RH0(α, β, P fa )with respect to β
as respectively:
∂RH0(α, β, P f a)
∂β =W1
(1 −α)(1 −α−β)
β(1 −β)(67a)
∂2RH0(α, β, P f a)
∂β2=−W1(1 −α)(1 −β)2+α(2β−1)
β2(1 −β)2,
(67b)
where W1=P(H0)(Nτs+M τr)r0
T.g1(Pf a ).
From (67a), it is clear that for the feasible values of α
and β,∂{RH0(α,β,P f a )}
∂β >0, which means that RH0(α, β, P fa )
monotonically increases with β.
As the term W1(1 −α)is always positive for any feasible
value of α, from (61), we can observe that the second
order derivative as given in (67b) is always negative for any
feasible value of αand β. Hence, we can conclude that
RH0(α, β, P f a)) is a concave function. Therefore, we can
conclude that RH0(α, β , P fa )) is a monotonically increasing
and concave function of β.
Now, for β=β, we prove that RH0(α, β , P fa )is a concave
function of α. The first and second order derivatives of
RH0(α, β, P f a )with respect to αmay be evaluated as follows:
∂RH0(α, β , P fa )
∂α =−P(H0)r0+ 2W1(1 −α) ln 1−α
β+
W1(1 −2α) ln 1−β
α
(68a)
∂2RH0(α, β, P fa )
∂α2=−W12 ln (1 −α)(1 −β)
αβ +1
α.
(68b)
We analyse the second order derivative as given in (68b),
where it may be observed that, the term multiplied with −W1is
always positive for the feasible values of αand β. It means that
the second order derivative as given in (68b) is always negative
for any feasible values of αand β. Hence, we can conclude
that RH0(α, β, P f a )is a concave function of α, which may be
monotonically increasing or may have an inflection point in
between α∈(0,1−β). We analyse the first order derivative
as given in (68a) to check this.
We evaluate the limits of ∂{RH0(α,β ,P fa )}
∂α with respect to α
at the two boundary points of the feasible set as:
lim
α→(1−β)
∂RH0(α, β, P f a )
∂α =−P(H0)r0<0(69a)
lim
α→0
∂RH0(α, β, P f a )
∂α = +∞.(69b)
From (69a) and (69b), we can say that RH0(α, β , P fa )in-
creases at first with αand then decreases again, which im-
plies an inflection point of RH0(α, β, P f a)within the interval
(0,1−β). Therefore, we conclude that RH0(α, β , P fa )is con-
cave over α.
APPENDIX F
PROOF:f1(α, ζ )AND f2(α, ζ)ARE MONOTONICALLY
DECREASING AND CONVEX FUNCTIONS
From (63a) and (60a), we can see that for β=ζ, the second
order derivatives for both of f1(α, ζ )and f2(α, ζ)with respect
to αbecome positive for any feasible value of α. It means that
f1(α, ζ)and f2(α, ζ )are convex functions of α.
Now, to prove that f1(α, ζ )and f2(α, ζ)monotonically
decrease with respect to α, we derive the first order derivatives
with respect to αas respectively:
∂{f1(α, ζ)}
∂α = ln αζ
(1 −α)(1 −ζ)(70a)
∂{f2(α, ζ)}
∂α =α+ζ−1
ζ(1 −ζ).(70b)
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at http://dx.doi.org/10.1109/TSP.2018.2838575
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15
∂2D(pi
1||pi
0)
∂P 2
fa
=¨
Pdlog2Pd(1 −Pf a)
Pfa (1 −Pd)−˙
Pd
(1 −Pd)−˙
Pd(1 −Pfa )
(1 −Pd)(1 −Pfa )log2(e)+
(1 −Pd)−˙
Pd(1 −Pfa )
(1 −Pfa )2log2(e) + ˙
Pd
Pfa ˙
Pd−Pd
Pfa Pd
log2(e)−Pf a ˙
Pd−Pd
P2
fa
log2(e).(66)
It may be observed that for feasible values of α, (70a) and
(70b) yields negative values always. Therefore, we conclude
that f1(α, ζ)and f2(α, ζ )are monotonically decreasing and
convex functions of α.
APPENDIX G
OPTIMIZATION PROBLEM FOR HSS-NNC
In Fig. 11, we show the corresponding sensing frame
structure for HSS-NNC. The FC, instead of collecting all CR
nodes’ hard decisions, collects single CR node’s hard decision
and performs SPRT. This process continues until the global
decision is taken at the FC. Therefore, the FC may not require
all CR nodes’ decisions to take the global decision, which
is shown in Fig. 11. The notations, i.e., N,Ts,and Tr, have
been defined earlier in this paper. We denote the expected
Figure 11: Frame Structure of HSS-NNC for sensing
number of hard decisions at the FC to terminate the SPRT by
NW ALD
Ht(α, β, Pf a )(under hypothesis Ht), such that, CR nodes
perform the local sensing procedure for NW ALD
Ht(α, β, Pf a )/M
times. So, following Fig. 11, we can write the required sensing
duration under hypothesis Htas:
τNN C
Ht=NW ALD
Ht(α, β, Pf a )
M·NTs+NW ALD
Ht(α, β, Pf a )·Tr.
(71)
As the FC collects single hard decision rather than Mnumber
of hard decisions to perform the SPRT, the average number of
hard decisions under hypothesis H1and H0becomes:
NW ALD
H1(α, β, Pf a )≥1
D(pi
1||pi
0)βln β
1−α+ (1 −β) ln 1−β
α
(72a)
NW ALD
H0(α, β, Pf a )≥1
−D(pi
0||pi
1)(1 −α) ln β
1−α+αln 1−β
α,
(72b)
where D(pi
1||pi
0)and D(pi
0||pi
1)have been defined in (11a) and
(11b) respectively. Like the HSS-BC, here also we consider
lower bounds of the number of decisions. Average throughput
for HSS-NNC under hypothesis H0is:
RNN C
H0(α, β, Pf a ) = T−τN N C
H0
T·P(H0)·(1 −α)·r0,(73)
We can frame the corresponding optimization problem for
HSS-NNC as:
P6 : maximize
(α,β,Pf a )RN NC
H0(α, β, Pf a )(74a)
subject to: (17b) −(17e)
NW ALD
H0(α, β, Pf a )
T≤Ψbits/sec. (74b)
NW ALD
H1(α, β, Pf a )
T≤Ψbits/sec.. (74c)
It can be observed that the structure of the optimization
problem P6 resembles with the optimization problem P1.
Therefore, we can follow the same approach like the HSS-
BC to get the optimal solution for P6.
ACKNOWLEDGEMENT
We would like to express our sincere gratitude to the
Associate Editor and the anonymous reviewers for taking their
time into reviewing this manuscript and providing constructive
comments which have helped us in improving the quality and
readability of the manuscript.
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The final version of record is available at http://dx.doi.org/10.1109/TSP.2018.2838575
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