Optimal Power Allocation Strategies for Fading Cognitive Radio Channels with Primary User Outage Constraint.

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374IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 2, FEBRUARY 2011
Optimal Power Allocation Strategies for Fading
Cognitive Radio Channels with Primary User
Outage Constraint
Xin Kang, Student Member, IEEE, Rui Zhang, Member, IEEE, Ying-Chang Liang, Senior Member, IEEE, and
Hari Krishna Garg, Senior Member, IEEE
Abstract—In this paper, we consider a cognitive radio (CR)
network where a secondary (cognitive) user shares the spectrum
for transmission with a primary (non-cognitive) user over block-
fading (BF) channels. It is assumed that the primary user has a
constant-rate, constant-power transmission, while the secondary
user is able to adapt transmit power and rate allocation over
different fading states based on the channel state information
(CSI) of the CR network. We study a new type of constraint
imposed over the secondary transmission to protect the primary
user by limiting the maximum transmission outage probability
of the primary user to be below a desired target. We derive
the optimal power allocation strategies for the secondary user
to maximize its ergodic/outage capacity, under the average/peak
transmit power constraint along with the proposed primary user
outage probability constraint. It is shown by simulations that the
derived new power allocation strategies can achieve substantial
capacity gains for the secondary user over the conventional
methods based on the interference temperature (IT) constraint
to protect the primary transmission, with the same resultant
primary user outage probability.
Index Terms—Cognitive radio, interference temperature, out-
age probability, power control, spectrum sharing.
I. INTRODUCTION
R
spectrum or the so-called “white spaces” to the new, unli-
censed, and sophisticatedly designed wireless devices. This
milestone change of policy by the FCC indicates that cognitive
radio (CR) [1] is fast becoming one of the most promising
technologies for the future radio spectrum utilization. Basi-
cally, there are two paradigms for the operation of CRs: op-
portunistic spectrum access (OSA) and spectrum sharing (SS)
(see, e.g., [2]–[4] and references therein). OSA is a sensing-
based technology, which allows a secondary user (SU) in the
CR network to opportunistically access the frequency band
ECENTLY, the Federal Communications Commission
(FCC) agrees to open the licensed, unused television
Manuscript received 1 December 2009; revised 15 May 2010. This work
was supported in part by research grants from National University of Singa-
pore (project number: R-263-000-436-112 and R-263-000-421-112). Part of
this paper has been presented in IEEE ICC’2009 and IEEE GlobeCom’2009.
X. Kang and H. K. Garg are with the Department of Electrical &
Computer Engineering, National University of Singapore, Singapore (e-mail:
kangxin@nus.edu.sg, eleghk@nus.edu.sg).
R. Zhang is with Institute for Infocomm Research, A*STAR, and De-
partment of Electrical & Computer Engineering, National University of
Singapore, Singapore (e-mail: rzhang@i2r.a-star.edu.sg).
Y. -C. Liang is with Institute for Infocomm Research, A*STAR, and School
of Electrical and Electronic Engineering, Nanyang Technological University,
Singapore (e-mail: ycliang@i2r.a-star.edu.sg).
Digital Object Identifier 10.1109/JSAC.2011.110210.
allocated to a primary user (PU) when the PU transmission
is detected to be inactive, while SS allows the SU to transmit
simultaneously with the PU over the same frequency even
when the PU transmission is active as long as the quality of
service (QoS) degradation of the PU transmission due to the
SU interference is tolerable.
For SS-based CR networks, a common method to protect the
PU transmission is to impose an interference power constraint
to the SU transmission, which requires the interference power
received at the PU terminal due to the SU transmission to be
below a prescribed threshold, also known as the interference
temperature (IT) constraint [2], [5]. Motivated by this idea,
a great deal of valuable scholarly work has recently been
done on the design of optimal transmission strategies for CRs
subject to the IT constraint. In [6], the author first studied
the capacities of different types of single-user and multiuser
additive white Gaussian noise (AWGN) SU channels subject
to a received-power constraint at the PU terminal. For fading
PU and SU channels, with perfect channel state information
(CSI) on the channels from the SU transmitter to the SU
and PU receivers, the optimal power allocation strategy to
achieve the ergodic capacity of the SU fading channel was
derived in [7] subject to an average/peak interference power
constraint at the PU receiver. Under the same system setup,
the optimal power allocation strategies to achieve the ergodic
and outage capacities of the SU fading channel subject to
both the SU transmit power constraint and the IT constraint
at the PU receiver were thoroughly studied in [8]. Transmit
optimizations for CRs operating under the IT constraint have
also been studied in, e.g., [9] for multi-antenna based CRs and
[10], [11] for multiuser CR channels.
Although applying the IT constraint is a practical method
for the SU to protect the PU transmission, it is worth in-
vestigating more advanced techniques over the IT to obtain
improved SS capacities for both the PU and SU. In [12], from
the information-theoretic perspective, the authors proposed
the use of the PU’s capacity requirement instead of the IT
constraint as a new criterion to regulate the SU transmission.
Clearly, this new constraint is more directly related to the PU
transmission, and is thus expected to result in improved SS
capacities. However, to realize this new type of constraint,
additional knowledge on the PU CSI is required at the SU
transmitter, while such a knowledge is not needed for applying
the IT constraint. Following the idea in [12], power alloca-
0733-8716/11/$25.00 c ? 2011 IEEE

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KANG et al.: OPTIMAL POWER ALLOCATION STRATEGIES FOR FADING COGNITIVE RADIO CHANNELS WITH PRIMARY USER OUTAGE CONSTRAINT375
tion strategies for the SU have been studied in [13], where
several suboptimal schemes were proposed by converting the
minimum-PU-outage-capacity constraint into an approximate
IT constraint. In addition, the authors in [14] have recently
addressed a special case of the minimum-PU-outage-capacity
constraint where the PU outage capacity loss due to the
SU transmission is limited to be zero. It is noted that the
approach of exploiting the additional PU CSI to improve the
SS capacities over the conventional IT constraint has also been
investigated in [15] for uncoded PU transmissions.
In this paper, we study the design of power allocation
strategies for the SU fading channel under the constraint on the
maximum acceptable PU transmission outage probability, in a
manner similar to the minimum-PU-outage-capacityconstraint
considered in [13], [14]. Based on convex optimization tech-
niques, we derive the power allocation strategies for the SU
to achieve the ergodic/outage capacity under the proposed PU
outage probability constraint, along with the SU average/peak
transmit power constraint. Note that the proposed power
allocation strategies are optimal for achieving various forms
of fading channel capacities for the SU, and are thus different
from the suboptimal strategies proposed in [13] based on a
heuristic approach. Moreover, as will been shown later in this
paper, our proposed strategy includes that in [14] as a special
case.
The rest of this paper is organized as follows. Section II
introduces the system model of PU and SU transmissions.
Section III and IV present the optimal power allocation
strategies for the SU to achieve the ergodic and outage
capacities, respectively, under the newly proposed PU outage
constraint along with the SU average/peak transmit power
constraint. Section V provides numerical results to corroborate
our proposed studies. Finally, Section VI concludes the paper.
II. SYSTEM MODEL
A. Channel Model
For the purpose of exposition, we consider in this paper
a simplified CR network with one primary link and one
secondary link. The primary link consists of a PU transmitter
(PU-TX) and a PU receiver (PU-RX), while the secondary link
consists of a SU transmitter (SU-TX) and a SU receiver (SU-
RX). All the terminals involved are assumed to be equipped
with a single antenna each. We assume that the PU and SU
links share the same narrow-band frequency for transmission.
All the channels involved in the PU-SU network are assumed
to be independent block fading (BF) channels. As shown in
Fig. 1, the instantaneous channel power gains for the primary
link, the secondary link, the link from PU-TX to SU-RX, and
the link from SU-TX to PU-RX at fading state ν are denoted
by gpp(ν), gss(ν), gps(ν), and gsp(ν), respectively, where ν
denotes the joint fading state for all the channels involved.
All the channel power gains are assumed to be independent
and identically distributed (i.i.d.) random variables (RVs) each
having a continuous probability density function (PDF). The
additive noises at PU-RX and SU-RX are assumed to be
independent circularly symmetric complex gaussian (CSCG)
RVs, each of which is assumed to have zero mean and
variance, N0.
PU-TX
SU-TXSU-RX
PU-RX
gpp
gps
gsp
gss
Fig. 1. Channel model.
In this paper, we assume that all the instantaneous channel
power gains in the PU-SU network are available to the SU
at each fading state, for studying the capacity limits of SU
fading channels. In practice, CSI of the secondary link can be
obtained at SU-TX by the classic channel training, estimation,
and feedback mechanisms, while CSI of the channel between
PU-TX and SU-RX can be obtained by SU-RX via estimating
the received signal power from PU-TX provided that it knows
a priori the transmit power of PU-TX. Furthermore, CSI of
the primary link and the link between SU-TX and PU-RX
can be obtained at SU-TX via the cooperation of the PU
[12]–[14]. For instance, PU-RX can estimate the PU channel
and the channel between SU-TX and PU-RX, and then feeds
them back to SU-TX. In the case that PU-RX is not willing
to cooperate with SU, it may be difficult to obtain perfect
CSI of the PU channel and the channel between SU-TX and
PU-RX. For such scenarios, we investigate how the channel
estimation errors will affect the performance of the proposed
optimal power allocation strategy and the capacity limits via
simulations. Besides, a modified power allocation strategy
is also proposed to mitigate the negative effect of channel
estimation errors. It is worth pointing out that the proposed
power allocation strategies in this paper is not suitable for
TV-band white-space CR applications, since CR applications
in TV-band white spaces have adopted the sensing-based OSA
model rather than IT-based spectrum sharing model adopted
in this paper.
B. Primary User Transmission
We assume that PU-TX has no CSI on the PU link and thus
transmits with constant power and rate denoted by Pp and
rp, respectively. Supposing that the SU link is not present or
inactive, the transmission outage probability of the PU channel
due to its own fading is given by
?gpp(ν)Pp
where Pr{·} denotes the probability, and γp is the desired
signal-to-noise ratio (SNR) at PR-RX. If the optimal Gaussian
codebook is used for the PU link, the receiver-aware outage
capacity [16] can be achieved, which is equal to log2(1 +
γp) corresponding to the receiver outage probability, εp. In
most practical situations with non-Gaussian codebooks, rpis
usually chosen to be smaller than the outage capacity, and εpis
in general approximately equal to the average block decoding
error probability experienced by PR-RX [17].
Let ε0denote the desired outage probability for the PU link.
It may be possible that because of superior fading channel
εp= Pr
N0
< γp
?
(1)

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376IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 2, FEBRUARY 2011
conditions of the PU link, the actual outage probability, εp, is
smaller than the desired target ε0, thus resulting in an outage
margin, denoted by Δε = ε0−εp≥ 0. Note that for the given
γp, Pp, and the distribution of gpp, from (1) it follows that εp
is a fixed value. Thus, given the outage probability target ε0,
Δε can be obtained correspondingly. As will be shown later in
this paper, the non-zero outage margin, i.e., Δε > 0, provides
the SU more opportunities to transmit and thereby improves
the SU link throughput.
Now, consider the effects of the SU transmission on the PU
link. Supposing that the SU transmits with powers {ps(ν)} for
different fading states of ν, the transmission outage probability
of the PU channel due to both its own fading and the additional
interference from the SU becomes
?
To protect the PU, we require that the resultant outage
probability be no greater than its tolerable target ε0, i.e.,
ε(c)
p
= Pr
gpp(ν)Pp
gsp(ν)ps(ν) + N0
< γp
?
.(2)
ε(c)
p
≤ ε0. (3)
We thus name (3) as the PU outage constraint.1
C. Secondary User Transmission
The SU link is assumed to be cognitive in the sense that
it is able to adapt its transmit power and rate allocation over
different fading states based on the CSI of all the channels
involved in the PU-SU network to maximize the SU link
throughput and yet ensure certain protection to the PU by
satisfying the PU outage constraint given by (3). We consider
two types of information-theoretic limits for the fading SU
channel, namely, the ergodic (expected) capacity [18] and the
transmitter-aware outage capacity [19]. The ergodic capacity
is defined as the maximum achievable rate averaged over
all the fading states, while the outage capacity is defined as
the maximum constant rate that can be reliably transmitted
with a prescribed outage probability. The ergodic and outage
capacities can be considered as the throughput limits of a
fading channel under no and stringent transmission delay
requirements, respectively.
In addition to the PU outage constraint imposed on the SU
transmission, we also consider the SU’s own transmit power
constraint, which can be categorized into two general types:
average transmit power constraint and peak transmit power
constraint. The former limits the average transmit power over
all the fading states while the instantaneous transmit power at
each particular fading state can be variable, i.e.,
E{ps(ν)} ≤ Pav
(4)
where E{·} denotes the expectation, and Pavis the prescribed
threshold for the average transmit power of the SU. In contrast,
the peak power constraint limits the instantaneous transmit
power at each of the fading states and is thus expressed as
ps(ν) ≤ Ppk, ∀ν(5)
1It is noted that to fulfill this constraint by the SU, in addition to the CSI
knowledge, the PU transmit power, and receiver SNR and outage targets are
also required to be known a priori to the SU.
where Ppk is the prescribed threshold for the peak transmit
power of the SU.
In this paper, we will derive the optimal power allocation
strategies for the SU to achieve the ergodic capacity and the
outage capacity, respectively, under the combined PU outage
and SU average/peak transmit power constraints, as presented
next.
III. ERGODIC CAPACITY OF SU UNDER PU OUTAGE
CONSTRAINT
In this section, we derive the optimal power control policies
for the SU to maximize the SU ergodic capacity under the
PU outage constraint, along with the average or peak transmit
power constraint of the SU in the following two subsections,
respectively.
A. Average Power Constraint
First, we consider the average transmit power constraint for
the SU. By assuming that the PU uses a Gaussian codebook
and the PU interference is treated as additional Gaussian noise
at SU-RX, the ergodic capacity of the SU can be obtained by
solving the following problem (P1):
?
s.t.
(3),(4).
max
ps(ν)≥0,∀ν
E
log2
?
1 +
gss(ν)ps(ν)
gps(ν)Pp+ N0
??
For the convenience of analysis, we introduce the following
indicator function for the outage event of the PU at fading state
ν:
?
1,otherwise.
χp(ν) =
0,
gpp(ν)Pp
gsp(ν)ps(ν)+N0≥ γp
(6)
Thus, the PU outage constraint in (3) can be rewritten as
E{χp(ν)} − εo≤ 0.(7)
Next, we apply the Lagrange duality method [20] to solve
P1. Let λ and μ be the non-negative dual variables associated
with (4) and (7), respectively. The Lagrangian of P1 can be
written as
?
−λ(E{ps(ν)} − Pav) − μ(E{χp(ν)} − ε0).
Then, the Lagrange dual function of P1 can be expressed as
L(ps(ν),λ,μ) = E
log2
?
1 +
gss(ν)ps(ν)
gps(ν)Pp+ N0
??
(8)
g(λ,μ) = max
ps(ν)≥0,∀νL(ps(ν),λ,μ).(9)
By applying the Lagrange dual-decomposition method in a
manner similar to [11], the maximization problem given by
(9) can be decoupled into parallel subproblems all having the
same structure and each for one fading state. For a particular
fading state, the associated subproblem can be shown to be
?
Note that in the above we have dropped the index ν for the
fading state for brevity, and express χpas an explicit function
of ps. P1 can then be solved by iteratively solving (10) for all
max
ps≥0
log2
1 +
gssps
gpsPp+ N0
?
− λps− μχp(ps). (10)

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KANG et al.: OPTIMAL POWER ALLOCATION STRATEGIES FOR FADING COGNITIVE RADIO CHANNELS WITH PRIMARY USER OUTAGE CONSTRAINT377
f(ps)
f(ps)−μ χp
(a)
(b)
(c)
(d)
0
0
0
0
(ps)
z
z
0 x<z, f(x)<f(z)-μ
>
zz
x
>
x
x
x z
0 x<z, f(x) f(z)-μ
>
x<0
ps
ps
ps
ps
>
Fig. 2.Illustration of different forms of function f(ps) − μχp(ps).
the fading states with fixed λ and μ, and updating these dual
variables via subgradient-based methods, e.g., the ellipsoid
method [20], for which the details are omitted here for brevity.
In the following, we derive the solution to each subproblem
expressed in (10). Define
?
It is easy to show that f(ps) is a concave function with respect
to ps, and attains its maximum value when psis equal to
?
where (·)+= max{·,0}. Note that the objective function in
(10) now becomes f(ps) − μχp(ps). Also note that χp(ps),
ps≥ 0, given in (6) is in general a step function of ps. Let
x denote the critical value of psover which the value of χp
changes from 0 to 1. It thus follows that
?gppPp
Notice that if x < 0, then we have χp(ps) = 1, for any ps≥ 0.
Let p∗
discussions are then made on p∗
Case 1: x ≥ z. In this case, f(ps) attains its maximum value
at ps= z, and at this value of ps, χp(ps) is at its minimum
value of zero. Thus, as shown in Fig. 2 (a), the objective
function of (10), f(ps)−μχp(ps), attains its maximum value
at z. Therefore, p∗
Case 2: 0 ≤ x < z. In this case, there are two subcases
for the objective function as shown in Fig. 2 (b) and (c),
respectively, where whether x or z maximizes the objective
function depends on the values of the objective function at
these two points. We thus have
?
f(ps) ? log2
1 +
gssps
gpsPp+ N0
?
− λps.(11)
z =
1
λln2−gpsPp+ N0
gss
?+
(12)
x =
1
gsp
γp
− N0
?
. (13)
sdenote the optimal solution of (10). The following
s:
s= z.
p∗
s=
z,
x,
f(x) < f(z) − μ
otherwise.
Case 3: x < 0. In this case, χp(ps) = 1 for any ps≥ 0.
As shown in Fig. 2 (d), f(ps) −μχp(ps) only has a constant
(equal to μ) gap from f(ps), indicating that χp(ps) has no
effects on the optimal solution. Therefore, we have p∗
Define the following three regions: R1? {x ≥ z},R2?
{0 ≤ x < z}, and R3? {x < 0}. The results obtained in the
above cases lead to the following theorem:
Theorem 1: The optimal solution of P1 is given by
⎧
⎪
where f(·), z, and x are given in (11), (12), and (13),
respectively.
Next, we provide further elaborations on the optimal power
control policy given by (14).
(1). Region R1: In this region, since x ≥ z, it follows that
χp(z) = 0. This indicates that there will be no outage of
the PU even when the SU transmits with the optimal power
z. In practice, this may happen when the channel condition
of the primary link is sufficiently good, and/or the channel
power gain of the interference link from SU-TX to PU-RX is
sufficiently small. As such, the interference power from SU-
TX is not yet large enough to render the PU into an outage.
Therefore, the SU can allocate transmit powers solely based
on its own CSI.
(2). Region R2: In this region, the PU channel conditions
are not as favorable as those in R1. As a result, whether the
PU is in an outage depends on the interference power from
the SU. If the SU transmits with the optimal power z, it will
cause an outage to the PU; if the SU transmits with the power
below or equal to x, it will not do so. Obviously, with the
transmit power z over x, the SU will have a larger transmit
rate. However, the SU cannot transmit with power z for each
fading state, since it cannot violate the PU outage constraint.
Therefore, the SU has to balance the SU capacity increment
and the cost to the PU outage constraint for allocating its
transmit power to be x or z. If we regard μ as the cost or the
penalty that the SU has to pay when it causes an outage to
the PU, then when the SU transmits with power x, there will
be no cost incurred and thus the total reward for the SU is
f(x), while when the SU transmits with power z, it causes a
penalty μ in the objective function and thus the total reward
for the SU is f(z) − μ. Obviously, when f(z) − μ is larger
than f(x), SU should transmit with power z; otherwise, the
SU should transmit with power x.
(3). Region R3: In this region, the PU is always in outage
regardless of ps. This happens when the channel condition of
the primary link is very poor, and the PU will be in outage
even if the SU does not transmit. In this case, the transmission
of the SU will not cause any additional outage loss to the
PU transmission. Therefore, the SU can allocate the transmit
power solely based on its own channel condition.
Remark 1: It is seen from (14) that if μ goes to infinity,
f(x) ≥ f(z)−μ will always be true. Therefore, for the whole
region of R2, the SU should transmit with power x, and it will
not cause any additional outage probability loss to the PU. In
s= z.
p∗
s=
⎪
⎪
⎪
⎩
⎨
z,
z,
x,
z,
R1
R2, f(x) < f(z) − μ
R2, f(x) ≥ f(z) − μ
R3
(14)

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378IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 29, NO. 2, FEBRUARY 2011
this case, (14) reduces to the optimal power allocation for the
case with a zero PU outage margin, i.e., Δε = 0.
B. Peak Power Constraint
Next, we consider the peak transmit power constraint for
the SU. In this case, the ergodic capacity of the SU can be
obtained by solving the following problem (P2):
?
s.t. (7),(5).
max
ps(ν)≥0,∀ν
E
log2
?
1 +
gss(ν)ps(ν)
gps(ν)Pp+ N0
??
Similar to P1, we introduce μ as the non-negative dual
variable associated with the constraint given by (7); and it can
be shown that P2 is decomposable into parallel subproblems
each for a different fading state, expressed as
?
Note that we have dropped the fading state index ν for brevity.
Similar to P1, P2 can be solved by iteratively solving (15) for
all the fading states with fixed μ, and then updating μ via a
subgradient-based method, e.g., the bisection search [20], for
which the details are omitted here for brevity. Define
?
The objective function in (15) then becomes h(ps)−μχp(ps).
The problem (15) can be solved based on the following
observations:
Case 1: x ≥ Ppk. In this case, the indicator function
χp(ps) = 0 for any ps ∈ [0,Ppk]. Obviously, (15) is
maximized when ps = Ppk due to the fact that h(ps) is an
increasing function of ps. Therefore, p∗
Case 2: 0 ≤ x < Ppk. In this case, the value of χp(ps)
depends on x. If ps > x, χp = 1; otherwise, χp = 0.
Hence, there are two possible values of ps at one of which
the maximum value of h(ps) − μχp(ps) is attained: One is
ps = x and in this case, the objective function is equal to
h(x); the other is ps= Ppkwith which the objective function
equals h(Ppk) − μ. Therefore, whether p∗
x or Ppkdepends on the inequality between the values of the
objective function at these two points. We thus have
?
Case 3: x < 0. In this case, χp(ps) = 1 for any ps ≥
0. Since h(ps) is an increasing function of ps, it is easy to
observe that (15) is maximized by p∗
Define the following three regions:˜
{0 ≤ x < Ppk}, and˜
in the above cases, the following theorem can be obtained:
Theorem 2: The optimal solution of P2 is given by
⎧
⎪
where h(·) and x are given in (16) and (13), respectively.
max
ps≥0,ps≤Ppklog2
1 +
gssps
gpsPp+ N0
?
− μχp(ps). (15)
h(ps) ? log2
1 +
gssps
gpsPp+ N0
?
. (16)
s= Ppk.
stakes the value of
p∗
s=
Ppk,
x,
h(x) < h(Ppk) − μ
otherwise.
s= Ppk.
R1? {x ≥ Ppk},˜
R2?
R3? {x < 0}. With the results obtained
p∗
s=
⎪
⎪
⎪
⎩
⎨
Ppk,
Ppk,
x,
Ppk,
˜
˜
˜
˜
R1
R2, h(x) < h(Ppk) − μ
R2, h(x) ≥ h(Ppk) − μ
R3
(17)
The elaborations on different cases of (17) are similar to
those for the case of average transmit power constraint and
are thus omitted here.
Remark 2: It is observed that when μ goes to infinity,
h(x) ≥ h(Ppk) − μ will always be true in region
this case, the SU will not cause any additional outage loss to
the PU, and (17) reduces to the power allocation strategy for
Δε = 0, which was also shown in [14].
˜
R2. In
IV. OUTAGE CAPACITY OF SU UNDER PU OUTAGE
CONSTRAINT
In this section, we consider the scenario where the sec-
ondary link transmission is delay-sensitive. Therefore, the
outage capacity is more suitable than the ergodic capacity
to be the performance indicator. Instead of characterizing
the outage capacity for a given outage probability target, a
normal practice in the literature (see, e.g., [19]) considers
the equivalent problem of minimizing the outage probability
for a given constant transmit rate (outage capacity), since the
outage capacity is a non-decreasing function of the outage
probability target. Consequently, we derive the optimal power
control policies for the SU to minimize the outage probability
for a given outage capacity under the PU outage constraint,
along with the average or peak transmit power constraint of
the SU in the following two subsections, respectively.
A. Average Power Constraint
In this part, we consider the average transmit power con-
straint for the SU. The minimum outage probability of the SU
can be obtained by solving the following problem (P3):
?
s.t. (7),(4).
min
ps(ν)≥0,∀ν
Pr
log2
?
1 +
gss(ν)ps(ν)
gps(ν)Pp+ N0
?
< rs
?
where rsis the prescribed constant transmission rate (outage
capacity) of the SU.
Similarly as for the PU, we introduce the following indicator
function for the outage event of the SU at fading state ν:
?
0, otherwise.
χs(ν) =
1,
log2
?
1 +
gss(ν)ps(ν)
gps(ν)Pp+N0
?
< rs
(18)
Let λ and μ be the non-negative dual variables associated
with the constraints given in (4) and (7), respectively. Similar
to the techniques applied in Section III, by adopting the
indicator functions χs(ν) in (18) and χp(ν) in (6), it can be
shown that P3 can be decomposed into parallel subproblems
each for a different fading state, expressed as
min
ps≥0χs(ps) + λps+ μχp(ps).
Here, we have dropped the index of the fading state for brevity,
and expressed χs and χp as explicit functions of ps. Define
q(ps) = χs(ps) + λps. Then, the objective function in (19)
becomes the sum of q(ps) and μχp(ps). Note that χs(ps) is
a step function where the transition occurs at ps= y, with y
equal to
y =(gpsPp+ N0)(2rs− 1)
(19)
gss
.(20)