Difference Antenna Selection and Power Allocation for Wireless Cognitive Systems
ABSTRACT In this paper, we propose an antenna selection method in a wireless cognitive radio (CR) system, namely difference selection, whereby a single transmit antenna is selected at the secondary transmitter out of $M$ possible antennas such that the weighted difference between the channel gains of the data link and the interference link is maximized. We analyze mutual information and outage probability of the secondary transmission in a CR system with difference antenna selection, and propose a method of optimizing these performance metrics of the secondary data link subject to practical constraints on the peak secondary transmit power and the average interference power as seen by the primary receiver. The optimization is performed over two parameters: the peak secondary transmit power and the difference selection weight $\delta\in [0, 1]$. We show that, difference selection using the optimized parameters determined by the proposed method can be, in many cases of interest, superior to a so called ratio selection method disclosed in the literature, although ratio selection has been shown to be optimal, when impractically, the secondary transmission power constraint is not applied. We address the effects that the constraints have on mutual information and outage probability, and discuss the practical implications of the results. Comment: 29 pages, 9 figures, to be submitted to IEEE Transactions on Communications

Conference Paper: SEPoptimal antenna selection for average interference constrained underlay cognitive radios
[Show abstract] [Hide abstract]
ABSTRACT: In the underlay mode of cognitive radio, secondary users can transmit when the primary is transmitting, but under tight interference constraints, which limit the secondary system performance. Antenna selection (AS)based multiple antenna techniques, which require less hardware and yet exploit spatial diversity, help improve the secondary system performance. In this paper, we develop the optimal transmit AS rule that minimizes the symbol error probability (SEP) of an average interferenceconstrained secondary system that operates in the underlay mode. We show that the optimal rule is a nonlinear function of the power gains of the channels from secondary transmit antenna to primary receiver and secondary transmit antenna to secondary receive antenna. The optimal rule is different from the several ad hoc rules that have been proposed in the literature. We also propose a closedform, tractable variant of the optimal rule and analyze its SEP. Several results are presented to compare the performance of the closedform rule with the ad hoc rules, and interesting interrelationships among them are brought out.Global Communications Conference (GLOBECOM), 2012 IEEE; 01/2012  [Show abstract] [Hide abstract]
ABSTRACT: Cooperative relay technology has recently been introduced into cognitive radio (CR) networks to enhance the network capacity, scalability, and reliability of endtoend communication. In this paper, we investigate an underlay cognitive network where the quality of service (QoS) of the secondary link is maintained by triggering an opportunistic regenerative relaying once it falls under an unacceptable level. Analysis is conducted for two schemes, referred to as the channelstate information (CSI)based and faulttolerant schemes, respectively, where different amounts of CSI were considered. We first provide the exact cumulative distribution function (cdf) of the received signaltonoise ratio (SNR) over each hop with colocated relays. Then, the cdf's are used to determine a very accurate closedform expression for the outage probability for a transmission rate R. In a highSNR region, a floor of the secondary outage probability occurs, and we derive its corresponding expression. We validate our analysis by showing that the simulation results coincide with our analytical results in Rayleigh fading channels.IEEE Transactions on Vehicular Technology 01/2013; 62(2):721734. · 2.06 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Transmit antenna selection (AS) is a popular, low hardware complexity technique that improves the performance of an underlay cognitive radio system, in which a secondary transmitter can transmit when the primary is on but under tight constraints on the interference it causes to the primary. The underlay interference constraint fundamentally changes the criterion used to select the antenna because the channel gains to the secondary and primary receivers must be both taken into account. We develop a novel and optimal joint AS and transmit power adaptation policy that minimizes a Chernoff upper bound on the symbol error probability (SEP) at the secondary receiver subject to an average transmit power constraint and an average primary interference constraint. Explicit expressions for the optimal antenna and power are provided in terms of the channel gains to the primary and secondary receivers. The SEP of the optimal policy is at least an order of magnitude lower than that achieved by several ad hoc selection rules proposed in the literature and even the optimal antenna selection rule for the case where the transmit power is either zero or a fixed value.Wireless Communications and Networking Conference (WCNC), 2013 IEEE; 01/2013
Page 1
arXiv:1010.0200v2 [cs.IT] 15 Oct 2010
1
Difference Antenna Selection and Power Allocation for
Wireless Cognitive Systems
Yue Wang, Member, IEEE, and Justin P. Coon, Senior Member, IEEE
Telecommunication Research Laboratory (TRL),
Toshiba Research Europe Limited, UK, BS1 4ND
Email: {yue.wang, justin}@toshibatrel.com
Abstract
In this paper, we propose an antenna selection method in a wireless cognitive radio (CR) system, which we
term difference selection, whereby a single transmit antenna is selected at the secondary transmitter out of M
possible antennas such that the weighted difference between the channel gains of the data link and the interference
link is maximized. We analyze the mutual information and the outage probability of the secondary transmission
in a CR system with difference antenna selection, and propose a method of optimizing these performance metrics
subject to practical constraints on the peak secondary transmit power and the average interference power as seen
by the primary receiver. The optimization is performed over two parameters: the peak secondary transmit power
and the difference selection weight δ ∈ [0,1]. Furthermore, we show that the diversity gain of a CR system
employing difference selection is an impulsive function of δ, in that a value of δ = 1 yields the full diversity order
of M and any other value of δ gives no diversity benefit. Finally, we demonstrate through extensive simulations
that, in many cases of interest, difference selection using the optimal values of these two parameters is superior
to the socalled ratio selection method disclosed in the literature.
Index Terms
Cognitive radio, interference mitigation, antenna selection, power allocation
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I. INTRODUCTION
Cognitive radio (CR) is a promising technology that facilitates efficient use of the radio spectrum.
Tremendous efforts have been made to study CR in recent years [1]–[9]. In particular, considering
coexisting CR systems where the secondary user is allowed to transmit as long as it causes a tolerable
level of interference to the primary receiver, multipleantenna techniques that can potentially exploit
spatial diversity have been investigated under the context of CR networks [4]–[8]. To this end, the
capacity of the secondary link in multipleinput multipleoutput (MIMO) systems was studied in [4],
[5]. Moreover, by employing all of the available antennas simultaneously at the transmitter, it was shown
in [6] and [7] that the interference caused by the secondary user can be controlled by using beamforming
techniques. However, this approach incurs a high computational cost due to the calculations and feedback
required to obtain the beamforming vector [10].
Antenna selection is an alternative to fullcomplexity beamforming that can be used to exploit spatial
diversity in an efficient manner [11], [12]. With transmit antenna selection, instead of transmitting data
from all available antennas (say, M), a subset of antennas is selected to meet a given criterion, and these
antennas are connected to the available radio frequency (RF) chains, which may be fewer in number
than the available transmit antennas. A key benefit of antenna selection lies in the reduction in the
associated implementation costs [11]. Additionally, antenna selection systems achieve the full diversity
gain of M [11].
The attractive features of antenna selection have motivated research on this technology within the
framework of CR networks. For example, in [10], [13]. In [10] and [13], a single antenna at the secondary
transmitter was selected such that the ratio between the channel gains of the secondarytosecondary
(s → s) link and the secondarytoprimary (s → p) link is maximized1. In [13], this approach, known as
ratio selection, was shown to offer a good tradeoff between the ergodic capacity of the secondary link
and the interference caused to the primary link when a fixed transmit power is used. In [10], using the
1Note that it is assumed that the secondary user must have knowledge of the channel gains for the s → s link and the s → p link,
which can be obtained from feedback in frequencydivision duplex (FDD) systems or from channel reciprocity in timedivision duplex
(TDD) systems.
October 18, 2010DRAFT
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3
same selection method, the ergodic capacity of the s → s link was maximized subject to a constraint on
the peak power of the interference caused to the primary link (herein referred to as a peak interference
power constraint (PIC)). Specifically, it was shown that ratio selection combined with power loading
based on instantaneous knowledge of the s → p channel yields optimal performance. However, the work
in [10] failed to consider the case where the secondary transmission power should also be limited. Such
a constraint is usually essential considering the practical poweremission rules such as those stipulated
by the Federal Communications Committee (FCC) [14].
In this paper, we propose an alternative antenna selection approach for use in CR systems, which we
term difference selection, where a single antenna is selected according to a weighted difference between
the channel gains for the s → s and s → p links. Based on this selection method, we optimize the
mutual information and the outage probability of the secondary link subject to a secondary transmission
power constraint and an interference power constraint. Optimization is performed over two parameters:
the peak secondary transmit power and the difference selection weight δ ∈ [0,1]. In contrast to [10]
where the peak interference power is constrained, we apply an average interference power constraint
(AIC) at the primary receiver, which is preferable to implementing a PIC in practice in terms of both
protecting the quality of the primary link and maximizing the throughput of the secondary link [15].
The main contributions of the paper are:
• a difference antenna selection method for CR systems is proposed;
• closedform expressions for the mutual information and the outage probability of the secondary
link of a CR system using difference selection are derived as functions of the difference selection
weight and the secondary transmit power;
• the diversity order of a secondary system employing difference selection is analyzed, and it is
shown that this is an impulsive function of δ;
• the mutual information and the outage probability of the secondary link are optimized subject to a
secondary transmission power constraint and an interference power constraint, where optimization
is performed over the weight δ and the secondary transmit power;
• extensive simulation results illustrating the mutual information and the outage probability of CR
October 18, 2010 DRAFT
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4
systems using difference selection and ratio selection are given, and it is shown that difference
selection often yields superior performance in practical scenarios.
The rest of the paper is organized as follows. In Section II, the model of the CR system with
difference selection is described, and the optimization problem is formulated. Section III derives the
mutual information and the outage probability of the secondary link with difference selection. The
power allocation strategy and the selection weight that optimizes the mutual information and the outage
probability are presented in Section IV. Finally, results and comparisons between difference selection
and ratio selection are given in Section V, and conclusions are drawn in Section VI.
Notations: The probability density function (p.d.f.) and cumulative distribution function (c.d.f.) of
a random variable X are denoted as fX(x) and FX(x), respectively; the probability of an event A
is denoted as P(A), and the conditional probability of an event A given B is denoted as P(AB);
In addition, E1(x) denotes the exponential integral function given by E1(x) =?∞
denotes the complementary error function given by erfc(x) =
x
e−u
udu, and erfc(x)
2
√π
?∞
xe−t2dt; max{a1,··· ,aM} and
min{a1,··· ,aM} denote the maximum and minimum number among M real numbers a1,··· ,aM,
respectively; and F[·] and F(·) denote a functional and a function of real arguments, respectively.
Finally, E denotes expectation, and x ∈ [a,b] denotes that a number x is in the closed interval of a and
b.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. Preliminaries and Optimal Problem Formulation
We consider a CR system with one primary link and one secondary link. The primary and secondary
receivers have one receive antenna, while the secondary transmitter has M transmit antennas. Such a
system model is illustrated in Fig. 1. We consider a coexisting CR system where the secondary user is
allowed to transmit subject to a peak transmission power constraint as long as the average interference
power caused to the primary system is below a given threshold. We assume that the channel coefficients
of the s → s link and the s → p link fade according to independent Rayleigh distributions. The
instantaneous channel gains of the s → s link and the s → p link corresponding to the ith transmit
October 18, 2010 DRAFT
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5
antennas, denoted as γs,i and γp,i, respectively, are exponentially distributed random variables. The
p.d.f.’s of γs,iand γp,iare given by [16]
fγs,i(γ) =1
¯ γse−γ/¯ γs
(1)
and
fγp,i(γ) =1
¯ γpe−γ/¯ γp
(2)
respectively, where ¯ γsand ¯ γpare the corresponding average channel gains.
At each time interval, the secondary transmitter selects one of the M antennas according to a
certain criterion to transmit data. Suppose the ˆ ıth antenna is selected, and the channel gains related to
transmission from this secondary transmit antenna to the secondary and primary receivers are denoted
by γs,ˆ ı and γp,ˆ ı, respectively. Let ℘ be the average interference power limit allowed at the primary
receiver, and Pmaxbe the maximum allowable transmission power at the secondary transmitter. Ideally,
one would determine the optimal antenna selection criterion and the secondary transmission power
allocation strategy by solving the following optimization problem
optimize C[Ps,fγs,ˆ ı(γ)]
s.t.
E(Psγp,ˆ ı) ≤ ℘
Ps≤ Pmax
(3)
where C is the objective functional, which can be mutual information or outage probability. Note that the
optimization is performed over the p.d.f. of the channel gain between the selected transmit antenna and
the secondary receiver fγs,ˆ ı(γ). In addition, Psis also a functional dependent on this function. Solving
the optimization problem stated above requires knowledge of fγs,ˆ ı(γ) and fγp,ˆ ı(γ), which are determined
by the selection criterion. Without knowing what this selection criterion is, the optimization problem
is intractable. One could perform the optimization over an ensemble of p.d.f.’s, but this is not possible
in a practical implementation. Here, we take a twostep approach where we first propose a selection
criterion, and then determine the value of Psthat maximizes the mutual information or minimizes the
outage probability of the secondary link with antenna selection based on such a criterion.
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B. Alternative Formulation Based on Difference Selection
Various criteria can be used to select the transmit antenna for the secondary user. For example, one
can select the antenna that yields the largest s → s channel gain, i.e., select the ˆ ıth antenna such that
γs,ˆ ı = max{γs,1,··· ,γs,M} [13]. Alternatively, one can select the antenna that yields the minimum
interference to the primary user, i.e., select the ˆ ıth antenna such that γp,ˆ ı= min{γp,1,··· ,γp,M} [13].
These two selection methods are referred to as maximum data gain selection and minimum interference
selection, respectively [13]. Ratio selection, proposed in [10] and [13], selects the ˆ ıth antenna such that
the ratio of the channel gains of the s → s and s → p links is maximized, i.e., the ˆ ıth antenna satisfies
?
In this paper, we propose an alternative antenna selection method for CR systems, which will be shown
γs,ˆ ı
γp,ˆ ı= max
γs,1
γp,1,··· ,γs,M
γp,M
?
.
to be superior to ratio selection in many practical cases. The proposed selection method, referred to as
difference selection, selects the antenna at the secondary transmitter such that the weighted difference
between the channel gains for the s → s link and the s → p link is maximized. Denote the selection
weight as δ ∈ [0,1]. Difference selection selects the ˆ ıth antenna such that Zˆ ı= max{Z1,··· ,ZM},
where Zi(i = 1,··· ,M) is given by
Zi= δγs,i− (1 − δ)γp,i.
(4)
Note that difference selection becomes minimum interference selection when δ = 0, and maximum data
gain selection when δ = 1.
With difference selection, the mutual information and outage probability for the secondary link are
dependent upon δ. In the following, we formulate optimization problems to jointly optimize δ and the
secondary transmission power, such that the mutual information and outage probability are optimized
subject to constraints on the peak transmission power for the secondary user and the average interference
power affecting the primary receiver.
Now, suppose difference selection selects the ˆ ıth antenna to transmit data at a given time slot. The
October 18, 2010DRAFT
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optimization problem is formulated as follows:
optimize C(Ps,δ)
s.t.
E(Psγp,ˆ ı) ≤ ℘
Ps≤ Pmax
(5)
where C(Ps,δ) is the objective function, which can be mutual information or outage probability, and Ps
is the secondary transmission power that is to be determined, which is a function of the instantaneous
channel gains γs,ˆ ı and γp,ˆ ı, and thus a function of δ. Note that in the problem considered here, the
optimization is performed over two variables: the secondary transmission power Psand and selection
weight δ. Therefore, in contrast to the optimization problem given in (3), the objective function C in
this case is a function, not a functional, and the optimization process can be significantly simplified. In
fact, for a given δ, the optimal power loading strategy for such an optimization problem is addressed in
[8], where the secondary transmission power Psis defined as a function of the instantaneous channel
gains γs,ˆ ıand γp,ˆ ı, and is given by
Ps(γs,ˆ ı,γp,ˆ ı) =
0,γp,ˆ ı≥
log2e
λN0γs,ˆ ı
log2e
λγp,ˆ ı−N0
γs,ˆ ı,
log2e
λN0γs,ˆ ı> γp,ˆ ı>
log2e
Pmax+N0
λ
?
γs,ˆ ı
?
Pmax,γp,ˆ ı≤
log2e
Pmax+N0
λ
?
γs,ˆ ı
?
(6)
where N0is the noise power, and λ is determined by substituting (6) into the average interference power
constraint given in (5), i.e., λ is defined implicitly by the equation
?∞
0
?∞
0
Ps(γs,ˆ ı,γp,ˆ ı)γp,ˆ ıfγs,ˆ ı,γp,ˆ ı(γs,ˆ ı,γp,ˆ ı)dγs,ˆ ıdγp,ˆ ı= ℘
(7)
where fγs,ˆ ı,γp,ˆ ı(γs,ˆ ı,γp,ˆ ı) is the joint p.d.f. of γs,ˆ ıand γp,ˆ ı. These two random variables are dependent due
to the selection process. The key to solving the optimization problem therefore lies in the derivation of
fγs,ˆ ı,γp,ˆ ı(γs,ˆ ı,γp,ˆ ı). Unfortunately, the expression for this joint p.d.f. can be complicated, thus making it
difficult to determine the optimal transmit power analytically.
October 18, 2010DRAFT
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8
C. Practical Formulation
The problem given in (3) can be made practical by considering power allocation based on channel
statistics rather than instantaneous channel knowledge. In such a case, the optimization problem becomes
optimize C(Ps,δ)
s.t.
PsE(γp,ˆ ı) ≤ ℘
Ps≤ Pmax
(8)
where, in contrast to (5), Psis taken out of the expectation function in (8) because it is not a function
of the instantaneous channel gains γp,ˆ ı. With the optimization problem considered above, to determine
the optimal transmission power, only the mean of γp,ˆ ı is required. Even so, when ratio selection is
considered, the calculation of this mean does not have a closed form. Fortunately, by using difference
selection at the secondary transmitter, a closedform expression for the mean of the s → p link gain
can be obtained, based on which the mutual information and the outage probability can be optimized.
Not only does the use of difference selection facilitate the mathematical tractability of the optimization
problem, but we further show that, with the practical peak secondary transmission power and average
interference power constraints, difference selection using optimal values of Psand δ is, in many cases
of interest, superior to ratio selection with respect to performance. The advantages of using difference
selection will be detailed later. In the following, we first provide an analysis of the mutual information
and the outage probability of the secondary link for CR systems with difference selection, and then
solve for selection weight δ and the secondary transmission power Ps that optimize these objective
functions.
III. MUTUAL INFORMATION AND OUTAGE PROBABILITY ANALYSIS
We now derive the mutual information and the outage probability for the secondary transmission in
CR systems with difference antenna selection. In the ensuing analysis, we make use of the following
lemma, the proof of which is given in Appendix A.
Lemma 1: The c.d.f. of the s → s channel gain due to the selection of theˆ ıth antenna using difference
October 18, 2010DRAFT
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9
antenna selection, denoted as γs,ˆ ı, is given by
Fγs,ˆ ı(x) =
?M−1
?M−1
k=0ρk(x) −
̟M
p
¯ γMe−
̟M
p
¯ γMe−
δ¯ γ
̟s̟px+
?
?
1 −̟s
¯ γe−x
¯ γs
?M
?M
, ̟s?= g̟p
k=0
k?=g
ρk(x) −
δ¯ γ
̟s̟px+1 −̟s
¯ γe−x
¯ γs
−M
¯ γδxµg(x) , ̟s= g̟p
(9)
where
ρk(x) = M
?M − 1
k
?
(−̟s)k+1̟p
¯ γk+1(̟s− k̟p)
?
e−(k+1)
¯ γs
x− µk(x)
?
(10)
µg(x) =
?M − 1
g
?(−̟s)g
¯ γg
e−g+1
¯ γsx.
(11)
In the equations above, ̟s= (1 − δ)¯ γs, ̟p= δ¯ γp, ¯ γ = ̟s+ ̟p, which are all functions of δ, and
g ∈ [0,M − 1] is an integer.
A. Mutual Information
Assuming Gaussian signaling is employed, the maximum mutual information2of the secondary link
for CR systems with difference selection is given by
Rmax= log2emax
Ps,δR(Ps,δ)
(12)
where Psand δ are subject to the constraints given in (8), and
R(Ps,δ) =
?∞
0
log
?
1 +Psx
N0
?
dFγs,ˆ ı(x).
(13)
In the equation above, N0 is the noise power and Fγs,ˆ ı(·) is the c.d.f. of γs,ˆ ı, given in Lemma 1.
Substituting (9) into (13), we have
R(Ps,δ) =
?M−1
?M−1
k=0(Ψk(Ps,δ) − Φk(Ps,δ)) + Υ(Ps,δ),
Ψk(Ps,δ) −?M−1
̟s?= g̟p
k=0
k?=g
k=0Φk(Ps,δ) + Υ(Ps,δ) + Θg(Ps,δ), ̟s= g̟p
(14)
where
Ψk(δ,Ps) = M
?M − 1
k
?
(−̟s)k+1̟p
¯ γk+1(̟s− k̟p)
?
−e
(k+1)N0
¯ γsPs E1
?(k + 1)N0
¯ γsPs
?
+ e
δ¯ γN0
̟s̟pPsE1
?
δ¯ γN0
̟s̟pPs
??
2In the unit of Bits/Sec/Hz.
October 18, 2010DRAFT
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10
Υ(Ps,δ) =̟M
p
¯ γMe
δ¯ γN0
̟s̟pPsE1
?
δ¯ γN0
̟s̟pPs
?
Φk(δ,Ps) = M
?M − 1
k
??
−̟s
¯ γ
?k+1
1
k + 1e
(k+1)N0
¯ γsPs E1
?(k + 1)N0
¯ γsPs
?
and
Θg(δ,Ps) = −
?
M
g + 1
??−̟s
¯ γ
?g+1
+ M
?M − 1
g
??−̟s
¯ γ
?g+1N0
¯ γsPse
(g+1)N0
¯ γsPs E1
?(g + 1)N0
¯ γsPs
?
.
From (14), it is known that R is a function of the secondary transmission power Psand the selection
weight δ. The method of determining Psand δ to maximize R will be discussed later. For now, assume
Psis given. It can be verified that, in such a case, R is a monotonically increasing function of δ. In
particular, the maximum R is achieved when δ = 1, which is given by
lim
δ→1R(Ps,δ) =
M−1
?
k=0
?
M
k + 1
?
(−1)ke
(k+1)N0
¯ γsPs E1
?(k + 1)N0
¯ γsPs
?
.
(15)
The expression of R when δ → 1 coincides with that given for a conventional nonCR system with
selection combining (c.f. (44) in [17]). This is intuitively correct because when δ = 1, for CR systems
with difference selection, data from the secondary user is transmitted as if the primary user does not
exist, and difference antenna selection essentially becomes the maximum data gain selection.
Similarly, when δ = 0, from the secondary user’s perspective, antenna diversity is not exploited, and
the data is effectively transmitted through a Rayleigh fading channel from a single (random) antenna.
In such a case, R reaches its minimum and it remains the same regardless of the number of transmit
antennas employed at the secondary transmitter. Following (14), one can verify that when δ = 0, R
indeed becomes the capacity of a single antenna transmission through a Rayleigh fading channel (c.f.
(34) in [17]), which is given by
lim
δ→0R(Ps,δ) = e
N0
¯ γsPsE1
?N0
¯ γsPs
?
.
(16)
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B. Outage Probability Analysis
The outage probability of the secondary link for a CR system with difference selection and a given
outage transmission rate r0is given by [18]
Pout= min
Ps,δP(Ps,δ)
(17)
where Psand δ are subject to the constraints given in (8), and P(Ps,δ) is the probability that the rate
of the secondary transmission is smaller than or equal to a given r0, given by [18]
P(Ps,δ) = Fγs,ˆ ı
?(2r0− 1)N0
Ps
?
(18)
and Fγs,ˆ ı(·) is given in Lemma 1.
1) Asymptotic Analysis: It is interesting to study the diversity order of the secondary system to obtain
insight into the secondary link performance at high SNR. Following (18), when Ps→ ∞, the diversity
order can be gleaned from the expression for Fγs,ˆ ı(x) when x → 0. Applying the first order Taylor
expansion of exponential functions ex= 1 + x + O(x2) to (9), one can confirm that
?M
̟M−1
p
¯ γM−1¯ γsx + O(x2)
1
(g+1)M−1¯ γsx + O(x2) , δ ?= 1,̟s= g̟p
Recall that in the equation above, ̟s= δ¯ γs, ̟p= (1−δ)¯ γp, and ¯ γ = ̟s+̟p, which are all functions
Fγs,ˆ ı(x → 0) =
?
x
¯ γs
+ O(xM+1) , δ = 1
, δ ?= 1,̟s?= g̟p
.
(19)
of δ, and g is an integer where g ∈ [0,M − 1].
Equation (19) indicates that the secondary transmission of a CR system with difference antenna
selection achieves a diversity order of M when δ = 1. Indeed, when δ = 1, a CR system with difference
antenna selection is essentially a conventional antenna selection system from the secondary user’s point
of view. Such a conventional system has been investigated in [19], and our result for δ = 1 agrees with
the expression for the outage probability given there (c.f. (11) in [19]). For any other δ ?= 1, however,
the secondary user only achieves a diversity order of 1. The same results on the diversity order of the
secondary link with difference selection are also verified in [20] via bit error rate (BER) analysis.
It is worth considering the implications of the impulselike nature of the diversity order detailed in the
calculations above. Effectively, this result suggests that any consideration made to reduce the interference
October 18, 2010DRAFT
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to a primary user via antenna selection negates all beneficial effects of employing multiple antennas in
the secondary link at high SNR. Therefore, if the secondary transmit power is constant, there is little
purpose in setting δ close to, but strictly less than one, since all diversity gain is surrendered in doing
so and only a minor emphasis is placed on reducing interference to the primary user. However, if the
secondary transmit power is allowed to vary with changing channel statistics (i.e., mean channel gains),
then there turns out to be an intricate relationship between the optimal power level and the weight δ at
finite SNR. This relationship and how it can be exploited to optimize the mutual information and the
outage probability is considered in the next section.
IV. POWER ALLOCATION STRATEGY AND OPTIMAL SELECTION WEIGHT
Having obtained the expressions for mutual information and outage probability given in (14) and
(18), respectively, we are in a position to determine the optimal selection weight and power allocation
strategy. First, we introduce the following lemma:
Lemma 2: The average s → p link channel gain due to the selection of theˆ ıth antenna using difference
antenna selection, denoted as E(γp,ˆ ı), is given by
E(γp,ˆ ı) =M(̟s+ ̟p)M−1̟s+ ̟M
M(̟s+ ̟p)M
p
¯ γp
(20)
where M is the number of secondary transmit antennas, ̟s= (1 − δ)¯ γs, and ̟p= δ¯ γp.
The proof of Lemma 2 is given in Appendix B. Applying (20), the left hand side of the first inequality
constraint given in (8) can be rewritten as
I(Ps,δ) = PsE(γp,ˆ ı) = PsM(α + 1)M−1α + 1
M(α + 1)M
¯ γp
(21)
where α =
̟s
̟p=
δ¯ γs
(1−δ)¯ γp, which is a function of δ. One can confirm that I(Ps,δ) is monotonically
increasing in α ∈ (0,+∞) by verifying that the first derivative of α is greater than 0. Therefore, I
reaches its maximum of Ps¯ γpwhen α → ∞, or equivalently when δ = 1. In addition, it reaches its
minimum of Ps¯ γp/M when α = 0, or equivalently when δ = 0. In other words, using the proposed
difference selection method, the average interference power is always guaranteed to be in a range of
I ∈ [Ps¯ γp/M,Ps¯ γp]. In particular, compared to the maximum data gain selection (δ = 1), the interference
October 18, 2010DRAFT
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13
power can be reduced by a factor of M when minimum interference power selection (δ = 0) is performed
for antenna selection.
Having noticed that, for a given δ, R(Ps) and P(Ps) are monotonically increasing and decreasing
functions of Ps, respectively, the maximum of R and the minimum of P are reached when Pstakes
the maximum possible value. In the case where ¯ γp is sufficiently small3such that Pmax¯ γp ≤ ℘, the
interference constraint is inactive and one should let Ps = Pmax. When Pmax¯ γp > ℘, the average
interference constraint is active, and the optimal secondary transmission power is the minimum between
Pmaxand the transmission power that satisfies the interference constraint with equality. Applying (21),
we have the following power allocation strategy:
Proposition 1: In a CR system with difference antenna selection, for a given selection weight δ, the
optimal secondary transmission power P∗
sthat maximizes the mutual information and minimizes the
outage probability, subject to the constraints that 1) the average interference power is at most ℘, and 2)
the peak secondary transmission power is at most Pmax, is given by
P∗
s=
min
?
℘M(α+1)M
(M(α+1)M−1α+1)¯ γp,Pmax
?
, ¯ γp≥ ℘/Pmax
Pmax
, ¯ γp< ℘/Pmax
(22)
where ¯ γsand ¯ γpare the average channel gains of the s → s and s → p links, respectively, M is the
number of antennas, and α =
δ¯ γs
(1−δ)¯ γp.
Note that P∗
sis a function of δ because α is a function of δ. Substituting (22) into (14) and (18), and
noticing the fact that R(δ) and P(δ) are concave and convex functions of δ, respectively, the optimal δ
that maximizes mutual information or minimizes outage probability can be determined by using existing
numerical techniques (see, e.g., [21]).
V. RESULTS AND DISCUSSIONS
In this section, we present simulation results for the mutual information and the outage probability of
the secondary link for a CR system using difference antenna selection, and compare them with a similar
3For example, when the secondary transmitter is far away from the primary receiver, or there is obstacle material that yields deep fading
channels between the two.
October 18, 2010 DRAFT
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14
system using ratio selection. The results are generated by using Rayleigh fading channels for both the
s → s link and the s → p link, with the mean of the channel gains being ¯ γs and ¯ γp, respectively.
Unless otherwise specified, we assumed that the interference power threshold ℘ = 1, which is the same
for both AIC and PIC, and is equivalent to the noise power [10], [15]. In addition, all the results for
outage probability are generated by assuming r0= 1. By way of example, we used M = 2 or 4 transmit
antennas at the secondary transmitter to study mutual information and outage probability. Using larger
numbers of transmit antennas leads to results and trends that are similar to those shown here.
We first show the results of mutual information and outage probability for the secondary link of a CR
system with difference antenna selection using the optimal δ, where M = 4 antennas are used at the
secondary transmitter. These results are presented in Figs. 2 and 3, where the xaxis is ξ = ¯ γs/¯ γp, and
the maximum allowed transmission power is Pmax= 10 dB, to follow the convention of [10]. Mutual
information and outage probability with the optimal δ are obtained by first substituting (22) into (14)
and (18), solving for the optimal δ, and then applying this δ to the respective objective functions.
Note that the optimal δ is also a function of ξ. For δ = 1 and δ = 0, the secondary transmission
?
both figures that, comparing to mutual information and outage probability for the secondary link with
power is Ps= min
℘
¯ γp,Pmax
?
and Ps= min
?
M℘
¯ γp,Pmax
?
, respectively, as given by (22). It is shown in
difference antenna selection using δ = 1 and δ = 0, significant gains can be observed when the optimal
δ is used.
Next, we present the mutual information and the outage probability of CR systems with difference
selection by using the optimal δ and the power allocation strategy proposed in this paper, and compare
the results with those obtained by using ratio selection with PIC or AIC and different power allocation
strategies. In all figures that follow, we assumed the number of antennas at the secondary transmitter is
M = 2. For difference selection, we considered an average interference constraint (DSAIC), where the
results are obtained by using the power allocation strategy and the optimal δ presented in this paper.
For ratio selection, when AIC is considered (RSAIC), the power allocation strategy is the same as that
given in this paper except that the mean of the s → p channel is simulated by observing a sufficiently
large number of channels since it cannot be calculated in closed form. For ratio selection with PIC
October 18, 2010DRAFT
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15
and power allocation based on instantaneous channel knowledge (RSPIC), the results are obtained by
using a secondary transmission power of P∗
s= min
?
Pmax,
℘
γp,ˆ ı
?
. When Pmax → ∞, the transmission
power becomes P∗
s=
℘
γp,ˆ ı, which is essentially the power allocation method given in [10]. We must
emphasize that in practice, a peak transmission power constraint shall be applied to the secondary
transmitter. Therefore, the results for systems without a peak transmission power constraint shown here
are impractical, and are used as benchmarks only.
We first show, in Figs. 4 and 5, the results for mutual information and outage probability as a function
of ξ = ¯ γs/¯ γp. It is observed from Figs. 4 and 5 that, for the impractical case where no peak transmission
power constraint is applied, i.e., when Pmax= +∞, RSPIC is indeed optimal in the sense that it provides
the maximum mutual information and the minimum outage probability among the systems considered.
However, in practice, when the secondary transmission power is constrained, the performance of the
secondary link with RSPIC degrades considerably. For example, it is observed from Fig. 5 that about a 3
dB degradation occurs at an outage probability of 10−2when a stringent transmission power constraint of
Pmax= 0 dB is applied. One can consider AIC and apply the same power allocation strategy described in
this paper to ratio selection. In such a case, RSAIC yields a slightly better performance in the secondary
link compared to the case where RSPIC is employed. This results from the fact that AIC is a more
relaxed constraint compared to PIC from the perspective of secondary transmission. A comparison
between the performance of ratio selection and difference selection shows that difference selection
yields inferior mutual information and outage probability without a peak transmission power constraint.
However, considering the practical case when such a constraint is applied, performance of a CR system
employing DSAIC significantly outperforms systems using RSAIC or RSPIC.
The results in the figures shown above apply when Pmax= 0 dB. In practice, the maximum allowable
transmission power at the secondary transmitter can vary. To study the effect of the secondary trans
mission power constraint, we show in Figs. 6 and 7 mutual information and outage probability of the
aforementioned five systems, where in all simulations it is assumed that ¯ γs= ¯ γp= 1. It is observed
from Fig. 6 that, when the impractical case is considered where no peak transmission power constraint
is applied, RSPIC outperforms all other systems as it is optimal in such a case. When this practical
October 18, 2010DRAFT
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