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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}@toshiba-trel.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 so-called 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, multiple-antenna 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 multiple-input multiple-output (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 full-complexity 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 secondary-to-secondary

(s → s) link and the secondary-to-primary (s → p) link is maximized1. In [13], this approach, known as

ratio selection, was shown to offer a good trade-off 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 frequency-division duplex (FDD) systems or from channel reciprocity in time-division duplex

(TDD) systems.

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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 power-emission 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;

• closed-form 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

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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(A|B);

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 co-existing 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

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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 two-step 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

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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.

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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 closed-form 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

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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.

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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 non-CR 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 impulse-like nature of the diversity order detailed in the

calculations above. Effectively, this result suggests that any consideration made to reduce the interference

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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

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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, 2010DRAFT

Page 14

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 x-axis 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 (DS-AIC), 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 (RS-AIC), 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, 2010 DRAFT

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15

and power allocation based on instantaneous channel knowledge (RS-PIC), 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= +∞, RS-PIC 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 RS-PIC 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, RS-AIC yields a slightly better performance in the secondary

link compared to the case where RS-PIC 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 DS-AIC significantly outperforms systems using RS-AIC or RS-PIC.

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, RS-PIC outperforms all other systems as it is optimal in such a case. When this practical

October 18, 2010 DRAFT