linear precoding for orthogonal space-time

7/28/2019 Linear Precoding for Orthogonal Space-Time

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011 767

Linear Precoding for Orthogonal Space-TimeBlock Coded MIMO-OFDM Cognitive Radio

A. Punchihewa, Student Member, IEEE, Vijay K. Bhargava, Fellow, IEEE,and Charles Despins, Senior Member, IEEE

AbstractThe paper presents design of a linear precoderfor orthogonal space-time block coded orthogonal frequencydivision multiplexing (OFDM)-based multiple-input multiple-output (MIMO) antenna cognitive radio (CR) when operatingin correlated Rayleigh fading channels. Unlike previous studieson precoder design for CR, this proposed linear precoder iscapable of handling both transmit and receive correlation in amulti-carrier based CR system. The linear precoder is designedto minimize an upper bound on the average pairwise errorprobability, constrained to a set of per subcarrier transmit powerconstraints at the CR transmitter and a set of interference power

thresholds at primary user receivers. The CR transmitter exploitsthe knowledge of transmit and receive correlation matrices whiledesigning the precoder. It is shown that the linear precoder designproblem is convex with these constraints, and convex optimizationtechniques are exploited to derive an efficient algorithm toobtain the optimal precoder matrices. Computer simulations areperformed to investigate the performance of the proposed linearprecoder in a CR system.

Index TermsCognitive radio, convex optimization, orthog-onal space-time block coding, multiple-input multiple-outputantennas, orthogonal frequency division multiplexing.

I. INTRODUCTION

THE electromagnetic radio spectrum is a precious resourceavailable for wireless communications, which demands

efficient usage. However, it has become increasingly scarcedue to a wide deployment of wireless services. According tothe Federal Communications Commissions spectrum policytask report [1], the usage of allocated spectrum varies fromfifteen to eighty-five percent at specific time and geographicallocation. This low spectrum utilization coupled with spectrumscarcity motivates the development of novel spectrum-sharingtechnologies with the aim of improving spectrum utilization.Cognitive radio (CR) has emerged as a promising technologyto improve spectrum utilization, while accommodating thegrowing amount of services and applications in wirelesscommunications [2]. CR is capable of dynamically sensingand identifying unoccupied spectrum bands that are initiallyallocated to licensed (primary) users (PUs), and allowingunlicensed (secondary) users (SUs) to communicate through

Paper approved by I. Lee, the Editor for Wireless Communication Theory ofthe IEEE Communications Society. Manuscript received September 8, 2009;revised June 2, 2010 and August 16, 2010.

A. Punchihewa and V. K. Bhargava are with the Department of Electricaland Computer Engineering, University of British Columbia, Vancouver, BC,Canada (e-mail: [email protected], [email protected]).

C. Despins is with the Universit du Qubec, Montral, QC, Canada (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCOMM.2011.011811.090548

these available spectrum segments without causing harmfulinterference to PUs, thus having the potential to efficientlyimprove spectrum utilization. Since CR operates with oppor-tunistic spectrum sharing in dynamically changing environ-ments, managing the quality of services (QoS) offered by a CRsystem while maintaining the QoS of the PUs, is challenging.Hence, proper design of a transmission scheme for CR tofacilitate high data rate access and better performance alongwith high spectral efficiency is very important. To achieve

this objective, it is crucial to integrate recent physical layertechnical advances into the CR systems.

Multiple-input multiple-output (MIMO) antenna systemsand space-time block coding (STBC) in wireless commu-nications have attached considerable attention due to theirability to increase capacity and improve system performanceover hostile wireless channels [3]-[5]. Orthogonal frequencydivision multiplexing (OFDM) is a promising transmissiontechnique in CR systems due to its several advantages suchas scalability, robustness against multipath fading, multipleaccess mechanisms, simplicity in channel equalization andcoding [6]. Therefore, with these valuable features, incor-

porating MIMO, STBC and OFDM into CR would promiseenhanced performance in terms of spectral efficiency, capacityand bit error rate over hostile wireless channels.

It is shown in previous studies that the performance ofconventional MIMO systems is degraded in spatially corre-lated channels based on the available channel state information(CSI) at the transmitter [7], [8]. However, efficient precodingtechniques in combination with STBC can be used to furtherimprove the system performance in such channel conditions,when the knowledge of CSI is available at the transmitter[9]-[12]. Therefore, linear precoding is a vital technique tocombat the correlation effect of MIMO channels. In practice

a perfect CSI is seldom available and is difficult to obtain atthe transmitter. Thus, a common practice is to assume partialchannel knowledge at the transmitter, for example, in termsof transmit or both transmit and receive correlation matrices[9]-[12]. In conventional MIMO systems [9]-[12], the linearprecoder is designed with the knowledge of transmit or bothtransmit and receive correlation matrices at the transmitter byminimizing a metric related to average error probability andconstrained only to total transmit power.

Although this topic has been extensively studied for con-ventional MIMO systems, less attention is given in previousstudies for design of a linear precoder for MIMO-based CR,

where additional constraints need to be incorporated in pre-0090-6778/11$25.00 c 2011 IEEE


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768 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011

coder design. The linear precoder designed in [13] for CR withthe intention of improving the error rate performance assumesonly the SU transmit antenna correlation, single antenna atthe PU receiver and considers the single carrier transmissionscheme. However, this paper extend the linear precoder designfor CR in several ways. The theoretical analysis of the linearprecoder design is based on a comprehensive signal model thattakes into account of multiple antennas at both SU and PUs,

multi-carrier transmission scheme and the correlation effectsof SUs both transmit and receive antennas. A linear precoderis designed for orthogonal space-time block coded (OSTBC)MIMO-OFDM based CR, when operating in frequency-flatcorrelated Rayleigh fading channels. The linear precoder isdesigned to minimize an upper bound on the average pairwiseerror probability (PEP) when the SU transmitter has theknowledge of transmit and receive correlation matrices, whileimposing a set of interference power constraints at the PUsand a set of per subcarrier transmit power constraints at theSU transmitter. It is shown that the precoder design problemfor CR is convex with these constraints. Furthermore, an

efficient algorithm based on the Lagrange dual-decompositionis proposed to obtain the linear precoder. The individualeffects of the SU transmit and receive antenna correlationon the linear precoder design for CR is also addressed inthis paper. A closed-form solutions for power loading in eachOFDM subcarrier for simplified correlation scenarios are alsopresented.

The rest of the paper is organized as follows. The systemmodel and OSTBC MIMO-OFDM transmission scheme areintroduced in Section II. The optimal linear precoder designproblem is formulated in Section III. The linear precoderdesign with SUs different correlation scenarios are investi-

gated in Section IV, and the Lagrangian dual-decomposition-based efficient algorithm is proposed to obtain the linearprecoder. Simulation results are provided in Section V. Finally,conclusions are drawn in Section VI. Proofs of the theoremsare given in the Appendix.

The following notations are used throughout the paper.Vectors are denoted by boldfaced lowercase letters, e.g., a,b, and matrices are denoted by boldface uppercase letters,e.g., A, B. The superscripts ()1, (), (), and ()1/2 standfor inverse, transpose, conjugate transpose, and square root,respectively. {} is the trace of a square matrix, det[] isthe determinant of a matrix, E{} is the expectation operation,

vec() is the vectorization operation, is the Frobeniusnorm of a matrix and is the Kronecker product. I is the identity matrix and A 0 indicates that the squarematrix A is positive semi-definite.

II. SYSTEM MODEL

A. System Description

A CR network as illustrated in Fig. 1 is considered, wherea single pair of SU transmitter and receiver coexist with the PUs receivers. We assume that PUs and SU share thesame bandwidth. The SU pair are equipped with and transmit and receive antennas, respectively, while thePUs have receive antennas each. The SU transmitteremploys subcarriers to modulate the signal using OFDM.

P U R e c e i v e r 1 P U R e c e i v e r 2

P U R e c e i v e r L

S U R e c e i v e r

S U T r a n s m i t t e r

ssH

1spH

2spH

LspH

Fig. 1. A cognitive radio network.

Furthermore, we assume that only the transmit and receivecorrelation matrices for the MIMO channels, between the SU

transmitter and PU receivers, and between the SU transmitterand SU receiver, are available at the SU transmitter. Thesematrices are obtained at the SU transmitter by periodicallysensing the transmitted signals from the PU and SU receivers.In addition, perfect CSI is assumed at the SU receiver.

B. Correlated Channel Model

Quasi-static frequency-flat correlated Rayleigh MIMOchannels between the SU transmitter and PU receivers, andbetween the SU transmitter and SU receiver are considered.Under the assumption of SU transmit and SU, PUs receivescattering radii are large compared to the distance between the

SU transmitter and SU, PUs receivers, the MIMO channelsbetween the SU transmitter and SU receiver and between theSU transmitter and PU receivers for the th subcarrier can berespectively written as [14]

H() = R1/2,()H,()R

1/2,(), = 1, . . . , , (1)

and

H() = R1/2,

()H,()R1/2,

(),

= 1, . . . , , = 1, . . . , ,(2)

where R,(), R,() are the transmit and receivecorrelation matrices of sizes and ,

respectively, for the MIMO channels between SU transmitterand SU receiver; and R,(), R,(), are the transmitand receive correlation matrices of sizes and , respectively, for the MIMO channels betweenSU transmitter and PU receivers. H,() and H,() arematrices of sizes and , respectively,with independent and identically distributed (i.i.d.) zero-meancircular symmetric complex Gaussian (CSCG) entries withunit variance. The transmit and receive correlation matricesfor the MIMO channels between the SU transmitter and SUreceiver, and between the SU transmitter and PU receivers canbe respectively written as [14]

R,() = R1/2,()R1/2,() = {

H()H()}

, = {,}, = 1, . . . , , = 1, . . . , ,

(3)


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PUNCHIHEWA et al.: LINEAR PRECODING FOR ORTHOGONAL SPACE-TIME BLOCK CODED MIMO-OFDM COGNITIVE RADIO 769

and

R,() = R1/2,()R

1/2,()

= {

H()H()}

,

= {,}, = 1, . . . , , = 1, . . . , .(4)

The full autocorrelation matrices, R() and R(), can beobtained in terms of the Kronecker product of the transmitand receive correlation matrices:

R() = R,()

R,(), = 1, . . . , , (5)R() = R,()

R,(),

= 1, . . . , , = 1, . . . , .(6)

By applying the vectorization operation to (1) and (2), we havevec (H()) = R

1/2 ()vec (H,()), and vec(H()) =

R1/2 ()vec(H,()), respectively. This Kronecker model

has been widely exploited in the previous studies for correlatedMIMO systems [10]-[12], [15], [16].

C. Transmission Scheme

A block diagram of the proposed OSTBC MIMO-OFDM

based CR transmission scheme with linear precoding is shownin Fig. 2. The SU transmitter includes an OSTBC encoder anda linear precoder followed by an OFDM modulator. At theSU transmitter, the input symbols are first serial to parallel(S/P) converted and fed into the OSTBC encoder. Secondly,the output symbols of the OSTBC encoder are multiplied by precoder matrices F(), = 1, . . . , . Third, theprecoded data to be transmitted by each transmitting antennaare subjected to typical OFDM transmit processing, suchas S/P conversion, inverse fast Fourier transform operation,parallel to serial (P/S) conversion and the addition of a cyclicprefix (CP). Fourth, the precoded data are transmitted over

the wireless MIMO channel. Then at the SU receiver, thereceived signal in each antenna is subjected to typical OFDMprocessing such as removal of CP, S/P conversion, fast Fouriertransform operation and P/S conversion. Finally, the maximumlikelihood (ML) detector recovers the received data symbols.

For this system, the received signal at the SU receiver forthe th subcarrier can be written as

Y() = H()F()C() + N(), = 1, . . . , , (7)

where C() is the transmitted OSTBC matrix of size ofdm, with ofdm as the total OFDM symbols transmitted in ablock of data. Individual data symbols ofC() are drawn from

a finite complex signal constellation with unit energy. N()1

is the complex additive white Gaussian noise matrix of size ofdm with zero-mean and variance 2I . In addition,the same noise statistics are assumed for all subcarriers.

Under the assumption of perfect CSI at the SU receiver, theML decoding of codeword C from the received signal matricesY(), = 1, . . . , , yields

C = arg minC

=1

Y() H()F()C()2

. (8)

1Note that the noise at the SU receiver also contains the interference fromthe PU transmitters in the network and therefore non-white in general. By

applying the noise-whitening filter at the SU receiver and incorporating thefilter effects into MIMO channel matrix between the SU transmitter and SUreceiver, the effective noise is assumed to be approximately white Gaussian.

S / P P r e c o d e r

F ( k )

S T B C

E n c o d e r

S / P , I F F T

P / S , C P A d d i t i o n

C P R e m o v a l , S / P

F F T , P / S

M L

D e c o d e r

ss( )kH

I n p u t

B i t s

R e c e i v e d

B i t s

S / P , I F F T

P / S , C P A d d i t i o n

C P R e m o v a l , S / P

F F T , P / S

Fig. 2. System block diagram of the precoded OSTBC MIMO-OFDM basedCR transmission scheme.

III. OPTIMAL LINEAR PRECODER DESIGN PROBLEMFORMULATION

The main objective is to find a set of linear precoder

matrices F(), = 1, . . . , , at the SU transmitter, so as tominimize an upper bound on the average PEP under a set ofper subcarrier power constraints at the SU transmitter and a setof PUs interference power thresholds. Upper bounds on theaverage PEP have been extensively exploited as design criteriaof the linear precoder in conventional MIMO systems [9]-[12].In our framework, an upper bound on the average PEP is alsoadopted as a design criteria, but for OSTBC MIMO-OFDMbased CR transmission scheme. The PEP, (C C ) is theprobability that ML decoding decides in favor of the codewordC instead of the actually transmitted codeword C.

Theorem 1: An upper bound of the average PEP of

the OSTBC MIMO-OFDM, when SUs transmit and receiveantennas are correlated, can be written as:

(C C )

=1

=1

det

[I + ,() F(),()

]1,

(9)

where is a factor that depends on the codeword pair C andC , ,() is the th eigenvalue of the receive eigenvaluematrix ,(), F() = F()F(), and ,() is thetransmit eigenvalue matrix of the transmit correlation matrix

R,().Proof: See the Appendix.In CR networks, CR users may coexist with PUs either on

a non-interfering basis or on an interference tolerance basis[17]. Therefore, one fundamental challenge of the CR is tomaintain the QoS of the PUs while maximizing the SUsperformance. Since PUs have a higher priority than the SUswhile opportunistically sharing the spectrum in an interferencetolerance basis, SUs have to maintain interference introducedto the PUs by SUs below a certain threshold, known as theinterference temperature constraint and defined by regulatorybodies. Therefore, the QoS of the PUs in the CR network ismaintained by introducing the additional interference powerconstraints, measured at the PUs receivers [18], [19] into theprecoder design problem for CR. The average interference


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power measure is appropriate for delay-insensitive commu-nications and has been extensively exploited in the previousstudies to limit the interference from SU transmission to thePUs [18], [19]. Following considers the average interferencepower introduced to the PU receivers by SU transmission. Theinterference power introduced by SU transmission at the thPU receiver, conditioned on the input signal constellation andthe channel realization, can be written as

(

C(), H())

=

H()F()C()C()

F()H()

, = 1, . . . , , = 1, . . . , .(10)

Therefore, the total average interference power introducedby the SU transmission to each PU can be obtained underthe assumption of independent channel realization for eachsubcarrier and by taking the expectation of (10) with respectto input signal and the channel realization as

, =

=1

F()E

C()C()

F()

E{

H()H()

}, = 1, . . . , ,

=

=1

F()R,()

, = 1, . . . , ,

(11)

where the unit variant input signal constellation is assumed(i.e., E{C()C()} = I ).

In OFDM transmission schemes, the transmitted signalpower should be limited to avoid generating strong inter-ference to other active users and the systems, and to avoidrequirement of linear amplifiers with large dynamic range.Furthermore, decreasing the transmit power will prolong the

battery lifespan. Therefore, in this CR network per subcarrierpower constraint is imposed at the SU transmitter. Underthe assumption of unit variant constellation, the transmitpower from the th subcarrier can be written as P() = {F()}, = 1, . . . , .

An upper bound on the average PEP of OSTBC is exploitedto obtain a set of linear precoder matrices F(), = 1, . . . , at the SU transmitter. By exploiting the property that thelogarithmic function is monotonic increasing for nonnegativevalues, we can obtain the optimum linear precoder matricesF(), = 1, . . . , , that minimize an upper bound on theaverage PEP by solving the following optimization problem(P1) as presented in (12)-(15). In the previous equations(12)-(15), I, is the interference power threshold specifiedby the th PU and P() is the transmit power available

for the th subcarrier at the SU transmitter. Equations (13)and (14) represent the average interference power constraintover all receive antennas for th PU receiver and the persubcarrier transmit power constraint at the SU transmitter,respectively. The third constraint implies that the matricesF(), = 1, . . . , , should be positive semi-definite. Notethat this linear precoder design problem is different from theprecoder design for conventional systems due to the additional

interference power constraints in (13). Therefore, the precoderobtained by standard multi-level water-filling is not optimalfor this problem. Based on the convexity of the optimizationproblem P1 and the structure of the optimal precoder, thefollowing theorem and the lemma can be stated.

Theorem 2: The linear precoder design problem P1 isconvex with constraints (13)-(15).

Proof: See the Appendix.Lemma: IfF(), = 1, . . . , is an optimal solution to the

problem P1, then the linear precoder F()U(), = 1, . . . , ,where U(), = 1, . . . , is a unitary matrix isalso optimal.

Proof: Since U(), = 1, . . . , is a unitary matrix andby insertion of F()U(), = 1, . . . , , into the objectivefunction (12) and the constraints (13)-(15) remain unchanged.Therefore, F()U(), = 1, . . . , , is also an optimal solu-tion to the problem P1.

Since the optimization problem P1 is convex, standardnumerical optimization techniques, e.g., the interior-pointmethod [20] can be employed to obtain the optimal linearprecoder. The details of this method are omitted for brevity.However, to get more insight into the system performance,Lagrange dual-decomposition-based algorithm is proposed inthe next section to obtain the set of linear precoder matrices.

IV. PRECODER DESIGNS FOR CR IN CORRELATED MIMOCHANNEL

Standard convex optimization techniques are exploited toderive efficient algorithms to obtain the optimal precodermatrices for SUs different transmit and receive correlationscenarios. This section will present the solution to the pre-coder design problem P1 using eigen-beamforming. In thisscenario, the linear precoder F() functions as a multi-modebeamformer based on the knowledge of the transmit andreceive correlation matrices of the MIMO channels betweenthe SU transmitter and SU receiver and between the SUtransmitter and PU receivers. The optimal precoder F() hasits orthogonal beams directions with the left eigenvectors

minimize

F(), =1,...,

=1

=1

log det[

I + ,() F(),()]

, (12)

subject to

=1

F()R,()

I,, = 1, . . . , , (13)

F()

P(), = 1, . . . , , (14)

F() 0, = 1, . . . , (15)


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of the SUs transmit correlation matrix R,(), and thepower loading across the beams as the square values of theeigenvalues of the matrix F(). Thus, by taking eigenvalue de-composition ofF() = UF()F()UF()

, choosing optimaleigen beam directions to be UF() = U,(), and usingthe properties of eigenvalue, the optimization problem (P2)in eigen-beamforming can be rewritten as in (16)-(19). Inequations (16)-(19), ,(), F() are the th eigenvalues

of,() and F(), respectively; and u,() is theth eigenvector of the U,(). Next, the Lagrange dual-decomposition method is applied to obtain the optimal powerallocation across each antenna and for each subcarrier indifferent SUs transmit and receive antenna correlation sce-narios. In previous studies, the Lagrangian dual-decompositionmethod has been extensively exploited for resource allocationin communication systems [21], [22].

A. Precoder Design With the SUs Both Transmit and Receive

Correlation

The general scenario, considering both SU transmit andreceive antenna correlation is presented. This is the commonsituation encountered in the uplink of a communication link,where a multi-antenna access point is located high above themulti-antenna subscriber transmit units. Since the access pointis installed at a high location, the receive antennas are having asmall spread of angle of arrival due to the less scatters aroundit. Therefore, causing a high receive antenna correlation at thereceive side. In the case of multi-antenna subscriber transmitunit, the possible causes of transmit correlation are lackof spacing between antennas, antenna arrangement and theantenna configurations. It can be noticed from the optimization

problem P2 that the SUs transmit and receive antenna cor-relations have different effects on the linear precoder designfor CR. Particularly, the objective function in (16) dependson the SUs both transmit and receive correlation matricesR,() and R,() through the eigenvalues ,(),and ,() , respectively. Furthermore, the interferencepower constraint depends on the transmit correlation matricesR,().

The Lagrange dual-decomposition method is applied hereto obtain the optimal values of the

F(). The optimal

power allocation policy for the th subcarrier and the optimalprecoder matrix, F(), can be obtained according to the

following theorem.

Theorem 4: The optimal power allocation for the thtransmit antenna and for the th subcarrier,

F()+ , =

1, . . . , , with + = max{0, } can be obtained by solvingthe following set of equations:

=1

,()

,()

1 + ,()

F

()

=

+

=1

(

u,() R,()u,()

), = 1, . . . , ,

(20)

where and are the non-negative Lagrange multipliersassociated with per subcarrier power constraint and the inter-ference power constraint, respectively. Therefore, the optimumlinear precoder matrix F() for the th subcarrier can beobtained as F() = U,()F().

Proof: See the Appendix.It can be noticed from equation (20) that for this scenario,

the optimal power loading across the th transmit antenna and

for th subcarrier depends on the eigenvalues of the SUstransmit and receive correlation matrices, eigenvectors of theSUs transmit correlation matrix and the transmit correlationmatrices of the MIMO channels between the SU transmitterand the PU receivers. Furthermore, the interference powerintroduced at the PU receivers are controlled by and ,which are calculated based on the constraints (17) and (18).

Normally, the cross-correlation between pairs of antennasis much smaller than one and, as a result for a well behavedreceive correlation matrix, the values of ,() are closeto each other and can be approximated as ,() =1 {,()}. In this scenario, the optimal solutionF()

+ , = 1, . . . , , for the th subcarrier can be obtained

from the following closed-form solution

F() =

( +

=1

u,()

R,()u,()

)

( {,()})1

+, = 1, . . . , .

(21)

To summarize, the complete algorithm for precoder designwith both transmit and receive correlation is given below.

Note that the solution derived for the linear precoderproblem in this paper is for a general scenario. The solution

for conventional MIMO systems can be straightforwardly

minimize{F()}

=1

=1

=1

log

1 + ,() ,()F()

, (16)

subject to

=1

=1

(

F()u,() R,()u,()

) I,, = 1, . . . , , (17)

=1

F() P(), = 1, . . . , , (18)

F

() 0, = 1, . . . , , = 1, . . . , (19)


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Algorithm 1 Computation of optimum precoder matrix

1: Given (0) , (0) , initial ellipsoids centered at (0), (0),

which contain the optimal dual solutions and ,respectively.

2: Set = 0.3: repeat

4: For each subcarrier = 1, . . . , obtain the optimalpower allocation

F

(), = 1, . . . , by solving(20).

5: Update (+1) , (+1) using

() ,

() and the sub-

gradients of() , = 1, . . . , , and () , = 1, . . . , ,

respectively [23].6: Set (+1) and (+1) as the centers of new ellipsoids

(+1) and

(+1) , respectively.

7: Set + 1.8: until the stopping criteria of the ellipsoid method is

satisfied [23].

obtained by simply setting the Lagrange multipliers corre-sponding to average interference power constraints to zero(i.e., = 0). Furthermore, the linear precoder design forthe single carrier systems in a CR system can be obtainedwith = 1. In addition, in the case of a single PU, thereis only one Lagrange multiplier for the interference powerconstraint. In such a scenario, this Algorithm 1 can be furthersimplified and the can be updated by the bisection method[20].

B. Precoder Design With Only the SU Transmit Correlation

The linear precoder design considering only the SU transmitantenna correlation is presented in this subsection. In practice,this situation is encountered in a downlink of a communicationsystem when the multi-antenna subscriber units have sufficientantenna spacing between them. In this scenario, the multi-antenna base station is situated high above the ground in alow scattering environment and thus, results a high transmitantenna correlation. The multi-antenna subscriber unit is situ-ated in rich scattering environment and with sufficient spacingbetween antennas, thus no receive antenna correlation. For thisscenario, the SU receive correlation matrix is equivalent to

the identity matrix, i.e., R,() = I , or equivalently,() = I . Therefore, using the properties of thedeterminant, the Kronecker product, and applying some math-ematical manipulations to P1, the optimization problem (P4)

in eigen-beamforming for this scenario can be written as

minimize{F()}

=1

=1

log

1 + ,()F()

,

subject to (17), (18), and (19). (22)

It can be seen that the problem P4 has a similar structureas problem P2. Therefore, the Lagrange dual-decompositionmethod can be exploited to solve problem P4 and derive anefficient algorithm to obtain the optimal precoder matricesF() for = 1, . . . , . A similar analysis can be performed asdescribed in Section IV.A. However, the analysis is presentedto obtain the optimal precoder matrix for a OFDM subcarrieras described in subproblem P3, in order to obtain the dualfunction (,), for given and . Therefore, subproblemP5 for the th OFDM subcarrier can be written as in (23).The optimal power allocation,

F(), = 1, . . . , , for this

scenario can be obtained as in the subproblem P3, by applyingthe KKT condition for the convex subproblem P5. Therefore,the optimal power allocation,

F(), = 1, . . . , , for

the th subcarrier with no receive correlation is given by thefollowing closed-form water-filling like solution for given and :

F() =

(

+=1

u,()

R,()u,()

)1

(,())1

+, = 1, . . . , .

(24)

Algorithm 1 proposed in Section IV.A to obtain the precoder

matrices F() for = 1, . . . , now needs to be slightly mod-ified for this scenario. In Step 4, the optimal solution

F(),

= 1, . . . , , for each OFDM subcarrier = 1, . . . , ,can now be obtained by (24). Therefore, the main advantageof this method in multicarier transmission scheme is that thesame computational routine can be simultaneously applied toall subcarriers in order to obtain the optimal F()s. Thus,the overall computational time will be maintained regardless ofthe number of subcarriers . This is a significant advantage inCR networks since the number of subcarriers are dynamicallyassigned and the overall computational time of the linearprecoder design algorithm will remain approximately the same

as the single carrier systems. Furthermore, for this scenario theconvergence time of the overall algorithm is further improveddue to the closed-form solution on

F(), = 1, . . . , in

(24).

minimizeF(), =1,...,

=1

log

1 + ,()F()

+

=1

=1

(

F()u,() R,()u,()

)

+

=1

F(), (23)

subject to F

() 0, = 1, . . . ,


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C. Precoder Design With Only the SU Receive Correlation

In this subsection, the linear precoder design consideringthe SU receive correlation and no SU transmit correlation isexplored. This situation is encountered in an uplink of a com-munication system when the multi-antenna subscriber units arelocated in a rich scattering environment with sufficient antennaspacing. There are two possible cases involving the receive-

side correlation, i.e., either R,() = I or R,() =R,() = I , = 1, . . . , . The latter is consideredto be the case when only the SUs receive-side correlationis considered, with no transmit side correlation. The formercase is considered first. For this scenario, R,() = I , orequivalently ,() = I . Therefore, using the propertiesof the determinant, the Kronecker product, and some trivialcalculations to problem P1, the optimization problem (P6) ineigen-beamforming for this scenario can be written as

minimize{

F

()}

=1

=1

=1

log 1 + ,()F(),subject to (17), (18), and (19). (25)

It is apparent from problem P6 that it has a similar structure toproblems P2 and P4. Therefore, as previously discussed, theLagrange dual-decomposition method can also be exploitedfor this scenario, to solve and derive an efficient algorithm toobtain the precoder matrices F(), = 1, . . . , . However,only the procedure to obtain the precoder matrix F() for aOFDM subcarrier, similar to the subproblem P3, is exploredhere in order to obtain the dual function (,), for given and . Therefore, the subproblem (P7) for the th subcarrier

in this scenario can be written as in (26). Applying theLagrangian multiplier method to subproblem P7, one can findthe optimal

F(), = 1, . . . , for the th subcarrier by

solving the following system of equations:

=1

,()

1 + ,()F

()1

=

+

=1

(

u,() R,()u,()

), = 1, . . . , .

(27)

Similarly, the optimal F

(), = 1, . . . , for the thsubcarrier for the latter case where, R,() = R,() =I , = 1, . . . , , can be obtained by solving the following

system of equations:

=1

,()

1 + ,()F

()1

=

+

=1

, = 1, . . . , .

(28)

It can be noticed from the equations (27) and (28) that theoptimal matrix F() for the th subcarrier has equal diagonalelements

F(), = 1, . . . , for given and . Since this

is the uncorrelated scenario and as CSI is not available at theSU transmitter, an equal diagonal precoder is expected. Thealgorithm proposed in Section IV.A, to obtain the precodermatrices F(), = 1, . . . , now needs to be modified forthese scenarios. Step 4 ofAlgorithm 1 can now be obtained by(27) and (28), for the cases R,() = I and R,() =R,() = I , = 1, . . . , , respectively.

V. NUMERICAL RESULTS

In this section, numerical results are presented to illustratethe performance of the proposed linear precoder in an OSTBCMIMO-OFDM based CR system. Throughout the simulations,the bit error rate (BER) is used as the performance measure.

In all the simulations, quasi-static frequency-flat correlatedRayleigh fading MIMO channels and zero-mean uncorrelatedCSCG noise with variance 2 are assumed. The elementsof the MIMO channel matrices H,() and H,1(),for = 1, . . . , , are generated as i.i.d. samples ofCSCG distribution with zero-mean and unit variance.Furthermore, it is assumed that the transmit and receiveantennas form linear arrays for both SUs and PUs. The

correlation coefficient between the th and the th transmitantennas for the th subcarrier with a small angle spreadcan be approximately obtained as [7]: [R,()], 2,

20 exp

2 sin,(),1 sin

= 2,0(2 sin,(),1 ), = {,},

= 1, . . . , , = 1, . . . , . Here, 0() is the zeroth-orderBessel function of the first kind, ,() is the transmitangle spread for the th subcarrier, , is the spacingbetween the transmit antennas, is the wavelength of thecarrier and 2, is the transmit antenna array power gain. The,(), = {,}, for each subcarrier is generated froma uniform distribution in the range [, /2, , + /2]

with ,, = {,} as the mean transmit angle spreadand = 60. The receive correlation matrices R,() andR,(), = 1, . . . , , = 1, . . . , are also obtainedsimilarly as the transmit correlation matrices. Furthermore,

minimizeF(), =1,...,

=1

=1

log

1 + ,()F()

+

=1

=1

(

F()u,() R,()u,()

)

+

=1

F(), (26)

subject to F

() 0, = 1, . . . ,


8/13


0 2 4 6 8 10 12 14 1510

4

103

102

101

100

SNR (dB)

BER

2 1, No Precoding

2 1, Precoding

2 2, No Precoding

2 2, Precoding

4 2, No Precoding

4 2, Precoding

tx,ss = 300

Iint = 100 mW

Fig. 3. The average BER versus SNR for the CR transmission scheme withand without precoding.

0 2 4 6 8 10 12 14 1510

4

103

102

101

100

SNR (dB)

BER

Iint = 100 mWIint = 200 mW

Iint = 300 mW

Iint = 400 mW

tx,ss = tx,spl = 200

Fig. 4. The BER performance of the precoded OSTBC MIMO-OFDM basedCR transmission scheme with different interference power thresholds.

antennas at transmitter and receiver are assumed to beuniformly spaced with half wavelength distance betweenthem for both SUs and PUs. Two PUs are assumed to bepresent in the system. The number of subcarriers is set to 16.The number of receive antennas for both PUs and SU areeither set to 1 and 2. The total available transmit power for asubcarrier at the SU transmitter is

1/W. The PU receiver

interference temperature power threshold is set to 100 mW[13]. Binary phase shift keying symbols are generated withunit variant signal constellation for signal transmission. TheAlamouti code was exploited as the OSTBC for the SUtransmission [5]. Simulation results are obtained by averaging1000 trials, with each block consisting of 10000 OSTBCOFDM symbols. Finally, the signal to noise ratio (SNR) isdefined as the signal power to noise power at the SU receiver.

The average BER of the CR system is plotted versus SNRin Fig. 3 for proposed precoded system and with non-precodedsystem. In order to have a fair compression, the transmitpower is uniformly loaded in non-precoding such that theinterference power constraint satisfied. In this scenario, thetransmit power from the th transmit antenna is obtained

0 2 4 6 8 10 12 14 1516

14

12

10

8

6

4

2

SNR (dB)

ReceivedInterfe

rencePowerataPU(

dB)

Iint = 100 mW

Iint = 200 mW

Iint = 300 mW

Iint = 400 mW


Fig. 5. The average received interference power versus SNR for differentinterference power thresholds.

by F,u() = I,

1 (u,()

R,()u,()),

= 1, . . . , , = 1, . . . , . This will ensure in uniformpower loading that the total interference introduced is belowthe prescribed threshold value. The BER results are presentedfor 2 1, 2 2 and 4 2 SU antenna configurations witha 100 mW interference power threshold. From the plots, itcan be seen that the proposed precoder in CR outperformsthe non-precoding scheme. The precoding gain is around 1.2dB for the presented SNR range. Similar performance resultsare achieved for all three antenna configurations. Furthermore,it is noticed that significant improvement in performance byadding more antennas to the SU receiver with the same inter-ference power constraint. Thus, employing multiple antennas

at the CR transmitter and receiver, system error performancecan be improved significantly for a fixed interference powerthreshold.

In Fig. 4, the average BER is plotted against SNR for the2 2 CR system with different interference power thresholds.As expected, BER performance improvement can be seenwith increased interference power thresholds. The increaseof the interference power limit allows the CR transmitterto allocate higher power to the precoder matrices. However,interference power cannot be increased significantly, since itcould increase an unacceptable interference receive at the PUreceivers. In Fig. 5, the average received interference power

at a PU receiver is plotted against the SNR for differentinterference power thresholds. It is apparent from Fig. 5 thatas the interference power threshold increases the amount ofreceived interference also increases. In addition, it can be seenfrom Fig. 5 that for all the cases, the received interferencepower at the PU is below or equal to the respective interferencepower thresholds, thus allowing CR and PU to coexist in thesame frequency band.

The average received interference power at a PU receiver isplotted against SNR in Fig. 6 for different number of subcar-riers. From Fig. 6 it is apparent that for all the subcarriers thereceived interference power at the PU remain below or equal tothe interference power threshold. Furthermore, as the numberof subcarriers increases, the received interference power at thePU receiver reach the interference power threshold at a higher


9/13


0 2 4 6 8 10 12 14 158

7

6

5

4

3

2

SNR (dB)

ReceivedInterfe

rencePowerataPU(

dB)

K = 8K = 1 2K = 1 6


Fig. 6. The average received interference power versus SNR for differentnumber of subcarriers.

0 2 4 6 8 10 12 14 15103

102

101

100

SNR (dB)

BER

tx,ss = tx,spl = 2 00

tx,ss = tx,spl = 2 00 and rx,ss = 3 0

0

Fig. 7. The average BER performance of a CR system for SUs transmitcorrelation and both transmit and receive correlation scenarios.

SNR value. In Fig. 7, the average BER is plotted against SNRfor a 22 CR system for SU transmit and receive correlations,and for only SU transmit correlation, at a 100 mW interferencepower threshold. Comparing the plots in Fig. 7 clearly showsthat the precoding gain decreases due to SU receive antennacorrelation. Fig. 8 depicts the average BER versus SNR fora 2 2 CR system with different , for a 100 mW

interference power threshold. As expected, BER performanceimproves significantly with decreased , at higher SNRregimes.

VI. CONCLUSION

This paper presents a design of a linear precoder for orthog-onal space-time block coded orthogonal frequency divisionmultiplexing-based multiple-input multiple-output (MIMO)antenna cognitive radio (CR). The CR coexists with theprimary user (PU) network by opportunistically sharing theoriginally allocated PU spectrum in correlated Rayleigh fadingchannels. The optimum linear precoder with respect to errorprobability performance is obtained by exploiting the partialchannel information in the form of transmit and receive

0 2 4 6 8 10 12 14 1510

3

102

101

100

SNR (dB)

BER




Iint = 100 mW

Fig. 8. The average BER versus SNR of the precoded OSTBC MIMO-OFDM based CR transmission scheme with different ,.

correlation matrices at the CR transmitter and by minimizing

an upper bound on the average pairwise error probability,while imposing a set of per subcarrier transmit power con-straints at the CR transmitter and a set of interference powerconstraints specified by the PUs. We have shown that theprecoder design problem with these constraints is convex andhave proposed a Lagrange dual-decomposition-based efficientalgorithm to obtain the optimal precoders. Furthermore, theindividual effects of the transmit and/or receive correlationon the linear precoder design for CR were investigated. Thecurrent study further reveals that for uncorrelated CR transmitantennas, the precoder consists of equally weighted diagonalelements. Simulation results illustrate that the proposed linear

precoder outperforms uniform power loaded systems in acorrelated MIMO channel even with the presence of additionalinterference power constraints.

APPENDIX

A. Proof of Theorem 1

An upper bound of the average PEP of OSTBC MIMO-OFDM, when the SUs transmit and receive antennas arecorrelated is derived here. Applying the Chernoff bound to(8), a tight upper bound on the PEP conditioned on channelmatrices H(), = 1, . . . , , can be obtained as [24]

(C C H(1), . . . , H()) 1

2exp

d2(CC)

2 , (29)

where

d2(C C ) =

=1

1

22

H()F()(C() C ())2 ,is the Euclidian distance between two codeword matrices. Dueto the orthogonality of the codeword error product matrix,E() = C() C (), OSTBC has the appealing propertythat E()E() = ()I , = 1, . . . , , where ()is the codeword distance that depends on the two code-words C() and C (). Therefore, exploiting these propertiesand the standard relations, {AB} = vec(A)vec(B) and


10/13


vec(ABC) = (C A)vec(B) for any arbitrary matrices A, B,and C [25], the conditional PEP of OSTBC can be written as

(C C H(1), . . . , H())

1

2exp

=1

()

42vec(H()

)(IF()F())vec(H()

).

(30)

An upper bound on the average PEP of OSTBC can then

be obtained by taking the expectation of (30) with respect toH(), = 1, . . . , , as

(C C )

exp

(1)

42h(1)(IF(1)F(1)

)h(1)(h(1))h(1) . . .

exp

()

42h()(IF()F()

)h()(h())h(),

(31)

where h() = vec(H()), = 1, . . . , . Note that theprobability distribution function (pdf) of the MIMO channelfor the th subcarrier is a complex Gaussian and can bewritten as (h()) = 1

det[R()]exph

()R

1 ()h(),

where R() = R,() R,(), = 1, . . . , , is thecovariance matrix of h(). Therefore, an upper bound on theaverage PEP can be rewritten as in (32).

The integrals in (32) can be easily solved by making useof the fact that 1 det[1()]

exph()

()h() h(),

where () = ()42 ((I F()F()) + R

1

()), =1, . . . , , is the integral of a complex Gaussian pdf and thusequals one. Therefore, an upper bound on the average PEP ofthe OSTBC MIMO-OFDM can be written as in (33). It can

be noticed from (33) that the dependence on the codewordpair is now only through the codeword distance ()s.Furthermore, we can notice that the (33) is a decreasingfunction of ()s and PEP is dominated by the codewordpairs corresponding to minimum (). Consequently, onlyone such pair is considered in the linear precoder optimiza-tion procedure, i.e., = min(),= { ()I =(C() C ()(C() C ())

}, = 1, . . . , . The depends on several factors, such as the modulationformat of the input signal constellation, the variance of theinput signal constellation and the OSTBC generator matrix[4]. Therefore, using this property of OSTBC, the equal-

ity (A B)(C D) = (AC) (BD), and the eigen-decomposition of R,() = U,(),()U,(),

R,() = U,(),()U,(), = 1, . . . , ,an upper bound on the average PEP in (33) can be rewrittenas (34), where = 42 ; F() = F()F()

; and U,(),U,() are the matrices of the eigenvectors; and ,(),,() are the diagonal eigenvalue matrices of the SUstransmit and receive correlation matrices, respectively. Usingthe properties of Kroneker product, one can easily writedet[()] = det [I + ,() F(),()]as in (35). In equation (35), () = F(),().

Furthermore, by using the property that det

A 0

C D

=

det[A]det[D], an upper bound on the PEP can be furtherwritten as in (9).

B. Proof of Theorem 2

The convexity of the linear precoder design problem P1given in (12)-(15), for OSTB MIMO-OFDM based CR isproven here. First, we define the function (()), forsubcarrier as (()) = logdet[()], = 1, . . . , ,

where

() = I + ,()F()

,(). One canprove the convexity of (12) by showing that (()) is

convex over the set of positive definite matrices. Thus, theconvexity of(()) is proven using the following theorem[26].

Theorem 3 [26]: If1 is Hermitian and 2 ispositive definite, then there exists a nonsingular matrix A such that A2A = I, and A

1A = D, where D is a

diagonal matrix with all diagonal elements, > 0.The function (()) is convex, if (1() + (1

)2()) (1()) + (1 )(2()) for any twopositive definite Hermitian matrices in and for any 0 1. Equality holds when 1() = 2(). Using theTheorem 3 and the properties of the logarithm, one canshow that (()) is convex, if (I + (1 )D()) (1 )(D()) for all 0 1 and for any diagonalmatrix D() with positive diagonal entities. This can be easilyproven by using the properties of the determinant and the strictconcavity of the logarithmic function itself as

(I + (1 )D()) =

=1

log( + (1 )())

(1 )log

=1

()

(1 )(D()).(36)

(C C ) 1

det[R()]

exp

h(1)((

I(1)

42F(1)F(1)

)+R1 (1)

)h(1)

h(1) . . .

1

det[R()]

exp

h()((

I()

42F()F()

)+R1 ()

)h()

h()

(32)

(C C )

=1

det

I +

()

42

I F()F()

(R,() R,())1

(33)


11/13


(C C )

=1

det

[I + ,()

1/2,()

F()1/2,()

]1(34)

det[()] = det

I + ,()1() 0...

. . ....

0 I + ,()()

(35)

Thus, (()) is convex over the set . Furthermore, usingthe property that the sum of convex functions is also a convexfunction [20], one can easily show that (12) is convex. Inaddition, it can be easily verified that the constraints (13)-(15)are convex [20]. Therefore the entire optimization problem P1is convex.

C. Proof of Theorem 4

The proof of the Theorem 4 is presented here. First, thenon-negative Lagrange multipliers = [1, . . . , ] and =[1, . . . , ]

associated with the average interference powerconstraints and per subcarrier power constraints in (17) and(18) are introduced and the Lagrangian of the primal problemis written as [20], [22]

{F

()},,

=

=1

=1

=1

log

1 + ,() ,()F()

+

=1

=1

F() P()+

=1

=1

=1

(

F()u,() R,()u,()

)

I,

,

(37)

where {F()} = {F(), = 1, . . . , , = 1, . . . , }is the set of eigenvalues of the eigenvalue matrices F(), =1 . . . , . The Lagrange dual function is then defined as [20],[22]

(, ) =minimize{F()}

{F()},,

, (38)

subject to

F() 0, = 1, . . . , ,

= 1, . . . , .

Therefore, the Lagrange dual problem of the primal problem isdefined as maximize0, 0 (,) [20], [22]. The optimalvalue of the dual problem is achieved by the optimal dualvariables and . Furthermore, the dual function (,)provides a lower bound of the optimal value, , of the primalproblem. Since the original problem P1 is indeed convex andalso satisfies Slaters condition, the duality gap ( )is zero [20], [22]. Therefore, these interesting results suggestthat the optimal solution, , for the primal problem can be

obtained by first minimizing the Lagrangian ({F()},,)in order to obtain the dual function (,) for some givendual variables , , and then maximizing (,) over allnon-negative values of and .

Therefore, consider the minimization of ({F

()},,)with respect to the variables {F()} and for some given fixedvalues of, to obtain the dual function (,). One cannotice from (37) and (38) that the dual function (,) hasthe following form:

(,) =

=1

(,)

=1

P()

=1

I,, (39)

where (,) is given as in (40).Therefore, it is interesting to note that the dual func-

tion (,) can be obtained by solving independentsubproblems (P3)s, (,), each for OFDM subcarrier = 1, . . . , . This implies that the same computationalroutine can be repeatedly applied for solving each subproblemP3. Thus, the convergence time of the overall algorithm canbe dramatically improved. Now it is required to find theoptimal solution for a subproblem P3. Since the subproblemP3 is convex, the globally optimal solution can be found bysolving the system of Karush-Kuhn-Tucker (KKT) conditions[20]. Therefore, the Lagrangian of the subproblem P3 can beformulated as

,

F(), ()

=

=1

=1

log(1 + ,() ,()F())

+

=1

=1

(

F()u,() R,()u,()

)

+

=1

F()

=1

()F(),

(41)

where () are the Lagrange multipliers associated with theinequality constraints in (40). Then, the KKT conditions forthe subproblem P3 can be written as in (42)-(44). In equations(42)-(44),

F() represents the optimal solution. Therefore,

the optimal solution F

()+ , = 1, . . . , , with + =

max{0, } for the th subcarrier can be obtained by solvingthe set of equations as given in (20).

Once the dual function (,), is obtained for given and , the next step of the Lagrange dual-decompositionis to maximize the dual function (,), over all possi-ble values of and . Searching for the optimal value


12/13


(,) =minimize

F(), =1,...,

=1

=1

log

1 + ,() ,()F()

+

=1

F()

+

=1

=1

(F()u,()

R,()u,()

),

subject to F() 0, = 1, . . . , (40)

F

(),

F(), ()

= =1

,() ,()1 + ,() ,()

F

() + ()

+

=1

(u,()

R,()u,()

)= 0, = 1, . . . , , (42)

()F

() = 0, = 1, . . . , , (43)

() 0, = 1, . . . , (44)

of () and () can be done by using, for exam-ple, the ellipsoid method [23], which exploits the factthat

=1 (F()u,()

R,()u,()) I, and

=1 F() P() are sub-gradients of , = 1, . . . , and , = 1, . . . , , respectively.

ACKNOWLEDGMENT

This research is supported by the Natural Science andEngineering Research Council of Canada (NSERC) under astrategic project grant. The authors like to acknowledge the

valuable feedback given by Dr. Zouheir Rezki.REFERENCES

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[9] H. Sampath and A. Paultry, Linear precoding for space-time codedsystems with known fading correlations, IEEE Commun. Lett., vol. 6,no. 6, pp. 239241, June 2002.

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[13] M. H. Islam and Y. C. Liang, Space-time coding in MIMO cognitivenetworks with known channel correlations, in Proc. IEEE EuWiT, Oct.2008, pp. 97102.

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[15] K. Yu, M. Bengtsson, B. Ottersten, D. McNamara, P. Karlsson, andM. Beach, Modeling of wide-band MIMO radio channels based onNLoS indoor measurements, IEEE Trans. Veh. Technol., vol. 53, no. 3,pp. 655665, May 2004.

[16] R. Stridh, K. Yu, B. Ottersten, and P. Karlsson, MIMO channel capacity

and modeling issues on a measured indoor radio channel at 5.8 GHz,IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 895903, May 2005.

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[18] G. Bansal, M. J. Hossain, and V. K. Bhargava, Optimal and suboptimalpower allocation schemes for OFDM-based cognitive radio systems,

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[23] R. G. Bland, D. Goldfarb, and M. J. Todd, The ellipsoid method: asurvey, Operations Research, vol. 29, no. 6, pp. 10391091, 1981.

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Anjana Punchihewa received the B.Sc. degreein electrical and electronics engineering from theUniversity of Peradeniya, Peradeniya, Sri Lanka,in 2004 and the M.Eng. degree in electrical en-gineering from Memorial University of Newfound-land, Canada, in 2007. He is currently pursuing thePh.D. degree with the Department of Electrical andComputer Engineering at the University of BritishColumbia, Canada. His research interests lie inthe areas of wireless communications and signal

processing, with particular emphasis on space-timecoding, cognitive radio, cooperative network, blind parameter estimation, andmodulation classification.

Vijay Bhargava an IEEE volunteer for threedecades, is a professor in the Department of Elec-trical and Computer Engineering at the Universityof British Columbia in Vancouver, where he servedas department head from 2003-2008. Previously, hewas with the University of Victoria (1984-2003)and Concordia University (1976-84). He received hisPh.D. from Queens University. As a distinguishedspeaker for the IEEE Communications Society andthe IEEE Information Theory Society, and as asenior level IEEE volunteer, he has lectured in 66

countries and assisted IEEE Presidents in negotiating sister society agreementsin India, Japan, and Russia. He has rudimentary knowledge of severallanguages and is an eager student of different cultures and societies.

Vijay served as the founder and president of Binary Communications Inc.(1983-2000). He has provided consulting services to several companies andgovernment agencies. He is a co-author (with D. Haccoun, R. Matyas, andP. Nuspl) ofDigital Communications by Satellite (New York: Wiley: 1981),which was translated into Chinese and Japanese. He is a co-editor (with S.Wicker) ofReed Solomon Codes and their Applications (IEEE Press: 1994);a co-editor (with V. Poor, V. Tarokh, and S. Yoon) ofCommunications, Infor-mation and Network Security (Kluwer: 2003); a co-editor (with E. Hossain) ofCognitive Wireless Communication Networks (Springer: 2007); and a co-editor(with E. Hossain and D. I. Kim) ofCooperative Wireless Communications

Networks, a forthcoming Cambridge University Press Publication.Vijay has served on the Board of Governors of the IEEE Information

Theory Society and the IEEE Communications Society. He has held importantpositions in these societies and has organized conferences such as ISIT95,ICC99, and VTC 2002 Fall. He has served as an editor of the IE EE

TRANSACTIONS ON COMMUNICATIONS. He played a major role in thecreation of the IEEE Wireless Communications and Networking Conference(WCNC) and IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, forwhich he served as editor-in-chief during 2007, 2008, and 2009. In 2010,he was appointed for a two year term as the IEEE Communications SocietyDirector of Journals. He is a past President of the IEEE Information TheorySociety. Vijay Bhargava has been elected to serve as IEEE CommunicationsSociety President-Elect during 2011 and as President during 2012 and 2013.

Charles Despins received a bachelors degree inelectrical engineering from McGill University, Mon-tral, QC, Canada, in 1984, and masters and Ph.D.degrees from Carleton University, Ottawa, ON,Canada, in 1987 and 1991, respectively.

He was, from 1992 to 1996, a Faculty Memberof the Institut National de la Recherche Scientifique(INRS), Universit du Qubec, Montral, Canada,following employment in 1984-1985 with CAEElectronics as a Member of the Technical Staff,

and in 1991-1992 with the Department of Electricaland Computer Engineering, cole Polytechnique de Montral, Canada, as aLecturer and a Research Engineer. From 1996 to 1998, he was with MicrocellTelecommunications Inc., a Canadian GSM operator, and was responsible forindustry standard and operator working groups, as well as for technology trialsand technical support for joint venture deployments in China and India. From1998 to 2003, he was Vice-President & Chief Technology Officer of BellNordiq Group Inc., a wireless and wireline network operator in northern andrural areas of Canada. Since 2003, he has been President and CEO of PromptInc., a university-industry research consortium in the field of informationand communications technologies. He remains a faculty member (on leave)at cole de Technologie Suprieure (Universit du Qubec), Montral, QC,Canada, with research interests in wireless communications. He is also a guestlecturer in the Master of Business Administration (MBA) program at McGillUniversity, Montreal, QC, Canada.

Dr. Despins was awarded the IEEE Vehicular Technology Society BestPaper of the Year prize in 1993, as well as the Outstanding Engineer Awardin 2006 from IEEE Canada. He is a Member of the Order of Engineers ofQubec and is also a Fellow (2005) of the Engineering Institute of Canada.

linear precoding for orthogonal space-time

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