Dynamic Resource Allocation in Cognitive Radio Networks: A Convex Optimization Perspective

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by  Rui Zhang

1
Dynamic Resource Allocation in Cognitive Radio
Networks: A Convex Optimization Perspective
Rui Zhang, Ying-Chang Liang, and Shuguang Cui
Abstract—This article provides an overview of the state-of- Opportunistic Spectrum Access (OSA) vs. Spectrum Sharing
art results on communication resource allocation over space, (SS).Inthe OSA model,the SUsareallowedto transmitover
time, and frequency for emerging cognitive radio (CR) wire-
the band of interest when all the PUs are not transmitting
less networks. Focusing on the interference-power/interference-
at this band. One essential enabling technique for OSA-
temperature (IT) constraint approach for CRs to protect pri-
0
mary radio transmissions, many new and challenging problems based CRs is spectrum sensing, where the CRs individually
1
regarding the design of CR systems are formulated, and some or collaboratively detect active PU transmissions over the
0
2 of the corresponding solutions are shown to be obtainable by band, and decide to transmit if the sensing results indicate
restructuringsomeclassicresultsknownfortraditional(non-CR)
that all the PU transmitters are inactive at this band with
n wireless networks. It is demonstrated that convex optimization
a high probability. Spectrum sensing is now a very active
a plays an essential role in solving these problems, in a both
J rigorous and efficient way. Promising research directions on area for research; the interested readers may refer to, e.g.,

9 interference management for CR and other related multiuser [4], [5], [6], [7] for an overview of the state-of-art results in
1 communication systems are discussed. this area. As a counterpart, the SS model allows the SUs to
transmit simultaneously with PUs at the same band even if
]
T they are active, provided that the SUs know how to control
I. INTRODUCTION
I their resultant interference at the PU receivers such that the
s. In recent years, cognitive radio (CR) networks, where CRs performancedegradationofeach active primarylink is within
c or the so-called secondary users (SUs) communicate over a tolerable margin.Thus, OSA and SS can be regardedas the
[
certain bandwidth originally allocated to a primary network, primary-transmitter-centric and primary-receiver-centric dy-

1 havedrawngreatresearchinterestsintheacademic,industrial, namicspectrumaccesstechniques,respectively.Consequently,
v and regulation communities. Accordingly, there is now a there will be an inevitable debate on which operation model,
7
rapidly growing awareness that CR technology will play an OSA or SS, is better to deploy CRs in practical systems;
8
essentialroleinenablingdynamicspectrumaccessforthenext however,arigorouscomparativestudyforthesetwomodels,in
1
3 generation wireless communications, which could hopefully terms of spectrum efficiency and implementation complexity
. resolve the spectrum scarcity vs. under-utilization dilemma tradeoffs, is still open. Generally speaking, SS utilizes the
1
caused by the current static spectrum management polices. spectrummoreefficientlythanOSA,sincetheformersupports
0
0 Specifically, the users in the primary network, or the so- concurrent PU and SU transmissions over the same band
1 calledprimaryusers(PUs),couldbe licensedusers, whohave while the latter only allows orthogonaltransmissionsbetween
v: the absolute right to access their spectrum bands, and yet them. Moreover,the receiver-centric approach for SS is more
i would be willing to share the spectrum with the unlicensed effective for CRs to manage the interference to the PU links
X
SUs. Alternatively, both the PUs and SUs could equally than the transmitter-centric approach for OSA.
r coexist in an unlicensed band, where the PUs are regarded Hence,theSSmodelforCRswillbefocusedinthisarticle.
a
asexistingactivecommunicationlinkswhilethe SUsare new It is worth noting that the optimal design approach for SS-
links to be added. A unique feature of CRs is that they are based CR networks should treat all coexisting PU and SU
able to identify and acquire useful environmentalinformation linksasagiantinterferencenetworkandjointlyoptimizetheir
(cognition) across the primary and secondary networks, and transmissions to maximize the SU network throughputwith a
thereby adapt their transmit strategies to achieve the best prescribedPU networkthroughputguarantee.Fromthisview-
performance while maintaining a required quality of service point,recentadvancesin networkinformationtheory[8] have
(QoS) for each coexisting active primary link. Depending provided promising guidelines to approach the fundamental
on the type of cognitive knowledge collected (e.g., on/off limits of such networks.However,froma practicalviewpoint,
statuses of primary links, PU messages, interference power thecentralizeddesignapproachforPUandSUnetworksisnot
levels at PU receivers, or primary link performance margins) desirable,sincePUandSUsystemsusuallybelongtodifferent
and the primary/secondary network models of interests (e.g., operators and thus it is difficult, if not infeasible, for them to
infrastructure-based vs. ad hoc), many new and challenging cooperate. Consequently, a decentralized design approach is
problemsonthe designofCR networkscanbe formulated,as morefavorable,wherethePUnetworkisdesignedwithoutthe
will be reviewed in this article. awareness of the existence of the SU network, while the SU
To date, quite a few operation models have been proposed network is designed with only partial knowledge (cognition)
forCRs;however,thereisnoconsensusyetontheterminology of the PU network.
used for the associated definitions [1], [2], [3]. Generally Followingthis(simplified)decentralizedapproach,thereare
speaking, there are two basic operation models for CRs: furthermoretwodesignparadigmsproposedforSS-basedCRs.
2
One is based on the “cognitive relay” concept [9], where S-BS
the SU transmitter allocates only part of its power to deliver
the SU messages, and uses the remaining power to relay
the PU messages so as to compensate for the additional SU
interference experienced at the PU receiver. However, this
technique requiresnon-causalknowledge of the PU messages
at the SU transmitter, which may be difficult to realize in
practice. In contrast, a more feasible SS design for the SU
(a) (b)
to protect the PU is to impose a constraint on the maximum
SU interference power at the PU receiver, also known as the
“interference temperature (IT)” constraint [10], by assuming Secondary terminal Primary terminal
that the SU-to-PU channels are either perfectly known at the
SU transmitters, or can be practically estimated. Fig.1. CRnetworks:(a)infrastructure-based; (b)adhoc.
In this article, we will focus our study on the IT-based SS
model for CRs, namely the IT-SS, as it is a more feasible II. CR NETWORK MODELS
approach compared with other existing ones. In a wireless
WeconsidertwogeneraltypesofCRnetworks,whichareof
communication environment, channels are usually subject to
both theoretical and practical interests: One is infrastructure-
space-time-frequency variation due to multipath propagation,
based, as shown in Fig. 1(a), where multiple secondary ter-
mobility, and location-dependent shadowing. Thus, dynamic
minals communicatewith a common secondarynode referred
resourceallocation(DRA)becomesanessentialtechniquefor
to as the secondary base station (S-BS); the other is ad hoc,
CRs to optimally deploytheir transmit strategies to maximize
as shown in Fig. 1(b), which consists of multiple distributed
the secondary network throughput, where the transmit power,
secondary links. In both types of CR networks, there are
bit-rate, bandwidth, and antenna beam should be dynamically
coexisting primary terminals operating in the same spectrum
allocated based upon the available channel state information
band. For the IT-SS modelof CRs, the exactoperationmodel
(CSI) of the primary and secondary networks. In particular,
oftheprimarynetworkisnotimportanttoourstudy,provided
this article will focus on DRA problem formulations unique
that all the secondary terminals satisfy the prescribed IT
toCRsystemsundertheIT-SSmodel,andtheassociatedsolu-
constraints to protect the primary terminals. Without loss of
tions that are non-obviousin comparisonwith existing results
generality,we assume that there are K secondarylinks and J
[11] known for the traditional (non-CR) wireless networks.
primary terminals in each type of the CR networks.
More importantly, we will emphasize the key role of various
Consider first the infrastructure-based secondary/CR net-
convex optimization techniques in solving these problems.
work with the S-BS coordinating all the CR transmissions,
Theremainderofthisarticleisorganizedasfollows.Section
which usually corresponds to one particular cell in a CR
IIpresentsdifferentmodelsoftheSUnetworkcoexistingwith
cellular network. The uplink transmissions from the SUs to
the PU network, and various forms of transmit power and
the S-BS are usually modeled by a multiple-access channel
interferencepower(IT)constraintsovertheSUtransmissions.
(MAC), while the downlink transmissions from the S-BS
SectionIIIisdevotedtothespatial-domaintransmitoptimiza-
to different SUs are modeled by a broadcast channel (BC).
tion at the SUs for different SU networks subject to transmit
For the MAC, the equivalent baseband transmission can be
and interference power constraints. Section IV extends the
represented as
results to the more generalcase of joint space-time-frequency
transmitoptimizationof the SUs, and addressesthe important K
y= H x +z (1)
issueonhowtooptimallysettheITthresholdsinCRsystems k k
to achieve the best spectrum sharing performance. Finally, kX=1
conclusions are drawn and future research directions are wherey∈CM×1denotesthereceivedsignalattheS-BS,with
discussed in Section V. M denoting the number of antennas at S-BS; Hk ∈ CM×Nk
Notation: Lower-case and upper-case bold letters denote denotes the channel from the kth SU to S-BS, k =1,··· ,K,
vectors and matrices, respectively. Rank(·), Tr(·), |·|, (·)−1, withN denotingthenumberofantennasatthekthSU;x ∈
k k
(·)H,and(·)1/2denotetherankofamatrix,trace,determinant, CNk×1 denotes the transmitted signal of the kth SU; and z∈
inverse, Hermitian transpose, and square-root, respectively. CM×1 denotes the noise received at S-BS. We assume that
I and 0 denote an identity matrix and an all-zero matrix, x ’s are independent over k.
k
respectively.Diag(a) denotesadiagonalmatrixwithdiagonal Similarly, the BC can be represented as
elements given in a. E(·) denotes the statistical expectation.
y =HHx+z , k =1,··· ,K (2)
A circularlysymmetriccomplexGaussian(CSCG) distributed k k k
random vector with zero mean and covariance matrix S is where yk ∈ CNk×1 denotes the received signal at the kth
denoted by CN(0,S). Cm×n denotes the space of m × n SU; for convenience, we have used the Hermitian transposed
complex matrices. k · k denotes the 2-norm of a complex uplinkchannelmatrixforthecorrespondingdownlinkchannel
vector. Re(·) and Im(·) denote the real and imaginary parts matrix,i.e.,HH denotesthechannelfromtheS-BStothekth
k
of a complex number, respectively.The base of the logarithm SU; x ∈ CM×1 denotes the transmitted signal from S-BS;
function log(·) is 2 by default. and zk ∈ CNk×1 denotes the receiver noise of the kth SU.
3
In the case that x carries informationcommonto all SUs, the • Peak transmit power constraint (PTPC):
associated downlink transmission is usually called multicast,
Tr(S [l])≤P (4)
while if x carries independent information for different SUs, k k
it is called unicast. whereS [l]denotesthetransmitcovariancematrixforthe
k
Next, consider the ad hoc secondary/CR network, which lthtransmitdimensionofthekthSU,l∈{1,··· ,L},k∈
is usually modeled as an interference channel (IC). For {1,··· ,K}; P denotes the kth SU’s peak power con-
k
convenience, we assume that for the kth secondary link, straint that applies to each of the L transmit dimensions.
k = 1,··· ,K, the transmitter is denoted as SU-TXk and • Average transmit power constraint (ATPC):
the receiver is denoted as SU-RX , although in general a
k
L
secondary terminal can be both a transmitter and a receiver. 1 Tr(S [l])≤P¯ (5)
The baseband transmission of the IC can be represented as L k k
l=1
X
K where P¯k denotes the kth SU’s average transmit power
y˜k =Hkkx˜k+ Hikx˜i+˜zk, k =1,··· ,K (3) constraint over the L transmit dimensions.
i=X1,i6=k • Peak interference power constraint (PIPC):
where y˜k ∈ CBk×1 denotes the received signal at SU-RXk, K Tr G [l]S [l]GH[l] ≤Γ (6)
with B denoting the number of receiving antennas; x˜ ∈ kj k kj j
k k
CAk×1 denotes the transmitted signal of SU-TXk, with Ak Xk=1 (cid:0) (cid:1)
denotingthenumberoftransmittingantennas;Hkk ∈CBk×Ak where Gkj[l] denotes the realization of channel Gkj for
denotes the direct-link channel from SU-TXk to SU-RXk, a given l; and Γj denotes the peak interference power
whileHik ∈CBk×Ai denotesthecross-linkchannelfromSU- constraint for protecting the jth PU, j ∈ {1,··· ,J},
TXi to SU-RXk, i6=k; and˜zk ∈CBk×1 denotesthe noise at which limits the total interference power caused by all
SU-RX . It is assumed that x˜ ’s are independentover k. the K SUs across all the receiving antennas of the jth
k k
PU, for each of the L transmit dimensions.
Furthermore, we assume that the jth PU, j = 1,··· ,J, in
each type of the CR networks is equipped with D antennas, • Average interference power constraint (AIPC):
j
Dj ≥ 1. We then use Gkj ∈ CDj×Nk to denote the channel 1 L K
fromthekthSUtothejthPUintheCRMAC,Fj ∈CDj×M L Tr Gkj[l]Sk[l]GHkj[l] ≤Γ¯j (7)
to denotethe channelfromS-BS tothe jthPU inthe CR BC, l=1k=1
and Ekj ∈ CDj×Ak to denote the channel from SU-TXk to where Γ¯ dXenoXtes the(cid:0)average interferen(cid:1)ce power con-
j
the jth PU in the CR IC. Moreover, the receiving terminals
straintforthejthPUtolimitthetotalinterferencepower
in the secondary networks may experience interference from
from the K SUs, which is averaged over the L transmit
active primary transmitters. For simplicity, we assume that
dimensions.
suchinterferenceistreatedasadditionalnoiseatthesecondary
Note that DRA for traditional (non-CR) wireless networks
receivers, and the total noise at each secondary receiving
under PTPC and/or ATPC has been thoroughlystudied in the
terminal is distributed as a CSCG random vector with zero
literature[12],whilethestudyofDRAsubjecttoPIPCand/or
mean and the identity covariance matrix.
AIPC as well as their variouscombinationswith PTPC/ATPC
Note that the (spatial) channels defined in the above CR
isuniquetoCRnetworksandisrelativelynew.Inordertogain
network models are assumed constant for a fixed transmit
moreinsightsintotheoptimalDRA designsforCR networks,
dimensionsuchasonetime-blockinatime-division-multiple-
we will first study the case of a single transmit dimension
access(TDMA)systemoronefrequency-bininanorthogonal-
(L = 1) with PTPCs and PIPCs by focusing on the spatial-
frequency-division-multiplexing (OFDM) system. In a wire-
domaintransmitoptimizationformulti-antennaCRsinSection
less environment, these channels usually change over time
III, and then study the general case of L>1 for joint space-
and/or frequency dimensions as governed by an underlying
time-frequencyDRAinCRnetworksunderATPCsandAIPCs
joint stochastic process. As such, DRA becomes relevant to
in Section IV.
schedule SUs into different transmit dimensions based on
their CSI. In general, the secondary transmitting terminals
need to satisfy two types of power constraints for DRA: III. COGNITIVE BEAMFORMING OPTIMIZATION
One is due to their own transmit power budgets; the other In this section, we consider the case of L = 1, where the
is to limit their resulting interference level at each PU to be DRA for CR networks reduces to the spatial-domain transmit
belowaprescribedthreshold.Theseconstraintscanbeapplied optimization under PTPCs and PIPCs to maximize the CR
over each fixed dimension as peak power constraints, or over network throughput.We term this practice as cognitive beam-
multiple dimensions as average power constraints. Without forming. In order to investigate the fundamental performance
lossofgenerality,weconsiderDRAforthesecondarynetwork limitsofcognitivebeamforming,westudytheoptimaldesigns
overLtransmitdimensionswithdifferentchannelrealizations, with the availability of perfect knowledge on all the channels
with L≥1. In total, four differenttypes of power constraints intheSU networks,andthosefromallthesecondarytransmit
can be defined for the secondary network. By taking the CR terminals to PUs. For convenience, we drop the dimension
MACasanexample(similarlyasfortheCRBC/IC),wehave index l for the rest of this section given L=1.
4
First, it is worth noting that the PIPC given in (6) can be optimality conditions of (P1) that transmit beamforming is
unified with the PTPC given in (4) into a formof generalized capacity optimal, i.e., Rank(S) = 1 [13]. Thus, without loss
linear transmit covariance constraint (GLTCC): of generality we could write S = vvH, where v ∈ CN×1
denotestheprecodingvector.Accordingly,(P1)forthespecial
K
Tr(W S )≤w (8) case of MISO CR channel is simplified as (P1-S) [13]:
i i
i=1
X Max. khvk
whereW ’sandwareconstants.Forexample,witheachPIPC v
i
given in (6), W = GHG ,∀i, and w = Γ , while for each s. t. kvk2 ≤P
i ij ij j
PTPC given in (4), Wi = I if i = k and 0 otherwise, with kGjvk2 ≤Γj, j =1,··· ,J,
w =P . Previous studies on transmit optimization for multi-
k
antenna or multiple-input multiple-output (MIMO) systems which is non-convexdue to the non-concavityof its objective
havemostlyadoptedsomespecialformsofGLTCCsuchasthe function. However, by observing the fact that if v is the
user individual power constraints and sum-power constraint. solution of (P1-S), so is ejθv for any arbitrary θ, we thus
However, it remains unclear whether such existing solutions assumewithoutlossofgeneralitythathvisarealnumberand
areapplicabletothegeneralformofGLTCC,whichiscrucial modify (P1-S) by rewriting its objective function as Re(hv)
to the problem of CR MIMO transmit optimization with the and adding an additional linear constraint Im(hv) = 0.
newlyaddedPIPCs. Inthefollowing,weprovideanoverview Thereby, (P1-S) can be converted into a second-order cone
of the state-of-art solutions for this problem under different programming(SOCP)[14]problem,whichisconvexandthus
CR networkmodels,while the developedsolutionsalso apply can be efficiently solved by available convex optimization
to the case with the general form of GLTCCs as in (8). softwares [15]. Alternatively, (P1-S) can be shown equiva-
From a convex optimization perspective, we next divide our lent to its Lagrange dual problem [13], which is a convex
discussionsintotwoparts,whichdealwiththecasesofconvex semi-definite programming (SDP) [14] problem and is thus
and non-convexproblem formulations, respectively. efficiently solvable [15]. For (P1-S) in the case of one single-
antenna PU, a closed-form solution for the optimal precoding
vector v was derived in [13] via a geometric approach.
A. Convex Problem Formulation
In order to reveal the structure of the optimal S for (P1),
First, consider the case where the associated optimization
we consider its Lagrange dual problem defined as (P1-D):
probleminatraditionalMIMOsystemwithoutPIPCisconvex.
In such cases, since the extra PIPCs are linear over the SU
Min. d(η)
transmitcovariancematrices,theresultingtransmitcovariance η(cid:23)0
optimization problem for CR systems remains convex; and
where η =[η ,η ,··· ,η ] denotes a vector of dual variables
thus,it canbe efficientlysolvedbystandardconvexoptimiza- 0 1 J
for (P1) with η associated with the PTPC, and η associated
0 j
tion techniques.
with the jth PIPC, j = 1,··· ,J, while we have the dual
CR Point-to-Point MIMO Channel: We elaborate this
function defined as
casebyfirstconsideringtheCRpoint-to-pointMIMOchannel,
which can be treated as the special case with only one active d(η),max log I+HSHH −η (Tr(S)−P)
0
SU link in the MAC, BC, or IC based CR network. Without S(cid:23)0
loss of generality, we will use the notations developed for (cid:12)(cid:12)J (cid:12)(cid:12)
− η (Tr G SGH −Γ ). (9)
the CR MAC with K = 1 in the following discussions. j j j j
Specifically,theoptimaltransmitcovariancetoachievetheCR Xj=1 (cid:0) (cid:1)
point-to-point MIMO channel capacity under both the PTPC Since (P1) is convex with Slater’s condition satisfied [14],
and PIPCs can be obtained from the following problem [13]: the duality gap between the optimal values of (P1) and (P1-
Max. log I+HSHH (P1) D) is zero, i.e., (P1) can be solved equivalently as (P1-D).
S Accordingly,an iterative algorithm can be developed to solve
s. t. Tr(S(cid:12))≤P (cid:12) (P1-D)byalternatingbetweensolvingd(η) fora givenη and
(cid:12) (cid:12)
Tr G SGH ≤Γ , j =1,··· ,J updating η to minimize d(η). At each iteration, η can be
j j j
updated by a subgradient-based method such as the ellipsoid
S(cid:23)0
(cid:0) (cid:1) method[16],accordingtothesubgradientsofd(η),whichcan
whereforconcisenesswehaveremovedtheSUindexk inthe beshownequaltoP−Tr(S⋆)andΓ −Tr G S⋆GH forη
j j j 0
symbol notations since K = 1, while S (cid:23) 0 means that S is andη ,j 6=0,respectively,whereS⋆ denotestheoptimalSto
j
(cid:0) (cid:1)
a positive semi-definite matrix [14]. obtain d(η) for a given η. From (9), it follows that S⋆ is the
We see that (P1) is a convex optimization problem since solution of the following equivalent problem (by discarding
its objective function is concave over S and its constraints irrelevant constant terms):
define a convex set over S. Thus, (P1) can be efficiently
max log I+HSHH −Tr(TS) (10)
solved by, e.g., the interior point method [14]. In the special
S(cid:23)0
case of CR multiple-inputsingle-output(MISO) channel, i.e.,
(cid:12) (cid:12)
H degrades to a row-vector denoted by h ∈ C1×N, it where T = η I+ J (cid:12) η (GHG )(cid:12) is a constant matrix for
0 j=1 j j j
can be shown by exploiting the Karush-Kuhn-Tucker (KKT) a given η. In order to solve Problem (10), we introduce an
P
5
auxiliary variable: Sˆ = T1/2ST1/2. Problem (10) is then re- “water-filling” algorithm [13]. Note that the partial channel
expressed in terms of Sˆ as projection works for any values of N and D ’s.
j
InFig.2,weplottheachievablerateofaCRMIMOchannel
max log I+HT−1/2SˆT−1/2HH −Tr(Sˆ). (11)
under the PTPC and PIPCs with the optimal transmit covari-
Sˆ(cid:23)0
(cid:12) (cid:12) ance solution for (P1) via the convex optimization approach,
The above prob(cid:12)lem can be shown equiva(cid:12)lent to the standard
(cid:12) (cid:12) against those with suboptimal covariance solutions via the
point-to-point MIMO channel capacity optimization problem
partial channel projection method with different values of b.
subjectto a single sum-powerconstraint[17], and its solution
The system parameters are given as follows: M = N = 4,
canbeexpressedasSˆ⋆ =VΣVH, whereV isobtainedfrom
J = 2, D = D = 1, and Γ = Γ = 0.1. The
1 2 1 2
the singular-value decomposition (SVD) given as follows:
SU achievable rate is plotted vs. the SU PTPC, P. It is
HT−1/2 = UΘVH, with Θ = Diag([θ ,...,θ ]) and
1 T observed that the optimal covariance solution obtained via
T = min(M,N), while Σ = Diag([σ ,...,σ ]) follows
1 T the convex optimization approach yields notable rate gains
the standard water-filling solution [17]: σ = (1/ln2 −
i over suboptimal solutions via the heuristic method, for which
1/θ2)+,i = 1,...,T, with (·)+ , max(0,·). Thus, the
i the optimal value of b (the number of SU-to-PU channel
solution of Problem (10) for a given η can be expressed as
dimensions to be nulled) to maximize the SU achievable rate
S⋆ =T−1/2VΣVHT−1/2.
increases with the SU PTPC.
Next,wepresentaheuristicmethodforsolving(P1),which
CR MIMO-MAC: The solutions proposed for the CR
leads to a suboptimal solution in general and could serve
point-to-point MIMO channel shed insights to transmit opti-
as a benchmark to evaluate the effectiveness of the above
mization for the CR MIMO-MACdefined in (1) with K >1.
two approaches based on convex optimization. To gain some
Assume that in the CR MIMO-MAC, the optimal multiuser
intuitions for this method, we first take a look at two special
detection is deployed at the S-BS to successively decode
cases of (P1). For the first case, supposing that all the PIPCs
differentSU messages from the receivedsum-signal. We then
are inactive (e.g., by setting Γ = ∞,∀j) and thus can be
j consider the problem for jointly optimizing SU transmit co-
removed,(P1)reducestothestandardMIMOchannelcapacity
variancematricestomaximizetheirweightedsum-ratesubject
optimization problem under the PTPC only, for which the
toindividualPTPCsandjointPIPCs.Thisproblemisreferred
optimalsolution of S is knownto be derivablefrom the SVD
to as weighted sum-rate maximization (WSRMax). Without
of H [17]. For the second case, assuming that Γ = 0,∀j,
j loss of generality, we assume that the given user rate weights
the solution for (P1) is then obtained by the “zero-forcing
satisfy that µ ≥ µ ≥ ··· ≥ µ ≥ 0; thus, the optimal
1 2 K
(ZF)” algorithm [18], which first projects H into the space
decodingorderofusersattheS-BStomaximizetheweighted
orthogonal to all G ’s, and then designs the optimal S based
j sum-rate is in accordance with the reverse user index [19].
on the SVD of the projected channel. Note that the (non-
Accordingly, the WSRMax for the CR MIMO-MAC can be
trivial) ZF-based solution exists only when N > J D .
j=1 j expressed as
From the above two special cases, we observe that as Γ ’s
j
decrease, the optimal S should evolve along with aPsequence K I+ k H S HH
i=1 i i i
of subspaces of H with decreasing dimensions as a result of Max. µ log (P2)
keepingcertain orthogonalityto Gj’s, which motivates a new S1,···,SK kX=1 k (cid:12)(cid:12)(cid:12)I+ Pik=−11HiSiHHi (cid:12)(cid:12)(cid:12)
Tdchehaseinngn,nedlmepfierntohejoedGc¯tifoo,nr c[[oG1¯g3T1n].i,t·iSv·pe·e,cbGi¯efiaTJcma]Tlfloy.r,mDlieentngoG,¯tenjatmh=eedSGVajsD/Γpaoj,rft∀iGa¯jl. s. t. TKr(STkr)≤GP(cid:12)(cid:12)(cid:12)kS, kGP=H1,≤..Γ.,K, j =(cid:12)(cid:12)(cid:12)1,··· ,J
as G¯ = U Λ VH. Without loss of generality, assume that kj k kj j
G G G k=1
the singular values in Λ are arranged in a decreasing order. X (cid:0) (cid:1)
G S (cid:23)0, k=1,··· ,K.
k
Then, we propose a generalizedchannel projectionoperation:
Reordering terms in the objective function of (P2) yields
H
H =H I−V(b) V(b) (12)
⊥ G G K−1 k
(cid:18) (cid:16) (cid:17) (cid:19) (µ −µ )log I+ H S HH
where V(b) consists of the first b columns of V corre- k k+1 (cid:12) i i i (cid:12)
G G kX=1 (cid:12) Xi=1 (cid:12)
sponding to the b largest singular values in ΛG, 1 ≤ b ≤ K (cid:12)(cid:12) (cid:12)(cid:12)
min(N−1, J D ).Notethatbcouldalsotakeazerovalue +µ log I+ H(cid:12)S HH (cid:12) (13)
j=1 j K i i i
forwhichVPG(0) ,0.Now,wearereadytopresentthetransmit (cid:12)(cid:12)(cid:12) Xi=1 (cid:12)(cid:12)(cid:12)
covariancematrixforthepartialprojectionmethodintheform From the above new(cid:12) form of the obje(cid:12)ctive function, it can
(cid:12) (cid:12)
of its eigenvalue decomposition (EVD) as S = V Σ VH, be verified that (P2) is a convex optimization problem over
⊥ ⊥ ⊥
whereV isobtainedfromthe SVD of the projectedchannel S ’s. Thus, similarly as for (P1), (P2) can be solved by an
⊥ k
H , i.e., H = U Λ VH. By substituting this new form interior-point-methodbasedalgorithmoraniterativealgorithm
⊥ ⊥ ⊥ ⊥ ⊥
of S into (P1), it can be shown that the problem reduces to via solving the equivalent Lagrange dual problem, for which
maximizing the sum-rate of a set of parallel channels (with the details are omitted here for brevity.
channel gains given by Λ ) over their power allocation Σ It is noted that (P2) is for the case with the optimal non-
⊥ ⊥
subject to (J + 1) linear power constraints, for which the linear multiuser decoder at the S-BS, while in practice the
optimal power allocation can be obtained by a generalized low-complexitylineardecoderisusuallymorepreferable.The
6
12 can be formulated as:
K I+HH K Q H
k i=k i k
10 Max. µ log (P3)
OSupbtiompatilm Saoll uStoioluntion (b=0) Q1,···,QK kX=1 k I(cid:12)(cid:12)(cid:12)+HHk (cid:16)PKi=k+1Q(cid:17)i H(cid:12)(cid:12)(cid:12)k
Rate (bits/sec/Hz)68 SSuubbooppttiimmaall SSoolluuttiioonn ((bb==12)) s. t. kXK=1Tr(QkK)(cid:12)(cid:12)(cid:12)≤P (cid:16)P (cid:17) (cid:12)(cid:12)(cid:12)
SU Achievable 4 TQrk (cid:23)F0j, kkX==1Q1,k·!··F,HjK!≤Γj, j =1,··· ,J
where Q ∈ CM×M denotes the covariance matrix for
2 k
the transmitted signal of S-BS intended for the kth SU,
k = 1,··· ,K; µ ’s are the given user rate weights; and P
k
0 denotes the transmit power constraint for the S-BS. Without
0 2 4 6 8 10 12 14 16 18 20
SU Transmit Power Constraint (dB) loss of generality, we assume that µ ≥µ ≥···≥µ ≥0;
1 2 K
thus, in (P3) the optimal encoding order of users for DPC
Fig.2. ComparisonoftheachievableratesfortheCRMIMOchannelunder to maximize the weighted sum-rate is in accordance with
the PTPCand PIPCs with the optimal transmit covariance solution for (P1)
the user index [22]. Note that (P3) is non-convex with or
viathetheconvexoptimizationapproachvs.suboptimalcovariancesolutions
viathepartial channel projection methodwithdifferent values ofb. without the PIPCs due to the fact that the objective function
is non-concave over Q ’s for K ≥ 2. As a result, unlike
k
use of linear instead of non-linear decoder at the receiver (P1)forthe point-to-pointCR channel,the standardLagrange
will change the user achievable rates for the CR MIMO- duality method cannot be applied for this problem. For (P3)
MAC,thusresultinginnewproblemformulationsfortransmit in the case withoutthe PIPCs, a so-called“BC-MAC duality”
optimization.Forexample,in[20],theauthorshaveconsidered relationshipwasproposedin[23]totransformthenon-convex
the CR SIMO-MAC (single-antenna for each SU transmitter) MIMO-BC problem into an equivalent convex MIMO-MAC
with a linear decoder at the receiver, where the power alloca- problem, which is solvable by efficient convex optimization
tion across the SUs is optimized to maximize their signal-to- techniques such as the interior point method. In [24], another
interference-plus-noise ratios (SINRs) at the receiver subject form of BC-MAC duality, the so-called “mini-max duality”
to both transmit and interference power constraints. wasexploredto solvethe MIMO-BCproblemundera special
case of GLTCC: the per-antenna transmit power constraint.
However, these existing forms of BC-MAC duality are yet
B. Non-Convex Problem Formulation unableto handlethe case with arbitrarynumbersof GLTCCs,
which is the case for (P3) with both the PTPC and PIPCs.
Next,weconsiderthecasewheretheoptimizationproblems
In [25], a general method was proposed to solve various
intheassociatedtraditional(non-CR)MIMOsystemsarenon-
MIMO-BC optimization problems under multiple GLTCCs,
convex. It thus becomes more challengingwhether these non-
thusincludingtheCR MIMO-BCWSRMax problemgivenin
convex problems with the addition of convex PIPCs in the
(P3). For this method, the first step is to combine all (J +1)
correspondingCR MIMO systems can be efficiently solvable.
power constraints in (P3) into a single GLTCC as shown in
In the following, we present some promising approaches to
the following optimization problem:
solve these problems for the CR MIMO-BC and MIMO-IC.
CR MIMO-BC: First, considerthe CR MIMO-BC defined K I+HH K Q H
k i=k i k
in (2) under both the PTPC at the S-BS and J PIPCs each Max. µ log
for one of the J PUs, which can be similarly defined as for Q1,···,QK Xk=1 k I(cid:12)(cid:12)(cid:12)+HHk (cid:16)PKi=k+1Q(cid:17)i H(cid:12)(cid:12)(cid:12)k
the MAC case in (4) and (6), respectively. We focus on the
unicast downlink transmission for the CR BC, while for the s. t. Tr A K (cid:12)(cid:12)(cid:12)Q ≤Q(cid:16)P (cid:17) (cid:12)(cid:12)(cid:12)
k
case of multicast,the interestedreadersmay referto [21]. For !
k=1
thepurposeofexposition,weconsidertwocommonlyadopted X
Q (cid:23)0, k =1,··· ,K (14)
k
design criteria for the traditional multi-antenna Gaussian BC
in the literature: One is for the MIMO-BC deploying the whereA=λ I+ J λ FHF ,andQ=λ P+ J λ Γ
0 j=1 j j j 0 j=1 j j
non-linear“dirty-paper-coding(DPC)” at the transmitter [22], withλ ,λ ,··· ,λ beingnon-negativeconstants.Foragiven
0 1 J
P P
which maximizes the weighted sum-rate of all the users set of λ ’s, i = 0,··· ,J, let the optimal value of the
i
(i.e., the WSRMax problem); the other is for the MISO-BC above problem be denoted by F(λ ,λ ,··· ,λ ). Clearly,
0 1 J
(single-antenna for each SU receiver) deploying only linear F(λ ,λ ,··· ,λ ) is an upper bound on the optimal value
0 1 J
encoding at the transmitter, which maximizes the minimum of (P3) since any feasible solutions for (P3) must satisfy
SINR among all the users, referred to as “SINR balancing”. the constraints of Problem (14) for a given set of λ ’s.
i
Specifically, the WSRMax problem for the CR MIMO-BC Interestingly, it can be shown that the optimal value of (P3)
7
worthnotingthatwithK =1,theabovemethodcanbeshown
z1
HH1 y1 x1 H1 z MeqIuMivOalecnhtatnontehlactadseevbealospededofnorth(eP1L)aignratnhgeeCdRuapliotiyn.t-to-point
z2 Consider next the SINR balancing problem for the CR
HH y2 x2 H2 y MISO-BC, which can be expressed as:
x 2
M
M
Max. α (P4)
zK α,v1,···,vK
khHv k2
HHK yK xK HK s. t. 1+ k kkhHv k2 ≥α, k =1,··· ,K
i6=k k i
(a) (b) K
P
kv k2 ≤P
k
Fig.3. GeneralizedMIMOMAC-BCDuality:(a)PrimalMIMO-BCchannel Xk=1
with downlink channels HHk and receiver noise vectors zk ∼ CN(0,I), K
k = 1,...,K, and a GLTCC: Tr(APKk=1Qk) ≤ Q; (b) Dual MIMO- kFjvkk2 ≤Γj, j =1,··· ,J
MAC with uplink channels Hk,k = 1,...,K and receiver noise vector
z ∼ CN(0,A), and a sum-power constraint: PKk=1Tr(Sk) ≤ Q. The Xk=1
MIMO-BCanddualMIMO-MAChavethesameachievablerateregion[25]. where α denotes an achievable SINR for all the SUs; v ∈
k
CM×1 denotesthe precodingvectorfor the transmitted signal
is equal to the minimum value of functionF(λ ,λ ,··· ,λ )
0 1 J of S-BS intended for the kth SU; and h represents H for
k k
over all non-negative λ ’s [25]. Therefore, (P3) can be re-
i theMISO-BCcase. Similarlyasfor(P1-S),bytreatinghHv
solved by iteratively solving Problem (14) for a given set k k
on the left-hand side (LHS) of each SINR constraint in (P4)
of λ ’s and updating λ ’s towards their optimal values to
i i as a positivereal number[26], itcan be shown that(P4) fora
minimizefunctionF(λ ,λ ,··· ,λ ).Specifically,λ ’scanbe
0 1 J i given α is equivalent to a SOCP feasibility problem and thus
updatedviatheellipsoidmethodaccordingtothesubgradients
efficientlysolvable[15].Foragivenα,iftheassociatedSOCP
of F(λ ,λ ,··· ,λ ), which can be shown [25] equal to
0 1 J problemisfeasible,weknowthattheoptimalsolutionof(P4)
P − Kk=1Tr(Q⋆k) and Γj −Tr(Fj( Kk=1Q⋆k)FHj ) for λ0 forα,denotedbyα⋆,mustsatisfyα⋆ ≥α;otherwise,α⋆ <α.
and λPj (j 6= 0), respectively, where QP⋆k’s are the solution of Based on this fact, α⋆ can be found by a simple bisection
Problem (14) for the given λk’s. search [14]; with α⋆, the corresponding optimal solution for
Furthermore, Problem (14) with a given set of λ ’s can be
k v ’s in (P4) can also be obtained. The above technique has
k
solvedbyapplyingthegeneralizedBC-MACdualityproposed
also been applied in [27] for (P4) without the PIPCs.
in [25], which extendsthe existingformsof BC-MAC duality
The SINR balancing problem for the conventional MISO-
[23], [24] to transform the MIMO-BC problem subject to
BCwithoutthePIPCs hasalso beenstudiedin[28],wherean
a single GLTCC as in Problem (14) to an auxiliary (dual)
algorithm was proposed using the virtual uplink formulation
MIMO-MAC problem subject to a corresponding sum-power
and a fixed-pointiteration. However,this algorithm cannot be
constraint. Specifically, it is shown in [25] that the MIMO-
extended directly to deal with multiple PIPCs for the case of
BCasinProblem(14)andthedualMIMO-MAC,asdepicted
CRMISO-BC.Similarlyasforthepreviousdiscussionsonthe
in Fig. 3, have the same achievable rate region. Accordingly,
WSRMaxproblemfortheCRMIMO-BCwhereageneralized
the optimal objective value (weighted sum-rate) of Problem
MIMO MAC-BC duality holds, a counterpart beamforming
(14) for the primal MIMO-BC can be obtained as that of the
duality also holds for the MISO-BC and SIMO-MAC [25].
following equivalent problem for the dual MIMO-MAC:
With this duality result, the SINR balancing problem (P4)
K−1 k for the CR MISO-BC can be converted into an equivalent
Max. (µk−µk+1)log A+ HiSiHHi problemforthedualSIMO-MAC,wheretheefficientiterative
S1,···,SK (cid:12) (cid:12)
Xk=1 (cid:12) Xi=1 (cid:12) algorithm in [28] can be directly applied. The interested
K (cid:12)(cid:12) (cid:12)(cid:12) readers may refer to [25] for the details of this method.
+µ log A+ H(cid:12) S HH (cid:12)
K i i i CR MIMO-IC: Second, consider the CR MIMO-IC given
(cid:12) (cid:12)
(cid:12) Xi=1 (cid:12) in (3), subject to both the PTPCs for the K SU-TXs and the
K (cid:12) (cid:12)
(cid:12) (cid:12) PIPCsfortheJ PUs,whichcanbesimilarlydefinedasforthe
s. t. Tr(S(cid:12))≤Q (cid:12)
k MAC case in (4) and (6), respectively. From an information-
k=1
X theoretic perspective, the capacity region for the Gaussian IC
S (cid:23)0, k =1,··· ,K. (15)
k underPTPCs,whichconsistsofallthesimultaneouslyachiev-
Similarto (P2),theaboveproblemisa WSRMax problemfor able rates of all the users, still remains unknown in general
the MIMO-MAC subject to a single sum-power constraint, evenforthecaseofK =2andA =B =1,k=1,2[29].A
k k
which is convex and thus can be efficiently solvable by, e.g., pragmatic approach that leads to suboptimal achievable rates
the interior point method. After solving Problem (15), the in the Gaussian IC is to restrict the system to operate in a
optimal user transmit covariance solutions for the MIMO- decentralizedmanner,i.e., allowing onlysingle-user encoding
MAC, S⋆’s, can be transformed to the corresponding ones anddecodingbytreatingtheco-channelinterferencesfromthe
k
for the original MIMO-BC, Q⋆’s, via a MAC-BC covariance other users as additional Gaussian noises. For this approach,
k
transformation algorithm given in [25]. Furthermore, it is transmitoptimizationfortheCRMIMO-ICreducestofinding
8
a set of optimal transmit covariance matrices for the K SU underitsPTPCandJ interference-powerconstraintsgivenby
links, denoted by Rk ∈CAk×Ak,k =1,··· ,K, to maximize (16), with all other Ri’s, i 6= k, fixed. It is observed that
the secondary network throughput under both the PTPCs and the resulting problem is in the same form of our previously
PIPCs. More specifically, the WSRMax problem for the CR studied (P1) for the CR point-point MIMO channel; thus,
MIMO-IC can be expressed as: similar techniques developed for (P1) can be applied. Note
that a suboptimal method for this problem in the same spirit
K
Max. µ log I+(I+ H R HH)−1H R HH of the partial channel projection method to reduce the design
k ik i ik kk k kk
R1,···,RKk=1 (cid:12) i6=k (cid:12) complexity for each SU transmit covariance matrix has also
X (cid:12) X (cid:12) been proposed in [37]. Moreover, it is noted that Γ(k)’s,
(cid:12) (P5) (cid:12) j
(cid:12) (cid:12) j = 1,··· ,J, k = 1,··· ,K, can be searched over the SUs
s. t. Tr(R )≤P , k =1,··· ,K
k k to further improve their weighted sum-rate.
K Alternatively, assuming that a centralized optimization is
Tr E R EH ≤Γ , j =1,··· ,J
kj k kj j feasible with the global knowledge of all the channels in the
kX=1 (cid:0) (cid:1) SU network, as well as those from different SU-TXs to all
R (cid:23)0, k =1,··· ,K
k PUs,anotherheuristicalgorithmfor(P5)wasproposedin[38].
By rewriting the SU transmit covariance matrices into their
where µ ’s are the given non-negative user rate weights. We
k
equivalent precoding vectors and power allocation vectors,
see that (P5) is non-convexwith or without the PIPCs due to
the fact that the objective function is non-concave over R ’s this algorithm iteratively updates the SU transmit precoding
k
vectors (based on the “network duality” [39]) or the power
for K > 1. As a result, there are no efficient algorithms yet
allocation vectors (by solving geometric programming (GP)
to obtain the globally optimal solution for this problem. For
problems [40]), with the others being fixed.
the same problemsetup, there have been recentprogresseson
characterizing the maximum achievable “degrees of freedom Itis worth pointingoutthat thereare otherproblemformu-
(DoF)” for the user sum-rate (i.e., µ =1,∀k) [30]. lationsdifferentfrom(P5)toaddressthetransmitoptimization
k
Next, we discuss some feasible solutions for (P5). First, it for the CR MIMO-IC. In [41], a new criterion was proposed
is worth noting that for (P5) in the case without the PIPCs, a to design the SU link transmission in a CR MISO-IC via an
commonlyadopted suboptimal approachis to iteratively opti- alternative decentralized approach, where each SU-TX inde-
mize each user’s transmit covariance subject to its individual pendently designs its transmit precoding vector to maximize
PTPC with the transmit covariances of all the other users theratiobetweenthereceivedsignalpoweratthedesiredSU-
fixed. This decentralized approach has been first proposed RX and the resulted total interference power at all the PUs,
in [31], [32] to obtain some local optimal points for (P5) inorderto regulatethe interferencepowersatPUs. Moreover,
with the PTPCs only, where they differ in that the one in the above discussions are all based on the assumption that
[31] maximizes the user individual rate at each iteration, each SU-RX treats the interferences from all the other SU
while the one in [32] maximizes the user weighted sum-rate. links as additional noises, which is of practical interest since
It is also noted that a parallel line of works with similar it simplifies the receiver design for each SU link. However,
iterative user optimizations has been pursued in the single- duetoindependentcross-linkchannelsbetweenSUterminals,
antenna but multi-carrier based interference channels such as it may be possible that a SU-RX could occasionally observe
the wired discrete-multi-tone (DMT) based digital subscriber “strong” interference signals from some co-existing SU-TXs
line (DSL) network [33], and the wireless OFDM based ad andthusbeabletodecodetheirmessagesviamultiuserdetec-
hoc network [34]. One important question to answer for such tion techniques and then cancel the associated interferences.
iterativealgorithmsisunderwhatconditionsthealgorithmwill Withsuch“opportunistic”multiuserdetectionateachSU-RX,
guarantee to converge to a local optimal point. This problem the achievable rate of each SU link becomes a function of
has been addressed in the contexts of both multi-carrier and not only its own transmit covariance, but also those of the
multi-antenna based interference channels in, e.g., [35], [36], other SUs as well as their instantaneous transmit rates. Thus,
via game-theoretic approaches. thecorrespondingtransmitoptimizationfortheCRMIMO-IC
However, the above iterative approach cannot be applied leadsto newandmorechallengingproblemformulationsthan
directly to solve (P5) with both the PIPCs and PTPCs, since (P5); the interested readers may refer to [42], [43].
each PIPC involves all the user transmit covariances and is
thus not separable over the SUs. Thus, a feasible approach IV. JOINT SPACE-TIME-FREQUENCY DRA OPTIMIZATION
for (P5) is to decompose each of the J PIPCs into a set of
In the previous section, we have studied DRA for different
interference-power constraints over the K SU-TXs, i.e., for
CRnetworksatasingletransmitdimensionintime/frequency,
the jth PIPC, j ∈{1,··· ,J},
byfocusingonspatial-domaintransmitoptimizationunderthe
Tr E R EH ≤Γ(k), k =1,··· ,K (16) peak transmit and interference power constraints (PTPC and
kj k kj j
PIPC). In this section, we bring the additional time and/or
where Γj(k) is(cid:0) a constant(cid:1), and all Γ(jk)’s, k = 1,··· ,K, frequency dimensions into the DRA problem formulations,
satisfy Γ(k) ≤ Γ such that the jth PIPC is guaranteed. by applyingthe average transmitand interferencepowercon-
k j j
Then, the iterative algorithm works here, where each SU link straints(ATPCandAIPC)inCRnetworks.ConsidertheDRA
P
independently optimizes R to maximize its achievable rate over L time/freqeuncy dimensions, for which all the required
k
9
channel knowledge is assumed to be known. Taking the CR equivalent problems. Furthermore, each subproblem in (18)
MAC as an example (similar argumentscan be developedfor is also convex. Thus, the dual decomposition method solves
theCRBC/IC),underboththeATPCsandAIPCsgivenin(5) (P6) via its dualproblem(P6-D), which is decomposableinto
and (7), respectively, a generic problem formulation for DRA L convex subproblems. For the second case, as a counter-
optimization can be formulated as: part, consider that U (·) is non-concave over S [l]’s (e.g.,
l k
the weighted sum-rate for the CR MIMO-BC/MIMO-IC in
Max. C({S [l]}) (P6)
k (P3)/(P5)).Asa result,(P6)isnon-convexandthedualitygap
Sk[l](cid:23)0,∀k,l
between (P6) and (P6-D) may not be zero. Furthermore, the
s. t. (5),(7)
subproblem(18) is also non-convex.For this case, evenwhen
where {S [l]} denotes the set of S [l]’s, k = 1,··· ,K, theoptimalsolutionsoftheLsubproblemsareobtainable,the
k k
and l = 1,··· ,L, while C(·) is an arbitrary utility func- optimal value of (P6-D) in general only serves as an upper
tion to measure the CR network performance. We as- bound on that of (P6). However, in [45] it is pointed out that
sume that C(·) is separable over l’s, i.e., C({S [l]}) = if a set of so-called “time-sharing” conditionsare satisfied by
k
1 L U (S [l],··· ,S [l]) with U (·)’s denotingindividual a non-convex optimization problem, the duality gap for this
L l=1 l 1 K l
utility functions. Since both the ATPC and AIPC involve L problemandits dualproblemis indeedzero.Furthermore,for
P
transmit covariance matrices, the Lagrange dual decomposi- theclassofDRA problemsinthe formof(P6),the associated
tion(see,e.g.,atutorialpaper[44])isageneralmethodtodeal time-sharing conditions are usually satisfied asymptotically
with this type of average constraints for optimization over a as L → ∞ under some cautious considerations on the
numberof paralleldimensions,which is explainedasfollows. continuity of channel distributions [46]. Therefore, the dual
By introducing a set of dual variables, ν ’s, each for one of decomposition method could still be applied to solve (P6) in
k
the K ATPCs, and δ ’s, each for one of the J AIPCs, the thenon-convexcaseforsufficientlylargevaluesofL,provided
k
Lagrange dual problem of (P6) can be written as (P6-D): that the optimal solutions for the subproblems in (18) are
obtainable (e.g., a variation of (P3) for the CR MIMO-BC).
Min. d(ν,δ)
However,with finitevaluesofL,howtoefficientlysolve(P6)
ν(cid:23)0,δ(cid:23)0
in the case of non-concave objective functions is still open.
with ν = [ν ,··· ,ν ],δ = [δ ,··· ,δ ], and the dual
1 K 1 J With the above discussions on the general approaches to
function
design joint space-time-frequencyDRA for CR networks, we
K L nextpresentsomeexamplesofuniqueintereststoCRsystems.
1
d(ν,δ), max C({S [l]})− ν ( Tr(S [l])
k k k
Sk[l](cid:23)0,∀k,l k=1 L l=1
X X
J L K A. TDMA/FDMA Constrained DRA: When Is It Optimal?
1
−P¯ )− δ ( Tr G [l]S [l]GH[l] −Γ¯ ).
k j L kj k kj j Time-/frequency-division multiple-access (TDMA/FDMA),
j=1 l=1k=1
X XX (cid:0) (cid:1) (17) which schedules only one user for transmission at each
time/frequencydimension,isusuallypreferableinpracticedue
Since the dual problem (P6-D) is convex regardless of the
to their implementation ease. For the TDMA/FDMA based
convexity of the primal problem (P6) [14], (P6-D) can be
CR MAC (similar arguments hold for the CR BC/IC), the
efficiently solved by the ellipsoid method according to the
optimal DRA over L transmit dimensions to maximize the
subgradients of the dual function d(ν,δ), similarly as in our
sum-capacity of the SUs can be formulated as (P6) with
previousdiscussions, providedthatthe maximizationproblem
properly chosen functions for U (·)’s, where for any given l,
l
in(17)issolvableforanygivensetofν andδ.Itisinteresting
U (·) is expressed as (l is dropped for conciseness)
l
toobservethatthismaximizationproblemcanbedecomposed
intoLparallelsubproblemseachforoneoftheLdimensions, log I+H S HH S =0,∀i6=k
U(S ,··· ,S )= k k k i
and all of these subproblems have the same structure and 1 K 0 otherwise.
(cid:26) (cid:12) (cid:12)
are thus solvable by the same algorithm, a practice known (cid:12) (cid:12) (19)
as “dual decomposition”. Without loss of generality, we drop
the dimension index l and express each subproblem as Note that U(·) defined above implies the TDMA/FDMA
constraint, i.e., only scheduling one user for transmission at
K
a given dimension with a positive contribution to the sum-
max U(S ,··· ,S )− Tr(B (ν ,δ)S ) (18)
Sk(cid:23)0,∀k 1 K k k k capacity. However, it can be shown that U(·) is non-concave
k=1
X over S ’s in this case and as a result, the corresponding (P6)
k
where B (ν ,δ) = ν I + J (δ GHG ) is a constant isnon-convex.Nevertheless,accordingtoourpreviousdiscus-
k k k j=1 j kj kj
matrix for the given ν and δ, k=1··· ,K. sions, since the time-sharing conditions hold approximately
k
P
We then discuss the followingtwo cases. For the first case, whenL→∞, thedualdecompositionmethodcan be applied
consider that U (·) is a concave function over S [l]’s, ∀l to solve (P6) for this case with very large values of L, where
l k
(e.g., the point-to-point CR channel capacity in (P1), or the the optimal solution of the associated subproblem at each
weighted sum-rate for the CR MIMO-MAC in (P2)). Then, dimension given in (18) can be obtained by finding the SU
(P6) is convex and thus the duality gap between the optimal (selected for transmission) with the largest objective value of
values of (P6) and (P6-D) is zero, i.e., (P6) and (P6-D) are the followingproblem(whichis ofthe same formas Problem
10
(10) and thus solvable in a similar way): andisassumedtoholdwithequality,i.e.,forCaseI,I(I) =Γ,
sp
for all the fading states, while for Case II, E(I(II)) = Γ.
max log I+H S HH −Tr(B (ν ,δ)S ). (20) sp
Sk(cid:23)0 k k k k k k Taking the PU ergodic capacity as an example, which can be
(cid:12) (cid:12) expressed as (assuming unit-power receiver Gaussian noise):
An importan(cid:12)t question to (cid:12)investigate for TDMA/FDMA
based DRA is how much the performance is degraded as h Q
C =E log 1+ p . (21)
compared with the optimal DRA that allows more than one p 1+I
(cid:18) (cid:18) sp(cid:19)(cid:19)
users to transmit at a given dimension. From an information-
theoretic viewpoint, it is thus pertinent to investigate the Let Cp(I) and Cp(II) denote the values of Cp in Cases I and II,
conditions for the optimality of TDMA/FDMA, i.e., when respectively. The following equalites/inequalities then hold
they are optimal to achieve the system sum-capacity. For the
h Q
traditional single-antenna fading MAC under the user ATPCs Cp(I) =Ehp log 1+ 1+p Γ
over time, it has been shown in [47] that TDMA is optimal (cid:18) (cid:18) (cid:19)(cid:19)
for achieving the ergodic/long-term sum-capacity. This result hpQ
=E log 1+
has been shown to hold for the fading CR MAC and CR BC hp 1+E(Is(pII))!!
underboththeATPCsandAIPCsin[48],wherebyexploiting
h Q
the KKT optimality conditions of the associated optimiza- ≤E E log 1+ p (22)
tion problems, the optimality conditions for TDMA in other hp Isp 1+Is(pII)!!!
cases of combined peak/average transmit/interference power =C(II)
p
constraintshavebeencharacterized.Forthe traditionalsingle-
where(22)isduetotheJensen’sinequality(see,e.g.,[17])and
antenna IC with interference treated as noise, the optimality
of TDMA/FDMA for the sum capacity has been investigated the convexityof the function f(x)=log 1+ κ where κ
1+x
undertheATPCsin[49],[50].Itwouldbeinterestingtoextend is anypositiveconstantand x≥0. Thus,(cid:16)it follows(cid:17)thatgiven
these results to the case of CR IC under the additional PIPCs the same average power of the interference, Γ, it is desirable
and/or AIPCs. for the PU to have the instantaneous interference power I
sp
fluctuate over fading states (Case II) rather than stay constant
(Case I), to achieve a larger ergodic capacity.
B. Peak vs. Average Interference Power Constraints: A New
In general, the results in [51] reveal a new interference
Interference Diversity
diversity phenomenon for SS-based CR networks, i.e., the
From a SU’s perspective,it is obviousthatthe ATPC/AIPC
randomized interference powers from the secondary network
ismoreflexiblethanthePTPC/PIPCforDRAunderthesame
can be more advantageous over deterministic ones across
power threshold and thus results in a larger SU link capac-
different transmit dimensions over space, time, or frequency
ity. However, from a PU’s perspective, it remains unknown
for minimizing the resulted primary network capacity losses.
whether the AIPC or PIPC causes more PU link performance
Further investigations are required on interference diversity
degradation. Intuitively speaking, the PIPC should be more
driven DRA for CR or other spectrum sharing systems.
favorable than the AIPC since the former limits the interfer-
ence power at the PU to be below certain threshold at each
C. BeyondInterferenceTemperature:ExploitingPrimaryLink
time/frequency dimension, while the latter results in varia-
Performance Margins
tions of interference power levels over different dimensions
although their average level is kept below the same threshold So far, we have studied DRA for CR networks based on
as that for the PIPC. the IT constraints for protecting the PU transmissions. Given
Somehow surprisingly, in [51] it is shown that for the that the IT constraints in general conservatively lead to an
single-antenna PU fading channel subject to the interference upper bound on the PU capacity loss due to the interference
from a SU transmitter, the AIPC is in fact better than its from the SUs [13], [52], it would be possible to improve
PIPC counterpart under the same average power threshold in the spectrum sharing capacities for both the SUs and PUs
terms of minimizing the PU capacity losses, which holds for over the IT-based methods if additional cognition on the PU
the cases of both ergodic and outage capacities of the PU transmissionsisavailableattheCRtransmitters.Forexample,
channel, with/without power control. To illustrate this result, by exploiting CSI of the PU links, the CRs could allocate
weconsiderforsimplicitythecasewithoutthePU linkpower transmit/interferencepowersmoreflexiblyoverthedimensions
control, i.e., the PU transmits with a constant power, Q, over where the PU channels exhibit poor conditions, without de-
all the fading states. Suppose that the PU link channel power gradingtoo muchthe PU link performances.These PU “null”
gainisdenotedby h , andthatfromthe SU transmitterto the dimensionscouldcomeupin time,frequency,or space.Thus,
p
PU receiverdenotedbyh . Next,considerthe followingtwo the IT constraints could be replaced by the more relevant
sp
cases, where the interference power from the SU transmitter primary link performance margin constraints [52], [53] for
at the PU receiver, denoted by I =h p , with p denoting the design of DRA in CR networks, in order to optimally
sp sp s s
the SU transmit power, is fixed over all the fading states in exploit the available primary link performance margins to
Case I (correspondingto the case of PIPC), and is allowed to accommodate the interference from the SUs. Following this
be variable in Case II (corresponding to the case of AIPC). newparadigm,manynewandchallengingDRAproblemscan
Forbothcases,aconstantinterferencepowerthresholdΓisset be formulated for CR networks. As an example, consider the