resource allocation and adaptive modulation in uplink sc-fdma systems
TRANSCRIPT
Wireless Pers CommunDOI 10.1007/s11277-013-1464-6
Resource Allocation and Adaptive Modulation in UplinkSC-FDMA Systems
Ayaz Ahmad
© Springer Science+Business Media New York 2013
Abstract In this paper, we study joint power and sub-channel allocation, and adaptive mod-ulation in Single Carrier Frequency Division Multiple Access (SC-FDMA) which is adoptedas the multiple access scheme for the uplink in the 3GPP-LTE standard. A sum-utility maxi-mization problem is considered. Unlike OFDMA, in addition to the restriction of allocating asub-channel to one user at most, the multiple sub-channels allocated to a user in SC-FDMAshould be consecutive as well. This renders the resource allocation problem prohibitivelydifficult and the standard optimization tools (e.g., Lagrange dual approach widely used forOFDMA, etc.) can not help towards its optimal solution. We propose a novel optimizationframework for the solution of this problem which is inspired from the recently developedcanonical duality theory. We first formulate the optimization problem as binary-integer pro-gramming problem, and then transform this binary-integer programming problems into acontinuous space canonical dual problem that is a concave maximization problem. Based onthe solution of the continuous space dual problem, we derive joint power and sub-channelallocation algorithm whose computational complexity is polynomial. We provide conditionsunder which the proposed algorithms are optimal. We also propose an adaptive modulationscheme which selects an appropriate modulation strategy for each user. We compare theproposed algorithm with the existing algorithms in the literature to assess their performance.The results show a tremendous performance gain.
Keywords 3GPP LTE · SC-FDMA ·Adaptive resource allocation · Integer programming ·Canonical duality theory
A. Ahmad (B)Department of Electrical Engineering, COMSATS Intitute of Information Technology,Wah Cantt 44000, Pakistane-mail: [email protected]
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A. Ahmad
1 Introduction
Single Carrier Frequency Division Multiple Access (SC-FDMA) is currently attracting alot of attention as an alternative to OFDMA in the uplink. Its low PAPR feature hasthe potential to benefit the mobile terminals in term of transmit power efficiency. Infact, SC-FDMA is a single carrier multiple access technique which utilizes single car-rier modulation and frequency domain equalization. Its overall structure and performanceare similar to that of OFDMA system. Unlike the parallel transmission of the orthogo-nal sub-channels in OFDMA, the sub-channels are transmitted sequentially in SC-FDMA.This sequential transmission of sub-channels considerably reduces the envelope fluctua-tion in transmitted waveform and results in low PAPR [1]. There are two types of SC-FDMA: localized-FDMA (L-FDMA) in which the sub-channels assigned to a user areadjacent to each other, and interleaved-FDMA (I-FDMA) in which users are assignedwith sub-channels distributed over the entire frequency band [1]. In 3GPP-LTE [2],OFDMA is adopted for the downlink transmission and localized SC-FDMA for the uplinktransmission.
1.1 SC-FDMA Versus OFDMA from a Resource Allocation Perspective
Most of the previous work on resource allocation has focused on power and sub-channelsallocation in downlink OFDMA systems [3–8]. Resource allocation in OFDMA while usingit as an uplink multiple access scheme is also considered. A recent work on joint subcar-rier and power allocation with fairness in uplink OFDMA systems that is based on antcolony optimization is presented in [9]. Moreover, a survey of resource allocation andscheduling schemes in OFDMA wireless networks for the uplink is carried out in [10].In addition to traditional OFDMA systems, resource allocation in OFDMA-based cogni-tive radio systems has also been studied. In [11], the authors propose a resource alloca-tion algorithm for multiuser OFDM-based cognitive radio systems with proportional rateconstraints.
One of the well known approaches for solving the OFDMA resource allocation prob-lem is exploiting its time-sharing property [12]. Based on this property, it is shown in[5], and [12] that for practical number of sub-channels, the resource allocation problemin OFDMA systems can be solved by Lagrange multipliers method with zero duality gap.However, none of above is directly applicable to uplink SC-FDMA. This is due to the factthat in localized SC-FDMA in addition to the restriction of allocating a sub-channel toone user at most, the multiple sub-channels allocated to a user should be adjacent to eachother as well. Furthermore, a frequency domain equalizer is used in SC-FDMA over all thesub-channels allocated to the user which makes the SNR expression much more compli-cated than in OFDMA where the SNR on each sub-channel is independent from the othersub-channels.
The common approach used for resource allocation in OFDMA is to formulate the mutualexclusivity restriction on sub-channels allocation as binary-integer constraint, solve the prob-lem to get an approximated solution in continuous domain, and then discretize the continuousvalues into the closest binary values. But in SC-FDMA resource allocation, this approachcannot be employed. The reason is that if the problem is solved by relaxing the 0–1 constraint,then, during discretization of the continuous domain solution, the adjacency constraint onsub-channels allocation cannot be assured. This necessitates the design of a framework thatalso ensures the adjacency constraint on sub-channels allocation which is a very difficulttask.
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1.2 Related Work
In most of the previous work on SC-FDMA, the implementation problems in the physicallayer are studied (e.g., [13–16]). The resource allocation problem in uplink SC-FDMA hasalso been addressed in a number of publications. In [17], a heuristic opportunistic schedulerfor allocating frequency bands to the users in the uplink of 3G LTE systems is proposed.In [18], the authors have proposed a greedy sub-optimal schedular for uplink SC-FDMAsystems that is based on marginal capacity maximization. In [19], the authors revise the sameframework used in [18] for developing a proportional fair scheduling scheme. However,in addition to being sub-optimal, the proposed schedulers in both [18] and [19] do not con-sider the sub-channels adjacency constraint which is an important physical layer requirementfor localized SC-FDMA. In [20], a set of greedy sub-optimal proportional fair algorithmsfor localized SC-FDMA systems is proposed in the frequency-domain setting. This workrespects the sub-channels adjacency constraint but does not consider any constraint on thepower. In [21], a weighted-sum rate maximization in localized SC-FDMA systems is consid-ered where the problem is formulated as a pure binary-integer program. Though the proposedbinary-integer programming framework captures all the basic constraints of the localized SC-FDMA and allows to perform resource allocation without resorting to exhaustive search, itis still not the best solution as the 0–1 requirement turns the problem into combinatorialwith exponential complexity. Thus, keeping in view the computational complexity of thebinary-integer programming, the authors have also proposed a greedy sub-optimal algo-rithm that is similar in spirit to the approach in [18] with an additional constraint on theadjacency of the allocated sub-channels. In [22], we addressed the same problem and usedcanonical duality theory to convert the binary-integer programming problem into a continu-ous domain concave maximization framework, which reduces the computational complexityof the solution from exponential to polynomial. In [23], the authors consider a Machine-to-Machine(M2M)/Human-to-Human (H2H) coexistence scenario in LTE cellular networksand propose a resource allocation algorithm for maximizing total utility of the network wherethe utility is a function of rate. In general, most of the previous work is based on rate/capacitymaximization and very less work to the best of our knowledge has considered power min-imization in uplink SC-FDMA systems. Since the mobile terminals have limited energy,energy-economization is needed and fast power control should be considered while allocat-ing the resources to the users in the uplink. In our previous work [24], power minimizationunder per-user target data rate constraint using canonical duality theory is presented. Thesame problem with hydrid automatic repeat request (HARQ) was considered in [25,26]. Inthese papers, the impact of retransmissions is considerd to ensure that the uplink users donot experience automatic repeat request (ARQ) blocking.
1.3 Motivation and Contributions
In this paper, we consider joint power and sub-channel allocation, and adaptive modulationin localized SC-FDMA systems. We formulate our framework as a sum-utility maximization(SUmax) problem. This problem is combinatorial in nature whose optimal solution hasexponential computational complexity in general. The performance metric considered in theSUmax problem is the total utility of the system. Utility is basically an economics conceptthat reflects the user satisfaction in the system. We assume that each user in the system has anassociated utility function, and the objective is to propose a polynomial-complexity resourceallocation framework that could maximize the sum-utility while respecting all the constraintsof localized SC-FDMA systems specific to the LTE uplink. The user utility function specific
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to this paper is defined as an arbitrary function that is monotonically increasing in user’s SNR.The performance of the system can be further enhanced by choosing an efficient modulationscheme for each user. Therefore, based on the resource allocation, we also propose an adaptivemodulation scheme, wherein an appropriate modulation is chosen for each user dependingupon its effective SNR.
In this paper, we propose a novel framework for the solution of the aforementioned prob-lem. In our optimization framework, first we formulate the optimization problem as binary-integer programming problem. We then transform the binary-integer programming probleminto canonical dual problems [27] in continuous space that are concave maximization prob-lems under ceratin conditions. We provide the global optimality conditions under which thesolution to the dual canonical problem is identical to the solution of the original/primal prob-lem. We also explore some bounds on the sub-optimality of the proposed framework whenthe optimality conditions are not satisfied. Our proposed framework has polynomial timecomplexity which is a significant improvement over exponential complexity.
The novelties of this paper in comparison to our previous works [22,24] are outlined asfollows. The major contribution of this paper in comparison to [22] lies in the enhancedanalytical analysis of the proposed canonical dual approach with the addition of an appendixcontaining the detailed proofs of all the theorems. The proofs of the lemmas are also pro-vided in this paper. In addition, this paper contains a subsection dedicated to the study ofthe optimality gap of the proposed algorithm which was not studied in the previous work.Furthermore, in this paper, we added the exploration of conditions under which the proposedalgorithm is very close to optimal. The adaptive modulation scheme that is a significantcontribution presented in this paper is also novel. Moreover, in [22], a weighted sum-ratemaximization problem was considered whereas in this paper, a more general framework thatis based on sum-utility maximization is formulated. Contrary to the fixed framework consid-ered in [22], the framework presented in this paper is more flexible where the utility functioncan be replaced by any other desired monotonic function of SNR e.g., rate, minus of BER,etc. A comprehensive and detailed system model for SC-FDMA uplink is also includedin this paper which is an important addition. The work done in [24] is different from thecurrent work not only as described above but also in two additional perspectives. First, thework in [24] considers a sum-power minimization problem under constraint on users ratesthat is completely different from the problem studied in the current paper. Second, in [24],no constraints on per-user and per-subchannel transmit power are considered whereas thesechallenging constraints are considered in the current work.
The rest of this paper is organized as follows: Sect. 2 provides the system model, andSect. 3 presents the problem formulation. The canonical dual optimization framework for thesolution of the problem is provided in Sect. 4, and the joint power and sub-channel allocation,and adaptive modulation algorithms are derived in Sect. 5. Section 6 illustrates the numericalresults, and Sect. 7 concludes the paper.
The following notations are used in this paper. Superscripts (.)T , and (.)H stand fortranspose, and Hermitian of a vector or a matrix respectively. Uppercase and lowercaseboldface letters denote matrices, and vectors respectively. The word “dual” used in this paperrefers to “canonical dual”.
2 System Model
We consider the uplink of a single cell model that utilizes localized SC-FDMA. The gener-alization to multi-cell scenario is straightforward by considering the inter cell interference
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in the signal-to-interference-plus-noise ratio (SINR) expression. We make it clear that thispaper does not study inter-cell interference reduction/mangement but aims to optimize theresources in each cell by an efficient resource allocation algorithm. In the cell, K users aresummed to be simultaneously active. The total bandwidth B is divided into N sub-channelseach having 12 sub-carriers. The channel is assumed to be slowly fading or in other wordsassumed to exhibit block fading characteristics. The coherence time of the channel is greaterthan the transmission-time-interval (TTI) so that the channel stays relatively constant duringthe TTI (in 3GPP-LTE, TTI= 0.5 ms). The users’ channel gains are assumed to be perfectlyknown.
In the following, all signals are represented by their discrete time equivalents in thecomplex baseband. Assume that Nk be the number of consecutive sub-channels allocatedto user k (since a sub-channel cannot be allocated to more than one user simultaneously,∑K
k=1 Nk = N ). Let sk = [sk,1, . . . , sk,Nk ]T be the modulated symbol vector of the kthuser, and FN and FH
N denote an N -point DFT and an N -point Inverse DFT (IDFT) matri-ces respectively. The assignment of the data modulated symbols sk to the user specific setof Nk sub-channels can be described by a Nk-point DFT precoding matrix FNk , a N ∗ Nk
mapping matrix Dk and an N -point IDFT matrix FHN . The mapping matrix Dk represents the
blockwise sub-channel allocation where the elements Dk(n, q) for n = 0, . . . , N − 1 andq = 0, . . . , Nk − 1 are given by
Dk(n, q) ={
1 n =∑k−1j=1 N j + q
0 elsewhere(1)
The transmitted signal is then
xk = FHN DkFNk sk (2)
At the receiver, the received signal is transformed into the frequency domain via a N -pointDFT. The received signal vector for user k assuming perfect sample and symbol synchro-nization, is given as
yk = HkFHN DkFNk sk + zk (3)
where Hk = diag(hk,1, . . . , hk,N ) and zk = [zk,1, . . . , zk,N ]T are respectively the diagonalchannel response matrix and the diagonal Additive White Gaussian Noise (AWGN) vectorin the frequency domain. A frequency domain equalizer is then used in order to mitigatethe ISI. The equalized symbols are transformed back to the time domain via an Nk-pointIDFT, and the detection takes place in the time domain. Let Pk,n , and σ 2
z denote the transmitpower of user k on sub-channel n, and the ambient noise variance at the receiver for userk respectively. After several manipulations, the effective SNR for user k can be obtained asfollows [16]:
γ Z Fk =
⎛
⎝ 1
Nk
Nk∑
n=1
1
Pk,nGk,n
⎞
⎠
−1
, γ M M SEk =
⎛
⎝ 11
Nk
∑Nkn=1
Pk,n Gk,n1+Pk,n Gk,n
− 1
⎞
⎠
−1
(4)
where γ Z Fk is the SNR when ZF equalizer is used and γ M M SE
k is the SNR when MMSE
equalizer is used, and where Gk,n = |hk,n |2σ 2
z. The optimization framework proposed in this
paper assumes an MMSE frequency domain equalization at the receiver. Nevertheless, theproposed framework is equally applicable for ZF equalization at the receiver.
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Unlike OFDMA where a different constellation can be adopted for each sub-channel,in SC-FDMA a single constellation is chosen for each user depending upon its channelquality. This is due to the fact that the transmit symbols directly modulate the sub-channelsin OFDMA whereas in SC-FDMA, the transmit symbols are first fed to the FFT block andthe output discrete Fourier terms are then mapped to the sub-channels. In 3GPP LTE, theconstellation for each user is chosen from the set M = {QPSK, 16QAM, 64QAM}.
3 Problem Formulation
In this section, we formulate the optimization problem and its equivalent binary-integerprogramming (BIP) problem. The formulation of the problem as an equivalent binary integerprogram is an intermediate step towards its solution which is then approached by the canonicaldual method.
3.1 Sum-Utility Maximization (SUmax)Problem Formulation
We want to maximize the sum-utility subject to constraint on the total transmit power ofeach individual user Pmax
k . We also have per sub-channel peak power constraint, P peakk,n
i.e., the peak power transmitted on each sub-channel by any user should not exceed P peakk,n
so that the PAPR is kept low [2]. In addition, in SC-FDMA for LTE uplink, the power onall the sub-channels allocated to a user should be equal [2], so that the low PAPR benefitscould retain [1]. The utility of user k denoted as Uk(γk) is an arbitrary function that ismonotonically increasing in user’s SNR γk . The overall resource allocation problem can beformulated as
maxK∑
k=1
Uk(γk) (5)
s.t.∑
n∈Nk
Pk,n ≤ Pmaxk , ∀k
Pk,n ≤ P peakk,n , ∀k, n
Pk,n = Pk,l , ∀k, n, l
Nk ∩N j = ∅,∀k �= j⎧⎨
⎩n ∩
⎛
⎝K⋃
j=1, j �=k
N j
⎞
⎠ = ∅ |n ∈ {n1, n1 + 1, . . . , n2 − 1, n2}⎫⎬
⎭,∀k
where Nk with cardinality Nk is the set of sub-channels allocated to users k, n1 = min(Nk)
and n2 = max(Nk). The fourth constraint determines that each sub-channel is allowed to beallocated to one user at most while the last constraint ensures that the sub-channels includedin the set Nk are consecutive.
The optimization problem (5) is combinatorial in nature. There is a twofold difficultyin solving this problem, that is in addition to the exclusivity restriction on the sub-channelallocation, the allocated sub-channels to any user should be adjacent as well. For example,for K = 10 users and N = 24 sub-channels, the optimal solution requires a search across5.26× 1012 possible sub-channel allocations [21], which is not practical.
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3.2 Equivalent BIP Problem for SUmax problem
As an intermediate step towards its solution, we transform the problem to a binary-integerprogramming where the decisions are made on the basis of feasible set of sub-channel allo-cation patterns that satisfies the exclusivity and adjacency constraints and not on the basis ofindividual sub-channels. In other words, we form groups of contiguous sub-channels whichwill be optimally allocated among the users while respecting the exclusive sub-channels allo-cation constraint. The idea of allocation of sub-channel patterns is the same as in [21]. Weelaborate the general idea of forming the feasible sub-channel patterns with a small example.Let us suppose that we have K = 2 users and N = 4 sub-channels. In any allocation pattern,we put 1 if a sub-channel is allocated to a user, and put 0 if it is not allocated to the user.Thus, keeping in view the sub-channel adjacency constraint, the feasible set of sub-channelpatterns for user k can be summarized in the following matrix.
Ak =
⎡
⎢⎢⎣
0 1 0 0 0 1 0 0 1 0 10 0 1 0 0 1 1 0 1 1 10 0 0 1 0 0 1 1 1 1 10 0 0 0 1 0 0 1 0 1 1
⎤
⎥⎥⎦
where each row corresponds to the sub-channel index, and each column corresponds to thefeasible sub-channel allocation pattern. Note that all the K users have the same allocation pat-terns matrix. We define a K J indicator vector i = [i1, . . . , iK ]T where ik = [ik,1, . . . , ik,J ]T ,and where J is the total number of allocation patterns. Each entry ik, j ∈ {0, 1}which indicateswhether a sub-channel pattern j is allocated to a user k or not. Since a single sub-channelpattern can be allocated to each user, maximizing the users’ sum-utility is equivalent to maxi-mizing the sum-utility of all users over all sub-channel allocation patterns such that each useris assigned a single pattern while respecting the exclusive sub-channel allocation constraint.Based on this analysis we have the following lemma.
Lemma 3.1 The sum-utility maximization problem can be written as the following binary-integer programming problem:
maxi
⎧⎨
⎩P(i) =
K∑
k=1
J∑
j=1
ik, j Uk, j
(γ
e f fk, j
)⎫⎬
⎭(6)
s.t.K∑
k=1
J∑
j=1
ik, j Akn, j = 1, ∀n (6a)
J∑
j=1
ik, j = 1, ∀k (6b)
ik, j ∈ {0, 1}, ∀k, j (6c)
where Uk, j (γe f fk, j ), a monotonically increasing function of the effective SNR γ
e f fk, j is the
utility of user k when allocation pattern j is chosen, and Akn, j denotes the element of matrix
Ak corresponding to nth row and j th column.
Proof The proof is simple and follows from the following illustration. The effective SNRγ
e f fk, j of user k for pattern j is defined as:
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γe f fk, j =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
1
1Nk, j
∑n∈Nk, j
min
(
P peakk,n ,
Pmaxk
Nk, j
)
Gk,n
1+min
(
P peakk,n ,
Pmaxk
Nk, j
)
Gk,n
− 1
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
−1
(7)
where Nk, j is the number of sub-channels allocated to user k when allocation pattern j ischosen. The constraint (6a) ensures the exclusive sub-channel allocation i.e., any two sub-channel patterns allocated to two different users must not have any sub-channel in common.The constraint (6b) means that at most one allocation pattern is chosen for each user. Theper-user total power, the per sub-channel peak power and the allocated sub-channels powerequality constraints are all implicitly accommodated in γ
e f fk, j .
Although the BIP problem may look simple compared to the primal problem but unfor-tunately, due to its combinatorial nature, the computational complexity of its solution isexponential. A similar binary-integer programming solution was proposed for weighted-sum rate maximization problem in [21] but as mentioned before it is exponentially com-plex which is not practical. In the following section, we propose a polynomial-complexityframework for the solution of both the above problems that is inspired from the canoni-cal dual transformation method. The main idea of our proposed approach is to transformeach binary-integer programming problem into a canonical dual problem in the continuousspace whose solution is identical to the corresponding binary integer program under certainconditions.
4 Canonical Dual Approach for Solving the BIP Problem
Under certain constraints/conditions, the canonical duality theory [27] can be used to reformu-late some non-convex/non-smooth constrained problem into certain convex/smooth canon-ical dual problems with perfect primal/dual relationship. However, this theory does notprovide any general strategy for the solution of non-convex/non-smooth problems. Theconstraints under which the canonical dual problem could be perfectly dual to its pri-mal problem is purely dependent on the nature of the primal problem under consider-ation and should be studied for each specific problem anew. This theory comprises ofcanonical dual transformation, an associated complementary-dual principle, and an asso-ciated duality theory. The canonical dual transformation can be used to convert the non-smooth problem into a smooth canonical dual problem; the complementary-dual prin-ciple can be used to study the relationship between the primal and its canonical dualproblems; and the associated duality theory can help to identify both local and globalextrema. Comprehensive details about this theory, and its application to an unconstrained0–1 quadratic programming problems can be found in [27], and [28] respectively. Due tothe presence of additional constraints, our problem is far more difficult compared to thatdescribed in [28].
By using the aforementioned theory, we transform the SUmax primal problem into acontinuous space canonical dual problem in the following. We then study the optimalityconditions, and prove that under these conditions, the solution of the canonical dual problemis identical to that of the corresponding primal problem.
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4.1 Canonical Dual Problem and Optimality Conditions
The objective function, P(i) in problem (6) is a real valued linear function defined on Ia =i ⊂ R
K×J with feasible space defined by
I f =⎧⎨
⎩i ∈ Ia ⊂ R
K×J∣∣∣∣
K∑
k=1
J∑
j=1
ik, j Akn, j = 1,∀n;
J∑
j=1
ik, j = 1,∀k; ik, j ∈ {0, 1}∀k, j
⎫⎬
⎭
(8)
We start our development by introducing new constraints ik, j (ik, j − 1) = 0,∀k, j whichmeans that any ik, j can only take an integer value from the set {0, 1}. This approach is usedfor the solution of a 0–1 quadratic programming problem in [28]. However, the problemconsidered in [28] is a simple unconstrained 0–1 quadratic programming problem while ourproblem is combinatorial in nature with additional constraints. In other words, in addition tothe binary-integer constraint on ik, j ’s, we have the mutual exclusivity restriction on the alloca-tion of sub-channel patterns
(i.e., {ik, j × il, j = 0|k �= l, ∀k, l ∈ {1, ..., K }}), and the mutual
exclusivity constraint on the sub-channel allocation i.e.,∑K
k=1∑J
j=1 ik, j Akn, j = 1,∀n. Fur-
thermore, at most one sub-channel pattern can be allocated to a user i.e.,∑J
j=1 ik, j = 1,∀k.Note that the mutual exclusivity restriction on the sub-channel patterns allocation is accom-modated implicitly in the formulation of the primal problem and does not show up explicitly.We temporarily relax the new constraints ik, j (ik, j−1) = 0,∀k, j , and the equality constraints(6a–6b) to inequalities and transform the primal problem with these inequality constraintsinto continuous domain canonical dual problem. We will then solve the canonical dual prob-lem in the continuous space and chose the solution which lies in I f as defined by (8).Furthermore, for our convenience, we reformulate our primal problem as an equivalent min-imization problem. The primal problem with these inequality constraints can now be writtenas follows.
mini
⎧⎨
⎩f (i) = −
K∑
k=1
J∑
j=1
ik, j Uk, j
⎫⎬
⎭(9)
s.t.K∑
k=1
J∑
j=1
ik, j Akn, j ≤ 1, ∀n
J∑
j=1
ik, j ≤ 1, ∀k
ik, j(ik, j − 1
) ≤ 0, ∀k, j
k, j ∈ {0, 1}, ∀k, j
where Uk, j is used to denote Uk, j (γe f fk, j ) and will be used in the remainder of the paper.
The temporary relaxation of the constraints to inequalities is needed for developing thecanonical dual framework. We prove later that the solution of the canonical dual problemachieves the binary-integer constraints i.e., ik, j
(ik, j − 1
) = 0,∀k, j and all the other con-straints with equality. As a first step towards its transformation into a canonical dual problem,we relax the primal problem [27,28]. To this end, we define the so-called canonical geomet-rical operator x = �(i) for the above primal problem as follows:
x = �(i) = (ε,λ, ρ) : RK J → RN × R
K × RK J (10)
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A. Ahmad
which is a vector-valued mapping and where ρ = [iT1 (i1−1), . . . , iTK (iK−1)]T is a KJ-vector
with iTk (ik−1) = [ik,1(ik,1−1), . . . , ik,J (ik,J−1)]T , ε=[(∑K
k=1∑J
j=1 ik, j Ak1, j − 1
), . . . ,
(∑Kk=1
∑Jj=1 ik, j Ak
N , j − 1)]T
is an N-vector, and λ =[(∑J
j=1 i1, j − 1)
, . . . ,(∑J
j=1
iK , j − 1)]T is a K-vector. Let χa be a convex subset of χ = R
N × RK × R
K J defined asfollows
χa ={
x = (ε,λ, ρ) ∈ RN × R
K × RK J | ε ≤ 0,λ ≤ 0, ρ ≤ 0
}(11)
We introduce an indicator function V : χ → R ∪ {+∞}, defined as
V (x) ={
0 if x ∈ χa,
+∞ otherwise.(12)
Thus, the inequality constraints in the primal problem (9) can now be relaxed by the indicatorfunction V (x), and the primal problem can be written in the following canonical form [28]:
mini
⎧⎨
⎩V (�(i))−
K∑
k=1
J∑
j=1
ik, j Uk, j | ik, j ∈ {0, 1}∀k, j
⎫⎬
⎭(13)
We now define the canonical dual variables and the canonical conjugate function associatedto the indicator function in order to proceed with the transformation of the primal probleminto canonical dual. Since V (x) is convex, lower semi-continuous on χ , the canonical dualvariable x∗ ∈ χ∗ = χ = R
N × RK × R
K J is defined as:
x∗ ∈ ∂V (x) ={(
ε∗,λ∗, ρ∗)
if ε∗ ≥ 0 ∈ RN ,λ∗ ≥ 0 ∈ R
K , ρ∗ ≥ 0 ∈ RK J ,
∅ otherwise.(14)
By the Legendre-Fenchel transformation, the canonical super-conjugate function of V (x) isdefined by
V �(x∗) = supx∈χ
{xT x∗ − V (x)
}= sup
ε≤0supλ≤0
supρ≤0
{εT ε∗ + λT λ∗ + ρT ρ∗
}
={
0 if ε∗ ≥ 0,λ∗ ≥ 0, ρ∗ ≥ 0,
+∞ otherwise.(15)
The effective domain of V �(x) is given by
χ∗a ={(
ε∗,λ∗, ρ∗) ∈ R
N × RK × R
K J |ε∗ ≥ 0 ∈ RN ,λ∗ ≥ 0 ∈ R
K , ρ∗ ≥ 0 ∈ RK J
}
(16)
Since both V (x) and V �(x) are convex, lower semi-continuous, the Fenchel sup-dualityrelations
x∗ ∈ ∂V (x)⇔ x ∈ ∂V �(x∗)⇔ V (x)+ V �(x∗) = xT x∗ (17)
hold on χ × χ∗. The pair (x, x∗) is called the extended/Legendre canonical dual pair onχ × χ∗, and the functions V (x) and V �(x) are called canonical functions [27]. The optimalsolution of our primal problem can be obtained if and only if x = I f ∈ χa , i.e., along withthe satisfaction of the binary-integer constraints, all the other constraints must be achievedwith equality. Thus, we need to study the conditions under which the canonical dual variables
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Resource allocation and adaptive modulation
x∗ ∈ χ∗a can ensure that x = I f ∈ χa . By the definition of sub-differential, the canonicalsup-duality relations (17) are equivalent to the following:
x ≤ 0, x∗ ≥ 0, xT x∗ = 0 (18)
From the complementarity condition xT x∗ = 0, for x∗ > 0, we have x = 0 (i.e., ε = 0,
λ = 0, ρ = 0) and consequently x = I f ∈ χa . This means that for x∗ > 0, all the constraintsof the primal problem (9) are achieved by equality (with ik, j ∈ {0, 1},∀k, j which comesfrom ρ = 0). Thus, the dual feasible space for the primal problem is an open positive conedefined by
χ∗� ={(
ε∗,λ∗, ρ∗) ∈ χ∗a | ε∗ > 0,λ∗ > 0, ρ∗ > 0
}(19)
The so-called total complementarity function (see [27,28] for definition), �(i, x∗) : χ ×χ∗� → R associated with the primal problem (9) can be defined as follows.
�(i, x∗) = �(i)T x∗ − V �(x∗)−K∑
k=1
J∑
j=1
ik, j Uk, j (20)
which is obtained by replacing V (�(i)) in (13) by �(i)T x∗ − V �(x∗) from Fenchel sup-duality relations (17). From the definition of �(i) and V �(x∗), the total complementarityfunction takes the form:
�(i, ε∗,λ∗, ρ∗) =K∑
k=1
J∑
j=1
{
ρ∗k, j i2k, j +
(
λ∗k − ρ∗k, j −Uk, j +N∑
n=1
ε∗n Akn, j
)
ik, j
}
−N∑
n=1
ε∗n −K∑
k=1
λ∗k (21)
Similar to [28], the canonical dual function fd(ε∗,λ∗, ρ∗) associated to our primal problemfor a given (ε∗,λ∗, ρ∗) ∈ χ∗� can be defined as
fd(ε∗,λ∗, ρ∗) = sta{�(i, ε∗,λ∗, ρ∗) | i ∈ Ia
}(22)
where sta{ f (x)} stands for finding the stationary points of f (x). The complementarity func-tion is a quadratic function of i ∈ Ia , and has therefore a unique stationary point with respectto it for a given (ε∗,λ∗, ρ∗) ∈ χ∗a . The stationary points of �(i, ε∗,λ∗, ρ∗) over i ∈ Ia
occurs at i(x∗) with
ik, j (x∗) = 1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)
, ∀k, j (23)
Replacing ik, j by ik, j (x∗) in (21), we have
fd(ε∗,λ∗, ρ∗) = −1
4
K∑
k=1
J∑
j=1
⎧⎪⎨
⎪⎩
(Uk, j + ρ∗k, j − λ∗k −
∑Nn=1 ε∗n Ak
n, j
)2
ρ∗k, j
⎫⎪⎬
⎪⎭
−N∑
n=1
ε∗n −K∑
k=1
λ∗k (24)
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A. Ahmad
which is a concave function in χ∗� . The canonical dual problem associated with the primalproblem (9) can now be formulated as follows
ext{
fd(ε∗,λ∗, ρ∗) | (ε∗,λ∗, ρ∗) ∈ χ∗�}
(25)
where the notation ext { f (x)} stands for finding the extremum values of f (x).We have the following canonical duality theorem (Complementary-Dual Principle) on the
perfect dual relationship between the primal and its corresponding canonical dual problem.
Theorem 4.1 If (ε∗,λ∗, ρ∗) ∈ χ∗� is the stationary point of fd(ε∗,λ∗, ρ∗), such that
i = [i1,1, . . . , i K ,J ]T with ik, j = 1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −
N∑
n=1
ε∗n Akn, j
)
,∀k, j
(26)
is the KKT point of the primal problem, and
f(i) = fd(ε∗,λ∗, ρ∗). (27)
then the canonical dual problem (25) is perfectly dual to the primal problem (6).
Proof See Appendix 1 for the proof. The above theorem shows that the binary-integer programming problem (6) is converted
into a dual problem in continuous domain which is perfectly dual to it. Furthermore, theKKT point of the dual problem provides the KKT point for the primal problem. However, asthe KKT conditions are necessary but not sufficient for optimality in general, we need someadditional information on the global optimality. Based on the properties of the primal anddual problems, we have the following theorem on the global optimality conditions.
Theorem 4.2 If (ε∗,λ∗, ρ∗) ∈ χ∗� , then i defined by (26) is a global minimizer of f(i) over
I f and (ε∗,λ∗, ρ∗) is a global maximizer of fd(ε∗,λ∗, ρ∗) over χ∗� , and
f(i) = mini∈I f
f(i) = max(ε∗,λ∗,ρ∗)∈χ∗�
fd(ε∗,λ∗, ρ∗) = fd(ε∗,λ∗, ρ∗). (28)
Proof See Appendix 2. Based on the above mathematical analysis, we provide joint power and sub-channel allo-
cation algorithm in the following section. An adaptive modulation scheme. The proposedadaptive modulation is based on the powers and sub-channels allocated to each user by theproposed resource allocation algorithm.
5 Resource Allocation and Adaptive Modulation Algorithms
5.1 Joint Power and Sub-channel Allocation Algorithm
The proposed algorithm is based on the solution of canonical dual problem which according totheorem 4.2 provides the optimal solution to the primal problem if the given global optimalityconditions are met. Since the dual problem is a concave maximization problem over χ∗� , it
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Resource allocation and adaptive modulation
is necessary and sufficient to solve the following system of equations for finding the optimalsolution [30].
∂fd
∂ε∗n=
K∑
k=1
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)
Akn, j
}
− 1 = 0, ∀n (29)
∂fd
∂λ∗k=
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)}
− 1 = 0, ∀k (30)
∂fd
∂ρ∗k, j=
(Uk, j − λ∗k −
∑Nn=1 ε∗n Ak
n, j
ρ∗k, j
)2
− 1 = 0, ∀k, j (31)
We propose a sub-gradient based iterative algorithm for the above system of non-linearequations that is equivalent to solving fd(ε∗,λ∗, ρ∗) using gradient-decent method [30]. Theinterest of using the sub-gradient method is its ability to use the decomposition techniquethat allows to simplify the solution by using a distributed method. The iterative algorithmis given in Table 1 where each of q , s and t denotes the iteration number and βρ∗ , βλ∗ andβε∗ denote the step sizes for the sub-gradient update. For an appropriate step size, the sub-gradient method is always guaranteed to converge [30]. The algorithm starts by initializingthe variables. Then, for the given ε∗(0) and λ∗(0), the solution to the set of equations (31) i.e.,ρ∗(q) is obtained in step 1. The operation
ρ∗(q)← χρ∗(ρ∗(q−1) + βρ∗ζ
(q−1))
:=⎧⎨
⎩
ρ∗k, j(q)=ρ∗k, j
(q−1) + sgn(ρ∗k, j
(q−1))
η if (ρ∗(q−1) + βρ∗ζ (q−1))=0,∀k, j
ρ∗k, j(q)=ρ∗(q−1) + βρ∗ζ (q−1) otherwise.
(32)
Table 1 Resource allocation algorithm
Initialize (ε∗(0), λ∗(0), ρ∗(0)) ∈ χ∗�1. Compute ζ (q) = ∂fd
∂ρ∗ |ρ∗(q) . If |ζ (q)| ≤ δ, go to step 2.
• Set ρ∗(q+1) ← χρ∗(ρ∗(q) + βρ∗ζ (q)
).
• Set q ← q + 1, and repeat step 1.
2. Compute η(s) = ∂fd
∂λ∗ |λ∗(s) . If |η(s)| ≤ δ, go to step 3.
• Set λ∗(s+1) ←(λ∗(s) + βλ∗η(s)
).
• Set s ← s + 1, and repeat step 2.
3. Compute υ(t) = ∂fd
∂ε∗ |ε∗(t) . If |υ(t)| ≤ δ, go to step 4.
• Set ε∗(t+1) ←(ε∗(t) + βε∗υ(t)
).
• Set t ← t + 1, and repeat step 3.
4. Recompute η(s) = ∂fd
∂λ∗ |λ∗(s) .
5. Repeat steps 2 through 4 until |η(s)| ≤ δ, and |υ(t)| ≤ δ
6. Recompute ζ (q) = ∂fd
∂ρ∗ |ρ∗(q)
7. Repeat steps 1 through 6 until |ζ (q)| ≤ δ, |η(s)| ≤ δ, and |υ(t)| ≤ δ
8. Compute i according to (26).
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A. Ahmad
in step 1 is the projection of ρ∗ onto the space χ∗ρ∗ = {ρ∗ ∈ RK J |ρ∗ �= 0}, since the canonical
dual objective function is not defined at ρ∗ = 0. In (32), sgn stands for sign/signum functionand 0 < η << 1. According to the above projection, if the updated value of ρ∗ in the currentiteration occurs to be zero, it is projected to the negative domain if its value was positivein the previous iteration, and vice versa. This projection has no impact on the convergence,since the sign of ρ∗ does not change the direction of the gradient (see Eq. (31)). Step 2 findsλ∗(s) that solves equations’ set (30) for the given ε∗(0) and ρ∗(q). These values of ρ∗(q) andλ∗(s) are then used to solve the set of equations (29) by updating ε∗(0) to ε∗(t) in step 3.
Step 4 checks whether | ∂fd
∂λ∗ | ≤ δ for ρ∗(q), λ∗(s) and the updated ε∗(t) where δ → 0 is the
stopping criterion for sub-gradient update. If | ∂fd
∂λ∗ | > δ, steps 2 through 4 are repeated until
both | ∂fd
∂λ∗ | ≤ δ and | ∂fd
∂ε∗ | ≤ δ. In step 6, ζ (q) is recomputed for ρ∗(q), and the updated λ∗(s)
and ε∗(t). If |ζ (q)| ≤ δ, the algorithm is stopped otherwise steps 1 through 6 are repeateduntil convergence. The resource allocation vector i is then obtained from the dual optimalsolution (ε∗,λ∗, ρ∗) in step 8.
5.2 Adaptive Modulation Scheme
By knowing perfectly the effective SNR of each user from the powers and sub-channels allo-cation performed according to the previous subsection, we propose an adaptive modulationscheme in this subsection. The proposed adaptive modulation scheme is based on the crite-rion of target Target Block Error Rate (BLER) at the receiver. According to this approach,for a modulation m ∈ M to be chosen, the effective SNR of the user should not be less thana minimum value �∗m that guarantees a target BLER at the receiver. Since the effective SNRof users are perfectly known from the the powers and sub-channels allocation performedaccording to the previous subsection, we adopt the modulation for each user which maxi-mizes its individual utility. Thus, depending upon γ
e f fk , the efficient modulation for user k
is determined as follows:
m∗(k) = arg minm∈M
{(γ
e f fk − �∗m
) ∣∣∣�∗m≤γ
e f fk
}
(33)
Note that the above approach is similar in spirit to the approach used in [29] where adaptivemodulation in OFDM system is considered and an efficient constellation is chosen for eachsub-channel.
5.3 Complexity of the Algorithm
In each iteration for ρ∗, we compute K J variables. The number of variables computed ineach iteration for λ∗ is K and that for ε∗ is N . Assume that the number of iterations requiredfor optimal ρ∗, λ∗ and ε∗ are Iρ∗ , Iλ∗ and Iε∗ respectively, then the algorithm has an overallcomplexity of O(Iρ∗K J+Iλ∗K+Iε∗N ). By ignoring the number of iterations, the complexityin terms of K , J and N becomes O(K J + K + N ).
5.4 On the Optimality of the Algorithm
The canonical dual problem is a concave maximization problem over χ∗� , the proposed
algorithm is then surely optimal if (ε∗,λ∗, ρ∗) ∈ χ∗� . However, if (ε∗,λ∗, ρ∗) is not inside thepositive cone χ∗� , then the canonical problem is not guaranteed to be concave. Consequently,the proposed algorithm may not find the optimal solution. From our simulation results, we
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Resource allocation and adaptive modulation
have observed that the proposed algorithm works well, and the canonical dual solution isvery close to the optimal solution.
In this subsection, we analyze the gap between the optimal solution and the solutionobtained by using our proposed sub-gradient based algorithm. We start the analysis by intro-ducing a modified problem whose optimal solution is not necessary and will not replace ouractual problem but is used only to study the optimality gap of our proposed algorithm. Inour analysis, first we find the solution of the modified problem (which is a stationary pointand may not be necessarily the optimal solution of this modified problem). Then, we show inTheorem 5.1 that there exist a primal problem with a slightly different values of the utilitiesUk, j ’s whose optimal solution is equal to the solution of this modified problem. Finally, inCorollary 5.1 we show that under certain conditions, the solution of the canonical dual prob-lem obtained using the algorithm in Table 1 provides solution to the primal problem whichis very close to optimal solution.
Let us consider the following modified problem
(P1) : maxε∗,λ∗,ρ∗
fd(ε∗,λ∗, ρ∗) (34)
s.t. ε∗ ≥ c (34a)
λ∗ ≥ d (34b)
where (c, d) ∈ (RN+ , RK+ ). We solve this problem using the standard Lagrangian technique.
Let (ε∗,λ∗, ρ∗) be the obtained solution. The corresponding Lagrangian can be defined as
L(ε∗,λ∗, ρ∗, σε∗ , σλ∗ , σρ∗) = fd(ε∗,λ∗, ρ∗)− (ε∗T − cT )σ ε∗ − σλ∗(λ∗T − dT )
(35)
where (σ ε∗ , σλ∗) ∈ (RN , RK ) are the Lagrange multipliers associated to the constraints
(34a–34b) respectively. The corresponding KKT conditions are:
∂L
∂ε∗= 0 ⇒
K∑
k=1
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)
Akn, j
}
= 1+ σ ε∗n ,∀n
(36)
∂L
∂λ∗= 0 ⇒
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)}
= 1+ σ λ∗k , ∀k (37)
∂L
∂ρ∗= 0 ⇒
(Uk, j − λ∗k −
∑Nn=1 ε∗n Ak
n, j
ρ∗k, j
)2
− 1 = 0, ∀k, j (38)
The above equation can be solved using the sub-gradient based algorithm in Table 1. More-over, in order to ensure that the solution of (38) is obtained for positive ρ∗, we can use thefollowing projection in the update of ρ∗:
ρ∗(q) ← χρ∗((q−1)
)
:=⎧⎨
⎩
ρ∗k, j(q) = arg minρ∗k, j∈χ∗� ‖�
(q−1)k, j − ρ∗k, j‖ if �k, j
(q−1) ≤ 0,∀k, j
ρ∗k, j(q) = �k, j
(q−1) otherwise.(39)
where (q−1) = ρ∗(q−1) + βρ∗ ∂L∂ρ∗ |ρ∗(q−1) denotes the sub-gradient update, and where βρ∗
is the step size. The above projection ensures the positivity of ρ∗.
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A. Ahmad
Theorem 5.1 For Uk, j = Uk, j − 2θk, jρ∗k, j with θk, j = {−1, 0, 1},∀k, j ; there exists a
primal problem f(i) with utilities Uk, j replaced for Uk, j that can be solved optimally usingthe algorithm in Table I. The solution (ε∗,λ∗, ρ∗) of f(i) obtained using Table I is equal tothe solution of the modified problem (34).
Proof See Appendix 3 for the proof. Moreover, we have the following result which is the corollary of Theorem 5.1.
Corollary 5.1 If ρ∗k, j << Uk, j ,∀k, j ; then the solution of the canonical dual problemobtained using the sub-gradient based algorithm (Table I) provides a solution to the primalproblem which is very close to the optimal solution.
Proof See Appendix 4.
5.4.1 Analysis of the Algorithm’s Results for N →∞
When the number of sub-channel N is very high, the utilities Uk, j will also be very high andthe ρ∗k, j will be very small compared to Uk, j .
6 Simulation Results
We consider a system with 5 MHz of bandwidth (i.e. LTE) divided into N = 25 sub-channelseach having a bandwidth of 180 kHz. We assume that K = 10 uniformly distributed users aresimultaneously active in a cell of 500 m. The scenario assumed is urban canyon macro whichexists in dense urban areas served by macro-cells. A frequency selective Rayleigh fadingchannel is simulated where the channel gain has a small-scale Rayleigh fading componentand a large-scale path loss and shadowing component. Path losses are calculated accordingto Cost-Hata Model and shadow fading is log-normally distributed with a standard deviationof 8 dBs. Time is divided into slots where the duration of each slot is 0.5 ms. The carrierfrequency is assumed to be 2.6 GHz. The power spectral density of noise is assumed to be−174 dBm/Hz. The power spectral density of noise is assumed to be −174 dBm/Hz. Theper sub-channel peak power constraint is P peak
k,n =10 mW, and the per user maximum powerconstraint is Pmax
k =200 mW. Simulation parameters are summarized in Table 2.In simulations, we assume that the utility of the user is equal to its weighted rate where
the rate is defined by Shannon’s formula. In other words, the SUmax problem is equivalent toweighted-sum rate maximization. Figure 1 plots the empirical cumulative distribution func-tion (CDF) of sum-utility for different resource allocation algorithms. The figure illustratesthe comparison of the CDF’s corresponding to our proposed algorithm, both the binary-integer programming solution and the greedy algorithm proposed in [21], and the roundrobin scheme in which an equal number of consecutive sub-channels are allocated to eachuser in turn. The figure shows that although the greedy algorithm proposed by Wong et al.is efficient in comparison to the round robin scheme, its performance is far away from theproposed solution. Moreover, it can be seen from the figure that the results of the proposedalgorithm are very close to that obtained by solving the binary-integer program which is theoptimal solution.
Figure 2 illustrates the the sum-utility of the system versus number of users. The sum-utility for all the algorithms increases as the number of users increases. The constraints on
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Resource allocation and adaptive modulation
Table 2 Simulation parameters Parameter Value/type
System bandwidth, W 5 Mhz
Total number of sub-channels 25
Bandwidth of each sub-channel 180 Khz
Carrier frequency 2.6 Ghz
Transmit time interval(TTI) duration
0.5 ms
Noise power spectral density −174 dBm/Hz
Radius of the cell 500 m
per sub-channel peak power, P peakk,n 10 mW
per user maximum power, Pmaxk 200 mW
Assumed wireless scenario Dense urban area servedby macro-cells
Path loss model used Cost-Hata Model
Shadow fading Log-normally distributed withstandard deviation of 8 dB
Fig. 1 Empirical CDF of sum-utility for K = 10 and N = 25
per-user power limits the transmission rate of the users. Due to this reason, even if all thesub-channels are used for transmission, the sum-utility of the system for smaller numberof user is less. As the number of users increases, the sub-channels are efficiently utilizedresulting in increased sum-utility. However, with the increasing number of users the slopeof the curve (rate of increase of sum-utility) is decreasing. This is due to that fact that asthe number of users increases, the number of sub-channels allocated to each user decreaseswhich due to the per sub-channel peak power constraint reduces the transmission rate of eachuser. It is clear from that the figure that the proposed algorithm outperforms both the round
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A. Ahmad
Fig. 2 Sum-utility versus number of users
Fig. 3 Spectral efficiency versus number of users
robin scheme and the greedy algorithm. It can also be observed that the performance of theproposed algorithm is very close to the optimal performance.
The spectral efficiency versus number of users for different algorithms is presented inFig. 3. The figure shows that the the proposed scheme achieves higher spectral efficiencycompared to the greedy algorithm [21] and the traditional round robin scheme. Furthermore,
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Resource allocation and adaptive modulation
Table 3 Consecutivesub-channels allocation amongusers by different algorithms forK = 4 and N = 12
Algorithm type Sub-channel allocation among users
User Index, k Indices of allocatedsub-channels to user, n
Proposed 1 8, 9, 10, 11
2 12
3 1
4 2, 3, 4, 5, 6, 7
Greedy (Wong. et al.) 1 Non
2 Non
3 1, 2
4 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Round Robin 1 1, 2, 3
2 4, 5, 6
3 7, 8, 9
4 10, 11, 12
it can be observed from the figure that the spectral efficiency curve of the proposed algorithmis almost overlapping that of the binary-integer programming based optimal algorithm.
Table 3 illustrate the performance of the proposed algorithm in terms of sub-channelallocation among users. Each sub-channel is allocated to a single user and the multiplesub-channels allocated to any user are contiguous for all the algorithms. It can be observedthat for the greedy algorithm, all the sub-channels are distributed among users 3 and 4without allocating any sub-channel to the other users. On the other hand, for the proposedalgorithm, every user has got a share in the sub-channel allocation in an efficient manner.This efficient sub-channel allocation by the proposed algorithm results in higher sum-utility.For the round robin scheme though all the users get equal number of sub-channels, thesub-channel allocation is inefficient and this leads to reduced sum-utility.
7 Conclusion
This paper studies joint power and sub-channel allocation, and adaptive modulation in uplinkSC-FDMA systems. A sum-utility maximization problem is considered whose optimal solu-tion is exponentially complex in general. A polynomial-complexity optimization frameworkthat is inspired from the recently developed canonical duality theory is derived for the solu-tion of the problem. Based on the resource allocation performed by the proposed framework,an adaptive modulation scheme is also proposed that determines the best constellation foreach user. The optimization problem is first formulated as binary-integer programming prob-lem and then, the binary-integer problem is transformed into a canonical dual problem inthe continuous space which is a concave maximization problem. The transformation of theproblem in continuous space significantly improves the performance of the system in termsof computational complexity. The proposed continuous space optimization framework has apolynomial time complexity that is a significant improvement over exponential complexity.It is proved analytically that under certain conditions, the solution of the canonical dual prob-lem is identical to the solution of the primal problem. However, if the dual solution does notsatisfy these conditions then the optimality can not be guaranteed. Therefore, some bounds
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A. Ahmad
on the sub-optimality of the proposed framework when these conditions are not satisfied arealso explored. The performance of the proposed canonical dual framework is assessed bycomparing it with the existing algorithms in the literature. The numerical results show thatthe proposed framework provides integer solution to the problem. In addition, the resultsshow the obtained solution is very close to the optimal solution.
Acknowledgments This work was supported by French Systematic Project RAF.
Appendix
Proof of Theorem 4.1
Note that the proof of this theorem can be directly obtained from the proof given in [28] butwe provide it for the completeness of the paper. We introduce Lagrange multipliers to relaxthe strict inequality constraints (ε∗,λ∗, ρ∗) > 0 in χ∗� . We recall that the canonical dualmethod is completely different from the Lagrange dual method and the Lagrange multipliershas nothing to do with the formulation of the canonical dual problem but are used here toprove that the primal and the corresponding conical dual problem have the same KKT points.Let (δε∗ , δλ∗ , δρ∗) ∈ (RN , R
K , RK J ) be the Lagrange multipliers associated to the inequality
constraints (ε∗,λ∗, ρ∗) > 0, then the Lagrangian associated to the complementarity function�(i, ε∗,λ∗, ρ∗) can be defined as follows:
L(i, ε∗,λ∗, ρ∗, δε∗ , δλ∗ , δρ∗) = �(i, ε∗,λ∗, ρ∗)+ ε∗Tδε∗ + λ∗T
δλ∗ + ρ∗Tδρ∗ (40)
The KKT conditions of the primal problem are:
∂L
∂i= 0 ⇒ 2ρ∗k, j i k, j +
(
λ∗k − ρ∗k, j −Uk, j +
N∑
n=1
ε∗n Akn, j
)
= 0, ∀k, j (41)
∂L
∂ε∗= 0 ⇒
K∑
k=1
J∑
j=1
Akn, j i k, j − 1+ δε∗
n = 0, ∀n (42)
∂L
∂λ∗= 0 ⇒
J∑
j=1
i k, j − 1+ δλ∗k = 0, ∀k (43)
∂L
∂ρ∗= 0 ⇒ i k, j (i k, j − 1)+ δ
ρ∗k, j = 0, ∀k, j (44)
(δε∗ , δλ∗ , δρ∗) ≤ 0, (ε∗,λ∗, ρ∗) > 0, ε∗Tδε∗ = 0, λ
∗Tδλ∗ =0, ρ∗T
δρ∗ = 0
(45)
From the KKT condition (41), we get i k, j = 12ρ∗k, j
(Uk, j + ρ∗k, j − λ
∗k −
∑Nn=1 ε∗n Ak
n, j
).
According to complementarity conditions (45), the Lagrange multipliers (δε∗ , δλ∗ , δρ∗) = 0for (ε∗,λ∗, ρ∗) > 0 and conditions (42–44) become
K∑
k=1
J∑
j=1
Akn, j i k, j − 1 = 0, ∀n (46)
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Resource allocation and adaptive modulation
J∑
j=1
i k, j − 1 = 0, ∀k (47)
i k, j (i k, j − 1) = 0, ∀k, j (48)
Replacing 12ρ∗k, j
(Uk, j + ρ∗k, j − λ
∗k −
∑Nn=1 ε∗n Ak
n, j
)for i k, j in (46–48) leads to
K∑
k=1
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −
N∑
n=1
ε∗n Akn, j
)
Akn, j
}
− 1 = 0, ∀n (49)
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −
N∑
n=1
ε∗n Akn, j
)}
− 1 = 0, ∀k (50)
1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −
N∑
n=1
ε∗n Akn, j
)
×{
1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −
N∑
n=1
ε∗n Akn, j
)
− 1
}
= 0,∀k, j (51)
which are in fact the KKT conditions of the canonical dual problem, fd(ε∗,λ∗, ρ∗). Thisproves that for (ε∗,λ∗, ρ∗) ∈ χ∗� being the KKT point of fd(ε∗,λ∗, ρ∗), i given by (26) isthe KKT point of the primal problem. This establishes the first part of the theorem.
According to (20), the total complementarity function at the KKT point (i, x∗) can bewritten as
�(i, ε∗,λ∗, ρ∗) = �(i)T x∗ − V �(x∗)−K∑
k=1
J∑
j=1
i k, j Uk, j
= −K∑
k=1
J∑
j=1
i k, j Uk, j = f(i) (52)
which is obvious from the fact that V �(x∗) = 0 for (ε∗,λ∗, ρ∗) > 0, and where Eqs. (46–48)imply that �(i) = 0. Similarly from (21), we have
�(i, ε∗,λ∗, ρ∗) =K∑
k=1
J∑
j=1
⎧⎨
⎩ρ∗k, j i
2k, j +
⎛
⎝λ∗k − ρ∗k, j −Uk, j +
N∑
n=1
ε∗n Akn, j
⎞
⎠ ik, j
⎫⎬
⎭
−N∑
n=1
ε∗n −K∑
k=1
λ∗k
= −1
4
K∑
k=1
J∑
j=1
⎧⎪⎨
⎪⎩
(Uk, j + ρ∗k, j − λ
∗k −
∑Nn=1 ε∗n Ak
n, j
)2
ρ∗k, j
⎫⎪⎬
⎪⎭−
N∑
n=1
ε∗n −K∑
k=1
λ∗k
= f d (ε∗,λ∗, ρ∗) (53)
This shows that the canonical dual problem is perfectly dual to the primal problem. Thiscompletes the proof.
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A. Ahmad
Proof of Theorem 4.2
The total complementarity function �(i, ε∗,λ∗, ρ∗) is convex in i and concave (linear) in ε∗,λ∗ and ρ∗. Therefore, the stationary point (i, ε∗,λ∗, ρ∗) is a saddle point of �(i, ε∗,λ∗, ρ∗).Furthermore, fd(ε∗,λ∗, ρ∗) is defined by �(i, ε∗,λ∗, ρ∗) with i being a stationary point of�(i, ε∗,λ∗, ρ∗) with respect to i ∈ Ia . Consequently, fd(ε∗,λ∗, ρ∗) is concave on χ∗� and
the KKT point (ε∗,λ∗, ρ∗) ∈ χ∗� must be its global maximizer. Thus, by the saddle mini–maxtheorem:
fd(ε∗,λ∗, ρ∗) = maxε∗>0
maxλ∗>0
maxρ∗>0
fd(ε∗,λ∗, ρ∗)
= maxε∗>0
maxλ∗>0
maxρ∗>0
mini∈Ia
�(i, ε∗,λ∗, ρ∗)
= maxε∗>0
maxλ∗>0
maxρ∗>0
mini∈Ia
{f(i)+ εT ε∗ + λT λ∗ + ρT ρ∗
}
= maxε∗>0
maxλ∗>0
mini∈Ia
⎧⎨
⎩f(i)+ εT ε∗+ λT λ∗+ max
ρ∗>0
⎧⎨
⎩
K∑
k=1
J∑
j=1
ρ∗k, j ik, j (ik, j−1)
⎫⎬
⎭
⎫⎬
⎭
= maxε∗>0
mini∈Ia
⎧⎨
⎩f(i)+ εT ε∗ + max
λ∗>0
⎧⎨
⎩
K∑
k=1
λ∗k
⎛
⎝J∑
j=1
ik, j − 1
⎞
⎠
⎫⎬
⎭
⎫⎬
⎭
s.t. ik, j (ik, j − 1) = 0,∀k, j
= mini∈Ia
⎧⎨
⎩f(i)+max
ε∗>0
⎧⎨
⎩
N∑
n=1
ε∗n
⎛
⎝K∑
k=1
J∑
j=1
ik, j Akn, j − 1
⎞
⎠
⎫⎬
⎭
⎫⎬
⎭
s.t. ik, j (ik, j − 1) = 0,∀k, j;J∑
j=1
ik, j = 1,∀k
= mini∈Ia
f(i) s.t.
⎧⎨
⎩ik, j (ik, j − 1) = 0,∀k, j;
J∑
j=1
ik, j = 1,∀k;
K∑
k=1
J∑
j=1
ik, j Akn, j = 1,∀n
⎫⎬
⎭
= mini∈I f
f(i) (54)
Note that the linear programming
maxρ∗>0
⎧⎨
⎩
K∑
k=1
J∑
j=1
ρ∗k, j ik, j (ik, j − 1)
⎫⎬
⎭
has a finite solution in the open domain χ∗� if and only if ik, j (ik, j−1) = 0,∀k, j . By a similar
argument, the solution of maxλ∗>0
{∑Kk=1 λ∗k
(∑Jj=1 ik, j − 1
)}and maxε∗>0
{∑Nn=1 ε∗n
(∑Kk=1
∑Jj=1 ik, j Ak
n, j − 1)}
leads to the last Eq. (54). This shows that the KKT point
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Resource allocation and adaptive modulation
(ε∗,λ∗, ρ∗) maximizes fd(ε∗,λ∗, ρ∗) over χ∗� if and only if i is the global minimizer of f(i)over I f . This completes the proof.
Proof of Theorem 5.1
Using sub-gradient method with projection defined by (39) ensures the positive solution ofKKT Eq. (38) which implies that the corresponding ik, j is binary integer. However, respectingthe positivity constraint on λ∗, Eq. (37) can not ensure that a single sub-channel pattern isallocated to each user but 1 + σ λ∗
k number of patterns will be allocated to each user k.Similarly, ensuring that ε∗ > 0, Eq. (36) means that a sub-channel can be allocated to morethan one users.
In the following, we discuss that we can find another approximate problem for which theabove KKT equations not only provide binary integer solution but also ensure that a user willbe assigned with a single sub-channel pattern and a sub-channel will be allocated to a singleuser. To this end, we proceed as follows. The KKT Eq. (38) can be written as
Uk, j − λ∗k −N∑
n=1
ε∗n Akn, j = ± ρ∗k, j , ∀k, j (55)
We introduce K J new variables θk, j ’s defined as follows
θk, j =⎧⎨
⎩
{1, 0} if Uk, j − λ∗k −∑N
n=1 ε∗n Akn, j = −ρ∗k, j
{−1, 0} if Uk, j − λ∗k −∑N
n=1 ε∗n Akn, j = +ρ∗k, j
(56)
From the above definition of θk, j , equations (55) can be written as
Uk, j − 2θk, jρ∗k, j − λ∗k −
N∑
n=1
ε∗n Akn, j = ± ρ∗k, j , ∀k, j (57)
Let Uk, j = Uk, j − 2θk, jρ∗k, j , then the above equations take the form:
Uk, j − λ∗k −N∑
n=1
ε∗n Akn, j = ± ρ∗k, j , ∀k, j (58)
Although the utilities are changed from Uk, j to Uk, j = Uk, j − 2θk, jρ∗k, j , the solution of the
above equations provide integer solution to ik, j ’s. We now apply this change in utilities tothe Eqs. (36–37). The KKT equations (37) can be written as
J∑
j=1
{1
2ρ∗k, j
(
Uk, j − 2θk, jρ∗k, j + 2θk, jρ
∗k, j + ρ∗k, j − λ∗k −
N∑
n=1
ε∗n Akn, j
)}
=1+ σ λ∗k , ∀k
(59)
Replacing Uk, j for Uk, j − 2θk, jρ∗k, j , the above equations become:
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)}
+J∑
j=1
θk, j = 1+ σ λ∗k , ∀k (60)
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A. Ahmad
If there exist θk, j ’s such that∑J
j=1 θk, j = σ λ∗k , then we have
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)}
= 1, ∀k (61)
This implies that there exist another problem with a different set of utilities for which theabove solution ensures that a single pattern will be allocated to each user. By using a similarprocedure for the KKT equations (36), we get
K∑
k=1
J∑
j=1
{1
2ρ∗k, j
(
Uk, j + ρ∗k, j − λ∗k −N∑
n=1
ε∗n Akn, j
)
Akn, j
}
= 1,∀n (62)
which enures that a sub-channel will be allocated to a single user at most when∑Kk=1
∑Jj=1 θk, j Ak
n, j = σ ε∗n , and the utilities are changed from Uk, j to Uk, j = Uk, j −
2θk, jρ∗k, j .
The above analysis shows that the solution of the problem P1, namely (ε∗,λ∗, ρ∗) that liesin the positive cone, is the solution of the above KKT Eqs. (58,61,62). Moreover, the KTT Eqs.(58,61,62) give the stationary point of a slightly modified problem fd(ε∗,λ∗, ρ∗) which is thecanonical dual of a slightly modified primal problem with utilities Uk, j = Uk, j − 2θk, jρ
∗k, j .
Since the solution (ε∗,λ∗, ρ∗) is positive, according to Theorems 4.1 and 4.2, the proposedsub-gradient based solution proposed in Table 1 optimally solves a corresponding primalproblem with utilities Uk, j ’s and an objective function f(i). Note also that the canonical dualfd(ε∗,λ∗, ρ∗) is concave (since the KKT solution is in the positive cone). However, how farthe solution of the modified problem will be from that of the primal problem (5) dependsupon the values of ρ∗k, j ’s.
Proof of Corollary 5.1
If ρ∗k, j << Uk, j ,∀k, j , then Uk, j ≈ Uk, j ,∀k, j , f(i) ≈ f(i), and
max(ε∗,λ∗,ρ∗)
fd ≈ max(ε∗,λ∗,ρ∗)
fd (63)
Forρ∗k, j << Uk, j ,∀k, j , the solution of the Eqs. (58,61,62) is very close to that of Eqs. (29,30,31). Furthermore, the solution of (58,61,62) is the optimal solution of the correspondingprimal problem with utilities Uk, j (which is very close to the optimal solution of the primalproblem with utilities Uk, j ). Consequently, the dual canonical problem obtained using thesub-gradient based algorithm (Table 1) will provide solution to the primal problem which isvery close to the optimal solution. This completes the proof.
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Author Biography
Ayaz Ahmad is currently serving as an Assistant Professor in theDepartment of Electrical Engineering, COMSATS, Wah Campus, Pak-istan. He obtained his Ph.D. on Resource Optimization in WirelessCommunication Systems from Supélec, France in 2011. He has doneM.S. in Wireless Communication Systems from Supélec, France in2008. He did his B.Sc. in Electrical Engineering from University ofEngineering and Technology, Peshawar, Pakistan in 2006. His researchinterests include resource optimization in wireless communication sys-tems, signal processing for wireless systems, and stochastic optimiza-tion, and its application to wireless networks.
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