Single- and multi- user uplink energy-efficient resource allocation
algorithms with users’ power and minimum rate constraint
in OFDMA cellular networks
Chieh Yuan Ho•Ching-Yao Huang
Published online: 30 August 2012
Springer Science+Business Media, LLC 2012
Abstract In this paper, single- and multi- user Resource Allocation (RA) optimization problems considering trans-mit power and minimum rate constraint of Mobile Station (MS) for maximizing MS’ energy efficiency, measured as bits-per-joule (bpj), are addressed. Assume channel state information of all MSs is known by base station. We propose uplink RA algorithms, performing subcarrier assignment and power allocation, for optimizing bpj of MS in a single-cell OFDMA-based cellular network for both single- and multi- user scenarios. In the single-user case, we propose RA algorithms, which utilize the closed-form solution derived by applying Lambert-W function and an iterative approach based on Karush–Kuhn–Tucker condi-tions respectively to achieve optimal bpj of MS. In the multi-user case, centralized iterative multi-user RA algo-rithms for maximizing sum of MS’ bpj, performing joint subcarrier assignment and power allocation iteratively, are proposed by utilizing the proposed single-user RA schemes. In particular, tradeoffs between energy efficiency and spectral efficiency are fully investigated, and the influence of MS’ power and minimum rate constraints on bpj performance is also studied. The effectiveness of pro-posed algorithms is presented by numerical experiments. Numerical results demonstrate the proposed algorithms can enhance bpj significantly with limited loss of total throughput compared to the sum-rate maximization algo-rithm (in Moretti et al., IEEE Trans Veh Technol 60(4):1788–1798,2011).
Keywords Resource allocation OFDMA Energy-efficient Uplink Power allocation Bits-per-joule
1 Introduction
In order to fulfill the fast-growing demand of high data-rate wireless applications as well as the increasing needs and desire for being able to access those applications ubiqui-tously for a substantial period of time, the wireless cellular technologies are required to continuously evolve as fast and good as possible. For many reasons, orthogonal fre-quency division multiple access (OFDMA) has been selected as the multiple access technologies for state-of-the-art wireless systems such as LTE and WiMAX. OF-DMA is considered as a promising technology for wireless broadband systems due to many of its advantages, e.g. robustness against inter-symbol interference and multipath fading and relatively simple equalizations. With the use of OFDMA technology, there are still technical challenges needed to be solved to meet requirements of subscribers and wireless applications in different kind of scenarios, e.g. high-rate requirements, low power consumption, and etc. With the fact that many wireless applications are highly energy-consuming and require large bandwidth, one of the major challenges is to fulfill the needs of every user, e.g. QoS requirements, in the system while keeping them with good energy utilization which directly leads to longer battery lifetime. In order to resolve it, the resource allo-cation (RA) technique can be applied to achieve substantial improvement on the related performance of the OFDMA cellular system. In this paper, we focus on the uplink RA with the objective of optimizing the overall uplink energy efficiency of the network. The multi-user RA problem in the uplink could be more complex than that in the downlink
C. Y. Ho (&) C.-Y. Huang
Department of Electronics Engineering, National Chiao Tung University, HsinChu, Taiwan, Republic of China
e-mail: [email protected] C.-Y. Huang
e-mail: [email protected] DOI 10.1007/s11276-012-0494-4
due to the distributed power sources. Briefly speaking, in contrast to downlink RA with a single power source from the base station (BS), each Mobile Station (MS) has its own power budget in uplink RA. Figure1 illustrates the underlying framework of the uplink RA in an OFDMA network, where the central BS, aiming to optimize the uplink energy efficiency, is in charge of assigning radio resources and allocating transmit power of users who provides BS the knowledge of necessary information, such as channel status and other particular required parameters, and their own QoS requirements, e.g. rate and power constraint. Under the control of BS, each user transmits on the assigned resources with the specified transmit power for each assigned resource.
The OFDMA RA problems can be classified into downlink RA and uplink RA in a single- or multi- cell network. Most studies of RA problems in OFDMA systems aim at optimizing the following aspects: (weighted) sum-rate maximization with users’ power constraint, transmit power minimization, sum-rate maximization with fairness constraints, and min–max problems. In [1–8], downlink RA algorithms are proposed in OFDMA networks to maximize various objectives, e.g. sum-capacity, weighted sum of minimal rates, and etc., under different constraints. Uplink RA problems with similar objectives as in the downlink are investigated in [9–15]. Besides the various objectives studied in the downlink and uplink RA problems [1–15], another crucial factor dramatically affecting per-formances of wireless devices, energy efficiency, becomes an attractive topic for researchers. Bits-per-energy effi-ciency has been studied in a notable literature, [16], where an information-theoretic characterization for single-user, multiple-access, and interference, is presented. In [17–25], various RA problems are formulated and solved for enhancing energy utilization of wireless transmission with
different definition of energy efficiency, assumptions, and system architecture. A more detailed review is given in Section II. Compared with the above existing works, the focus and advantages of this paper have its uniqueness, and the distinct contributions of this paper lie in the following aspects:
• In the uplink of an OFDMA-based network, we address uplink RA optimization problems imposed with multi-ple MS’ maximum transmit power constraint and minimum rate requirement for maximizing MS’ energy efficiency, measured as total bits transmitted per joule of energy consumed (namely instantaneous bpj), and sum of MS’ bpj in the single- and multi- user scenario respectively. Several centralized single- and multi- user RA algorithms are proposed to resolve the formulated problem. In the proposed algorithms, BS with the knowledge of MSs’ channel information is in charge of deciding which MS can transmit on each subcarrier as well as allocating the radio transmit power of the MS on each subcarrier for the purpose of optimizing the formulated objective, i.e. instant energy efficiency of single- or multi- user in bpj.
• In the case of single-user network, the closed-form solution of MS’ optimal total transmit power for maximizing MS’ bpj is firstly derived by applying Lambert-W function [27]. Based on that, a power allocation (PA) scheme is developed with the water-filling algorithm. Another PA scheme with a low-complexity iterative approach is proposed by applying Karush–Kuhn–Tucker (KKT) conditions. These two schemes resolves the bpj maximization problem with MS’ maximum power constraint. Finally, with the derived minimum required power for achieving a certain rate, an extended PA scheme is proposed to
Fig. 1 The uplink resource allocation framework in an OFDMA-based network. The assigned resources for each user are pointed by the arrow lines, and p1–p18means the specified transmit power of users on each resource
solve the same problem imposed with both MS’ power and rate constraints.
• In the case of multi-user network, the bpj optimization problem becomes complex to solve due to the involve-ment of subcarrier assigninvolve-ment along with transmit power allocation (PA) for all MSs which results in the non-convex objective function. Therefore, two effec-tive iteraeffec-tive RA algorithms are proposed to achieve the sub-optimal solution, where the proposed SA and PA optimization are executed iteratively. In each iteration, a better RA is guaranteed to be found to improve the system energy efficiency computed in last iteration, and the RA result will finally converge and approach the sub-optimal solution.
• In particular, tradeoffs between energy efficiency and spectral efficiency are illustrated for both the single-and multi-user scenario. The influence of MS’ maximal transmit power, minimal rate constraint, and circuit power consumption on performance of bpj is fully investigated. Additionally, the advantage of the pro-posed schemes is shown via performance comparison with other related multi-user RA scheme and RA scheme for maximizing sum capacity in terms of bpj and throughput. It shows the enhancement of bpj is significant with respect to many parameters while the decrease of rate is relatively marginal.
The rest of this paper is organized as follows. In Sect.2, a review of related works is given. In Sects.3and4, single-user bpj optimization problems with MS’ transmit power constraint and minimal rate requirement are addressed, and three PA schemes are developed to obtain optimal bpj. In Sects. 5 and6 multi-user bpj optimization problems with MS’ power and minimal rate constraint are addressed with proposed iterative joint SA and PA algorithms. Finally, numerical experiments illustrate tradeoffs between energy efficiency and spectral efficiency in both single- and multi-user scenarios, and show the enhancement of the bpj per-formance by comparing with the other multi-user RA algorithm maximizing sum of users’ rate in Sect. 7. The paper is concluded in Sect.8.
2 Related works
In [1], authors propose a subcarrier, bit, power allocation algorithm to minimize the total transmit power subject to users’ rate requirement in the downlink. Similar problems in the downlink of a multi-cell OFDMA system are also addressed in [2,3]. In [4], a downlink sub-optimal RA algo-rithm is proposed to maximize sum capacity while main-taining proportional fairness. In [5], an iterative RA algorithm is proposed to maximize the weighted sum of the minimal user
rates of coordinated cells by applying duality-based approa-ches. In [6], a max–min RA problem is addressed to achieve the maximum fairness among users. In [7], capacity maxi-mization RA scheme applying water-filling PA is proposed in the downlink of an OFDMA system. In [8], authors proposed a resource allocation structure which performs iterative RA in a distributed manner to minimize interference and packet scheduling to guarantee fairness of resource sharing between users in the multi-cell downlink OFDMA system. The studies of [1–8] address RA problems in the downlink OFDMA system where the methodologies and ideas cannot be applied to the uplink transmission directly due to the multiple access nature of uplink OFDMA where each MS has its individual requirements rather than the BS-centralized control in the downlink system. In [9,10], authors propose a sub-optimal uplink RA scheme for maximizing sum capacity which greedily allocates subcarriers to users achieving maximal Signal-to-Noise Ratio (SNR) by performing water-filling for power allocation. In [11], utility maximization uplink RA schemes are proposed for optimizing sum rate, proportional fairness, and max–min fairness. In [12], the additional fairness constraint is added into the approach of [9] to maximize sum of users’ rate. A dual decomposition approach is proposed to solve the uplink instantaneous rate maximization problem in [13]. In [14], the uplink PA problem is modeled as a non-cooperative game while subcarrier assignment (SA) is done similarly as that in [9]. An uplink ergodic weighted sum-rate maximization RA problem is solved in [15]. The literatures [9–15] studying uplink RA problems focus on typical objec-tives without considering any type of energy efficiency opti-mization. However, performance of these kinds of schemes in terms of spectral and energy efficiency will be compared to show the advantages of the proposed scheme. In [17–19], game-theoretic approaches are applied for maximizing energy efficiency. In [17] and [18], authors proposed a game-theoretic model of joint power and rate control with packet-delay constraints for maximizing bits-per-joule (bpj) in the uplink of CDMA systems. In [19], an energy-efficient power control scheme modeled as a non-cooperative game in mul-ticarrier CDMA systems is presented. The proposed methods in [17–19] cannot be applied to the OFDMA network since there has no issue of subcarrier-based allocation involved in CDMA systems where users transmit in the same band with different codes. Additionally, the transmit power and rate in [17–19] has no inter-relationship and can be separately cho-sen. That is quite different from the model used in the paper, where the achievable rate from Shannon-Capacity formula is applied. Tradeoffs between energy efficiency and delay in the single-user case are studied in [20], where only a single transmitter sending fixed amount of data is considered in the one-dimensional time-varying channel. The assumptions and system model are totally different from the multi-user OF-DMA network assumed in this paper, and no QoS requirement
is considered. In [21], authors proposed a two-user game-theoretic PA approach for maximizing bits-per-energy in user-cooperation networks, where it assumes two-user uplink transmission, and each user can decide whether to relay the packet for the other or not. The transmission architecture assumed is simple, and no subcarrier allocation and QoS requirement are involved. In [22], a link adaptation scheme for single-user uplink transmission is proposed to optimize bpj in an OFDM system where the only one user uses the whole bandwidth. The subcarrier allocation is not available for a single-user OFDM system, and no QoS requirement is imposed. In [23], energy-efficient schemes for improving average energy utilization of MS in the uplink of OFDMA networks are investigated, where the energy efficiency in bpj is defined as the ratio of time-averaged rate to time-averaged power including transmit power and circuit power con-sumption. In [24], the RA algorithm satisfying additional MSs’ fairness constraints with specially-designed subcarrier ordering and assignment methods is proposed for optimizing time-averaged bpj in OFDMA networks. In [25], another downlink RA approach using standard optimization methods is proposed to maximize instantaneous bpj with the assump-tion of flat fading across subcarriers in an OFDMA network. In addition to the fact that the scheme in [25] is addressed in the centralized downlink OFDMA system which is very dif-ferent from the uplink case and no constraint is given, the assumption that each user experiences flat-fading across all subcarriers is also too simple. In [26], a joint RA and relay selection scheme is proposed to maximized time-averaged bpj in cooperative-relay OFDMA-based networks where the cooperative relaying with maximum ratio combining is uti-lized to further improve energy efficiency. Although in [23–
26], the time-averaged bpj optimization RA problems are investigated ([26] is for cooperative-relay network) in multi-user uplink OFDMA networks, they do not consider MS’ power constraint and QoS requirement in the problem while both MSs’ maximal transmit power and minimum rate con-straint are considered in this paper. Furthermore, there are some disadvantages of the consideration of time-averaged bpj instead of instantaneous bpj, which is shown and discussed in Section of numerical results and discussion.
3 Single-user energy-efficient resource allocation with MS’ transmit power constraint
3.1 System model
Consider the uplink transmission of a single-cell OFDMA network with K subcarriers, indexed by k2 K ¼
1; 2; 3; . . .; K
f g. Assume Channel State Information (CSI) is perfectly known by both MS and BS. Let skðtÞ denote the received signal transmitted by MS on subcarrier k at time t,
PkðtÞ and xkðtÞ denote the transmit power and the transmitted signal with unit energy on subcarrier k at time t respectively, and hkðtÞ and n0ðtÞ denote the channel fading coefficient (gain) of the link between MS and BS on subcarrier k and the Addictive White Gaussian Noise (AWGN) at time t respec-tively. Then, the received signal can be written as:
skðtÞ ¼pffiffiffiffiffiffiffiffiffiffiPkðtÞhkðtÞxkðtÞ þ n0ðtÞ ð1Þ With Shannon Capacity formula, we can express the achievable rate on subcarrier k at time t as:
rkðtÞ ¼ log2 1þPkðtÞ hkðtÞj j 2 N0B
!
ð2Þ
where N0 and B denotes the noise power spectral density and subcarrier bandwidth respectively.
3.2 Problem formulation
In this paper, the instantaneous energy efficiency is defined as number of bits transmitted by MS when consuming one joule of its energy, namely bits-per-joule, expressed as:
EE¼number of bits transmitted consumed energy ¼ DB DE¼ RDt PDt¼ R P ¼ R Ptþ Pc ð3Þ
where Dt, R, and P denote transmit time duration, transmit rate, and consumed power of MS respectively. The con-sumed power includes transmit power, denoted by Pt, and circuit power, Pc, which is assumed as a constant. From Eq. (3), it is clear to see that bits-per-joule is equivalent to the ratio of MS’ transmit rate to MS’ consumed power.
Thus, the energy efficiency of MS at time t can be expressed as the ratio of sum of MS’ achievable rate on each subcarrier to MS’ consumed power:
EEðtÞ ¼ RðtÞ PtðtÞ þ Pc¼ PK k¼1rkð Þt PK k¼1PkðtÞ þ Pc ¼ PK k¼1log2 1þ PkðtÞ hjkðtÞj2 N0B PK k¼1PkðtÞ þ Pc ð4Þ
The single-user RA optimization problem is formulated as follows: max PkðtÞ;8k f gEEðtÞ s:t: XK k¼1 PkðtÞ Pn; PkðtÞ 0; 8k ð5Þ
Our goal is to optimize MS’ bits-per-joule in the current OFDMA frame through allocating the transmit power on each subcarrier. The total transmit power can’t exceed the upper bound, Pn, and power on each subcarrier must be positive. Note that since the proposed schemes are performed on a per-frame basis, the time domain expression is neglected
in the following sections, which means, for instance, EE(t) and Pk(t) are expressed as EE and PKrespectively.
3.3 Optimal RA with Lambert-W function
Firstly, we simplify Problem (5) into Problem (6) by con-sidering fixed transmit power constraint:
max Pk;8k f gEE s:t XK k¼1 Pk¼ P ð6Þ
According to Eq. (4), the solution of Problem (6) is equivalent to that of Problem (7), aiming to maximizing throughput with a total power constraint:
max R¼X K k¼1 log2 1þPkj jhk 2 N0B ! s:t:X K k¼1 Pk¼ P ð7Þ
Problem (7) can be resolved by applying the conventional water-filling scheme:
Pk¼ 1 k ln 2 N0B hk j j2 !þ ; where xð Þþ¼ x for x [ 0 0 otherwise ð8Þ where k is the Lagrange Multiplier, and the term, 1
k ln 2, in
Eq. (8) denotes the water-level in the water-filling scheme, which must be chosen to satisfyPKk¼1Pk¼ P. Let Kþ K denotes the resultant subcarrier index set after performing the standard water-filling scheme, in which all the subcarriers are allocated positive power in (8), namely Pk0[ 0, for 8k
0
2 Kþ. With Kþ, we can obtain k by solvingPKk¼1Pk¼ P as: k¼ K 0 PþPk02Kþ N0B h k0 j j2 ln 2 ð9Þ
where K0 denotes number of subcarriers in Kþ. By substituting Eq. (8) into Eq. (7), we can have the maximum achievable rate as follows:
R¼X K k¼1 rk¼X K k¼1 log2 1þ Pkj jhk2 N0B ! ¼ X k02Kþ log2 hk0 2 N0BK0 Pþ X k02Kþ N0B hk0 2 ! " # ð10Þ
By substituting Eq. (10) into Eq. (4), the solution of Problem (6) can be expressed as:
EE¼ P k02Kþlog2 h k0 j j2 N0BK0 Pþ P k02Kþ N0B h k0 j j2 ðP þ PcÞ ð11Þ
In order to solve Problem (5) where the total power constraint is not fixed and an upper bound of that is given,
we can utilize Eq. (11) and treat the total transmit power, P, in Eq. (11) as a variable. It can be shown that MS’ energy efficiency, Eq. (11), is a strictly quasi-concave function with respect to P, plotted in Fig.2. Thus, a globally optimal P exists for maximizing EE. The proof of the strict quasi-concavity of Eq. (11) is given in ‘‘Appendix 2’’.
Therefore, we first derive the optimal total transmit power by solving: oEE oP ¼ 0 ð12Þ Assume H0 ¼Pk02Kþ N0B h k0
j j2, and we can have:
oEE oP ¼ K0 PþH0 ð Þln 2 PþPc ð Þ P k02Kþlog2 h k0 j j2 r2 K0 þK0log2ðPþH0Þ ðPþPcÞ2 ¼ 0,
and then have 1
ðPþH0ÞðP þ PcÞ lnðP þ H 0 Þ ¼ðln 2ÞK0 Pk0 2Kþ log2 h k0 j j2 r2K0
, where r2¼ N0B. Let X¼ P þ H0, and with some mathematical minipulations, we can have:
Pc H0 X ln X ¼ ln 2 ð Þ K0 X k02Kþ log2 hk0 2 r2K0 ! 1 ð13Þ Let a¼ Pc H0 and b¼ðln 2ÞK0 Pk0 2Kþlog2 h k0 j j2 r2K0 1, and substitute a and b into Eq. (13).
Then, we have ln X¼a
X b, which can also be written as: eb¼1 Xe a X ð14Þ 0 5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Total transmit power (dBm)
Bits-per-Joule
Fig. 2 EE, bits-per-joule, of MS by performing water-filling with different total transmit power, P
Multiply the left and right side of Eq. (14) by parameter a, and then we have:
aeb¼a Xe
a
X ð15Þ
With Eq. (15), the closed-form solution can be obtained by applying Lambert-W function [27]:
X¼ a W aeð bÞ; if ae b2 1 e S 0; ½ 1Þ X¼ a W aeð bÞ a W1ðaebÞ ; if aeb2 1 e; 0 N=A; if aeb2 1; 1 e 8 > > > < > > > : ; ð16Þ where WðxÞ ¼P1n¼1ðnn!Þn1xn represents Lambert-W function.
From Eq. (16) the optimal total transmit power, denoted as P, for maximizing energy efficiency, bpj, can be written as:
whereðÞþmeans to choose the positive value. Therefore, if P Pn, the maximum bits-per-joule of Problem (5) can be found by substituting P into Eq. (11), expressed as:
EE¼ P k02Kþlog2 h k0 j j2 N0BK0 P þP k02Kþ N0B h k0 j j2 ðPþ PcÞ ð18Þ
The corresponding transmit power on each subcarrier can be obtained from (8), where
k¼ K 0 PþP k02Kþ N0B h k0 j j2 ln 2 ð19Þ
It can be shown that Eq. (11) is monotonically increasing with respect to the total transmit power (shown in Fig.2) while the power is smaller than the optimal transmit power. Hence, in the case of P[ Pn, the optimal total transmit power achieving maximum bits-per-joule is Pn, so by applying water-filling, PA for subcarriers can be expressed as:
Pk¼ 1 k ln 2 N0B hk j j2 !þ ð20Þ
where k ln 21 is chosen such that PKk¼1Pk¼ Pn. We can
obtain: k¼ K0 Pnþ P k02Kþ N0B h k0 j j2 ln 2 .
The proposed PA algorithm applying the closed-form solution to solve Problem (5) is described as follows:
MaxEE Algorithm 1
Step 1:With given channel information, hf kjk 2 1; K½ g, examine the availability of Eq. (17). (The initialKþ¼ K)
Step 2:If the solution is available, compute the optimal total power, P, by Eq. (17).
Step 3:Compare the resultant Pwith the power constraint, P n. If P P
n, compute the final power allocation by Eqs. (8and19). Otherwise, PA can be obtained from Eq. (20). (If the resultant PA withKþcontains zero power on any subcarrier, according to standard water-filling procedure, those should be removed from Kþand restart from Step 1.)
3.4 Optimal RA with an iterative approach
The defect of the Lambert-W method is the optimal power, P, isn’t always available if the required condition, Pc\Pk0
2Kþ N0B
h k0
j j2, is not met. Hence, we use KKT condi-tions to further develop an iterative algorithm iteratively computing the transmit power on each subcarrier until it converges. The formulated RA problem is the same as Problem (5). The Lagrangian function can be written as:
L Pkð Þ ¼ PK k¼1log2 1þ Pkj jhk2 N0B PK k¼1Pkþ Pc þ k X K k¼1 Pk Pn ! lkPk ð21Þ
Thus, the KKT conditions can be written as:
oL oPk¼ 1 ln 2 ð ÞPKk¼1Pkþ Pc 1 r2= hkj j2þPk PK R k¼1Pkþ Pc 2þ k lð kÞ ¼ 0 ð22Þ P ¼ a W aeð bÞ P k02Kþ N0B hk0 j j2; if ae b2 1 e S 0; ½ 1Þ P ¼ max Pð Þþ ; P¼ a W aeð bÞ P k02Kþ N0B h k0 j j2 a W1ðaebÞ P k02Kþ N0B h k0 j j2 8 > < > : ; if ae b2 1 e; 0 N=A; if Pc\ P k02Kþ N0B h k0 j j2 8 > > > > > > > > < > > > > > > > > : ð17Þ
k X K k¼1 Pk Pn ! ¼ 0; k 0 ð23Þ XK k¼1 lkPk¼ 0; lk 0; 8k ð24Þ
where k and lk are non-negative Lagrangrian multipliers. From Eq. (22), we can derive:
R PK k¼1Pkþ Pc k lkð Þ X K k¼1 Pkþ Pc ! ¼ 1 ln 2 1 r2= hkj j2þPk EE¼ PK R k¼1Pkþ Pc ¼ 1 ln 2 1 r2= hkj j2þPkþ k lkð Þ XK k¼1 Pkþ Pc ! ð25Þ
From Eq. (24) and Pk 0; 8k, we can have: lk¼ 0; 8k. Furthermore, because the optimal transmit power may not be the maximal power according to Eq. (11), k must be zero. Thus, from Eq. (25) the maximum EE and the cor-responding optimal transmit power on subcarrier k can be derived as: EE¼ 1 ln 2 1 r2= hkj j2þPk Pk¼ 1 EE ln 2 r2 hk j j2 ð26Þ
Equations (25) and (26) show that the optimal energy efficiency and transmit power on subcarrier k intervene with each other, so the maximal EE, bpj, and corresponding optimal PA, fPk;8kg, cannot be computed directly. Therefore, an iterative approach, MaxEE Algorithm 2, to jointly solve Eqs. (25and26) is proposed and depicted in the following, and the effectiveness of the proposed algorithm is shown in Lemma 1.
MaxEE Algorithm 2
Initialization:Allocate equal power on subcarriers while satisfying the power constraint. Compute the initial energy efficiency, EE0, by Eq. (4). Set the iteration index, i, to be 1.j While
Step 1:Compute the power of subcarrier k of the ith iteration, denoted as Pi
k, with EEi1for8k by Eq. (26)
Step 2:Compute EEiwith the resultant power allocation of Step 1, Pi
k;8k
, by Eq. (4)
Step 3:Compute the difference between the bpj in the current and previous iteration, denoted as DEEi, which can be expressed as:
ifDEEi\eðe [ 0Þ
MaxEE Algorithm 2 continued
Then, the optimal bpj is EEi, and Popt¼PKk¼1Pik, and P¼ P
kjPk¼ Pik;8k
.
(Popt and P denote the optimal total transmit power and optimal power allocation respectively.)
break while
end of if (i = i?1 for the next iteration count) End of while
Step 4:If Popt P
n, the optimal solution is the final result. Otherwise, the final PA can be obtained by Eq. (20).
Lemma 1 (MaxEE Algorithm 2) With sufficient itera-tions, the energy efficiency, EEi, will converge to the optimum solution of Problem (5), which is the solution of Eq. (25).
Proof See ‘‘Appendix1’’.
4 Single-user energy-efficient resource allocation with MS’ power and minimum rate constraint
4.1 Problem formulation and proposed algorithm
In fact, achieving higher bpj might result in reduction of the data rate possibly causing violation of user’s QoS. Therefore, in addition to the power constraint, MS’ mini-mum transmit rate is also a key factor to be considered. The problem of bpj optimization is formulated as follow:
max Pk;8k f gEE:s:t: XK k¼1 Pk Pn; Pk 0; 8k; XK k¼1 rkðtÞ Rn ð28Þ The rate constraint implies that sum of the achievable rate on each subcarrier, rkðtÞ for 8k, must be equal or larger than the minimum rate requirement, denoted as Rn. Firstly, the min-rate constraint can be converted into the minimum transmit power requirement by exploiting the water-filling solution. Equation (10) can be illustrated as the maximum rate which can be achieved with the total transmit power P. Thus, we can obtain the minimum required transmit power, denoted as Pr, achieving Rn by solving Eq. (10) with the input of Rn, which is written as:
Pr ¼ 2Rn , Y k02Kþ hk0 2 !1 K0 K0N0B X k02Kþ N0B hk0 2 ð29Þ
Pr denotes the minimum required transmit power for achieving Rn. In other words, Rnis just achieved by water-filling PA withPKk¼1Pk¼ Pr. Note that during the compu-tation of Prif the resultant PA, computed from (8) withKþ used in (29) and P¼ Pr, contains zero power on any
subcarrier inKþ, they are removed fromKþ, and then Pris re-computed with updatedKþ. The final resultant setKþcan only include the subcarriers with positive transmit power in Eq. (8). Figure3 illustrates the relationship between the optimal achievable rate computed with the water-filling scheme and its corresponding bpj with respect to MS’ total transmit power, described in Eqs. (10) and (11), respectively. It shows that the bpj curve has a peak (maximal) value with a specific transmit power, namely P, while the maximum achievable rate is monotonically increasing with respect to MS’ total transmit power. The derived Pr and the maximal transmit power, Pn, can be either smaller or larger than P. Thus, to solve Problem (28), MaxEE Algorithm 3 is devel-oped with the joint consideration of Pr, P, and Pn.
MaxEE Algorithm 3
Step 1:Use MaxEE Algorithm 1 or 2 to calculate the optimal transmit power, P¼PKk¼1Pk.
Step 2:Calculate Pr by Eq. (29)
Step 3: if P\Pr\Pn, then P¼ Pr, else if Pr\P\Pn, then P¼ P, else if P
r\Pn\P, then P¼ Pn, else, Problem (28) has no solution. [P denotes the resultant optimal total transmit power for Problem (28)]
Step 4:Compute the final PA with P by applying the standard water-filling scheme.
5 Multi-user energy-efficient resource allocation with MS’ power constraint
5.1 Problem formulation
In this section, the problem of single-user RA for maxi-mizing energy efficiency is extended to the multi-user case
in a single-cell OFDMA network. Firstly, we measure the energy efficiency of the uplink multi-user network as sum of MS’ bits-per-joule, which can be written as:
EE¼X N n¼1 EEn¼X N n¼1 Rn Pt;nþ Pc ¼X N n¼1 PK k¼1Ck;nlog2 1þ Pk;njhk;nj 2 N0B PK k¼1Pk;nþ Pc ð30Þ
where EEn denotes the energy efficiency of MS n, and Ck;n denotes the assignment indicator which represent 1 if subcarrier k is assigned to MS n, otherwise it is set to 0. The optimization problem can be formulated as follows:
max Pk;n;Ck;n f gEE s:t: XK k¼1 Pk;n Pn; 8n; Pk;n 0; 8k; 8n;X N n¼1 Ck;n 1; 8k; Ck;n2 0; 1f g; 8k and 8n ð31Þ Our goal is to maximize the so-called System Energy Efficiency (SEE), sum of MS’ bpj, subject to MS’ transmit power budget. The last constraint implies that each subcarrier must be assigned to one MS at most to prevent from intra-cell interference. We can see that Problem (31) is a problem of constrained non-linear programming which includes both integer and continuous variables. In addition, the objective function is a non-concave function, which means that the global optimum is quite difficult to find because it might have more than one local optimum.
5.2 Iterative multi-user energy-efficient RA
Therefore, by applying the single-user bpj optimization addressed previously, we proposed an iterative multi-user RA algorithm performing subcarrier assignment and power allocation optimization alternately to approach the opti-mum. The overall algorithm is described as follows:
MU-MaxEE Algorithm 1
Initialize 1: (Initial subcarrier allocation)Randomly assign subcarriers to MSs. One subcarrier can only be assigned to one MS. Set the initial iteration time, j, to 1.
Initialize 2: (Initial PA optimization)With the given initial SA, use either MaxEE Algorithm 1 or 2 to obtain the initial optimal PA and bpj, EE0
n;8n
, for each MS, and then compute the initial SEE, denoted as SEE0. j
While for i = 1:K
Step 1:Choose a subcarrier k randomly from the subcarrier set, K. K ¼ K kf g. -30 -20 -10 0 10 20 30 0 10 20 30 40
Total transmit power (dBm)
Bits-per-Joule -30 -20 -10 0 10 20 30 0 200 400 600 800 1000
Total transmit power (dBm)
Maximum Achievable
Rate (bps/Hz)
Fig. 3 R, Eq. (10), and EE, Eq. (11), with respect to P. Assume Pc= 10 dBm, K = 50, and N0B = -120 dB
MU-MaxEE Algorithm 1 continued
Step 2:With subcarrier k, execute Joint SA and PA Algorithm 1, depicted in the following Sub-Section C, to obtain SEEij, denoting SEE of the ith iteration in the inner loop and the jth iteration in the outer loop, and the new subcarrier and power allocation for the (i ? 1)th iteration.
end of for
Step 3:Compute SEEj by Eq. (38), and if SEE j SEEj1\d or j ¼ J
(NOTE:d [ 0. SEEj denotes the SEE of the jth iteration in the outer loop, and J is the pre-defined maximum times of iteration to prevent the infinite loop situation.)
break while
end of if (j = j?1 for the next iteration count) end of while
5.3 Joint SA and PA optimization
The main idea of this approach is to re-assign subcarriers based on the current subcarrier and power allocation in order to improve the SEE gradually. Each subcarrier is re-assigned to the MS which provide the maximum margin for SEE. Firstly, let In¼ i jCi;n ¼ 1; 8i 2 K denote the index set of assigned subcarriers for MS n. We define the incremental set, In0, which adds subcarrier k chosen in Step 1 into In for 8n, expressed as:
I0n¼ In[ kf g; 8n 6¼ m ð32Þ
For MS m, which subcarrier k is originally assigned to, namely k2 Im, Im0 is obtained by removing subcarrier k from Im. That can be written as:
I0m¼ Im kf g ð33Þ
The difference between the optimal bpj with In, denoted as EEn, and that with In0, denoted as EE0n, can be written as:
DEEn¼ EE0n EEn ð34Þ
where DEEn represents the changed amount of bpj of MS n after adding the subcarrier to In. EE0n and EEn can be obtained by using MaxEE Algorithm 1 or 2 with the input of In and I0n respectively. Therefore, after re-assigning subcarrier k, originally assigned to MS m, to MS n, the overall difference of SEE, denoted as gk
n, can be written as:
gkn¼ DEEnþ DEEm; 8n 6¼ m ð35Þ
gk
m denotes subcarrier k is re-assigned to the original MS, so gk
m¼ 0. Consequently, subcarrier k should be re-assigned to MS n achieving maximum gn among all in order to obtain the maximum improvement on SEE. The decision rule can be expressed as:
In¼ In[ kf g; Im¼ Im kf g Ck;n¼ 1; Ck;m¼ 1 if n¼ arg max s g k s ð36Þ
After re-assigning subcarrier k in the ith iteration of the inner loop, SEEij, denoting SEE of the ith iteration in the inner loop and the jth iteration in the outer loop, can be expressed as: SEEij¼ SEEi1j þ gk
n; i2 1; K½ ð37Þ
Note that when i¼ 0, SEE0j¼ SEEj1. Thus, after all subcarriers are re-assigned, the SEE of the jth iteration, SEEj, can be written as:
SEEj¼ SEEj1þX K
k¼1 gk
n ð38Þ
Joint SA and PA Algorithm 1
Step 1:Obtain In0 for8n 6¼ m and Im0 by Eqs. (32and33) Step 2:Obtain EE0nand EEn by MaxEE Algorithm 1 or 2, and
compute DEEnand gknfor8n by Eqs. (34and35) Step 3:Re-assign the subcarrier by Eq. (36)
Step 4:Update bpj of MS n and MS m as EE0nand EEm0 respectively, and compute SEEijby Eq. (37).
From Eq. (38), we know that SEEj SEEj1 due to gk
n 0; 8k. Thus, it is clear to see the proposed MU-MaxEE Algorithm 1 can at least converge to the sub-optimal solution, which will be shown in the numerical results.
6 Multi-user energy-efficient resource allocation scheme with MS’ power and minimum rate constraint
6.1 Problem formulation
In this section, the user’s minimum rate constraint is included in the multi-user bpj optimization problem. The achievable rate of MS n can be written as:
rn¼X K k¼1 Ck;nrn;k¼X K k¼1 Ck;nlog2 1þ Pk;nhk;n 2 N0B ! ð39Þ
The problem is formulated as follows:
max Pk;n;Ck;n f gEE:s:t: XK k¼1 Pk;n Pn; 8n; Pk;n 0; 8k; n; XN n¼1 Ck;n 1; 8k; Ck;n2 0; 1f g; 8k; n; rn Rn; 8n ð40Þ With the rate constraints included, Problem (40) becomes more complicated than Problem (31). The global optimum solution is difficult to find with common optimization approaches. Thus, an iterative RA algorithm with a similar
idea as MU-MaxEE Algorithm 1 is proposed to resolve it. The proposed algorithm can be divided into the initialization phase and the RA optimization phase. In the initialization phase, a RA approach is proposed to perform subcarrier assign-ment and allocate transmit power on each subcarrier in order to satisfy all MS’ rate constraint. In the phase of RA optimization, an iterative approach performing joint subcarrier assignment and power allocation iteratively is proposed to obtain the optimum RA solution of Problem (40).
6.2 Initial resource allocation
The initial RA algorithm, which gives initial subcarrier and power allocation satisfying all MSs’ rate constraint, per-forms subcarrier exchanging iteratively between MSs to compensate the rate of MSs who have not yet met the rate constraint until the rate constraint of each MS is met. It is described as follows:
Initial Resource Allocation Algorithm Step 1:Randomly assign subcarriers to MS.
Step 2:For each MS, with channel gain information of the assigned subcarriers, compute the minimum required power, Pr;n, by Eq. (29) and compare with the maximum transmit power, Pn. Find the set of MSs which cannot meet the rate constraint, written as:U ¼ njPn\Pr;n;8n
, whereU denotes the index set of MSs violating their rate constraint.
Step 3:For each MS n2 U, additional subcarriers from other MSs are given to it to compensate its achievable rate for just meeting the rate constraint. Note that we have to make sure that the chosen candidates of MSs, which are going to give away their assigned subcarriers to MS n2 U, still meet their rate constraint after giving their subcarriers away. Besides, the given subcarrier is selected randomly from candidates of MSs. This process continues until all MSs meet their minimum rate constraint. The pseudo code of this process is listed as follows:
WhileU 6¼ ;
Step 1:Select a MS n2 U
Step 2:Randomly select a subcarrier kðk 2 K, k 62 In8n 2 U). Remove it fromK, namely K ¼ K kf g.
Step 3:Examine whether the original owner of subcarrier k, denoted as MS q, can still meet the rate constraint with the rest of subcarriers, denoted as ^Iq¼ Iq kf g.
If ^Pr;q Pq, (Re-assign subcarrier k)
In¼ In[ kf g, Ck;n=1; Iq¼ Iq kf g, Ck;q¼ 0 Else ifK ¼ ;
Break whileand go back to Step 1 of the initial RA algorithm to re-assign subcarriers again.
Else, (It means the chosen subcarrier isn’t able to be re-assigned) go back to Step 2.
End of if
Step 4:Compute the new minimum required power, Pr;nand compare with Pn.
If Pn Pr;n, (The rate constraint of MS n is now met.) Remove MS n fromU, U ¼ U nf g, and go back to Step 1.
Initial Resource Allocation Algorithm continued
Else, (It means the rate of MS n is still insufficient) go back to Step 2
End of if End of While End of Algorithm
6.3 Iterative joint subcarrier and power allocation with MS’ minimal rate constraints
We propose an iterative RA algorithm by applying similar philosophy of MU-MaxEE Algorithm 1 which optimizes SEE without MS’ rate constraint. The proposed algorithm taking advantage of MaxEE Algorithm 3 can gradually enhance SEE by performing joint SA and PA iteratively while maintaining the satisfaction of all MS’ rate constraint which is initially met by Initial RA Algorithm. The whole algorithm is described as follows:
MU-MaxEE Algorithm 2
Initialization: Perform Initial Resource Allocation Algorithm to obtain initial subcarrier and power allocation. Set the initial iteration time, j, to 1. j
While for i = 1:K
Step 1:Choose a subcarrier k randomly fromK. Remove subcarrier k fromK, namely K ¼ K kf g.
Step 2:With the chosen subcarrier k, execute Joint SA and PA Algorithm 2 described below to obtain SEEij and the new subcarrier and power allocation for the (i ? 1)th iteration. end of for
Step 3:Compute SEEjby Eq. (38). If SEEj SEEj1\d or j¼ J, break while. end of if (j = j?1 for the next iteration count) end of while
Joint SA and PA Algorithm 2
Step 1:Obtain In08n 6¼ m and Im0 by Eqs. (32and33). Compute the minimum required power for In0 and I0m,8n 6¼ m, denoted as P0r;n and P0r;mrespectively.
If P0r;m[ Pm, (That means subcarrier k can’t be re-assigned because the rate constraint cannot be met if subcarrier k, originally assigned to MS m, is re-assigned to other MS.) Don’t re-assign the subcarrier and go back to Step 1 of MU-MaxEE Algorithm 2 for the next iteration.
End of if
Step 2:Obtain EE0n, EE0m, EEn, and EEmby MaxEE Algorithm 3. Compute DEEn and gknfor8n by Eqs. (34and35)
Step 3:Re-assign the subcarrier by Eq. (36)
Step 4:Update bpj of MS n and MS m as EE0nand EEm0 respectively, and compute SEEijby Eq. (37).
7 Numerical result and discussion
We consider a single cell OFDMA network with 1,000-m radius. MSs are randomly distributed within the cell. Assume the noise power spectral density, N0, and subcar-rier bandwidth, B, are -174 dBm/Hz and 10 kHz respec-tively. Each subcarrier suffers from i.i.d. Rayleigh fading. Lognormal shadowing with zero mean and the standard deviation of 8 dB is considered. The transmitted signal experiences path loss modeled as: PLðdBÞ ¼ 30:18þ 26 log10d, where PL and d denote the path-loss coefficient and the distance (meter) between MS and BS respectively. Monte-Carlo simulations are performed in all figures. Figures4 and 5 compare the proposed single-user RA algorithm with/without MS’ rate constraint (MaxEE 1, 2, and 3) and the water-filling scheme, which optimizes the
total achievable rate with given transmit power, in terms of performance of bits-per-joule and bps/Hz. Additionally, it also illustrates the tradeoff between energy efficiency and spectral efficiency. In Figs.4 and 7, when the maximum transmit power, Pn, is less than the optimal power, P, performance of bpj and achievable rate of the water-filling and proposed scheme w/o MS’ rate constraint are the same. That is because of the feature of quasi-concavity of bits-per-joule and the monotonic increase of optimum rate with respect to the transmit power, P, as shown in Fig.3. Since the proposed schemes keep using P as its total transmit power even if the given power budget is larger than the optimal one, the performance of bpj and bps/Hz remain unchanged. For MaxEE 3 considering Rn, while Pn\Pr, the rate constraint cannot be met even when MS transmits at Pn. In Figs.4and7, because Pr for the two cases, Rn¼ 150; 170 bps/Hz, is larger than P, their best transmit power for optimizing bpj is Pr. Thus, they have worse bpj but better achievable rate than MaxEE 1 and 2. Figure6
compares the optimal bpj of four different schemes, the two proposed schemes and water-filling with different Pn, with respect to MS’ circuit power, Pc. The two proposed schemes overwhelmingly outperform the water-filling scheme while Pcis small. However, when the circuit power increases, it dominates the total power consumption, which results in the decrease of the performance gap between the proposed schemes and water-filling. Figure 7 provides an overall view on how the bpj performance varies with MS’ maximum transmit power, Pn, and minimum rate con-straint, Rn. While the optimal transmit power, P, falls between Pn and Pr, the optimal bpj can be achieved. Per-formance of bpj decreases while Pn\P, or Rnbecomes too large, resulting in Pr exceeds P. Note that the area where
0 2 4 6 8 10 12 14 16 18 20 80 100 120 140 160 180 200 220 240
Maximum transmit power (dBm)
Achievable Rate (bps/Hz) MaxEE 1 MaxEE 2 Water-Filling MaxEE 3 (Rn=150 bps/Hz) MaxEE 3 (Rn=170 bps/Hz)
Fig. 4 Achievable rate in bps/Hz with respect to Pnin dBm. Assume Pc= 10 dBm, K = 20. The corresponding Prfor the line in cyan and black are 9 and 12 dBm respectively
0 2 4 6 8 10 12 14 16 18 20 2 3 4 5 6 7 8 9 10
Maximum transmit power (dBm)
Bits-per-Joule MaxEE 1 MaxEE 2 Water-Filling MaxEE 3 (Rn=150 bps/Hz) MaxEE 3 (Rn=170 bps/Hz)
Fig. 5 Performance of bits-per-joule with respect to Pn in dBm. Assume Pc= 10 dBm, K = 20 0 2 4 6 8 10 12 14 16 18 20 0 5 10 15 20 25 30 35 40 45 Circuit Power Pc (dBm) Bits-per-Joule MaxEE 1 MaxEE 2 Water-Filling, Pn=15 dBm Water-Filling, Pn=20 dBm
Fig. 6 Performance of bpj with respect to Pc in dBm. Assume K = 20 and Pn= 15 dBm for MaxEE 1 and 2
there’s no value of bpj (Z-axi) indicates MS cannot meet the minimum rate constraint with the given Pn, Pn\Pr.
For the multi-user case, we compare performances of bpj and total throughput of MU-MaxEE 1 and 2 with/ without minimum rate constraints and the algorithm pro-posed in [9] which can achieve the sub-optimal solution for maximizing sum of users’ rate without considering users’ minimum rate constraint in the uplink of an OFDMA network. In Fig.8, it shows that Algorithm in [9] performs better in total throughput but worse in bits-per-joule than the proposed algorithms. Figure9 reveals that the channel coefficient (gain) of the assigned MS on each subcarrier for the proposed algorithm is lower than that for Algorithm in [9]. That’s because the proposed algorithm allocates
resources based on the capacity of MS’ energy efficiency determined by its channel characteristics and the ratio of rate to power while the sum-rate optimization method (Algorithm in [9]) assigns resources by taking users’ SNR as the main consideration. This phenomenon clearly explains the tradeoff between energy efficiency and spec-tral efficiency. By comparing performance of the proposed algorithms and Algorithm in [9], it is observed that the improvement on bpj is much larger than the degradation of total throughput in percentage (%). In addition, MU-MaxEE 1 achieves almost same results as the optimum (Exhaustive Search) while MU-MaxEE 2 can still achieve about 96–99 % of the optimum. In Fig.8, both the per-formance of total throughput and bits-per-joule of the proposed algorithm decreases with the increase of the minimum rate constraint, Rn. Briefly speaking, that is because the proposed algorithm has to sacrifice the opportunity of assigning a subcarrier to a better MS to enhance bpj for meeting every MS’ rate constraint. MU-MaxEE 2 must satisfy all MS’ rate constraint even if some of them have poor channel quality, which results in the decrease of both sum of bpj and total throughput. More-over, bpj performance of MU-MaxEE 2 rises at first, and starts to decrease while the number of MSs is greater than 7. Similarly, that is because to satisfy every MS’ min-rate requirement results in reduction of the total throughput especially when N increases. In Fig. 10, the bpj curves of MU-MaxEE 2 intersects with that of Algorithm in [9] since many more subcarriers are needed in order to meet Rnof each MS, which might lead to huge reduction of bpj when Pnbecomes too low. Figure 11demonstrates similar con-cepts as the single user case, which indicates that the gap of bpj performance between the proposed algorithms and Algorithm in [9] reduces when the circuit power becomes
-5 0 5 10 15 100 120 140 160 180 200 220 240 10 11 12 13 14 15 16 Maximum transmit power (dBm) Bits-per-Joule Minimum Rate Constraint (bps/Hz)
Fig. 7 bpjof MaxEE 2 with respect to Pnand Rn. Assume Pc= 10 dBm 2 4 6 8 10 12 14 16 700 800 900 1000 1100 1200 1300 Number of Users Total Throughput (bps/Hz) 2 4 6 8 10 12 14 16 20 40 60 80 100 Number of Users SEE (bpj/Hz) MU-MaxEE 2 (Rn=40 bps/Hz) MU-MaxEE 2 (Rn=20 bps/Hz) Algorithm in [8] MU-MaxEE 1 Exhaustive Search (Rn=0 bps/Hz) Exhaustive Search (Rn=20 bps/Hz)
Fig. 8 Performance of total throughput and SEE with respect to N. Assume K = 50, Pc= 10 dBm, and Pn= 15 dBm 0 5 10 15 20 25 -60 -50 -40 -30 -20 -10 0 Subcarrier Index Channel Gain (dB) Prposed MU-MaxEE 1 Algorithm in [8]
Fig. 9 Channel gain of the assigned MS for each subcarrier averaged through Monte-Carlo simulation
more and more dominant in total power consumption of MS. Note that bpj of the proposed algorithms is still about 12 % better than that of Algorithm in [9] even when Pc= 20 dBm. Figure12shows how much different initial SA influences the final bpj result and demonstrates the effectiveness of the proposed iterative algorithms with and without Rn(MU-MaxEE 1 and 2 respectively). Each point in Fig.12represents the variance of bpj computed from the numerous results generated by Monte-Carlo simulation with different initial SA. It illustrates that MU-MaxEE 1 is almost irrelevant to initial SA with respect to both K/N and Pc while MU-MaxEE 2 have reasonably small variance, which rapidly decreases when Pcincreases. In Figs.13,14,
15, performance comparison with the scheme proposed in [23], where the similar methodology and definition of energy efficiency have also been applied in [24–26], is
2 4 6 8 10 12 14 16 18 20 800 1000 1200 1400 1600 1800 2000
Maximum transmit power (dBm)
Total throughput (Bits/Hz)
2 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 120
Maximum transmit power (dBm)
SEE (bpj/Hz)
MU-MaxEE 1 Algorithm in [8]
MU-MaxEE 2 (Rn=40 bps/Hz) MU-MaxEE 2 (Rn=60 bps/Hz)
Fig. 10 Total throughput and SEE with respect to Pn. Assume K = 100, N = 7, and Pc= 10 dBm 0 2 4 6 8 10 12 14 16 18 20 700 800 900 1000 1100 1200 1300 Circuit power (dBm) Total throughput 0 2 4 6 8 10 12 14 16 18 20 0 200 400 600 800 Circuit power (dBm) SEE (bpj/Hz) MU-MaxEE 1 Algorithm in [8] MU-MaxEE 2 (Rn=40 bps/Hz) MU-MaxEE 2 (Rn=10 bps/Hz) (Bps/Hz)
Fig. 11 Performance of total throughput and SEE with respect to Pc. Assume K = 50, N = 10, and Pn= 15 dBm 3 4 5 6 7 8 9 10 0 0.02 0.04 0.06 0.08 0.1 K/N Variance of SEE 0 2 4 6 8 10 12 14 16 18 20 0 0.5 1 1.5 2 2.5 3 Circuit Power (dBm) Variance of SEE MU-MaxEE 2, Rn=40 bps/Hz MU-MaxEE 2, Rn=20 bps/Hz MU-MaxEE 1, w/o Rn
Fig. 12 Variance of SEE with respect to Pcand the ratio of number of subcarrier to number of MSs 10 15 20 25 30 35 40 45 50 11 12 13 14 15 16 17 18 Number of Users SEE (bpj/Hz) MU-MaxEE 1
Scheme in [22] with 200 frame averaging Scheme in [22] with 500 frame averaging Scheme in [22] with 700 frame averaging
Fig. 13 SEE comparison with the scheme in [23], Pc= 10 dBm, Pmax= 15 dBm 10 15 20 25 30 35 40 45 50 20 40 60 80 100 120 140 Number of Users SEE (bpj/Hz) MU-MaxEE 1
Scheme in [22] with 200 frame averaging Scheme in [22] with 500 frame averaging Scheme in [22] with 700 frame averaging
Fig. 14 SEE comparison with the scheme in [23], Pc= 0 dBm, Pmax= 15 dBm
presented. While our proposed scheme can optimize instantaneous energy efficiency by tracking users’ channel condition on a per-frame basis, the major defect of the scheme in [23] which optimizes time-averaged energy efficiency, defined as the ratio of time-averaged rate to time-averaged power, instead of instantaneous one needs a long period, ranging from roughly 100 to 500 frames depending on the averaging window size, to converge from the initial bpj to a stable (sub-optimal) value. Therefore, we can see that in Figs.13 and14, where we compare bpj of proposed MU-MaxEE 1 with bpj of Scheme in [23], averaging over the first 200, 500 and 700 OFDMA frames respectively, the proposed scheme outperforms Scheme in [23] about 6–24 %. In Fig.15, MU-MaxEE 1 is compared directly with the converged stable value achieved by Scheme in [23] w.r.t. Pcand N. Even we compare with the converged result of [23], the proposed scheme can still outperform [23] by 5–14 %. In addition, the proposed scheme performs even better when number of users is very
high due to the advantage of instant bpj optimization. Note that since [23] doesn’t have any power or QoS constraint, we don’t enforce MSs’ min-rate constraint for fair comparison.
8 Conclusion
In OFDMA-based cellular networks, the uplink RA prob-lems for maximizing MS’ bits-per-joule subject to MS’ transmit power constraint and minimum rate requirement are addressed in both single- and multi-user scenario. Two single-user RA algorithms based on the derived closed-form solution and an iterative approach applying KKT conditions are proposed to achieve optimal transmission in terms of bpj. The RA algorithm considering additional minimum rate requirement is proposed. For the multi-user case, we propose two iterative RA algorithms, which per-form joint RA and SA optimization iteratively and achieve sub-optimal solution. Numerical results present the tradeoff between energy efficiency and spectral efficiency, and show great improvement on SEE with limited loss of total throughput compared to the sum-rate maximization scheme [9]. It also highlights how the power and minimal rate constraint affect performance of bpj and demonstrates the effectiveness of proposed algorithms.
Appendix 1
Proof By Eq. (26), the transmit power on subcarrier k of the ith iteration can be written as:
Pik¼ 1 EEi1ln 2 r2 hk j j2 ð41Þ -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 1 1.05 1.1 1.15 1.2 Circuit Power (dBm) Number of Users
SEE Ratio of Proposed
Scheme to Scheme in [22]
Fig. 15 The ratio of SEE of ‘‘MU-MaxEE 1’’ to SEE of the scheme in [23], Pmax= 15 dBm. For each simulation point, the SEE of [23] is its stable value which is converged from its initial 0 1 2 3 4 5 6 7 8 9 10 -4 -3 -2 -1 0 1 2 3 4 EEi-1 EE i
By Eq. (4), the energy efficiency of the ith iteration can be written as: EEi¼ PK k¼1log2 1þ Pi kj jhk2 N0B PK k¼1Pikþ Pc
By substituting Eq. (28) into EEi, we can express EEias a function of EEi1: EEi¼ K log2 1 EEi1ln 2 þPKk¼1log2 j jhk2 N0B K EEi1ln 2 PK k¼1Nh0B k j j2þ Pc ð42Þ
Then, by substituting Eq. (42) into Eq. (27), we can have DEEi as: DEEi¼ K log2 EEi11ln 2 þPKk¼1log2 hk j j2 N0B K EEi1ln 2 PK k¼1 N0B hk j j2þ Pc EEi1 ð43Þ Figure3 shows the curve of Eq. (43) as a function of EEi1. It indicates that the intersection point of Eq. (43) and the x-axi, meaning EEi¼ EEi1, represents the optimal solution. Therefore, from Fig.16, it is clear to see that no matter where the initial value located, the value of energy efficiency, EEi, will be getting closer to the intersection point at each iteration. As a result, we can conclude that EEi will finally converge to the optimal solution, which achieves DEEi¼ 0, with sufficient iterations. j
Appendix 2
Proof The function of MS’ energy efficiency function (11) is written as: EEðPÞ ¼ P k02Kþlog2 h k0 j j2 N0BK0 Pþ P k02Kþ N0B hk0 j j2 Pþ Pc ð Þ ;
where P denotes MS’ total transmit power. We know from Proposition C.9 in [28] that EEðPÞ is a strictly quasi-concave function with respect to P if and only if the upper contour set, defined as Ud¼ P 0jEEðPÞ df g, is strictly convex for any real number d. For d\0, it is obvious that no solution of EE Pð Þ\0 exists. For d ¼ 0, the only solution is P¼ 0 which makes Kþ¼ ; and EE Pð Þ ¼ 0. Thus, we can see that Ud is strictly convex for d 0. For d [ 0, we can first express Ud as:
Ud¼ P 0jdðP þ PcÞf X k02Kþ log2 hk0 2 N0BK 0 Pþ X k02Kþ N0B hk0 2 ! " # 0 ) ð43Þ Let f Pð Þ ¼ d P þ Pð cÞ P k02Kþ log2 h k0 j j2 N0BK0 Pþ P k02Kþ N0B h k0 j j2 !# " , and then we can obtain the second order derivative of f Pð Þ w.r.t. P as: o2f Pð Þ oP2 ¼ f Pð Þ 00 ¼ K 0 ln 2 Pþ X k02Kþ N0B hk0 2 !2 ð44Þ
From (44), we can see that f Pð Þ00[ 0. Therefore, Ud is also strictly convex for d [ 0, and then the strict quasi-concavity of (11) is proved.
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Author Biographies
Chieh Yuan Ho has his B.S. degree in Department of Com-munication Engineering, National Chiao Tung Univer-sity, in 2005, and received his M.S. and Ph.D. degree in Insti-tute of Electronics Engineering, National Chiao Tung Univer-sity, in 2007 and 2012 respec-tively. Mr. Ho’s research interests include radio resource allocation, machine-to-machine (M2 M) communications, energy-efficient communica-tions, cross-layer optimization in wireless communication systems, especially in OFDMA-based wireless cellular systems.
Ching-Yao Huanghas his B.S. degree in Physics from National Taiwan University, Taiwan, in 1987 and then the Master and Ph.D. degrees in Electrical and Computer Engineering from NJIT and Rutgers University (WINLAB), the state university of New Jersey, in 1991 and 1996 respectively. Dr. Huang jointed AT&T Whippany, New Jersey and then Lucent Tech-nologies in 1995 as a consultant and Member of Technical Staff in 1996. In the years of 2001 and 2002, Dr. Huang was an adjunct professor at Rutgers University and NJIT. Since 2002, Dr. Huang joins the department of Electronics Engineering, National Chiao Tung University, Taiwan and currently is an associate professor and director of Technology Licensing Office and Incubation Center. Dr. Huang is the recipient of ‘‘Bell Labs Team Award’’ from Lucent in 2003, ‘‘Best Paper Award’’ from IEEE VTC Fall 2004, and ‘‘Outstanding Achievement Award’’ from National Chiao Tung University from 2007 to 2011. Dr. Huang’s research areas include medium access controls, radio resource management, and machine to machine communications for wireless systems. Dr. Huang has been published more than 60 technical memorandums, journal papers, and conference papers and has 20 patents. Dr. Huang has also served as Editor for ACM WINET and Recent Patents on Electrical Engineering.
![Fig. 15 The ratio of SEE of ‘‘MU-MaxEE 1’’ to SEE of the scheme in [ 23 ], P max = 15 dBm](https://thumb-ap.123doks.com/thumbv2/9libinfo/7897316.155877/14.892.303.816.85.370/fig-ratio-mu-maxee-scheme-p-max-dbm.webp)
