Adaptive Resource Allocation Algorithm Based on Cross ... - IEEE Xplore

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IEEE TRANSACTIONS ON BROADCASTING, VOL. 60, NO. 3, SEPTEMBER 2014

Adaptive Resource Allocation Algorithm Based on Cross-Entropy Method for OFDMA Systems Kuan-Chou Lee, Sen-Hung Wang, Chih-Peng Li, Ho-Hsuan Chang, and Hsueh-Jyh Li

Abstract—The cross-entropy (CE) method was proposed to estimate the probability of rare events. Afterwards, it was applied to solve combinatorial optimization problems. This paper presents a CE-based method that adaptively allocates sub-carrier, bit, and power in orthogonal frequency division multiple access systems to obtain minimum total transmit power under given quality of service requirements, that is, the minimum data rate and the bit error rate. The proposed scheme first converts the sub-carrier and bit allocations of various users into a binary sequence. Then, a CE-based method is proposed to minimize the total transmission power. Compared with the well-known genetic algorithm and the traditional heuristic algorithm, the proposed scheme has achieved the lowest total transmission power with a few iterations. Index Terms—Orthogonal frequency division multiple access (OFDMA), resource allocation, cross entropy (CE), genetic algorithm (GA).

I. I NTRODUCTION RTHOGONAL frequency division multiple access (OFDMA) is one of the most promising techniques for the next generation wireless networks, such as the Long Term Evolution Advanced (LTE-Advanced) [1]–[10]. In OFDMA systems, each sub-carrier is assigned to at most one user and the channel conditions for various sub-carriers of each user experience different fading conditions. Therefore, the optimization of the sub-carrier, bit, and power allocation in OFDMA systems has received considerable attention. In general, the methods proposed in literature can be categorized into three major approaches. The first approach allocates sub-carriers and power to maximize the system spectral efficiency under the total transmission power constraint [11]–[18]. The second approach obtains the minimum total transmission power under the quality of service (QoS) constraints, such as data rate and bit error rate (BER) requirements [19]–[21]. The third approach maximizes the total system utility using a cross-layer design [22], [23]. This study adopts the second approach to investigate the resource allocation problem. Determining the optimal

O

Manuscript received March 19, 2014; revised June 17, 2014; accepted June 30, 2014. Date of publication August 12, 2014; date of current version September 3, 2014. (Corresponding author: C.-P. Li.) K.-C. Lee and H.-J. Li are with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan. S.-H. Wang is with the Intel-NTU Connected Context Computing Center, National Taiwan University, Taipei, Taiwan. H.-H. Chang is with the Department of Communications Engineering, I-SHOU University, Kaohsiung, Taiwan. C.-P. Li is with the Department of Electrical Engineering, Institute of Communications Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TBC.2014.2339551

sub-carrier, bit, and power allocation can be translated into a non-deterministic polynomial time (NP)-hard combinatorial optimization problem. However, the globe optimal solution cannot be found within the polynomial time. Therefore, various methods employing evolutionary algorithms have been adopted to obtain a sub-optimal solution. In particular, a joint sub-carrier and bit allocation method, which utilizes the genetic algorithm (GA), was proposed [24]. However, the algorithm converges slowly. Thus, another evolutionary algorithm, the ant colony optimization (ACO) algorithm, was proposed [25]. The algorithm obtains the same performance as the GA and achieves a faster convergent speed. However, the ACO algorithm needs to create a three-dimension table which requires a large memory space. In addition, a real-time sub-optimal algorithm was proposed where only the subcarrier assignment is investigated without considering the bit assignment problem [19]. Several researchers proposed heuristic algorithms to further enhance the performance of the real-time sub-optimal algorithm by simultaneously considering the sub-carrier, bit, and power assignment problems [20]–[21]. The cross-entropy (CE) method was first proposed by Rubinstein [26] to estimate the probability of rare events. Subsequently, the method was applied to solve combinatorial optimization problems [27], [28]. The CE method is an iterative algorithm and has two main phases in each iteration. In the first phase, random samples are generated according to the probability mass function. In the second phase, the probability mass function is updated based on the acquired information. In the present study, the CE method is adopted to solve the sub-carrier, bit, and power allocation problems in OFDMA systems, which eventually should minimize the total transmission power under minimum data rate and BER constraints. In the proposed scheme, the assignment information is encoded into a binary sequence and the CE method is employed to search for the sequence that corresponds to the minimum total transmission power. In the first phase of the CE method, a number of binary sample sequences are randomly generated to allocate the sub-carriers and modulation orders for various users. However, a number of the randomly generated sample sequences may not provide sufficient sub-carriers for all the users to fulfill the minimum data rate constraint even when the highest order of possible modulations is adopted. In addition, the corresponding transmission rate may not yield the minimum transmission power required even for the random sequences that provide sufficient sub-carriers for all the users. To address these issues, the proposed scheme introduces a

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LEE et al.: ADAPTIVE RESOURCE ALLOCATION ALGORITHM BASED ON CROSS-ENTROPY METHOD

sequence modification procedure which reassigns sub-carriers and modulation orders for all users to achieve the minimum total transmission power. The second phase of the proposed scheme is performed in the same way as the traditional CE method. The simulation results demonstrate that the proposed scheme substantially outperforms the heuristic algorithm [21] and the well-known GA [24]. This paper is organized as follows. Section II describes the system model and formulates the optimization problem. Section III presents the proposed algorithm. Section IV demonstrates the simulation results. Finally, concluding remarks are presented in Section V.

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users is assumed to be perfectly known at the base station. Consequently, the total transmission power is given by: Ptotal =

N−1 K−1

Pk,n βk,n .

(5)

n=0 k=0

Finally, the minimum total transmission power under the data rate and BER requirements for OFDMA systems can be formulated as min

N−1 K−1

Pk,n βk,n

(6)

n=0 k=0

subject to II. S YSTEM M ODEL AND P ROBLEM F ORMULATION An OFDMA system with N sub-carriers and K users is considered in this paper. Each sub-carrier in the OFDMA system is assigned to at most one user. Let βk,n be an assignment indicator defined as: 1, n ∈ ϒk , (1) βk,n = 0, otherwise, where ϒk is the set of sub-carriers assigned to the k-th user. Given that each sub-carrier is assigned to at most one user, ϒk ∩ ϒk is an empty set for any k, k ∈ , k = k , where = {0, 1, . . . , K − 1} is the set of all user indexes. To guarantee that the system can support the data rate requirements for all users, this study assumes that the total required number of sub-carriers is less than or equal to N when the highest order of possible modulation schemes is utilized. In this study, an adaptive sub-carrier, bit, and power allocation algorithm is investigated with the aim of minimizing the total transmission power under the minimum data rate and BER constraints. To achieve a given BER requirement, the minimum received power of the k-th user at the n-th sub-carrier is given by [20] r ξBER,k σ2 Q−1 2 k,n − 1 , (2) k,n (rk,n , ξBER,k ) = 3 4 where ξBER,k denotes the BER requirement of the k-th user, rk,n is the number of bits carried by the n-th sub-carrier for n ∈ ϒk , σ 2 is the noise variance, and Q−1 (·) represents the inverse Q function, where the Q function is given by ∞ 1 2 Q(x) = √ e−t /2dt. (3) 2π x M = 4 types of modulation schemes, namely, binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), 16-quadrature amplitude modulation (QAM), and 64-QAM are adopted in this paper. Therefore, rk,n ∈ r ≡ {1, 2, 4, 6} and r is the set of all possible numbers of bits carried by a sub-carrier. If the channel effect is taken into consideration, the transmission power of the n-th sub-carrier assigned to the k-th user has the following form: k,n rk,n , ξBER,k , (4) Pk,n =

Hk,n 2 where Hk,n is the channel gain of the n-th sub-carrier for the k-th user. The channel state information (CSI) of all

K−1 k=0 N−1 n=0 K−1

βk,n ≤ 1, ∀n ∈ {0, 1, . . . , N − 1},

(7)

rk,n βk,n ≥ Rk , ∀k ∈ ,

(8)

αk ≤ N,

(9)

k=0

where Rk is the minimum data rate requirement of the k-th user and αk denotes the minimum required number of sub-carriers for the k-th user, that is,

Rk , k = 0, 1, . . . , K − 1, (10) αk = max{r} where · is the ceiling function. Note that max{r} = 6 in this study. III. P ROPOSED S UB -C ARRIER , B IT, AND P OWER A LLOCATION A LGORITHM BASED ON THE C ROSS E NTROPY M ETHOD The CE method is a simple, efficient, and general method used to estimate the probability of rare event, which has been adopted to solve combinatorial optimization problems [26]. The CE method is an iterative algorithm and has two main phases in each iteration. In the first phase, random samples are generated according to the probability mass function. In the second phase, the probability mass function is updated based on the acquired information. The details of the proposed resource allocation algorithm are presented in the following discussion. A. Assignment Information In this study, the assignment information is encoded as a binary sequence. In particular, an OFDMA system with N sub-carriers, K users, and M modulation schemes is considered. The assignment information of one sub-carrier requires ( log2 K+ log2 M) bits, where the first log2 K bits and the successive log2 M bits denote the assigned user index and the modulation type, respectively. For example, considering an OFDMA system with 8 users and 4 modulation schemes, if a sub-carrier is assigned to user 4 (encoded as [1 0 0]) and utilized the third kind of modulation scheme (encoded as [1 0]), the assignment information is encoded as [1 0 0 1 0]. In addition, the binary sequences of all the sub-carriers are

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concatenated one after another to form a binary vector s of length B = N · ( log2 K + log2 M) bits. The b-th element of s is denoted as s[b] for b = 0, 1, . . . , B − 1. If there are 256 sub-carriers, 8 users, and 4 modulation schemes, 256 · (log2 8 + log2 4) = 1280 bits are required to encode assignment information for all sub-carriers. B. Random Sequences Generation In each iteration, the first phase of the CE method represents the generation of random sequences according to the probability mass function. In particular, a number of random sequences are generated to represent various assignment information vectors by using the Bernoulli distribution. Note that a random sequence is simply termed as a sample hereafter. The c-th sample of the t-th iteration is denoted as st,c , where st,c [b], b = 0, 1, . . . , B − 1, are Bernoulli random variables with the probability P(st,c [b] = 1) = pt,b and P(st,c [b] = 0) = 1 − pt,b for all b, 0 ≤ pt,b ≤ 1. The probabilities of various sample elements are written in the t-th iteration as an 1 × B vector pt = pt,0 pt,1 · · · pt,B−1 . Ns samples, st,c , c = 0, 1, . . . , Ns − 1, at the t-th iteration are generated according to the probability vector of the previous iteration pt−1 . As a result, the probability mass function of st,c can be represented as f (st,c ; pt−1 ) =

B−1

pt−1,b

st,c [b] 1−s [b] · 1 − pt−1,b ( t,c ) . (11)

b=0

Initially, the iteration counter t is set to 1 and all the elements of the probability vector p0 are set to 0.5. C. Valid and Invalid Sets In each iteration, Ns samples are randomly generated according to the probability mass function given in (11). These samples are classified into two sets, namely, valid and invalid sets. A sample is classified into the valid set if the corresponding sub-carrier allocation and modulation schemes meet the minimum data rate requirements for all users, i.e., (8). By contrast, the sample is designated to the invalid set if at least one user’s data rate requirement is not fulfilled.

Fig. 1.

Flowchart of the proposed sample modification scheme.

the valid set, only bit reassignment is required. The procedure of sample modification is depicted in Fig. 1. i) Sub-carrier reassignment Sub-carrier reassignment is performed on samples belonging to the invalid set. The sub-carrier reassignment is conducted to guarantee that all users have the minimum number of sub-carriers required when the highest possible order of modulation scheme is adopted, as demonstrated in (10). The principle of sub-carrier reassignment is to select a sub-carrier from the sub-carrier set of the user that has the largest number of extra sub-carriers and to reassign this sub-carrier to the user needing the most sub-carriers. If the user that has the most extra sub-carriers is identified, the channel quality of all the sub-carriers assigned to this user is evaluated and the subcarrier with the worst channel gain is selected. Subsequently, the selected sub-carrier is reassigned to the user requiring the largest number of sub-carriers. This process iterates until all users have the minimum number of sub-carriers required. The amount of extra sub-carriers of the k-th user in the c-th sample, gk,c , is defined as gk,c ≡ uk,c − αk ,

D. Sample Modification Sample modification has two main operations, namely, the sub-carrier reassignment and the bit reassignment. The principle of sub-carrier reassignment is to reallocate sub-carriers to ensure that all users have the minimum number of subcarriers required. Bit reassignment has two objectives. One, to increase the available data rate to fulfill the minimum data rate requirements of all users specific to the samples in the invalid set where at least one user does not have sufficient data rate. To achieve this objective, the modulation order is increased. Two, to decrease the transmission power while fulfilling the minimum data rate requirement for all the users having sufficient data rate. To attain the objective, the modulation order is decreased. Both sub-carrier and bit reassignments have to be performed for the samples in the invalid set. However, for the samples in

(12)

where uk,c is the amount of sub-carriers of the k-th user in the c-th sample. A positive gk,c indicates that the number of sub-carriers assigned to the k-th user exceeds the minimum number of sub-carriers required. By contrast, a negative gk,c implies that the k-th user does not have the minimum number of sub-carriers required. In addition, the amount of extra bits, Dk,c , of the k-th user in the c-th sample is defined as the difference between the available bits and the minimum required bits: Dk,c =

N

rk,n,c · βk,n,c − Rk ,

(13)

n=1

where rk,n,c denotes the number of bits carried by the n-th sub-carrier assigned to the k-th user in the c-th sample, and


Fig. 2.

527

Flowchart of the proposed sub-carrier reassignment algorithm.

βk,n,c is the sub-carrier assignment indicator. βk,n,c = 1 if and only if the c-th sample indicates that the n-th sub-carrier is assigned to the k-th user. If two or more users have the same amount of extra subcarriers gk,c in each iteration of the sub-carrier reassignment, the user having the largest Dk,c is selected for the reassignment of the sub-carriers. However, if two or more users require the same number of sub-carriers to meet the minimum number of sub-carriers required, the user with the smallest Dk,c is selected. The procedure of the sub-carrier reassignment is illustrated in Fig. 2. The details are described as follows: • Step 1: Calculate the number of extra sub-carriers, gk,c , ∀ k. If gk,c ≥ 0 for all k, the algorithm is terminated. Otherwise, go to Step 2. • Step 2: Calculate the number of extra bits, Dk,c , ∀ k. • Step 3: Identify the ka -th user having the most number of extra sub-carriers (gk,c > 0). If two or more users have the same number of extra sub-carriers, the user with the largest extra bits Dk,c is selected (Dk,c > 0). • Step 4: Identify the kb -th user requiring the most number of sub-carriers (gk,c < 0). If two or more users need the same number of sub-carriers, the user with the smallest extra bits Dk,c is selected (Dk,c < 0). • Step 5: Identify the sub-carrier having the worst channel gain among all the sub-carriers assigned to the ka -th user, ϒka , and reassign the sub-carrier to the kb -th user. • Step 6: Go to Step 1. ii) Bit reassignment Bit reassignment is performed on all the samples. On the one hand, modulation orders are decreased for the users having extra bits Dk,c > 0, to save transmission power. On the other hand, the modulation orders are increased for the users with negative Dk,c until the minimum data rate requirement is met.

Fig. 3.

Flowchart of the proposed bit reassignment algorithm.

To illustrate, a user having extra bits (i.e., Dk,c > 0) is considered. All the sub-carriers of this user are evaluated and the sub-carrier with the highest order modulation is identified. The modulation of this sub-carrier is decreased to the second highest order. If two or more sub-carriers have the same highest order modulation, the sub-carrier with the worst channel gain is selected. This process iterates as long as Dk,c > 0, and the same process has to be performed for all the users. Next, a user that does not have sufficient bits (i.e., Dk,c < 0) is considered. All the sub-carriers of this user are evaluated and the sub-carrier with the lowest order modulation is identified. Subsequently, the modulation of this sub-carrier is increased to the second lowest order. If two or more sub-carriers have the same lowest order modulation, the sub-carrier with the best channel gain is selected. This process iterates until Dk,c > 0, and the same process has to be performed for all the users. Fig. 3 illustrates the bit reassignment algorithm. Detailed operations of the bit reassignment for the k-th user, which have to be performed for all users, are described as follows: • Step 1: Calculate Dk,c in (13). If Dk,c > 0, go to Step 2. If Dk,c < 0, go to Step 5. If Dk,c = 0, the algorithm is terminated. • Step 2: Identify the highest order modulation of all the sub-carriers assigned to the k-th user and obtain the number of bits per symbol for this modulation: rˆk,c = arg

max {rk,n,c },

rk,n,c ∈rk,c , n∈ϒk

(14)

528

•

•

•


where rk,c is a set of all possible numbers of bits carried by the sub-carriers assigned to the k-th user in the c-th sample. If rˆk,c = min{r}, the algorithm is terminated. Otherwise, go to Step 3. Step 3: Identify the sub-carrier that has the highest order modulation. If two or more sub-carriers have the same highest order modulation, the sub-carrier nˆ with the worst channel gain is selected. Step 4: Obtain the number of bits per symbol, rˆ , for the second highest order modulation. If rˆk,c − rˆ ≤ Dk,c , decrease the modulation type of the selected nˆ -th subcarrier to the second highest order modulation and go to Step 1. Otherwise, the algorithm is terminated. Step 5: Identify the lowest order modulation of all the sub-carriers assigned to the k-th user and obtain the number of bits per symbol for this modulation: r¯k,c = arg

•

•

min {rk,n,c }.

(15)

rk,n,c ∈rk,c , n∈ϒk

Step 6: Identify the sub-carrier that has the lowest order modulation. If two or more sub-carriers have the same lowest order modulation, the sub-carrier n¯ with the best channel gain is selected. Step 7: Increase the modulation of the n¯ -th sub-carrier to the second lowest order modulation and go to Step 1.

E. Fitness Function This study aims to minimize the total transmission power of OFDMA systems under the QoS requirements. Based on (4) and (5), the fitness function of the c-th random sample, c = 0, 1, . . . , Ns − 1, at the t-th iteration is defined as K−1 k rk,n,c , ξBER,k N−1 βk,n,c . (16) ψ st,c = |Hk,n |2

Finally, instead of updating pt,b directly through (17) as a measure to prevent fast convergence to a local optimum, the updated probability is given by pt,b = λ · pt,b + (1 − λ) · pt−1,b , b = 0, 1, . . . , B − 1,(19) where λ denotes a smoothing factor and 0 < λ ≤ 1. G. Summary The proposed resource allocation scheme based on CE method is summarized as follows: • Step 1: Set the iteration counter t : = 1 and initialize the probability vector p0 with p0,b = 0.5 for b = 0, 1, . . . , B − 1. Ns −1 • Step 2: Generate Ns random samples, {st,c }c=0 , from the probability mass function f (s; pt−1 ) shown in (11). • Step 3: Classify the samples into two sets, namely, the valid and the invalid sets. The samples in the invalid set are modified using the proposed sub-carrier and bit reassignment algorithms to guarantee that all the users have the minimum required data rate. For the samples belonging to the valid set, the bit reassignment algorithm is applied to ensure that all the users can transmit using the minimum power. • Step 4: Calculate the fitness function according to (16) Ns −1 and to obtain a set of performance values {ψ(st,c )}c=0 rank them in ascending order for ψ0 ≤ ψ1 · · · ≤ ψNs −1 . Then, set μt = ρNs , where ρ denotes the ratio of elite samples and · represents the ceiling function. • Step 5: Update pt by using (17) to (19). • Step 6: If the stop criterion is not reached, that is, t is smaller than a predefined value, set t := t + 1 and go to Step 2. Otherwise, the algorithm is terminated.

n=0 k=0

The samples, st,c , consist of binary bits that need to convert into the resource assignment information, that is, rk,n,c and βk,n,c . F. Updating the Probability of Binary Samples From the fitness function given in (16), a set of fitness values are obtained, namely, ψ(st,0 ), ψ(st,1 ), . . . , ψ(st,Ns −1 ). These values are then ranked from the smallest to the largest, that is, ψ0 ≤ ψ1 · · · ≤ ψNs −1 . In addition, the number of elite samples is defined as μ = ρNs , where ρ represents the ratio of elite samples and 0 < ρ ≤ 1. Consequently, the updated probability of the samples can be obtained through the following equation N s −1

pt,b =

c=0

I ψ st,c ≤ ψμ · st,c [b]

N s −1 c=0

I ψ st,c ≤ ψμ

,

where the indicator function I{·} is defined as 1, ψ st,c ≤ ψμ I ψ st,c ≤ ψμ = 0, otherwise.

(17)

(18)

IV. S IMULATION R ESULTS An OFDMA system comprising K = 4 users and N = 48 sub-carriers is considered in the simulation experiments. The frequency-selective Rayleigh fading channel with a channel length of L = 4 is adopted, where the exponentially decayed power delay profile is utilized and the decay factor is set as 0.1. The additive white Gaussian noise is assumed to have a zero mean and a variance of σ 2 = 10−2 . In addition, the minimum data rate requirement Rk is set as 24 for various k and the BER requirement of each user ξBER,k is 10−2 . The performance of the GA is also obtained for comparison purpose. The parameters adopted in the simulation experiments are summarized as follows: For the GA [24]: • Penal value: 0.2 • Number of samples: 500 • Number of iterations: 50 • Number of elite samples: 10 • Crossover probability: 0.9 • Mutation probability: 0.05 For the proposed CE-based algorithm: • Number of samples: Ns = 500 • Number of iterations: 50


Fig. 4.

Total transmission powers for various schemes (K = 4).

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Fig. 6. Total transmission power of the proposed CE-based scheme for various ρ (λ = 0.9).

Fig. 5. Total transmission powers for the proposed algorithm with various computational complexity (K = 4, λ = 0.9, ρ = 0.05).

Fig. 7. Total transmission power of the proposed CE-based scheme for various λ (ρ = 0.05).

Fig. 4 shows the total transmission power of various algorithms, where the smooth factor and the ratio of elite samples for the proposed CE-based scheme are set as λ = 0.9 and ρ = 0.05, respectively. The GA has the largest total transmission power. By contrast, the proposed scheme achieves the smallest total transmission power after 7 iterations. The total transmission power of the proposed scheme is 71% of that of Chen’s algorithm after 16 iterations only. Therefore, the proposed CE-based scheme has a fast convergence rate and achieves the smallest total transmission power. Fig. 5 demonstrates the simulation results of various numbers of samples in each iteration for the proposed CEbased algorithm using λ = 0.9 and ρ = 0.05, where the computational complexity is defined as the number of iterations multiplies the number of samples in each iteration. When the computational complexity is less than 15000, Ns = 500 results in the lowest total transmission power given the same computational complexity. Simulation results also indicate that, when the value of Ns increases, a lower total transmission power can be obtained. However, the computational complexity becomes relatively high, and the

improvement is only marginal. Therefore, Ns = 500 in each iteration is adopted in the following simulations. The total transmission power of the proposed algorithm for various ratios of elite samples ρ using the same smooth factor λ = 0.9 is delineated in Fig. 6. A tradeoff exists between the total transmission power and the convergence rate. A larger ρ results in a lower total transmission power, but a slower convergence speed is attained. In addition, the total transmission power of the proposed algorithm for various smooth factors λ given the same ratio of elite samples ρ = 0.05 is plotted in Fig. 7. Simulation results demonstrate the tradeoff between convergence speed and the total transmission power. Although a smaller λ yields a smaller total transmission power, the improvement is not significant. Therefore, the simulation results suggest that a large smooth factor (λ = 1) should be adopted to obtain a fast convergence speed. Fig. 8 illustrates the total transmission power of the proposed algorithm for various minimum data rate requirements, where the smooth factor is set as λ = 0.9 and the ratio of elite samples is set as ρ = 0.05. For both cases of K = 4

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improvement in the total transmission power. Therefore, λ = 1 should be adopted. R EFERENCES

Fig. 8. Total transmission power of the proposed CE-based scheme for various minimum data rate requirements.

Fig. 9. Total transmission power of the proposed CE-based scheme for various BER requirements.

and K = 8, the total transmission power increases with the minimum data rate requirement. Finally, Fig. 9 delineates the total transmission power of the proposed algorithm for various BER requirements, where the smooth factor is set as λ = 0.9, the ratio of elite samples is set as ρ = 0.05, and the number of users is set as 4 and 8. As expected, the total transmission power increases when the minimum required BER decreases. The total transmission power of the proposed algorithm V. C ONCLUSION This paper presents a novel sub-carrier, bit and power allocation scheme based on the CE method for OFDMA systems. Compared with the well-known GA scheme [24], the proposed algorithm obtains a significantly lower total transmission power. In addition, the proposed scheme outperforms Chen’s scheme [21] in seven iterations. The simulation results demonstrate that a larger ratio of elite samples results in a lower total transmission power, although the convergence speed is slower. This study also shows that a smaller smooth factor significantly decreases the convergence speed with marginal

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Kuan-Chou Lee received the B.S. degree in electrical engineering from the Fu Jen Catholic University, Taipei, Taiwan, in 2007, the M.S. degree in communications engineering from the National Sun Yat-sen University, Kaohsiung, Taiwan, in 2009. He is currently pursuing the Ph.D. degree in communication engineering from the National Taiwan University, Taipei. His current research interests include wireless communication, OFDM, cooperative, and signal processing.

Sen-Hung Wang received the B.S. degree in electrical engineering from National Dong Hwa University, Hualien, Taiwan, in 2004, the M.S. and the Ph.D. degrees in communications engineering and electrical engineering from National Sun Yat-sen University, Kaohsiung, Taiwan, in 2006 and 2010, respectively. He is currently a Post-Doctoral Research Fellow at the Intel-NTU Connected Context Computing Center, National Taiwan University, Taipei, Taiwan. His current research interests include wireless communication, signal processing, sequence design, multiple access technology, machineto-machine communication, and the fifth generation mobile communication technology.

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Chih-Peng Li received the B.S. degree in physics from National Tsing Hua University, Hsinchu, Taiwan, in 1989, and the Ph.D. degree in electrical engineering from Cornell University, Ithaca, NY, USA, in 1998. From 1998 to 2000, he was a member of Technical Staff with the Lucent Technologies, Murray Hill, NJ, USA. From 2001 to 2002, he was the Manager at the Acer Mobile Networks. In 2002, he joined the Faculty at the Institute of Communications Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan, as an Assistant Professor, and then promoted as a Professor in 2010. He is currently the Chairman of the Department of Electrical Engineering and the Director of the International Master’s Program in Electric Power Engineering. His current research interests include wireless communications, baseband signal processing, and data networks. He is currently the President of Taiwan Institute of Electrical and Electronics Engineering, the Chapter Chair of the IEEE Tainan Section Communications Society, the Chapter Chair of the IEEE Tainan Section Broadcasting Technology Society, and the Vice Chair of Chapter Coordination Committee for IEEE Asia Pacific Board. He also serves as an Editor for the IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS, the Associate Editor for the IEEE T RANSACTIONS ON B ROADCASTING, and the General Chair of 2014 IEEE Vehicular Technology Society Asia Pacific Wireless Communications Symposium. He was the Lead Guest Editor of the Special Issue on Advances in Antenna Design and System Technologies for Next Generation Cellular Systems, International Journal of Antennas and Propagation. He was also the recipient of the 2014 Outstanding Professor Award of the Chinese Institute of Electrical Engineering Kaohsiung Branch.

Ho-Hsuan Chang received the Ph.D. degree in electrical engineering from Syracuse University, Syracuse, NY, USA, in 1997. From 1997 to 2003, he joined the Faculty of the Department of Electrical Engineering, Chinese Military Academy, Kaohsiung, Taiwan, as an Associated Professor. He is currently with the Department of Communication Engineering, I-Shou University, Kaohsiung. His current research interests include wireless communication, signal processing, space-time coding, and sequence design.

Hsueh-Jyh Li was born in Yun-Lin, Taiwan, in August 11, 1949. He received the B.S.E.E. degree from National Taiwan University (NTU), Taipei, Taiwan, in 1971, the M.S.E.E. and the Ph.D. degrees from the University of Pennsylvania, Philadelphia, PA, USA, in 1980 and 1987, respectively. From 1973 to 2012, he was with the Department of Electrical Engineering, NTU, where he was retired in 2012. He was the Director of the Communication Research Center at NTU from 1995 to 2000 and the Chairman of the Graduate Institute of Communication Engineering at NTU from 1997 to 2000. He was the Joint Professor at the Department of Electrical Engineering and the Graduate Institute of Communication Engineering from 1997 to 2012. He was the recipient of the Distinguished Research Award from the National Science Council, Republic of China. His current research interests include microstrip antennas, radar scattering, microwave imaging, radio channel characteristics, and wireless communications.