Home ยป Low Complexity Transmit Antenna Selection with Power Balancing in Ofdm Systems

Low Complexity Transmit Antenna Selection with Power Balancing in Ofdm Systems

3018 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 10, OCTOBER 2010 Low Complexity Transmit Antenna Selection with Power Balancing in OFDM Systems Ki-Hong Park, Student Member, IEEE, Young-Chai Ko, Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE Abstract—In this paper, we consider multi-carrier systems with multiple transmit antennas under the power balancing constraint, which is de? ned as the constraint that the power on each antenna should be limited under a certain level due to the linearity of the power ampli? er of the RF chain.

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Applying transmit antenna selection and ? xed-power variable-rate transmission per subcarrier as a function of channel variations, we propose an implementation-friendly antenna selection method which offers a reduced complexity in comparison with the optimal antenna selection scheme. More speci? cally, in order to solve the subcarrier imbalance across the antennas, we operate a two-step reallocation procedure to minimize the loss of spectral ef? ciency. We also provide an analytic lower bound on the spectral ef? ciency for the proposed scheme.

From selected numerical results, we show that our suboptimal scheme offers almost the same spectral ef? ciency as the optimal one. Index Terms—Transmit antenna selection, orthogonal frequency division multiplexing, power balancing constraint, adaptive modulation. T I. I NTRODUCTION HE design of the transmitter or receiver with multiple antennas is an important problem since it has a major impact on the performance of wireless communication systems. In order to obtain considerable improvements, perfect channel state information should be used at the transmitter.

However, in practice, the limitation on the feedback overhead and the need of calibrations for channel reciprocity lead to the development of transmit diversity systems with partial channel knowledge [1]. Transmit antenna selection (TAS) can signi? cantly reduce the feedback load since only the information of the selected antenna should be known at the transmitter [2]. On another front, state-of-the-art wireless systems have been developed and deployed with orthogonal frequency division multiplexing (OFDM) to obtain high spectral ef? ciency Manuscript received May 1, 2010; accepted August 2, 2010.

The associate editor coordinating the review of this paper and approving it for publication was C. Tellambura. This is an expanded version of work that was presented at the IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC’09), Perugia, Italy, in June 2009. K. -H. Park and Y. -C. Ko are with the Department of Electronics and Computer Engineering, Korea University, Seoul, Korea (e-mail: {grn552; koyc}@korea. ac. kr). M. -S. Alouini was with the Electrical and Computer Engineering Program of TAMU-Qatar, Doha, Qatar.

He is currently with the Division of Physical Science and Engineering, KAUST, Thuwal, Makkah Province, Saudi Arabia (e-mail: mohamed. [email protected] edu. sa). This research was supported in part by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program, supervised by the NIPA (National IT Industry Promotion Agency) (NIPA-2010-(C1090-1011-0011)), and funded in part by QNRF (The Qatar National Research Fund; a member of QF). Digital Object Identi? er 10. 1109/TWC. 2010. 09. 100736 nd robustness against intersymbol interference (ISI) caused by multipath propagation [3]. High peak-to-average power ratio (PAPR) has been the main drawback of OFDM systems since the transmit signals of subcarriers which are overlapped constructively in the time domain may make the power ampli? er of the RF chain operate in its non-linear region. Accordingly, this often leads to an undesirable and inef? cient performance degradation [4]. The issue becomes more problematic in the case the system involves multiple antennas, especially when TAS is used.

For instance, when the antennas are selected and are assigned to subcarriers solely based on the subcarrier channel quality, the subcarriers end up being randomly allocated to each antenna and a power imbalance across the antennas may occur. Indeed if too many subcarriers are assigned to a particular antenna, the power ampli? er ends up operating in the non-linear region and this of course causes a certain performance degradation. To this effect, TAS problem under a power balancing constraint in OFDM has been formulated in [5] and an optimal TAS has been proposed by using linear programming (LP) in [6].

In this paper, under the power balancing constraint, we propose an implementation-friendly TAS that uses only comparators but is still spectrally ef? cient. Starting from a conventional TAS, our proposed scheme ful? lls two steps to solve the subcarrier imbalance by (i) reallocation without loss of spectral ef? ciency and (ii) reallocation with minimum loss of spectral ef? ciency. The lower bound of average spectral ef? ciency (ASE) is derived to verify that the proposed scheme has a negligible loss in comparison with the optimal scheme in [6]. The remainder of this paper is organized as follows.

Section II presents the system model. Section III brie? y reviews the previous work and describes our proposed scheme. The performance of our scheme is analyzed in Section IV, and its complexity is discussed in Section V. Numerical results for different cases are then given and interpreted in Section VI. Finally, a brief conclusion summarizing the main results of the paper is given in Section VII. Throughout the paper, we use the following notations. ?????? [? ] denotes an expectation operator and ??? is a ceil function. ???????? denotes the cardinality of set ?????? and ?????? means a null set. For any functions ?????? ?????? ) and ?????? (?????? ), ?????? (?????? ) = ?????? (?????? (?????? )) is equivalent to lim?????? >? ??????? (?????? )/?????? (?????? )? < ? , and this notation is used to measure the computational complexity. We( ) denote the number ??????! of ?????? -combinations of an ?????? -element set as ?????? = ??????! (????????????? )! . ?????? c 1536-1276/10$25. 00 ? 2010 IEEE IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 10, OCTOBER 2010 3019 II. P ROBLEM F ORMULATION A. System and Channel Model In this paper, we consider multiple input single output (MISO) OFDM systems with ?????? ransmit antennas and ?????? subcarriers which is modeled as ???????????? = ?????? ? ?????? =1 TABLE I SNR T HRESHOLDS TO SATISFY 0. 1% BER FOR 2???????????? – ARY QAM ?????????????????? [dB] for BER0 = 10? 3 9. 64 13. 32 16. 63 19. 79 22. 86 25. 91 28. 94 ? ?????? 1 2 3 4 5 6 7 (= ?????? ) 8 Rate (???????????? ) 2 3 4 5 6 7 8 QAM 4 8 16 32 64 128 256 ??????? ,?????? ???????????? ,?????? + ???????????? , ?????? = 1, 2, . . . , ?????? , (1) where ??????? ,?????? and ???????????? ,?????? are the channel gain and transmit signal of the ?????? -th antenna for the ?????? -th subcarrier. ???????????? s a white Gaussian noise with zero mean and unit variance. Although we assume for the purpose of the performance analysis of our proposed scheme that each channel gain, ??????? ,?????? , is an independent and identically distributed (i. i. d. ) complex Gaussian random variable with ??????? ,?????? ? ???????????? (0, 1), we apply the proposed scheme to practical frequency selective channel and show its performance by some selected simulations in Section VI. Applying TAS, only one antenna is activated for each subcarrier. Denoting the index of active antenna for the ?????? th subcarrier as ???????????? we set ?????????????????? ,?????? = ???????????? for the active antenna and ???????????? ,?????? = 0 for all ?????? except the active antenna. Then, the received signal can be rewritten as ???????????? = ????????????? ,?????? ???????????? + ???????????? , ?????? = 1, 2, . . . , ??????. (2) III. T RANSMIT A NTENNA S ELECTION Prior to describing the proposed TAS strategy, we brie? y review the existing TAS strategies. A. Unconstrained Transmit Antenna Selection If we do not consider the balancing constraint, the maximum spectral ef? ciency can be achieved by selecting the antenna of the best SNR among ?????? ntennas for each subcarrier. The index of active antenna for the ?????? -th subcarrier can be computed by ???????????? = arg max ???????????? ,?????? . We note that this ?????? ? conventional TAS procedure does not guarantee ?????? for each antenna. B. Optimal Transmit Antenna Selection with Power Balancing In [6], the optimal TAS strategy has been given to achieve the maximum/minimum of the cost functions such as capacity, BER and SNR. The cost functions are optimized for antenna selection constraint as well as power balancing constraint using LP such as the simplex methods or interior point methods [9].

We can apply the optimal TAS whose cost function is the discrete rate supported by adaptive modulation. C. Proposed Transmit Antenna Selection We now propose a TAS strategy with sequential reallocation for the subcarriers of the antennas which have subcarriers ? exceeding the balanced number of subcarriers, ??????. In Fig. 1, we illustrate our proposed method as per the two following steps. The following algorithm explains the mode of operation of the proposed TAS. 1. (Initialization) Initialize the unconstrained TAS process, ???????????? = arg max ???????????? ,?????? , to maximize the spectral ef? ciency ?????? and ? d all the needed sets. For ?????? ? {1, 2, . . . , ?????? }, ???????????? = {??????????????????? = ?????? } and for ?????? ? {1, 2, . . . , ?????? }, ?????? + = ? ? {??????? ????????????? ? > ?????? }, ??????? = {??????? ????????????? ? < ?????? } and ?????? 0 = ? ???????????? denotes the set of indices of {??????? ????????????? ? = ?????? }. the selected subcarriers for the ?????? -th antenna and ?????? + , ??????? , and ?????? 0 are the sets of indices of the antennas with overloaded subcarriers, underloaded subcarriers and the ? same subcarriers as ?????? , respectively. 2. (step 1) For the antennas in ?????? , execute a reallocation process without rate loss. We assume that the transmit power for each subcarrier is the [ ] same as ?????? , i. e. , ?????? ????????????? ?2 = ?????? , ???????. We note that adaptive power ? ? loading per subcarrier is not an option in this paper due to the complexity of power loading and the strict regulation of spectral mask, while it provides a small spectral ef? ciency gain over equal power loading. As such, the channel SNR for the 2 ? ?????? -th antenna of the ?????? -th subcarrier, i. e. , ???????????? ,?????? ? ???????? ,?????? ? ?????? , has a probability density function (PDF) and cumulative ?????? ? distribution function (CDF) given by ?????????????????? ,?????? (?????? ) = ?????? ??????? ?????? and ? ?????? ? ?????? ?????????????????? ,?????? (?????? ) = 1 ? ?????? ? for ?????? ? 0, respectively. For convenience, we de? ne a common PDF and CDF as ???????????? (?????? ) ? ?????????????????? ,?????? (?????? ) and ???????????? (?????? ) ? ?????????????????? ,?????? (?????? ), respectively. B. Antenna Power Balancing The goal is to equally allocate the aggregate subcarriers for each antenna so that the total transmit power can be evenly distributed among the transmit antennas [6].

Denoting the ? balanced number of subcarriers as ?????? , this constraint can be written by ] ? ? [ ? ?????? ????????????? ?2 ?????? ? ? ? ?????? , ?????? = 1, 2, . . . , ??????. (3) ?????? ? ?????? {??????????????????? =?????? } C. Adaptive Modulation We consider adaptive modulation [7], the objective of which is to obtain higher spectral ef? ciency in a practical scenario. In this paper, we use multi-level quadrature amplitude modulations (QAM) and each modulation size is decided if the instantaneous SNR meets one of the SNR thresholds to satisfy the target bit error rate (BER).

Using the exact BER expression for multi-level QAMs [8], we can obtain multiple SNR thresholds to satisfy 0. 1% BER in Table I. 3020 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 10, OCTOBER 2010 Overloaded subcarriers Reallocated subcarrier without loss Reallocated subcarrier with loss KB # of Allocated Subcarriers KB # of Allocated Subcarriers KB # of Allocated Subcarriers 1 2 3 4 1 1 2 3 4 2 3 4 Antenna Index Step 1 Fig. 1. Antenna Index Step 2 Antenna Index Final Example of the proposed TAS illustrated step by step for the ?????? = 4 case. a) For the arbitrary subcarrier of the antennas in ?????? + , i. e. ?????? ? ???????????? , where ?????? ? ?????? + , ? nd the index of the alternative antenna by using ??????? = arg max ???????????? ,?????? and check if the reallocation is feasible, i. e. , ?????? (?????????????????? ,?????? ) = ?????? (????????????? ,?????? ), where ?????? (???????????? ,?????? ) is the achievable rate of the SNR, ???????????? ,?????? . b) If ?????? (?????????????????? ,?????? ) = ?????? (????????????? ,?????? ), reallocate the ?????? -th subcarrier to the ??????? -th antenna and update all the sets to be needed, ???????????? , ?????? + , ??????? and ?????? 0 by changing ???????????? as ???????????? ??????? . c) Repeat a)-b) until all the antennas satisfy the power balancing condition, i. e. , ?????? + = ?????? , or all the antennas in ?????? + are fully examined. 3. (step 2) For the antennas in ?????? + , complete a reallocation process with rate loss. a) Among the subcarriers in ???????????? , where ?????? ? ?????? + , ? nd the index of the subcarrier with the minimum of rate difference, i. e. , ?????? ? = arg min ?????? (?????????????????? ,?????? ) ? ?????? (????????????? ,?????? ), where ??????? = arg max ???????????? ,?????? . Reallocate the ?????? ? -th subcarrier to the ??????? -th antenna and update ???????????? ?????? + , ??????? and ?????? 0 by altering ????????????? = ??????? . b) Repeat a) until the reallocations for all the antennas in ?????? + are ? nished, i. e. , ?????? + = ??????. The proposed TAS may be terminated after ? nishing the step 1 as the case may be. It is noted that the reallocation may be completed without loss of spectral ef? ciency according to the instantaneous channel condition. ?????????????? ??????????????????? ?????????????? A. Unconstrained Transmit Antenna Selection 1) Average Number of Overloaded Subcarriers per Antenna: Averaging the number of overloaded subcarriers in ???????????? the average number of overloaded subcarriers per antenna can be computed as ( ) ???????????????????????? = ?????? ? ? (?????? ? ?????? ) ?????? ? ?????? Pr [ ???????????? = ???????????? ] (4) ?????? ? ( ?????? ) ? = (?????? ? ?????? ) ?????? ?????? (1 ? ?????? )????????????? , ?????? ? ?????? =?????? +1 ? ?????? =?????? +1 ?????? =1 where ???????????? is one of the sets the cardinality of which is ?????? by selecting arbitrary ?????? subcarriers among ?????? ones. Using the i. i. d. assumption, we can calculate Pr [???????????? = ???????????? ] = ?????? ?????? (1 ? ?????? )????????????? for all ?????? , where ?????? s the probability that the ?????? -th subcarrier is allocated to the ?????? -th antenna and can be written by ?????? 1 ?????? = ?????? (1 ? [???????????? (???????????? 1 )] ). We note that the subcarrier is not allocated to any antennas if the best SNR is below the minimum threshold, ???????????? 1 . 2) Average Spectral Ef? ciency per Antenna: The ASE for each antenna can be obtained as a sum of the weighted spectral ef? ciencies as SE = = ?????? ?????? [ ] 1 ?? ???????????? Pr ?????????????????? ? ?????????????????? ,?????? < ?????????????????? +1 ?????? ?????? =1 ?????? =1 ?????? ] [ ?????? ? ????????????

Pr ?????????????????? ? ???????????? 1 ,1 < ?????????????????? +1 , ?????? ?????? =1 (5) IV. P ERFORMANCE A NALYSIS In this section, we analyze the ASE for the proposed scheme. We note that the average number of overloaded subcarriers per antenna is derived to verify the effect caused by reallocation of the subcarriers onto one of the antennas after the proposed scheme ? nishes the step 1. where the second equality is possible due to the homogenous nature for all the subcarriers. For the convenience of analysis, we omit the indices of the antenna and subcarrier and de? ne the random variables of the ordered SNRs as ?????? 1) ? ?????? (2) ? . . . ? ?????? (?????? ) , where ???????????? 1 ,1 = ?????? (1) in this case. Then, the ASE per antenna can be computed as ) ( ?????? ?[ ]?????? ?????? ???????????? (?????????????????? ) SE?????????????????? = ? [???????????? (???????????? 1 )]?????? . (6) ???????????? ? ?????? ?????? =1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 10, OCTOBER 2010 3021 B. Proposed Transmit Antenna Selection 1) Average Number of Overloaded Subcarriers per Antenna: The exact average number of overloaded subcarriers per antenna is not easily tractable because the state of the sets, ???????????? ?????? + , ??????? and ?????? 0 , changes each time an overloaded subcarrier is reallocated. For this reason, we consider an upper bound of the average number of residual overloaded subcarriers after ? nishing the step 1. Similar to (4), the average number of overloaded subcarriers per antenna can be represented by ?????????????????? = = ? ?????? ? integrating the joint PDF of ordered SNRs given by [10] ???????????? (1) ,?????? (?????? ) (?????? , ?????? ) = ?????? ! [???????????? (?????? )]????????????? ???????????? (?????? ) (?????? ? ?????? )! (?????? ? 2)! ??????? 2 ? [???????????? (?????? ) ? ??????????? (?????? )] ???????????? (?????? ), ?????? > ??????. (9) ? (?????? ? ?????? ) Pr [ ????????????? ? = ?????? ] ? (?????? ? ?????? ) ?????? ? ?????? =?????? Finally, the average number of overloaded subcarriers after the step 1 can be bounded by ?????? ?????? ? ? ( ?????? ) ? (?????? ? ?????? ) ?????? ?????? (1 ? ?????? )????????????? ?????????????????? ? ?????? ? ?????? =?????? ?????? =?????? +1 ( ) ?????? ? ??????????????????? (1 ? ?????? )?????? ? ???????????? . (10) ?????? ? ?????? 2) Average Spectral Ef? ciency per Antenna: Now we analyze the lower bound of the ASE for the proposed scheme.

After completing the step 1, the residual overloaded subcarriers are forced to be reallocated at the expense of a spectral ef? ciency loss in the step 2. To ? nd a tractable lower bound, we relax the selection criterion by randomly selecting a subcarrier among ???????????? , while actually we ? nd the subcarrier with minimum rate gap. With this assumption, we analyze an upper bound of the ASE loss per subcarrier, which is given by ] [ ? ? ? = ?????? ?????? (?????? (1) ) ? ?????? (?????? ) ?????? (?????? (1) ) > ?????? (?????? ), ?????? (1) ? ???????????? 1 [ ] Pr ?????????????????? ? ?????? (1) < ?????????????????? 1 , ?????? (?????? (1) ) > ?????? (?????? ? ) [ ] Pr ?????? (1) ? ???????????? 1 , ?????? (?????? (1) ) > ?????? (?????? ? ) ?????? =1 [ ] ??????? 1 ? Pr ?????????????????? ? ?????? ? < ?????????????????? +1 , ?????? (?????? (1) ) > ?????? (?????? ? ) [ ] ? ???????????? Pr ?????? (1) ? ???????????? 1 , ?????? (?????? (1) ) > ?????? (?????? ? ) ?????? =1 ? ?????? ? ???????????? ?????? ?????? (?????? , ?????? , ?????????????????? , ?????????????????? +1 ) ?????? =2 = ??????? ??????? ? =1 ?????? ?????? =2 ?????? (?????? , ?????? , ??????????????????? , ??????????????????? +1 ) ?????? 1 ??????? ??????? 1 ? ???????????? ?????? =2 ?????? (?????? , ?????? , ?????????????????? , ?????????????????? +1 ) ? , ??????? ??????? ?????? ? =1 ?????? =2 ?????? (?????? , ?????? , ??????????????????? , ??????????????????? +1 ) ?????? =1 = ?????? ? ? ?????? =?????? +1 ?????? ? ? ?????? =?????? +1 ? Pr [????????????? ? [ ] ? Pr ????????????? ? = ?????? ????????????? ? = ?????? (7) = ?????? ] , ? where ???????????? is the set of the subcarriers allocated to the ?????? -th antenna before starting the step 1 and, as presented in the previous section, the probability that the cardinality is equal to ?????? fter the unconstrained TAS can be given ( ) ? by Pr [????????????? ? = ?????? ]= ?????? ?????? ?????? (1 ? ?????? )????????????? . When there exist ?????? ?????? subcarriers allocated to the ?????? -th antenna, ?????? ? ?????? subcarriers are reallocated in the step 1 since the subcarriers in ???????????? are reduced to ?????? subcarriers. In other word, we can ? nd ?????? ? ?????? subcarriers which can be reallocated at the same spectral ef? ciency as the best one. In order to ? nd a tractable upper bound for lossless reallocation, we assume that the reallocation is achieved by randomly selecting an antenna in ??????? while the antenna with the best SNR among ??????? is selected as a candidate in the step 1 of the proposed scheme. For brevity, noting that ?????? ? is the alternative SNR by the step 1 and ?????? (1) is the best SNR among all ?????? antennas, its probability can ? ?????? be given by Pr [????????????? ? = ?????? ? ????????????? ? = ?????? ] = ( ?????? ? ?????? )??????????????????? (1 ? ?????? )?????? , ? ? where ?????? ? Pr[?????? (?????? ) = ?????? (?????? (1) ) ? ????????????? ? = ?????? ]. Then, ?????? can be derived to be given as (?????? ) ?????? ? ?????? =1 ???????????? (11) ?????? = Pr ?????????????????? ?????? ? ? ?????? (1) < ?????????????????? +1 ?????? (1) ? ???????????? 1 [ ] where ?????? (?????? , ?????? , ?????? , ?????? ) ? ] [ ?????? ?????? (?????? ) ? ? Pr ?????????????????? ? ?????? (?????? ) ? ?????? (1) < ?????????????????? +1 ] [ = (?????? ? 1)Pr ?????? (1) ? ???????????? 1 ?????? =1 ?????? =2 (?????? ) ??????? 2 ( ? ?????? ? 2 ) ?????? ! ?????? (?????? ? ?????? )! (?????? ? 2)! ?????? =0 ) ( (?????? ? 1) 1 ? [???????????? (???????????? 1 )]?????? ??????? 2 ( ? ?????? ? 2 ) (? 1)?????????????? 2 { 1 [ ]?????? ? ???????????? (?????????????????? +1 ) ?????? ?????? ? ?????? ? 1 ?????? ????? =0 ( ) ]?????? 1 1 [ ? + ???????????? (?????????????????? ) ?????? + 1 ?????? } ]?????????????? 1 [ ]?????? +1 1 [ ???????????? (?????????????????? ) ???????????? (?????????????????? +1 ) ? , (8) ?????? + 1 = ?????? =1 ?????? =2 ?????? ?????? ?? ??????! (????????????? )! (??????? 2)! (? 1)?????????????? 2 ?????????????? 1 [???????????? (?????? )] ? (?????? ? ?????? ? 1)(?????? + 1) ) ( ?????? +1 ?????? +1 (12) ? [???????????? (?????? )] ? [???????????? (?????? )] and ??????? 2 ( ? ?????? ? 2 ) ?????? ! ?????? (?????? ? ?????? )! (?????? ? 2)! ?????? =0 ( ) ?????????????? 2 (? 1) ?????? +1 1 ? ???????????? (?????? )] ? (?????? ? ?????? ? 1)(?????? + 1) ) ( ?????????????? 1 ?????????????? 1 . (13) ? [???????????? (?????? )] ? [???????????? (?????? )] ?????? (?????? , ?????? , ?????? , ?????? ) ? where (a) holds since the best SNRs of all the subcarriers are above the minimum threshold for allocation, (b) holds due to the random selection for reallocation and (c) holds by Finally, the lower bound of ASE can be calculated by using (6), (10) and (11), yielding SE???????????? ? SE?????????????????? ? ????????????? ? SE?????? . (14) 3022 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 0, OCTOBER 2010 1. 4 Average Spectral Efficiency per Antenna [bits/OFDM symbol] Average Number of Overloaded Subcarriers per Antenna 140 1. 2 120 Optimal TAS? PB on Capacity (sim) [6] No const TAS on SE (6) Optimal TAS? PB on SE (sim) [6] Proposed TAS? PB on SE (sim) Lower bound on SE of Proposed TAS? PB (14) 1 No const TAS (4) No const TAS (sim) Upper bound of Proposed TAS? PB (10) Upper bound of Proposed TAS? PB (sim) Proposed TAS? PB step 1 (sim) 100 0. 8 80 0. 6 60 75 0. 4 40 73 0. 2 20 71 0 0 5 10 15 20 SNR per Subcarrier [dB] 25 30 0 17 18 0 5 10 15 20 SNR per Subcarrier [dB] 25 30 Fig. 2.

Comparison of the residual overloaded subcarriers between the proposed TAS (after the step 1) and the unconstrained TAS for i. i. d. fading per subcarrier. Fig. 3. Comparison of ASE among the proposed TAS and the previous methods for i. i. d. fading per subcarrier. V. C OMPLEXITY A NALYSIS Compared to the optimal algorithm, the complexity of the proposed TAS is quanti? ed in terms of the number of subcarriers and antennas, ?????? and ?????? . In case of the optimal scheme in [6], assuming that the interior point method is used to optimize the cost function, we note v the theoretical that complexity for LP can be lowered to ?????? ???????????? ?????? ) iterations, requiring a total of ?????? ((???????????? )3 / log(?????? ?????? )?????? ) arithmetic operations [9], where ?????? is the length of a binary encoding of the input data, i. e. , a cost vector, a matrix and a vector for linear constraint in an LP problem. Although it is not easy to compare the complexity between our scheme and the optimal scheme, we note that our scheme does not require several iterations but only 2 steps as well as ?????? (???????????? ?????? ) operations, where ?????? is not an important factor for this comparison.

Therefore, our proposed TAS can perform the reallocation with a much lower complexity. VI. N UMERICAL R ESULTS In this section, we present some selected simulation results to compare the performance of the proposed scheme with the other schemes. They also show that the reallocation for the overloaded subcarriers reduces the level of power imbalance and validate our analysis. We assume that ?????? = 4 and ?????? = 64 respectively. First, we assume i. i. d. fading for each subcarrier in Fig. 23. In Fig. 2, we show the effect of the reallocation when ? the subcarriers more than ?????? are reallocated to an arbitrary antenna.

The analytic results completely agree with the simulation curves and the number of overloaded subcarriers is reduced signi? cantly through the greedy search in the step 1 of the proposed scheme. For the low SNR region, the probability for the subcarriers to be loaded into an antenna over the balancing level is low because there are many subcarriers with outage. On the other hand, in the high SNR range, the number of overloaded subcarriers is reduced as SNR increases. This is because the highest modulation size is ? xed and the probability that the reallocation is terminated in the step 1 without rate loss becomes larger.

In this region, we can easily reallocate the overloaded subcarriers without using complicated optimization techniques. In Fig. 3, we present the ASE per antenna for three different schemes, which is enlarged in the sub-? gure, especially for the medium SNR region. We also present the simulation result for the optimal TAS whose cost function is not a discrete rate but the capacity, log2 (1 + ?????? ) in [6]. As a benchmark, this curve shows that three other schemes using adaptive modulation can exploit the same degrees of freedom as the capacity based optimal scheme. As shown in this ? gure, we can ? d that the lower bound is very tight with the difference of nearly 0. 22 dB for a ? xed ASE. Compared to the optimal scheme in [6], our proposed scheme has a negligible loss, i. e. , about 0. 03 dB for a ? xed ASE. Now, in order to see the performance of our proposed scheme in more practical channel environment, we adopt the IEEE 802. 11n channel D [11] and present the ASE per antenna and the BER in Figs. 4-5, respectively. To verify the performance degradation by power imbalance, we assume that the transmit signals pass through the non-linear power ampli? er which is modeled as Rapp model [12] with the smoothness, ?????? 3, and 7 dB back-off from 1 dB compression point. Compared to i. i. d. case, this channel is relatively insensitive in the frequency domain and the band consisting of the successive subcarriers is often allocated to an antenna at once. Accordingly, the variance of the number of subcarriers allocated to an antenna increases when we exploit unconstrained TAS. It increases the number of subcarriers to be reallocated in the proposed scheme and leads to the degradation of ASE since the probability that the reallocation with rate loss is performed becomes higher, which is the reason that the ASE of the proposed scheme is degraded

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 10, OCTOBER 2010 3023 140 Average Spectral Efficiency per Antenna [bits/OFDM symbol] of power ampli? er signi? cantly affects the transmit signals of subcarriers which are excessively allocated to an antenna. No const TAS on SE (sim) Proposed TAS? PB on SE (sim) 120 VII. C ONCLUSIONS In this paper, we proposed a low complexity TAS scheme when OFDM systems with multiple antennas need to satisfy a per-antenna power balancing constraint. In order to overcome the subcarrier imbalance problem across the antennas, we use a two-step reallocation procedure (i) ? st without loss of spectral ef? ciency and (ii) then with minimum loss of spectral ef? ciency. We also offered some analytic results to prove the effect of reallocation and performance of our proposed scheme. We show from some selected numerical results that the proposed lower complexity scheme offers almost the same spectral ef? ciency as the optimal scheme. 100 80 60 40 20 0 5 10 15 20 25 SNR per Subcarrier [dB] 30 35 ACKNOWLEDGMENT The authors would like to thank the associate editor, Chintha Tellambura, and anonymous reviewers for the careful reading of the paper and for the constructive, focused, and detailed comments and suggestions.

R EFERENCES Fig. 4. Comparison of ASE between the proposed and unconstrained TAS for IEEE 802. 11n channel D with the modeled power ampli? er. 10 ?1 No const TAS (sim) Proposed TAS? PB (sim) Target BER 10 ?2 10 ?3 10 ?4 10 ?5 5 10 15 20 25 SNR per Subcarrier [dB] 30 35 Fig. 5. Comparison of BER between the proposed and unconstrained TAS for IEEE 802. 11n channel D with the modeled power ampli? er. in Fig. 4. However, as shown in Fig. 5, we should notice that the BER of the unconstrained TAS cannot satisfy target BER at all for most of the SNR region, while the proposed scheme has good BER performance below target BER.

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