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IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 9, SEPTEMBER 2011
Partial Transmit Sequences for PAPR Reduction Using Parallel Tabu Search Algorithm in OFDM Systems Necmi Tas¸pınar, Adem Kalınlı, and Mahmut Yıldırım, Member, IEEE
Abstract—In this letter, partial transmit sequences (PTS) based on parallel tabu search (Parallel TS-PTS) scheme is proposed to reduce the peak-to-average power ratio (PAPR) of orthogonal frequency division multiplexing (OFDM) signals. PTS is a distortionless PAPR reduction technique, but its high search complexity for finding optimal phase factors must be reduced for usable applications. Parallel TS-PTS is compared to different PTS schemes for PAPR reduction and search complexity performances. The simulation results show that the proposed parallel TS-PTS method provides good PAPR reduction and biterror-rate (BER) performances.
PAPR reduction performance compared to conventional PTS, TS-PTS, ABC-PTS and IDPSO-PTS. In addition, parallel TSPTS is compared with original OFDM signals, ABC-PTS and optimum-PTS for bit-error-rate (BER) performances of the OFDM system using the HPA. In the simulations, solid-state power amplifier (SSPA), which is a kind of HPA, was used.
Index Terms—OFDM, PAPR, PTS, parallel tabu search, tabu search, high power amplifier (HPA).
The discrete-time transmitted OFDM signal with 𝑁 subcarriers is given by
I. I NTRODUCTION
O
RTHOGONAL frequency division multiplexing (OFDM) is a very attractive multicarrier modulation technique for various wireless communication standards [1]. Despite the advantages, high PAPR value of the signals is a major drawback of the OFDM systems. The high PAPR value causes in-band distortion and out-of-band radiation due to unwanted saturation in the high power amplifier (HPA) [2], and this leads to performance degradation of the OFDM systems. In order to eliminate degradation of the high PAPR of the OFDM signals, many PAPR reduction methods have been proposed [1]. From these methods, partial transmit sequences (PTS) is the most popular technique because of good PAPR reduction performance and distortionless structure [1]. However, the conventional PTS requires an exhaustive search, which causes high search complexity to find the optimal phase factors. In the literature, there are some recently proposed methods to reduce PAPR with low complexity in OFDM systems [3-5]. Among these methods, we compared the improved discrete particle swarm optimization (IDPSO) [4] and artificial bee colony (ABC) algorithm [5] with our proposed method to reduce the PAPR. In this letter, a hybrid method based on the PTS technique and parallel tabu search (parallel TS) algorithm is introduced to reduce the PAPR of the OFDM signals with low search complexity. The parallelism helps the TS algorithm to find the promising regions of the search space very quickly [6-7]. The simulation results show that the parallel TS-PTS gives better Manuscript received May 13, 2011. The associate editor coordinating the review of this letter and approving it for publication was A. Panagopoulos. N. Tas¸pınar is with the Department of Electrical and Electronics Engineering, Erciyes University, Kayseri, Turkey (e-mail:
[email protected]). A. Kalınlıis with the Kayseri Vocational Technical School, Erciyes University, Kayseri, Turkey (e-mail:
[email protected]). M. Yıldırım is with the Department of Electrical and Electronics Engineering, Bozok University, Yozgat, Turkey (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2011.072911.110999
II. S YSTEM M ODEL A. PAPR of the OFDM Signal
𝑁 −1 1 ∑ 𝑥𝑘 = √ 𝑋𝑛 𝑒𝑗2𝜋𝑛𝑘/𝐿𝑁 , 𝑘 = 0, 1, ⋅ ⋅ ⋅ , 𝐿𝑁 − 1, (1) 𝑁 𝑛=1 𝑇
where X = [𝑋0 , 𝑋1 , . . . , 𝑋𝑁 −1 ] is the input signal vector with each symbol modulated by PSK or QAM, and 𝐿 is the oversampling factor where 𝐿 = 4, which is enough to provide an accurate approximation of the PAPR [1]. The PAPR of the discrete-time OFDM signal 𝑥𝑘 is defined as 𝑃 𝐴𝑃 𝑅 (𝑥) =
max ∣𝑥𝑘 ∣ [ ] 2 𝐸 ∣𝑥𝑘 ∣
0≤𝑘≤𝐿𝑁 −1
2
(dB) ,
(2)
B. PTS for PAPR reduction In the PTS, input signal vector 𝑿 is partitioned into 𝑀 disjoint sub-blocks, and oversampled by inserting (𝐿 − 1) ⋅ 𝑁 zeros, such that ⎤ ⎡ (𝑚)
𝑿 (𝑚) = ⎣𝑋0
(𝑚)
(𝑚)
(𝑚)
, . . . , 𝑋𝑁 ╱2−1 , 0 ⋅ ⋅ ⋅ 0 , 𝑋𝑁 ╱2 , . . . , 𝑋𝑁 −1 ⎦ , (𝐿−1)⋅𝑁
(3)
where 𝑚 = 0, 1, . . . , 𝑀 − 1; therefore, 𝑿 =
𝑀−1 ∑
𝑿 (𝑚) .
(4)
𝑚=0
Next, sub-blocks are transformed from the frequency domain to the time domain by inverse fast Fourier transform (IFFT) with 𝐿𝑁 point. Then, each sub-block is rotated to minimize the value of the PAPR by phase factors 𝒃 = [𝑏0 , 𝑏1 , . . . , 𝑏𝑀−1 ], where 𝑏𝑚 = 𝑒𝑗𝜙 , and 𝜙 ∈ [0, 2𝜋) .Finally, the sub-blocks are summed up. After the PTS optimization, the OFDM signal is given by
c 2011 IEEE 1089-7798/11$25.00 ⃝
′
𝑥𝑘 =
𝑀−1 ∑ 𝑚=0
} { 𝑏𝑚 𝐼𝐹 𝐹 𝑇 𝑿 (𝑚) .
(5)
TAS¸PINAR et al.: PARTIAL TRANSMIT SEQUENCES FOR PAPR REDUCTION USING PARALLEL TABU SEARCH ALGORITHM IN OFDM SYSTEMS
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III. PARALLEL TS-PTS T ECHNIQUE FOR PAPR R EDUCTION The TS algorithm is a form of iterative heuristic search based on intelligent problem solving principles. In this letter, the parallel TS algorithm is used to find the phase factors with the aim of minimizing the PAPR of the OFDM signals. The problem of searching the optimal phase factors 𝒃 that makes 𝑃 𝐴𝑃 𝑅 (𝒃) minimum is called the optimization problem of the 𝑃 𝐴𝑃 𝑅 (𝒃) and can be mathematically expressed as 𝑚𝑖𝑛𝑖𝑚𝑖𝑠𝑒 𝑃 𝐴𝑃 𝑅 (𝒃) (6) 𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝒃 ∈ {+1, −1} TS has a flexible memory that retains the information about previous steps of the search, and uses it to create and exploit new solutions in the search space. Main steps of the TS algorithm are described as follows: Step 1:Get a randomly generated initial solution 𝒃𝑛𝑜𝑤 . Step 2:Select the best admissible solution: 𝒃𝑏𝑒𝑠𝑡 (𝒃𝑏𝑒𝑠𝑡 is the best of all 𝒃∗ ∈ 𝐴 (𝒃𝑛𝑜𝑤 ) : 𝒃∗ is not in tabu list). Step 3:Update the current solution 𝒃𝑛𝑜𝑤 ←− 𝒃𝑏𝑒𝑠𝑡 , and update the tabu list. Step 4:Repeat Step 2 and Step 3 until a stopping criterion (𝑚𝑎𝑥𝑖𝑡) is satisfied. A step of the TS starts with a present solution 𝒃𝑛𝑜𝑤 . 𝒃𝑛𝑜𝑤 has an associated set of feasible solutions 𝐴 which can be obtained by applying a simple modification to 𝒃𝑛𝑜𝑤 . This modification is called a move. In order to be able to get rid of a local minima, a move to the neighbor 𝒃∗ , is created even if 𝒃 is worse than 𝒃𝑛𝑜𝑤 . This would cause the cycling of the search. In order to avoid the cycling problem, a tabu list 𝑇 is introduced. The tabu list stores all the tabu moves that can not be applied to the present solution, 𝒃𝑛𝑜𝑤 . The moves stored in the tabu list are those carried out most frequently and recently, according to certain criteria called tabu restrictions. The use of tabu list decreases the possibility of cycling because it prevents returning, in a certain number of iterations, to a solution visited recently. After a subset of feasible solutions 𝐴∗ is produced according to the tabu list and evaluated for 𝑃 𝐴𝑃 𝑅 (𝒃), the next solution is selected from it. The highest evaluation solution is selected as the next solution 𝒃𝑛𝑒𝑥𝑡 . This loop is repeated until a stopping criterion is satisfied. In the parallel TS approach considered in this letter, the information exchange process between the basic TS algorithms executed in parallel is based on the crossover operation used in genetic algorithm (GA). The crossover operator employed by GA is used to create two new solutions (children) from two existing solutions (parents) in the population formed by the selection operation. The potential phase factors solutions are represented with binary strings in the parallel TS algorithm. Therefore, a solution is created from the binary string by changing the value of a bit, 0 to −1 and 1 to 1. Since the problem is represented in the binary string form, the crossover operation is applied as follows: Two solutions are selected as parent solutions from the population and cut at a randomly selected point. The tails, which are the parts after the cutting point, are swapped and two new solutions are produced. A crossover operation can thus yield better solutions by combining the good features of parent solutions. The flowchart of the parallel TS is depicted in Fig. 1.
Fig. 1. Flowchart of the parallel TS algorithm for phase factors optimization.
IV. S IMULATION R ESULTS In the simulations, we consider an OFDM system with 𝑁 = 256 subcarriers, and data symbols are modulated using the 16 QAM constellation. In order to generate the complementary cumulative distribution function (𝐶𝐶𝐷𝐹 = Pr [𝑃 𝐴𝑃 𝑅 > 𝑃 𝐴𝑃 𝑅0 ]) of the 𝑃 𝐴𝑃 𝑅, 104 OFDM blocks are generated randomly, where the transmitted signal is oversampled by a factor of 𝐿 = 4. In the PTS optimization, the 256 subcarriers are divided into 𝑀 = 16 sub-blocks, and two allowed phase factors 𝑏𝑚 ∈ {+1, −1} (𝑊 = 2) are used. The SSPA is used with input back-offs 𝐼𝐵𝑂 = 0, 1, . . . , 12 dB and smoothness factor 𝑝 = 0.5, 2 [2]. Fig. 2(a) and 2(b) compare the PAPR reduction performance of the parallel TS-PTS with the conventional PTS, TS-PTS, IDPSO-PTS [4] and ABC-PTS [5]. The PAPR value of the original OFDM signal is measured before the PTS optimization so its search number (𝑆) is 0. The optimum-PTS tests all the phase factors, and requires 𝑊 𝑀 = 216 = 65536 search numbers. Parallel TS-PTS is composed of parallel structures of the TS algorithms and crossover operator. The number of TS algorithms running in parallel is ℎ = 4 and each TS algorithm searches the phase factors until the 𝑚𝑎𝑥𝑖𝑡 as mentioned in the main steps of the TS-PTS algorithm. Parallel TS-PTS uses 𝑚𝑎𝑥𝑖𝑡 = 11. Therefore, the search cost of parallel structures of the TS algorithms is 𝑚𝑎𝑥𝑖𝑡 × ℎ. Crossover is applied for all candidate solutions of the TS algorithms with each( )other. Therefore, the search cost of the crossover is 𝑐 = 42 = 6, where 4 and 2 are equal to ℎ and the number of phase factors to which the crossover operation is applied, respectively. The total search cost of the parallel TSPTS for one cycle is equal to (𝑚𝑎𝑥𝑖𝑡 × ℎ) + 𝑐. The cycles are repeated until they reach the search number 𝑆. In the ABC-PTS, the search number is equal to 𝑆 = 𝑀 𝐶𝑁 ⋅ 𝑆𝑁 , where 𝑀 𝐶𝑁 is the maximum cycle number and 𝑆𝑁 is the size of the population. ABC-PTS uses 𝑀 𝐶𝑁 ⋅ 𝑆𝑁 = 5 ⋅ 10 and 𝑀 𝐶𝑁 ⋅ 𝑆𝑁 = 100 ⋅ 10 for 𝑆 = 50 and 𝑆 = 1000, respectively with limit value 𝐿𝑉 = 10. In the IDPSO-PTS, the search number is equal to 𝑆 = 𝐾0 ⋅ 𝑄, where 𝐾0 is
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IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 9, SEPTEMBER 2011
Fig. 2. PAPR reduction performances of the PTS schemes: (a) parallel TSPTS, ABC-PTS and IDPSO-PTS; (b) parallel TS-PTS, TS-PTS and RS-PTS.
Fig. 3. BER performances of the OFDM system using linear amplifier and SSPA (a): effect of SNR with 𝑝 = 2; (b): effect of 𝐼𝐵𝑂.
TABLE I C OMPUTATIONAL C OMPLEXITY OF THE PTS S CHEMES
seen from the zoom window in Fig. 3(a), parallel TS-PTS has better performance than ABC-PTS for 𝐼𝐵𝑂 = 6 dB. Fig. 3(b) shows effect of the 𝐼𝐵𝑂 on the BER performance using the parallel TS-PTS with 𝑆 = 1000 and the original OFDM signal with 𝑝 = 0.5, 2 on AWGN channel. As shown in Fig. 3(b), 𝐼𝐵𝑂 and 𝑝 must be increased for better BER performance of the OFDM system. As expected, parallel TS-PTS shows better BER performance compared to the original OFDM signal.
Methods
Number of Searches (𝑆)
PAPR [dB]
Original
0
11.26
Optimum-PTS
6.74 6.93
ABC-PTS
𝑊 𝑀 = 216 = 65536 [(𝑚𝑎𝑥𝑖𝑡 × ℎ) + 𝑐]⋅𝑐𝑦𝑐𝑙𝑒 = [(11 × 4) + 6]⋅20 = 1000 𝑀 𝐶𝑁 ⋅ 𝑆𝑁 = 100 ⋅ 10 = 1000
TS-PTS
𝑚𝑎𝑥𝑖𝑡 = 1000
7.06
IDPSO-PTS
𝐾0 ⋅𝑄 = 100 ⋅ 10 = 1000 number of randomly selected phase factors = 1000
7.09
Parallel TS-PTS
RS-PTS
6.98
7.13
the maximum number of iterations and 𝑄 is the number of particles. IDPSO-PTS uses 𝐾0 ⋅𝑄 = 5⋅10 and 𝐾0 ⋅𝑄 = 100⋅10 for 𝑆 = 50 and 𝑆 = 1000, respectively. As shown in Fig. 2(a), parallel TS-PTS shows better PAPR reduction performance than IDPSO-PTS and ABC-PTS. In the random search (RS)PTS [8], the search number is equal to the number of randomly selected phase factors. The search number of the TS-PTS is equal to the number of 𝑚𝑎𝑥𝑖𝑡 = 𝑆. As shown in Fig. 2(b), parallel TS-PTS with 𝑆 = 250 and RS-PTS with 𝑆 = 1000 show the nearly same PAPR performance. Therefore, parallel TS-PTS has about 1 ∖ 4 search complexity of the RS-PTS. Computational complexity and PAPR reduction performances of the PTS schemes at 𝐶𝐶𝐷𝐹 = 10−3 are shown in Table I. Fig. 3(a) shows effect of the SNR on the BER performance of the OFDM system using the ABC-PTS with 𝑆 = 1000, parallel TS-PTS with 𝑆 = 1000, optimum-PTS and original OFDM signal on AWGN channel. BER performance of the system with linear amplifier, which does not cause any signal distortion, is 𝐵𝐸𝑅 = 10−5 when 𝑆𝑁 𝑅 = 13.5 dB. In the case of high values of the IBO, SSPA works near the linear region of the amplification. But in the case of low values of the 𝐼𝐵𝑂, SSPA works near the nonlinear region of the amplification. At 𝐵𝐸𝑅 = 10−5 , parallel TS-PTS needs 𝑆𝑁 𝑅 = 17.54 dB for 𝐼𝐵𝑂 = 6 dB, and this value is only 0.18 dB higher than the optimum-PTS. In addition, as it can be
V. C ONCLUSION In this letter, we propose a parallel TS-PTS scheme to improve the PAPR reduction performance with low search complexity for the OFDM signals. The proposed method is compared with the conventional PTS, IDPSO-PTS, TS-PTS and ABC-PTS. The simulation results show that the proposed parallel TS-PTS method provides good PAPR reduction and bit-error-rate (BER) performances. R EFERENCES [1] T. Jiang and Y. Wu, “An overview: peak-to-average power ratio reduction techniques for OFDM signals,” IEEE Trans. Broadcast., vol. 54, no. 2, pp. 257–268, June 2008. [2] H. G. Ryu, J. Sok Park, and J. S. Park, “Threshold IBO of HPA in the predistorted OFDM communication system,” IEEE Trans. Broadcast., vol. 50, pp. 425–428, Dec. 2004. [3] V. P. G. Jim´enez, Y. Jabrane, A. G. Armada, B. A. E. Said, and A. A. Ouahman, “Reduction of the envelope fluctuations of multicarrier modulations using adaptive neural fuzzy inference systems,” IEEE Trans. Commun., vol. 59, no. 1, pp. 19–25, Jan. 2011. [4] J. Gao, J. Wang, and B. Wang, “Improved particle swarm optimization for PAPR reduction of OFDM systems,” in Proc. Int. Conf. on Networking, Sensing and Control, vol. 1, pp. 621–624, 2010. [5] Y. Wang, W. Chen, and C. Tellambura, “A PAPR reduction method based on artificial bee colony algorithm for OFDM signals,” IEEE Trans. Wireless Commun., vol. 9, no. 10, pp. 2994–2999, Oct. 2010. [6] A. Kalinli and D. Karaboga, “Training recurrent neural networks by using parallel tabu search algorithm based on crossover operation,” Eng. Appl. of Artificial Intelligence, vol. 17, no. 5, pp. 529–542, Aug. 2004. [7] Y. Isik, A. Kalinli, and N. Taspinar, “Multiuser detection in asynchronous CDMA systems based on parallel tabu search algorithm,” Frequenz, vol. 59, no. 7-8, pp. 166–170, July-Aug. 2005. [8] L. J. Cimini, Jr. and N. R. Sollenberger, “Peak-to-average power ratio reduction of an OFDM signal using partial transmit sequences,” IEEE Commun. Lett., vol. 4, pp. 86–88, Mar. 2000.
