- Email: [email protected]
PAPR reduction for pilot-aided OFDM systems with the parametric minimum cross-entropy method
Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371
Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
SHORT COMMUNICATION
PAPR reduction for pilot-aided OFDM systems with the parametric minimum cross-entropy method Wei-Wen Hu ∗ The Department of Electrical Engineering, Southern Taiwan University of Science and Technology, Taiwan
a r t i c l e
i n f o
Article history: Received 12 May 2014 Accepted 29 November 2015 Keywords: Peak-to-average power ratio (PAPR) Pilot sequences Parametric minimum cross-entropy
a b s t r a c t This letter considers the selection of the optimal pilot symbol to decrease the peak-to-average power ratio (PAPR) in pilot-aided orthogonal frequency division multiplexing systems. The conventional pilot-aided PAPR reduction technique necessitates an exhaustive search of all combinations of possible pilot symbol configurations, resulting in high computational complexity. To reduce computational complexity while maintaining PAPR reduction performance, we propose the parametric minimum cross-entropy (PMCE) method for determining near-optimal pilot symbols. Simulation results show that performance of the PMCE-based PAPR reduction is very close to that obtained using the exhaustive search algorithm, with low computational complexity. Compared with the standard cross-entropy method, PMCE has the advantages of better PAPR reduction performance at the same computational complexity. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Orthogonal frequency division multiplexing (OFDM) is extensively used in wireless communications because of its high spectral efficiency and robustness against the interference inherent in multipath fading. However, OFDM systems suffer from a large peak-to-average power ratio (PAPR) of transmitted signals, causing signal distortion at the power amplifier output [1]. To reduce the high PAPR, various methods have been presented in the literature, including companding [2], clipping [3], selected mapping [4–6], and partial transmit sequences [7–9]. The pilot symbols applied in OFDM systems for channel estimation and synchronization has recently been recognized as equally applicable to PAPR reduction. The governing principle of this scheme is the reversal of a subset of subcarriers for pilot tones and the selection of one sequence from a set of candidate pilot symbols; thus, the OFDM signal with the selected pilot symbols yields the lowest PAPR [10–12]. In [10,11], the exhaustive search algorithm (ESA) was adopted to identify the optimum pilot symbol from a set of MNp candidate pilot symbols, where Np and M are the length of the pilot symbol and number of allowed phases in each pilot tone, respectively. The search complexity of ESA scheme then exponentially increases with the length of pilot symbols. The techniques
∗ Tel.: +886 63847422. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.aeue.2015.11.008 1434-8411/© 2015 Elsevier GmbH. All rights reserved.
proposed for reducing complexity accomplish such reduction at the cost of PAPR performance [12]. In this letter, we propose the application of parametric minimum cross-entropy (PMCE) [13] to identify the optimal pilot symbol required to reduce the PAPR of pilot-aided OFDM systems. Simulation results demonstrate that the PMCE-based PAPR reduction exhibits a performance that is very close to that obtained using the ESA scheme; however, the computational complexity of the latter is considerably lower than that of the ESA scheme. In addition, the proposed PAPR reduction outperforms cross-entropy (CE) methods [14] under the same level of complexity. The remainder of this letter is organized as follows. Section 2 describes the signal models for pilot-aided OFDM systems. The proposed PMCE algorithm for the design of optimal pilot symbols is developed in Section 3. Section 2 demonstrates the simulation results. Finally, Section 6 ends with some concluding remarks. 2. System models For an OFDM system with N subcarriers, an inverse discrete Fourier transform (IDFT) is adopted on frequency domain data vector X to obtain time-domain signal vector x, where the nth element of x is expressed as 1 xn = √ LN
LN−1
k=0
Xk ej2kn/LN ,
n = 0, 1, . . ., LN − 1,
(1)
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W.-W. Hu / Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371
where Xk is the kth element of X and L represents the oversampling factor adopted to provide a sufficiently accurate PAPR estimate. The PAPR of the OFDM signal is defined as max0≤n≤LN−1 |xn |2 , E[|xn |2 ]
PAPR(x) =
(2)
where max |xn |2 denotes the maximum peak power, and E[|xn |2 ] denotes the average power. In pilot-aided OFDM systems, frequency domain data vector X can be divided into two components, namely,
Xk =
Dk ,
k ∈ ϒ⊥
Sk ,
k ∈ ϒ,
(3)
where ϒ denotes the set of pilot tone indices, and ϒ ⊥ denotes the complement of ϒ. In addition, Sk and Dk are the kth elements of frequency domain pilot symbol S and data symbols D, respectively. The nth element of the time-domain signal vector x is given by xn = sn + dn ,
(4)
where dn and sn are the nth elements of time-domain data symbols and pilot symbols, respectively. Fig. 1 depicts the idea of pilot-aided PAPR reduction scheme in OFDM systems. The governing principle of this scheme is the reversal of a subset of subcarriers for pilot tones and the selection of one sequence from a set of candidate pilot symbols; thus, the OFDM signal with the selected pilot symbols yields the lowest PAPR. Su denotes the uth frequency-domain pilot symbol in vector form and U is the number of candidate signals. The pilot-aided PAPR reduction scheme primarily involves the identification of the pilot symbol that minimizes the PAPR, i.e., PAPR(˜x)
= min{PAPR(xu )}, Su
= min Su
0≤u≤U−1
max0≤n≤LN−1 |xn,u |2 E[|xn,u |2 ]
(5) ,
where xn,u = sn,u + dn is the uth time-domain candidate signal comprising of the data symbol dn and the uth pilot symbol sn,u . Consequently, the optimal pilot symbol design for minimizing the PAPR of an OFDM signal is related to the combinatorial optimization problem, i.e., minimize PAPR(Q(D + S))
(6)
subject to Sk ∈ {Ak ejk }Np , k ∈ ϒ, k ∈ {0 ∼ 2}
where Q denotes the IDFT matrix, Np is the length of the pilot symbol. In addition, Ak and k is the amplitude and phase of kth pilot signal, respectively. The optimal solution of (5) is to perform ESA over all possible parameter combinations of pilot symbols. Sameamplitude pilot symbols are optimal for channel estimation, but the optimal solution of (5) remains a difficult problem with the search complexity of MNp , where M is the number of allowed phases in each pilot tone; that is, k = 2l/M, l = 0, 1, . . ., M − 1. Thus, ESA possess exponential search complexity with Np . 3. Pilot-aided PAPR reduction using parametric minimum cross-entropy (PMCE) The parametric minimum cross-entropy (PMCE) method which was originally proposed by [13] for solving combinatorial optimization problems is adopted to determine the near-optimum pilot signal. It can be summarized as follows: (1) It generates random data samples according to a specified probability distribution function. (2) It updates the parameters of probability distribution to amend samples in the succeeding iteration. The detailed definition and expressions of the PMCE method and its applications are provided in [13]. In order to employ the PMCE method to find the pilot sequences that minimize the PAPR in pilotaided OFDM systems, we have to define the fitness function for the proposed PMCE scheme. The fitness function can be expressed as F(S) = PAPR(Q(D + S)).
(7)
In this letter, the optimization problem in the proposed PMCE method is the minimization of the fitness function. The selection of the pilot symbols is generally limited to a set with a finite number of candidate pilot symbols. In this letter, phases k = {0, } (i.e., M = 2) and amplitude |Sk | = Ak = 1 are chosen. Thus, the minimization of the fitness function is transformed into the following equations: minimize F(S) = PAPR(Q(D + S))
(8)
subject to Sk ∈ {1, −1}Np , k ∈ ϒ Notably, minimizing the fitness function defined by (8) necessitates 2Np candidate pilot symbols. Inspired by the efficiency of the PMCE method in solving the combinatorial optimization problem, we propose the PMCE method to determine the near-optimal pilot symbol. The procedure of the proposed PMCE-based PAPR reduction is described as follows: Np −1
Step (1) Iteration counter i = 1 is set and probability p0 = {p0m }m=0 is initialized with p0m = 0.5 for all m, where p0m denotes the probability of the mth element of the pilot symbol. Step (2) Each element of S is modeled as an independent Bernoulli random variable with probability mass function (PMF) P(Sk = 1) = pk and P(Sk = 0) = 1 − pk for k = 0, 1, . . ., Np − 1. The PMF is defined as
Np −1
f (S; p) =
pSkk · (1 − pk )1−Sk
(9)
k=0
Using the PMF f(S ; pi−1 ) to generate K random samples i−1 i−1 and then calculating the fitness funcSi−1 1 , S2 , . . ., SK
Fig. 1. Block diagram of pilot-aided PAPR reduction scheme in OFDM systems.
i−1 i−1 tion according to (8) yields F(Si−1 1 ), F(S2 ), . . ., F(SK ).
W.-W. Hu / Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371 i−1 i−1 Step (3) F(Si−1 1 ), F(S2 ), . . ., F(SK ) is sorted in ascending order so
that
F1i−1
i =
≤
K˜ 1
K˜
F2i−1
≤ ··· ≤
FKi−1 .
Define
Fki−1
(10)
k=1
where K˜ = ceil(K), ceil(·) is the ceiling operation and is the rarity parameter defined by [13]. i−1 i−1 Step (4) Samples Si−1 are used to update parameter 1 , S2 , . . ., SK i p according to
K pim =
I
k=1 {Si−1 =1}
K
k,m
k=1
exp(−F(Si−1 )i ) k
exp(−F(Si−1 )i ) k
(11)
k,m
I{Si−1 =1} = k,m
1,
if Si−1 =1 k,m
0,
otherwise.
(12)
In addition, parameter i is computed from the equation
K i =
k=1
F(Si−1 )exp(−F(Si−1 )i ) k k
K
k=1
exp(−F(Si−1 )i ) k
(13)
Step (5) Parameter pi is updated using a smoothing factor ˛ to prevent a fast convergence to a local optimum: pi = ˛ · pi + (1 − ˛) · pi−1
Table 1 Computational complexity of various schemes.
SLM PTS ESA SO CE PMCE
Number of complex multiplications
Number of complex additions
Y · (LN/2) · log2 (LN) WV−1 · (LN/2) · log2 (LN) 2Np · (LN/2) · log2 (LN) Np · (LN/2) · log2 (LN) · (LN/2) · log2 (LN) · (LN/2) · log2 (LN)
Y · (LN) · log2 (LN) WV−1 · (LN) · log2 (LN) 2Np · (LN) · log2 (LN) Np · (LN) · log2 (LN) · (LN) · log2 (LN) · (LN) · log2 (LN)
5. Simulation results
where I{Si−1 =1} is an indicated variable defined by
369
Simulations were performed to verify the PAPR performance of the PMCE-based PAPR reduction scheme in pilot-aided OFDM systems. The OFDM system has N = 128 subcarriers with length Np = 16 of the pilot symbol, and the data symbols were modulated using QPSK modulation. 106 OFDM symbols are randomly generated to obtain the complementary cumulative distribution function of the PAPR defined by CCDF = Prob[PAPR > ], where the transmitted signal is over-sampled by a factor of 4 to obtain accurate PAPR. The pilot symbols are equally spaced in the frequency domain. Although the pilot symbols are used for PAPR reduction, the power allocation factor (defined as the ratio between the power of the pilot symbol and total transmitted power) should be also considered for channel estimation. Therefore, minimizing the mean square errors of data determines the optimal power allocated to pilot symbols [15]: ˇ =1−
(14)
where 0 < ˛ < 1. Step (6) Repeat step 2 to step 6 for i = i + 1 until the predefined number of iteration is reached. The PMCE method is similar to the standard CE approach, except the former uses entire samples whereas CE uses elite samples to update pi . In our scheme, side information must be transmitted to the receiver to indicate the index of the optimum pilot sequence. Side information is usually embedded in the transmitted signal and protected by error control codes so that the probability of erroneous detection can be neglected.
1+
1
(15)
Np /(N − Np )
Fig. 2 compares the PAPR performance of the proposed PMCE scheme with that of the sub-optimal (SO) [12], CE [14], and original OFDM schemes. Note that the complexity of CE-based is identical to that of and PMCE-based PAPR reduction scheme. In this letter, smoothing factor ˛ = 0.8, = 0.1, K = 300, and iteration number i = 15 are used in the PMCE and CE methods. The proposed PMCE scheme exhibits a PAPR reduction performance that is approximately equal to that of the ESA scheme, while reducing required search by about 14 (216 /(15 × 300) 14) times. In addition, the CE scheme performs slightly worse than the PMCE method under the same complexity level. Although the side information is not required for transmission in the SO scheme, the PAPR reduction performance is sub-optimal.
4. Analysis of computational complexity 0
10
Original OFDM SO CE PMCE ESA −1
10 Prob(PAPR>γ)
This section evaluates the computational complexity of the proposed PMCE scheme and different PAPR reduction methods. The computational complexity of PMCE, ESA, CE, PTS, SLM and SO methods depends on the number of candidate signals. Note that in performing the following analysis, the complexity of calculating PAPR process is excluded since the complexity is the same in all schemes. In SO method [12], the number of candidate signals is identical to the number of orthogonal pilot sequences, which is equal to the sequence length Np . In the CE method and proposed PMCE method, the number of candidate signals is express as = K × It, where K is the number of samples in both methods and It is the number of iteration. In ESA scheme, the number of candidate signals is calculated as 2Np . The traditional SLM scheme requires a total of Y IFFT operation to generate Y candidate signals. For traditional PTS scheme, the number of candidate signals is calculated as WV−1 , where W and V are the number of allowed phase weighting factors and disjoint subblocks, respectively. As discussed above, each IDFT operation involves (LN/2) · log2 (LN) complex multiplications and (LN) · log2 (LN) complex additions. Table 1 summarizes the computational complexities of various schemes.
−2
10
−3
10
4
5
6
7
8 γ[dB]
9
10
11
Fig. 2. Comparison of the PAPR reduction performance of various methods.
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W.-W. Hu / Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371 2
1
10
CE PMCE PTS SLM
0.95 0.9 Detection Probability
Average number of searchers
RS CE PMCE
1
10
0.85 0.8 0.75 0.7 0.65
0
10 6.75
7
7.25
7.5
7.75
−10
8
−8
−6
γ[dB]
−4
−2 SNR[dB]
0
2
4
Fig. 5. Detection probability of various methods.
Fig. 3. Comparison of the average numbers of searches for various methods with predefined thresholds. 0
10
Perfect SI CE PMCE PTS SLM
0
10
Original OFDM SO RS PMCE PTS SLM
−1
−1
10
Pr[PAPR>γ]
BER
10
−2
10
−3
10
−2
10
−4
10
0
5
−3
10
4
6
8
10 γ[dB]
12
14
10 SNR(dB)
15
20
16
Fig. 4. Comparison of the PAPR reduction performance of various methods with same number of candidate signals.
Fig. 3 plots the average number of searchers of random search (RS), CE, and PMCE for different thresholds . The stop criterion for all the schemes is when a pilot symbol resulting in a PAPR below threshold is found. The RS scheme involves randomly choosing one pilot symbol out of a set of 216 candidate pilot symbols during PAPR calculation. Compared with the RS scheme, the PMCE and CE methods present lower complexity for all the thresholds. However, Fig. 3 also reveals that the computational complexity of PMCE and CE schemes is almost identical for thresholds from 7.25 dB to 8 dB, whereas the PMCE method has lower complexity for thresholds lower than 7.25 dB. For fairness of comparison of the PAPR reduction performance, Fig. 4 shows the CCDF of the various methods with the identical number of candidate signals. With 64 candidates, the results shown that the PAPR performance of proposed PMCE scheme are marginally poorer than that of the traditional SLM and PTS schemes, which is better than the RS and SO schemes. Fig. 5 illustrates the detection probability of side information for SLM, PTS, PMCE, and CE schemes over multipath channel. To avoid inter-symbol interference, a cyclic prefix of length N/4 is employed. The channel comprises of NL = 5 statistically
Fig. 6. Comparison of BER performance of various methods.
independent taps, each being a zero-mean complex-valued Gaussian random variable with an exponential power decay profile, i.e.,
h2 = E[|hl |2 ] = exp(−l/10), 0 ≤ l ≤ NL − 1, where is chosen l
such that
NL −1 l=0
h2 = 1. From Fig. 5, we can observe that the prol
posed PMCE method can achieve the same detection probability as the traditional SLM and PTS scheme. In addition, BER performance as shown in Fig. 6 is consistent with the results provided in Fig. 5. 6. Conclusion We have proposed a PMCE-based scheme to determine the near-optimal pilot symbol for PAPR reduction in pilot-aided OFDM systems. The simulation results demonstrate that the PMCE-based scheme not only obtains better PAPR reduction performance but also obtains computational complexity advantages compared with the existing methods. References [1] Han SH, Lee JH. An overview of peak-to-average power ratio reduction techniques for multicarrier transmission. IEEE Wirel Commun 2005;12(2):56–65. [2] Huang X, Lu J, Zheng J, Letaief KB, Gu J. Companding transform for reduction in peak-to-average power ratio of OFDM signals. IEEE Trans Wirel Commun 2004;3(6):2030–9.
W.-W. Hu / Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371 [3] Chen Y, Zhang J, Jayalath ADS. Estimation and compensation of clipping noise in OFDMA systems. IEEE Trans Wirel Commun 2010;9(2):523–7. [4] Breiling M, Muller-Weinfurtner SH, Huber JB. SLM peak-power reduction without explicit side information. IEEE Commun Lett 2001;5(6): 239–41. [5] Yang L, Siu YM, Soo KK, Leung SW, Li SQ. Low-complexity PAPR reduction technique for OFDM systems using modified widely linear SLM scheme. Int J Electron Commun (AEU) 2012;66(12):1006–10. [6] Le Goff SY, Al-Samahi SS, Khoo BK, Tsimenidis CC, Sharif BS. Selected mapping without side information for PAPR reduction in OFDM. IEEE Trans Wirel Commun 2009;8(7):3320–5. [7] Yang L, Chen RS, Soo KK, Siu YM. An efficient sphere decoding approach for PTS assisted PAPR reduction of OFDM signals. Int J Electron Commun (AEU) 2007;61(10):684–8. [8] Guan L, Jiang T, Qu D, Zhou Y. Joint channel estimation and PTS to reduce peakto-average-power ratio in OFDM systems without side information. IEEE Signal Proc Lett 2010;17(10):883–6.
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[9] Jayalath ADS, Tellambura C. SLM and PTS peak-power reduction of OFDM signals without side information. IEEE Trans Wirel Commun 2005;4(5):2006–13. [10] Hosokawa S, Ohno S, Teo KAD, Hinamoto T. Pilot tone design for peak-toaverage power ratio reduction in OFDM. In: IEEE ISCAS 05, vol. 6. 2006. p. 6014–7. [11] Venkatasubramanian P, Ilow J. Opportunistic configurations of pilot tones for PAPR reduction in OFDM systems. In: IEEE PIMRC 07, vol. 36. 2007. p. 1–5. [12] Fernandez-Getino Garcia MJ, Edfors O, Paez-Borrallo JM. Peak power reduction for OFDM systems with orthogonal pilot sequences. IEEE Trans Wirel Commun 2006;5(1):47–51. [13] Rubinstein RY, Dolgin A. Fast parametric entropy-based and the parametric MinxEnt method for counting of #P-complete problems; 2007. [14] Rubinstein RY, Kroese DP. The cross-entropy method. Berlin, Germany: Springer; 2004. [15] Ohno S, Giannakis GB. Optimal training and redundant precoding for block transmissions with application to wireless OFDM. IEEE Trans Commun 2002;50(12):2113–23.
Contents lists available at ScienceDirect
International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue
SHORT COMMUNICATION
PAPR reduction for pilot-aided OFDM systems with the parametric minimum cross-entropy method Wei-Wen Hu ∗ The Department of Electrical Engineering, Southern Taiwan University of Science and Technology, Taiwan
a r t i c l e
i n f o
Article history: Received 12 May 2014 Accepted 29 November 2015 Keywords: Peak-to-average power ratio (PAPR) Pilot sequences Parametric minimum cross-entropy
a b s t r a c t This letter considers the selection of the optimal pilot symbol to decrease the peak-to-average power ratio (PAPR) in pilot-aided orthogonal frequency division multiplexing systems. The conventional pilot-aided PAPR reduction technique necessitates an exhaustive search of all combinations of possible pilot symbol configurations, resulting in high computational complexity. To reduce computational complexity while maintaining PAPR reduction performance, we propose the parametric minimum cross-entropy (PMCE) method for determining near-optimal pilot symbols. Simulation results show that performance of the PMCE-based PAPR reduction is very close to that obtained using the exhaustive search algorithm, with low computational complexity. Compared with the standard cross-entropy method, PMCE has the advantages of better PAPR reduction performance at the same computational complexity. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Orthogonal frequency division multiplexing (OFDM) is extensively used in wireless communications because of its high spectral efficiency and robustness against the interference inherent in multipath fading. However, OFDM systems suffer from a large peak-to-average power ratio (PAPR) of transmitted signals, causing signal distortion at the power amplifier output [1]. To reduce the high PAPR, various methods have been presented in the literature, including companding [2], clipping [3], selected mapping [4–6], and partial transmit sequences [7–9]. The pilot symbols applied in OFDM systems for channel estimation and synchronization has recently been recognized as equally applicable to PAPR reduction. The governing principle of this scheme is the reversal of a subset of subcarriers for pilot tones and the selection of one sequence from a set of candidate pilot symbols; thus, the OFDM signal with the selected pilot symbols yields the lowest PAPR [10–12]. In [10,11], the exhaustive search algorithm (ESA) was adopted to identify the optimum pilot symbol from a set of MNp candidate pilot symbols, where Np and M are the length of the pilot symbol and number of allowed phases in each pilot tone, respectively. The search complexity of ESA scheme then exponentially increases with the length of pilot symbols. The techniques
∗ Tel.: +886 63847422. E-mail addresses: [email protected], [email protected] http://dx.doi.org/10.1016/j.aeue.2015.11.008 1434-8411/© 2015 Elsevier GmbH. All rights reserved.
proposed for reducing complexity accomplish such reduction at the cost of PAPR performance [12]. In this letter, we propose the application of parametric minimum cross-entropy (PMCE) [13] to identify the optimal pilot symbol required to reduce the PAPR of pilot-aided OFDM systems. Simulation results demonstrate that the PMCE-based PAPR reduction exhibits a performance that is very close to that obtained using the ESA scheme; however, the computational complexity of the latter is considerably lower than that of the ESA scheme. In addition, the proposed PAPR reduction outperforms cross-entropy (CE) methods [14] under the same level of complexity. The remainder of this letter is organized as follows. Section 2 describes the signal models for pilot-aided OFDM systems. The proposed PMCE algorithm for the design of optimal pilot symbols is developed in Section 3. Section 2 demonstrates the simulation results. Finally, Section 6 ends with some concluding remarks. 2. System models For an OFDM system with N subcarriers, an inverse discrete Fourier transform (IDFT) is adopted on frequency domain data vector X to obtain time-domain signal vector x, where the nth element of x is expressed as 1 xn = √ LN
LN−1
k=0
Xk ej2kn/LN ,
n = 0, 1, . . ., LN − 1,
(1)
368
W.-W. Hu / Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371
where Xk is the kth element of X and L represents the oversampling factor adopted to provide a sufficiently accurate PAPR estimate. The PAPR of the OFDM signal is defined as max0≤n≤LN−1 |xn |2 , E[|xn |2 ]
PAPR(x) =
(2)
where max |xn |2 denotes the maximum peak power, and E[|xn |2 ] denotes the average power. In pilot-aided OFDM systems, frequency domain data vector X can be divided into two components, namely,
Xk =
Dk ,
k ∈ ϒ⊥
Sk ,
k ∈ ϒ,
(3)
where ϒ denotes the set of pilot tone indices, and ϒ ⊥ denotes the complement of ϒ. In addition, Sk and Dk are the kth elements of frequency domain pilot symbol S and data symbols D, respectively. The nth element of the time-domain signal vector x is given by xn = sn + dn ,
(4)
where dn and sn are the nth elements of time-domain data symbols and pilot symbols, respectively. Fig. 1 depicts the idea of pilot-aided PAPR reduction scheme in OFDM systems. The governing principle of this scheme is the reversal of a subset of subcarriers for pilot tones and the selection of one sequence from a set of candidate pilot symbols; thus, the OFDM signal with the selected pilot symbols yields the lowest PAPR. Su denotes the uth frequency-domain pilot symbol in vector form and U is the number of candidate signals. The pilot-aided PAPR reduction scheme primarily involves the identification of the pilot symbol that minimizes the PAPR, i.e., PAPR(˜x)
= min{PAPR(xu )}, Su
= min Su
0≤u≤U−1
max0≤n≤LN−1 |xn,u |2 E[|xn,u |2 ]
(5) ,
where xn,u = sn,u + dn is the uth time-domain candidate signal comprising of the data symbol dn and the uth pilot symbol sn,u . Consequently, the optimal pilot symbol design for minimizing the PAPR of an OFDM signal is related to the combinatorial optimization problem, i.e., minimize PAPR(Q(D + S))
(6)
subject to Sk ∈ {Ak ejk }Np , k ∈ ϒ, k ∈ {0 ∼ 2}
where Q denotes the IDFT matrix, Np is the length of the pilot symbol. In addition, Ak and k is the amplitude and phase of kth pilot signal, respectively. The optimal solution of (5) is to perform ESA over all possible parameter combinations of pilot symbols. Sameamplitude pilot symbols are optimal for channel estimation, but the optimal solution of (5) remains a difficult problem with the search complexity of MNp , where M is the number of allowed phases in each pilot tone; that is, k = 2l/M, l = 0, 1, . . ., M − 1. Thus, ESA possess exponential search complexity with Np . 3. Pilot-aided PAPR reduction using parametric minimum cross-entropy (PMCE) The parametric minimum cross-entropy (PMCE) method which was originally proposed by [13] for solving combinatorial optimization problems is adopted to determine the near-optimum pilot signal. It can be summarized as follows: (1) It generates random data samples according to a specified probability distribution function. (2) It updates the parameters of probability distribution to amend samples in the succeeding iteration. The detailed definition and expressions of the PMCE method and its applications are provided in [13]. In order to employ the PMCE method to find the pilot sequences that minimize the PAPR in pilotaided OFDM systems, we have to define the fitness function for the proposed PMCE scheme. The fitness function can be expressed as F(S) = PAPR(Q(D + S)).
(7)
In this letter, the optimization problem in the proposed PMCE method is the minimization of the fitness function. The selection of the pilot symbols is generally limited to a set with a finite number of candidate pilot symbols. In this letter, phases k = {0, } (i.e., M = 2) and amplitude |Sk | = Ak = 1 are chosen. Thus, the minimization of the fitness function is transformed into the following equations: minimize F(S) = PAPR(Q(D + S))
(8)
subject to Sk ∈ {1, −1}Np , k ∈ ϒ Notably, minimizing the fitness function defined by (8) necessitates 2Np candidate pilot symbols. Inspired by the efficiency of the PMCE method in solving the combinatorial optimization problem, we propose the PMCE method to determine the near-optimal pilot symbol. The procedure of the proposed PMCE-based PAPR reduction is described as follows: Np −1
Step (1) Iteration counter i = 1 is set and probability p0 = {p0m }m=0 is initialized with p0m = 0.5 for all m, where p0m denotes the probability of the mth element of the pilot symbol. Step (2) Each element of S is modeled as an independent Bernoulli random variable with probability mass function (PMF) P(Sk = 1) = pk and P(Sk = 0) = 1 − pk for k = 0, 1, . . ., Np − 1. The PMF is defined as
Np −1
f (S; p) =
pSkk · (1 − pk )1−Sk
(9)
k=0
Using the PMF f(S ; pi−1 ) to generate K random samples i−1 i−1 and then calculating the fitness funcSi−1 1 , S2 , . . ., SK
Fig. 1. Block diagram of pilot-aided PAPR reduction scheme in OFDM systems.
i−1 i−1 tion according to (8) yields F(Si−1 1 ), F(S2 ), . . ., F(SK ).
W.-W. Hu / Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371 i−1 i−1 Step (3) F(Si−1 1 ), F(S2 ), . . ., F(SK ) is sorted in ascending order so
that
F1i−1
i =
≤
K˜ 1
K˜
F2i−1
≤ ··· ≤
FKi−1 .
Define
Fki−1
(10)
k=1
where K˜ = ceil(K), ceil(·) is the ceiling operation and is the rarity parameter defined by [13]. i−1 i−1 Step (4) Samples Si−1 are used to update parameter 1 , S2 , . . ., SK i p according to
K pim =
I
k=1 {Si−1 =1}
K
k,m
k=1
exp(−F(Si−1 )i ) k
exp(−F(Si−1 )i ) k
(11)
k,m
I{Si−1 =1} = k,m
1,
if Si−1 =1 k,m
0,
otherwise.
(12)
In addition, parameter i is computed from the equation
K i =
k=1
F(Si−1 )exp(−F(Si−1 )i ) k k
K
k=1
exp(−F(Si−1 )i ) k
(13)
Step (5) Parameter pi is updated using a smoothing factor ˛ to prevent a fast convergence to a local optimum: pi = ˛ · pi + (1 − ˛) · pi−1
Table 1 Computational complexity of various schemes.
SLM PTS ESA SO CE PMCE
Number of complex multiplications
Number of complex additions
Y · (LN/2) · log2 (LN) WV−1 · (LN/2) · log2 (LN) 2Np · (LN/2) · log2 (LN) Np · (LN/2) · log2 (LN) · (LN/2) · log2 (LN) · (LN/2) · log2 (LN)
Y · (LN) · log2 (LN) WV−1 · (LN) · log2 (LN) 2Np · (LN) · log2 (LN) Np · (LN) · log2 (LN) · (LN) · log2 (LN) · (LN) · log2 (LN)
5. Simulation results
where I{Si−1 =1} is an indicated variable defined by
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Simulations were performed to verify the PAPR performance of the PMCE-based PAPR reduction scheme in pilot-aided OFDM systems. The OFDM system has N = 128 subcarriers with length Np = 16 of the pilot symbol, and the data symbols were modulated using QPSK modulation. 106 OFDM symbols are randomly generated to obtain the complementary cumulative distribution function of the PAPR defined by CCDF = Prob[PAPR > ], where the transmitted signal is over-sampled by a factor of 4 to obtain accurate PAPR. The pilot symbols are equally spaced in the frequency domain. Although the pilot symbols are used for PAPR reduction, the power allocation factor (defined as the ratio between the power of the pilot symbol and total transmitted power) should be also considered for channel estimation. Therefore, minimizing the mean square errors of data determines the optimal power allocated to pilot symbols [15]: ˇ =1−
(14)
where 0 < ˛ < 1. Step (6) Repeat step 2 to step 6 for i = i + 1 until the predefined number of iteration is reached. The PMCE method is similar to the standard CE approach, except the former uses entire samples whereas CE uses elite samples to update pi . In our scheme, side information must be transmitted to the receiver to indicate the index of the optimum pilot sequence. Side information is usually embedded in the transmitted signal and protected by error control codes so that the probability of erroneous detection can be neglected.
1+
1
(15)
Np /(N − Np )
Fig. 2 compares the PAPR performance of the proposed PMCE scheme with that of the sub-optimal (SO) [12], CE [14], and original OFDM schemes. Note that the complexity of CE-based is identical to that of and PMCE-based PAPR reduction scheme. In this letter, smoothing factor ˛ = 0.8, = 0.1, K = 300, and iteration number i = 15 are used in the PMCE and CE methods. The proposed PMCE scheme exhibits a PAPR reduction performance that is approximately equal to that of the ESA scheme, while reducing required search by about 14 (216 /(15 × 300) 14) times. In addition, the CE scheme performs slightly worse than the PMCE method under the same complexity level. Although the side information is not required for transmission in the SO scheme, the PAPR reduction performance is sub-optimal.
4. Analysis of computational complexity 0
10
Original OFDM SO CE PMCE ESA −1
10 Prob(PAPR>γ)
This section evaluates the computational complexity of the proposed PMCE scheme and different PAPR reduction methods. The computational complexity of PMCE, ESA, CE, PTS, SLM and SO methods depends on the number of candidate signals. Note that in performing the following analysis, the complexity of calculating PAPR process is excluded since the complexity is the same in all schemes. In SO method [12], the number of candidate signals is identical to the number of orthogonal pilot sequences, which is equal to the sequence length Np . In the CE method and proposed PMCE method, the number of candidate signals is express as = K × It, where K is the number of samples in both methods and It is the number of iteration. In ESA scheme, the number of candidate signals is calculated as 2Np . The traditional SLM scheme requires a total of Y IFFT operation to generate Y candidate signals. For traditional PTS scheme, the number of candidate signals is calculated as WV−1 , where W and V are the number of allowed phase weighting factors and disjoint subblocks, respectively. As discussed above, each IDFT operation involves (LN/2) · log2 (LN) complex multiplications and (LN) · log2 (LN) complex additions. Table 1 summarizes the computational complexities of various schemes.
−2
10
−3
10
4
5
6
7
8 γ[dB]
9
10
11
Fig. 2. Comparison of the PAPR reduction performance of various methods.
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W.-W. Hu / Int. J. Electron. Commun. (AEÜ) 70 (2016) 367–371 2
1
10
CE PMCE PTS SLM
0.95 0.9 Detection Probability
Average number of searchers
RS CE PMCE
1
10
0.85 0.8 0.75 0.7 0.65
0
10 6.75
7
7.25
7.5
7.75
−10
8
−8
−6
γ[dB]
−4
−2 SNR[dB]
0
2
4
Fig. 5. Detection probability of various methods.
Fig. 3. Comparison of the average numbers of searches for various methods with predefined thresholds. 0
10
Perfect SI CE PMCE PTS SLM
0
10
Original OFDM SO RS PMCE PTS SLM
−1
−1
10
Pr[PAPR>γ]
BER
10
−2
10
−3
10
−2
10
−4
10
0
5
−3
10
4
6
8
10 γ[dB]
12
14
10 SNR(dB)
15
20
16
Fig. 4. Comparison of the PAPR reduction performance of various methods with same number of candidate signals.
Fig. 3 plots the average number of searchers of random search (RS), CE, and PMCE for different thresholds . The stop criterion for all the schemes is when a pilot symbol resulting in a PAPR below threshold is found. The RS scheme involves randomly choosing one pilot symbol out of a set of 216 candidate pilot symbols during PAPR calculation. Compared with the RS scheme, the PMCE and CE methods present lower complexity for all the thresholds. However, Fig. 3 also reveals that the computational complexity of PMCE and CE schemes is almost identical for thresholds from 7.25 dB to 8 dB, whereas the PMCE method has lower complexity for thresholds lower than 7.25 dB. For fairness of comparison of the PAPR reduction performance, Fig. 4 shows the CCDF of the various methods with the identical number of candidate signals. With 64 candidates, the results shown that the PAPR performance of proposed PMCE scheme are marginally poorer than that of the traditional SLM and PTS schemes, which is better than the RS and SO schemes. Fig. 5 illustrates the detection probability of side information for SLM, PTS, PMCE, and CE schemes over multipath channel. To avoid inter-symbol interference, a cyclic prefix of length N/4 is employed. The channel comprises of NL = 5 statistically
Fig. 6. Comparison of BER performance of various methods.
independent taps, each being a zero-mean complex-valued Gaussian random variable with an exponential power decay profile, i.e.,
h2 = E[|hl |2 ] = exp(−l/10), 0 ≤ l ≤ NL − 1, where is chosen l
such that
NL −1 l=0
h2 = 1. From Fig. 5, we can observe that the prol
posed PMCE method can achieve the same detection probability as the traditional SLM and PTS scheme. In addition, BER performance as shown in Fig. 6 is consistent with the results provided in Fig. 5. 6. Conclusion We have proposed a PMCE-based scheme to determine the near-optimal pilot symbol for PAPR reduction in pilot-aided OFDM systems. The simulation results demonstrate that the PMCE-based scheme not only obtains better PAPR reduction performance but also obtains computational complexity advantages compared with the existing methods. References [1] Han SH, Lee JH. An overview of peak-to-average power ratio reduction techniques for multicarrier transmission. IEEE Wirel Commun 2005;12(2):56–65. [2] Huang X, Lu J, Zheng J, Letaief KB, Gu J. Companding transform for reduction in peak-to-average power ratio of OFDM signals. IEEE Trans Wirel Commun 2004;3(6):2030–9.
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