IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 5, MAY 2014
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Curve Fitting Based Tone Reservation Method with Low Complexity for PAPR Reduction in OFDM Systems Tao Jiang, Chunxing Ni, Chang Xu, and Qi Qi
Abstract—In this letter, a novel curve fitting based tone reservation (CF-TR) method is proposed to reduce the peakto-average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems, and its key idea is to utilize the clipping noise introduced by clipping to generate an ideal peakcanceling signal. To effectively eliminate the peaks of the OFDM signal, the CF-TR method generates the peak-canceling signal by directly fitting the waveform of the peak-canceling signal to the waveform of clipping noise. The proposed CF-TR method obtains significant PAPR reduction with very few iterations, and only one IFFT operation is required at each iteration, resulting in dramatic reduction of the computational complexity. Index Terms—Orthogonal frequency division multiplexing, peak-to-average power ratio, tone reservation, curve fitting.
I. I NTRODUCTION
A
S an attractive technology for wireless communications, orthogonal frequency division multiplexing (OFDM) has been widely used in many standards. However, one major drawback of OFDM systems is the high peak-to-average power ratio (PAPR) of the transmitted signal. Recently, various methods have been proposed to reduce the PAPR for OFDM systems in literature, such as subcarrier shifting [1], selected mapping (SLM) [2], [3] and tone reservation (TR) [4], [5]. The TR technique is attractive, and it reserves a small subset of subcarriers, called as peak reduction tones (PRTs), to generate a peak-canceling signal to minimize the PAPR. As the simplest TR scheme, the clipping control TR (CC-TR) [6] method utilizes the filtered clipping noise as the peak-canceling signal to reduce the PAPR. However, the CCTR method suffers from slow convergence rate. The adaptive amplitude clipping TR (AAC-TR) method modifies the CCTR method by adaptively controlling both the convergence rate and the clipping threshold [7]. However, AAC-TR cannot control the final PAPR, since the clipping threshold increases when the number of the iteration increases. The least squares approximation based TR (LSA-TR) [8] method converges fast, but its PAPR reduction performance is not good. Moreover, these methods need a pair of inverse fast Fourier transform/fast
Manuscript received September 26, 2013. The associate editor coordinating the review of this letter and approving it for publication was A. Ghassemi. This work was supported in part by the National Science Foundation of China with Grants 61172052 and 60872008, and the Open Research Fund of National Mobile Communications Research Laboratory in Southeast University with Grant number 2012D08. T. Jiang, C. Ni, and C. Xu are with the Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China (e-mail:
[email protected];
[email protected];
[email protected]). Q. Qi is with LSI Corporation, 1320 Ridder Park Dr., San Jose, CA 95131 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/LCOMM.2014.032014.132174
Fourier transform (IFFT/FFT) operations during each iteration, resulting in high complexity. In this letter, we propose a novel curve fitting based tone reservation (CF-TR) method to reduce the PAPR of OFDM signals, and its key idea is to iteratively generate the peakcanceling signal by fitting the peak-canceling signal waveform to the clipping noise waveform. Moreover, we derive the optimal clipping threshold for the proposed CF-TR method. Therefore, significant PAPR reduction could be achieved with very few iterations due to the fully use of the clipping noise and the optimal predefined clipping threshold, and only one IFFT operation is required during each iteration, resulting in significant reduction of the complexity. Compared with the CC-TR, AAC-TR and LSA-TR methods, the proposed CF-TR method achieves better PAPR reduction with lower complexity. The rest of this letter is organized as follows. In Section II, a typical OFDM system is given and the problem of generating peak-canceling signal is formulated. Then, we propose the CFTR method in Section III. Simulation results are presented in Section IV, followed by conclusions in Section V. II. P ROBLEM F ORMULATION For OFDM systems, an input data symbol vector X = [X0 , X1 , · · · , XN −1 ]T in the frequency domain is modulated by N orthogonal subcarriers to generate a discrete time OFDM signal x = [x(0), x(1), · · · , x(N − 1)]T , where x can be obtained by taking an N -point inverse fast Fourier transform (IFFT) of X, i.e., N −1 j2πkn 1 Xk e N , n = 0, 1, · · · , N − 1. x(n) = √ N k=0
(1)
Generally, the PAPR of an OFDM signal x is defined as max
PAPR =
[|x(n)|2 ]
0≤n≤N −1
E[|x(n)|2 ]
,
(2)
where E[·] represents the expectation. For the TR technique, M PRTs are utilized to generate peak-canceling signal c = [c0 , c1 , · · · , cN −1 ]T . Note that, these M PRTs do not carry any data information, and they are only used for reducing the PAPR. Then, c is added to x to reduce the PAPR, and the peak-reduced signal x = [ x(0), x (1), · · · , x (N − 1)]T can be expressed as N −1 j2πkn 1 x (n) = x(n) + c(n) = √ (Xk + Ck )e N , N k=0
c 2014 IEEE 1089-7798/14$31.00
(3)
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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 5, MAY 2014
where C = [C0 , C1 , · · · , CN −1 ]T is the peak-canceling signal vector in frequency domain. For distortionless data transmission, the data vector X and the peak reduction vector C are restricted to lie in disjoint frequency tones, i.e., Xk , if k ∈ Rc , (4) Xk + Ck = Ck , if k ∈ R where R = {k0 , k1 , · · · , kM−1 } represents the positions of the PRTs, and Rc is the complement set of R in N = {0, 1, · · · , N − 1}. In addition, Ck = 0 only if k ∈ R. For PAPR reduction, c is designed to minimize the maximum of x as x∞ = arg min x + c∞ , copt = arg min c
c
(5)
where · ∞ denotes the ∞-norm of a vector. Obviously, (5) can be reformulated as a quadratically constrained quadratic program (QCQP) problem [4], and the complexity of obtaining its optimal solution is extremely high.
j2πkm n ˆ = and the vector C where rm (n) = √1N e N T [Ck0 , Ck1 , · · · , CkM −1 ] corresponds to the subset of C. Although the optimization problems (8) and (9) are not equivalent, and (9) can be seen as a simplified and approximate form of (8). Then, we denote ⎤ ⎡ ⎤ ⎡ ··· rM−1 (n0 ) f (n0 ) r0 (n0 ) ⎥ˆ ⎢ ⎥ ⎢ .. .. .. .. R=⎣ ⎦ ,f = ⎣ ⎦. . . . .
r0 (nL−1 )
···
rM−1 (nL−1 )
Substituting (10) to (9), we have
2
ˆ − ˆf ˆ = min J(C)
.
RC 2
A. CF-TR method For the CF-TR method, the peak-canceling signal generated by the reserved tones is added to the original time domain signal to reduce the PAPR of the OFDM signal. Moreover, clipping is introduced to obtain the peak-canceling signal. Given the clipping threshold A, the clipped OFDM signal x = [ x(0), x (1), · · · , x (N − 1)]T can be obtained by x(n), if |x(n)| < A , (6) x (n) = Aejθn , if |x(n)| ≥ A where x(n) = |x(n)| ejθn and θn is the phase of x(n). Then, the clipping noise f = [f (0), f (1), · · · , f (N − 1)]T is defined as f (n) = x (n) − x(n). (7) Substituting (7) into (5), we have x∞ = arg min x + (c − f)∞ . copt = arg min c
c
(8)
According to (8), it is obvious that the PAPR is minimized when the peak-canceling signal c equals to the clipping noise f. However, the peak-canceling signal c hardly equals to f, because Ck = 0 only if k ∈ R. Therefore, we propose the CF-TR method to efficiently reduce the PAPR by fitting the waveform of the peak-canceling signal c to the waveform of clipping noise f. Define the positions of non-zero values of f (n) as the time index set S = {n0 , n1 , · · · , nL−1 }, where L is the number of non-zero values. Thus, for n ∈ S, f (n) contains all the information about the peaks of the OFDM signal. The problem of fitting c(n) to f (n) is equivalent to choosing the parameters Ckm to minimize the Euclidean distance between the points c(nl ) and f (nl ) for l = 0, 1, · · · , L − 1, which can be expressed as ˆ = min J(C)
L−1 l=0
= min
|c(nl ) − f (nl )|
L−1 M−1 l=0
m=0
2
2 Ckm rm (nl ) − f (nl ) ,
(9)
(11)
ˆ denoted as C ˆ opt , could Therefore, the optimal value of C, be found by solving the following problem
2
ˆ − ˆf ˆ opt = arg min C (12)
.
RC ˆ C
III. C URVE F ITTING BASED T ONE R ESERVATION M ETHOD
f (nL−1 ) (10)
2
1) When M = L, (12) has a unique solution, i.e., ˆ opt = R−1ˆf. C
(13)
2) When M = L, (12) has the solution as ˆ opt = R+ˆf, C
(14)
+
where R is the Moore-Penrose generalized inverse of R [9]. Therefore, the CF-TR method can be summarized as follows Algorithm 1 : The CF-TR method 1: Set the maximal iteration number K and the reserved tone set R. Then, set the initial clipping level A. 2: Set i = 0, the time-domain signal of the 0th iteration x0 = x, where x is the original time-domain signal obtained by (1), and x0 = [x0 (0), x0 (1), · · · , x0 (N − 1)]. i 3: Calculate the clipping noise f using (6) and (7). If no i clipping noise, choose x as the transmitted signal and terminate the program. ˆ opt using (12), then, calculate the peak4: Calculate C i ˆ opt }. by copt = IF F T {C canceling signal copt i i i opt 5: Update the signal as x(i+1) = xi + ci . 6: Set i = i + 1, if i < K, go to Step 3; Otherwise, choose x(i+1) as the transmitted signal and terminate the program. 7: End
B. Optimal Clipping Threshold Obviously, when the clipping threshold A is too large, the peak-canceling signal c is close to 0, leading to very little PAPR reduction during each iteration. Moreover, when A is too small, each iteration can also contribute to very little PAPR reduction since the peak-canceling signal c does not approximate to the clipping noise f. Thus, the clipping threshold A directly affects the PAPR reduction of the CFTR method. Moreover, it is very important to obtain the optimal clipping threshold A to improve the PAPR reduction performance and to reduce the computational complexity.
JIANG et al.: CURVE FITTING BASED TONE RESERVATION METHOD WITH LOW COMPLEXITY FOR PAPR REDUCTION IN OFDM SYSTEMS
Note that, the value of L varies with different input OFDM signals, and the value of L is different during each iteration. Therefore, we discuss the average value of L instead. Denote ¯ as the average value of L, which is calculated as [10] L
0
(15)
where σ 2 is the mean power of the OFDM signal. As discussed in Section III-A, according to (11), (13), ˆ = 0 and the peak-canceling signal c equals and (14), J(C) to the clipping noise f, when M ≥ L. However, when ˆ > 0 and the peak-canceling signal c does not M < L, J(C) approximate to the clipping noise f. Therefore, the clipping threshold should be chosen as M ≥ L. ¯ into (15), we have Substituting M ≥ L N A ≥ σ ln . (16) M
Original CF−TR A=4.5 CF−TR A=5.0 CF−TR A=5.3 CF−TR A=6.0 CF−TR A=6.5
−1
10
0
Complementary Cumulative Density Function Pro{PAPR>PAPR }
10
−2
10
−3
10
−4
10
7
8
9
10
11
12
PAPR (dB) 0
Fig. 1.
PAPR reductions when 16-QAM is employed with different A.
0
10
Complementary Cumulative Density Function Pro{PAPR>PAPR0}
¯ = N e−A2 /σ2 , L
807
According to (16), A is closely related with the mean power of the OFDM signal σ and the percentage of the PRTs M N. However, when the clipping threshold A increases, less signal would be clipped. It means that with the increasing of A, the peak-canceling signal c will approximate to 0, resulting in the PAPR reduction degradation. Therefore, the optimal clipping threshold should be N (17) A = σ ln . M
AAC−TR CC−TR CF−TR LSA−TR Original iter=1 iter=2 iter=3
−1
10
−2
10
−3
10
−4
10
5
6
7
8
9
10
11
12
13
PAPR (dB) 0
C. Computational Complexity For the CF-TR method, the clipping noise is calculated with (6) and (7), which needs 5L real multiplications and 2L real additions [10]. The frequency domain peak-canceling ˆ opt is obtained with (13) or (14), and the complexity signal C of (14) is the highest. In addition, converting C to the time domain requires a N -point IFFT operation with 2N log2 N real multiplications and 3N log2 N real additions. As a result, the total number of the real multiplications M U LCF and real additions ADD CF can be expressed as 4(M 3 + M 2 ) +5L, 3 (18) ADD CF =3N log2 N + (8L − 2)M 2 + (4L − 1)M (19) 4(M 3 + M 2 ) + 2L. + 3 For comparison, we also provide the complexity analysis for the CC-TR and AAC-TR methods. Apart from calculating the clipping noise, the CC-TR method needs a pair of N point IFFT/FFT operations during each iteration, which needs 4N log2 N real multiplications and 6N log2 N real additions. Consequently, the total computational complexity of the CCTR method is M U LCF = 2N log2 N +8LM 2 +4LM +
M U LCC = 4N log2 N + 5L,
(20)
ADDCC = 6N log2 N + 2L.
(21)
During each iteration, the AAC-TR method updates the clipping level and adaptively scales the filtered clipping noise as the peak-canceling signal. For each update, it requires
Fig. 2.
PAPR reductions with different methods for 16-QAM.
6L + 2N + 2 real multiplications and 5L + 2N real additions [10]. Therefore, the total complexity of the AAC-TR method complexity is M U LAAC = 4N log2 N + 11L + 2N + 2,
(22)
ADD AAC = 6N log2 N + 7L + 2N.
(23)
Based on the above analysis, the CF-TR method requires one IFFT operation while both the CC-TR and AAC-TR methods require a pair of IFFT/FFT operations during each iteration. As mentioned before, L approximately equals to M , and M N . Thus, the complexity is largely depended on the IFFT/FFT part. Therefore, the CF-TR method has much lower complexity than the CC-TR and AAC-TR methods. IV. S IMULATION R ESULTS In this section, 105 OFDM data blocks are randomly generated with N = 1024 and M = 64, where the PRTs set R is randomly generated. 16-QAM modulation with constellation points {±1 ± j, ±3 ± j, ±1 ± 3j, ±3 ± 3j} are employed, and complementary cumulative distribution function (CCDF) is employed to show the statistical properties of PAPR. We compare the CF-TR method with some promising TR-based methods, including the CC-TR, AAC-TR and LSA-TR methods. The loss in data rate is M/N = 6.25% for these methods. Fig. 1 shows the PAPR reduction of the CF-TR method for different clipping threshold A, with one iteration. When
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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 5, MAY 2014
7.5 CF−TR AAC−TR CC−TR LSA−TR
7 6.5
Average PAPR (dB)
6 5.5 5 4.5 4 3.5 3 2.5
1
2
3
4
5
6
7
8
9
10
Iteration Numbers
Fig. 3. Relationship of PAPR reduction with iteration numbers for 16-QAM.
0
10
16QAM
−1
10
Bit error rate
Ideal Original CC−TR AAC−TR LSA−TR CF−TR
−2
10
V. C ONCLUSIONS
QPSK
−3
10
−4
10
0
5
model is employed with the nonlinearity parameter to be 2 and the input backoff (IBO) to be 0dB [8]. Note that, the curves labeled “Ideal” show the BERs of the OFDM signal without SSPA and PAPR reduction. Seen from Fig. 4, the proposed CFTR scheme offers better BER performance, compared with the CC-TR and AAC-TR methods. For example, at BER = 10−4 , SN R = 16.5dB with the CF-TR method, SN R = 18dB with the AAC-TR method and 18dB with the CC-TR method, respectively, when QPSK is employed. When the PAPR reduction is equal or larger than 4dB at CCDF = 10−4 , the CC-TR, AAC-TR, LSA-TR, and CFTR methods require 12, 2, 6, 2 iterations, respectively. Thus, compared with the power on data subcarriers, the average increased power (AIP) of the CC-TR, AAC-TR, LSA-TR, and CF-TR methods are 0.102dB, 0.294dB, 0.150dB, and 0.607dB, respectively. Although the CF-TR method slightly increases the AIP, the CF-TR method provides better average PAPR reduction as shown in Fig. 3. Since the BER performance is mainly affected by the PAPR reduction, the CF-TR method can provide the best BER performance due to its good PAPR reduction.
10
15
20
25
30
E /N (dB) b
0
Fig. 4. BER performance comparison of different schemes over AWGN channel.
CCDF = 10−4 , the PAPR is 9.7dB, 9.3dB, 8.8dB, 9.0dB, and 9.5dB, respectively, with A = 4.5, 5.0, 5.3, 6.0, and 6.5. In particular, the CF-TR method achieves the best PAPR reduction when A = 5.3. As analyzed in Section III, substituting √ N = 1024, M = 64 and σ = 10 into (17), we have A = 5.3, which is consistent with the results shown in Fig. 1. Fig. 2 shows the PAPR reduction of the CC-TR, AACTR, LSA-TR, and CF-TR methods, respectively. Seen from Fig. 2, it is obvious that the CC-TR method converges slowly. The LSA-TR method converges very fast, however, its PAPR reduction performance is not very good. Furthermore, the proposed CF-TR method achieves better PAPR performance than the CC-TR, AAC-TR, and LSA-TR methods. For example, when CCDF = 10−4 and the iteration number is 2, the PAPR reduction of the CC-TR, AAC-TR, LSA-TR, and CFTR methods are 0.8dB, 4.2dB, 3.8dB, and 4.6dB, respectively. Fig. 3 depicts the average PAPR reduction of the CC-TR, AAC-TR, LSA-TR, and CF-TR methods for the same iteration number, respectively. Obviously, the average PAPR reduction performance of the CF-TR method is better than those of the CC-TR, AAC-TR, and LSA-TR methods. The bit error rate (BER) performances with different methods over additive white Gaussian noise (AWGN) channel are shown in Fig. 4, where the number of iterations is 3. For the simulations, the Rapp’s solid state power amplifier (SSPA)
In this letter, we proposed an effective TR method based on the curve fitting algorithm to generate peak-canceling signals to reduce the PAPR in the OFDM systems. Theoretical derivation of the optimal clipping threshold for the proposed CF-TR method was also provided. Simulation results showed that the proposed method could offer better PAPR reduction with much lower computational complexity, compared with the CC-TR, AAC-TR and LSA-TR methods. R EFERENCES [1] B. Wang, P. H. Ho, and C. H. Lin, “OFDM PAPR reduction by shifting null subcarriers among data subcarriers,” IEEE Commun. Lett., vol. 16, no. 9, pp. 1377–1379, Sep. 2012. [2] A. Ghassemi and T. A. Gulliver, “Partially selective mapping OFDM with low complexity IFFTs,” IEEE Commun. Lett., vol. 12, no. 1, pp. 4–6, Jan. 2008. [3] J. Park, E. Hong, and D. Har, “Low complexity data decoding for SLMbased OFDM systems without side information,” IEEE Commun. Lett., vol. 15, no. 6, pp. 611–613, Jun. 2011. [4] J. Tellado, “Peak to average power reduction for multicarrier modulation,” Ph.D. dissertation, Stanford University, Stanford, CA, 2000. [5] S. Gazor and R. AliHemmati, “Tone reservation for OFDM systems by maximizing signal-to-distortion ratio,” IEEE Trans. Wireless Commun., vol. 11, no. 2, pp. 762–770, Feb. 2012. [6] A. Gatherer and M. Polley, “Controlling clipping probability in DMT transmission,” in Proc. 1999 Asilomar Conference on Signals, Systems and Computers, pp. 1076–1079. [7] Y. Wang, W. Chen, and C. Tellambura, “Genetic algorithm based nearly optimal peak reduction tone set selection for adaptive amplitude clipping PAPR reduction,” IEEE Trans. Broadcast., vol. 58, no. 3, pp. 462–471, Sep. 2012. [8] H. Li, T. Jiang, and Y. Zhou, “An improved tone reservation scheme with fast convergence for PAPR reduction in OFDM systems,” IEEE Trans. Broadcast., vol. 57, no. 4, pp. 902–906, Dec. 2011. [9] G. H. Golub and C. F. VanLoan, Matrix Computations. The Johns Hopkins University Press, 1996. [10] L. Wang and C. Tellambura, “Analysis of clipping noise and tone reservation algorithms for peak reduction in OFDM systems,” IEEE Trans. Veh. Technol., vol. 57, no. 3, pp. 1675–1694, May 2008.
