PAPR Reduction of OFDM Signals Using Deliberate Clipping and Pre-scrambling Technique Lei Wang, Kyongkuk Cho, Dongweon Yoon, and Sang Kyu Park Department of Electronics and Computer Engineering Hanyang University Seoul, Korea firstname.lastname@example.org Abstract—Orthogonal Frequency Division technique. Through the analysis in PAPR reduction capability, Multiplexing (OFDM) is considered to be a promising system complexity and error performance, we demonstrate technique against the multipath fading channel for that the system has proper performance in real applications. wireless communications. One of the disadvantages of OFDM is peak to average power ratio (PAPR) problem. II. OFDM SIGNALS AND PAPR REDUCTION In this paper, we analyze a PAPR reduction system which OFDM symbols can be given as the sum of a numbers of combines a selected mapping technique (SLM) and a independent symbols which are modulated onto subchannels deliberate clipping technique. The numerical analysis and of equal bandwidth. Let Xk ( k =0,1 N−1) denote the input data ,..., computer simulation show that the system has effective PAPR reduction capability, moderate system complexity symbol whose period is T . Then the complex representation and reasonable bit error rate (BER) performance. of an OFDM symbol is given as N −1 x ( t ) = ∑ X k ⋅ e j 2π k ft , 0 ≤ t < NT (1) Index Terms—Selected Mapping, Clipping, PAPR, k =0 OFDM where N is the number of subcarriers, and f = 1/ NT is the subcarrier spacing. The samples are denoted by I. INTRODUCTION xn ( n = 0,1,..., LN − 1) for the OFDM symbols with the Orthogonal Frequency Division Multiplexing (OFDM) is considered to be a promising technique against the multipath sampling rate L . In the following, we consider the sampling fading channel for wireless communications. However, rate to be the Nyquist rate which corresponds to the case of OFDM has a serious peak to average power ratio (PAPR) L = 1 . The amplitude of the n th sample of an OFDM symbol problem. The simplest and most effective method to reduce is given as rn xn . As N is a sufficiently large number, rn PAPR might be the clipping and filtering , but the error performance of clipped OFDM signal is degraded due to the is considered to be approximately equal to a Rayleigh random distortion of the original signals. In non-distortion techniques, variable with probability density function several symbol selection schemes, such as partial transmit (pdf) given as  sequence (PTS)  and selected mapping (SLM) , are 2r f R (rn ) = n e − rn / Pin , rn ≥ 0 2 (2) widely used. Symbol selection schemes can obtain a moderate Pin PAPR reduction ability but increase the complexity of OFDM where Pin = 2σ 2 is the input power of the OFDM signal. system. To obtain an effective PAPR reduction ability and a moderate system complexity,  proposed an OFDM system The PAPR of the OFDM symbol is defined as the ratio of the which combines the deliberate clipping and the symbol peak power and the average power as selection schemes. max ⎡ rn 2 ⎤ ⎣ ⎦ max ⎡ rn 2 ⎤ ⎣ ⎦ In the paper, base on , we analyze the OFDM system PAPR = , n ∈ [ 0, N − 1] (3) E ⎡ rn 2 ⎤ ⎣ ⎦ Pin which combines deliberate clipping technique and SLM where φn is a random variable which has uniformly where E [⋅] denotes the statistical expectation function and distribution on [ 0, 2π ) . As derived in , the SNDR of the max [⋅] gives the highest value among the samples. For one clipped OFDM signal xn can be presented as OFDM symbol, the probability of the peak amplitude being α 2 Pin smaller than a given threshold W can be obtained by  SNDR = (11) Pout − α 2 Pin + N 0 ( Fl (W ) = Pr max rn < W 0≤ n< N ) where N 0 is the total variance of the AWGN. Therefore, the = Pr ( r0 < W ) ⋅ Pr ( r1 < W ) ⋅⋅⋅ Pr ( rN −1 < W ) . (4) BER of QPSK OFDM signal after deliberate clipping can be ( ) N −W 2 / Pin calculated by = 1− e Thus, the cumulative distribution and the complementary Pb = Q ( SNDR ) (12) cumulative distribution of the peak amplitude can be ( ) ( )∫ ∞ where Q ( x ) = (1/ 2 ) erfc x / 2 = 1/ 2π e−t 2 /2 dt . respectively given as x ( ) N A. B Selected Mapping Technique Fl ( l ) = 1 − e − l 2 / Pin (5) The SLM for PAPR reduction is a non-distortion and technique. In this approach, the transmitter generates a set of ( ) N sufficiently different candidate data symbols, all representing Fl c ( l ) 1 − Fl ( l ) = 1 − 1 − e − l 2 / Pin . (6) the same information as the original data symbol. Among To solve the PAPR problem, an OFDM system combining these symbols, the symbol which has the smallest PAPR deliberate clipping and SLM can be used. The system model value is selected and the information of the selected data is shown in Fig. 1. symbol is transmitted as the side information. Assume P A Deliberate Clipping candidate symbols are generated in the transmitter, then the Deliberate clipping might be the simplest method to transmitter needs P IFFT operations and the bits number of reduce PAPR. This method limits the samples’ amplitudes of required side information is larger than log 2 P for each the input OFDM signal to a predetermined value. The amplitude of the n th output sample of the clipped OFDM symbol. As given in , the probability of the peak amplitude signal is given as of the selected OFDM symbol exceeding the given threshold W can be given as ⎧rn , for rn ≤ W ⎨ ( ) rn , (7) ⎩W , for rn ≥ W Fl c (W ) = Pr min l p > W 1≤ p ≤ P and the power of the clipped OFDM signal therefore becomes = Pr ( l1 > W ) Pr ( l2 > W ) ⋅⋅⋅ Pr ( lP > W ) (13) ∞ ( = 1 − Fl p (W ) ) P Pout = E ⎡ rn2 ⎤ = ∫ rn2 f R (rn )drn . ⎣ ⎦ (8) 0 Equation (7) shows that deliberate clipping is a memoryless where l p is the peak amplitude of the p th OFDM symbol nonlinear transformation. The output of the memoryless nonlinear transformation of OFDM signal xn can be and Fl p ( l ) is the complementary cumulative distribution of expressed as xn = α xn + d n (9) l p . Therefore, the cumulative distribution of the peak where d n is the distortion term uncorrelated with xn , and α is an attenuation factor that can be calculated as  amplitude of the selected OFDM symbol can be given as Ern ,φn [ rn cos φn rn cos φn ] ( Fl ( l ) = 1 − Fl c ( l ) = 1 − 1 − Fl p ( l ) ) P α= . (14) σ 2 ∞ 2π 1 (10) For the selected OFDM symbol without over-sampling, there ∫ rn rn f R ( rn )drn ⋅ ∫ cos 2 φn ⋅ d φn = 0 0 2π are N statistically independent samples for one OFDM σ2 symbol period. Just as in (4), (14) can be expressed by (a) (b) Fig. 1. The proposed OFDM system combining clipping and SLM: (a) transmitter and (b) receiver multiplying N cumulative density functions. Since these C Combination of Deliberate Clipping and SLM cumulative density functions are considered to be same for all The implementation of deliberate clipping is quite simple samples of one OFDM symbol, the cumulative density and effective in PAPR reduction, but larger clipping ratio function of the amplitude of the n th sample can be results in the severe BER performance degradation. On the calculated from (5) and (14) as other hand, the SLM technique does not cause distortion on 1/ N the error performance if there are no errors in the ⎛ ⎞ ⎞ ( ) P = ⎜ 1 − ⎛ 1 − 1 − e − l / Pin N Pr ( rn < l ) = ( Fl (l ) ) 1/ N 2 ⎜ ⎟ ⎟ . (15) sideinformation; however, in order to obtain effective PAPR ⎝ ⎝ ⎠ ⎠ reduction ability like clipping method, the system complexity By calculating the derivative of the function (15) and becomes challenging as the number of the candidate symbols increases. Thus, the use of only deliberate clipping or only substituting rn for l , we obtain the pdf of rn by SLM technique cannot obtain satisfactory error performance 1/ N −1 and moderate system complexity simultaneously. However, if ⎛ ⎞ ⎞ ( ) P f R (rn ) = P ⋅ ⎜1 − ⎛1 − 1 − e− rn / Pin 2 N ⎜ ⎟ ⎟ these two approaches are combined, the effective PAPR ⎝ ⎝ ⎠ ⎠ (16) reduction can be achieved with reasonable BER performance P −1 ( ⋅ ⎛1 − 1 − e− rn / Pin ) ⎞ ( ) N N −1 2r − rn2 / Pin ⋅ n e− rn / Pin . 2 2 ⎜ ⎟ ⋅ 1− e ⎝ ⎠ Pin and suitable system complexity. Applying f R ( rn ) instead of Fig. 2 shows the pdf of the samples’ amplitudes of the f R ( rn ) in the BER analysis by substituting (17) into (8), original OFDM signal and the samples’ amplitudes of the (10)-(12), the BER performance for clipped selected OFDM selected OFDM signal. We can see that the probability signals can be calculated. distribution for the non-selected OFDM signal can be approximated as the Rayleigh distribution, and the probability III. SIMULATION RESULTS distribution for the adaptively selected OFDM signal shows In the simulation, QPSK systems with 256 subcarriers little difference from the Rayleigh distribution. were used. OFDM symbols were transmitted over an AWGN channel. Fig. 3 shows the error performance of the proposed 0.8 system with different numbers of selective mapping symbols non-selected OFDM signal 0.7 adaptively selected OFDM signal, P=16 and clipping ratio (CR). CR is given as the ratio of the maximum permissible amplitude and root mean square power 0.6 Relative number of samples of the OFDM signal. Because the selected OFDM signal has 0.5 smaller PAPR than non-selected OFDM signal, the distortion 0.4 caused by clipping on the selected OFDM signals is more 0.3 serious than that on the non-selected OFDM signals. From the 0.2 figure, we can see that the BER performance of the clipped-selected OFDM signal is improved as the number of 0.1 selective mapping symbols increases. Fig. 4 shows the PAPR 0 0 0.5 1 1.5 2 2.5 3 3.5 4 reduction ability of the system combining SLM and deliberate Magnitude clipping. It is clear that the clipping method is much more Fig. 2. Probability distribution of samples’ amplitudes for non-selected effective in PAPR reduction than SLM with smaller value P . OFDM symbols and selected OFDM symbols (QPSK, 256 subcarriers ). Therefore, performing clipping after taking SLM can significantly reduce PAPR and system complexity. -1 10 CR = 1.2 -2 10 IV. CONCLUSION no clipping In this paper, an OFDM system which combines SLM -3 10 technique and deliberate clipping technique was discussed. CR = 1.6 -4 bit error rate 10 The effect of symbol selection scheme on the deliberate clipping was analyzed by deriving the pdf of the samples’ -5 10 P= 1 amplitude of the adaptively-selected OFDM symbol. From -6 10 P= 4 the analysis and the computer simulation, we can see that P= 8 -7 there is the tradeoff between the PAPR reduction ability, 10 P=16 system complexity and the BER performance. -8 10 6 8 10 12 14 16 18 20 Es/N0 [dB] REFERENCES  Eetvelt, P.V., Wade, G. and Tomlinson, M. “Peak to average power Fig. 3. BER perfromance of clipped-selected OFDM signal, N = 256 reduction for OFDM schemes by selective scrambling,” IEE Electronics Letters, vol. 32, no. 21, pp. 1962-1964, Oct. 1996. 0 10  S. H. Müller and J.B. Huber, “OFDM with reduced peak-to-average P=1 P=4 power ratio by optimum combination of partial transmit sequences,’ P=8 no clipping P =16 IEE Electronics Letters, vol. 33, no. 5, pp. 368-369, Feb. 1997. 10 -1  Baml, R. 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