PAPR Reduction of OFDM Signals Using Deliberate Clipping and Pre

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					  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

    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 [1], 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 [4]
sequence (PTS) [2] and selected mapping (SLM) [3], are                                            2r
                                                                                       f R (rn ) = n e − rn / Pin , rn ≥ 0
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, [4] 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 [4], 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 [4], 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 [5]                                                                                      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

                 (                         )
                              −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
                                                                                                                                                                                                             dt .
respectively given as                                                                                                                                                                 x

                                                       (                        )
                                                                                    N                               A.   B Selected Mapping Technique
                                    Fl ( l ) = 1 − e − l
                                                                        / Pin
                                                                                                                              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
                                                                                            / 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 [5], 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 )        )
                        Pout = E ⎡ rn2 ⎤ = ∫ rn2 f R (rn )drn .
                                 ⎣ ⎦                                                                             (8)

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 [4]                                                                 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 )               )

               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


                                                                 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
                                           ⎛                                                            ⎞ ⎞
                                                                   (                            )
                                         = ⎜ 1 − ⎛ 1 − 1 − e − l / Pin
    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
                   ⎛                                            ⎞ ⎞
                              (                         )
   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
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
serious than that on the non-selected OFDM signals. From the
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
                                                                                                                                                                                  CR = 1.2
                    IV. CONCLUSION
                                                                                                                                                                                      no clipping
     In this paper, an OFDM system which combines SLM                                                                       -3

technique and deliberate clipping technique was discussed.                                                                                                                                       CR = 1.6
                                                                                     bit error rate

The effect of symbol selection scheme on the deliberate
clipping was analyzed by deriving the pdf of the samples’                                                                   -5
                                                                                                                                                        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
there is the tradeoff between the PAPR reduction ability,                                                                  10
system complexity and the BER performance.                                                                                  -8
                                                                                                                                 6          8           10             12        14              16           18        20
                                                                                                                                                                        Es/N0 [dB]

[1]   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.
[2]   S. H. Müller and J.B. Huber, “OFDM with reduced peak-to-average                                                                                                                                          P=1
      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

[3]   Baml, R. W., Fischer, R.F.H. and Huber, J.B., “Reducing the peak to

      average power ratio of multicarrier modulation by selected mapping,”                                                  -2
      IEE Electronics Letters, vol. 32, no. 22, pp. 2056-2057, Sep. 1997.                                                                                              CR = 1.6

[4]   Hideki Ochiai and Hideki Imai, “Performance of the deliberate clipping
      with adaptive symbol selection for strictly band-limited OFDM
      systems,” IEEE Journal on Selected Areas in Communications, vol.18,
                                                                                                                                                                 CR = 1.2
      no.11, pp. 2270-2277, Nov. 2000.                                                                                      -4
                                                                                                                                 2      3       4            5          6      7             8            9        10   11
[5]   Hideki Ochiai and Hideki Imai, “On the distribution of the                                                                                                       PAPR0 [dB]

      peak-to-average power ratio in OFDM symbols,” IEEE Trannsaction
      on Communications, vol.49, no.2, pp. 282-289, February. 2001.                                           Fig. 4. Complementary cumulative distribution functions of PAPR of
                                                                                            an OFDM signal with 256 subcarriers for QPSK modulation.

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