PAPR REDUCTION FOR OFDM SYSTEMS USING CLIPPING AND SQUARE ROOTING TECHNIQUES Deepa.T
1
Kumar R
Department of Telecommunication Engineering SRM University, Kattankulathur - 603203, Tamil Nadu,India. 1 Phone no: 9884028949 ,
[email protected]
2
Department of Electronics and Communication Engineering SRM University, Kattankulathur - 603203 Tamil Nadu, India. 2 Phone no: 9940036036
1
[email protected]
2
ABSTRACT Orthogonal Frequency Division Multiplexing (OFDM) is an efficient modulation technique for high data rate communication systems. Peak to Average Power Ratio (PAPR) is one of major problems in OFDM system. In this paper, a joint clipping and square rooting (SQRT) method has been suggested and investigated for PAPR reduction of OFDM signal. The SQRT process changes the statistical characteristics of the OFDM signal from Rayleigh distribution to Gaussian like distribution after the clipping process. This change in statistical distribution results changes of both the peak and average power values of OFDM signals, and consequently reduces significantly the PAPR value. This technique decreases the bit error rate (BER) slightly while significantly reduces the PAPR value. For 64-QAM OFDM system with 1024 subcarriers, up to 7 dB PAPR reductions with fixed degradation in bit error rate (BER) equal to 2.5 dB can be achieved by combining both clipping and square rooting scheme.
The primary advantage of OFDM is no Inter Symbol Interference (ISI) [2].However, high Peak to Average Power Ratio (PAPR) is a major drawback of OFDM system. High PAPR requires a highresolution digital to analog convertor (DAC), analog to digital convertor (ADC), a wide linear region of the amplifier in order to avoid signal waveform distortion. As the number of subcarriers increases, the maximum possible peak power becomes higher than the average power. High PAPR leads to inter modulation products among the subcarriers causing In Band Interference (IBI) and Out of Band Interference (OBI) to signal. Hence, it is undesirable. The paper is structured as follows. Section 2 describes OFDM system model and formulates the problem of high PAPR. Section 3 introduces the Clipping and Square Rooting schemes for PAPR reduction. In section 4, the CCDF and BER performance of the PAPR reduction scheme is compared with the conventional OFDM systems through simulation. Finally, conclusions are presented in section 5.
Keywords Complementary cumulative distribution function (CCDF), OFDM, Clipping, Square Rooting.
2. OFDM SYSTEM MODEL In an OFDM system, the input data stream is converted into N parallel data streams each with symbol period Ts through a serial to parallel convertor. When the parallel symbol streams are generated, each data stream would be modulated and carried at different center frequencies. Then the N data symbols are mapped to bins of an inverse fast Fourier transform (IFFT) and then followed by cyclic prefix (CP).An IFFT transform converts the frequency components of the spectrum into the time domain OFDM symbols, adds a prefix and transmits the resulting signal over the AWGN channel [3].
1. INTRODUCTION Nowadays, Mobile networks need modern technology for the fast & efficient communication. OFDM is a multi carrier modulation method of transmitting data by splitting it into several components and sending each of these components over separate carrier signals [1]. In OFDM, the data are divided into several parallel data streams or Channels, one for each subcarrier then the subcarrier transmitted simultaneously at different frequencies to the receiver. A large number of closely spaced orthogonal subcarriers are used to carry data in a system.
If ‘N’ number of subcarriers is used and each sub carrier is modulated, the OFDM signal can be described as [4],
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x n
1 N 1 ¦ X k e Nk 0
j 2Snk N
Where, X(k) are the transmitted symbols on each subcarrier N is the number of subcarriers
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Serial to Parallel Convertor
Modulator Input data
M-QAM
N-point IFFT & Add
CP
Clipping
Parallel to Serial Convertor
D/A Convertor
& Square Rooting
AWGN CHANNEL
Output data
Parallel to Serial Convertor
Demodulator M-QAM
CP removal &
Serial to Parallel Convertor
Inverse Square Rooting
A/D Convertor
N-point FFT
Figure 1. Block diagram of OFDM system with PAPR reduction techniques At the receiver section, the receiver cancels the prefix and uses a Fast Fourier Transform (FFT) to transform the received signal back into the frequency domain. N 1
y ( m)
¦ y(n) exp
j 2Smn N
n(m)
For large values of subcarriers (N>64),the complex representation of OFDM signal x(n) become Gaussian distributed. Therefore the envelope of the OFDM signal has a Rayleigh distribution with a cumulative distribution and can be written by,
F ( z ) 1 e PAPR0
n 0
Where,
(5)
y(m) is the received symbol on the mth subcarrier.
The complementary cumulative distribution function (CCDF) of OFDM signals can be expressed as [7],
n(m) is the additive noise added to mth subcarrier.
CCDF
Pr PAPR ! PAPR
2.1 Peak to Average Power Ratio (PAPR) The PAPR for the discrete time OFDM signal x(n) is defined as the ratio of maximum signal power to average power of signal[5],[6].
PAPR 10 log
max x (n)
2
Mean x (n)
2
(3)
The peak power of the OFDM signal, when all the subcarriers are added-up constructively, is the sum of all the N subcarriers. The mean power of the OFDM signal is the sum of all the values of the signal, which is actually N, divided by the total number of subcarriers, which is also N. Therefore the maximum PAPR is,
PAPRm
N2 N
N
Pr PAPR ! PAPRO
(4)
The PAPR reduction capability is measured by the Complementary Cumulative Distributive Function (CCDF) which indicates the probability that the PAPR exceeds a certain threshold. The CCDF of PAPR can be applied to determine the bounds for the minimum number of redundancy bits required to identify the PAPR sequences and evaluate the performance of any PAPR reduction schemes.
0
1 1 e PAPR 0
(6)
N
(7)
PAPR0 is the threshold level.
3. PAPR REDUCTION TECHNIQUES 3.1 Clipping The simplest technique for PAPR reduction is amplitude clipping. This scheme limits the peak amplitude of the input signal to a predetermined value for some desired signal level. Clipping reduces large peaks by nonlinearly distorting the signal [8],[9].It does not require any additional information to the signal and too large peaks occur with low probability, so the signal is seldom distorted. In this technique, maximum peak amplitude A is chosen so that the OFDM signal does not exceed the limits of this region, symbols that exceed this maximum amplitude, will be clipped. The clipping function is performed in digital time domain, before square rooting and the Digital to Analog conversion as shown in Figure 1. The clipping process is described by the following expression [10],
xc n
® ¯
jI ( n ), x ( n ) ! A
½ ¾ x( n), x ( n ) d A ¿ Ae
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(8)
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Where, xc(n) is the clipped signal
Subcarriers
64
Cyclic Prefix
1/4
3.2 Square Rooting Technique (SQRT)
Techniques
Clipping and Square rooting
The SQRT – OFDM signals Xn is processed by,
Noise Channel
AWGN
A
is the clipping threshold
x c (n) exp( jI n ), n
X SQRT
0,1,.....N 1
(9)
Where, x(n) is the nth OFDM output signal. XSQRT is the nth SQRT–OFDM signal. ĭn is the phase of x(n) In SQRT process, the real and imaginary part of XSQRT is denoted by Re {XSQRT} &Im{XSQRT}, are independent & identically distributed Gaussian random variables with zero mean & a common variance. According to central limit theorem , so the amplitude (or) modulus of OFDM signal XSQRT is given by the large number of input samples , the imaginary and real parts of IFFT outputs will follow Gaussian distributions.
The Simulation Parameters of the OFDM system is shown in table 1. We have randomly generated OFDM signal shown in figure 2. A joint clipping and square rooting is applied to the same randomly generated OFDM signal of figure 2,as it seen in figure 3, the peak amplitude of the input signal is limited by amplitude clipping with clipping threshold ‘A=0.7’and followed by square rooting. It changes the Rayleigh distribution of OFDM output signal into Gaussian like distribution. 1 0.9 0.8 0.7
Re 2 ^X SQRT ` Im 2 ^X SQRT `
(10)
Magnitude
X SQRT
The power of OFDM signal can be expressed as,
X SQRT
2
N1 N 1
1 § j2S (m k)n · x(m)x(k) exp¨ ¸ ¦¦ N m 0k 0 N ¹ ©
0.6 0.5 0.4 0.3
(11)
0.2 0.1
Where,
0
|XSQRT|2 denotes the power value of OFDM Output symbol. During the entire signal processing in OFDM system, the phases of the OFDM output sigQDOV ɮn are kept unchanged while only the amplitudes of the OFDM signals are treated and changed. SQRT technique, changes not only the statistical distribution of
0
2
4
6 Time sample
8
10
12 x 10
4
Figure 2. Illustration of conventional OFDM signals 0.9
2
0.8 0.7 0.6 Magnitude
OFDM signals, but the values of the mean μ and variance V of the OFDM signals are also changed. As an example, the Rayleigh distribution of any signal will change into Gaussian distribution if the square root operation is applied to that signal. Also, the chi-square distribution can be transformed into Rayleigh distribution by applying square rooting [11, [12]. The impact of this operation on the average power is higher than that on the peak power value, which is always a lead to reduction in the PAPR value.
0.5 0.4 0.3 0.2
4
SIMULATION RESULTS
To show the PAPR reduction capacity, BER performance, and other features of the SQRT technique we considered an OFDM system using 1024 subcarriers and a 64-QAM modulation scheme (M=64, M is the modulation order), simulated by randomly generated data. Table 1. Simulation Parameters for OFDM transceiver Parameter
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Value
Modulation
64-QAM
IFFT /FFT size
1024
0.1 0
0
2
4
6 Time sample
8
10
12 x 10
4
Figure 3. Effect of Clipping and Square rooting on a OFDM signal. Figure. 4 shows the influence of the SQRT process on the statistical characteristics of the OFDM output signals. The figure shows clearly the change achieved in the statistical distribution for both the amplitude and power of the OFDM output signals resulting from the SQRT signal processing. A Rayleigh
International Conference on Advances in Computing, Communications and Informatics (ICACCI-2012)
distribution of amplitude of the conventional OFDM output (Figure. 4.a) changed into a Gaussian-like distribution (Figure. 4.b); and at the same time, the Chi-square distribution of the OFDM output power signals (Figure. 4.c) changed to Rayleigh distribution (Figure. 4.d).
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CCDF plots for PAPR
0
CCDF
Conventional OFDM Square rooting method
10
10
-1
-2
5
6
7
8
9 PAPRO(dB)
10
11
12
13
Figure 6. PAPR reduction performance of SQRT-OFDM Figure 6 shows the CCDF performance of SQRT OFDM system about the reduction in PAPR. It can be seen that the SQRT scheme provides better performance than that of the clipping method. At CCDF=10-2, the PAPR of SQRT scheme is 2 dB smaller than that of clipping method. Table 2. PAPR of OFDM signals with different schemes
Figure 4. Impact of SQRT technique on statistical distribution of OFDM output signals 10
CCDF
Conventional OFDM
Clipping
Square Rooting
Clipped -SQRT
PAPR(dB)
12.380
9.116
6.710
5.078
CCDF plots for PAPR
0
Conventional OFDM Clipping method
10
Schemes
The measured values of PAPR with different schemes are tabulated in table 2, and performance results are plotted in figure 7. It can be seen that the PAPR of the clipped SQRT-OFDM system is 5.078dB while the original OFDM is 12.380dB. This result shows that up to 7dB reduction in PAPR of signal was achieved by combining both clipping and square rooting method compared to conventional OFDM system.
-1
10
CCDF plots for PAPR
0
Conventional OFDM Clipped OFDM Clipped SQRT-OFDM
-2
6
7
8
9 PAPR0(dB)
10
11
12 CCDF
10
10
-1
Figure 5. PAPR reduction performance of clipping method Figure 5 shows CCDF curves of OFDM signals before and after the application of the Clipping method. The PAPR measures at different probability of occurrence can be extracted from the CCDF curves for both the conventional OFDM & clipped OFDM systems. It can be noticed that the PAPR obtained by clipping method at CCDF=10-2 is reduced to about 3dB reduction in PAPR obtained from the conventional OFDM system. The effect of square rooting scheme after the clipping process on the PAPR reduction of the OFDM system is considered.
10
-2
4
5
6
7
8 PAPR0(dB)
9
10
11
12
Figure 7. Comparisons of CCDF of PAPR Figure 8 shows the BER performances as a function SNR over AWGN channel. For low SNR, the BER performance of both clipped SQRT-OFDM scheme and conventional OFDM system
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are close to each other. At 10-1 BER level, the difference in SNR is about 2.5dB only. 10
0
[7] H. Rohling, T. May, K. Bruninghaus, and R. Grunheid, “Broadband OFDM radio transmission for multimedia applications,”Proc. IEEE, vol.87, pp.1778-1789, Oct.1999.
Conventional OFDM Clipped SQRT-OFDM
BER
10
10
10
10
[6] Xiaodong zhu, Guangxi Zhu, Tao Jiang, “Reducing the peak-to-average power ratio using unitary matrix transformation. IET Communications, 2009, pp.161-171.
-1
[8] R. Bahai et al, “A new approach for evaluating clipping distortion in multicarrier systems,” IEEE J. Selec. Area Commun., vol. 20, no. 5, pp.1037-1046, June 2002. -2
[9] H. Ochiai and H. Imai, “Performance of the deliberate clipping with adaptive symbol selection for strictly bandlimited OFDM systems,” IEEE J. Select. Areas. Comm., vol. 18, no. 11, pp. 2270–2277, Nov.2000.
-3
-4
0
2
4
6 EbNo(dB)
8
10
12
[10] H. Chen and M. Haimovish, “Iterative estimation and cancellation of clipping noise for OFDM signals,” IEEE Commun. Lett., vol. 7, no. 7,pp. 305–307, July 2003.
Figure 8. BER performance of clipped SQRT-OFDM system
[11] Wisam F. Al-Azzo, Borhanuddin M. Ali, Sabira Khatun, and Syed M.Bilfagih, “Time domain statistical control for PAPR reduction in OFDM system”, Proc. IEEE APCC 2007, Bangkok, Thailand, 2007 pp.141-144.
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[12] J. G. Proakis, Digital Communications, Mc-Graw Hill, 2001.
CONCLUSION
In this paper, the problem of PAPR in OFDM signals is addressed. An efficient PAPR reduction method for the OFDM system by combining both clipping and square rooting has been suggested and investigated. Simulation results of 64-QAM OFDM systems show that PAPR was reduced effectively by using clipping and square rooting operation of the OFDM signals. The results of computer simulation show that about 7dB PAPR reduction at 10-2 CCDF was achieved with a penalty of only 2.5dB difference in SNR at 10-1 BER level.
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ACKNOWLEDGEMENTS
The authors would like to thank the anonymous reviewers for their valuable comments which helped to improve the presentation of the paper.
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REFERENCES
[1] R. Van Nee and R. Prasad, ‘’OFDM for Wireless Multimedia Communications’’, Artech House, London, UK, 1st edition, 2000. [2] Ahmad R. S. Bahai and Burton R. Saltzberg, ‘’MultiCarrier Digital Communications: Theory and Applications of OFDM’’, 2nd ed. New York: Kluwer Academic Publishers, 2002. [3] A.R.Bahai and B.R.Saltzberg, “Multi-Carrier Digital Communications: Theory and Applications of OFDM,” Kluwer Academic/Plenum Publishes, New York, NY, USA, 1999. [4] T. Jiang, Y. Yang, and Y. Song, “Exponential companding transform for PAPR reduction in OFDM systems,” IEEE rans. Broadcast, vol. 51,no. 2, pp. 244–248, Jun. 2005. [5] T. Jiang and Y. Imai, “An overview: peak-to- average power ratio reduction techniques for OFDM signals,” IEEE Trans. On WirelessCommunications, June 2008.
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