A New

Abstract

Distribution Bound and Reduction Scheme for OFDM PAPR Xuefu Zhou James Caffery, Jr.

Department of ECECS, University of Cincinnati Cincinnati, OH, USA 45220

xzhou, jcaffery@ececs.uc.edu

A new simple upper bound on the PAPR distribution of an OFDM signal is proposed and simulations show that it is tight when the subcarrier number is greater than 64. In addition, a PAPR reduction scheme named selective mapping-dynamic constellation (SLM-DC) is proposed. Simulations show this combination of SLM- DC achieves good PAPR reduction while it extends DC's ability to reduce PAPR and has less computa- tional complexity. The average power increase in SLM- DC is estimated by the proposed upper bound.

Keywords OFDM, PAPR, Distribution Bound, PAPR Reduction.

INTRODUCTION The spectacular growth of voice, video and data commu- nication justify great expectations for mobile multimedia. The bit rate under investigation for some systems is in the range of 2-155 Mbit/s [l]. Orthogonal frequency division multiplexing (OFDM) is a promising technique for achiev- ing such a high data rate in a mobile environment because of its high spectral efficiency, robustness to channel fading, immunity to impulse interference, capability to handle very strong echoes (i.e., multipath fading) together with certain implementation advantages over single-canier systems [2]. However, a major obstacle to its practical application is that an OFDM signal exhibits a very high peak to average power ratio (PAPR). Therefore, RF power amplifiers should oper- ate in a very large linear region. Otherwise, if signal peaks move into the nonlinear region of the power amplifier, sig- nal distortion results by introducing intermodulation among the subcarriers and out-of-band radiation. Thus, it is highly desirable to reduce the PAPR. Over the past decade, consid- erable attention has been given to PAPR reduction methods. In order to get a deeper insight into OFDM PAPR, evaluate the capability of PAPR reduction schemes, provide bounds for the minimum number of redundancy bits required to iden- tify high PAPR sequences and design OFDM systems in- volving nonlinear devices, it is of importance to understand PAPR stochastic characteristics. It is difficult to get an ex- act closed-form expression for the PAPR distribution. Thus, several approximations and bounds for the PAPR distribution have been proposed recently. A pair of lower bounds were given in [3], an approximation to the CDF was developed in [4] and an upper bound as well as analysis on the asymptotic PAPR behavior were proposed in [5 ] . An upper bound of

the PAPR distribution and the PAPR estimation error based on oversampled signals were presented in [6]. However, all of these bounds or approximations are complex to use or exhibit some discrepancy with simulations. Thus, based on level crossing rates, we will give a new upper bound in a simple closed-from expression and simulations are given to test its tightness. In addition, based on the analysis of selective mapping [7] (SLM) and dynamic constellation (DC) [SI for PAPR reduc- tion, we propose SLM-DC which extends DC's capability. The overall computational complexity of SLM-DC is less than that of DC without SLM. Simulations show that this combination achieves large PAPR reduction and negligible average power increase as determined by the proposed upper bound.

SYSTEM MODEL OFDM can be simply defined as a form of multicarrier modu- lation (MC) where its subcarrier spacing is carefully selected as the reciprocal of the useful OFDM symbol period T so that orthogonality is archived. The continuous time OFDM signal, in one symbol period, is given by

where Sk is the QAM value of the kth subcarrier, g( t ) is the rectangular window of unit height over the OFDM symbol interval [0, T] and N is the subcarrier number. The Nyquist- rate sampled OFDM signal is given by

Therefore, the input symbols [SO, SI, ..., S N - 1 I T are trans- formed into TIN-spaced samples [SO, SI, ..., ~ ~ - 1 1 ~ . The PAPR of the OFDM signal sr, where T is used to represent both the continuous time index t and discrete time index n, is defined as

(3)

where 5 denotes either T or N . For the purpose of analysis, we make several observations and assumptions. Since the guard time is just a partial replica of the signal during [0, TI, it changes neither the average nor peak power of the signal and thus it does not affect the PAPR and is removed from our analysis. Second, since the RF frequency is much higher

0-7803-7442-81021$17.00 0 2002 IEEE 158

than the subcarrier's frequency, the baseband OFDM signal has the same PAPR as its passband equivalent, so we use the baseband signal to analyze the PAPR. Third, we assume an ideal bandlimited OFDM signal where the bandwidth B = N/T and its power spectral density (PSD) is constant over

Now we consider the baseband OFDM signal s ( t ) and as- sume that s ( t ) can be written in complex form as s ( t ) = ~ ( t ) + j y ( t ) . According to central limit theorem, z( t ) and y(t) can be approximated as two independent Gaussian ran- dom processes when the subcarrier number N is large and S k , 5 = 0,1, ... N - 1, are i.i.d random variables, so the en- velope of s ( t ) can be approximated as Rayleigh process. For convenience we define the normalized OFDM signal as

[-B/2, BPI.

where PA" = E[ls(t)12] denotes the average power over the entire signal. Thus, Ir(t) 1 is also approximated by a Raleigh process. The PAPR cumulative CDF is defined as

where T denotes one OFDM symbol period. It is straight-

forward that CPAPR(Y) = P r oytyT Ir(t)l 2 a .

BOUNDS AND APPROXIMATIONS

Previous Results First, we give an overview of several results from the litera- ture on bounds and approximations of the PAPR distribution. According to the Rayleigh distribution and Nyquist sampling rate, a PAPR distribution bound is given in [3] as

~pApR(y) = 1 - (1 - .-TIN. (6)

Using the level crossing rate approach, an approximation of the PAPR CCDF is given in [4] as

. , 7 5 5

(7) where 7 stands for the reference PAPR value whose proba- - bility is negligible (close to 0), and 7 2 9. An upper bound based on level-crossing rates is proposed in [ 5 ] . However, that group of expressions is complex to use. Another upper bound based on oversampled signals is proposed in [6] as

* (8) - T( 1 -"/kept 1

CPAPR(PAPR > Y) I koptNe

where kept > 7r and "(1 - ") = 1.

A New Simple Upper Bound Consider r ( t ) in one OFDM symbol period. We assume that the probability that the PAPR is greater than y2 is equivalent

kept kopt Y

to the probability that r( t ) will cross y at least once during one OFDM symbol period T1. This is expressed by

where Cr(y, T) denotes the number of times that r( t ) crosses level y during time period T. By using the Markov inequality which converts (9) into an inequality, we obtain

Thus, E[C,(y, T ) ] , the mean number of crossings at level y, denotes an upper bound of the crest factor CCDF. Next we derive E[C,(7,T)] for an OFDM signal. The level crossing rate of a Rayleigh or Ricean process v,(y) is given by [9, 101

v,(y) = /-T$2 - - --

where $(r), the autocovariance of r( t ) , is defined as

and the nth spectral moment b, is defined as

Since x ( t ) and y ( t ) are assumed unconelated and have the same autocorrelation function, we have &(r) = Ry(7). For the Rayleigh process, we have

IE[r(t>l12 = 0 (14) and

E [ T * ( t ) T ( t + T)] = E[z(t)x(t + .)I + E[Y(t)Y(t + .)I = 2R,(r).

(15)

Thus, we obtain

4(T) = M T ) . (16) If we use S,(f) to denote the PSD of z(t), then we have

(17)

Therefore, we can express the first and second order deriva- tives of $(T) as

--m

and

-- d42(T) - j2(27rf)2 SZ(f)ej2"f'df. (19) d r 2

--m

'In this paper, the level crossing adopted is the one with positive dope.

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By letting ‘T = 0, we obtain bo, bl and b2 from (1 6), (1 8) and (19) as

-M

(20) -m

--M

b z = 4 7 r 2 J S z ( f ) f 2 d f . (22) --M

For an ideal bandlimited OFDM signal, the PSD of z( t ) is expressed by

where H is a constant. Integrating (20), (21) and (22), we obtain bl = 0 and

Therefore, the level crossing rate of r( t ) at level y is given by

According to the assumption of B = N I T , we obtain the mean number of level crossings at y during time period T as

E[C,(y,T)] = (26)

Thus, the PAPR CCDF bound is given by

cPAPR(Y) 5 @fie-’. (27)

An important consequence of the bound is to provide a bound for the minimum number of redundancy bits p required to identify high PAPR sequences. The redundancy bits p is the optimum redundancy required to limit the PAPR [6]. By using our upper bound and the lower bound in [3], we introduce bounds for the minimum number of redundancy bits p to limit PAPR below A. These bounds are

logz (1 - < p < - log, (1 - &&e-*> . (28)

Simulation Results of PAPR Distribution We choose 4-ary QAM and lo7 OFDM symbols to simu- late the distribution of PAPR and compare it to those bounds and approximations described by (6), (7), (8) and (27). We have simulated the PAPR of 32,64, 128,256,512 and 1024 subcarriers, but only comparisons of 64 and 1024 subcar- riers are given in Figure 1. The simulation shows that the

6 7 8 9 10 11 12 13 14 Peak To Average Power Ratio (dB)

Figure 1. Comparison of bounds and approximation (64 and 1024 subcarriers).

proposed bound matches the simulation results well when the subcarrier number 2 64. The lower bound in (6) and the upper bound in (8) are looser compared to the bound in (27). The approximation in (7) approximates the PAPR well if the reference ;U is carefully chosen. It also approaches (27) asymptotically as shown in Figure 1 so this confirms our bound’s tightness from another point of view. In addi- tion, a higher oversampling rate will approximate the actual PAPR more accurately, but an oversampling rate of 8 does not indicate much difference from an oversampling rate of 4. Thus, in our simulation, we use an oversampling rate of 4. It is important to note that when the subcarrier number is 32, none of these bounds or approximations are tight. The explanation is that z(t ) and y(t) are not approximated well as Gaussian random processes for N 5 32. As indicated by the normally cited max PAPR 5 N, the PAPR of the OFDM signals does grow linearly with N be- cause, for some OFDM symbols, all of the subcarrier wave- forms add in phase at some point. This will be the dominant effect in systems with a small number of carriers. However, when N is large, the probability that such an OFDM symbol will be sent is extremely low so PAPR 5 N is not adequate. Instead of the actual PAPR being directly proportional to the subcarrier number N, we find that the probabili ty that the PAPR is greater than a given value y is directly proportional to the subcarrier number N . When the subcarrier number is large, the dominant effect is the stochastic behavior of the OFDM signal and a stochastic description of PAPR will be more suitable.

SLM-DC FOR PAPR REDUCTION In this section, we present the SLM-DC scheme. First, we give a briefly introduction of SLM. The key idea of SLM is that we have D statistically independent sequences to repre- sent the same information [7]. They are generated by using D

160

pseudo-random but fixed sequences to multiply the original information sequence. The sequence resulting in the lowest PAPR is selected for transmission. Thus, the probability that the PAPR of all these D independent sequences exceeds y is expressed by Pr(PAPR > Y ) ~ , which is the PAPR prob- ability distribution after using SLM and Pr(PAPR > y) is the PAPR probability before using SLM and D is the SLM order. Second, we give a brief description of the dynamic constellation method [8]. This algorithm intentionally adds some noise to the input symbols so that the PAPR is reduced. The noise is added so that each symbol in 4-ary QAM will increase either its real or its imaginary value so that the min- imum distance between two signal points is increased. For 4-ary QAM mdulation, we can use the projection between the frequency domain and time domain to implement DC as follows [8]:

(1) Take an OFDM symbol U of N signal points which are taken from the 4-ary QAM constellation.

(2) Compute the FFT of the OFDM symbol U to obtain the time domain signal samples V = [VI, Vz, . -.Vj]*.

(3) Examine V and clip each sample point that exceeds the threshold $, so that we get a modified time domain signal 9.

(4) Compute the FFT of the modified time domain signal 9 to get the modified frequency domain signal U.

(5 ) Compare 6 and U, and perform the following opera- tion:

Re U [i] if (Ree[i]l < 1, then Ree[i] =

(6) Go to step (2) until either the iteration reaches a maxi- mum limit or the PAPR falls under the threshold.

Limitations of DC and SLM First, the iterative projections onto the time and frequency domains have a very high computational cost. Second, when a sequence’s symbols are unbalanced (all symbols in a se- quence are not equally located at the four signal points when using 4-ary QAM), DC will fail. For example, if all of the symbols are 1 +j, the largest peak in time domain will occur at t = 0 or n = 0. Clipping the time domain signal is equiv- alent to moving all of the input symbols towards the origin. By the restrictions given in (29) and (30), the symbols will not be changed in this circumstance. These two limitations are improved if we combine SLM with DC. SLM is a promising non-distortion PAPR reduction scheme. However, there are no good methods for generating ideal independent sequences to represent the same information and increasing the SLM order higher than D = 7 does not achieve much improvement. Therefore, SLM’s capability is

1.51

-0.5 O t

-1.5’

Figure 2. SLM extends DC‘s ability by making all symbols balanced.

also limited.

SLM-DC The characteristics of SLM make it a good candidate to com- bine with DC. SLM-DC simply cascades DC to SLM. In its implementation, we use an adaptive mechanism, to set a threshold value to invoke SLM or DC. If a sequence’s orig- inal PAPR is below this threshold, we do not use SLM-DC and we transmit the sequence directly. Otherwise, we pass this sequence to SLM. If after using SLM, the PAPR is be- low the threshold, we transmit the sequence directly without DC. Otherwise we further employ DC to reduce the PAPR. This method reduces the overall computational complexity compared to DC only. Since SLM can reduce the probability of large peaks, the overall computational complexity of SLM-DC is less than that of DC only. In addition, SLM makes the input sequence more randomly located in the signal space so that DC’s abil- ity to reduce PAPR is extended. For example, if there is a sequence with all 1 + j symbols (256 subcarriers), SLM (D = 7) first maps this sequence to a new one whose symbols are balanced as shown in Figure 2, where noise is intention- ally added to help us distinguish symbols at the same signal point. At the same time, SLM reduces the PAPR from 24 dB to 8 dB. After SLM, we employ DC to this new bal- anced sequence and we finally achieve a PAPR of 4.829 dB (if the threshold is set to 4.77 dB). This example illustrates that DC’s ability is extended and the overall computational complexity is reduced (SLM reduces PAPR from 24 dB to 8 dB by only 7 IFFT computations).

PAPR Reduction Capacity of SLM-DC Through simulation, we have investigated SLM-DC’s PAPR reduction ability. SLM-DC achieves more PAPR reduction than both SLM and DC as shown in Figure 3. In this example of a 128-subcarrier OFDM system, it reduces the PAPR at a

161

I -I I T - I-- 100 [ -

lo-’

- ; lo-2

D

A U a d 10-~ t -

1 o4 1 o4

I b I

2 4 6 8 10 1.2 14 16 18 Peak To Average Power Ratio (dB)

Figure 3. Effect of DC combined with SLM.

probability of is 2 dB more than SLM and 0.5 dB more than DC.

Average Power Increase in SLM-DC We can use (27) to estimate the average power increase. If a peak is suppressed, it is similar to adding another signal in the time domain. From the principle of DC, the signal points move in specific areas so that the average power increases. The increased power will be the average power of the peaks added to the original time domain signal, so we can estimate an upper bound of the average power increase as

from 11.8 dB to 5.2 dB. The reduction

N

AP 5 / Pr PAPR > y . (y - p ) dy (31) P

where p is the threshold. In the case of Dth order SLM, PrPAPR > y can be approximated by ( G N f i e - 7 ) ” . Thus, the overall average power increase will be bounded by

P

However, (32) works only under the condition that the PAPR is greater than some value where the original CCDF of PAPR is not equal to 1. If we take the threshold to be 7.7 dB for 128 subcarriers, the average power increase will be about 0.25% for 7th order SLM-DC. In the case without SLM, DC will have an average power increase of 40%.

CONCLUSION A new simpe bound on the PAPR distribution in OFDM is proposed in this paper and our simulations show that this bound is tight when the subcarrier number N is greater than 64. We find that the probability that the PAPR is greater than a given value is directly proportional to the subcarrier number N .

In addition to this bound, we proposed SLM-DC for PAPRre- duction and simulations show that this combination achieves good PAPR reduction. An estimate of the average power in- crease in SLM-DC via the proposed bound shows that, in the case of 128 subcarrier OFDM and a threshold value of 7.7 dB, the average power increase will be about 0.25%, which is negligible compared to using DC only.

REFERENCES Korea TeleCom project on wireless ATM transmission techniques, available at: http://irctr.et.tudelft.nl/projects/kwatt/frames.html. N. Rajatheva, “Orthogonal frequency division multiplexing application for wireless communications with coding”, available at: http://www.comlab.hut.fi/opetus/3 1 l/ofdmmod.pdf. R. van Nee and A.de Wild, “Reducing the peak-to-average power ratio of OFDM’, IEEE Vehicular Technology Conference, Vo1.3, 1998, pp.

H. Ochiai and H. Imai, “On the distribution of the peak-to-average power ratio in OFDM signals”, IEEE Transactions on Communications, Vo1.49, Feb 2001,

N. Dinur and D. Wulich, “Peak-to-average power ratio in high-order OFDM’, IEEE Transactions on Communications, Vo1.49, June 2001, pp. 1063-1072. M. Sharif and B.H. Khalaj, “Peak to mean envelope power ratio of oversampled OFDM signals: an analytical approach”, IEEE International Conference on Communications, Vo1.5, June 2001, pp. 1476-1480. R.W. Bauml, R.F.H. Fischer and J.B. Huber, “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping”, IEEE Electronics Letters, Vo1.32,Oct.1996, pp. 2056-2057. D.L. Jones, “Peak power reduction in OFDM and DMT via active channel modification”, Conference Record of the Thirty-Third Asilomar Conference on Signals, Systems and Computers, V01.2, 1999, pp. 1076-1079. W.C.Jakes, Jr, “Multipath interference”, in Microwave Mobile Communicatios, New York, Wiley 1974, pp. 11-78.

2072-2076.

pp. 282-289.

[lo] A. Abdi, K. Wills, H.A. Barger, M.-S. Alouini and M. Kaveh, “Comparison of the level crossing rate and avergae fad duration of Rayleigh, Rice and Nakagami Fading Models with Mobile Channel Data”, IEEE Vehicular Technology Conference, Vo1.4, 2000, pp. 1850-1857.

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