Nous donnons un apperçu de l'état de l'art en réduction du rapport puissance maximale à puissance moyenne (PAPR) pour les systèmes OFDM, puis, nous présentons une nouvelle méthode permettant une bonne réduction du PAPR tout en ayant une complexité de calculs et une redondance relativement faibles. Cette méthode introduit de l'interférence entre porteuses (ICI). Nous montrons analytiquement comment réduire l'ICI et donnons quelques résultats de simulations

BONACCORSO, Mario

BUZENAC, Véronique

MAOUCHE, Karim

MHIRSI, Karim

I-Revues

Reducing the Peak to Average Power Ratio in OFDM
Systems
Mario Bonaccorso
1
, KarimMhirsi
1
, KarimMaouche
1
, Veronique Buzenac
2
1
Institut Eurecom
2229 Route des Cre^tes, BP 193, 06904 Sophia-Antipolis
2
Centre de Recherche Motorola Paris
Espace Technologique Saint-Aubin, 91193 Gif-Sur-Yvette
bonaccor@eurecom.fr, mhirsi@eurecom.fr, maouche@eurecom.fr, buzenac@crm.mot.com
Resume { Nous donnons un appercu de l'etat de l'art en reduction du rapport puissance maximale a puissance moyenne (PAPR)
pour les systemes OFDM, puis, nous presentons une nouvelle methode permettant une bonne reduction du PAPR tout en ayant
une complexite de calculs et une redondance relativement faibles. Cette methode introduit de l'interference entre porteuses (ICI).
Nous montrons analytiquement comment reduire l'ICI et donnons quelques resultats de simulations.
Abstract { We review some techniques for reducing the Peak to Average Power Ratio (PAPR) in OFDM systems and present a
new method called Discrete Clipping. This method achieves a good reduction of the PAPR with a low computational complexity
and a small redundancy at the expense of some Inter Carrier Interference (ICI). Through analysis, we show how to reduce the
amount of ICI. Finally, some simulations are given.
1 Introduction
Although ignored during several years, multi-carrier mo-
dulation has become a strong competitor to classical single
carrier modulations, mainly because of their very attrac-
tive properties. Indeed, besides exibility and bandwidth
eciency which are inherent qualities of OFDM (Orthogo-
nal Frequency Division Multiplexing) systems, they also
allow communications in severe channels without heavy
equalization. That is the reason why OFDM is now used
in a wide range of standards such as DAB (Digital Au-
dio Broadcasting), ADSL (Asymmetric Digital Subscri-
ber Line), HIPERLAN/2 (HIgh PErformance Radio Lo-
cal Area Network).
However, one of the main drawbacks of OFDM systems
is that they can present a high Peak to Average Power
Ratio (PAPR) when the phases of the dierent carriers
add constructively to form large peaks. When such peaks
occur, they may be cut-o by the amplier nonlineari-
ties, which leads to out of bands radiations and Inter
Carrier Interference (ICI) at the receiver, which may de-
grade considerably the performances. To avoid this degra-
dation, one has either to use ampliers with large back-
os, which are both expensive and less ecient, or to re-
duce the PAPR before the transmission. This paper fo-
cuses on the second solution. Namely, we rst provide a
review of the already existing PAPR reduction techniques
for OFDM systems. Afterwards, we propose a new PAPR
reduction method that we have called discrete clipping,
whose performances approach those of the best known
methods while being much more simple. This new me-
thod introduces ICI that we have analyzed by deriving an
upperbound on the Signal-to-Interference Ratio (SIR).
2 OFDM systems and PAPR
In this section, useful denitions on OFDM systems and
their PAPR are provided. The complex envelope of an
OFDM system may be written as:
x(t) =
+1
X
= 1
K=2
X
n= K=2
A
;n
g(t   (T   T
g
))e
2jnt=T
; (1)
where the A
;n
are the complex informative symbols (cho-
sen in an alphabet ofM symbols) to transmit (for example
QPSK symbols), T is the inverse of the carrier spacing
and T
g
is the guard interval duration. Let us note T
s
=
T+T
g
the whole OFDM symbol duration. Remember that
a guard interval is needed to ensure the classical OFDM
easy equalization scheme.
It can be easily shown that sampling x(t) with a period
T=N;N  K; leads to the digital baseband implementa-
tion of the OFDM modulator represented on gure 1. N
is the number of carriers, and K the number of useful car-
riers. The N   K carriers containing no information are
called virtual carriers.
The PAPR of the signal x(t) on an interval [T
s
; ( +
1)T
s
], that is on one single OFDM symbol, is:

cont;
=
max
t2[T
s
;(+1)T
s
]
jx(t)j
2
R
(+1)T
s
T
s
jx
2
(t)jdt
: (2)
A PAPR can also be dened in the discrete-time do-
main:

disc;
=
max

ja
;
j
2
Efja
;
j
2
g
; (3)
where a

= [a
;0
  a
;N 1
] is obtained by pre and post-
concatenating the vector A

=
h
A
; 
K
2
  A
;
K
2
i
with
zeros to obtain a N length vector and by taking its Inverse
Discrete Fourier Transform (IDFT). In [1], it is proved
that the dierence between the PAPR of the continuous
and discrete signals is marginal. In the sequel, we will focus
on the discrete-time denition of the PAPR. The PAPR
of one single symbol can be as high as N , that is the
maximum PAPR increases with the number of carriers.
3 State of the art
Several techniques have been proposed for reducing the
PAPR in OFDM systems. The common idea of many of
these methods is the introduction of redundancy. Intuiti-
vely, ifR
ac
bits of the OFDM symbols do not contain infor-
mation but redundancy in some way, it is obvious that the
same information is represented by 2
R
ac
dierent OFDM
symbols. Then the OFDM symbol having the smallest
PAPR can be chosen for the transmission. This conside-
ration is used in many PAPR reduction methods.
The dierent techniques can be divided in three fami-
lies, as follows.
3.1 Scaling techniques
Scaling techniques such as Block Scaling and Peak Win-
dowing [2] are used to reduce the continuous-time PAPR.
In such methods, the maximum value of the continuous-
time signal corresponding to an OFDM symbol is com-
puted and compared to one or several thresholds. If this
value exceeds one of the thresholds, the whole signal is
scaled by a predened coecient so that the ampliers
work in linearity. As an alternative, it can also be multi-
plied by a window. These methods are quite simple and
do not introduce redundancy, but in both cases, the main
drawback is that not only the peaks but the whole signal
is attenuated, resulting in an increase of the bit error rate
3.2 Coding techniques
The basic idea of these techniques is to avoid the trans-
mission of symbols that present a high PAPR, which in
fact, is equivalent to use redundant codes. Many coding
techniques have been proposed [3], [4], [5], [6], [7]. In some
of them the introduced redundancy is also interestingly
used for error detection or correction. Among all the used
codes, one may mention Rudin-Shapiro codes [4], Golay
sequences [5], M-sequences [7] and Reed-Muller codes [6].
The main drawback of all these coding schemes is that the
computational complexity which can be very high when
the number of carriers is relatively large.
3.3 Miscellaneous techniques
Some of the methods do not really belong to a speci-
c family, and are described in this section. All of them
introduce redundancy. In the Selective Mapping method
(SLM) [1], U dierent OFDM symbols representing the
same information are produced using U xed rotation vec-
tors supposed to be known at the receiver. Each OFDM
symbol PAPR is computed and the symbol having the lo-
west PAPR is selected for transmission. A variant called
Partial Transmit Sequence (PTS) [1] consists to split the
OFDM symbol in V disjoint subsets of carriers and to
choose V rotational factors to apply to the subsets. An
exhaustive search of the rotational factors that minimize
the PAPR is done. In these two methods, the introdu-
ced redundancy is in fact the information on the chosen
rotation vectors that has to be transmitted to the recei-
ver as side information. Both methods present really good
performances and work for high number of carriers. Their
main drawback is their complexity, since several IDFTs
have to be computed for each transmitted OFDM symbol.
One may also cite the selective scrambling technique where
the coecients of the OFDM symbol A

are scrambled
(using scrambling sequences) and the sequence with the
lowest PAPR is sent with the corresponding scrambling
sequence. This technique behaves nearly like the the SLM
method but due to the fact that scrambling is easier to
perform than computing DFTs, it is much easier to im-
plement.
In what follows, we present a new technique for PAPR
reduction which is relatively simple and which introduce
a small amount of redundancy.
4 Discrete Clipping
Our method is based upon the following idea: rather
than clipping or scaling the continuous time signal x(t)
like is the case for some systems, we propose to clip or
scale the discrete signal x[k], which is equivalent to clip or
scale the coecients a
;
. However, in order to reduce the
distortion of the OFDM signal, we constrain the number
of the modied coecient to be relatively small.
From eq.(3), it appears that if some coecients a
;
that
are beyond a certain threshold are scaled then the numera-
tor can be signicantly reduced. Meanwhile, the denomi-
nator remains nearly the same if the number of modied
coecient is small. Hence, clipping a small part of the
most important coecients a
;
can reduce signicantly
the PAPR.
This is the principle of the method. The complete process
is described as follows :
At the Transmitter
{ Consider one symbol A

of length K corresponding
to one time interval T
s
, add the virtual carriers to
obtain a N length vector whose IDFT is a

.
{ Compare a

to a xed threshold  and apply a
xed scaling function f to all the coecients ex-
ceeding the threshold. The new vector is denoted
~a

: 8; 80   < N ; if ja
;
j   ) ~a
;
=
a
;
else ~a
;
= f(a
;
):
{ Modulate the continuous signal x(t) with ~a

and
transmit it.
{ Transmit as side information the indices of the sca-
led coecients : R

= f=ja
;
j >  g:
At the Receiver
{ From the received signal y(t) = x(t)  c(t) (c(t) is
the channel impulse response), sample at t =  +
T and compute the vector
~
b

corresponding to the
transmitted vector ~a

:
~
b
;
= y( + T ):
{ Compute the vector b

corresponding to a

at the
transmitter using the received side information: 8 2
R

; b
;
= f
 1
(
~
b
;
):
{ Compute B

which corresponds to the vector A

at
the transmitter. A simple DFT is required : B

=
DFT(b

).
This technique oers a signicant PAPR reduction for
all values of N . Unlike coding techniques, it is very ef-
fective for large values of N and M . Performances are
comparable to those of SLM and PTS if enough scaling is
introduced. Unlike SLM and PTS which are computatio-
nally heavy, this technique is very simple to implement.
At the transmitter only one IDFT, N comparisons and
very few divisions (case of the scaling function being a
simple division) need to be computed. At the receiver, de-
modulation is even simpler. It requires one DFT, and few
multiplications (applying f
 1
on the scaled coecients).
In the discrete clipping technique, redundancy is due to
the coding of the indices of the scaled coecients. The
number of redundant bits per symbol isR
ac
= log
2

N
N

,
where  is the average percentage of scaled coecients.
Henceforth, the ratio of redundant bits is R
ac
=2N log
2
M
which is low even Redundancy is low (compared to coding
techniques) even for small alphabet sizes.
The PAPR is bounded if f is bounded. Indeed, since 8 ; 
j~a
;
j  f
max
where f
max
= max
a
jf(a)j, the PAPR is en-
sured to be within a xed limit:

max
'
f
2
max
R
f
max
0
u
2
p
jaj
(u)du
: (4)
This property is not present in both SLM and PTS which
only reduce the probability of exceeding a certain thre-
shold but never ensure that this PAPR will be under a
certain value. However, and this is the strongest drawback
of this technique, scaling introduces ISI between symbols
of dierent carriers. One of the most important advan-
tages of OFDM transmission is lost. However, as only a
few number of coecients are modied, the ISI may be
small. In fact, there is a trade o between the permitted
amount of ISI and the PAPR reduction as is shown in
the next section. Finally, note that unlike usual clipping
(where this phenomenon is also present), this method does
not cause out of band radiation.
5 ICI Analysis
The important predictable drawback of this technique
is the introduction of ISI between symbols of dierent car-
riers. This is a direct consequence of the modications ap-
plied to vectors a

. In this section we propose to measure
this ISI. Our aim is to compute the relation between the
end vectors A and B. We assume that the side informa-
tion R is safely transmitted. Using the result obtained in
classic OFDM transmission we have:
~
b = ~a 
 c : (5)
Then, assuming that f is a linear contracting function
(f(x) = x where  2]0; 1[), the scaling operation can be
written analytically as :
~a = a (1  (1  )1
R
) ; (6)
where 1 denotes the all one vector, and 1
R
denotes the
vector containing ones on the indices where scaling is per-
formed, and zeros elsewhere. The rescaling operation can
similarly be written as:
b =
~
b (1  (1=   1)1
R
) : (7)
Combining equations (5), (6) and (7), we obtain:
b = a 
 c
  (1  ) (a 1
R
) 
 c
+ (1=   1) f(a
 c)  1
R
g
  (1=   1) (1  ) f[(a 1
R
) 
 c] 1
R
g :
(8)
Finally by taking the DFT, we obtain:
B = AC + I
1
+ I
2
+ I
3
: (9)
The rst term in equation (9) corresponds to the zero-ISI
received signal. The three other terms , denoted by I
1
, I
2
and I
3
, which are the DFT of the second, third and forth
terms in (8), correspond to ISI. Using Parseval's identity
and Cauchy-Schwarz inequality we are able to bound the
energies of these signals:
kI
1
k
2
 (1  )
2
kCk
2
ka 1
R
k
2
kI
2
k
2

jRj
N
0
(1=   1)
2
kACk
2
kI
3
k
2

jRj
N
0
(1=   1)
2
(1   )
2
ka 1
R
k
2
kCk
2
:
(10)
Note that the signal energy is given by kSk
2
= kACk
2
.
Considering ISI as noise, we dene the ISI-SNR, where
only noise due to ISI is considered, by :
SNR
 1
ISI
=
kI
1
k
2
+ kI
2
k
2
+ kI
3
k
2
kSk
2
(11)
This noise-to-signal ratio can be bounded by :
SNR
 1
ISI
 (1  )
2
kCk
2
ka1
R
k
2
kACk
2
+
jRj
N
(1=   1)
2
+
jRj
N
(1=   1)
2
kCk
2
ka1
R
k
2
kACk
2
:
(12)
Note that  =
jRj
N
is the percentage of scaled coecients
and is of the order of 1%, both second and third terms of
equation (12) are kept within reasonable limits. However
the rst term is in general much larger due to the fraction
kCk
2
ka1
R
k
2
kACk
2
. However, by keeping  close to one, the total
ISI can also be kept within reasonable bounds.
Discrete clipping resorts a trade-o between PAPR re-
duction and level of ICI. The level of ICI can be adjusted
by manipulating two parameters that are :  (the scaling
factor), and  (the percentage of scaled factors). ICI in-
creases when both  and  are xed, and the number of
carriers increases (in fact the amount of ICI depends on
the number of scaled coecients), so in order to make fair
comparisons at constant ICI, we take  inversely propor-
tional to N : 1    = const=N: Figures 2 and 3 compare
the performances of discrete clipping and selective map-
ping techniques. In the rst gure, SIR = 20dB and one
sees that the performances of discrete clipping are not far
from those of SLM. In the second gure, we see that the
performances are similar to those of SLM at the expense
of an increase in ICI (SIR = 6dB).
6 Conclusions
Various techniques for reducing the PAPR in OFDM si-
gnals were reviewed. Basic scaling techniques reduce PAPR
but decrease the SNR. Coding techniques present good
reduction performances but generally introduce a lot of
redundancy. Other miscellaneous techniques such as SLM
and PTS methods provide both good reduction perfor-
mances and introduce low redundancies. However, they
are still too computationally expensive to be implemen-
ted. We have proposed a method called discrete clipping
which provides a signicant reduction of the PAPR at low
computational complexity and with a little a mount of re-
dundancy. In conclusion, the great advantage of discrete
clipping remains its simplicity. This new method allows to
get the same PAPR reduction as the best other methods,
at the expense of an ICI introduction, whose amount may
be chosen by tuning two parameters.
References
[1] S. Muller, R. Bauml, R. Fisher and J. Huber, \OFDM
with reduced peak to average power ratio by mul-
tiple signal representation," Annals of telecommuni-
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[2] R.D.J. van Nee and A. de Wild, \Reducing the peak-
to-average power ratio of OFDM," Proc. 1998 Vehi-
cular Technology Conf. (VTC'98), pp. 2072-2076.
[3] D. Wullich, \Reduction of peak to mean ratio of mul-
ticarrier modulation using cyclic coding," Electronic
Letters, vol 32, no. 5, pp. 432-433, Feb. 1996.
[4] S.J. Shepherd, P.W.J. Van Eetyelt, C. W. Wyatt-
Millington, S.K.Barton, \Simple coding scheme to re-
duce peak factor in QPSK multicarrier modulation,"
Electronic Letters, vol. 31, pp. 1131-1132, July 1995.
[5] R.D.J. van Nee, \OFDM codes for peak-to-average
power reduction and error correction," Proc. 1996
Global Telecommun. Conf. (Globecom '96), pp. 740-
744.
[6] J.A. Davis and J. Jedwab, \Peak-to-mean power
control and error correction for OFDM transmission
using Golay sequences and Reed-Muller codes," Elec-
tronic Letters, vol. 33, pp. 267-268, Feb. 1997.
[7] X. Li and J.A. Ritcey, \M-sequences for OFDM peak-
to-average power ratio reduction and error correc-
tion," Electronic Letters, vol. 33, pp.554-555, March
1997.
sampling
DAC
0
0
0
0
conversion
P=S
modulation
parallel to serial
digital to analog
converter
guard interval
insertion
A
; K=2
A
;K=2
A

x(t)
a
;0
a
;1
a
;2
a
;N T
g
a
;N 1
a

IFFT
rate T=N
Fig. 1: OFDM modulator digital baseband implementa-
tion.
4 5 6 7 8 9 10 11 12
2
3
4
5
6
7
8
9
log2(N)
P
A
P
R
smart clipping
no reduction
SLM
Fig. 2: Mean PAPR in discrete clipping for SIR = 20dB:
4 5 6 7 8 9 10 11 12
2
3
4
5
6
7
8
9
log2(N)
P
A
P
R
smart clipping
SLM
no reduction
Fig. 3: Mean PAPR in discrete clipping for SIR = 6dB:


Reducing the peak to average power ratio in OFDM systems

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