Selected mapping without side information for PAPR reduction in OFDM

Abstract

Selected mapping (SLM) is a technique used to reduce the peak-to-average power ratio (PAPR) in orthogonal frequency-division multiplexing (OFDM) systems. SLM requires the transmission of several side information bits for each data block, which results in some data rate loss. These bits must generally be channel-encoded because they are particularly critical to the error performance of the system. This increases the system complexity and transmission delay, and decreases the data rate even further. In this paper, we propose a novel SLM method for which no side information needs to be sent. By considering the example of several OFDM systems using either QPSK or 16-QAM modulation, we show that the proposed method performs very well both in terms of PAPR reduction and bit error rate at the receiver output provided that the number of subcarriers is large enough.

Full text

3320 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009
Selected Mapping without Side Information for
PAPR Reduction in OFDM
St´ephane Y. Le Goff, Samer S. Al-Samahi, Boon Kien Khoo, Charalampos C. Tsimenidis, and Bayan S. Sharif
Abstract—Selected mapping (SLM) is a technique used to
reduce the peak-to-average power ratio (PAPR) in orthogonal
frequency-division multiplexing (OFDM) systems. SLM requires
the transmission of several side information bits for each data
block, which results in some data rate loss. These bits must
generally be channel-encoded because they are particularly
critical to the error performance of the system. This increases the
system complexity and transmission delay, and decreases the data
rate even further. In this paper, we propose a novel SLM method
for which no side information needs to be sent. By considering
the example of several OFDM systems using either QPSK or 16-
QAM modulation, we show that the proposed method performs
very well both in terms of PAPR reduction and bit error rate at
the receiver output provided that the number of subcarriers is
large enough.
Index Terms—Orthogonal frequency-division multiplexing
(OFDM), peak-to-average power ratio (PAPR), selected mapping,
side information.
I. INTRODUCTION
HIGH peak-to-average power ratio (PAPR) is a well-
known drawback of orthogonal frequency-division mul-
tiplexing (OFDM) systems. Among all the techniques that
have been proposed to reduce the PAPR (see, e.g., [1] - [7]),
selected mapping (SLM) is one of the most promising ones
because it is simple to implement, introduces no distortion
in the transmitted signal, and can achieve significant PAPR
reduction [2]. The idea in SLM consists of converting the
original data block into several independent signals, and then
transmitting the signal that has the lowest PAPR. The selected
signal index, called side information index (SI index), must
also be transmitted to allow for the recovery of the data
block at the receiver side, which leads to a reduction in data
rate. This index is traditionally transmitted as a set of bits
(the SI bits). The probability of erroneous SI detection has a
significant influence on the error performance of the system
since the whole data block is lost every time the receiver does
not detect the correct SI index. In practice, a channel code
must thus be used to protect the SI bits. This further reduces
the data rate, makes the system more complex, and increases
the transmission delay.
It may therefore be worth trying to implement the SLM
method without having to explicitly send any SI bit. A few
techniques for doing so have already been proposed such as,
for example, the scrambling method described in [8] or the
maximum-likelihood decoding scheme introduced in [9]. More
Manuscript received May 3, 2007; revised August 22, 2007; accepted
November 29, 2007. The associate editor coordinating the review of this paper
and approving it for publication was X. Wang.
The authors are with the School of Electrical, Electronic and Computer
Engineering, Newcastle University, UK (e-mail: stephane.le-goff@ncl.ac.uk).
Digital Object Identifier 10.1109/TWC.2009.070463
recently, another method was proposed in [10] that combines
channel estimation using high-power pilot tones and PAPR
reduction via SLM. One of the key ideas in [10] consists of
choosing the location of the pilot tone used inside each data
sub-block depending on the SI index, and then exploit the
power disparity between this pilot tone and the data symbols
in the same sub-block in order to recover this index at the
receiver side. The technique in [10] assumes the use of several
pilot tones in each data block, which is not a suitable option for
fading channels that change slowly. The practical limitations
of the work in [10] can however be suppressed by leaving
out the pilot tones and replacing them with data symbols. By
doing so, we remove the restrictions regarding the possible
locations of the high-power subcarriers inside the data block,
and are therefore left with a problem which is much more
general than that addressed in [10]. In this paper, we propose
to address this problem by describing a novel SLM method
without side information and studying its performance in terms
of probability of erroneous SI detection, bit error rate (BER),
and PAPR reduction. In our SLM method, each SI index is
associated with a particular set of locations inside the data
block at which the modulation symbols have been extended.
In the receiver, a SI detection block attempts to determine the
locations of the extended symbols.
The paper is organized as follows: In Section II, the
proposed SLM technique is presented. Then, in Section III, we
consider an example for which we provide various computer
simulation results. Finally, conclusions are drawn in Section
IV.
II. THE PROPOSED SLM TECHNIQUE WITHOUT SIDE
INFORMATION
In this section, a notation in the form V=(vq)Qshall be
used to denote a vector Vcomposed of Qscalar quantities
vq,q∈{0,1, ..., Q −1}.
A. The proposed SLM transmitter
Consider an OFDM system using Northogonal subcarriers.
A data block is a vector X=(xn)Ncomposed of N
complex symbols xn, each of them representing a modulation
symbol transmitted over a subcarrier. In the classical SLM
technique, Xis multiplied element by element with Uvectors
Bu=(bu,n)Ncomposed of Ncomplex numbers bu,n ,u∈
{0,1, ..., U −1},defined so that |bu,n |=1,where|·| denotes
the modulus operator. Each resulting vector Xu=(xu,n)N,
where xu,n =bu,n ·xn, produces, after inverse discrete Fourier
transform, a corresponding vector X
ucomposed of Nsymbols
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x
u,q ,q∈{0,1, ..., N −1},givenby
x
u,q =1
√N
N−1

n=0
xu,n ·exp j2πnq
N.(1)
Among the Uvectors X
u, the one leading to the lowest
PAPR, denoted as X
v, is selected for transmission. To allow
for the recovery of the original vector Xat the receiver side,
one needs to transmit the index vof the selected vector. This
index, hereafter called SI index, is generally transmitted as a
set of log2(U)bits.
In this paper, we propose a novel SLM technique that
allows for the SI index vto be reliably embedded in the
transmitted vector X
vso that no additional bits need to be sent
to the receiver. In this technique, the vectors Bu=(bu,n)N,
u∈{0,1, ..., U −1}, are such that, for each Bu, the moduli
of Kelements bu,n are set to a constant C>1, called
extension factor, whereas the moduli of the other (N−K)
elements bu,n remain equal to the unit. Note that the phases
of these elements can be set to any random values as in
classical SLM. For a given vector Bu, the locations of the
elements bu,n for which |bu,n|=Cform a set Sucomposed
of Kintegers. For instance, if a vector Buis associated with
the set Su={0,7,25,37}, it means that only the complex
elements bu,n in positions n=0,7,25,and37 in Buhave a
modulus equal to C.ThesetS
uis actually representative of
the index u. To allow for SI recovery in the receiver, there
must be a one-to-one correspondence between an index uand
asetS
u. In other words, two distinct vectors Bumust NOT
be associated with identical sets. The number of distinct sets
Suthat can be generated is given by the binomial coefficient
N
K=N!
K!·(N−K)! . Since the SI index can take Uvalues, we
only need to use Usets Suamong the N
Kavailable sets. In
the next section, we will describe a simple method to select
the Usets Su.
Once all vectors Buhave been generated, our SLM method
works exactly like the classical one, i.e. the data block Xis
multiplied element by element with each Buso as to produce
Uvectors Xu=(xu,n)N, with xu,n =bu,n ·xn,aswellasU
corresponding vectors X
u. In a given vector Xu, the symbols
xu,n with n∈Suhave an average energy C2times greater
than that of the other symbols, and are hereafter referred
to as extended symbols. This disparity in average energies
between extended and non-extended symbols is what allows
the receiver to recover the SI index after transmission. Finally,
the vector X
uwith the lowest PAPR is transmitted. We recall
that, throughout this paper, this particular vector is denoted
as X
vand is associated with the vectors Xv=(xv,n)Nand
Bv=(xv,n)N. The index vrepresents the SI index to be
transmitted.
In the proposed SLM technique, the average energy per
transmitted symbol is increased when the data block Xis
multiplied by the vectors Bubecause the fact that |xu,n |=
|bu,n|·|xn|, with |bu,n|=1or C, implies that E[|xu,n |2]>
E[|xn|2],whereE[·]designates the expectation operator. Note
that any energy increase in Xutranslates into an identical
energy increase in X
u(Parseval’s theorem). This energy
increase G, expressed in decibels (dB), is given by
G=10·log10 1+δ·(C2−1),(2)
where δ=K
N. In practice, the energy increase is compensated
for by decreasing the average energy per complex symbol
xn, i.e. reducing the minimum Euclidean distance between
signal points in the constellation from which the symbols xn
are drawn. Remarkably, this does not necessarily result in
some significant error performance degradation at the receiver
output, as will be seen later.
B. The proposed SLM receiver
In this paper, we assume transmission over a quasi-
static frequency-selective Rayleigh fading channel with Z
equal-power taps. For each transmitted symbol x
v,q,q∈
{0,1, ..., N −1}, the corresponding received sample y
qis thus
given by
y
q=
Z−1

z=0
h
z·x
v,q−z+n
q,(3)
where h
zis a complex zero-mean Gaussian sample repre-
senting the fading experienced by the zth tap. In (3), n
q
denotes a complex Gaussian noise sample with zero mean
and variance σ2=N0,whereN0denotes the one-sided
power spectral density of the additive white Gaussian noise
(AWGN). We also assume that the Zfading samples h
zare
independent and perfectly known at the receiver side, i.e.
perfect channel state information (CSI) is considered. Under
these assumptions, we can show that, after discrete Fourier
transform, the received sample yncorresponding to the nth
subcarrier, n∈{0,1, ..., N −1}, is given by [11]
yn=hn·xv,n +nn,(4)
where
nn=1
√N
N−1

q=0
n
q·exp −j2πnq
N(5)
is a complex zero-mean Gaussian noise sample with variance
σ2=N0and
hn=
Z−1

z=0
h
z·exp −j2πnz
N(6)
is a complex Gaussian noise sample with zero mean and unit
variance.
The role of the SI detection block is to recover the index v
by processing both vectors Y=(yn)Nand H=(hn)N,
the latter being computed using (6). This block has the
knowledge of the Upossible vectors Buand the one-to-one
correspondence between an index uand a set Su. Under these
conditions, the optimal performance in terms of SI detection
is obtained by applying the well-known maximum-likelihood
(ML) detection algorithm: the receiver selects as SI index the
value of uthat minimizes the term
N−1

n=0 |yn−hnbu,nxn|2,u∈{0,1, ..., U −1}.(7)
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3322 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009
When 2p-ary complex symbols xnare employed, the num-
ber of terms to compute is equal to U·2pN , which leads to
a prohibitive computational complexity for practical values of
N. In fact, for each vector Buto be considered, there are 2pN
possible sequences of symbols xn,i.e.2pN possible vectors
Xu(or X). In practice, it is thus necessary to use a suboptimal
algorithm instead of ML detection. In this sub-section, we
propose an algorithm that exploits the disparity in average
energy between the Kextended symbols and the (N−K) non-
extended symbols in vector Xv=(xv,n)Nso as to recover the
SI index v. Assume a particular location n∈{0,1, ..., N −1}
in the received vector and its corresponding fading sample hn.
If the SI index was u∈{0,1, ..., U −1}, the average received
energy at location nin the absence of additive noise would
be given by
E[|hn|2|xu,n|2]=|hn|2|bu,n |2γ, (8)
where γdenotes the average energy per complex symbol xn.
At the same time, we can show that the average energy of the
received sample ynis given by
E[|yn|2]=|hn|2|bv,n|2γ+σ2.(9)
At this stage, we can introduce a metric αu,n defined as
αu,n =
E[|yn|2]−σ2−|hn|2|bu,n |2γ
.(10)
By combining (9) and (10), we can show that
αu,n =|hn|2γ|bv,n|2−|bu,n|2
.(11)
The minimal value of the metric αu,n is equal to 0 and
obtained for |bv,n|=|bu,n|. Extending this reasoning to the
whole received vector rather than focusing on a particular
location nleads us to introduce another metric βudefined
as
βu=
N−1

n=0
αu,n.(12)
The minimal value of this metric βuis also equal to 0 and
obtained when the condition |bv,n|=|bu,n |is satisfied for all
values of n,i.e.Bv=Buor equivalently u=v. This shows
that the SI index can be recovered by determining, using (10)
and (12), the value of uthat minimizes the metric βu.Itis
important to mention that, in the computation of (10), the
receiver uses the sample ynto approximate the term E[|yn|2]
as follows:
E[|yn|2]≈|yn|2.(13)
Evaluating an average by only considering a single sample
may obviously lead to a gross estimate of this average. In case
we use too many unreliable estimates of the terms E[|yn|2],
n∈{0,1, ..., N −1}, in the computation of (10), the minimal
value of the metric βumay not be obtained for u=v, thus
leading to an erroneous detection of the SI index. In order to
minimize the probability of occurence of such an error event,
we must ensure that the Uvectors Buare as different as
possible, i.e. the number of locations nin two distinct vectors
Buand Bufor which |bu,n| =|bu,n |is maximised. This
should, in most cases, prevent a metric βu,u∈{0, ..., v −
1,v+1, ..., U −1}, from satisfying the condition βu<β
v,even
if we have αu,n <α
v,n for several locations n∈{0,1, ..., N −
1}.
III. EXAMPLE
To illustrate the proposed SLM technique, we consider in
this section the example of several OFDM systems based on
either QPSK or 16-QAM modulation with Gray mapping. Our
goal is to achieve PAPR reduction using U=10vectors
Bu. All results given in this section are obtained by computer
simulations considering the frequency-selective fading channel
model previously described with Z=4taps. We also assume
the use, at the transmitter output, of a nonlinear solid-state
power amplifier (SSPA) simulated using Rapp’s model [12]
with a smoothness parameter p=3and an input backoff
(IBO) of 7 dB.
A. Construction of the vectors Bu
In this sub-section, we present a technique to construct the
set of U=10vectors Bu=(bu,n)N,u∈{0,1, ..., U −1}.
First, for a given vector Bu, the phases of the complex
elements bu,n can be chosen randomly, as already suggested in
previous works (see, e.g., [13]). The first step of the procedure
used for defining the moduli |bu,n|consists of dividing the
vectors Buinto subvectors of length M,whereMis the
smallest possible integer satifying the condition M
k≥U,
kbeing any integer smaller than M. In our example, U=10,
and therefore M=5because 5
k=10when k=2or 3.IfU
was equal to 11 instead of 10, we should have taken M=6
because there is no integer kfor which 5
k≥11. Each vector
Buis thus viewed as a succession of L=N/M subvectors,
each of them composed of Mcomplex elements. Before
moving to the next step, we also have to select a particular
value of kamong all those satisfying the condition M
k≥U.
For reasons explained later, we keep the smallest value of the
integer kfor further considerations. In our example, we thus
find M=5and k=2.
Hereafter, we denote by bm,l the (m+1)th complex
element in the (l+1)th subvector, m∈{0,1, ..., M −1},
l∈{0,1, ..., L −1}. In each subvector, the moduli of k
elements are set to a constant C>1, whereas the moduli
of the other (M−k)elements remain equal to the unit. The
locations in each subvector of the elements bm,l for which
|bm,l|=Care identical for each subvector, i.e. the moduli
|bm,l|do not depend on the index l∈{0,1, ..., L −1}.This
implies that, once the Mmoduli have been defined for the first
subvector (corresponding to l=0), this pattern of moduli is
simply repeated (L−1) times in order to obtain the moduli for
all locations n∈{M, M +1, ..., N −1}in the corresponding
vector Bu. The fact that M
k≥Uimplies that there are
enough possible permutations of kextended elements in a
M-element subvector in order to generate the Udistinct sets
Suthat are needed. As an example, Fig. 1 shows the U=10
subvectors that can be generated when M=5and k=2.
Note that Fig. 1 only shows the moduli of the elements bm,l
since their phases can basically take any random values.
The set of Uvectors Bu,u∈{0,1, ..., U −1}, obtained
using the simple procedure described above is such that, when
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C 1 1
Subvector l
u = 0 1
0 1 2 3 m = 4
C
C 1 1
u = 1 1 C
C 1 1
u = 2 1 C
C 1 1
u = 3 1 C
C 1 1
u = 4 1 C
C 1 1
u = 5 1 C
C 1 1
u = 6 1 C
C 1 1
u = 7 1 C
C 1 1
u = 8 1 C
C 1 1
u = 9 1 C
Fig. 1. List of the 10 possible subvectors that can be used to apply our
SLM technique, when M=5 and k=2. We only show the moduli of the
elements bm,l since their phases can take any random values.
considering any pair of vectors Buand Bu,u=u,the
number of locations nfor which |bu,n| =|bu,n |is at least
equal to 2N
M. In other words, the probability of erroneous SI
detection can be lowered by increasing the ratio N
M.Thisis
why it is crucial to minimize, for given values of Nand U,the
length Mof the subvectors. With the construction procedure
proposed in this sub-section, the total number of extended
symbols in the transmitted vector Xvis given by K=Nk
M.
Note that δ=K
N=k
M. Hence, in order to minimize the energy
increase given by (2), we must ensure to use the smallest
possible value of kamong all those satisfying the condition
M
k≥U.
B. Probability of SI detection error
Fig. 2 shows the probability of SI detection error, Pde,asa
function of the extension factor C, for four different numbers
of subcarriers, N= 65, 125, 255, and 510. We consider two
OFDM systems using either QPSK or 16-QAM modulation.
All results are obtained for Eb/N0=10dB, where Eb
designates the average energy per bit and N0is the one-sided
power spectral density of white Gaussian noise. The parameter
Pde represents the probability that the receiver cannot recover
the SI index v, i.e. a complete OFDM frame (vector X)is
lost. Note that, for simplicity sake, the numbers of subcarriers
are chosen to be multiples of M=5close to the usual powers
of two used in practice. Finally, the suboptimal algorithm
introduced in the previous section is used for recovering the
SI index.
From Fig. 2, it is observed that the value of Pde depends
on the extension factor C, the number Nof subcarriers, and
the modulation scheme that is employed. As Cis increased,
the performance of our suboptimal algorithm improves simply
because a higher value of Callows for a better distinction
1.E -05
1.E -04
1.E -03
1.E -02
1.E -01
1.E +00
1 1.1 1.2 1.3 1.4 1.5
Extension Factor C
Probability of SI Detection Error P
de
N = 65, QPSK N = 125, QPSK
N = 255, QPSK N = 510, QPSK
N = 65, 16- QAM N = 125, 16 -QA M
N = 255, 16 -QA M N = 510, 16 -QA M
Fig. 2. Probability of side information detection error, Pde, obtained with
the proposed SLM technique as a function of the extension factor C,for
N=65, 125, 255, and 510 subcarriers. The OFDM systems use either
QPSK or 16-QAM modulation, with Gray mapping in both cases. The system
parameters are: Eb/N0=10dB, quasi-static frequency-selective Rayleigh
fading channel with Z=4equal-power taps and perfect CSI, suboptimal SI
detection algorithm, M=5,k=2, SSPA with p=3and IBO = 7 dB.
between extended and non-extended symbols after transmis-
sion through the channel, which makes the occurence of an
erroneous detection event less likely. Increasing the number N
of subcarriers also results in a lower probability of detection
error. We recall that, if we consider two distinct vectors Bu
and Bu, the minimum number of locations nfor which
|bu,n| =|bu,n |is equal to 2N
M. Hence, any increase in the
number of subcarriers results in an increase in this number of
locations, which then reduces the probability of an erroneous
detection event.
The results in Fig. 2 also indicate that our SI detection
algorithm performs better with QPSK than with 16-QAM. This
can be explained by the fact that, in QPSK, the energy per
symbol xnis constant, i.e. we always have |xn|2=γ.Thisis
not the case for 16-QAM where the term |xn|2can take three
different values: |xn|2=γ
5,γ,or9γ
5, meaning that the term
|xn|2may significantly differ from its average value γ. Hence,
the corresponding received sample yn, given by (4), has an
energy |yn|2that tends to deviate more from its average value
E[|yn|2]in 16-QAM than it does in QPSK. As a consequence,
the approximation E[|yn|2]≈|yn|2used in the computation
of (10) tends to be more accurate in QPSK than in 16-QAM,
which makes the occurence of an erroneous detection event
less likely with QPSK than with 16-QAM.
C. Bit error rate performance
It is also important to study the error performance degrada-
tion caused by the application of the technique proposed in this
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1.E -04
1.E -03
1.E -02
1.E -01
1.E +00
0 5 10 15 20 25 30 35
Eb/N 0(dB)
BER
N=65, QPSK N=125, QPSK
N=255, QPSK N=510, QPSK
PSI, QPSK N=65,16-QAM
N = 125, 16-QAM N=255,16-QAM
N = 510, 16-QAM PSI, 16-QA M
Fig. 3. BER performance of several OFDM systems using the proposed SLM
technique, for N=65, 125, 255, and 510 subcarriers. The OFDM systems use
either QPSK or 16-QAM modulation, with Gray mapping in both cases. The
system parameters are: C=1.2, quasi-static frequency-selective Rayleigh
fading channel with Z=4equal-power taps and perfect CSI, suboptimal SI
detection algorithm, M=5,k=2, SSPA with p=3and IBO = 7 dB.
For comparison purposes, the BER plots obtained with equivalent OFDM
systems based on classical SLM with perfect side information (PSI) are also
displayed.
paper. Such degradation is potentially due to both the energy
increase Gand the occasional SI detection error events. In Fig.
3, we have plotted the BER versus Eb/N0curves obtained for
the OFDM systems based on QPSK and 16-QAM, for N=
65, 125, 255, and 510 subcarriers. For all simulations, the
extension factor value is C=1.2. For comparison purposes,
we have also plotted the BER curves obtained with equivalent
OFDM systems using classical SLM with perfect, i.e. error-
free, side information (SLM-PSI). The latter scenario typically
implies the use of a channel code entirely dedicated to the
protection of the SI bits during transmission, which is probably
more an ideal situation than a practical one.
Fig. 3 shows that the performance gap between our SLM
technique and SLM-PSI can be made very small. When QPSK
is employed, this gap is actually only significant for low SNRs
and small numbers Nof subcarriers, i.e. in situations where
the probability of erroneous SI detection is sufficiently high
to have an impact on the overall BER. Some unpublished
simulation results indicated that, when using QPSK and a
small number of subcarriers, the performance gap at low SNRs
can be reduced by increasing the value of the extension factor
C. When QPSK is replaced by 16-QAM, the performance
difference between our method and SLM-PSI becomes sig-
nificantly larger due to the increased probability of erroneous
SI detection. However, as Nis increased, the gap diminishes
to the extent of actually becoming negligible for N= 510
subcarriers.
As a conclusion, if the probability of erroneous SI detection
can be made small enough (e.g., by increasing the SNR
and/or the number of subcarriers), the performance difference
between our SLM method and SLM-PSI is actually marginal.
This is apriorirather surprising given the fact that the
energy increase per symbol obtained with our SLM method
is G=0.70 dB in this example (for which C=1.2and
δ=k
M=2
5). At first glance, we would therefore expect to
have a performance difference of G=0.70 dB in the absence
of any erroneous SI detection event.
Hereafter, we propose to clarify this result by considering,
for simplicity sake and without loss of generality, the example
of BPSK and ignoring the presence of a nonlinear SSPA at
the transmitter output. As previously mentioned, throughout
our work, we actually compensate for the increase in average
energy per transmitted symbol xv,n (or equivalently x
v,q)by
decreasing the average energy per symbol xn,i.e.making
the constellation more compact. This results in a reduced
minimum Euclidean distance between signal points, and thus
some error performance degradation at the receiver output.
In the case of BPSK, the symbols xn∈{±
√Eb}are
simply replaced by the symbols xn∈{±
Eb/ω},where
ω=1+δ·(C2−1). It is easy to show that extending K
symbols by a factor Cand leaving (N−K) of them unchanged
in a frame of NBPSK symbols xn∈{±
Eb/ω}leads to
an average energy per symbol equal to Eb, and thus identical
to the energy per symbol obtained using classical SLM with
xn∈{±
√Eb}.
If the transmission channel over each BPSK subcarrier can
be modeled as AWGN, the bit error probability peb obtained
with the proposed SLM method in the absence of any SI
detection error can then be expressed as
peb =1−δ
2·erfc 1
ω·Eb
N0+δ
2·erfc ⎛
⎝C2
ω·Eb
N0⎞
⎠.
(14)
This equation indicates that, at high SNR, the performance
gap over AWGN channel between our SLM method and SLM-
PSI is equal to 10 ·log10ωdB, which corresponds exactly
to the energy increase Ggiven by (2). However, as shown by
(4), the transmission channel over each subcarrier can actually
be modeled more accurately as flat Rayleigh fading rather
than AWGN. Based on (14), we can demonstrate that the bit
error probability for BPSK over flat Rayleigh fading channel
is given by [14]
peb ≈1−δ
2ω·+∞
0
erfc √x·exp −x
ω·dx
+δ
2ω·C2·+∞
0
erfc √x·exp −x
ω·C2·dx,
(15)
where ω=1
ω·Eb
N0. This equation has the simple closed-form
expression
peb =1
2·1−(1 −δ)·1+ 1
ω−1
2
−δ·1+ 1
C2·ω−1
2,
(16)
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009 3325
1.E -04
1.E -03
1.E -02
1.E -01
1.E +00
5678 1211109
PAPR0(dB )
Pr[PAPR > PAPR0]
N = 65 - Classical SLM N=65-NewSLM
N=65-NoPAPR reduct. N=125-ClassicalSLM
N=125-NewSLM N=125-NoPAPRreduct.
N = 2 55 - Cl as si cal S LM N=255-NewSLM
N=255-NoPAPRreduct. N=510-ClassicalSLM
N=510-NewSLM N=510-NoPAPRreduct.
Fig. 4. CCDF of the PAPR obtained with the proposed SLM technique (New
SLM), for N=65, 125, 255, and 510 subcarriers. The OFDM systems use 16-
QAM modulation with Gray mapping. The system parameters are: C=1.2,
M=5,k=2, oversampling factor of 4. For comparison purposes, we
also display for each value of Nthe plots obtained with classical SLM as
well as those associated with an equivalent OFDM system without any PAPR
reduction.
which can be approximated at high SNR (ω→+∞)using
peb ≈1+δ·(C2−1)·C2−δ·(C2−1)
4C2·Eb
N0−1
.
(17)
Using (17), it is possible to evaluate the expected error
performance gap at high SNR between our SLM technique and
SLM-PSI in the absence of any erroneous SI detection event.
Note that the bit error probability expression for SLM-PSI is
obtained by computing (17) with C=1and δ=0.When
C=1.2and δ=2
5,wefind a gap of approximately 0.14 dB,
which is indeed significantly smaller than the energy increase
G=0.70 dB. This theoretical result is therefore coherent with
the observations made from the BER plots in Fig. 3.
D. PAPR reduction performance
Finally, Fig. 4 shows the complementary cumulative dis-
tribution function (CCDF) of the PAPR obtained with the
proposed SLM technique when using 16-QAM modulation,
for C=1.2and N= 65, 125, 255, and 510 subcarriers.
The plots corresponding to the ordinary OFDM without PAPR
reduction and the classical SLM method are also depicted for
comparison purposes. These results are obtained by using an
oversampling factor equal to 4 [15]. We observe that, for all
configurations, the performance of our SLM method in terms
of PAPR reduction is identical to that of classical SLM. Note
that the results obtained with the proposed SLM method take
into account the increase in average energy given by (2).
IV. CONCLUSION
We have proposed a simple SLM technique for PAPR reduc-
tion in OFDM that does not require the explicit transmission of
SI bits. Our investigations, performed by considering OFDM
schemes based on QPSK and 16-QAM modulations, have
shown that this technique is particularly attractive for systems
using a large number of subcarriers. In fact, the probability
of SI detection error can be made very small by increasing
the extension factor and/or the number of subcarriers. In
cases where this probability becomes sufficiently low, the
BER performance difference between the proposed technique
and classical SLM using error-free side information has been
shown to be negligible. In addition, the application of our
SLM technique leads to a reduction in PAPR which is iden-
tical to that obtained with classical SLM, for any number
of subcarriers. For OFDM systems with a large number of
subcarriers, the only significant price to pay for employing
the proposed technique instead of classical SLM is a slight
complexity increase at the receiver side due to the use of a SI
detection block.
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