IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008
257
An Overview: Peak-to-Average Power Ratio
Reduction Techniques for OFDM Signals
Tao Jiang, Member, IEEE, and Yiyan Wu, Fellow, IEEE
Abstract—One of the challenging issues for Orthogonal
Frequency Division Multiplexing (OFDM) system is its high
Peak-to-Average Power Ratio (PAPR). In this paper, we review
and analysis different OFDM PAPR reduction techniques, based
on computational complexity, bandwidth expansion, spectral
spillage and performance. We also discuss some methods of PAPR
reduction for multiuser OFDM broadband communication systems.
Index Terms—Complementary cumulative distribution function
(CCDF), high power amplifier (HPA), multiuser OFDM, OFDM,
peak-to-average power ratio (PAPR).
I. INTRODUCTION
A
S AN attractive technology for wireless communications,
Orthogonal Frequency Division Multiplexing (OFDM),
which is one of multi-carrier modulation (MCM) techniques,
offers a considerable high spectral efficiency, multipath delay
spread tolerance, immunity to the frequency selective fading
channels and power efficiency [1], [2]. As a result, OFDM has
been chosen for high data rate communications and has been
widely deployed in many wireless communication standards
such as Digital Video Broadcasting (DVB) and based mobile
worldwide interoperability for microwave access (mobile
WiMAX) based on OFDM access technology [3].
However, still some challenging issues remain unresolved in
the design of the OFDM systems. One of the major problems
is high Peak-to-Average Power Ratio (PAPR) of transmitted
OFDM signals. Therefore, the OFDM receiver’s detection efficiency is very sensitive to the nonlinear devices used in its signal
processing loop, such as Digital-to-Analog Converter (DAC)
and High Power Amplifier (HPA), which may severely impair
system performance due to induced spectral regrowth and detection efficiency degradation. For example, most radio systems
employ the HPA in the transmitter to obtain sufficient transmits
power and the HPA is usually operated at or near the saturation region to achieve the maximum output power efficiency,
and thus the memory-less nonlinear distortion due to high PAPR
of the input signals will be introduced into the communication
channels. If the HPA is not operated in linear region with large
power back-off, it is impossible to keep the out-of-band power
Manuscript received August 13, 2007; revised December 3, 2007.
T. Jiang is with the Department of Electronic and Information Engineering,
Huazhong University of Science and Technology, Wuhan 430074, P. R. China
(e-mail: Tao.Jiang@ieee.org).
Y. Wu is with the Communications Research Center, Ottawa, ON K2H 8S2,
Canada (e-mail: yiyan.wu@crc.ca).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TBC.2008.915770
below the specified limits. This situation leads to very inefficient
amplification and expensive transmitters.
Therefore, it is important and necessary to research on the
characteristics of the PAPR including its distribution and reduction in OFDM systems, in order to utilize the technical features
of the OFDM.
As one of characteristics of the PAPR, the distribution
of PAPR, which bears stochastic characteristics in OFDM
systems, often can be expressed in terms of Complementary
Cumulative Distribution Function (CCDF). Recently, some
researchers have reported on determination of the PAPR distribution based on different theoretics and hypotheses [4]–[10].
Moreover, various approaches also have been proposed to
reduce the PAPR including clipping [11]–[14], coding schemes
[15]–[21], phase optimization [22], [23], nonlinear companding
transforms [24]–[29], Tone Reservation (TR) and Tone Injection (TI) [30], [31], constellation shaping [32]–[34], Partial
Transmission Sequence (PTS) and Selective Mapping (SLM)
[35]–[51] and other techniques such as pre-scrambles proposed
in [52]. These schemes can mainly be categorized into signal
scrambling techniques, such as block codes and PTS etc., and
signal distortion techniques such as clipping.
Although some techniques of PAPR reduction have been
summarized in [53], it is still indeed needed to give a comprehensive review including some motivations of PAPR reductions,
such as power saving, and to compare some typical methods
of PAPR reduction through theoretical analysis and simulation
results directly. An effective PAPR reduction technique should
be given the best tradeoff between the capacity of PAPR reduction and transmission power, data rate loss, implementation
complexity and Bit-Error-Ratio (BER) performance etc.
In this paper, firstly we investigate the distribution of PAPR
based on the characteristics of the OFDM signals. Then, we analyze five typical techniques of PAPR reduction and propose the
criteria of PAPR reduction in OFDM systems in details. Finally,
we briefly discuss the issue of PAPR in some broadband communication systems correlative with OFDM technology, such as
multiuser OFDM systems.
II. CHARACTERISTICS OF OFDM SIGNALS
Let a block of symbols
is formed with each symbol modulating one of a set of subcarriers
, where is the number of subcarriers. The
subcarriers are chosen to be orthogonal, that
is,
, where
and is the original
symbol period. Therefore, the complex envelope of the transmitted OFDM signals can be written as
0018-9316/$25.00 © 2008 IEEE
(1)
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Fig. 1. Distribution of PAPR of OFDM signal samples oversampled by different L.
where
.
Suppose that the input data streams is statistically independent and identically distributed (i.i.d.), i.e. the real part
and imaginary part
are uncorrelated and
orthogonal. Therefore, based on the central limit theorem, when
is considerably large, the distribution of both
and
approaches Gaussian distribution with zero mean
and variance
, where
is the expected value of [27]. In other words, OFDM
signals with large become Gaussian distributed with Probability Density Function (PDF) as [25]
(2)
.
where is the variance of
Moreover, the Rayleigh nature of original OFDM signals’
amplitude can be gotten and its PDF can be expressed as [30]
is the average power of
and it can be comwhere
puted in the frequency domain because Inverse Fast Fourier
Transform (IFFT) is a (scaled) unitary transformation.
2) Discrete-time PAPR
The PAPR of the discrete time sequences typically determines the complexity of the digital circuitry in terms of
the number of bits necessary to achieve a desired signal
to quantization noise for both the digital operation and
the DAC. However, we are often more concerned with reducing the PAPR of the continuous-time signals in practice, since the cost and power dissipation of the analog
components often dominate.
To better approximate the PAPR of continuous-time
OFDM signals, the OFDM signals samples are obtained
by times oversampling. -times oversampled time-domain samples are
-point IFFT of the data block with
zero-padding. Therefore, the oversampled IFFT
output can be expressed as
(3)
where
(5)
is the amplitude of OFDM signals.
III. DEFINITION OF PAPR
A. Baseband PAPR
1) Continuous-time PAPR
In general, the PAPR of OFDM signals
is defined as
the ratio between the maximum instantaneous power and
its average power
Fig. 1 shows the distribution of the PAPR of the OFDM
and
. As shown, the
signals with
largest PAPR increase happens from
to
.
However, the PAPR does not increase significantly after
. It has shown that
is sufficient to get accurate
PAPR results [30]. The PAPR computed from the -times
oversampled time domain OFDM signal samples can be
defined as
(6)
(4)
where
denotes the expectation operator.
JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS
B. Passband PAPR
is large, an OFDM system usually does not
Note that, if
employ pulse shaping, since the power spectral density of the
band-limited OFDM signal is approximately rectangular. Thus,
the amplitude of OFDM RF signals can be expressed as
(7)
. Therefore, the
where is the carrier frequency and
peak of RF signals is equivalent to that of the complex baseband
signals.
Moreover, the average power of the passband signal is
259
Although, a high precision DAC supports high PAPR with a
reasonable amount of quantization noise, but it might be very
expensive for a given sampling rate of the system. Whereas,
a low-precision DAC would be cheaper, but its quantization
noise will be significant, and as a result it reduces the signal
Signal-to-Noise Ratio (SNR) when the dynamic range of DAC
is increased to support high PAPR. Furthermore, OFDM signals show Gaussian distribution for large number of subcarriers,
which means the peak signal quite rarely occur and uniform
quantization by the ADCs is not desirable. If clipped, it will introduce in band distortion and out-of-band radiation (adjacent
channel interference) into the communication systems.
Therefore, the best solution is to reduce the PAPR before
OFDM signals are transmitted into nonlinear HPA and DAC.
B. Power Saving
Therefore, the passband PAPR is approximately twice the
baseband PAPR, i.e.
When a HPA have a high dynamic range, it exhibits poor
power efficiency. It has been shown that PAPR reduction can
significantly save the power, in which the net power saving is
directly proportional to the desired average output power and it
is highly dependent upon the clipping probability level [54].
Suppose that an ideal linear model for HPA, where linear amplification is achieved up to the saturation point, and thus we
obtain
(9)
(10)
(8)
In this paper, we only consider the PAPR of the baseband
OFDM signals.
IV. MOTIVATION OF PAPR REDUCTION
A. Nonlinear Characteristics of HPA and ADC
Most radio systems employ the HPA in the transmitter
to obtain sufficient transmission power. For the proposed of
achieving the maximum output power efficiency, the HPA is
usually operated at or near the saturation region. Moreover,
the nonlinear characteristic of the HPA is very sensitive to the
variation in signal amplitudes.
However, the variation of OFDM signal amplitudes is
very wide with high PAPR. Therefore, HPA will introduce
inter-modulation between the different subcarriers and introduce additional interference into the systems due to high
PAPR of OFDM signals. This additional interference leads
to an increase in BER. In order to lessen the signal distortion
and keep a low BER, it requires a linear work in its linear
amplifier region with a large dynamic range. However, this
linear amplifier has poor efficiency and is so expensive. Power
efficiency is very necessary in wireless communication as it
provides adequate area coverage, saves power consumption
and allows small size terminals etc. It is therefore important to
aim at a power efficient operation of the non-linear HPA with
low back-off values and try to provide possible solutions to the
interference problem brought about. Hence, a better solution is
to try to prevent the occurrence of such interference by reducing
the PAPR of the transmitted signal with some manipulations of
the OFDM signal itself.
Large PAPR also demands the DAC with enough dynamic
range to accommodate the large peaks of the OFDM signals.
where
is the HPA efficiency and it is defined as
, where
is the average of the output
is the constant amount of power regardless of
power and
the input power.
To illustrate the power inefficiency of a HPA in terms of the
PAPR, we give an example of OFDM signals with 256 subcarriers and its CCDF has been shown in Fig. 1. In order to guarantee that probability of the clipped OFDM frames is less than
0.01%, we need to apply an input backoff (IBO) equivalent to
probability level, i.e.
the PAPR at the
( 25.235), referring to Fig. 1, and thus the efficiency of HPA be. Therefore, so low efficiency
comes
is a strong motivation to reduce the PAPR in OFDM systems.
V. DISTRIBUTION OF THE PAPR IN OFDM SYSTEMS
It is known that the CCDF of PAPR can be used to estimate
the bounds for the minimum number of redundancy bits required to identify the PAPR sequences and evaluate the performance of any PAPR reduction schemes. We can also determine
a proper output back-off of HPA to minimize the total degradation according to CCDF. Moreover, we can directly apply distribution of PAPR to calculate the BER and estimate achievable
information rates. In practice, we usually adjust these design
parameters jointly according to simulation results. Therefore, if
we can use an analytical expression to accurately calculate the
PAPR distribution for OFDM systems, it can greatly simplify
the system design process. Therefore, it is of great importance
to accurately identify PAPR distribution in OFDM systems.
Recently, some upper and lower bounds of the PAPR, which
is based on the Rayleigh distribution and Nyquist sampling rate,
have been derived. In the OFDM system with M-Phase-ShiftKeying (MPSK) modulation, signal constellation has the same
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IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008
amplitude level, and thus the power of each subcarrier is constant. Therefore, the PAPR of an MPSK-OFDM signal can be
expressed as [30]
In this case, the expression of the PAPR CCDF, as shown in [6],
can be simplified as
(15)
(11)
However, for the OFDM system with square M-Quadrature
Amplitude Modulation (MQAM), signal constellation has
varying signal power levels over different constellation points.
When all the subcarriers have the same phase, the maximum of
PAPR occurs. Therefore, according to the conclusion of [55],
the upper bound of PAPR in MQAM-OFDM systems can be
derived out
(12)
For a relatively large , the lower and upper bounds of the
distribution of the PAPR have been proposed in [7], which
were developed based on the previous works in conjunction
with some approximations and parameters obtained through
simulations. In [8], some bounds analysis has also been developed for both independent and dependent subcarriers in
OFDM systems. For independent subcarriers, a generic path for
bounding practical constellations was used and discussed. For
dependent subcarriers, some theoretical bounds of distributions
of the PAPR have been obtained in terms of the Euclidian
distance distributions, in which the focus was mainly on binary
codes, such as Bose-Chaudhuri-Hocquenghem (BCH) codes.
However, the lower and upper bounds can offer little help in
characterizing the distribution of the PAPR in practical OFDM
systems. In fact, the accurate statistical distribution of the PAPR
for generic OFDM system is what we want.
When the number of the subcarriers is relatively small, the
CCDF expression of the PAPR of OFDM signals can be written
as [4]
(13)
However, (13) does not fit well in OFDM systems with a very
large [4]. In [5], an empirical approximation expression of the
CCDF of the PAPR in OFDM systems has been given as
(14)
It should be noted that (14) lacks theoretical justification and
also yields some discrepancies with the simulation results for
large , which has been proven in [6].
In [6], an analytical PAPR CCDF expression has been developed, which is based on the level-crossing rate approximation of the peak distribution along with the exact distribution,
since the envelope of an OFDM signal can always be considered as an asymptotically Gaussian process in a band-limited
OFDM system. In fact, the theoretical results obtained in [6]
were based on the conditional probability of the peak distribution of the OFDM signals when the reference level is given.
When the constraint provides a lower bound of
, the
effect on the accuracy of the PAPR distribution can be numerically evaluated. Indeed, for high , the conditional probability
that the peak of the OFDM signals exceeds may be very small.
The approximation of (15) can be made relatively accurate
for a relatively large number of subcarriers by appropriately
adjusting the reference level of the PAPR. If the range of the
PAPR of interest is great, the distribution can be further simplified without loss of the accuracy. In [6], it also has been shown
that the statistical distribution of the PAPR of the OFDM signals
is not so sensitive to the increase of the number of subcarriers.
In coded OFDM systems, it has been proven that the complex envelope of the coded OFDM signals can converge weakly
to a Gaussian random process if the number of subcarriers goes
to infinity [9]. In [9], a simple approximation of the CCDF
of PAPR has been developed by employing the extreme value
theory, and the expression can be written as
(16)
However, all the above mentioned the expressions of the
CCDF have not been considered power distribution strategy in
OFDM systems. Similarly, with the help of the extreme value
theory for Chi-squared-2 process, a more accurate analytical
expression of the CCDF of PAPR for adaptive OFDM systems
with unequal power allocation to subcarriers has been derived
in [10]
(17)
where
if the subcarrier at DC is inactive, othif the subcarrier at DC is active,
erwise
in which
denotes the number of active subcarriers (cardenotes the number of inactive
rying information) and
subcarriers (idle). If the subcarrier at DC is nonzero, it is active
subcarrier; otherwise it is inactive subcarrier. Thus, the number
and
of the subcarriers should be equal to the sum of
.
denotes the transmission power allocated to the
-th subcarrier.
VI. PAPR REDUCTION TECHNIQUES IN OFDM SYSTEMS
In this section, we mainly discuss five typical techniques for
PAPR reduction in OFDM systems.
A. Clipping and Filtering
The simplest and most widely used technique of PAPR reduction is to basically clip the parts of the signals that are outside
the allowed region [11]. For example, using HPA with saturation
level below the signal span will automatically cause the signal
to be clipped. For amplitude clipping, that is
(18)
where
is preset clipping level and it is a positive real number.
JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS
Generally, clipping is performed at the transmitter. However,
the receiver need to estimate the clipping that has occurred and
to compensate the received OFDM symbol accordingly. Typically, at most one clipping occurs per OFDM symbol, and thus
the receiver has to estimate two parameters: location and size of
the clip. However, it is difficult to get these information. Therefore, clipping method introduces both in band distortion and
out of band radiation into OFDM signals, which degrades the
system performance including BER and spectral efficiency.
Filtering can reduce out of band radiation after clipping although it can not reduce in-band distortion. However, clipping
may cause some peak regrowth so that the signal after clipping
and filtering will exceed the clipping level at some points. To reduce peak regrowth, a repeated clipping-and-filtering operation
can be used to obtain a desirable PAPR at a cost of computational complexity increase.
As improved clipping methods, peak windowing schemes attempt to minimize the out of band radiation by using narrowband windows such as Gaussian window to attenuate peak signals.
B. Coding Schemes
When signals are added with the same phase, they produce
a peak power, which is times the average power. Of course,
not all code words result in a bad PAPR. Therefore, the good
PAPR reduction can be obtain when some measures are taken
to reduce the occurrence probability of the same phase of the
signals, which is the key idea of the coding schemes.
A simple block coding scheme was introduced by Jones et
al.[15], and its basic idea is that mapping 3 bits data into 4 bits
codeword by adding a Simple Odd Parity Code (SOBC) at the
last bit across the channels. The main disadvantage of SOBC
method is that it can reduce PAPR for a 4-bit codeword. Later,
Wulich applied the Cyclic Coding (CC) to reduce the PAPR
[56]. In 1998, Fragiacomo proposed an efficient Simple Block
Code (SBC) to reduce the PAPR of OFDM signals [57]. However, it is concluded that SBC is not effective when the frame
size is large. Subsequently, Complement Block Coding (CBC)
and Modified Complement Block Coding (MCBC) schemes
were proposed to reduce the PAPR without the restriction of
frame size [20], [58]. CBC and MCBC are more attractive due
to their flexibility on choosing the coding rate, frame size and
low implementation complexity. CBC and MCBC utilize the
complementary bits that are added to the original information
bits to reduce the probability of the peak signals occurrence.
To make comparisons, some results of the PAPR reduction
obtained with different coding schemes have been shown in
Table I, in which the number of subblock is 2 and the coding
for MCBC. About 3-dB PAPR reduction can be
rate
obtained when coding rate
by using CBC
with long frame size. It is also shown that the PAPR reductions
obtained with CBC when coding rate
are
almost the same as that when
. In addition,
when coding rate is 3/4, more than 3-dB more PAPR reduction
can be obtained using MCBC than the other schemes with any
frame size. The flexibility in coding rate choice and low complexity makes the proposed CBC and MCBC schemes attractive
for OFDM systems with large frame sizes and high coding rates.
261
TABLE I
PAPR REDUCTION COMPARISON WITH DIFFERENT CODING SCHEMES
In [59], [60], [62], authors used the Golay complementary
sequences to achieve the PAPR reduction, in which more than
3-dB PAPR reduction had been obtained. Codes with error correcting capabilities has been proposed in [61] to achieve more
lower PAPR for OFDM signals by determining the relationship
of the cosets of Reed-Muller codes to Golay complementary sequences. While these block codes reduce PAPR, they also reduce the transmission rate, significantly for OFDM systems with
large number of subcarriers.
be a code defined over an equal energy conIn fact, let
stellation, denotes the rate and denotes the length of the
, respectively, then
has
possible codewords. Therefore, it is possible to compute the codewords with large PAPR
by trying all the codewords of and computing the peaks of
the corresponding signals at some selected time points. However, it is little hope for computing the PAPR of an arbitrary
code when is large. Even if it is possible, the complexity is
still too high. Based on this motivates, authors of [22] proposed
a novel method of computation and reduction of the PAPR and
it mainly introduced a specific phase shift to each coordinate of
all possible codewords where phase shifts are independent of the
codewords and known both to transceiver, then it can be freely
obtained more 4.5-dB PAPR reduction by using the optimized
phase shifts. From this viewpoint, we also consider the coding
scheme of PAPR reduction as a special phase optimization.
In summarization, the inherent error control capability
and simplicity of implementation make coding method more
promising for practical OFDM systems design. However, the
main disadvantage of this method is the good performance of
the PAPR reduction at the cost of coding rate loss.
C. PTS and SLM
In a typical OFDM system with PTS approach to reduce the
PAPR, the input data block in is partitioned into
disjoint
subblocks, which are represented by the vectors
[35] as shown in Fig. 2. Therefore, we can get
(19)
with
or 0
. In general, for PTS scheme, the known
subblock partitioning methods can be classified into three categories [35]: adjacent partition, interleaved partition and pseudowhere
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IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008
Fig. 2. Block diagram of PTS technique.
Fig. 3. Block diagram of SLM technique.
random partition. Then, the subblocks
are transformed
into
time-domain partial transmit sequences
(20)
These partial sequences are independently rotated by phase
factors
. The objective
is to optimally combine the
subblocks to obtain the timedomain OFDM signals with the lowest PAPR
(21)
Therefore, there are two important issues should be solved in
PTS: high computational complexity for searching the optimal
phase factors and the overhead of the optimal phase factors as
side information needed to be transmitted to receiver for the
correct decoding of the transmitted bit sequence.
Suppose that there are phase angles to be allowed, thus
can has the possibility of
different values. Therefore, there
are
alternative representations for an OFDM symbol. To
reduce the searching complexity and avoid/reduce the usage of
side information, many extensions of PTS have been developed
recently [63]–[67]. In [66], authors proposed a novel scheme,
which is based on a nonlinear optimization approach named as
simulated annealing, to search the optimal combination
of
phase factors with low complexity. In general, PTS needs
IFFT operations for each data block, and the number of the re, where
denotes
quired side information bits is
the smallest integer that does not exceed .
Similarly, in SLM, the input data sequences are multiplied
by each of the phase sequences to generate alternative input
symbol sequences. Each of these alternative input data sequences is made the IFFT operation, and then the one with the
lowest PAPR is selected for transmission [51]. A block diagram
of the SLM technique is depicted in Fig. 3. Each data block
different phase factors, each of length ,
is multiplied by
, resulting
in different data blocks. Thus, the vth phase sequence after
multiplied is
. Therefore, OFDM signals becomes as
(22)
where
,
.
Among the data blocks
, only one
with the lowest PAPR is selected for transmission and the corresponding selected phase factors
also should be transmitted
to receiver as side information. For implementation of SLM
OFDM systems, the SLM technique needs
IFFT operation
and the number of required bits as side information is
for each data block. Therefore, the ability of PAPR reduction in
SLM depends on the number of phase factors and the design
of the phase factors. Some extension of SLM also have been
proposed to reduce the computational complexity and number
of the bits for side information transmission [36]. For example,
JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS
263
Fig. 4. The spectrums of original OFDM signals and companded signals.
an SLM scheme without explicit side information was proposed
in [68].
Although PTS and SLM are important probabilistic schemes
for PAPR reduction, it was already known that SLM can produce
multiple time domain OFDM signals that are asymptotically
independent, whereas the alternative OFDM signals generated
by PTS are interdependent. PTS divides the frequency vector
into some subblocks before applying the phase transformation.
Therefore, some of the complexity of the serval full IFFT operations can be avoided in PTS, so that it is more advantageous
than SLM if the amount of computational complexity is limited
[69]. Also it is demonstrated that the PAPR reduction in PTS
performs better than that of SLM. However, the required bits of
the side information in PTS is larger than that of SLM.
D. Nonlinear Companding Transforms
One of the most attractive schemes is nonlinear companding
transform due to its good system performance including PAPR
reduction and BER, low implementation complexity and no
bandwidth expansion.
The first nonlinear companding transform is the -law companding, which is based on the speech processing algorithm
-law, and it has shown better performance than that of clipping method [24]. -law mainly focuses on enlarging signals
with small amplitude and keeping peak signals unchanged, and
thus it increase the average power of the transmitted signals and
possibly results in exceeding the saturation region of HPA to
make the system performance worse.
In fact, the nonlinear companding transform is also an especial clipping scheme. The differences between the clipping and
nonlinear companding transform can be summarized as: 1) Clipping method deliberately clips large signals when the amplitude
of the original OFDM signals is larger than the given threshold,
and thus the clipped signals can not be recovered at the receiver. However, nonlinear companding transforms compand
original OFDM signals using the strict monotone increasing
function. Therefore, the companded signals at the transmitter
can be recovered correctly through the corresponding inversion
of the nonlinear transform function at the receiver; 2) Nonlinear
companding transforms enlarge the small signals while compressing the large signals to increase the immunity of small
signals from noise, whereas clipping method does not change
the small signals. Therefore, clipping method suffers from three
major problems: in-band distortion, out-of-band radiation and
peak regrowth after digital analog conversion. As a result, the
system performance degradation due to the clipping may not
be optimistic. However, nonlinear companding transforms can
operate well with good BER performance while keeping good
PAPR reduction [70].
The design criteria of nonlinear companding transform has
also been given in [70]. Since the distribution of the original
OFDM signals has been known, such as Rayleigh distribution
of the OFDM amplitudes written in (3), we can obtain the nonlinear companding transform function through theoretical analysis and derivation according to the desirable distribution of the
companded OFDM signals. For example, we transform the amplitude of the original OFDM signals into the desirable distribution with its PDF
,
. Therefore,
the nonlinear transform function can be derived as
(23)
Obviously, this nonlinear companding transform of (23) belongs to the exponential companding scheme. Based on this
design criteria, two types of nonlinear companding transform,
which are based on error function and exponential function, respectively, have been proposed in [25], [27].
It is well-known that original OFDM signals have a very
sharp, rectangular-like power spectrum as shown in Fig. 4. This
good property will be affected by the PAPR reduction schemes,
e.g. slower spectrum roll-off, more spectrum side-lobes, and
higher adjacent channel interference. Many PAPR reduction
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IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008
Fig. 5. Block diagram of TR/TI approaches for PAPR reduction.
schemes cause spectrum side-lobes generation, but the nonlinear companding transforms cause less spectrum side-lobes.
As seen in Fig. 4, error and exponential companding transforms have much less impact on the original power spectrum
comparing to the -law companding scheme. It is the major
reason that the error and exponential companding schemes not
only enlarge the small amplitude signals but also compress the
large amplitude signals, while maintain the average power unchanged by properly choosing parameters, which can increase
the immunity of small amplitude signals from noise. However,
the -law companding transform increases the average power
level and therefore requires a larger linear operation region in
HPA.
Nonlinear companding transform is a type of nonlinear
process that may lead to significant distortion and performance
loss by companding noise. Companding noise can be defined
that the noises are caused by the peak regrowth after DAC to
generate in-band distortion and out-band noise, by the excessive
channel noises magnified after inverse nonlinear companding
transform etc. For out-of-band noise, it needs to be filtered and
oversampled. For in-band distortion and channel noises magnified, they need to iterative estimation. Unlike Additive White
Gaussian Noise (AWGN), companding noise is generated by
a process known and that can be recreated at the receiver, and
subsequently be removed. In [28], the framework of an iterative
receiver has been proposed to eliminate commanding noise for
companded and filtered OFDM system.
E. TR and TI
TR and TI are two efficient techniques to reduce the PAPR of
OFDM signals [30]. Fig. 5 describes the block diagram of TR
and TI, in which the key idea is that both transmitter and receiver
reserve a subset of tones for generating PAPR reduction signals
. Note that these tones are not used for data transmission.
In TR, the objective is to find the time domain signal to be
added to the original time domain signal to reduce the PAPR.
Let
denote complex symbols for
tone reservation at reserved tones. Thus, the data vector changes
to
after tone reservation processing, and this results in a
new modulated OFDM signals as
(24)
where
. Therefore, the main aim of the TR is
to find out the proper to make the vector with low PAPR.
To find the value of , we must solve a convex optimization problem that can easily be cast as a linear programming
problem.
Similarly, TI also uses an additive correction to optimize in
(24). The basic idea of TI is to extend the constellation and thus
the same data point corresponds to multiple possible constellation points. One option is to replicate the original shaded constellation into serval alternative ones. Therefore, is a translation vector such that
. Note that TI needs not
require the extra side information and the receiver only needs
to know how to map the redundant constellations on the original one. An alternative strategy is to move the constellation
points by applying an FFT on the clipped time signals, and the
same operations are repeated till all the constellation points are
within specified boundaries and the PAPR specification of the
time signal is satisfied [71]. Some modifications of TI have been
proposed to obtain good performance including PAPR reduction
and low complexity [72].
The TI technique is more problematic than the TR technique
since the injected signal occupies the frequency band as the information bearing signals. Moreover, the alternative constellation points in TI technique have an increased energy and the
implementation complexity increases for the computation the
optimal translation vector.
VII. CRITERIA OF THE PAPR REDUCTION IN OFDM SYSTEMS
As above analyzed, we find most of existing solutions still
have some drawbacks and the obvious one is the trade-off between PAPR reduction and some factors such as bandwidth. The
criteria of the PAPR reduction is to find the approach that it can
reduce PAPR largely and at the same time it can keep the good
performance in terms of the following factors as possible.
1) High capability of PAPR reduction: It is primary factor to
be considered in selecting the PAPR reduction technique
with as few harmful side effects such as in-band distortion
and out-of-band radiation.
2) Low average power: Although it also can reduce PAPR
through average power of the original signals increase, it
requires a larger linear operation region in HPA and thus
resulting in the degradation of BER performance.
3) Low implementation complexity: Generally, complexity
techniques exhibit better ability of PAPR reduction. However, in practice, both time and hardware requirements for
the PAPR reduction should be minimal.
4) No bandwidth expansion: The bandwidth is a rare resource
in systems. The bandwidth expansion directly results in
the data code rate loss due to side information (such as
the phase factors in PTS and complementary bits in CBC).
Moreover, when the side information are received in error
unless some ways of protection such as channel coding employed. Therefore, when channel coding is used, the loss
in data rate is increased further due to side information.
Therefore, the loss in bandwidth due to side information
should be avoided or at least be kept minimal.
5) No BER performance degradation: The aim of PAPR reduction is to obtain better system performance including
BER than that of the original OFDM system. Therefore,
all the methods, which have an increase in BER at the receiver, should be paid more attention in practice. Moreover,
if the side information is received in error at the receiver,
which may also result in whole erroneous data frame and
thus the BER performance is reduced.
6) Without additional power needed: The design of a wireless
system should always take into consideration the efficiency
JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS
265
Fig. 6. Comparisons of CCDF based on different PAPR reductions.
of power. If an operation of the technique which reduces
the PAPR need more additional power, it degrades the BER
performance when the transmitted signals are normalized
back to the original power signal.
7) No spectral spillage: Any PAPR reduction techniques can
not destroy OFDM attractive technical features such as
immunity to the multipath fading. Therefore, the spectral
spillage should be avoided in the PAPR reduction.
8) Other factors: It also should be paid more attention on the
effect of the nonlinear devices used in signal processing
loop in the transmitter such as DACs, mixers and HPAs
since the PAPR reduction mainly avoid nonlinear distortion due to these memory-less devices introducing into the
communication channels. At the same time, the cost of
these nonlinear devices is also the important factor to design the PAPR reduction scheme.
We consider a typical OFDM system with 256 subcarriers
) and 16-QAM constellation in which over(namely
sampled OFDM sequences with the oversampling rate of 4 are
used to analyze PAPR reduction and BER performance based on
different schemes as shown in the following figures. The preset
clipping level has been selected to 80% of the maximum of the
original OFDM symbols in the clipping scheme and the number
of the reserved tone is 20 in TR scheme. For PTS scheme, the
and the number of
rotation vectors belong to the set
searches is for each opthe subblocks is 16. Therefore, the
timal PTS. Note that, for PTS and TR schemes, all the side information have not been submitted to the receiver. For nonlinear
companding transform, the error companding is that proposed
in [25], and the exponential companding is based on (23).
As shown in Fig. 6, different curves of the CCDF have been
random original OFDM symbols generated and
given for
different PAPR reduction schemes. From Fig. 6, it is very clear
that all schemes can reduce the PAPR largely in OFDM system.
However, their performances of the PAPR reduction are dif-
, the PAPRs are
ferent. For example, when
2.6 dB, 4.5 dB, 6.6 dB, 6.8 dB, 6.9 dB and 11.7 dB for the
exponential companding, error companding, PTS, TR, clipping
scheme and original OFDM signals, respectively. Obviously,
the signals companded by the nonlinear companding transform
with exponential function can reduce the PAPR largest and the
PAPR reduction of the clipping scheme is the smallest among
these typical methods. Although clipping scheme can improve
its performance of the PAPR reduction through reducing its
preset clipping level A. However, the performance of the BER
will be degraded largely when its preset clipping level is reduced [73].
Fig. 7 depicts the performance of BER versus SNR of actual
OFDM signals with PAPR reduction based on different schemes
over the AWGN channel, in which the typical HPA of the Solid
State Power Amplifier (SSPA) has been considered. Note that
SSPA produces no phase distortion and only the AM/AM conversion [25]. In Fig. 7, the performance bounds are obtained
by ignoring the effect of SSPA and directly transmitting the
original OFDM signals through the AWGN channels. Generally speaking, the performances of the BER with different PAPR
reduction schemes have some degradation from Fig. 7. Specifi, the minimum required SNR is
cally, to achieve a BER of
13.8 dB (performance bound). The required SNRs under the exponential companding, PTS, TR, error companding and clipping
schemes are 14.9 dB, 15.7 dB, 16.6 dB, 17.5 dB and 25.6 dB,
respectively.
Therefore, an efficient PAPR reduction should be the lowest
possible value of PAPR while keeping a minimal level BER. In
Table II, we summarize the five typical PAPR reduction techniques based on the theoretical analysis and simulation results.
VIII. PAPR REDUCTION FOR MULTIUSER OFDM SYSTEMS
Recently, multiuser OFDM also has received much attention
due to its applicability to high speed wireless multiple access
266
IEEE TRANSACTIONS ON BROADCASTING, VOL. 54, NO. 2, JUNE 2008
Fig. 7. Comparisons of BER based on different PAPR reductions.
TABLE II
COMPARISON OF DIFFERENT PAPR REDUCTION TECHNOLOGIES
communication systems. In multiuser OFDM system, data
streams from multiple users are orthogonally multiplexed onto
the downlink and uplink subchannels. In a multiuser OFDM
system, a group of carriers is assigned for each user with
adaptive modulation, bit and power allocation. Obviously, the
characteristics including distribution of the PAPR for each user
in uplink multiuser OFDM is the same as that of the PAPR
in single user OFDM system since the data of each user will
be transmitted to channels independently in uplink multiuser
OFDM system. Therefore, the PAPR can be reduced according
to these schemes mentioned above in the uplink multiuser
OFDM systems. However, the characteristics of the PAPR in
downlink multiuser OFDM is different from that of the PAPR
in single user OFDM system since the data composed from
different users will be transmitted to channels successively
in downlink multiuser OFDM system. Therefore, the PAPR
reduction is more complicated in an downlink than that in
OFDM uplink in multiuser OFDM systems. If downlink PAPR
reduction is achieved by some approaches which have been
designed for OFDM, each user has to process the whole data
frame and then demodulate the assigned subcarriers to extract
their own information. Thus, it introduces additional processing
for each user at the receiver. Therefore, we mainly describe
some modifications of PAPR reduction techniques for the
downlink multiuser OFDM systems.
1) PTS/SLM for PAPR reduction in multiuser OFDM systems: PTS and SLM techniques can easily be modified for
PAPR reduction in downlink of multiuser OFDM systems.
For PTS, subcarriers assigned to one user are grouped into
one or more subblocks, and then PTS can be applied to subblocks for all users. As side information, the selected phase
factor for each subblock can be embedded into the pre-reserved subcarrier in each subblock. Note that, the pre-reserved subcarrier does not undergo the phase rotation in
each subblock. Similarly, some of the subcarriers can be
used to transmit side information when the modified SLM
is applied to reduce the PAPR for multiuser OFDM systems. All users use the information carried by these subcarriers to obtain the phase sequence is used at the transmitter,
and thus the data for each user can be recovered correctly.
2) TR for PAPR reduction in multiuser OFDM systems: In the
TR technique for multiuser OFDM systems, the symbols
in peak reduction subcarriers are optimized for the whole
data frame in both amplitude and phase. At the same time,
some peak reduction subcarriers are assigned to each user
in the TR for PAPR reduction.
IX. CONCLUSIONS
OFDM is a very attractive technique for wireless communications due to its spectrum efficiency and channel robustness.
JIANG AND WU: OVERVIEW OF PEAK-TO-AVERAGE POWER RATIO REDUCTION TECHNIQUES FOR OFDM SIGNALS
One of the serious drawbacks of in OFDM systems is that
the composite transmit signal can exhibit a very high PAPR
when the input sequences are highly correlated. In this paper,
we described several important aspects, as well as provide a
mathematical analysis, including the distribution of the PAPR,
in OFDM systems. Five typical techniques to reduce PAPR
have been analyzed, all of which have the potential to provide
substantial reduction in PAPR at the cost of loss in data rate,
transmit signal power increase, BER performance degradation,
computational complexity increase, and so on. We also showed
that it is possible to reduce the PAPR of for multiuser OFDM
systems.
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Tao Jiang (M) received the B.S. and M.S. degrees
in applied geophysics from China University of
Geosciences, Wuhan, P. R. China, in 1997 and 2000,
respectively, and the Ph.D. degree in information
and communication engineering from Huazhong
University of Science and Technology, Wuhan,
P. R. China, in April 2004. Then, he joined the
Brunel University, UK. In Oct. 2006, he moved
to the University of Michigan, USA. He is now
with the Department of Electronics and Information
Engineering, Huazhong University of Science and
Technology, Wuhan, 430074, P. R. China. He has authored or co-authored over
40 technical papers in major journals and conferences and five books/chapters
in the areas of communications. His current research interests include the areas
of wireless communications and corresponding signal processing, especially
for OFDM, UWB and MIMO systems, cooperative networks, cognitive radio
and wireless sensor networks. He serves on the editorial boards of some
international journals, such as Wiley of Wireless Communications and Mobile
Computing (WCMC), and as a Technical Program Committee Member for
some major international conferences, including IEEE Globecom, ICC, VTC
and Chinacom. Dr. Jiang is a Member of IEEE Communication Society and
IEEE Broadcasting Society.
Yiyan Wu (Fellow) received the B.S. degree from
Beijing University of Posts and Telecommunications,
and M.S. and Ph.D. degrees in electrical engineering
from Carleton University, Ottawa, Canada, in 1986
and 1990, respectively. After graduation, he worked
at Telesat Canada as a senior satellite communication
systems Engineer. In 1992, He joined Communications Research Center Canada (CRC) and now is a
Principle Research Scientist. His research interests
include broadband multimedia communications,
digital broadcasting and communication systems
engineering. He is an IEEE Fellow, an adjunct professor of Carleton University,
Ottawa, Canada; Shanghai Jiaotung University; and Beijing University of
Posts and Telecommunications, China. He is a member of IEEE Broadcast
Technology Society Administrative Committee, and a member of the ATSC
Board of Directors, representing IEEE. He is the Editor-in-Chief of the IEEE
TRANSACTIONS ON BROADCASTING. He has more than 200 publications and
received many technical awards for his contribution to the research and development of digital broadcasting and broadband multimedia communications.