Evaluation of performance improvement
capabilities of PAPR-reducing methods
Marc Deumal†, Ali Behravan*, Thomas Eriksson‡ and Joan Lluís Pijoan†
†Department
of Communications and Signal Theory
La Salle School of Engineering, Ramon Llull University, Barcelona, Spain
*Qamcom
Technology, Göteborg, Sweden
‡Department
of Signals and Systems,
Chalmers University of Technology, Göteborg, Sweden
Outline
• Introduction
• Theoretical analysis
– OFDM system performance in NL environments
– Peak-to-average power ratio
• Considerations on PAPR-reduction
• Performance of the PAPR-reduced signals
• Conclusions
Introduction
• OFDM is a powerful modulation technique being used in
many new and emerging broadband communication
systems.
– Advantages:
• Robustness against frequency selective fading and time dispersion.
• Transmission rates close to capacity can be achieved.
• Low computational complexity implementation (FFT).
– Drawbacks:
• Sensitivity to frequency offset.
• Sensitivity to nonlinear amplification.
• Compensation techniques for nonlinear effects
– Linearization (digital predistortion).
– Peak-to-average power ratio (PAPR) reduction.
– Post-processing.
Introduction
• PAPR-reduction techniques:
– Varying PAPR-reduction capabilities, power, bandwidth and complexity
requirements.
– The performance of a system employing these techniques has not been
fully analyzed
• PAPR is a very well known measure of the envelope fluctuations of
a MC signal
– Used as figure of merit.
– The problem of reducing the envelope fluctuations has turned to
reducing PAPR.
• In this paper we ...
– present a quantitative study of PAPR and NL distortion
– simulate an OFDM-system employing some of these techniques
Motivation: evaluate the performance improvement capabilities of PAPRreducing methods.
Theoretical analysis of PAPR and system performance
Orthogonal Frequency Division Multiplexing
•
An OFDM signal can be expressed as
s t   1
N
N 1
 Sk e
j 2 kt / NT
,
k 0
t  0, NT 
S
 k

 N
Complex baseband modulated symbol
Number of subcarriers
If the OFDM signal is sampled at t  nT, the complex samples can be described as
sn  1
N
N 1
Sk e j 2 kn / N ,

k 0
n  0, N 1
Theoretical analysis of PAPR and system performance
Peak-to-average power ratio
•
Let sm be the m-th OFDM symbol, then
its PAPR is defined as
PAPR m 
s m 
E  s m

2

2


N
The CCDF of the PAPR of a nonoversampled OFDM signal is

Pr    0  1 1 e 0
•

N
CCDF of PAPR increases with the number of subcarriers in the OFDM
system.
– It is widely believed that the more subcarriers are used in a OFDM system, the
worse the distortion caused by the nonlinearity will be.
Theoretical analysis of PAPR and system performance
In-band and out-of-band distortion
•
If N is large enough, the OFDM signal can be
approximated as a complex Gaussian distributed
random variable. Thus its envelope is Rayleigh
distributed
2
 x2
2
x
f X  x  2 e  ,

with E  X     and var  X    2 1  ,
2

4
where the variance of the real and imaginary
parts of the signal is  2 2
•
Bussgang theorem
 x t 
1
Rx1x2   
 x2  t 

x1 t  
NL
y2 t 
 Rx1 y2


 
where Rx1y2     Rx1x2  
In particular if x1 t   x2 t  ,then Rxy     Rxx  
An interesting result is that the output of a NL with Gaussian input (OFDM) can be
written as:
Rxy 1 
y t    x t   d t  , where  
Rxx 1 
Theoretical analysis of PAPR and system performance
In-band and out-of-band distortion (cont.)
y  t    x t   d t  ,
where  
Rxy 1 
Rxx 1 
attenuation and rotation: compensated at the rx.
distortion: in the frequency domain:
in
out
D  D   D 
with
Dk(in)
D
  k
0

if k  0, N 1
otherwise
introduces an in-band noise that
increases the error probability

out   Dk if k   N , LN 1



Dk  
0
otherwise

is the out-of-band radiation
•
•
Both the distortion term and α are
independent of N.
The envelope of the OFDM signal
is also independent of N.
Considerations on PAPR reduction
•
In order to improve the system performance, PAPR should predict the
amount of distortion introduced by the nonlinearity
– PAPR increases with the number of subcarriers in the OFDM signal.
– The distortion term and the uniform attenuation and rotation of the constellation
only depend on the back-off.
The effect of a nonlinearity to an OFDM signal is not clearly related to its PAPR
•
The effective energy per bit at the input of the nonlinearity is
Eb(eff )  Eo  p
K
where Eo is the average energy of the signal at the input of the nonlinearity, K is the
number of bits per symbol and ηp is the power efficiency.
– There will only be a a BER performance improvement when the effect of reducing
the in-band distortion becomes noticeable and more important than the loss of
power efficiency.
– This is not taken into account in the majority of the PAPR reducing methods.
Performance of the PAPR-reduced signals
Active Constellation Extension (ACE)
•
In ACE, at each OFDM block, some of the outer signal constellation points
are extended towards outside of the constellation such that the PAPR of the
resulting block is reduced
 Advantages:
– It is transparent to receiver.
– There is no loss of data rate.
– No side information is required.
× Drawbacks:
– The increase in the average energy per bit might be higher than the NL
distortion reduction.
– The larger the constellation size is the lower the number of extensible
points will be.
Performance of the PAPR-reduced signals
Active Constellation Extension (ACE)
Bit Error Rate
Power Spectral Density
PSD rectangular window
Performance of the PAPR-reduced signals
Tone Reservation (TR)
•
TR consists on reducing the PAPR by reserving a few tones (PRT) within
the transmitted bandwidth and assign them the appropriate values
 Advantages:
– No distortion is introduced to the data bearing tones
– No side information is required.
× Drawbacks:
– Increase in the average energy per bit which might reduce the BER
performance improvement.
– Loss of spectral efficiency due to tone reservation
Performance of the PAPR-reduced signals
Tone Reservation (TR)
Bit Error Rate
Power Spectral Density
4.3% of the subcarriers are reserved for PAPR-reduction
Performance of the PAPR-reduced signals
Selected Mapping (SLM)
•
•
In SLM, from the original data block several candidate data blocks are generated and
the one with lowest PAPR is transmitted.
At the receiver the reverse operation is performed to recover the original data block.
 Advantage: No distortion is introduced
× Drawback: It requires transmitting log 2 U  bits of side information per OFDM symbol


» It is crucial that the side information is received without errors.
» The side information has to be heavily protected.
•
•
SLM has a complexity of U IFFT operations and U complex vector multiplications.
The amount of PAPR reduction depends on U and the design of the phase sequences.
Performance of the PAPR-reduced signals
Selected Mapping (SLM)
Bit Error Rate
Power Spectral Density
U=8 phase sequences are used
Performance of the PAPR-reduced signals
Partial Transmit Sequences (PTS)
•
•
The original data block is partitioned into V disjoint subblocks. The subcarriers in
each subblock are rotated by the same phase factor such the PAPR of the
combination is minimized.
At the receiver the reverse operation is performed to recover the original data block.
 Advantage: No distortion is introduced

V 1 
×
Drawback: It requires transmitting log 2 W  bits of side information per OFDM symbol.
» It is crucial that the side information is received without errors.
» The side information has to be heavily protected.
•
PTS has a complexity of V IFFT operations, V 1 W V 1 complex vector multiplications
and V 1 W V 1 complex vector sums.
The amount of PAPR reduction depends on V, W and the subblock partitioning.
•




Performance of the PAPR-reduced signals
Partial Transmit Sequences (PTS)
Bit Error Rate
Power Spectral Density
V=3 subblocks and W=4 phase factors are used
Conclusions
•
In this paper we presented a quantitative study of both the PAPR and the
performance of an OFDM system when a NL is present.
•
PAPR-reduction is meant to decrease the distortion introduced by the NL.
– We showed that the effect of a NL on an OFDM signal is not clearly related to
its PAPR.
– In some recent contributions other measures of the envelope fluctuations have
been proposed.
• The cubic metric [Motorola,TechReport,2005] relies on the fact that major distortion is
caused by the third order intermodulation product.
• The variance of the instantaneous power [Behravan,VTC,2006] directly reduces the
envelope fluctuations
 
VP  var sn
•
2
We also compared the BER performance and the PSD of a conventional
OFDM with that of a PAPR-reduced OFDM system.
– Spectral spreading is reduced when applying PAPR-reduction.
– BER performance improvement only occurs when the effect of reducing the inband distortion is more important than the loss of power efficiency.
Thank you!
Thank you!
High power amplifiers
• Baseband model:
bx  uxe j x
by1  G ux  e
HPA
j  x ux 


G 
Amplitude to amplitude modulation (AM/AM)
 
Amplitude to phase modulation (AM/PM)
• Operating point:
– Input back-off:
P
IBO  10log10 



 dB 

 

max,in 
Px
– Output back-off:
P
OBO  10log10  max,out
Py


  dB 

 

High power amplifiers
• We assume that predistortion is done at the transmitter side
– The idea of predistortion is to modify the input signal of the HPA so that
the output is as close as possible to the linearly amplified input signal
bx
bx
PD
 A,  
pd
HPA
 G,  
– The AM/AM and AM/PM characteristics
of the PD satisfy:
A u   G 1 u 

 u    G 1 u 

– After combining the PD and the NL we
obtain a SL:
 
(u)    A u  
G A u  
u

1
 0
if u  1
otherwise
by