Peak-to-Average Power Ratio
(PAPR)
Wireless Information Transmission System Lab.
Institute of Communications Engineering
National Sun Yat-sen University
2012/07/30
邱營棋
Multi-carrier systems
◊
The complex baseband representation of a multicarrier signal
consisting of N subcarriers is given by
1 N 1
j 2 nft
x t  
X

e
,0  t  T

n
N n 0
where f is the subcarrier spacing.
◊
In OFDM systems, the subcarriers are chosen to be orthogonal.
(i.e., f  1 T )
2
An example of the time-domain signals with
64 subcarriers
0.35
0.3
Amplitude
0.25
0.2
0.15
0.1
0.05
0
0
10
20
30
40
Sample Index
3
50
60
70
The effect of high PAPR
◊
Due to the large number of sub-carriers in typical OFDM systems,
the amplitude of the transmitted signal has a large dynamic range,
leading to in-band distortion and out-of-band radiation when the
signal is passed through the nonlinear region of power amplifier.
◊
Although the above-mentioned problem can be avoided by operating
the amplifier in its linear region, this inevitably results in a reduced
power efficiency.
◊
The PAPR of the transmit signal is defined as
2
max x  t 
PAPR  0t TT
2
1/ T   x  t  dt
0
4
AM/AM distortion
Soft limiter
5
Bandwidth regrowth
6
PAPR in discrete-time case
◊
If we sample x(t) by a sampling rate of 1/Ts (the sampling period Ts
= T/N ), we may miss some signal peaks and get optimistic results
for the PAPR.
◊
For better approximating the true PAPR in the discrete-time case,
we usually oversample x(t) by a factor of L , i.e., the sampling rate is
L/Ts .
◊
It was shown in [15] that an oversampling factor L=4 is sufficient to
approximate the true PAPR.
7
PAPR in discrete-time case
For an OFDM system with N sub-carriers, an oversampling rate of L
can be achieved by inserting (L－1)·N zeros in the middle of the
modulated symbol vector to form a 1×LN data vector X, i.e.


X   X  0 , X 1 ,
, X  N / 2  1 , 0,
, 0, X  N / 2  ,
, X  N  1


L

1
N




◊
◊
The PAPR computed from the L-times oversampled time-domain
signal samples is given by
2
max xk
PAPR  0k  LN 1 2
E  xk 


8
The CCDF of the PAPR
◊
The cumulative distribution function (CDF) of the PAPR is one of
the most frequently used performance measures for PAPR reduction
techniques. In the literature, the complementary CDF (CCDF) is
commonly used instead of the CDF itself.
◊
The CCDF of the PAPR denotes the probability that the PAPR of a
data block exceeds a given threshold.
◊
From the central limit theorem, the real and imaginary parts of the
time domain signal samples follow Gaussian distributions, assuming
each distribution with a mean of zero and a variance of 0.5 for a
multicarrier signal with a large number of subcarriers.
9
The CCDF of the PAPR
◊
◊
◊
Hence, the amplitude of a multicarrier signal has a Rayleigh
distribution, while the power distribution becomes a central chisquare distribution with two degrees of freedom.
The CDF of the instantaneous power of a signal sample is given by
F  z   1  exp   z  .
The CCDF of the PAPR of a data block with Nyquist rate sampling
is derived as
P  PAPR  z   1  P  PAPR  z 
 1 F  z 
N
 1  1  exp   z   .
N
This expression assumes that the N time domain signal samples are
mutually independent and uncorrelated.
10
PAPR Reduction Methods
◊
Distortion
◊
◊
◊
Distortionless
◊
◊
◊
Clipping
Companding
Selected Mapping (SLM)
Partial Transmit Sequence (PTS)
Others
◊
◊
Active Constellation Extension (ACE)
Tone Reservation (TR)
11
Clipping
◊
The simplest way to reduce the PAPR.
◊
The peak amplitude becomes limited to a desired level.
◊
Clipping
xn
, xn  A

yn  
 A  exp{ j  arg( xn )} , xn  A
◊
Clipping Ratio
CR  20log
A
X
dB
 X : RMS value of xn
12
Clipping
◊
By distorting the OFDM signal amplitude, a kind of selfinterference introduced that degrades the BER.
◊
Nonlinear distortion increases out-of-band radiation.
13
Companding
14
Companding
15
Selected mapping (SLM)
P (1)
Serial-toparallel
conversion of
user bit stream
Optionally:
Differential
encoding
(2)
Optionally:
Differential
encoding
IDFT
.
.
.
.
.
.
IDFT
a (1)
P (2)
A
A
Coding &
Interleaving
Mapping
A (1)
P
.
.
.
(U )

A (U )
Optionally:
Differential
encoding
IDFT
a (2)
Selection
of a
desirable
symbol
a
a (U )
If necessary:
Bit
source
Side information
16
Selected mapping (SLM)
1. A set of U markedly different, distinct, pseudo-random
but fixed
(u )
vectors P(u) = [P0(u) ,…, PN-1(u) ], with Pn(u )  e jn , n(u ) [0, 2 ),
0  n  N , 1  u  U must be defined.
2. The subcarrier vector A is multiplied subcarrier-wise with each one
of the U vectors P(u), then resulting to component An(u )  An  Pn(u ) ,
0  n  N, 1  u  U.
3. Then all U alternative subcarrier vectors are transformed into time
domain to get a(u )  IDFT {A( u ) } and finally that transmit sequence
a  a(u ) with the lowest PAPR is chosen.
◊
For implementation, the SLM technique needs U IDFT operations,
and the number of required side information bits is log 2 U ,  y 
denotes the smallest integer that exceed y.
17
PAPR reduction performance of SLM
◊
N = 256, L = 4, 16-QAM,
Pn(u ) {1,  1,  j,  j}.
18
Partial transmit sequence (PTS)
Serial-to-parallel
conversion of
user bit stream
Coding &
Interleaving
A (1)
IDFT
a (1)
b (1)
A (2)
IDFT
a (2)
b (2)
Mapping
Subblock
partationing
Optionally:
Differential
encoding
.
.
.
.
.
.
A (M )
+
IDFT
.
.
.
a( M )
b( M )
Pack value optimization
Bit source
19
If necessary :
side information
a
Partial transmit sequence (PTS)
1. In this scheme, the subcarrier vector A is partitioned into M pairwise
disjoint subblocks A ( m ) , 1  m  M . All subcarrier positions in A ( m )
which are already represented in another subblock are set to zero, so
M
that
A   A( m) .
m1
 j ( m )
,
2. We introduce complex-valued rotation factors b  e
( m) [0, 2 ), 1  m  M , and μ is index of all phase rotation of “Peak
value optimization”. Enabling a modified subcarrier vector
( m)
M
A    b( m )  A( m)
m 1
which represents the same information as A, if the set b( m ) ,1  m  M 
(as side information) is known for each μ.
20
Partial transmit sequence (PTS)
3. To calculate a   IDFT {A  } , the linearity of the IDFT is exploited.
Accordingly, the subblocks are transformed by M separate and
parallel N-point IDFTs, yielding
M
M
a    b  IDFT {A }   b( m)  a( m) .
( m)
( m)
m 1
m 1
4. Based on them a peak value optimization is performed by suitably
choosing the free parameters b( m ) such that the PAPR is minimized
for b( m ).
◊
It should be noted, that PTS can be interpreted as a structurally
modified special case of SLM.
21
Partial transmit sequence (PTS)
◊
◊
◊
◊
◊
In general, the selection of the phase factors is limited to a set with a
finite number of elements to reduce the search complexity.
The set of allowed phase factors is written as P  e j 2 n /W | n  0,1,
W  1 , where W is the number of allowed phase factors.
In addition, we can set b(1)  1 without any loss of performance.
Hence, W M 1 sets of phase factors are searched to find the optimum
set of phase factors.
PTS needs M IDFT operations for each data block, and the number
of required side information bits is log 2 W M 1  .
22
,
PAPR reduction performance of PTS
◊
N = 256, L = 4, 16-QAM, exhausted research for W M .1
W=2 means [+1,-1], W=4 means [+1, -1, +1j, -1j].
23
Active Constellation Extension (ACE)
◊
In this technique, some of the outer signal constellation points in the
data block are dynamically extended toward the outside of the
original constellation such that the PAPR of the data block is
reduced.
Im
Im
a1
Re
Re
QPSK
24
16-QAM
Active Constellation Extension (ACE)
clipping
X
IFFT
x
x
FFT
X
Iteration?
 x  n  , | x  n  |  A
x  n    j  n
 Ae , | x  n  |  A
25
PAPR
PAPR reduction performance of ACE
◊
N = 256, L = 4, QPSK, A = 4.86 dB.
26
PAPR
Tone Reservation (TR)
◊
The transmitter reserves a small number of unused subcarriers. These
subcarriers are referred to as peak reduction carriers (PRCs).
◊
Since PRCs do not carry data, this increment induces a severe
degradation of system’s power efficiency.
◊
In general, there are two approaches to reduce the PAPR in the TR
technique.
◊
◊
The first is to select the PRC indices for the TR technique to be used in
reducing the PAPR.
The second is to design the proper values on these PRCs to generate an
optimal peak-canceling signal that minimizes the PAPR of a transmitted
OFDM signal.
27
Tone Reservation (TR)
 X n , n  R
X n  Cn  
c

C
,
n

R
.
 n
28
Reference
◊
◊
◊
◊
◊
◊
◊
◊
[1] S. H. Han and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for
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29
Reference
◊
◊
◊
◊
◊
◊
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[9] S. G. Kang, J. G. Kim, and E. K. Joo, “A novel subblock partition scheme for partial transmit
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