PEAK-TO AVERAGE-POWER-RATIO (PAPR) REDUCTION IN
OFDMUSING VARIOUS CODING TECHNIQUES: A REVIEW
Samreen Kosser1, Vikas Chawla2
1. INTRODUCTION
OFDM is considered a good candidate for wireless systems because it offers diversity gain in frequency selective channels [1]. It uses the FFT to multicarrier modulate a signal and thus can take advantage of advances in DSP and digital circuitry. As in other multicarrier schemes, however, OFDM suffers from high PAPR. This is a major drawback of the scheme and ways of minimizing the PAPR have been researched through many years. This report covers the major developments. It first reviews the idea behind OFDM and the meaning of PAPR. It then presents some simple block coding schemes and moves on to the more elegant coding schemes using Complementary Sequences and Reed-Muller Codes.
OFDM is a form of multicarrier transmission that sends information simultaneously over N orthogonal carriers. It introduces frequency diversity by making the bandwidth of each carrier smaller than the coherence bandwidth of the channel. Each carrier may still suffer from flat-fading, however. OFDM is considered a good candidate for high data rate wireless systems and is currently used for the HyperLAN II standard [7]. The transmitted signal over a symbol duration T is [4].
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0
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(
2
exp(
Re
)
,
(
1 1 0
1
0
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N N
i
s i
c
c
c
c
T
t
t
if
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j
c
t
c
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The codeword c consists of N symbols chosen from a Mary modulation method. All of the codewords form the set C.For MPSK,
M i a
M j
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a
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) ( 2
The duration of an OFDM symbol T is N times the duration of the symbols ci plus the duration of the cyclic prefix or guard band. The complex envelope of the transmitted signal, sampled at 1/T, is:
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,
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~
10
N
ni
j
xp
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c
n
c
s
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This equation can be recognized as the IDFT of the sequence co …cN-1
2. PEAK-TO-AVERAGE POWER RATIO
An important limitation of OFDM is that it suffers from a high Peak-to-Average Power Ratio (PAPR) resulting from the coherent sum of several carriers. This forces the power amplifier to have a large input backoff and operate inefficiently in its linear region to avoid intermodulation products. High PAPR also affects D/A converters negatively and may lower the range of transmission. PAPR is defined as:
2 2)
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)
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max
t
s
E
t
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PAPR
1
Department of Electronics and Communication Engineering, Galaxy Global Group of Institutions Kurukshetra, Kurukshetra University, Haryana
2
Department of Electronics and Communication Engineering, Galaxy Global Group of Institutions Kurukshetra, Kurukshetra University, Haryana
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DOI: http://dx.doi.org/10.21172/1.103.22
e-ISSN:2278-621X
Abstract: OFDM is a transmission scheme that offers diversity in frequency selective fading environments but suffers from high PAPR. Many researchers have studied the appropriate coding scheme to reduce PAPR and offer good error control properties as well. This article reviews the major results obtained up to date for the reduction of PAPR. Various scrambling techniques like PTS and SLM have been combined with clipping-filtering and DCT to reduce PAPR. The various coding techniques like Block Coding and Reed Muller codes have been evaluated in this paper to obtain significant reduction in PAPR.CCDF graphs depict performance of proposed techniques
Theoretically, the PAPR can be as high as N, but the occurrence of such peaks is rare. The summation of a large number of carriers assumes a Gaussian distribution. The numerator, max|s(t)|2, is also known as the PEP (Peak Envelope Power). It is also equal to:
)
(
~
)
(
~
*t
s
t
s
PEP
Several methods have been devised throughout the years to limit the PAPR of a multicarrier signal. These methods include clipping, filtering, and coding. Clipping methods are the most widely used but at the cost of degradation of performance [10]. Some filtering and coding methods modify an OFDM symbol to lower its PAPR [3][4]. The more sophisticated methods form error-correcting codes with inherently low PAPR. [10]
3. BLOCK CODING
There are several simple block coding methods that can be used to reduce the PAPR. One way of coding for OFDM is to map 2k messages into 2N symbols In [2] the authors investigated numerically the effect of excluding codewords with high PAPR using QPSK modulation. They found that the set of codewords could be divided into cosets of identical PEP, and ranked in terms of PEP. By excluding those cosets with high PEP, the PAPR was lowered, but at the cost of coding rate k. They speculated that for k=1/2, the PEP e. This compromise between coding rate and PAPR affects all of the coding methods [7].
Searching for all codewords with minimum PEP can be done, but is only practical for a small codeword space. An alternative method proposed in [3][4] modifies a codeword c by adding a set of phase shifts to form the modified codeword c’.
1 1
0
1 1
0
1 1 0
...
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N
j N j j
N
e
c
e
c
e
c
c
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c
c
c
This modified codeword may have a lower PAPR, and preserves the error correction capability, coding rate, encoding and decoding complexity as C. The receiver needs to know the phase shifts to decode correctly. The set of phase shifts needs to be found and can become computationally intensive. Using the full range of 0-2 for the phase shifts i may yield a global minimum for the PAPR. If the application can tolerate a higher PAPR, however, this may not be necessary and a sup-optimum phasing will do by restricting the range of the phase shifts to a discrete set (i.e. I={1,-1}).. This lowers the computational complexity of the algorithm and the codeword will have a lower PAPR than the unmodified one. Using the Hiperlan-II standard for OFDM, the authors in [4] were able to lower the PAPR by 4.09 dB using QPSK type phase shifts. Another similar approach uses Partial Transmit Sequences[6]. The idea consists in dividing the symbols of c into M disjoint clusters. The clusters are also combined with a set of weights that consist of pure rotations to minimize the PAPR. The discrete signal partitioned into clusters is:
10 M
m m
x
c
The modified data set is:
10
'
M
m m m
x
b
c
Again, the rotations can be limited to a discrete set. These weights are sent to the receiver as side information. The algorithms for finding the set of weights vary but should not add too much computational complexity and improve the PAPR performance of the system. The authors in [6] were able to lower the PAPR by more than 3 dB with a simple algorithm. An alternative coding approach is to look for codewords that inherently have low PAPR and also offer error protection capabilities. Such a set of codes can be formed using Golay Complementary Pairs and are discussed in the next sections.
4. PAPR REVISITED
As mentioned previously, the PEP of an OFDM symbol can be found from its complex envelope. It can be shown that the PEP can also be written as (unit amplitude PSK symbols)[8].
i N
k j i
N
k
i k i c
N
i
T it j c
i k k
i
e
c
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R
e
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N
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0 *
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/ 2
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)
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Re
2
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Rcis known as the aperiodic autocorrelation function of the sequence co …cN-1. Sequences with low sidelobes will give a small PAPR. Golay binary complementary sequences are one of the sequences that give a PAPR of no more than 3 dB [10].
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5. COMPLEMENTARY SEQUENCES
Golay first introduced these sequences while studying infrared spectroscopy. Their basics are well described in [9]. A complementary sequence pair has the property that the sum of their aperiodic autocorrelation is zero everywhere except for the zero shifts. Therefore if S1 and S2 are two sequences of length N, they form a complementary pair if:
10
, 2 , 2 , 1 , 1 2
1
(
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Nn
n n n
n S
S
R
s
s
s
s
R
The pair (S1,S2)=(-1 –1 1 –1 1 1 1 –1, 1 1 –1 1 1 1 1 –1) form a binary complementary pair [9]. Among other properties, a new complementary pair (R1,R2) can be formed by the transformation from (S1,S2) ( m is any nonzero integer).
2
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1
/ 2 ,
,
s
e
l
r
ln ln j n mThe idea can be extended from pairs to sets of sequences. For a pair of sequences, their PAPR is at most 2 for any length N as shown below:
2
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2
1
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1
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/ 2 2
1
N
i
T it j S S
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i
e
R
N
N
S
PAPR
S
PAPR
Since, the PAPR is a positive quantity; the PAPR of each sequence must be no greater than 2. Recently, in [10], the authors made a major theoretical discovery by finding that large sets of binary complementary pairs of length 2m can be obtained from the 2nd order cosets of the well-known 1st order Reed-Muller code R(1,m). This discovery also introduces the error-correcting capabilities of these codes in addition to a low PAPR.
6. REED-MULLER CODES
These codes are among the oldest and best understood codes [12]. They can be simply described by using Boolean functions of m Boolean variables (v1,…,vm). A vector f is formed by the function f(v1,…,vm) as the variables go through all binary m- tuples. The rth order Reed-Muller code R(r,m) of length n=2m, 0rm, is the set formed by all vectors f, where f is a polynomial of degree less than or equal to r. R(1,2) consists of the length 4 codes shown below:
1001
1
1100
1
1010
1
1111
1
0110
0101
0011
0000
0
1
0
1
2 1 2 1 2 1 1 2
2 2 1 1 0
v
v
v
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v
or
a
v
a
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a
a
i
The general code R(r,m) consists of all the linear combinations of the vectors:
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,...,
,
,...,
,
1
v
1v
mv
1v
2v
1v
3v
m1v
mup
to
ree
r
The Reed-Muller codes can be easily encoded and decoded using majority logic. Their main properties are:
r m m
d
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dis
r
m
m
k
ension
n
length
2
tan
min
...
1
1
dim
2
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Except for R(1,m), these codes have a minimum distance lower than BCH codes. Half of the codes of R(r,m) are complements of the other half. R(1,m) is also known as a biorthogonal code since it can be obtained from thegenerator matrix of an orthogonal code by adding the all-ones codeword to it. They are also equivalent to cyclic codes with an overall parity check added.
7. REED MULLER CODES AND PAPR
In [10], the authors use a generalized Boolean function, mapping into Z2h instead of Z2 (h1), to describe Complementary Pairs over Z2hof length 2m. The two sequences a(v1…vm) and b(v1…vm) form a Complementary Pair and are given by ( is a permutation of the symbols {1,2,…,m) and c,c’ are members of Z2h)
m
k k k m
k
k k h
m
h m m
m m
v
c
v
v
v
v
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c
v
v
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c
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) 1 ( ) ( 1 1
) 1 ( 1 1
1
1 1
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)
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A new sequence formed from a by any permutation of the symbols {1,…,m} and adding any c also is a Complementary Sequence. This gives 2h(m+1)m!/2 Golay sequences.
The authors then generalize the Reed-Muller codes from binary (h=1) to h 1 by defining them over a ring instead of a field. RM2h(r,m) over Z2h is generated by the same way asbefore. In addition, the rth order linear code ZRM2h(r,m) is generated by the monomials in the vi of degree no greater than r-1 along with twice the monomials in the vi of degree r (monomials of degree –1 and m+1 equal to 0). Each of the m!/2 cosets of RM2h(1,m) that are in ZRM2h(1,m) make up 2h(m+1) Complementary Sequences of length 2m. The cosets have a representation of the form:
1
1
) 1 ( ) ( 1
2
m
k
k k h
v
v
Using these cosets in OFDM transmission limits the PAPR to at most 2. Using more cosets would increase the transmission rate, but would also increase the PAPR. The authors were able to partition the cosets into increasing values of PAPR for small values of h and m. In [11], the authors generalize these results by using q alphabets (an even number) instead of 2h an d consider general second order cosets of Rq(1,m)
The authors in [14] used these codes in an indoor wireless environment. The codes offered protection against amplifier nonlinearities and offered error protection, but fell short of the gain that could be provided by convolutional codes.
8. RESULTS AND DISCUSSIONS
Figure 1 shows PAPR reduction for Partial Transmit Scheme (PTS) in combination with clipping, filtering and Discreet Cosine Transform (DCT). The original signal is having PAPR of 11db. The PTS reduces PAPR to 7.1 db. The combination of PTS and DCT reduces PAPR to 6.9 db. Most of the reduction occurs for the combination of PTS, clipping-filtering and DCT in which PAPR reduces to 6.8db.
Figure 2 shows PAPR reduction for SLM scheme. The original signal is having PAPR of 10.9db. SLM reduces PAPR to 8.8db. The combination of SLM, clipping-filtering and DCT reduces PAPR to 6.7 db.
Figure 3 shows PAPR reduction for block coding for various m numbers. PAPR shows significant improvement with the increasing values of m. For m=8 PAPR reduces to 3.9db as compared to 9.5db for original signal
Figure 4 shows PAPR reduction for Reed Muller codes. For m=8 PAPR reduces to1.5 db. The original signal is having PAPR of 9db, thus there is 7.5 db decrease in PAPR. The results are more precisely explained in Tables 1,2,3,4.
9. CONCLUSIONS
This article has reviewed several coding techniques to reduce the PAPR in OFDM transmission. One approach to reduce the PAPR tries to minimize the PEP of a signal by combining it with a weight vector. This method can lower the PAPR and its complexity depends on the algorithm used. The more sophisticated approach uses sequences that inherently have low PAPR. Golay Complementary sequences are in this class and are related to the well-known Reed Muller codes. Like these, codes, using a higher number of carriers decreases the coding rate, but offers the gain obtained from these codes. There is a fundamental trade-off between coding rate and PAPR.
10. REFERENCES
[1] Bingham, John A. C. “ Multicarrier Modulation for Data Transmission: An Idea Whose Time has Come “ IEEE Communications Magazine May 1990, pp. 5-14.
[2] Sheperd, Orriss, and Barton “Asymptotic Limits in Peak Envelope Power Reduction by Redundant Coding In Orthogonal Frequency -Division Multiplex Modulation” IEEE Transactions on Communications, Vol 46, No 1, January 1998 pp 5-10.
[3] Wilkinson and Jones “ Combined Coding For Error Control and Increased Robustness to System Nonlinearities in OFDM “ Vehicular Technology Conference, 1996. 'Mobile Technology for the Human Race'. IEEE 46th , Volume: 2 , 28 April-1 May 1996, pp: 904 -908.
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[4] VahidTarokh and Hamid Jafarkhani “ On the Computation and Reduction of the Peak-to-Average Power Ratio in Multicarrier Communications “ IEEE Transactions of Communications, Vol 48, No 1, January 200 pp. 37 -44.
[5] Wilkinson and Jones “ Minimisation of the Peak-to-Mean Envelope Power Ratio of Multicarrier Transmission Schemes by Block Coding “Vehicular Technology Conference, 1995 IEEE 45th, Volume: 2, 25-28 July 1995 pp: 825 -829 .
[6] Cimini, L.J., Jr.; Sollenberger, N.R.; “ Peak-to-average power ratio reduction of an OFDM signal using partial transmit sequences” Communications, 1999. ICC '99. 1999 IEEE International Conference on, Volume: 1, 6-10 June 1999 pp: 511 -515.
[7] Paterson, Kenneth G. “ On the Existence and Construction of Good Codes with Low Peak-to-Average Power Ratios ” IEEE Transactions on Information Theory, Vol. 46, No. 6, Septermber 2000, pp. 1974-1987.
[8] Ochiai, H.; Imai, H.; “Performance of Block Codes with Peak Power Reduction for Indoor Multicarrier Systems “ Vehicular Technology Conference, 1998. VTC 98. 48th IEEE, Volume: 1, 18-21 May 1998 pp: 338 -342.
[9] Fan and Darnell Sequence Design for Communications Applications Copyright 1996 by Research Studies Press Ltd. Ch. 13.
[10] Davis and Jedwab “ Peak-to-Mean Power Control in OFDM, Golay Complementary Sequences, and Reed-Muller Codes “ IEEE Transactions on Information Theory, Vol 45, No. 7, November 1999, pp. 2397-2417.
[11] Paterson “ Generalized Reed-Muller Codes and Power Control in OFDM Modulation “ IEEE Transactions on Information Theory, Vol. 46, No. 1, January 2000, pp. 104-120.
[12] MacWilliams and Sloane The Theory of Error-Correcting Codes North-Holland Publishing Company 1977 Chapters 13-15. [13] Sweeny Error Control Coding John Wiley & Sons Copyright 2002.
