Ak25214217

Charanjit Singh, Gurpreet Singh Walia / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.214-217
A Modified Coding Technique For High Peak-To-Average Power
Ratio Reduction
Charanjit Singh1, Gurpreet Singh Walia 2
1
(ECE, University College of Engineering, Punjabi University, Patiala, India)
2
(ECE, University College of Engineering, Punjabi University, Patiala, India)

ABSTRACT
Orthogonal Frequency Division Multiplexing low PAPR or what the minimum distance properties
System (OFDM) suffers from the problem of high of the code has been discussed. However in [4], it is
peak to average power ratio reduction (PAPR). It observed that most of codes found, are Golay
is studied that Golay complementary codes (GCC) complementary sequences (GCS), which define a
can provide codewords with low PAPR. However, structured way of generating PAPR reduction codes.
the usefulness of this coding technique is limited to Golay complementary sequences are sequence pairs
the OFDM with a small number of subcarriers. for which the sum of autocorrelation functions is zero
With the help of certain theorems and definitions, for all delay shifts not equal to zero [5–8]. Davis and
this paper discusses a modified GCC technique to Jedwab in [9] obtained a large set of length 2m binary
combat the effect of high PAPR and achieve Golay Complementary Pairs (GCP) from certain
higher code rate and information rate for 16- second order cosets of the first order Reed-Muller
QAM OFDM system. (RM) codes. Using the correlation property of GCS
small PAPR of 3dB can be achieved. The usefulness
Keywords- GCC, OFDM, PA, PAPR, QAM, of these coding techniques is limited to the OFDM
RGCC. system with a small number of subcarriers. The
development of a good code for OFDM systems with
1. INTRODUCTION a large number of subcarriers is difficult, which limits
OFDM system is mainly designed to combat the the actual benefits of coding for PAPR reduction in
effect of frequency selective fading, by dividing the practical OFDM systems.
wide-band frequency selective fading channel into The rest of the paper has been organized as
many narrow-band flat fading sub-channels [1]. follows: Section 2 presents PAPR problem in OFDM
OFDM has many advantages such as high spectral system. Section 3 describes about Golay
efficiency, robustness to channel fading, immunity to complementary sequence. Section 4 has two
impulse interference, uniform average spectral subsections; first subsection presents important
density capacity of handling very strong echoes and definitions and theorems which helped in the
non-linear distortion, it suffers from major drawback, development of Recursive Golay complementary
i.e., high PAPR. The result of this high PAPR is that, code, second subsection presents the generation of
the transmitted signal may suffer significant spectral Recursive Golay 16-QAM signal from QPSK pairs.
spreading and in-band distortion as a consequence of And section 5 is conclusion.
inter-modulation effects induced by a non-linear
power amplifier (PA) [2]. One possible way to 2. PAPR PROBLEM
combat this problem is the use of a large power back- An OFDM signal is considered as the sum
off which allows the amplifier to operate in its linear of many independent signals modulated onto sub
region. However, this results into large decrease in channels of equal bandwidth. The complex baseband
power efficiency of the PA. representation of a OFDM signal consisting of 𝑁
This gives a clear motivation to find other subcarriers in continuous time domain is given by
N 1
methods of minimizing the PAPR of the transmitted x(t )  1  X n  e j 2 nft ,0  t  NT (1)
OFDM signal. A method which has been introduced N n0
in [3] and developed in [4] uses block coding to
transmit only those codewords which produces
where j  1 , f is the subcarrier spacing, and
OFDM symbols with PAPR below some threshold
level. In [3], it is observed that for eight channels, a NT denotes the useful data block period. In OFDM
code rate of 3/4 exists that provides a maximum the subcarriers are chosen to be orthogonal
PAPR of 3 dB. However for a large number of (i.e., f  1/ NT ) .
subcarriers, a reasonable coding rate larger than 3/4 The peak to average ratio of a zero-mean signal can
can be achieved for a PAPR level of 4dB. It is be defined in continuous as,
concluded that as the desired PAPR level decreases,
achievable code rate also decreases. There is no
structured way of getting these codewords having

214 | P a g e

max x(t )
2 schemes it is named as Recursive Golay
PAPR  0t  NT
complementary code (RGCC). Mathematically, (2)
1 NT
 x(t ) dt
2
NT 0 ( x, y)  ( x | y, x | y )
4.1 Recursive Golay Complementary Codes for Low
The Complementary CDF (CCDF) is used to PAPR
measure the probability that the PAPR of a certain Definition 1.Two N-valued complex sequences x and
data block exceeds the given threshold. y are called Recursive Golay Complementary Pairs
Let Z max denote the PAPR (i.e., (RGCP) if
 They are Golay complementary pairs.
( Z max  max n  0,1,2,3......... N 1 Z n ) . The CDF of Z max for the  If Pav is the average power of the
peak power per OFDM symbol having N subcarriers constellation,
is given by, Px  Py  2 NPav
FZ ( z )  P( Z max  Z )
max
where x  y
  Definition 2.The Cross-Correlation of two N-valued
N
 1  exp( Z )
2
(3)
complex codes x and y with replacement
 N  l  N  1 is defined as
The probability of exceeding the PAPR from
the value ‘PAPR0’ is given by Complementary  N l 1 xi l yi* if 0  l  N  1
 i
Cumulative Distribution Function (CCDF). Usually  N l 1 *
Cl  x, y    i xi yi l if  N  l  1 (6)
(CCDF) can be used to evaluate the performance of

any PAPR reduction schemes, 0 otherwise
 
FZ ( PAPR0 )  P( Z max  PAPR0 )
max

 1  P( Z max  PAPR0 ) Definition 3.The Auto-Correlation of an N-valued
 
N
 1  1  exp( PAPR0 ) complex codes x with nonzero replacement l is
2
(4)
defined as
 x x
 N l 1 * for l  0
3. GOLAY COMPLEMENTARY Al ( x)   i 0 i l i (7)
SEQUENCE  Al ( x),

*
for l  0
Golay complementary sequence is defined
as a pair of two sequences x and y whose aperiodic Theorem 1.The PAPR achieved by any RGCS is
autocorrelations sum equal to zero in all out-of-phase bounded up by 3dB.
positions [10]. Proof. By virtue of Theorem 1 and the fact that the
2 N if l  0 instantaneous power of each code is always non-
Ax x (l )  Ay y (l )   (5) negative,
 0 if l  0 max  p (t ) px (t )  py (t ) px  py
In [11] the possibility of using PAPR( x)  t x    2  3dB (8)
Px NPav NPav
complementary codes for both PAPR reduction and
forward error correction has been shown. Also, [9]
Following theorems represent a way to develop
shows that a large set of binary length 2m Golay
Recursive Golay Codes.
complementary pairs can be obtained from Reed-
Theorem 2.The property of being Recursive Golay
Muller codes. If the total number of bits assigned to
complementary pairs is invariant under the following
one OFDM symbol is Nm, where N is the number of
transformations:
subchannels, then N-valued sequence can be chosen
a. Reflection w.r.t the origin
from a codebook and this sequence is fed into the
b. Reflection w.r.t both axes.
FFT block.
c. Multiplication of one or both sequences by a
complex number with magnitude 1.
4. MODIFIED GOLAY d. Reflection w.r.t the bisectors of all regions.
COMPLEMETARY CODING SCHEME e. Rotation.
The need for low PAPR [12], high coding
rate has motivated us to investigate the non-equal Proof. Each item will be proved separately;
power codes that achieve low PAPR. To do this we a. Using Definition 3,
define a special case of GCSs using the recursive
construction. Using (5), many recursive constructions Al ( x)  Al ( x)
for GCPs can be obtained by simply performing the
algebraic manipulation. By doing so, it is observed
b. The reflection of x w.r.t the real axis is x*. Using
that 2N-valued Recursive Golay pairs can be created
Definition 3.
from N-valued GCP. As Golay complementary code
has been modified using recursive construction

215 | P a g e


Al ( x* )  [ Al ( x)]* l 1 (10)
Al ( x  y)  i 0 2 xi* y
N 1
The reflection of a sequence x w.r.t the imaginary l 1 (16)
i
axis is -x* 2

c. If α is an arbitrary complex number, then therefore Al ( x  y )  Al ( x   y )  0
If x and y are N-valued sequence then,
Al ( x)   Al ( x)
2 ˆ
x represents the inverse of x (11)
*
x represents the conjugate of x
Therefore if x  y and   1 , then x | y represents the concatenation of x and y
x   y and  x   y .
x  y represents the interleaving of x and y
d. The reflection of x w.r.t the bisector of the first By applying the transformations defined in
and third regions is jx*. Also, the reflection of x Theorems 2 and 3 to the statements of Theorem 4, a
w.r.t the bisector of the second and fourth set of structures can be generated that create 2N-
regions is –jx*. valued Recursive Golay pairs from N-valued ones.
e. Rotation of a sequence with the angle 𝜃 is Specifically, if x  y each with size N, then the
equivalent to multiplying the sequence by ejθ. following sequences are Recursive Golay pairs:
Note that, in all of these cases, the power of the 1)   j   x y     j   x  y 
sequences is preserved.
2)   j   x  y     j   x   y 
Theorem 3. If x  y then 3)   j   x y     j   y  x * 
ˆ ˆ

a. x '  y ' b. x  y
ˆ ˆ c. x  y
ˆ
4)   j   x  y     j   y   x  
ˆ ˆ
Proof. If x  y then
5)   j   x   y     j   y *  x * 
ˆ ˆ
a. The k of the sequence x is xk   1 xk ,
 th k
However, because of the special structure of
therefore using Definition 3. 16-QAM constellation, many of these constructions
yield similar sequences. For example reversing the
Al ( x)  (1)l Al (x ) (12) role of x and y will not yield new pairs. If the number
of N-valued pairs is M, the first structure yields 4M of
ˆ 2N-valued Recursive Golay pairs and this is true for
b. The kth member of the sequence x is
the second structure too. It has been found that in
xk  xN 1k , therefore using Definition 3.
ˆ general each pair with size N yields 32 pairs each
N l 1 N l 1 with size 2N.
Al ( x) 
ˆ  xˆ
i 0
ˆ
x
i l i x
i 0
x 
N l 1 N i 1

4.2 Recursive Golay 16 QAM pairs from QPSK
  k 0 xkl xk  ( Al ( x))
N l 1
(13) pairs [13]
To generate 16-QAM Recursive Golay sequences
c. Using (10) and (13) the statement is concluded.
from QPSK Golay sequences, let’s define QPSK
symbols as
Theorem 4. If xy then
   k    
x yx y b. x  y  x   y QPSK  exp  j    , k  Z 2h 
2 4 
a.
    
Proof. The items are proved separately:
Using Definitions 3 and 2, the following theorem can
a. Concatenation Property
l 1
be proved easily,
Al  x y   Al ( x )  Al ( y )   xN 1i yl 1i
* Theorem 5. For any two sequences x and y any two
i 0 complex number α and β
and therefore Al  x   y    Al ( x )   Al y   *C1  x, y 
2 2

Al ( x  y)  Al ( x  y)  2( Al ( x)  Al ( y))  0 (14)
  *  Cl ( y, x)
b. Interleaving Property
If l = 2k then Theorem 6. If x and y are N- valued QPSK Golay
complementary pairs, and α and β are two arbitrary
Al ( x  y)  Al ( x)  Al ( y)  0 (15)
2 2 complex numbers with   , then each of the
Using (9) and since x   y , Al ( x  y )  0 .Therefore, following pairs are 16-QAM Recursive Golay
sequences:
Al ( x  y )  Al ( x   y )  0 .If l  2k  1 then
1. c    x  2 y  and t   ( 2 x  y )

216 | P a g e

REFERENCES
2. c    x  2 y  and t   (2 x  y ) [1] Bingham J A C, Multicarrier Modulation for
data Transmission: An Idea Whose Time
3. c   ( x  2 jy ) and t    2 jx  y 
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 2 j  Cl  x, y   2 j  Cl  y, x 
2 2

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
Pc  Pt   x  2 jy  2 jx  y
2 2
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 x 2  4 y 2  2 jx H y  2 jy H x 
 
2   [6] Sivaswamy, R., Multiphase Complementary
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 y  4 x  2 jx y  2 jy x 
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Efficient Multitone Signals with Flat
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(17)
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