PAPR Reduction in OFDM & MC-CDMA System using Nonlinear Companding Techniques

ACEEE International Journal on Network Security, Vol 1, No. 1, Jan 2010

PAPR Reduction in OFDM & MC-CDMA Sys-
tem using Nonlinear Companding Techniques
R.Manjith1, Dr.(Mrs).M.Suganthi2
1
Thiagarajar College of Engineering/ECE Department,Tamilnadu,India
Email:manjithkmr@gmail.com
2
Thiagarajar College of Engineering/ECE Department,Tamilnadu,India
Email:msuganthi@tce.edu

.
Abstract - High peak to average power ratio (PAPR) of Original OFDM and MC-CDMA signals have very high
the transmitted signal is a major drawback of OFDM and Peak-to-Average Power Ratio (PAPR), which require
MC-CDMA systems. In this paper various existing nonli- sophisticated (expensive) radio transmitters with their
near companding transforms are analyzed and compared high power amplifiers operating in a very large linear
for the reduction of peak to average power ratio in OFDM
and MC-CDMA systems. Nonlinear companding transforms
range otherwise; nonlinear signal distortion occurs and
transform the amplitude or power of the original signals leads to high adjacent channel interference and poor sys-
into uniform distribution, which can effectively reduce the tem performance [2].
PAPR for different modulation formats and subcarrier sizes
without any complexity increase and bandwidth expansion SK sn tn
.Nonlinear companding technique adjust both large and
QAM/ S I H
small signals and can keep the average power at the same PSK to F P/S and P
level. Nonlinear companding transforms can significantly Mapping P F Companding A
improve the performance of OFDM and MC-CDMA system T
including bit-error-rate and PAPR reduction.
wn
Index Terms –Nonlinear companding transform, peak
SK ’ sn’ rn
to average power ratio (PAPR), orthogonal frequency divi- QAM/ P F
PSK De to
sion multiplexing (OFDM), Multicarrier code division mul- Mapping S
F S/P and De A/
T Companding D
tiple access (MC-CDMA)

I.INTRODUCTION Figure 1. OFDM system using non linear companding technique

1:M e(1)
As a multicarrier modulation technique, Orthogonal tn
Frequency Division Multiplexing (OFDM) [1] and Mul- User1 S I
F
ticarrier Code Division Multiple Access (MC-CDMA) d(1) to P/S &
To
P Com-
[2] has the following advantages: (1) robust to multi-path to F
pand-
fading, inter- symbol interference, co-channel interfe- T
i
rence and impulsive parasitic noise; (2) lower implemen- Channel
tation complexity compared with the single carrier solu-
tion; and (3) high spectral efficiency in supporting broad- e(k) S to P
User k S
band wireless communications. Therefore, OFDM is be- d (k) to
lieved to be a suitable technique for broadband wireless P
communications and has been used in many wireless
standards, such as Digital Audio Broadcasting (DAB),
Terrestrial Digital Video Broadcasting (DVB-T), Wire- Figure 2. MC-CDMA system using non linear companding technique
less Local Area Networks (WLAN). Multicarrier Code
Division Multiple Access (MC-CDMA), which is a com- Recently, various schemes have been proposed to re-
bination of two radio access techniques called Orthogonal duce the PAPR of OFDM and MC-CDMA signals [4-11].
Frequency Division Multiplexing (OFDM) and Code Among these schemes, nonlinear companding transforms
Division Multiple Access (CDMA), has attracted more are the most attractive schemes due to their good system
and more attentions as a very promising modulation performance, the simplicity of implementation, without
technique [3]. restriction on the number of subcarriers, the type of con-
The main idea behind MC-CDMA is to spread and stellation and any bandwidth expansion [12-15].
convert input signals into parallel data streams, which are In this paper, we analyze three nonlinear companding
then transmitted over multiple carriers. MC-CDMA can techniques, namely “C1 (.)”, “C2 (.)”[14] and “exponen-
realize high bit rate and large capacity transmission. tial companding- C(.)”, [15],to reduce the PAPR of
37
© 2010 ACEEE
DOI: 01.ijns.01.01.08


OFDM signals. These techniques effectively transform
the original Gaussian-distributed OFDM signals into uni- By using companding transform, sn are companded
form-distributed (companded) signals without changing before converted into analog waveforms and amplified by
the average power level. Unlike the µ-law companding the HPAs. sc(n) is given by sc(n) = C(sn), (n = 0, 1, · · ·,N
scheme, which mainly focuses on enlarging small signals, − 1),
nonlinear companding schemes adjust both small and where C(·) is the companding transform function.
large signals without bias so that it is able to offer better Then, OFDM and MC-CDMA signals are transmitted
performance in terms of PAPR reduction, Bit-Error-Rate into the radio channel. After passing through both
(BER) and phase error for OFDM and MC-CDMA sys- AWGN and the frequency selective fading channel, the
tems. The rest of this paper is organized as follows. received signal can be formulated as r(n) = sc(n) + w(n) =
C(sn) + wn, where wn =w(n)*hinv(n) with the variance of
In Section II, typical OFDM and MC-CDMA system is
σ2w.At the received end of the de-companded MCM sys-
given and the high PAPR problem is formulated. Then,
tem, r(n) has to be expanded according to Sn’ = C−1(sc(n)
nonlinear companding techniques, namely “C1(.)”,
+wn) = sn + C−1(wn), where C−1(·) is the de-companding
“C2(.)” and “exponential companding-”C(.),” is ana-
function, or the inverse function of C(·).
lyzed in Section III to reduce PAPR. In Section IV, the
performance of the three companding schemes are stu-
III.ANALYSIS OF NONLINEAR COMPANDING
died and compared through computer simulations, fol-
TRANSFORMS
lowed by conclusions in Section V.
In this section, we propose the design criteria of the
nonlinear companding transforms, which is based on de-
II.PROBLEM FORMULATION rivation and analysis of some existing novel nonlinear
companding transforms[14],[15]. As examples, the exist-
Figure 1&2 shows the block diagram [14], [10] of a ing nonlinear companding transforms can effectively
typical OFDM and MC-CDMA systems using the nonli- reduce the PAPR of the OFDM signals by transforming
near companding technique for PAPR reduction. Let N
the statistics of the amplitudes or power of the original
denote the number of sub-carriers used for parallel in- OFDM signals into uniform distribution. These novel
formation transmission and let SK(0<K<N-1) denote the schemes also have the advantage of maintaining a con-
Kth complex modulated symbol in a block of N informa-
stant average power level in the nonlinear companding
tion symbols. The outputs sn of the N -point Inverse Fast
operation. The strict linearity requirements on HPA can
Fourier Transform (IFFT) of SK are the OFDM signal
then be partially relieved.
samples over one symbol interval, or mathematically
Let us denote X and Y as random variables representing
the amplitudes of the inputs and outputs signals of the
1 .2 companding transform C1 (·) with the CDFs marked
exp 1 Fx(x) and Fy (y), respectively. Since Y is to be desired the
√ uniform distribution in the interval [0, h1] (h1 > 0), then
the CDFs of Y can be given by
According to the central limit theorem, it follows that for 1
large values of N, sn becomes Gaussian distribution with , 0 1 5
2 1 2
the probability density function (PDF)
Since the Fx(x) and Fy (y) are strictly monotone in-
1 creasing functions, they have corresponding inverse func-
fsn s exp 2 tions. At the same time, the companding transform C1(·)
√2πσ 2
is also restricted to be a strictly monotone increasing
where σ2 is the variance of the original MCM signals. function and has its inverse transform. When these condi-
Therefore, the signal sn has distribution with the cumu- tions are satisfied, we can educe these conclusions as
lative distribution function (CDF) as following following

1 1 1
1 erf 3 1 6
2 √2
where erf .
√
Therefore, the companding function C1 (·) is given by
Then, the PAPR of MCM signals sn in one symbol pe- the following identity
riod is defined as
1 7
| |
10 4 Substituting (3) and (5) into (7) shows that
38
© 2010 ACEEE
DOI: 01.ijns.01.01.08


transforms can compress large signals and expand small
1 1. erf ,0 1 8 signals simultaneously. However, μ-law transform only
√ enlarges small signals and does not change the signal
peaks, which increases average power of the companded
The positive constant h1 determines the average power signals.
of the output signals. In order to keep the input and out-
put signals at the same average power level during the 3

companding transform, companding transform C1(·) C1(x)
C2(x)
should satisfy E[(|sn|)2] = E[(|tn|)2]. Therefore, we can 2.5 µ-law
obtain
2
9

Output C(x)
1.5

Substituting (5) into (9), we can obtain that h1 = √3σ.
Therefore, nonlinear companding transform function 1
C1(x) can be expressed as
1 √3 . erf ,0 1 (10) 0.5
√

Considering the phase of input OFDM signals, we can 0
1 2 3 4 5 6 7 8 9 10 11
obtain another nonlinear companding transform function "x"
based on transforming the power of MCM signals into a Figure .2 (a) Profiles of C1 (.), C2 (.) and μ-law
uniform distribution as following
1
µ-law
| | 0.8 d=2
2 . 2. erf ) (11) d=1
√ 0.6

0.4
Similarly, in order to keep the average power of the
0.2
companded OFDM signals the same level with that of the
original OFDM signals, namely E[(|sn|)2] = E[(|tn|)2],
C(x)

0
and thus, the parameter h2 can be obtained h2 = 2σ. -0.2
Therefore, C2(x) can be expressed as
-0.4

| | -0.6
2 . 2 . erf (12)
√ -0.8

-1
In the same way, the nonlinear transform function C(x) 1 2 3 4 5 6 7 8 9 10 11
"x"
can be derived
Figure .2 (b) The exponential companding function C(x)
√6 1 exp (13)
Figure 2(b) shows exponential companding function
It belongs to the type of “Exponential Companding C(x) with degree‘d’ as a parameter. The exponent ‘d’ is
Transform”[15]. Considering the phase of input signals, called the degree of a specific exponential companding
the exponential companding function C(x) is given by scheme. The companded signals have uniformly distri-
buted amplitudes and powers, respectively, for the cases
d=1and d=2. When d 2, the proposed function can com-
1 exp (14) press large input signals and expand small signals simul-
taneously.
where sgn(x) is the sign function. The positive constant
determines the average power of output signals. The IV.PERFORMANCE EVALUATION
companded signals have uniformly distributed amplitudes
To verify the performance of the nonlinear compand-
and power, respectively for the cases d=1and d=2. At the
ing schemes in the PAPR reduction and BER perfor-
receiver side, the inverse function of C(x) is used in the
mance, the simulation parameters are set as follows. For
de-companding operation.
OFDM system: BPSK modulation, the number of data
Figure. 2(a) depicts the profiles of three types of nonli-
symbols per user N is 8, 16,32&64.For MC-CDMA sys-
near companding transforms C1 (·) C2 (·) and μ-law,
tem: BPSK modulation, the number of data symbols per
from that we can see the proposed nonlinear companding
user N is 8 and 16, the number of active users K is 2 and
39
© 2010 ACEEE
DOI: 01.ijns.01.01.08


3 . Hadamard transform is used as spreading code with curve marked with “Ideal” is obtained without nonlinear
length L=8.In order to obtain sufficient transmit power; transform and ignoring the effect of SSPA, which means
most radio systems often employ HPAs in the transmitter. directly transmitting the original MCM signals into chan-
nel. It has the best BER performance, but it has an ex-
tremely high PAPR compared with that of companded
signals.
Moreover, from Figure. 3 for OFDM system, compared
TABLE I
OFDM PAPR REDUCTION COMPARISON
to all companding schemes, C1 (·) can offer better BER
performance, and it is only about 4.0dB loss compared to
PAPR(db) the case without any companding scheme at BER = 10-2.
C1 (·) can obtain an improvement of BER performance
N Ideal C1(.) C2(.) C(.) C(.) μ-law
about 3.0 dB relative to C2 (·), but it has higher PAPR
d=1 d=2
than that of C2(·).
8 3.010 1.906 1.491 1.986 2.128 2.754
16 2.916 2.119 1.403 2.185 2.433 2.701 0
BER Performance of OFDM modulaton

32 5.016 3.920 2.589 4.014 4.376 4.676 10
ideal
64 8.304 6.683 4.151 6.826 6.983 7.762 C1(.)
C2(.)
C(.)d=1
C(.)d=2
-1
PAPR reduction comparison of OFDM for various 10

companding transforms are given in Table I. For different

BER
sub carrier sizes, the reduction in PAPR is calculated for
four companding transforms C1(.), C2(.) ,C(.) and μ-law. -2
10
Ideal value of PAPR is calculated without applying any
companding transform. The reduction in PAPR increases
with the increase in number of sub carriers. For various
subcarrier sizes (N = 8, 16 32 and 64) companding trans- 10
-3

form C2 (.) gives the best reduction in PAPR. The PAPR 0 5 10 15
SNR (db)
20 25 30

value calculated for N=64, BPSK shows a reduction of
Figure .3. BER versus SNR of OFDM in AWGN fading channel under
1.621db can be obtained for C1 (.) and a reduction of different companding transforms
4.153db can be obtained for C2 (.) end a reduction of 1
BER Performance of MC-CDMA(3 USER) System
10
1.478 db can be obtained for C(.),exponential compand- ideal
ing for d=1.Obviously, the signals companded by the C1(.)
nonlinear companding transforms C (.), C1 (·), C2 (·) can C2(.)

reduce the PAPR greater than that of μ-law companding
0
transform. 10

TABLE II
2 USER MC-CDMA PAPR REDUCTION COMPARISON
BER

PAPR(db)
-1

N Orig C1(.) C2(.) C(.) C(.) μ-law 10

inal d=1 d=2
8 12.4 6.44 4.74 6.65 6.55 11.1
PAPR(db)
16 9.40 3.26 1.72 3.46 3.25 8.03
N Ideal C1(.) C2(.) C(.) C(.) μ-law -2
10
0 5 10 15 20 25
d=1 d=2 SNR (db)

8 10.7 5.34 4.12 5.50 5.24 9.57 Figure 4. BER versus SNR of MC-CDMA in AWGN fading channel
16 7.68 2.13 0.96 2.28 1.99 6.46 under different companding transforms

TABLE III
3 USER MC-CDMA PAPR REDUCTION COMPARISON From Figure 4 for MC-CDMA system, compared to all
companding schemes, C1 (·) can offer better BER per-
PAPR reduction comparison of MC-CDMA for vari- formance, and it is only about 0.12dB loss compared to
ous companding transforms are given in Table II &III. the case without any companding scheme at BER = 10-2.
The companding transform C2 (.) gives the best reduction C1 (·) can obtain an improvement of BER performance
in PAPR among all other transforms for 2 user and 3 user about 3.0 dB relative to C2 (·), but it has higher PAPR
MC-CDMA. than that of C2 (·).
The wireless channel is assumed to be AWGN. Fig- Due to the high PAPR, ideal OFDM&MC-CDMA sig-
ure3&4 depicts the performance of BER versus signal-to- nals have a very sharp, rectangular-like power spectrum
noise ratio (SNR) of actual signals under different com- [14]. This good property will be affected by the PAPR
panding schemes. Note that, the performance bound reduction schemes. Non linear companding scheme has
40
© 2010 ACEEE
DOI: 01.ijns.01.01.08


much less impact on the original power spectrum [15]. timal approach based on dual layered phase sequencing,”
the major reason that the non linear companding scheme IEEE Trans. Broadcast., vol. 49, pp. 225–231, Jun. 2003.
not only enlarges the small amplitude signals but also [8] S. G. Kang, J. G. Kim, and E. K. Joo, “Anovel subblock
compresses the large amplitude signals, while maintains partition scheme for partial transmit sequence ofdm,” IEEE
Trans. Broadcast., vol. 45, pp.333–338, Sep. 1999.
the average power unchanged by properly choosing pa- [9] S. G. Kang, J. G. Kim, and E. K. Joo, “Anovel subblock
rameters, which can increase the immunity of small am- partition scheme for partial transmit sequence ofdm,” IEEE
plitude signals from noise. The signals companded by Trans. Broadcast., vol. 45, pp.333–338, Sep. 1999.
C1 (.) have a good spectrum characteristic and have al-
most no spectral regrowth caused by the PAPR reduction. [10] E Alsusa & Lin Yang, "MC-CDMA Specific PAPR Re-
For C2(·) nonlinear transform[14], it has the worst power duction Technique Utilising Spreading Code Redistribu-
spectrum, which may be caused by transforming the tion", in Proc. IEEE Vehicular Technology Conf ,Fall
power not amplitude of OFDM&MC-CDMA into a uni- 2006, Montreal, Canada, Sept. 2006.
form distribution, so that it can bring out-of-band distor- [11] R. W. Bami, R. F. H. Fischer, and J. B. Hber, “Reducing
the peak-to average power ratio of multicarrier modulation
tion and result in more severe inter-carrier interference. by selective mapping,” IEE Electron. Lett., vol. 32, pp.
2056–2057, Oct. 1996.
V.CONCLUSION [12] S. H. Han and J. H. Lee, “Modified selected mapping tech-
nique for papr reduction of coded ofdm signal,” IEEE
Non-linear companding transform is an effective Trans. Broadcast., vol. 50, pp. 335–341, Sep. 2004.
technique in reducing the PAPR of OFDM&MC-CDMA [[13] X. Wang, T. T. Tjhung, and C. S. Ng, “Reduction of peak-
signals. In addition, the schemes based on nonlinear to-average power ratio of ofdm system using a compand-
companding techniques have low implementation com- ing technique,” IEEE Trans.Broadcast., vol. 45, pp. 303–
307, Sep. 1999.
plexity and no constraint on modulation format and sub-
[14] Tao Jiang, Weidong Xiang, Paul C. Richardson, Daiming
carrier size. In this paper, three nonlinear companding Qu, and Guangxi Zhu “On the Nonlinear Companding
transforms based on the derivation and theoretical analy- Transform for Reduction in PAPR of MCM Signals” IEEE
sis of some existing nonlinear companding transforms Transaction on Wireless communications, Vol. 6, No. 6,
were analyzed and evaluated for reducing the PAPR of June 2007
OFDM&MC-CDMA signals. The BER performance of [15] Tao Jiang, Yang Yang, Yong-Hua Song “Exponential
the nonlinear companding transforms was also studied by Companding Technique for PAPR Reduction in OFDM
mat lab simulation. It is proved that the best tradeoff be- Systems” IEEE Transaction on Broadcasting, Vol. 51, NO.
tween BER performance and PAPR reduction can be 2, June 2005
.
achieved by C1 (·) among these nonlinear transforms but
C2 (·) has the best PAPR reduction. Simulation results
have shown that C1 (·) companding scheme could offer
better system performance in terms of PAPR reduction,
power spectrum, BER for MC-CDMA system than
OFDM system.

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41
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PAPR Reduction in OFDM & MC-CDMA System using Nonlinear Companding Techniques

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