DISCRETE COSINETRANSFORM-II FOR REDUCTION IN PEAK TO AVERAGE POWER RATIO OF OFDM SIGNALS THROUGH -LAW COMPANDING TECHNIQUE

International Journal of Wireless & Mobile Networks (IJWMN) Vol. 5, No. 2, April 2013
DOI : 10.5121/ijwmn.2013.5211 143
DISCRETE COSINETRANSFORM-II FOR
REDUCTION IN PEAK TO AVERAGE POWER RATIO
OF OFDM SIGNALS THROUGH -LAW
COMPANDING TECHNIQUE
NavneetKaur and Lavish Kansal
Department of Electronics and Communication Engineering
Lovely Professional University, Phagwara
Punjab, India
nv_neet@yahoo.com
Department of Electronics and Communication Engineering
Lovely Professional University, Phagwara
Punjab, India
lavish.15911@lpu.co.in
Abstract
Orthogonal frequency Division multiplexing (OFDM) is the most familiar word in telecommunication
and wireless communication systems as it provides enhanced spectral efficiency than Frequency division
multiplexing (FDM).Although it is sustaining an orthogonal relation betweencarriers but high peak to
average power ratio (PAPR) is one of the main disadvantages of OFDM system.Various PAPR reduction
techniques have been used, including techniques based on companding. Incompanding, -Law
companding has potential to reduce the PAPR of OFDMsignals. -Law Companding technique
preserves the dynamic range of samples at low amplitudes.A new method named as precoding which is
having less complexity compared to the other power reduction techniques is proposed to reduce PAPR.
This paper put forward combination of two existing techniques namely -Law Companding Transform
and Discrete Cosine Transform-II precoding technique. The simulation results show that, the proposed
combinedscheme gives better result for PAPR Reduction and results in no distortion.
Keywords
FDM, FDM, PAPR, FFT, IEEE, DCT, CCDF
1. INTRODUCTION
As the demand for high data rate wireless digital communication is rapidly expanding, several
approaches such as increasing modulation order or employing multiple antennas at both
transmitter and receiver that are reliable for wireless systems have been studied to enhance the
spectral efficiency [1,2]. In today’s communication systems Orthogonal Frequency Division
Multiplexing (OFDM), widespread modulation technique has been considered to achieve high
rate data transmission for wireless environment.OFDM system has high spectral efficiency,
ease of implementation using the fast Fourier transform (FFT) and simple equalization
techniques and also provides immunity to multipath fading, robustness against inter-symbol

144
interference. Therefore it is considered for digital communication systems and also for future
broadband applications.
Multicarrier networks such as Frequency Division Multiplexing (FDM) have been around since
the late 1950’s [3], however due to their implementation complexity transmitted in parallel.
Multicarrier systems in wireless applications are less susceptible to channel induced distortions
than single carrier systems at corresponding data rates and inefficient use of the frequency band
they were restricted to military applications.
In multicarrier transmission, a single data stream bearing a large number of narrow-band
subcarriers is transmitted over a number of lower rate subcarriers.Thesesubcarriers are regularly
spaced in frequency, domain to avoid the interference. The subcarriers are totally independent
of each other because thefrequency spacing and time synchronization of the subcarriers is
chosen in such a waythat the subcarriers are orthogonal. By selecting a special set of orthogonal
carrier frequencies, high spectral efficiency isobtained because the spectra of the sub-carriers do
not overlap. The orthogonality among subcarriers can be maintained by introducing a cyclic
prefix over a dispersive channel. The advantage of this structure over single carrier systems is
that the extended symbol time due to lower data rate makes the signal less susceptible to effects
of the channel such as multipath propagation which introduces Inter Symbol Interference (ISI).
Each sub-channel will therefore experience flat fading reducing the equalization complexity in
the receiver.
Fig. 1: Block Diagram of OFDM system
In an OFDM scheme shown in fig. 1, the orthogonal and overlapped subcarriers are a large
number of orthogonal, overlapping, narrow band sub-carriers are transmitted in
parallel.Maintaining total data rates similar to conventional single-carrier modulation schemes
in the same bandwidth to modulate each sub-carrier is with a conventional modulation scheme
such as Quadrature Amplitude Modulation (QAM) or Phase-Shift Keying (PSK) at a low
symbol rate than that required for the whole data stream, but still.In OFDM system a high speed
data is splits into a number of parallel lower data rate streams and these low rates data streams
are transmitted simultaneously over a number of orthogonal subcarriers.The amount of
dispersion in time caused due to multipath delay is reduced as, the symbol duration increases
for lower rate parallel sub-carriers. These carriers divide the available transmission bandwidth.
144
the receiver.
144
the receiver.

145
The sub-carriers are separated in such a way so that there is a very compact spectral utilization
and each being modulated at a low bit rate. In a conventional frequency division multiplex the
carriers are spaced apart to avoid spectral overlap and multipath distortion.
OFDM has progressed to the point where it has now been widely used for various
communication applications such as digital audio and video broadcasting, asymmetric digital
subscriber line modems, and wireless local area networks systems(WLAN).It has also been
adopted as the physical layer modulation scheme for wireless networking standards such as
Hiperlan2 in Europe and the Institute of Electrical and Electronic Engineers (IEEE) 802.11a, g
standards in the United States.
OFDM systems have a large dynamic signal range with a very high peak to average power ratio
(PAPR) which is a one of major drawbacks of the transmitted OFDM signal. As a result, the
OFDM signal will be clipped when passed through a nonlinear high power amplifier (HPA) at
the transmitter as it only operate in its linear region with sufficient back-off. Thus, HPA with a
large dynamic range is required for OFDM systems. These amplifiers are very expensive and
are major cost component of the OFDM systems. Thus reduction in the value of the PAPR not
only reduces the cost of OFDM systems and reduces the complexity of analog-to-digital (A/D)
and digital-to-analog (D/A) converters, but also increases the transmit power, thus, improving
received signal-noise-ratio (SNR).
To reduce the PAPR, several techniques have been proposed, which can be divided into three
groups.The first group of reduction techniques includes signal distortion techniques, which
reduce the peak amplitudes simply by nonlinearly distorting the OFDM signal at or around the
peaks. Examples of distortion techniques are clipping [5], peak windowing [6], and peak
cancellation. Second, there are coding techniques that use a special code set that excludes
OFDM symbols with a large PAPR. The special codes includes Golay complementary
sequence [7], Reed Muller code [8], and Hadamard code in this approach [9].The third
technique scrambles each OFDM symbol with different scrambling sequences and selecting the
sequence that gives the smallest PAPR such as selected mapping (SLM) [10] and partial
transmit sequence (PTS) [11].
The organization of this paper is as follow. Section 2 presents formulates the problem of PAPR
in OFDM system.The reduction technique based on µ-Law companding is explained in section
3.Section 4 describes the precoding based Discrete Cosine Transform-II technique. PAPR
schemes based on combined DCT-II and µ-Law companding technique are proposed in section
5. Computer simulations results are presented in Section 6. Section 6 draws conclusions
regarding the existing and proposed combined technique.
2. PAPR IN OFDM SIGNAL
Presence of large number of independently modulated sub-carriers in an OFDM system
increasesthe peak value of the system as compared to the average of the whole system. The
ratio of the peak to average power value is termed as Peak to Average Power Ratio.At some
instant in time coherent addition of independently modulated N sub-carriers same phase
produces a peak:
PAPR[ ( )]=

146
=10
| ( )|
[| | ]
(1)
Where represents peak output power, means average output power. E[ ·]
denotesthe expected value, represents the transmitted OFDM signals which are obtained by
takingIFFT operation on modulated input symbols is expressed as:
=
∑ (2)
WhereN is the total number of subcarriers and the OFDM symbol duration is T seconds.
An OFDM symbol consists of N subcarriers by the frequency spacing of ∆f. The total
bandwidthBof a symbol will be divided into N equally spaced subcarriers and thus, all the
subcarriers are orthogonal to each other within a time interval of length T=1 ∆⁄ . A complex
modulation symbol , , isused to modulate each sub-carrier independently where m is a time
index and n is a subcarrier index.Then within the time interval T the following signal of the m-
th OFDM block period can be described as:
( )=√
∑ , ( − ) (3)
Where, (t) is defined as:
(t)=
exp( 2 ∆ ) , 0 ≤ ≤
0,
(4)
Where, (t) is a rectangular pulse applied to each subcarrier. The total continuous time signal
x (t) consisting of all the OFDM block is given as:
( )=
√
∑ ∑ , ( ) (5)
When compared with traditional single carrier systems, the instantaneous output of an OFDM
system often has large fluctuations. To handle these large fluctuations the system devices, such
as power amplifiers, A/Dconverters and D/A converters, must have large linear dynamic
ranges. If the peak signal goes into the non-linearregion of devices at the transmitter,a series of
undesirable interference is encountered, such as high out of band radiation and inter-
modulationdistortion. Therefore PAPR reduction techniques are requiredfor OFDM systems.
The power output has large fluctuations due to which, the high power amplifier (HPA) should
have largedynamic range. This results in poor power efficiency.
OFDM systems combine a number of independent sub carriers which causes increase in PAPR.
The increase in PAPR is related to the number of sub carriers and their order of modulation. A
curve called as Complementary Cumulative Distribution Function (CCDF)is used to present the
vital information regarding the OFDM signal to be transmitted. These curves also provide the
PAPR data needed by component designer. The main use of power CCDF curves is to identify
the power characteristic of the signals which are amplified, mixed and decoded [12]. The plot
of relative power levels of signal against their probability of occurrence is called CCDF curve.

147
This curve illustrates the amount of time the signal remains at or above a given threshold level
[13, 14]. The ratio between power level and the average power is expressed in dBas:
CCDF=Prob(PAPR > ) (6)
where is the threshold level.
3. -LAW COMPANDING TECHNIQUE
In an OFDM system,large signals occur very infrequently therefore itis similar to thespeech
signal, so the same companding technique can be used to improve the OFDM transmission
performance. One of the companding technique named asµ-Law Companding can effectively
reduce the Peak to Average Power Ratio (PAPR) of the transmitted or the companded OFDM
signals by transforming the statistics of the amplitudes of these signals into uniform
distribution. The key idea of the µ-law companderis that it covers all the encountered
amplitudes of the OFDM signal. In µ-law companding, the uniform distribution of the signals
can be obtained by compressing the peak signals and expanding the small signals. In the
process of companding, the amplitudes of the small signalsget enlarged, while the peaks remain
unchanged. Therefore, the average power is increased and thus the PeaktoAverage Power Ratio
(PAPR) can be reduced. This technique overcomes the deficiencies that are caused by clipping
the original OFDM signal. The other advantage of the companding transform is that, it
eliminates the Out of Band Interference (OBI) and can also maintain a constant average power
level and also.
The original OFDM signal is converted into the companded signal by using the Mu
Companding Transform. The companded signal obtained by using the MuCompanding
Transform is given by the equation as:
( )=
( ) ( | |)
( )
(7)
Where, ( )=Companded Signal obtained byµ-LawCompandingTechnique,
( )=sign function,
=original OFDM signal,
µ=255 and it is the Mu-Law parameter of the compander, which controls the amount
ofcompression.
4. DISCRETE COSINE TRANSFORM-II
The discrete cosine transforms (DCT) is a Fourier-related transform similar to the discrete
Fourier transform (DFT). It is a member of a family of sinusoidal unitary transforms. This
transform is real, orthogonal and separable with fast algorithms for its computation. They have
abetter job of concentrating energy and compression of audio and data. The Discrete Cosine
Transform (DCT) was first proposed by Ahmed et al. (1974) [15]. It is a Fourier-like
transform, which equivalent to a DFT of roughly twice the length, operating on real data with
even symmetry. The discrete cosine transform-II is a linear, invertible function F:  ;
whereR denotes the set of real numbers, or equivalently an n × n square matrix. Then real
numbers ,…, are transformed into the n real numbers ,…, according to one of the
formulas:

148
The DCT-II implies the boundary conditions: is even around k=-1/2 and even around k=n-
1/2; is even around j=0 and odd around j=n.
The DCT II is important because it has better computational efficiency, superiority in
bandwidth compression of a wide range of signals, its powerful performance in the bit-rate
reduction and robust against inter carrier interference (ICI). The precoding based techniques are
simple and linear techniques with less complexity to implement without the need of any side
information.
5. PROPOSED SCHEME
A new technique named as precoding technique is also used in OFDM systems to
reduce the PAPR. Precoding techniques obtain a signal with lower PAPR than in the case of a
without precoded OFDM system and it also reduces the interference produced by multiple
users. The µ-LawCompanding Transform also causes less spectrum side-lobes as compared to
other reduction techniques.In proposed scheme µ-Law companding was combined with
precoding technique called Discrete Cosine Transform-II. The combined techniques based
OFDM system was shown in fig. 2. In the transmitter, firstly DCT-II is used to transform the
parallel data stream, and then the transformed data is processed by the companding unit. If data
block passed by DCT-II, before IFFT, the PAPR is reduced, then the PAPR of OFDM signal
could be further reduced by companding.
Fig. 2 Block scheme of DCT-II Precoding technique withµ-law companding in OFDM system

149
After applying the DCT-II on OFDM signal the signal get compressed decreasing the PAPR
value. Mucompanding is performed on the I and Q outputs of the IFFT after precoding. As the
word length at the IFFT output is decreased, the power consumption and complexity of the
DAC/ADC decreases. Also companding at the IFFT output increases the resolution giving a
better average signal.
6. SIMULATION RESULTS
To reduce the PAPR, the above discussed has been designed and results are shown in the form
of CCDF vs. PAPR plot for different modulations and different values of M. In simulation, an
OFDM system is considered under an additive white Gaussian noise (AWGN ) channel, where
data is randomly generated with subcarriers N=2400, then the signal is modulated by M-PSK
and M-QAM respectively. The PAPR reduction capability is measured by the complementary
cumulative distribution (CCDF = Prob (PAPR>PAPR0), which indicates the probability that
PAPR is above a certain threshold. We compared the simulation results of proposed system
with DCT-II precoded OFDM signal and precodedµ-Law companded signal. The results were
observed for precoded signal and for precodedµ-Law companded signal.
6.1 M-PSK Modulation
In this section µ-Law companding technique is applied over DCT-II precoded OFDM signal
with phase shift keying modulation. We assume CCDF clip rate of 10 .
using M-ary PSK
technique for N=2400 and M=16, 32, 64, 128,256, 512 and 1024.
Fig. 3: CCDF of µ-Law companding with proposed DCT-II technique for 16 PSK
For OFDM system CCDF vs. PAPR plots using M=16 over AWGN channel is shown in Fig. 3.
The graph gives the clear idea that at CCDF rate of 10 .
the PAPR value decreases by 0.53
dB using PSK modulation. The proposed precoded µ-Law companded technique provides
much reduction in PAPR value as compared to DCT-II precoded signal.
6 6.5 7 7.5 8 8.5 9
10
-2
10
-1
10
0
PAPR Reduction for PSK Modulation
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)

150
dB using PSK modulation. The proposed precodedµ-Law companded technique provides much
reduction in PAPR value as compared to DCT-II precoded signal.
the PAPR value decreases by 0.7 dB
using PSK modulation. The proposed precoded µ-Law companded technique provides much
5 5.5 6 6.5 7 7.5 8 8.5 9
10
-2
10
-1
10
0
PAPR0
CCDF
(W HT OFDM)
(Mu-Companded)
5 5.5 6 6.5 7 7.5 8 8.5
10
-0.9
10
-0.7
10
-0.5
10
-0.3
10
-0.1
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)

151
For OFDM system CCDF vs. PAPR plots using M=128 over AWGN channel is shown in Fig.
6. The graph gives the clear idea that at CCDF rate of 10 .
5 6 7 8 9 10 11
10
-0.9
10
-0.7
10
-0.5
10
-0.3
10
-0.1
PAPRReduction for PSK Modulation
PAPR0
CCDF
(WHTOFDM)
(Mu-Companded)
6 6.5 7 7.5 8 8.5 9
10
-0.8
10
-0.7
10
-0.6
10
-0.5
10
-0.4
10
-0.3
10
-0.2
10
-0.1
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)

152
Fig. 8: CCDF ofµ-Law companding with proposed DCT-II technique for 512 PSK
6.2 M-QAM Modulation
For M=16, 64, 256 and 1024 M-ary quadrature amplitude modulation (QAM) technique is used
here at CCDF rate of 10 .
.
6 6.5 7 7.5 8 8.5 9
10
-0.7
10
-0.6
10
-0.5
10
-0.4
10
-0.3
10
-0.2
10
-0.1
PAPR0
CCDF
(W HT OFDM)
(Mu-Companded)
5.5 6 6.5 7 7.5 8
10
-0.7
10
-0.6
10
-0.5
10
-0.4
10
-0.3
10
-0.2
10
-0.1
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)

153
Fig. 10: CCDF ofµ-Law companding with proposed DCT-II technique for 16 QAM
the PAPR value decreases by
0.87 dB using QAM modulation. The proposed precoded µ-Law companded technique
provides much reduction in PAPR value as compared to DCT-II precoded signal.
Fig. 11: CCDF of µ-Law companding with proposed DCT-II technique for 64 QAM
0.83 dB using QAM modulation. The proposed µ-Law companded technique provides much
4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
10
-2
10
-1
10
0
PAPR Reduction for QAM2 Modulation
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)
5.5 6 6.5 7 7.5 8 8.5 9 9.5
10
-0.9
10
-0.7
10
-0.5
10
-0.3
10
-0.1
PAPR Reduction for QAM Modulation
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)

154
1.23 dB using QAM modulation. The proposed precoded µ-Law companded technique
provides much reduction in PAPR value as compared to DCT-II precoded signal.
dB using QAM modulation. The proposed precoded µ-Law companded technique provides
7. CONCLUSIONS
In this paper, a new idea about the performance of the µ-Law companding technique combined
with precoded technique is presented for the reduction of PAPR value. The PAPR value can be
reduced for different modulation formats and for different values of M by using proposed
6 6.5 7 7.5 8 8.5 9
10
-0.8
10
-0.7
10
-0.6
10
-0.5
10
-0.4
10
-0.3
10
-0.2
10
-0.1
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)
5 5.5 6 6.5 7 7.5 8
10
-0.7
10
-0.6
10
-0.5
10
-0.4
10
-0.3
10
-0.2
10
-0.1
PAPR0
CCDF
(WHT OFDM)
(Mu-Companded)

155
scheme without increasing the system complexity. The scheme shows superiority in bandwidth
compression of signals and powerful performance in the bit-rate reduction. This system
produce the lowest PAPR and is efficient, signal independent, distortion less and does not
causes any spectrum side-lobes. The simulation results shows, that the proposed precoded
combinedµ-law companding method displays a better PAPR reduction performance than the
simple DCT-II precoded OFDM signal. Thus, it is concluded that the proposed scheme reduces
PAPR effectively compared with precoded DCT-II transform.
REFERENCES
[1] H. Jiang & P. A. Wilford, "A hierarchical modulation for up grading digital broadcasting systems,"
IEEE Transaction on Broadcasting, Vol. 51, pp. 222-229, 2005.
[2] P. W. Wolniansky, G. J. Foschini, G. D. Golden & R. A. Valenzuela, "V-BLAST: an architecture
for realizing very high data rates over the rich- scattering wireless channel," In Proceeding of
International symposium on Signals, Systems and Electronics, pp. 259-300, 1998.
[3] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come”,
IEEE Communication Magazine, Vol. 28, Issue: 1, pp. 5-14, 1990.
[4] R. Gross & D. Veeneman, “Clipping distortion DMT ADSL systems,” IEEE Electronics
Letters, Vol. 29, pp. 2080- 2081,1993
[5] T. Jiang & Y. Wu, “An Overview: Peak to Average Power Ratio Reduction Techniques for
OFDM Signals,” IEEE Transactions on Broadcasting, Vol. 54, Issue: 2, pp. 257-268, 2008
[6] F. Fiedler, J. Jedwab& M. G. Parker,“ A framework for the construction of Golay sequences,”
IEEE Information Theory, Vol. 54, pp. 3114-3129, 2008.
[7] J. A. Davis & J. Jedwab, “Peak to mean power control and error correction for OFDM
transmission using Golay sequences and Reed-Muller codes,” Electronics Letters, Vol. 33, pp.
267-268, 1997.
[8] M. Park, J. Heeyong, J. Cho, D. Hong & C. Kang,“PAPR Reduction in OFDM
Transmission Using Hadamard Transform,” IEEE International Conference on
Communications, Vol. 1, pp. 430-433, 2000.
[9] R. W. Bauml, R. F. Fischer & J. B. Huber, “Reducing the Peak to Average Power Ratio of
Multicarrier Modulation by Selected Mapping,” Electronics Letters, Vol. 32, Issue: 22, pp. 2050-
2057, 1996.
[10] H. Breiling, S.H. H. Muller Weinfurtner& J.B. Huber, “SLM peak power reduction without
explicit side information,” IEEE Communication Letters, Vol. 5, Issue: 6, pp. 239-241, 2001.
[11] S. H. Muller and J. B. Huber, “OFDM Ratio by Optimum Combination of Partial Transmit
Sequences,” IEEE Electronic Letters, Vol. 33, Issue: 5, pp. 368-369, 1997.
[12] Agilent Technologies Application Note, “Characterizing Digitally ModulatedSignal with CCDF
Curves”, 2000.
[13] R.Van Nee and A. de Wild, “Reducing the peaktoaverage power ratio of OFDM,”IEEE
Proceeding on Vehicular Technology Conference, pp. 2072-2076, 1998.
[14] H. Ochiai and H. Imai, “On the distribution of thepeak to average power in OFDM signals,” IEEE
Transactions on Communications, Vol.49, pp.282-289, 2001.
[15] N. T. Ahmed, Natarajan& K. R. Rao, “Discrete Cosine Transform,” IEEE Transactions on
Computer, Vol. 23, pp. 90-93,1974.

156
[16] L. Kansal, A. Kansal& K. Singh, “Analysis of Different High Level Modulation Techniques for
OFDM System,” International Journal of VLSI and Signal Processing Applications, Vol. 1, Issue:
2, 2011.
[17] L. Kansal, A. Kansal& K. Singh, “Performance Analysis of MIMO-OFDM System Using
QOSTBC Code Structure for M-PSK,” Signal Processing: An International Journal, Vol. 5, Issue:
2, 2011.
[18] N. Kaur& L. Kansal, “Reducing the Peak to Average Power Ratio of OFDM Signals through
Walsh Hadamard Transform,” Global Journal of Researches in Engineering, Vol. 13, Issue: 1,
2013.
[19] N. Kaur& L. Kansal, “Peak to Average Power Ratio Reduction of OFDM Signal by Combining
Clipping withWalsh Hadamard Transform,”International Journal of Wireless& Mobile
Networks,Vol. 5, Issue: 1, 2013.
Authors
NavneetKaur was born in Kapurthala .She has done her B.Tech degree in
Electronics and Communication Engineering from LovelyProfessional University,
Phagwara, India. She is currently pursuing her M.Tech degree from Lovely
Professional University, Phagwara, India. Her research area includes Digital
Communication & Wireless Communication.
Lavish Kansal was born in Bathinda.He received his B.Tech degree in
Electronics and Communication Engineering from PTU, Jalandhar in 2009 and
M.E. degree in Electronics and Communication Engineering from Thapar
University, Patiala in 2011. He is working as Assistant Professor in the department
of Electronics and communication Engineering, Lovely Professional University,
Phagwara, India. He has published 15 papers in International journals. His research
area includes Digital Signal Processing, Digital Communication & Wireless
Communication.
156
2, 2011.
2, 2011.
2013.
Authors
Communication.
156
2, 2011.
2, 2011.
2013.
Authors
Communication.

DISCRETE COSINETRANSFORM-II FOR REDUCTION IN PEAK TO AVERAGE POWER RATIO OF OFDM SIGNALS THROUGH -LAW COMPANDING TECHNIQUE

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