Journal of Signal and Information Processing, 2011, 2, 100-104
doi:10.4236/jsip.2011.22013 Published Online May 2011 (http://www.SciRP.org/journal/jsip)
Combined DCT and Companding for PAPR
Reduction in OFDM Signals
Zhongpeng Wang
School of Information and Electronic Engineering, Zhejiang University of Science and Technology, Hangzhou, China.
Received March 21st, 2011; revised April 12th, 2011; accepted April 21st, 2011.
ABSTRACT
The high peak-to-average (PAPR) is one of the serious problems in the application of OFDM technology. The com-
panding transform approach is a very attractive technique to reduce PAPR, but large PAPR reduction leads to a high
bit error rate (BER) by the available companding transform techniques. In this paper, a joint reduction in PAPR of the
OFDM signals based on combining the discrete cosine transform (DCT) with companding is proposed. In the first step
of the proposed scheme, the data are transformed by a DCT into new modified data. In the second step, the proposed
scheme utilizes the companding technique to further reduce the PAPR of the OFDM signal. The performance of the
PAPR is evaluated using a computer simulation. The simulation results indicate that the proposed scheme may obtain
about 1 dB PAPR reduction compared with the conventional compand ing algorithm.
Keywords: Companding, DCT Transform, PAPR, OFDM
1. Introduction
OFDM (orthogonal-frequency-division multiplexing) is a
promising technique that is able to provide high data
rates over multipath fading channels. However, OFDM
systems have the inherent problem of a high peak-to-
average power ratio (PAPR), which causes poor power
efficiency or serious performance degradation in the
transmitted signal. To reduce the PAPR, many tech-
niques have been proposed, such as clipping, coding,
partial transmit sequence (PTS), selected mapping (SLM)
[1-3], nonlinear companding transforms [4,5], and Ha-
damard transforms [6]. These schemes are primarily sig-
nal scrambling techniques, such as PTS, and signal dis-
tortion techniques such as the clipping and companding
techniques. Among those PAPR reduction methods, the
simplest scheme to use is the clipping process. However,
use of the clipping processing causes both in-band distor-
tion and out-of-band distortion, and causes an increased
bit error rate (BER) in the system. As an alternative ap-
proach, the companding technique shows better per-
formance than the clipping technique, because the in-
verse companding transform (expanding) can be applied
at the receiver end to reduce the distortion of signal. A
DCT may reduce the PAPR of an OFDM signal, but does
not increase the BER of system. Park et al. [6] proposed
a scheme for PAPR reduction in OFDM transmission
using a Hadamard transform. The proposed Hadamard-
transform scheme may reduce the occurrence of the high
peaks when compared the original OFDM system. The
idea is to use the Hadamard transform to reduce the au-
tocorrelation of the input sequence to reduce the peak to
average power problem. In addition, it requires no side
information to be transmitted to the receiver. Inspired by
the literature [6,7], we propose an efficient PAPR reduc-
ing technique based on a joint companding and DCT
method. The proposed scheme makes use of the character
domain. The data encoded in the OFDM signal are mod-
ulated by an IFFT (inverse fast Fourier transform) after
being processed with the DCT, which can reduce the
PAPR of OFDM signals. Then, companding algorithm is
applied further to reduce the PAPR of the OFDM signal
after the IFFT operation. This scheme will be compared
with the original system with companding technique for
reduction PAPR.
The organization of this paper is as follow. Section 2
presents the PAPR problem of OFDM signals. Com-
panding transform and DCT transform are introduced in
section 3 and section 4. In section 5, a PAPR reduction
scheme by combing companding transform and DCT
transform is proposed. Simulation results and perform-
ance analysis are reported in section 6 and conclusions
are presented in 7.

Combined DCT and Companding for PAPR Reduction in OFDM Signals101



2. PAPR Problem of OFDM Signals
An OFDM signal consists of N data symbols transmitted
over N distinct subcarriers. Let
be a block of N symbols formed by each symbol modu-
lating one of a set of subcarriers .
The N subcarriers are chosen to be orthogonal, that is,
k
 
,0,1,,1
k
Xk NX

,0,1,,1
k
fk N
f
kf, where

1
f
NT and T is the original
symbol period. Therefore, the complex baseband OFDM
signal can be written as

12π
0
1e0
k
Njft
k
k
x
tX t
N



NT (1)
In general, the PAPR of the OFDM signal,


x
t, is
defined as the ratio between the maximum instantaneous
power and its average power during an OFDM symbol

 
2
0
2
0
max
PAPR
1d
tNT
NT
xt
NTx tt







 (2)
Reducing

max
x
t is the principle goal of PAPR
reduction techniques. In practice, most systems deal with
a discrete-time signal. Therefore, we have to sample the
continuous-time signal

x
t.
To better approximate the PAPR of continuous-time
OFDM signals, the OFDM signals samples oversampled
by a factor of L. By sampling,

x
t defined in Equation
(1), at frequency s
f
LT, where L is the oversampling
factor, the discrete-time OFDM symbol can be given by:

2π
1
0
1e0
Njkn
NL
k
k
1
x
nX nN
N



L (3)
Equation (2) can be implemented using an IFFT op-
eration of length (NL). The new input vector, X, is ex-
tended from the original X by using the zero-padding
scheme, i.e. by inserting


1LN zeros in the middle
of X. The PAPR computed from the L-oversampled
time-domain OFDM signal can be defined as:
 

2
01
2
max
PAPR10lgnNL xn xn Exn
 





 



(4)
where the expectation is taken over all OFDM symbols.
3. Companding Transform
In this section, we review companding techniques [5] for
the reduction of the PAPR in an OFDM signal. In this
section we study a companding transform, in which
compression is used at the transmit end after the IFFT
process and is used expansion at the receiver end prior to
the FFT (fast Fourier transform) process.
For the discrete OFDM signal given by Equation (3),
the companded signal


s
n can be given by:
  

 
ln 1
ln 1
vx nu
s
n Cxnxn
v
uxn

 


(5)
where v is the average amplitude of the signal and u is
the companding parameter. Specifically, the companding
transform should satisfy the following two conditions:





2
Esn Exn2
. (6)




s
nxn, for

x
nv; (7)




s
nxn, for


x
nv. (8)
This transform reduces the PAPR of OFDM signal by
amplifying the small signal and attenuating the period of
high signal.
On the receiver end, the receiver signal must be ex-
panded by the inverse companding transform before it
can be sent to the FFT processing unit. The expanded
signal at the receiver is
  


 
1ln 1
exp 1
rn u
vr n
ynC rnv
ur n





 





(9)
4. Discrete Cosine Transform
Like other transforms, such as the Hadamard transform,
the DCT decorrelates the data sequence. To reduce the
PAPR in an OFDM signal, a DCT is applied to reduce
the autocorrelation of the input sequence before the IFFT
operation is applied [8]. In this section, we briefly review
the DCT. The formal definition of a one-dimensional
DCT of length N is given by the following formula:
 
1
0
π21
cos ,
2
for 0,,1
N
cn
nk
Xkk xnN
kN











(10)
Similarly, the inverse transformation is defined as
 
1
0
π21
cos ,
2
for 0,,1
N
c
k
nk
xnkX kN
nN











(11)
For both Equations (10) and (11) is defined as

k 

1,for 0
2,for 0
k
N
k
k
N








(12)
Equation (10) can be expressed in matrix form as:

Combined DCT and Companding for PAPR Reduction in OFDM Signals 102

cN
X
Cx (13)
where c
X
and x are both vectors of dimension 1N

,
and N is a DCT matrix of dimension CNN

. The
rows (or column) of the DCT matrix, N, are orthogo-
nal matrix vectors. We can use this property of the DCT
matrix and reduce the peak power of OFDM signals.
C
According to [9], there is a close relation between the
PAPR of an OFDM signal and the aperiodic autocorrela-
tion function (ACF) of an input vector. Assume


i

is
the ACF of a signal vector, X, then:

1*
0
Ni
kik
k
iX 



X for (14) 0,1,,1iN
where the superscript * denotes the complex conjugate.
Then, the PAPR of the transformed OFDM signal is
bounded by [9]:

1
1
2
PAPR 1
N
i
i
N



  (15)
Let

1
1
N
i
i 


, we found that an input vector with
a lower

yields a signal with a lower PAPR in OFDM
systems. It has been proved that if input vector passed by
DCT transform before IFFT, the

i

and thus PAPR
could be reduced [9].
5. Proposed Scheme
To reduce the PAPR an OFDM signal, we propose a
scheme involving the combination of a companding
transform and DCT. The input data stream is processed
with a DCT then with an IFFT signal processing unit. A
block diagram of the system is shown in Figure 1.
The key signal processing step is described as below:
Step 1: The sequence X is transformed using the DCT
matrix, i.e. . YHX
Step 2: An IFFT(Y) is applied, yielding:
 
12,, T yyyyN

.
Step 3: A companding transform is then applied to y,
Figure 1. OFDM system block with DCT-companding.
i.e.






s
nCyn.
Step 4: An inverse companding transform is applied to
the received signal,


rn, i.e. .
  
1
ˆ
ynC rn


Step 5: A FFT transform is applied to the signal,


ˆ
y
n,
i.e.


ˆˆ
FFT Yy
ˆ
, where .
 
ˆˆˆ ˆ
12,, yy yy

T
N 

Step 6: An inverse DCT transforms applied to the sig-
nal, , i.e.
ˆ
Yˆ
T

X
HY. Then, the signal, ˆ
X
, is de-
maped from the bit stream.
6. Simulation Results
In this section, we present the results of computer simu-
lations used to evaluate PAPR reduction capability and
BER of the proposed scheme. The channel was modeled
as additive white Gaussian noise (AWGN). In the simu-
lation, an OFDM system with a sub-carrier of N = 128,512
and QPSK modulation was considered. We can evaluate
the performance of the PAPR reduction scheme using the
complementary cumulative distribution (CCDF) of the
PAPR of the OFDM signal.
6.1. CCDF Performance
We can evaluate the performance of PAPR using the
cumulative distribution of PAPR of OFDM signal. The
cumulative distribution function (CDF) is one of the
most regularly used parameters, which is used to measure
the efficiency of and PAPR technique. The CDF of the
amplitude of a signal sample is given by


1exp

F
zz

(16)
However, the complementary CDF (CCDF) is used in-
stead of CDF, which helps us to measure the probability
that the PAPR of a certain data block exceeds the given
threshold. The CCDF of the PAPR of the data block is
desired is our case to compare outputs of various reduc-
tion techniques. This is given by
 

PAPR 1PAPR 11exp
N
PzPz z
(17)
Figure 2 shows the CCDF performance of a com-
panding algorithm for PAPR reduction. The values of the
companding factor, u, for the companding procedure of
the second step were fixed to 2, 3, and 5. With this com-
panding method, the peak power at CCDF = 10–3 is re-
duced by 3.5 dB, 5 dB and 5.5 dB when compared with
the case of original system.
Figure 3 shows the CCDF performance of the DCT
scheme compared with that of the original and Hadamard
transform techniques. At CCDF = 10–3, the DCT scheme
reduces the PAPR by 3 dB over original system, but the
Hadamard transform only reduced the PAPR by 1 dB.
Figure 4 shows the CCDF performance of the pro-
posed PAPR reduction scheme. In the simulation OFDM
system, the number of sub carrier is 128. At CCDF = 10–3,

Combined DCT and Companding for PAPR Reduction in OFDM Signals103
Figure 2. Comparisons of the CCDF of different compand-
ing factor u.
Figure 3. CCDFs of the matrix transformations.
Figure 4. Comparisons of the CCDF of different PAPR re-
duction schemes.
the proposed scheme reduces the PAPR 1 dB more than
the companding method and reduces PAPR 2.5 dB more
than the DCT method.
Figure 5 is the CCDF performance of proposed re-
duction PAPR scheme at difference subcarriers. We can
see from Figure 5, the effect of difference subcarriers to
PAPR performance of OFDM signals is very small.
6.2. Analysis of Algorithm Complexity
Compared to the ordinary companding algorithm, the
computational complexity of the proposed scheme is
increased because the DCT is used. However, like FFT,
there are many fast methods to computer DCT. In litera-
ture [10], a fast DCT algorithm is proposed and the algo-
rithm requires 2
log
2
NN multiplications and
2
3log 1
2
NNN

 additions for N-length sequence. So
the multiplications and
2
logNN 2
3
2log
2
NNN
1







additions are added in proposed PAPR scheme.
7. Conclusions
In this paper, while taking both PAPR performance and
BER performance into account, we proposed a combined
DCT and companding scheme for the reduction of the
PAPR of OFDM signals. The proposed scheme is com-
posed of the DCT transform followed by the companding
transform. The DCT, used in the first step, does not in-
fluence the BER. The PAPR reduction performance of
the proposed scheme was evaluated using a computer
simulation. The simulation results show that the PAPR
reduction is improved when compared with those of a
companding transform.
Figure 5. Comparisons of the CCDF of proposed scheme
with different subcarriers.

Combined DCT and Companding for PAPR Reduction in OFDM Signals 104
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