1. A Novel Particle Swarm Optimization for PAPR Reduction of OFDM Systems
Ali Asghar Parandoosh, Javad Taghipour and Vahid Tabataba Vakili
School of Electrical Engineering, Department of Telecommunications
Iran University of Science and Technology (IUST)
Tehran, Iran
aa.parandoosh@ee.iust.ac.ir, jtaghipour@elec.iust.ac.ir, vakily@iust.ac.ir
Abstract— Orthogonal frequency division multiplexing
(OFDM) is usually regarded as a spectral efficient multicarrier
modulation technique, yet it suffers from a high peak to
average power ratio (PAPR) problem. Partial transmit
sequences (PTS) is one of the most well-known schemes to
reduce the PAPR in OFDM systems. However, the
conventional PTS scheme requires an exhaustive searching
over all combinations of allowed phase factors. Consequently,
the computational complexity increases exponentially with the
number of the sub-blocks. Particle swarm optimization (PSO)
algorithm is a recently proposed approach to solve the
optimization problem of finding the phase factors of the PTS.
In this paper we propose a new method for reducing
computational complexity of the original PSO (OPSO)
technique. Simulation results show that the proposed PSO
(PPSO) compare to the original PSO can effectively reduce the
computational complexity of finding phase factors of the PTS.
Keywords- OFDM; PAPR; PTS; Particle Swarm
Optimization (PSO).
I. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is
an attractive technique for high speed data transmission in
fading channels. It has been used as a standard transmission
modulation for many digital transmission systems, such as
digital video broadcasting (DVB), the fourth generation of
mobile communication system and so on [1].
The high peak to average power ratio (PAPR) at the
transmitter is one of the major drawbacks of OFDM signals,
which causes signal distortion such as in-band distortion and
out-of band radiation due to the nonlinearity of the high
power amplifier (HPA) [2].
To deal with this problem, many PAPR reduction
schemes have been proposed, such as block coding [3],
selective mapping [4], and partial transmit sequence [5–6].
Among all of these methods PTS is considered as a
promising distortion less phase optimization scheme that
provides excellent PAPR reduction. In PTS scheme, several
replicas of the OFDM symbol of a given data frame is
formed by phase factors and the one with the minimum
PAPR is chosen for transmission. However, the
conventional PTS requires an exhaustive searching over all
combinations of allowed phase factors. In order to reduce
the computational complexity of solving this optimization
problem, several algorithms have been proposed [7–9],
among which particle swarm optimization (PSO) is a newly
proposed method which can reduce the computational
complexity of the PTS technique. Original PSO (OPSO) has
a tendency of being trapped in local minima and yet it
suffers from slow convergence and therefore high
computational complexity. In this paper, we propose a new
particle swarm optimization for reducing computational
complexity of the original PSO technique and achieving
nearly the same performance in reduction of the PAPR
compared to the OPSO. Basically PSO is a mathematically
approach for solving optimization problems which
optimized variables could be usually any values without any
limitation for their final answers. But in PPSO because in
PTS scheme weighting factors are limited to certain values
such as { }1± or { }1, j± ± or … and they should be selected
from these certain sets, so we can modify the equations of
OPSO in order to achieve lower computational complexity
with nearly the same performance compared to the OPSO.
The rest of this paper is organized as follows. In Section
II, we present the basic concepts of OFDM system, such as
OFDM signals, definition of PAPR, PTS and original PSO
technique. Section III introduces proposed PSO (PPSO)
method for solving optimization problem of the PTS scheme.
In Section IV, the performance of proposed method is
discussed, the simulation results are shown and finally some
conclusions for the proposed method are drawn in Section V.
II. SYSTEM MODEL
A. OFDM signals and PAPR
In OFDM systems, a block of transmitted signals,
0 1 1
[ , ,..., ]N
x x x −
=x is represented by
1
2 /
0
1
; 0 1,
N
j ik N
i k
k
x X e i N
N
π
−
=
= ≤ ≤ −¦ (1)
where N is the number of subcarriers and
0 1 1
[ , ,..., ]N
X X X −
=X denotes the input data symbols at
sub-bands. The PAPR of transmitted signal, can be
expressed as
2
0 1
2
max
,
[ ]
i
i N
i
x
PAPR
E x
≤ ≤ −
= (2)
2012 International Conference on Control Engineering and Communication Technology
978-0-7695-4881-4/12 $26.00 © 2012 IEEE
DOI 10.1109/ICCECT.2012.174
681
2012 International Conference on Control Engineering and Communication Technology
978-0-7695-4881-4/12 $26.00 © 2012 IEEE
DOI 10.1109/ICCECT.2012.174
681

2. where E[.] denotes the expected value operation. The
complementary cumulative distribution function (CCDF) of
the PAPR is the probability that the PAPR of an OFDM
symbol exceeds the given threshold (PAPR0), which can be
expressed as
{ }0
Pr .CCDF PAPR PAPR= > (3)
B. PTS Scheme
The principle structure of the PTS scheme is shown in
Figure 1. In the PTS technique, a data block is partitioned
into M disjoint sub-blocks, which are represented by the
vectors { } 1
M
m m=
X , therefore
1
.
M
m
m=
= ¦X X (4)
The sub-blocks are transformed into time domain partial
transmit sequences using inverse fast Fourier transform
(IFFT). Then these partial sequences multiply by phase
weighting factors { } 1
, [0, 2 )m
Mj
m m m
b e
θ
θ π
=
= ∈ . The goal of
the PTS approach is to find an optimal weighted
combination of the M sub-blocks to minimize the PAPR
value. The time domain transmitted signal after combination
can be expressed as
1
( ) ( ),
M
m m
m
b IFFT
=
′ = ¦x b X (5)
and the minimization of PAPR is related to the
minimization of following equation
2
0 1
max ( ) .i
i N
x
≤ ≤ −
′ b (6)
The phase factors are chosen in order to minimize the
PAPR of transmitted signal. The selection of the phase
factors is limited to a set with finite number of elements for
reducing search complexity. Assuming that, there are B
phase factors to be allowed:
{ }2
, 0,1,..., 1 .m
l
l B
B
π
θ = = − (7)
We can set 1
1b = without any loss of performance.
Therefore, in conventional PTS
1M
B
−
sets of phase factors
should be searched to find the optimum set of phase factors.
As we can see the search complexity increases exponentially
with the number of sub-blocks M.
C. PSO-based PTS
In order to reduce the computational complexity of
searching the optimum set of phase factors, the PSO
technique is proposed in [9]. The PSO is a randomized,
population based optimization method. In PSO algorithm,
each single solution is a particle in the search space. A
swarm of these particles moves through the search space to
find an optimal position. The position and velocity are two
parameters to characterize each particle.
In PSO based PTS for a K-dimensional optimization, the
position and velocity of the ith particle can be represented as
{ },1 ,2 ,
, ,...,i i i i K
b b b=b and { },1 ,2 ,
, ,...,i i i i K
v v v=v ,
respectively. PSO algorithm is initialized with a group of
random particles and then searches for optima by updating
generations. In each iteration, particle updates itself through
tracking two best positions. The first one is the local best
position ( )
p
i
b , which represents the position vector of the
best solution of this particle has achieved so far. The other
one is the global best position ( )
g
b , which represents the
best position obtained so far by any particle. After finding
the two best values, the update of velocity and position for
each particle are described as
( ) ( )1 1 2 2
( 1) ( ) ( ) ( ) ( ) ( ) ,
p g
i i i i i
t w t c r t t c r t t+ = + − + −v v b b b b
(8)
( 1) ( ) ( 1),i i i
t t t+ = + +b b v (9)
where ( )i
tv is the velocity of the ith particle and ( )i
tb is
current solution of the ith particle at the time t. The c1 and c2
are the acceleration terms, r1 and r2 are two random variables
with uniform distribution between [0,1] and w is the inertia
weight which shows the effect of the previous velocity
vector on the new position vector.
Figure 1. The structure of transmitter with PSO-based PTS scheme.
III. PROPOSED PSO (PPSO) ALGORITHM
Basically PSO is a mathematically approach for solving
optimization problems in which optimized variables could
be usually any values without any limitation for their final
answers. However, in the PTS scheme weighting factors are
limited to certain values such as { }1± , { }1, j± ± or … and
they should be selected from these certain sets, so we can
modify the equations of OPSO in order to achieve lower
computational complexity with nearly the same
performance compared to the OPSO.
For the case { }1= ±b , because the phase factors should
be just +1 or -1, therefore without loss of performance, we
can simply discard random variables r1 and r2 which are
between [0,1] and thus the computational complexity is
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3. reduced. We propose following algorithm (PPSO) in order
to find the optimal phase factors:
Step 1: Initialization of the particle swarm:
• Generate N different (1)i
b which means N
different phase vectors of the length of PTS sub-blocks; (N
is the size of the swarm population.)
• Initialize the velocity (1)i
v by zeros, note that the
size of v and b are the same;
• Calculate the fitness values of all particles, set the
local best position of each particle and its objective value
equal to its current position and objective value, and set the
global best position and its objective value equal to the
position and objective value of the best initial particle;
Step 2: Update particles (the (t + 1)th iteration):
Update velocity and position according to the following
equations:
( ) ( )1 2
( 1) ( ) ( ) ( ) ( ) ( ) ,
p g
i i i i i
t w t c t t c t t+ = + − + −v v b b b b
(10)
( 1) ( ) ( 1),i i i
t t t+ = + +b b v (11)
{ }( 1) sgn ( 1) ,i i
t t+ = +b b (12)
where sgn (.) is the signum function;
Step3: Calculate the objective values of all particles and
for each particle compare its current objective value with the
object value of its local best position. If current value is
better, then update the local best position and its object
value with the current position and objective value.
Moreover, determine the best particle of current swarm with
the best objective values. If the objective value is better than
the object value of the global best position, then update the
global best position and its objective value with the position
and objective value of the current best particle;
Step4: End if a pre-defined stopping criterion (such as
certain number of iteration) is met, otherwise go back to
theStep2;
For the case { }1, j= ± ±b , PPSO is used for phase
optimization where { }0, / 2, , / 2m
π π π= −ș .
So we can express following algorithm (PPSO) in order
to find the optimal phase factors:
Step 1: Initialization of the particle swarm:
• Generate N different (1)i
ș ;
• Initialize the velocity (1)i
v by zeros;
• Calculate the fitness values of all particles, set the
local best position of each particle and its objective value
equal to its current position and objective value, and set the
global best position and its objective value equal to the
position and objective value of the best initial particle;
Step 2: Update particles:
Update velocity and position:
( ) ( )1 2
( 1) ( ) ( ) ( ) ( ) ( ) ,
p g
i i i i i
t w t c t t c t t+ = + − + −v v ș ș ș ș
(13)
{ },1 ,2 ,3 ,4
( 1) ( ) ( 1); ( 1) , , , ,i i i i i i i i
t t t t θ θ θ θ+ = + + + =ș ș v ș
(14)
{ }, ,
0, / 2, , / 2
arg min ,
i
i j i j i
ϕ π π π
θ θ ϕ
= −
= − (15)
( )( 1) exp ( 1) .i i
t j t+ = +b ș (16)
Step3: this step is similar to the case b= {+1,-1};
Step4: End if a pre-defined stopping criterion is met,
otherwise go back to Step2.
IV. SIMULATION RESULTS
In this section, we present various simulation results to
demonstrate the performance of the PTS technique based on
PPSO in reducing PAPR of OFDM systems. in the
conducted computer simulations 5×105
independent OFDM
symbols are randomly generated, and correlative parameters
are preset as 256 subcarriers (N=256), inertia weight
(Ȧ=0.5) and QPSK modulation. The sampling rate for an
accurate estimation of PAPR needs to be increased by 4
times (L=4). When B=2, the acceleration terms c1 and c2 are
2 (c1=c2=2) and when B=4, in order to achieve better
performance, we set these acceleration terms as 0.5
(c1=c2=0.5).
In Figure 2 and Figure 3 some results of the CCDF of
the PAPR are simulated for OFDM system in which phase
weighting factors of PTS are selected from { }1
M
= ±b and
{ }1,
M
j= ± ±b respectively (B=2, B=4). We also set
iteration as 10, and the number of particle generations is 10
(Gn=10). As we can see for M=8, 16, 32 sub-blocks, the
performance of PPSO is nearly the same as OPSO but with
lower computational complexity. For example for Gn=10,
B=2 and M=16, in PPSO compare to the OPSO, the number
of 2×10×(16-1)=300 generations and multiplications of
random numbers are reduced in each iteration. So here, for
10 iterations, 3000 multiplications are reduced. We can also
see that as the number of sub-blocks (M) increases, the
performance of PPSO becomes better.
In Figure 2, when CCDF=Pr(PAPR>PAPR0) =10-3
, the
PAPR0 of the original OFDM is 11.3dB, OPSO-PTS (M=16)
is 8.2dB, and PPSO-PTS (M=16) is 8.4dB. It is evident that
the PPSO-PTS can provide nearly the same performance of
PAPR reduction compare to the OPSO-PTS while keeping
lower complexity.
Figure 4 illustrates some performance of the PTS
technique in PAPR reduction using PPSO for different
number of particle generations (Gn) with M=16 sub-blocks,
iteration=10 and B=4. It can be observed that probability of
very high peak power has been increased significantly if PTS
techniques are not used. As the number of particle
generations (Gn) is increased, the performance of the PAPR
reduction becomes better. The PAPR performance is
improved with Gn increasing. However, the degree of
improvement is limited for larger Gn’s. On the other hand,
the computational complexity is increased with Gn. In Figure
4 (for Gn=30), When CCDF=Pr (PAPR>PAPR0) =10-3
the
683683

4. PAPR0 of the original OFDM is 11.3dB, while PPSO-PTS is
7.4dB. Therefore, PPSO-PTS technique can offer good
PAPR reduction with lower complexity.
Figure 5 shows some comparisons of the PAPR
reduction performance with different number of iterations.
As iteration increases, the PAPR performance of PTS-PPSO
based becomes better, but the computational complexity
becomes high accordingly. Therefore, an appropriate
iteration number is needed to achieve the best tradeoff
between the PAPR reduction performance and complexity.
3 4 5 6 7 8 9 10 11 12 13
10
-4
10
-3
10
-2
10
-1
10
0
PAPR0
(dB)
Pr(PAPR>PAPR0
)
Original OFDM
OPSO M=8
PPSO M=8
OPSO M=16
PPSO M=16
OPSO M=32
PPSO M=32
Figure 2. CCDFs comparison of the PPSO-based PTS scheme with
different number of sub-blocks when iteration = 10, Gn = 10 and B= 2.
3 4 5 6 7 8 9 10 11 12 13
10
-4
10
-3
10
-2
10
-1
10
0
PAPR0
(dB)
Pr(PAPR>PAPR0
)
Original OFDM
OPSO M=8
PPSO M=8
OPSO M=16
PPSO M=16
OPSO M=32
PPSO M=32
Figure 3. CCDFs comparison of the PPSO-based PTS scheme with
different number of sub-blocks when iteration = 10, Gn = 10 and B= 4.
5 6 7 8 9 10 11 12
10
-4
10
-3
10
-2
10
-1
10
0
PAPR0
(dB)
Pr(PAPR>PAPR0
)
Original OFDM
PPSO M=16, B=4
Gn
= 5,10,15,20,25,30
Figure 4. PAPR reduction performance with different number of particle
generations (Gn), when M=16, B=4 and iteration=10.
5 6 7 8 9 10 11 12
10
-4
10
-3
10
-2
10
-1
10
0
PAPR0
(dB)
Pr(PAPR>PAPR0
)
Original OFDM
PPSO M=16, B=4
iteration = 2, 4, 6, 8, 10, 12
Figure 5. PAPR reduction performance with different number of
iterations, when M=16, B=4 and Gn=4.
V. CONCLUSION
This paper proposed a new particle swarm optimization
(PPSO) for finding the optimal phase factors of PTS.
Simulation results show that, compared with PTS-OPSO
based technique, the PTS using PPSO, can not only
dramatically reduce computational complexity but also have
nearly the same performance in PAPR reduction.
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