1. An Agent Based Particle Swarm Optimization for
PAPR Reduction of OFDM Systems
Javad Taghipour, Ali Asghar Parandoosh and Vahid Tabataba Vakili
School of Electrical Engineering, Department of Telecommunications
Iran University of Science and Technology (IUST)
Tehran, Iran
jtaghipour@elec.iust.ac.ir, aa.parandoosh@ee.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 an agent based particle swarm
optimization for reducing computational complexity of the
original PSO (OPSO) technique. Simulation results show that the
agent based PSO (APSO) 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 an
effective and 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 an agent
based particle swarm optimization for reducing computational
complexity of the original PSO technique and achieving nearly
the same performance in PAPR reduction compared to the
OPSO. In OPSO the velocity of the particle in the problem
space is governed by a central algorithm that gives the particle
limited flexibility and knowledge about the environment. Due
to these limitations, computational complexity of original PSO
is still high. We argue that elevating the particle to the status of
the agent will make the whole algorithm perform much more
effectively, especially in a large problem space with a complex
structure. In this paper we reduce the computational
complexity by viewing the swarm as an agent-based system.
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 agent based PSO (APSO)
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)
20th Telecommunications forum TELFOR 2012 Serbia, Belgrade, November 20-22, 2012.
978-1-4673-2984-2/12/$31.00 ©2012 IEEE 839

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 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)
Figure 1. The structure of transmitter with PSO-based PTS scheme.
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.
III. AGENT BASED PSO (APSO)
In APSO by elevating a particle to the status of agent and
setting the velocity of all particles equal to the velocity of
agent, we can dramatically reduce the computational
complexity of finding phase factors of PTS. This particle
should have the maximum fitness value (maximum PAPR)
and it could change in each iteration. Simulation results show
that PTS-APSO based compare to the PTS-OPSO based, has
nearly the same performance in PAPR reduction of OFDM
systems. In order to solve the optimization problem of the PTS
scheme, APSO is used for phase optimization. So we propose
following algorithm (APSO) for finding the optimal phase
factors:
Step 1: Initialization of the particle swarm:
840

3. • Generate N different (1)i
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 are the same;
• Calculate the fitness values of all particles.
Determine the one (current agent) with the maximum PAPR
and set its local best position and its fitness value equal to its
current position and fitness value.
• set the global best position and its fitness value
equal to the position and fitness value of the best initial
particle ;
Step 2: Update particles (the (t + 1)th iteration):
Update the velocity of the agent according to the following
equation:
( ) ( )1 1 2 2
( 1) ( ) ( ) ( ) ( ) ( ) ,
p g
agent agent agent
t w t c r t t c r t t+ = + − + −v v
(10)
• Set the velocity of all particles equal to the velocity of
the agent and update their position according to the
following equations:
( 1) ( ) ( 1),i i
t t t+ = + +v (11)
( )( 1) exp ( 1) .i i
t j t+ = +b (12)
Step3: Calculate the fitness values of all particles.
Determine the one (new agent) with the maximum fitness
value. For new agent compare its current fitness value with the
fitness value of the local best position ( )
p
agent
t .
If current value is better, then update the local best position
and its object value with the current position and fitness value.
• Determine the best particle of current swarm with the
best fitness values. If the fitness value is better than
the fitness value of the global best position, then
update the global best position and its fitness value
with the position and fitness 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 the
Step2;
In OPSO, according to (8) & (9), for each particle we should
calculate 5 additions and 5 multiplications. But in APSO,
according to (10) & (11), we should calculate these additions
and multiplications only for the agent. As we can see, in
APSO compare to the OPSO, the number of
4 ( 1) ( 1)n
G M× − × − additions and 5 ( 1) ( 1)n
G M× − × −
multiplications are reduced in each iteration of the algorithm.
IV. SIMULATION RESULTS
In this section, we present various simulation results to
demonstrate the performance of the PTS technique based on
APSO 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 ( =0.5), acceleration
terms (c1=c2=2) and QPSK modulation. The sampling rate for
an accurate estimation of PAPR needs to be increased by 4
times (L=4).
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
APSO is nearly the same as OPSO but with lower
computational complexity. For example for Gn=10, B=2 and
M=16, in APSO compare to the OPSO, the number of 5×(10-
1)×(16-1)=675 multiplications and 4×(10-1)×(16-1)=540
additions are reduced in each iteration. So here, for 10
iterations, 6750 multiplications and 5400 additions are
reduced. We can also see that as the number of sub-blocks
(M) increases, the performance of APSO becomes better.
In Figure 3, when CCDF=Pr(PAPR>PAPR0) =10-3
, the
PAPR0 of the original OFDM is 11.3dB, OPSO-PTS (M=32) is
7.3dB, and APSO-PTS (M=32) is 7.4dB. It is evident that the
APSO-PTS can provide nearly the same performance in PAPR
reduction compare to the OPSO-PTS while keeping lower
complexity.
Table I shows the number of additions and multiplications
of OPSO and APSO techniques for M=16, iteration = 10 and
Gn = 10. For this particular case, the number of additions in
APSO algorithm is almost three times lower than the number
of additions in OPSO algorithm, and the number of
multiplications in APSO algorithm is ten times lower than the
number of multiplications in OPSO algorithm.
2 4 6 8 10 12 14
10
-4
10
-3
10
-2
10
-1
10
0
PAPR0
(dB)
Pr(PAPR>PAPR0
)
Original OFDM
OPSO M=8
APSO M=8
OPSO M=16
APSO M=16
OPSO M=32
APSO M=32
Figure 2. CCDFs comparison of the APSO-based PTS scheme with different
number of sub-blocks when iteration = 10, Gn = 10 and B= 2.
841

4. 3 4 5 6 7 8 9 10 11 12 13
10
-4
10
-3
10
-2
10
-1
10
0
PAPR0
Pr(PAPR>PAPR0
)
Original OFDM
OPSO M=8
APSO M=8
OPSO M=16
APSO M=16
OPSO M=32
APSO M=32
Figure 3. CCDFs comparison of the APSO-based PTS scheme with different
number of sub-blocks when iteration = 10, Gn = 10 and B= 4.
TABLE I. THE COMPUTATIONAL COMPLEXITY OF OPSO AND APSO
ALGORITHMS FOR M=16, ITERATION = 10 AND GN = 10.
Algorithm # of additions # of multiplications
OPSO 7500 7500
APSO 2100 750
Figure 4 and Figure 5 illustrate some performance of the
PTS technique in PAPR reduction using APSO for different
number of particle generations (Gn) with M=16 sub-blocks,
iteration=10 and B=2, B=4 respectively. 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) and the set of phase weighting factor
are increased, the performance of the PAPR reduction becomes
better. Basically, 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 and Figure 5 (for Gn=25), When
CCDF=Pr (PAPR>PAPR0) =10-3
the PAPR0 of the original
OFDM is 11.3dB, while APSO-PTS with B=2 is 7.9dB and
APSO-PTS with B=4 is 7.5dB. Therefore, APSO-PTS
technique can offer good PAPR reduction with lower
complexity.
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
APSO M=16, B=2
Gn
= 5,10,15,20,25
Figure 4. PAPR reduction performance with different number of particle
generations (Gn), when M=16, B=2 and iteration=10.
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
APSO M=16, B=4
Gn
= 5,10,15,20,25
Figure 5. PAPR reduction performance with different number of particle
generations (Gn), when M=16,B=4 and iteration=10.
V. CONCLUSION
In this paper, we have proposed a new algorithm (APSO)
for finding the optimal phase factors of PTS. The new
algorithm combines the optimization technique with the feature
set of an agent-based system. Simulation results show that,
compared with PTS-OPSO based technique, the PTS using
APSO, can not only dramatically reduce computational
complexity but also have nearly the same performance in
PAPR reduction.
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