An Overview of Optimization Algorithm for Complexity Reduction in PTS technique
of PAPR Reduction
Renu Verma
Electronics & Telecommunication
Engineering
Chhatarpati Shivaji Institute of
Technology, Durg (C.G.) India
Mangal Singh
Electronics & Telecommunication
Engineering
Chhatarpati Shivaji Institute of
Technology, Durg (C.G.) India
Mangalsingh @csitdurg.in
Neelam Dewangan Electronics & Telecommunication
Engineering
Chhatarpati Shivaji Institute of
Technology, Durg (C.G.) India
Abstract— Partial Transmit Sequence technique is one of the
most popular technique for PAPR reduction in OFDM. But in
this scheme the complexity for searching the phase factor is
increased with increase in number of sub blocks. This paper
describes the different types of optimization methods used in PTS
scheme, for optimization of phase factor & reducing the
searching complexity .Also we can analyze their performance in
terms of PAPR reduction for OFDM.
Key words— Artificial Bee Colony(ABC), Cross Entropy
& Parametric Cross Entropy Optimization(CE & PCE),
Firefly Optimization algorithm(FA), Harmony Search
(HS), Orthogonal Frequency Division Multiplexing
(OFDM), Partial Transmit Sequence (PTS), Particle
Swarm Optimization(PSO), Peak to Average Power Ratio(PAPR).
1. INTRODUCTION
The required transmission rate is very high for broadband
multimedia mobile communication system. Multipath fading
& inter symbol interference is very common problem for .high
data rate mobile radio channels. Use of adaptive equalizer at
receiver end is one of the method to solve this problem. But
due to high cost & large complexity it is not so common
method which is used.
Orthogonal Frequency Division Multiplexing ( OFDM )
is one of the parallel transmission scheme that reduces the
effect of multipath fading .Due to use of Fast Fourier
Transform ( FFT ) in OFDM the hardware implementation
becomes easy .When compared to single carrier transmission
scheme ,the multicarrier OFDM system has high peak to
average power ratio (PAPR), because, the transmit signal in
an OFDM system will have the high peak value in the time
domain form. The high PAPR of OFDM system reduces the
efficiency of power amplifier, used in transmitter & it also
increases the complexity of analog to digital and digital to
analog converters. There are number of methods to reduce the
PAPR of OFDM system, like Clipping, Tone Reservation,
Tone Injection, Selective Mapping & partial transmit sequence
(PTS) etc [1-4].
Among these techniques PTS is one of the distortion less
technique. That means, this scheme does not introduce
spectral regrowth & also it gives better result for PAPR
reduction.
In PTS method input data block is divided into number of
disjoint sub blocks. Then each sub block signal is converted
into time domain & weighted by rotating phase factors. At last
we add all the sub block signals that gives the OFDM symbol
with reduced PAPR. The improvement in performance of
PAPR reduction is obtained when the number of sub-blocks
are increased in PTS. But, when we increase the number of
sub blocks the searching complexity of optimal phase factor
also increases. To reduce this searching complexity various
phase optimization methods are applied with PTS to improve
the performance of PTS scheme.
In this paper we describe some of the simplified
optimization method like Particle Swarm Optimization [12-
13], Artificial bee Colony[10-11], Cross Entropy &
Parametric Cross Entropy Optimization[8-9], Harmony
Search[18-22] and Firefly Optimization [23]algorithm along
with PTS to reduce the searching complexity of optimal phase
factors. This paper has been organized as follows: First we
have introduction of OFDM & PAPR of OFDM with CCDF.
Then we discuss about partial transmit sequence technique
with its complexity issues & at last we analyze different
optimization method used in PTS for PAPR reduction of
OFDM system.
2. ORTHOGONAL FREQUENCY DIVISION
MULTIPLEXING
Orthogonal frequency division multiplexing converts a
high rate data stream into number of low data rate streams in
Renu Verma et al, Int.J.Computer Technology & Applications,Vol 5 (2),479-484
IJCTA | March-April 2014 Available [email protected]
479
ISSN:2229-6093
various channels. Signal of each channel is modulated by
using different modulation scheme like QAM & QPSK. Then
IFFT is used which gives the OFDM samples. After the
Parallel to serial converter OFDM signal is obtained .Basic
block diagram of OFDM is shown in figure 1. When the data
block as a vector is
X= [X0X1 X2 ….XN-1] (1)
The discrete time OFDM signal is given by
𝑥 𝑛 =1
𝑁 𝑋𝑛𝑒
𝑗2𝜋𝑘𝐿𝑁
𝑛
𝑁−1
𝑘=0
,
𝑛 = 0, 1, 2, 3,… . , 𝐿𝑁 [5] (2)
Where N is number of sub carrier & L is oversampling factor.
Figure.1 Block Diagram of OFDM System
3. PEAK TO AVERAGE POWER RATIO (PAPR)
The PAPR for OFDM signal is given by Muller &
Hubber & equation is
𝑃𝐴𝑃𝑅 =max 0≤𝑛≤𝑁−1 𝑥𝑛 2
𝐸[ 𝑥𝑛 ]2 (3)
Where PAPR–Peak-to-Average Power Ratio
xn – Oversampled OFDM signal
max 0≤n≤N-1 - Peak Power
[│xn│]2
– Average Power
E{.} denotes the expected value[5].
When the oversampling factor is 4 then the PAPR for discrete
time & continues time is same.
4. COMPLEMENTARY COMMULATIVE DISTRIBUTIVE FUNCTION
( CCDF )
CCDF is one of the parameter which is used for
performance evaluation of PAPR reduction techniques. It
gives the probabilities that PAPR of input data blocks cross
the given threshold level [25]. The equation for CCDF is
Pr(PAPR> PAPR o) = 1- (1-e-PAPRo
)N
(4)
The CCDF of original OFDM signal is shown in figure.2.The
PAPR of original OFDM signal is 11.8 db with CCDF of 10 -2
.
Figure.2 PAPR Vs CCDF of Original OFDM Signal
5. PARTIAL TRANSMIT SEQUENCE TECHNIQUE
Partial Transmit Sequence technique is a probabilistic
(Scrambling) technique which scrambles an input data block
of the OFDM symbol & select one of them with the minimum
PAPR for transmission as shown in figure 3. In this method
the input data S of N symbol is partitioned into disjoint V sub
blocks.
S=[S0 S
1 …….S
V-1]T
(5)
Each sub block are of equal size. After that each sub block are
phase shifted separately. Let complex phase factor is
bv= e
jϕv (6)
Where v= 1 2 3….V
Subsequently taking its IFFT it gives
x=IFFT 𝑆𝑣𝑏𝑣𝑉𝑣=1 = 𝑠𝑣𝑉
𝑣=1 𝑏𝑣 (7)
where s v
is referred to as a partial transmit sequence (PTS).
The phase factor is selected so that the PAPR can be
minimized. So the received signal with lowest PAPR can be
given as
𝑠 = 𝑠𝑣𝑉𝑣=1 𝑏 v
(8)
The PTS performance evaluation shows that the PAPR is
reduced when we increased the number of sub blocks but at
the same time searching complexity is increased exponentially
with sub blocks as shown in figure 4 .Alternatively the
optimization methods are applied in which best transmit signal
is stored until better one is found .To select the optimum phase
weighting factor for each input sequence we have to check
WV-1
possible combination. At the receiver for decode the data
side information is required in the form of phase factor i.e. log
2WV-1
[5-7].
0 2 4 6 8 10 1210
-2
10-1
100
<------------------ PAPR in dB ------------------>
<--
----
----
----
----
---
CC
DF
---
----
----
----
----
-->
PAPR Vs CCDF of Original OFDM Signal
Original OFDM Signal
Renu Verma et al, Int.J.Computer Technology & Applications,Vol 5 (2),479-484
IJCTA | March-April 2014 Available [email protected]
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ISSN:2229-6093
There are three sub block partition schemes that are
pseudo-random, adjacent & interleaved .Pseudo random sub
block partition scheme gives the good result compared to
other two methods. But in terms of hardware complexity the
pseudo random is very complex compared to others.
Figure.3 Block Diagram of PTS Technique of PAPR
Reduction
6. PHASE OPTIMIZATION METHODS IN PTS
In this paper we give a overview of different type of
optimization used to optimized the phase factor in PTS
scheme. These sub optimal schemes give the best combination
of phase factor with very less computational complexity& at
the same time it gives good PAPR performance.
Figure.4 PAPR performance of a 16 QAM/OFDM system
with PTS technique when the number of sub block varies
A. PARTICLE SWARM OPTIMIZATION (PSO)
The optimization procedure of PSO is based on population
of particles (swarm) ,which fly in solution space with velocity
dynamically updated on basis of its own flying experience &
the experience of best swarm velocity. The name is so given
as the general behavior of swarm of bees to search the best
food position in a field is done in same manner. In PSO
scheme each particle have a position vector x & the solution is
given in terms of W phase factor and V is moving velocity.
For m dimensional optimization position & velocity of ith
particle can be given as
Wi= ( Wi,1 Wi,2……. Wi,m) &
Vi= (Vi,1 Vi,2 ……Vi,m). (9)
The two important factor here is defined Gb= WG & Pb,=W
Pi ,
which represents the global best particle respectively. &
individual best position depending upon best objective value .
WG = Wi,1 Wi,2 ….. Wi,m
WP
i= Wg,1 Wg,2……Wg,m (10)
At time t+1 the new velocity Vi(t+1) for particle i is updated
by
Vi(t+1) = wVi(t)+ C1r1(WP
i(t)- Wi(t))
+ C2r2(WG(t) – Wi(t)). (11)
Where Vi(t) is old velocity of particle i at time t. The C1 & C2
are called acceleration factor or rate to obtain best position &
w is inertia factor. The new position of particle i is calculated
on basis of updated velocity by equation
Wi(t+1)= Wi(t) +Vi(t+1). (12)
On basis of performance evaluation (PAPR) we can say that
this sub optimal PTS scheme is slightly degrade the PAPR
performance to the conventional PTS .However the
computational complexity is very less when we have the
threshold value for number of iteration [12-13].
B. ARTIFICIAL BEE COLONY OPTIMIZATION (ABC)
The ABC-PTS algorithm can reduce the PAPR efficiently
& at the same time the computational complexity is also very
less for large sub blocks. The ABC optimization method is
proposed by Karaboga .This process is inspired by the process
of searching of optimum food source by bees. Bees are
onlooker, Scout & employed type bees. In PTS scheme of
PAPR reduction of OFDM the phase factor represents the
food source position which have to be optimized. Firstly the
food source position is selected randomly .The employed bees
search for a new source near the current food source position
.If the nectar amount of new one is greater than the current
one, the new source position is memorized by employed bees.
The updated phase factor is given by
4 5 6 7 8 9 10 1110
-3
10-2
10-1
100
PAPR0[dB]
Pr(
PA
PR
>P
AP
R0)
16-QAM CCDF of OFDMA signal with PTS
original
PTS N=1
N=2
N=4
N=8
N=16
Renu Verma et al, Int.J.Computer Technology & Applications,Vol 5 (2),479-484
IJCTA | March-April 2014 Available [email protected]
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ISSN:2229-6093
bi = bi + ϕi( bi- bm) (13)
Where bi =[bi1 bi2 …..bi(v-1)]
i= 1 2 3….SN(population Size)
Where ϕi is random number within the range of [-1 1], & bm is
solution within the neighborhood of bi. The fitness of a
solution is calculated by
fit(bi) = 1/(1+ f(bi)) if f(bi)≥ 0
= 1+ abs( f(bi)) if f(bi)<0
(14)
Here f(bi ) is PAPR of transmit signal , which have to be
minimized. This information is shared to onlooker bees & then
they move toward position of new food source. The
probability of selecting the new food source by onlooker bee
is
Pi= fit(bi) / ( 𝑓𝑖𝑡(𝑏𝑖)𝑆𝑁𝑣=1 𝑓𝑖𝑡(𝑏𝑖)𝑆𝑁
𝑣=1 ) (15)
If the fitness value is not improved after the complete search
(up to maximum no. of iteration: limit) ,the employed become
the scout bee. The scout bee search for new source randomly
by
bi = min (bi) + rand ( 0 1) * (max( bi) – min( bi)) (16)
The min (bi) and max(bi) are lower & upper boundary of phase
factor. This process is repeated till the optimum phase factor is
not obtained [10-11].
C. CROSS ENTROPY OPTIMIZATION (CE)
For solving the rare event estimation problems Rubinstein
proposed the Cross Entropy optimization. In that all possible
solution are distributed & adaptively update this distribution
according to Kullback Leibler distance i. e. CE between the
associated density & optimal importance sampling density.
The optimization of phase factor in PTS method to reduce the
PAPR of OFDM is performed by using this CE scheme. The
score function is defined in terms of PAPR.
L(X’(ϕ)) = 1
(10* log10 (max [x’(ϕ)]
2/E[[x
’(ϕ)]
2]
(17)
The score function is inversely proportional to PAPR .So for
reducing PAPR we have to maximize the score function over
the set [0, 2π) for all ϕ. Such that
arg max L (X’(ϕ))
ϕ є[0, 2π) (18)
The stochastic sampling problem can be easily solved, So
in CE, the deterministic optimization problem is transform
into stochastic problem. It provides almost same PAPR
reduction like conventional PTS while maintaining low
complexity.
Some modification in CE method is called parametric CE
optimization (PM CE) .In PM CE the parameter is updated
according to entire samples where as in the CE it is updated by
only best scoring sample called elite sample. It gives the low
computation complexity & at the same time it gives the
improve PAPR reduction performance compared to
conventional PTS & even CE –PTS [8-9].
D.HARMONY SEARCH OPTIMIZATION (HS)
The musician wants to play the pleasing music, for that
they continue polishing the pitches which give the better state
of harmony, given by aesthetic standard. Similarly the
optimization algorithm seeks global optimum value
determined by evaluating objective function. The objective
function f(x) is defined as PAPR of OFDM in PTS scheme for
PAPR reduction. In the phase optimization process firstly we
define the number for parameters like number of phase factor,
pitch range (range of decision variable), harmony memory
size (HMS), harmony memory consideration rate (HMCR),
pitch adjustment rate (PAR), distance bandwidth (bw) &
stopping criterion (K). After the parameter initialization the
harmony memory is defined .Each row of HM is one possible
solution of optimum phase factor. The new phase factor is
searched on basis of three factors that are pitch adjustment,
memory consideration & random selection. If the new
harmony performed better than the worst harmony of HM is
replaced by new one. At last the stopping criterion is checked.
Initially HM is randomly selected in [b il biu] for i= 1 2 3….N
by
bij = bil +(q *( biu- bil)) j= 1 2….. HMS
where q is random number [0 1]
(19)
The new harmony bnew after improvisation based on memory
consideration is
bnew =bi ± (r*bw) (20)
where r is uniform random number in [0 1]
In HS method PAR & bw value are adjusted in initialization
step & they are fixed throughout the algorithm. An improved
HS method is proposed in literature in which variable PAR &
bw is used in improvisation step. The lower value of
bandwidth distance & higher value of PAR gives the best
solution. This method gives the good tradeoff between the
PAPR performance & searching complexity of phase factor in
PTS scheme [18-22].
E.FIREFLY ALGORITHM (FA)
In firefly algorithm the objective function depends on
light intensity. Fireflies are attracted towards the light & they
Renu Verma et al, Int.J.Computer Technology & Applications,Vol 5 (2),479-484
IJCTA | March-April 2014 Available [email protected]
482
ISSN:2229-6093
move toward the brighter location continuously. The objective
function contains the information related to brightness of
firefly. The attractiveness of firefly is proportional to its
brightness which is given by
β(r) = β0( e-γr
)m
m≥1 (21)
Brightness is inversely proportional to the distance between
two fireflies. And β0 is maximum attractiveness (at r=0).The
movement of firefly i is determined by
xi = xi+ β0( e-γr
)2
(xj- xi)+α (rand-0.5) (22)
Where the first term is current position of firefly i ,the second
term gives the fireflies attractiveness and the last term is used
for random movement if there are not any brighter firefly. The
r is distance between two firefly i & j.
The advantage of FA is that different fireflies will work
almost independently it is thus suitable for parallel
implementation. FA can find the global & local optimum
solution simultaneously & effectively.[23]
RESULT & CONCLUSION
In this paper a PAPR reduction method called Partial
Transmit Sequence technique is describe along with different
optimization schemes used for reducing the searching
complexity of phase factor .They provide the good tradeoff
between PAPR performance & computational complexity. In Harmony search algorithm control parameters are less so it is
very easy to adjust.
TABLE I. When 𝐶𝐶𝐷𝐹 = 10-3, comparison of computational
complexity among different methods for phase factor optimization in
QAM.𝑊 = 2(possible phase factor), 𝑀 = 16 (sub-blocks), size of
particle 𝑆 = 30 & maximal iterations 𝐺 = 𝐾 = 30[9,10,23]
Methods Computational Complexity PAPR
OPTS WM-1= 215= 32768 6.45dB
PSO-PTS 𝑆𝐺 = 30 * 30 = 900 7.1 dB
ABC-PTS 𝑆𝐾 = 30 * 30 = 900
6.8 dB
CE-PTS
22 searches for 8 sub-block &
QPSK 7.5 dB
FA-PTS No. of fireflies =10,Iteration=5
6,7dB
HS-PTS 30*8=240
6.7 dB
In improved HS algorithm the performance depends on value
of PAR & bw. ABC-PTS scheme is slightly degrade the
PAPR performance to the conventional PTS but the computational complexity is very less . FA can find the global
& local optimum solution of phase factor simultaneously & it
reduces the PAPR of OFDM signal effectively.
ACKNOWLEDGMENT
I am very grateful to the Chhatarpati Shivaji Institute of
Technology, Durg & also want to thank my guide Mr. Mangal
Singh and Neelam Dewangan for providing me the necessary
support.
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