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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

[email protected]

Mangal Singh

Electronics & Telecommunication

Engineering



Mangalsingh @csitdurg.in

Neelam Dewangan Electronics & Telecommunication

Engineering



[email protected]

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]

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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



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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



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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



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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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