Abstract – This work presents a novel approach to the reduction of Peak-to-Average Power Ratio (PAPR) in multi-carrier modulation: Partial Transmit Sequence (PTS) is optimized by opportune tailored Genetic Algorithms (GA)which allow achieving solution with pre-selected accuracy and reduced computational burden. A comparative study on both Orthogonal Frequency Division Multiplexing (OFDM) and Wavelet Packets Multi-Carrier Modulation (WP-MCM) has been conducted. Results show that although WP-MCM outperforms conventional OFDM schemes in terms of PAPR, GA applied to PTS are more effective in reducing PAPR for OFDM rather than for WP-MCM.

Index Terms – Multi-Carrier Modulation, Partial TransmitSequence, Genetic Algorithms.

I. INTRODUCTION

With the increasing spread of multimedia broadband services, multi-carrier modulations are becoming a key technology extensively deployed in wireless communication systems for broadband access such as WiFi and WiMAX. Classic Orthogonal Frequency Division Multiltiplexing (OFDM) based on Fourier transform [1] and more recent Wavelet Packets Multi-Carrier Modulation (WP-MCM) [2] are the current state-of-the art multi-carrier technologies, which suffer from high Peak-to-Average Power Ratio (PAPR). PAPR limits the performance of wireless communication systems: high power peaks in transmitted signals lead to a necessity for increasing the dynamic range of corresponding parts of the communication system in order to avoid clipping of signals. Several methods have been proposed to reduce PAPR [3]. Among them, Partial Transmit Sequence (PTS) [4] is known as distortionless scheme which require a small amount of redundancy. The PTS approach is based on merging signal subblocks which are opportunely phase shifted as to reduce PAPR. Adrawback of PTS is that requires multiple inverse fast Fourier/wavelet transforms (IFFT/IDWT), thus resulting in high computational complexity in practical systems. If M is the number of subblocks and W the number of phase shifts the computational burden of PTS increases with 1MW 1 . In general, the complexity of PTS is proportional to the number of the candidate signals. Therefore, to reduce the complexity, some simplified techniques have been proposed [5]–[8] aiming at reducing the number of candidate signals. In [5], the candidate signal can be transmitted directly without an exhaustive search when its PAPR is lower than a preset threshold. In [6] a new algorithm for computing the phase factors that achieves better performance than the Optimal Binary Phase Sequence search approach is presented. A new algorithm based on Fincke Phost Sphere Decoder is proposed in [7], where only those phase vectors

that guarantee that the PAPR is bounded are searched. In the technique proposed in [8] a gradient descent search isperformed to find the phase factors. In this work we proposea complexity reduction of PTS optimization using Genetic Algorithms (GA) [9]. GA are search techniques used to find true or approximate solutions to optimization problems. GAare extensively used in literature in different application fields of communication engineering such as for instance, network design, unicast and multicast routing, and adaptive modulation for wireless transmission [10], [11], [12]. GAallow finding iterated numerical solution to complex problems with an accuracy dependent on the number of iterations selected. They can deal with highly nonlinear problems and non-differentiable functions as well as functions with multiple local optima. Optimization by GAallows numerical solution, otherwise unfeasible by analytic approach. In our case PTS optimization is done “off-line” for the case of AWGN and phase shifts are a priori calculated. As to this, no real-time constraint needs to be guaranteed. In fact, during transmission, phase shifts are a priori known for each possible channel conditions. GA are chosen due to their properties to allow simple solutions respect to classic gradient-based numerical methods, (e.g., Lagrangian multipliers/Newton’s).

Extensive tests have been conducted on both OFDM andWP-MCM schemes as to evaluate the reduction of both PAPR and the complexity of PTS using GA. Results show the effectiveness of the proposed method in case of AWGN channel for both systems. However, it is noticeable that, although the joint use of PTS and GA allows reducing the absolute PAPR of both multi-carrier modulation schemes with a prevalence of WC-MCM, the relative decrease of the PAPR is higher for OFDM.

The remaining of the paper is organized as follows: first, a brief introduction to OFDM and WP-MCM is given insection II. Section III illustrates PTS algorithm. Then, the proposed system model and simulation results are presented in section IV. Conclusions are finally drown in section V.

II. MULTI-CARRIER MODULATION

In multi-carrier modulation systems the transmitted signal in the discrete domain, [ ]y k is composed of successive modulated symbols, each of which is constructed as the sum of N sub-carriers, [ ]m k[ ]m k[ ]k[ ] , individually modulated. It can be expressed in the discrete domain as:

1

,0

[ ] ( [ ])N

s n ns n

y k x k sN1

0

[ ] ( [ ])y k x k sN[ ] ( [ ])y k x k sN[ ] ( [ ])y k x k sN[ ] ( [ ])y k x k sN[ ] ( [ ])y k x k sN[ ] ( [ ])y k x k sN[ ] ( [ ])N 1

,s n n,s n n,0

s n n,s n n,y k x k sN[ ] ( [ ])y k x k sN[ ] ( [ ])s n ny k x k sNs n n,s n n,y k x k sN,s n n,[ ] ( [ ])s n n[ ] ( [ ])y k x k sN[ ] ( [ ])s n n[ ] ( [ ]),[ ] ( [ ]),s n n,[ ] ( [ ]),y k x k sN,[ ] ( [ ]),s n n,[ ] ( [ ]),s n ny k x k sN[ ] ( [ ])y k x k sN[ ] ( [ ])s n ny k x k sNs n n[ ] ( [ ])s n n[ ] ( [ ])y k x k sN[ ] ( [ ])s n n[ ] ( [ ]) (1)

PAPR reduction in Multicarrier Modulations using Genetic Algorithms

Marco Lixia*, Maurizio Murroni*, Vlad Popescu***Department of Electrical and Electronic Engineering

University of Cagliari, 09123 Cagliari, Italy**Department of Electronics and Computers

Transilvania University of Brasov, 500019 Brasov, Romaniamarco.lixia@diee.unica.it, murroni@diee.unica.it, vladpop@unitbv.ro

938978-1-4244-7020-4/10/$26.00 '2010 IEEE

2010, 12th International Conference on Optimization of Electrical and Electronic Equipment, OPTIM 2010

where ,s nx is a constellation encoded ths data symbol

modulating the thn waveform. To reduce probability oferror, the sub-carriers are made orthogonal. The orthogonality of the subcarriers is given as follows [1]:

[ ], [ ] [ ]i jk k i j[ ], [ ] [ ]i j[ ], [ ] [ ]i j[ ], [ ] [ ]k k i j[ ], [ ] [ ]k k i j[ ], [ ] [ ]i jk k i ji j[ ], [ ] [ ]i j[ ], [ ] [ ]k k i j[ ], [ ] [ ]i j[ ], [ ] [ ] (2)

where, represents an inner product operation and [ ][ ][ ] is the dirac-delta function. In OFDM, the discrete

functions [ ]n k[ ]n k[ ]k[ ] are the complex basis functions 2

[ ]j nk

Np k ej nk

limited in the time domain by the window function [ ]p k . In the WP-MCM scheme, the sub-carrier waveforms are obtained by successively decomposing a couple of elementary paraunitary filters [2]. Let 0g i be a unit-energy real causal FIR filter of length Q which is orthogonal to its even translates; i.e.,

0 0 2ng i g i j j2g i g i j j2g i g i j j2g i g i j j0 0ng i g i j j0 0g i g i j j0 0g i g i j j , and let 1g i be the

(conjugate) Quadrature Mirror Filter (QMF),

1 01 1ig i g Q i1 0ig i g Q i1 0g i g Q i1 01 1g i g Q i1 11 01 11 0g i g Q i1 01 11 0ig i g Q ii1 1i1 1g i g Q i1 1i1 1 . If 0g i satisfies some mild

technical conditions [2], we can use an iterative algorithm to find the function 01 0 01 02 2

it g i t iT01 0 01 0t g i t iT01 0 01 0t g i t iT01 0 01 001 0 01 0t g i t iT01 0 01 0t g i t iT01 0 01 001 0 01 02 2t g i t iTt g i t iTt g i t iTt g i t iT01 0 01 0t g i t iT01 0 01 001 0 01 0t g i t iT01 0 01 001 0 01 0t g i t iT01 0 01 001 0 01 0t g i t iT01 0 01 02 2t g i t iT2 201 0 01 02 201 0 01 0t g i t iT01 0 01 02 201 0 01 001 0 01 0i01 0 01 0i01 0 01 001 0 01 02 201 0 01 0i01 0 01 0t g i t iT01 0 01 0t g i t iT01 0 01 02 2t g i t iT2 201 0 01 02 201 0 01 0t g i t iT01 0 01 02 201 0 01 001 0 01 0i01 0 01 0t g i t iT01 0 01 0i01 0 01 0 for

an arbitrary interval 0T . Subsequently, we can define the

family of functions ,l ml m,l m, , 00l , lm 21 2m in the following (binary) tree-structured manner:

1,2 1 0

1,2 1

l m lm li

l m lm li

t g i t iT

t g i t iT

t g i t iT

1,2 1l m lm l1,2 1t g i t iTl m lm lt g i t iTl m lm l1,2 1l m lm l1,2 1t g i t iT1,2 1l m lm l1,2 1

1,2 1 01,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0t g i t iTl m lm lt g i t iTl m lm l1,2 1 0l m lm l1,2 1 0t g i t iT1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0t g i t iTl m lm lt g i t iTl m lm l1,2 1 0l m lm l1,2 1 0t g i t iT1,2 1 0l m lm l1,2 1 01,2 1 0l m lm l1,2 1 0t g i t iTl m lm lt g i t iTl m lm l1,2 1 0l m lm l1,2 1 0t g i t iT1,2 1 0l m lm l1,2 1 0l m lm ll m lm l1,2 1 0l m lm l1,2 1 0l m lm ll m lm l1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0l m lm ll m lm l1,2 1 0l m lm l1,2 1 0

l m lm lt g i t iTl m lm lt g i t iTl m lm l1,2 1l m lm l1,2 1t g i t iTl m lm lt g i t iTl m lm l1,2 1l m lm l1,2 1t g i t iT1,2 1l m lm l1,2 1t g i t iT1,2 1l m lm l1,2 1l m lm l1,2 1l m lm l1,2 1l m lm l1,2 1t g i t iTl m lm lt g i t iTl m lm l1,2 1l m lm l1,2 1t g i t iT1,2 1l m lm l1,2 11,2 1l m lm l1,2 1l m lm l1,2 1l m lm l1,2 1l m lm l1,2 1t g i t iTl m lm lt g i t iTl m lm l1,2 1l m lm l1,2 1t g i t iT1,2 1l m lm l1,2 1t g i t iT1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0i1,2 1 0i1,2 1 01,2 1 0l m lm l1,2 1 0i1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 0l m lm l1,2 1 01,2 1 0l m lm l1,2 1 0i1,2 1 0l m lm l1,2 1 0t g i t iTl m lm lt g i t iTl m lm l1,2 1 0l m lm l1,2 1 0t g i t iT1,2 1 0l m lm l1,2 1 01,2 1 0l m lm l1,2 1 0i1,2 1 0l m lm l1,2 1 0t g i t iT1,2 1 0l m lm l1,2 1 0i1,2 1 0l m lm l1,2 1 0

1,2 1l m lm l1,2 1l m lm l1,2 1i1,2 1i1,2 11,2 1l m lm l1,2 1i1,2 1l m lm l1,2 1l m lm l1,2 1l m lm l1,2 11,2 1l m lm l1,2 1i1,2 1l m lm l1,2 1t g i t iTl m lm lt g i t iTl m lm l1,2 1l m lm l1,2 1t g i t iT1,2 1l m lm l1,2 11,2 1l m lm l1,2 1i1,2 1l m lm l1,2 1t g i t iT1,2 1l m lm l1,2 1i1,2 1l m lm l1,2 1

i (3)

where 02 TT ll 2 . For any given tree structure, the functions

at the leafs of the tree form a wavelet packet. Within a packet waveforms have a finite duration, 1 lQ TlQ T1Q T1 lQ Tl , and are self- and mutually-orthogonal at integer multiples of dyadic intervals. Fig. 1 shows the wavelet packets tree structure and the corresponding spectrum.

III. PTS FOR PAPR REDUCTION

A. Peak-to-Average Power RatioFor multi-carrier modulation system with N subcarriers

and input 0 1 1[ , ,......, ]TNX X X X0 1 1[ , ,......, ]0 1 1[ , ,......, ]0 1 10 1 1[ , ,......, ]0 1 1[ , ,......, ]0 1 1X X X X[ , ,......, ]X X X X[ , ,......, ]0 1 1[ , ,......, ]0 1 1X X X X0 1 1[ , ,......, ]0 1 1 consisting of N

independent modulated signal points, the transmitted signal in the time domain is given by

1x F Xx F X1x F X1x F X1 (4)

in which 0 1 1[ , ,......, ]TNx x x x0 1 1[ , ,......, ]0 1 1[ , ,......, ]0 1 1[ , ,......, ]0 1 1[ , ,......, ]0 1 1[ , ,......, ]0 1 1x x x x[ , ,......, ]x x x x[ , ,......, ]0 1 1[ , ,......, ]0 1 1x x x x0 1 1[ , ,......, ]0 1 1 and 1F 1 denotes an N NN N

inverse fast Fourier transform (IFFT) or inverse digital wavelet transform (IDWT) matrix, in case of OFDM or WP-MCM respectively. Then the PAPR of the transmitted signal is defined as:

1 202

maxNk kx

PAPR N2

1 21 21 2xxmaxk kPAPR NPAPR Nk kPAPR Nk k2PAPR N2PAPR N2

k kk k0k k0 xk kxk kPAPR NPAPR NPAPR Nk kPAPR Nk kk kPAPR Nk k0k k0PAPR N0k k0 (5)

in which 22 is the average power of the transmitted signal.

B. Partial Transmit SequenceBasically, PTS is obtained by the following steps:1. Partition the input data block X into M disjoint

subblocks ,0 ,1 , 1[ , ,......, ]Tm m m m NX X X X,0 ,1 , 1[ , ,......, ],0 ,1 , 1[ , ,......, ],0 ,1 , 1[ , ,......, ]m m m m N,0 ,1 , 1m m m m N,0 ,1 , 1[ , ,......, ]m m m m N[ , ,......, ],0 ,1 , 1[ , ,......, ],0 ,1 , 1m m m m N,0 ,1 , 1[ , ,......, ],0 ,1 , 1X X X X[ , ,......, ]X X X X[ , ,......, ],0 ,1 , 1[ , ,......, ],0 ,1 , 1X X X X,0 ,1 , 1[ , ,......, ],0 ,1 , 1m m m m NX X X Xm m m m N[ , ,......, ]m m m m N[ , ,......, ]X X X X[ , ,......, ]m m m m N[ , ,......, ],0 ,1 , 1[ , ,......, ],0 ,1 , 1m m m m N,0 ,1 , 1[ , ,......, ],0 ,1 , 1X X X X,0 ,1 , 1[ , ,......, ],0 ,1 , 1m m m m N,0 ,1 , 1[ , ,......, ],0 ,1 , 1 , with

1,2,...,m M1,2,...,1,2,...,m M1,2,...,m M1,2,..., and in which only /N M signal points exist and others are padded by zeros. For instance, in adjacent subblocks partition style

/ ( 1), / 1( 1) / ( ) //

[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N Mm N M M m N MN M

X X X[ 0,...,0 , ,..., , 0,...,0 ]( 1) / ( ) /m N M M m N M( 1) / ( ) /m N M M m N M( 1) / ( ) /( 1) / ( ) /( 1) / ( ) /m N M M m N M( 1) / ( ) /

[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]X X X[ 0,...,0 , ,..., , 0,...,0 ]X X X[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N MX X Xm mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]X X X[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1m mN M m N M/ ( 1), / 1m mN M m N M/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1m mN M m N M/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1

N M/N M/

[ 0,...,0 , ,..., , 0,...,0 ]X X X[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1X X X/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]X X X[ 0,...,0 , ,..., , 0,...,0 ]m mN M m N M[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1m mN M m N M/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1X X X/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1m mN M m N M/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1/ ( 1), / 1m mN M m N M/ ( 1), / 1/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1m mN M m N M/ ( 1), / 1[ 0,...,0 , ,..., , 0,...,0 ]/ ( 1), / 1( 1) / ( ) /( 1) / ( ) /m N M M m N M( 1) / ( ) /( 1) / ( ) //( 1) / ( ) /m N M M m N M( 1) / ( ) //( 1) / ( ) /N M( 1) / ( ) /N M( 1) / ( ) //N M/( 1) / ( ) //( 1) / ( ) /N M( 1) / ( ) //( 1) / ( ) /( 1) / ( ) /m N M M m N M( 1) / ( ) /N M( 1) / ( ) /m N M M m N M( 1) / ( ) /( 1) / ( ) //( 1) / ( ) /m N M M m N M( 1) / ( ) //( 1) / ( ) /N M( 1) / ( ) //( 1) / ( ) /m N M M m N M( 1) / ( ) //( 1) / ( ) /

.

2. For each subblock, an N-points IFFT/IDWT is computed.

3. Introduce the phase shifts set 1 2, ,......, Wjj je e e Wj1 2j j1 2j j1 2e e e1 2e e e1 21 2j j1 2e e e1 2j j1 2, ,......,j je e e1 2e e e1 2, ,......,e e e, ,......,j je e ej jj j1 2j j1 2e e e1 2e e e1 2j je e ej j1 2j j1 2e e e1 2j j1 2

in which [0,2 )ww [0,2 ) .4. The time domain signal after phase shifts combining is

given by

1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]ll M l l l Nx x x x b x x x1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]ll M l l l N1 2 ,0 ,1 , 1l M l l l N1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]l M l l l N' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1l M l l l N1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1x x x x b x x x' [ , ,..., ] [ ' , ' ,..., ' ]x x x x b x x x' [ , ,..., ] [ ' , ' ,..., ' ]lx x x x b x x xl' [ , ,..., ] [ ' , ' ,..., ' ]l' [ , ,..., ] [ ' , ' ,..., ' ]x x x x b x x x' [ , ,..., ] [ ' , ' ,..., ' ]l' [ , ,..., ] [ ' , ' ,..., ' ]' [ , ,..., ] [ ' , ' ,..., ' ]l M l l l N' [ , ,..., ] [ ' , ' ,..., ' ]x x x x b x x x' [ , ,..., ] [ ' , ' ,..., ' ]l M l l l N' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1l M l l l N1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1x x x x b x x x1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1l M l l l N1 2 ,0 ,1 , 1' [ , ,..., ] [ ' , ' ,..., ' ]1 2 ,0 ,1 , 1 (6)

with 1,2,...,l L1,2,...,l L1,2,...,l L1,2,..., and in which 1 2[ , ,..., ]l l l l TMb b b bl l l l T

1 2[ , ,..., ]1 2[ , ,..., ]1 2b b b b[ , ,..., ]b b b b[ , ,..., ]1 2[ , ,..., ]1 2b b b b1 2[ , ,..., ]1 2l l l l Tb b b bl l l l T[ , ,..., ]l l l l T[ , ,..., ]b b b b[ , ,..., ]l l l l T[ , ,..., ] with

lmb and where 1ML W ML W 1 is the number of

candidate signals.5. Select the optimum candidate signal '

optlx with the

lowest PAPR as

0101

1111 1212

2121 2222

3131 3232 3333 3434

2323 2424

0[ ]g n 1[ ]g n

0[ ]g n

0[ ]g n

0[ ]g n

0[ ]g n

1[ ]g n

1[ ]g n 1[ ]g n

(a)

f

2424 3232 3232 3333 3434 2424

Bw / 2Bw

/ 4Bw /8Bw

(b)

Fig. 1. (a) WP-MCM tree structure (b) Symbolic subband structure of the system in (a).

939

1 20 ,

1 2

max 'arg min

Nn l nL

opt l

xl 2

1 21 2max 'max '1 2max '1 21 2max '1 2max 'xmax 'max 'max 'max '0 ,0 ,n l n0 ,n l n0 ,0 ,n l n0 ,max 'n l nmax 'max 'n l nmax '0 ,max '0 ,n l n0 ,max '0 ,0 ,max '0 ,n l n0 ,max '0 ,max 'xmax '0 ,n l n0 ,x0 ,n l n0 ,max 'n l nmax 'xmax 'n l nmax '0 ,max '0 ,n l n0 ,max '0 ,x0 ,max '0 ,n l n0 ,max '0 ,max 'max 'max 'xmax '1arg minL

opt larg minopt larg min (7)

In PTS, the computational complexity consists of the following parts:

a. M zero padded IFFT/IDWT;b. Phase shift combining in (6);c. PAPR computation and comparison among L

candidate signals in (7).In general, a. are fixed and kept constant in PTS. Therefore, the computational complexity is mainly due to b. and c.

C. Phase Shifts OptimizationIn order to reduce the computational burden of PTS, in this work we propose a solution based on Genetic Algorithms (GA). GA are implemented as a computers simulation in which a population of abstract representations (chromosomes) of candidate solutions (genes) to an optimization problem evolves toward better solutions. The evolution usually starts from a population of randomly generated chromosomes and happens in generations. In each generation, the fitness of every chromosome in the population is evaluated, multiple chromosomes are stochastically selected from the current population (based on their fitness), and modified (mutated or recombined) to form a new population. The new population is then used in the next iteration of the algorithm.

In the proposed system, the chromosomes are defined as arrays of M genes ib . An initial population {INIT} of L chromosomes is randomly selected. The fitness function is as defined as in (5). Two operations are allowed to determine the evolution of the initial population: crossover(with probability crossP ) used to interchange the elements of two chromosomes and mutation (with probability mutP )which modify the value of one o more genes within a chromosome with the aim of leading the search out of local optima. In particular, the most fitting part of the population {BEST} is selected and directly inserted in the new generation, while the rest of the population {WORST} is discarded and replaced by a sub-population created by means of the crossover and mutation operators. The termination condition is satisfied once either the algorithmreaches a selected number of iterations ( IT ) or the fitness function remains unchanged for MAXIT iterations. At the end of the process the chromosome with low score in the fitness function PAPR will be selected for the transmission.

Fig. 3 gives an example of the crossover and mutation operations. In this particular case chromosomes are composed by four genes: at iteration 1k 1 the crossoveroperator swaps the first two genes of the chromosomes pand q as they were at iteration k , whereas the mutation varies the chromosome r by multiplying the second and fourth genes for the quantity ii with 2, 4i 2, 4 ,respectively.

The flowchart of the proposed GA is shown in Fig. 4. The accuracy of such approach is strictly dependent on the values of IT and ITMAX, whereas the complexity of the algorithm depends also on the definition of chromosomes, on the size L of the initial population and on the crossP and

mutP probabilities. Chromosomes are arrays of genes which are real values. The higher the precision on the representation of the genes (i.e., the number of decimal digits used to approximate real values) the higher the accuracy achieved, but also, the higher the complexity of the algorithm. Similarly, big size populations guarantee higher performance, but also lead to time consuming processing.

A critical matter is the selection of crossP and mutPprobabilities: high values can determine instability of the

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

1

q q q q

p p p p

p p q q

q q p p

b b b bk

b b b b

b b b bk

b b b b

b b b bb b b b1 2 3 4q q q q1 2 3 4q q q q1 2 3 41 2 3 4q q q q1 2 3 4q q q q1 2 3 4b b b b1 2 3 4b b b b1 2 3 41 2 3 4q q q q1 2 3 4b b b b1 2 3 4q q q q1 2 3 4q q q q1 2 3 4q q q q1 2 3 4b b b b1 2 3 4b b b b1 2 3 4q q q qb b b bq q q q1 2 3 4q q q q1 2 3 4b b b b1 2 3 4q q q q1 2 3 4q q q q1 2 3 4q q q q1 2 3 4q q q q1 2 3 4q q q q1 2 3 4

b b b bb b b b1 2 3 4p p p p1 2 3 4p p p p1 2 3 41 2 3 41 2 3 4p p p p1 2 3 4b b b b1 2 3 4b b b b1 2 3 41 2 3 4p p p p1 2 3 4b b b b1 2 3 4p p p p1 2 3 4b b b bb b b bp p p pp p p p1 2 3 4p p p p1 2 3 4p p p p1 2 3 4p p p p1 2 3 4b b b b1 2 3 4b b b b1 2 3 4p p p pb b b bp p p p1 2 3 4p p p p1 2 3 4b b b b1 2 3 4p p p p1 2 3 4

b b b bb b b b1 2 3 4p p q q1 2 3 4p p q q1 2 3 41 2 3 41 2 3 4p p q q1 2 3 4b b b b1 2 3 4b b b b1 2 3 41 2 3 4p p q q1 2 3 4b b b b1 2 3 4p p q q1 2 3 41 2 3 4p p q q1 2 3 4b b b b1 2 3 4b b b b1 2 3 41 2 3 4p p q q1 2 3 4b b b b1 2 3 4p p q q1 2 3 41

p p q q1 2 3 4p p q q1 2 3 4p p q q1 2 3 4p p q q1 2 3 4b b b b1 2 3 4b b b b1 2 3 4p p q qb b b bp p q q1 2 3 4p p q q1 2 3 4b b b b1 2 3 4p p q q1 2 3 4p p q q1 2 3 4p p q q1 2 3 41 2 3 4p p q q1 2 3 4p p q q1 2 3 4p p q q1 2 3 4p p q q1 2 3 4p p q q1 2 3 4

b b b bb b b b1 2 3 4q q p p1 2 3 4q q p p1 2 3 41 2 3 41 2 3 4q q p p1 2 3 4b b b b1 2 3 4b b b b1 2 3 41 2 3 4q q p p1 2 3 4b b b b1 2 3 4q q p p1 2 3 4b b b bb b b bq q p pq q p p1 2 3 4q q p p1 2 3 4q q p p1 2 3 4q q p p1 2 3 4b b b b1 2 3 4b b b b1 2 3 4q q p pb b b bq q p p1 2 3 4q q p p1 2 3 4b b b b1 2 3 4q q p p1 2 3 4

1 2 3 4

1 2 2 3 4 41 ( ) ( )r r r r

r r r r

k b b b b

k b b b b1 2 2 3 4 41 ( ) ( )1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4r r r r1 2 2 3 4 4r r r r1 2 2 3 4 41 ( ) ( )r r r r1 ( ) ( )1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4r r r r1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4k b b b b1 ( ) ( )k b b b b1 ( ) ( )1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4k b b b b1 2 2 3 4 41 ( ) ( )1 2 2 3 4 41 ( ) ( )r r r r1 ( ) ( )k b b b b1 ( ) ( )r r r r1 ( ) ( )1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4r r r r1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4k b b b b1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4r r r r1 2 2 3 4 41 ( ) ( )1 2 2 3 4 4

crossover

mutation

Fig. 3. Example for crossover and mutation operators in case of chromosomes composed by four genes.

DataSource

serial toparallel

conversion

Divisioninto

subblocks

IDWT/IFFT

+

Genetic Algorithms optimization

IDWT/IFFT

IDWT/IFFT

X

X

XX

X1

X2

XM

b1

b2

X’lopt

bM

Fig. 2. PTS scheme for OFDM and WP-MCM

940

GA which could diverge, whereas, on the other side, low values likely lead to slow convergence.

Moreover, one of the points of strength of the GA is the reduced computational load which is independent from the number W of the possible values for the phase shift factors.It depends on size of the initial population L and on the number K of iterations executed before the terminationcondition is satisfied. The computational load can be expressed as .L K.L K

{INIT}

mutation, mutP

crossover, crossP

{BEST} {WORST}

FITNESS EVALUATION

TERMINATION CONDITION?

YES END

NO

SELECTION

DISCARD

11 12 1

21 22 2

1 2

. . .

. . .

.

.

.

. . .

M

M

L L LM

b b b

b b b

b b b

b b bb b b11 12 1M11 12 1M11 12 111 12 1. . .11 12 1Mb b b11 12 1b b b11 12 111 12 1. . .11 12 1b b b11 12 1. . .11 12 1

b b bb b b21 22 2M21 22 2M21 22 221 22 2. . .21 22 2Mb b b21 22 2b b b21 22 221 22 2. . .21 22 2b b b21 22 2. . .21 22 2

.

.

.

b b bb b bL L LM1 2L L LM1 2L L LM1 2L L LM1 2 . . .L L LM. . .b b bL L LMb b bL L LM1 2L L LM1 2b b b1 2L L LM1 2 . . .L L LM. . .b b b. . .L L LM. . .L L LMb b b1 2b b b1 2L L LMb b bL L LM1 2L L LM1 2b b b1 2L L LM1 2 . . .L L LM. . .b b b. . .L L LM. . .

1 2 . . . Mb b bb b b1 2 M1 2 . . . Mb b b1 2b b b1 2 . . .b b b. . . Mb b bM

Fig. 4. Flowchart of the proposed GA

IV. TESTING AND SIMULATION

Referring to the sketch of the system in Fig. 2, a binary periodic (period ) memoryless source : 1 ,S s: 1 ,: 1 ,

with a rate of 58 1058 10 bit/s and 16-QAM constellation

, 1, 2, 16,, ,...,s m m m mS x x x xs m m m m, 1, 2, 16,s m m m m, 1, 2, 16,, 1, 2, 16,, ,...,, 1, 2, 16,s m m m m, 1, 2, 16,, ,...,, 1, 2, 16,S x x x x, 1, 2, 16,S x x x x, 1, 2, 16,, 1, 2, 16,, ,...,, 1, 2, 16,S x x x x, 1, 2, 16,, ,...,, 1, 2, 16,s m m m mS x x x xs m m m m, 1, 2, 16,s m m m m, 1, 2, 16,S x x x x, 1, 2, 16,s m m m m, 1, 2, 16,, 1, 2, 16,, ,...,, 1, 2, 16,s m m m m, 1, 2, 16,, ,...,, 1, 2, 16,S x x x x, 1, 2, 16,, ,...,, 1, 2, 16,s m m m m, 1, 2, 16,, ,...,, 1, 2, 16, have been considered.Classic FFT algorithm for OFDM and Daubechies QMF of 5th order for WP-MCM have been used. 128N 128subcarriers and a subblocks size of 8M 8 have been set,whereas the number of possible phase shifts W was varied in the set {4, 20, 40, 60} and 2 /w Ww 2 /W .As regarding the GA several tests have been conducted with different combination of values of initial population, mutPand crossP , therefore the following considerations can be made: huge size initial population brings to better solution at the expense of a higher processing time; the mutPprobability is suggested to be set equal to or higher than 0.1 to avoid an excessive number of iterations; the crossPprobability does not sort significant effects in the used range. Finally in the simulation values of mut 0.3P 0.3 and

cross 0.7P 0.7 have been chosen. The initial population L was chosen within the set {150, 350, 500}, the number of iteration 70IT 70 and 5MAXIT 5 iterations after which the GA ends if the fitness function does not improve at least of 0.05 dB. Table I summarizes the parameter setting for the test.

TABLE IPARAMETERS SETTING FOR EXPERIMENTS

Symbol Definition Setting

S source 58*10 bit/s

M subblock size 8

W Number of possible phase shifts

4,20,40,60

Lsize of initial population

150,350,500

ib Phase Factor (gene) 1 2, ,..., Wj j jib e e e Wj j jb e e e1 2b e e e1 2j j jb e e ej j j1 2j j j1 2b e e e1 2j j j1 2, ,...,j j jb e e e1 2b e e e1 2, ,...,b e e e, ,...,j j jb e e ej j jj j jb e e e1 2b e e e1 2j j jb e e ej j j1 2j j j1 2b e e e1 2j j j1 2

mutP mutation probability 0.3

mutation quantity99

crossP crossover probability 0.7

IT number of iteration 70

MAXITmaximum number of

iteration with unchanged fitness

5

The Complementary Cumulative Distribution Function (CCDF) of the PAPR denotes the probability that the PAPR of a data block exceeds a given threshold and is expressed as follow:

0 0PrCCDF PAPR PAPR PAPR0 0CCDF PAPR PAPR PAPR0 0CCDF PAPR PAPR PAPR0 0PrCCDF PAPR PAPR PAPRPr0 0Pr0 0CCDF PAPR PAPR PAPR0 0Pr0 0 (8)

Fig. 5 shows the CCDF trends of PAPR for classic OFDM and OFDM with PTS for 500L 500 and confirms the effectiveness of PTS for OFDM systems in case of AWGN channel.

Fig. 5. PTS performance in terms of PAPR in the case of OFDM

Zooming around CCDF= 210 2 (fig. 6) it is possible to note that increasing the number of phase shifts allows reducingPAPR of 0.44 dB: 8.22PAPR 8.22 dB for 60W 60 whereas for 4W 4 8.66PAPR 8.66 dB. Moreover, an absolute reduction of about 2 dB respect to the classic OFDM is achieved. Figs. 7 and 8 show the CCDF trend of PAPR for classic WP-MCM and WP-MCM with PTS again for

500L 500 and a zoomed version around CCDF= 210 ,210 ,210 ,2

respectively for transmission on AWGN channel. Although, in this case the absolute performance of the PTS is higher with respect to the OFDM ( 7.36PAPR 7.36 dB and

8.03PAPR 8.03 dB for 60W 60 and 4W 4 , respectively) the improvement achieved is lower: in OFDM system the performance gain is 2 dB ( 60W 60 ) whereas in WP-MCMsystem the total gain is about 1.5 dB ( 60W 60 ).

0 2 4 6 8 10 12

10-2

10-1

100

PAPR0 [dB]

CC

DF=

Pr(P

APR

>PA

PR0)

W=4W=20W=40W=60Classic OFDM

941

Fig. 6. Zoomed version of Fig. 5 around CCDF=10-2

Table II shows the PTS performance in terms of PAPRcorresponding to a value of CCDF= 210 2 and depending on initial population for different values of W. It is worth noting that for values of L less than 350 there is nodifference between 4W 4 and 60,W 60, and results are not consistent. The better solution is 500.L 500. During the simulations, larger values of L have been used but the system did not improve enough to justify the increased computational complexity.

TABLE IIPTS PERFORMANCE DEPENDING ON INITIAL POPULATION

WOFDM WP-MCM

Initial Population Initial Population150 350 500 150 350 500

4 8.73 dB 8.67 dB 8.6 dB 8.17 dB 8.06 dB 8.03 dB20 8.48 dB 8.47 dB 8.42 dB 8.17 dB 7.98 dB 7.85 dB40 8.73 dB 8.57 dB 8.34 dB 7.91 dB 7.76 dB 7.67 dB60 8.72 dB 8.27 dB 8.22 dB 8.05 dB 7.8 dB 7.36 dB

For the sake of clarity, Table III summarizes all results showed above and adds the computational load of the simulations which allows some more considerations.Comparison between classic PTS and PTS with GAoptimization leads to underline the computational improvement guaranteed by the system presented in this work: for 60W 60 and 500L 500 the computational load of the PTS classic is 1 760MW 1 71 7601 7601 7601 7 whereas using GAoptimization the computational load is

3500 46 23 10 .L K 3500 46 23 10 .3500 46 23 10 .3L K

TABLE IIIPAPR AND NUMBER OF ITERATION FOR CCDF=10-2

ParameterOFDM WP-MCM

PAPR Number of iterations PAPR

Number of iterations

W=4 8.6 dB 18 8.03 dB 16W=20 8.42 dB 30 7.85 dB 23W=40 8.34 dB 42 7.65 dB 27W=60 8.225 dB 46 7.36 dB 36

Moreover, even if the parameter 60W 60 guarantees better performance, it involves also a higher processing time. As a consequence in a real case a difference choice can be made in order to obtain a trade-off between performance and computational load.

V. CONCLUSION

In this paper we have proposed the use of GA in conjunction with PTS to reduce PAPR in multi-carrier

modulation systems. Results show the effectiveness of the proposed method in reducing both the computational complexity of the PTS algorithm and the absolute value of the PAPR in case of AWGN channel.

Fig. 7. PTS performance in terms of PAPR in the case of WP-MCM

Fig. 8. Zoomed version of Fig. 7 around CCDF=10-2

REFERENCES

[1] I J. A. C. Bingham, “Multi-carrier modulation for data transmission: an idea whose time has come,” IEEE Communications Magazine, vol. 28, no. 5, pp. 5–14, May 1990.

[2] A. R. Lindsey, “Wavelet packet modulation for orthogonally multiplexed communication,” IEEE Transaction on Signal Processing, vol. 45, no. 5, pp. 1336–1339, May 1997.

[3] S. H. Han; J. H. Lee, "An overview of peak-to-average power ratio reduction techniques for multi-carrier transmission," Wireless Communications, IEEE , vol.12, no.2, pp. 56-65, April 2005.

[4] S. H. Müller and J. B. Huber, “A novel peak power reduction scheme for OFDM,” Proc. IEEE PIMRC, Helsinki, Finland, Sept. 1997, pp. 1090– 1094.

[5] A. D. S. Jayalath and C. Tellambura, “Adaptive PTS approach for reduction of peak-to-average power ratio of OFDM signal,” Electron. Lett., vol. 36, no. 14, pp. 1226–1228, Jul. 2000.

[6] C. Tellambura, “Improved phase factor computation for the PAR reduction of an OFDM signal using PTS,” IEEE Commun. Lett., vol. 5, no. 4, pp. 135–137, Apr. 2001.

[7] A. Alavi, C. Tellambura, and I. Fair, “PAPR reduction of OFDM signals using partial transmit sequence: An optimal approach using sphere decoding,” IEEE Commun. Lett., vol. 9, no. 11, pp. 982–984, Nov. 2005.

[8] S. H. Han and J. H. Lee, “PAPR reduction of OFDM signals using a reduced complexity PTS technique,” IEEE Signal Process. Lett., vol. 11, no. 11, pp. 887–890, Nov. 2004.

[9] D. Whitley, “A genetic algorithm tutorial” Statistics and Computing, vol. 4, pp. 65–85, 1994.

[10] L. Atzori and A. Raccis, “Network capacity assignment for multicast services using genetic algorithms”, IEEE Communications Letters, vol. 8, no. 6, pp. 403-405, June 2004.

[11] C.W. Ahn and R.S. Ramakrishna, “A genetic algorithm for shortest path routing problem and the sizing of population”, IEEE Transaction on Evolutionary Computation, vol. 6, no. 6, Dec. 2002.

[12] M. Murroni, ”Performance analysis of modulation with unequal power allocations over fading channels: A genetic algorithm approach,” 14th European Wireless Conference, 2008. EW 2008, pp: 1-6, 22-25 June 2008, Prague (CZ).

7 7.5 8 8.5 9

10-2

PAPR0 [dB]

CC

DF=

Pr(P

APR

>PA

PR0)

W=4W=20W=40W=60

0 2 4 6 8 10 12

10-2

10-1

100

PAPR0 dB

CC

DF

= Pr

(PA

PR >

P0)

Classic WP-MCMW=4W=20W=40W=60

6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6

10-2

PAPR0 dB

CC

DF

= Pr

(PA

PR >

P0)

W=4W=20W=40W=60

942