PAPR Reduction using Combined Coding, Weighting, and

PAPR Reduction using Combined Coding, Weighting, and Mappingin OFDM Systems1

Abul K. M. G. A. Akanda∗ and Raveendra K. Rao∗∗

Faculty of Engineering

Department of Electrical & Computer Engineering

The University of Western Ontario

Elborn College, 1201 Western Road

London, ON N6G 1H1, Canada

E-mail: ∗[email protected],∗∗[email protected]

Abstract– In this paper we present methods forreduction of Peak-to-Average Power Ratio (PAPR)in multi-carrier Orthogonal Frequency Division Mul-tiplexing (OFDM) through the joint use of coding,weighting, and signal mapping. Simple and easy-to-implement coding schemes have been identifiedthat when employed with suitable weighting func-tions can reduce PAPR considerably. Indeed it isshown that PAPR can be controlled by appropriatechoice of coding scheme and weighting function for agiven signal mapping. A thorough numerical investi-gation using computer simulations is presented andschemes that offer considerable reduction in PAPRare identified as a function of number of sub-carriersin OFDM systems.

I. Introduction

The demand for mobile communications andcomputing combined with increased growth forInternet access is growing rapidly. The key torealizing this potential is the development anddeployment of high performance radio systemsthat support high data rates with wide area cov-erage. In conventional single-carrier communi-cation systems for high data rates the effect ofmultipath propagation increases the equalizationcost due to short symbol durations and relativelylong channel delay times. OFDM [1]-[2] is an al-ternative technique for high bit rate transmissionin a radio environment in which data transmis-sion is parallel; unlike in the single-carrier sys-tem where it is serial. In such a system, thedeleterious effect of fading is spread over manybits; therefore, instead of a few adjacent bitscompletely destroyed by the fading, it is morelikely that several bits only be slightly affectedby the channel. Furthermore, when the data is

1This work was supported by Natural Sciences and En-gineering Research Council of Canada (NSERC) underGrant No.239042-01 RGPIN

transmitted at high bit rates over mobile radiochannels, there exists ISI due to the channel im-pulse response extending over several symbol du-rations. In such a situation OFDM can be gain-fully employed to mitigate the effects of ISI. Atechnique to reduce ISI is to increase the numberof sub-carriers by reducing the bandwidth of eachsub-channel while maintaining the overall band-width. In OFDM systems ISI can be eliminatedusing guard interval. Among the other advan-tages of OFDM are: a) Spectral efficiency due tooverlapping sub-channels; b) realization of mod-ulation and demodulation using FFT techniques;c) Less sensitivity to sample timing offsets thansingle carrier systems; d) Superior performanceover frequency-selective channels via the use ofchannel coding and interleaving; and e) protec-tion against co-channel interference.In recent years OFDM systems are included

in digital audio/video broadcasting (DAB/DVB)standards in Europe [3]-[4], while the discretemulti-tone (DMT) (its wire line counterpart inUSA) has been applied to high-speed digital sub-carrier (DSL) modems over twisted pair [5]-[6].OFDM is also a candidate proposed for digitalcable TV systems [7] and local area mobile wire-less networks such as IEEE 802.11a, the MMAC,and the HIPERLAN/2 [8]-[9]. Multi-carrier hy-brids with DS-CDMA have also been developedfor wideband cellular communications.In OFDM systems, the transmitted signal oc-

casionally exhibits very high peaks. Thus an in-herent difficulty is the fact that they have a verylarge PAPR. The principal drawback of OFDMis that the peak-transmitted power may be up toN times the average power, with N, the numberof sub-carriers [10]. A large PAPR introduces anincreased complexity of the analog-to-digital anddigital-to-analog converters and degrade the effi-

ciency of RF power amplifier. Linear amplifiersthat can handle the peak power are less efficient.Hard limiting of the transmitted signal gener-ates intermodulation among the sub-carriers andresults in out-of-band radiation. In the liter-ature [2],[10]—[23] various techniques have beenproposed and investigated to overcome the prob-lem of large peaks in the transmitted OFDM sig-nals. Recently, weighting functions [12]-[13] havebeen considered for reduction of PAPR. Appro-priate choice of coding and weighting thus stillremains an open option in order to jointly opti-mize PAPR and bit error rate (BER). While inthe literature, coding and weighting have beenconsidered separately for reduction of PAPR, ourobjective of this paper is, therefore, to investigatethe combined influence of coding and weightingon PAPR in OFDM systems. It is shown thatPAPR can be controlled by appropriate choice ofcoding scheme and weighting function for a givensignal mapping used in OFDM system. Whileweighting can be used to reduce PAPR, it in-creases the probability of bit error. However, bychoosing coding and weighting it is possible to re-duce PAPR and at the same time one can controlbit error rate. The objective therefore is to in-vestigate the problem of PAPR by examining theinfluence of coding, in particular, parity bit andhorizontal/vertical parity bit codings on PAPR.These codings have not been examined in the lit-erature. Next, we consider novel weighting func-tions in an attempt to reduce PAPR. Finally, Athorough numerical investigation using computersimulations is presented and schemes that offerconsiderable reduction in PAPR are identified asa function of number of sub-carriers in OFDMsystems.In Section II we briefly describe a typical

OFDM system. In section III we address theproblem of controlling PAPR in a OFDM systemby jointly using coding and weighting and arriveat expressions for its computation. In SectionIV the coding and weighting functions consid-ered are briefly explained. Section V is devotedto numerical results and discussion of techniquesfor controlling PAPR. The paper is concluded inSection VI.

II. Typical OFDM System

The block diagram of a typical OFDM systemis shown in Fig. 1 . The data source is assumedto be a sequence of discrete digits:

{a0, a1, a2, ...}

Serialto

ParallelConverter

IFFT

P/SGuard

IntervalInsertion

DataSource Encoder

Transmitter

Receiver

y(t)

SignalMapper

SignalMapper

D/A Modulator

Parallelto

SeriallConverter

FFT

S/PGuard

IntervalRemoval

DataSink Decoder

SignalDeMapper

SignalDeMapper

A/D Demodulator

x(t)

r(t)

Fig. 1. Block diagram of a typical OFDM system.

where,

ai = 1 or 0, i = 0, 1, 2, 3, ...

and

P (ai = 1) = P (ai = 0) =1

2

The output of the encoder is fed to a serial-to-parallel converter that partitions the input dataarriving at the rate R into N parallel informationsymbols each at a reduced data rate of RN . Thenumber of bits in each of the N output sequencesof a serial-to-parallel converter is determined bythe constellation of the signal mapper. For ex-ample, when a BPSK mapper is used, each out-put sequence carries one bit of information. ForQPSK and 16-QAM, each channel carries 2 and4 bits of information. The output of the signalmapper can be thought of as a discrete complexsignal which during any arbitrary OFDM symbolduration is a vector of N complex numbers givenby:

C = [C0, C1, ..., CN−1] (1)

This discrete signal representing the data is fedto an N -point IFFT block to obtain a trans-formed discrete signal given by:

D = [D0,D1, ...,DN−1] (2)

The relationship between the input and the out-put discrete signals of IFFT block is given by thewell known Discrete Time Fourier Series (DTFS).That is,

Dn = An + jBn

=N−1

m=0

Cmej 2πN mn;n = 0, 1, ..., N − 1(3)

where it is assumed that the input sequence C isperiodic with period equal to N .

x(t) =N−1

m=0

Cmej2πfmt (4)

where fm =mNTb

and t = nTb. Tb is the originalduration of bit and fm is the frequency of themth carrier. It can be shown that the discretesequenceD = [D0,D1, ...,DN−1] can be obtainedby sampling x(t) at t = nTb, n = 0, 1, ..., N − 1.That is ,

x(nTb) = Dn =N−1

m=0

Cmej2πfmnTb (5)

The complex continuous-time modulating signalgiven in (4) can be written as:

x(t) = mI(t) + jmQ(t) (6)

wheremI(t) = Re{x(t)} (7)

andmQ(t) = Im{x(t)} (8)

The relationship between discrete and continu-ous signals given by (3) and (4) is illustrated inFigs.2 and 3. The signals mI(t) and mQ(t) canbe written as:

mI(t) =N−1

m=0

(am cos 2πfmt− bm sin 2πfmt) (9)

and

mQ(t) =N−1

m=0

(am sin 2πfmt+ bm cos 2πfmt)

(10)where Cm = am + jbm, with am and bm repre-senting the in-phase and quadrature components,respectively. The modulating signal given by(6) is then modulated and transmitted using in-phase and quadrature-phase processing using acarrier frequency fc. The modulator structure isshown in Fig.4 and the modulated signal can bemathematically written as:

y(t) = mI(t) cos 2πfct−mQ(t) sin 2πfct (11)

Thus the objective at the receiver is to recovermI(t) and mQ(t) from y(t), assuming a noiselesssystem. This can be accomplished using the in-phase and quadrature-phase detector shown inFig.5. The demodulated baseband signals mI(t)

1 2 3 N-1. . .

1 2 3 . . . N-1

0

0

n

n

A0′

D/A mQ(t)

D/A mI(t)

A1′ A2

′

AN-1′

B0′

B1′

B2′

BN-1′

mI[n]

mQ[n]

Tb

Fig. 2. Discrete-time signals at the output of theparallel-to-serial converter

1 2 3 N-1. . .

A/0

A/1

A/N-1

mI(t)

1 2 3 . . . N-1

mQ(t)

0

0

B/0

B/1

B/N-1

t

t

A/2

B/2

Tb

Tb

Fig. 3. Continuous-time signals at the output of theD/A converter

mQ(t)900

cos2πfct

∑+

-

mI(t)

IFFT

D/A

D/A

P/S

P/S

[A′]

[B′]

[A′]=[A′0, A′

1,…, A′N-1]; [B′]=[B′

0, B′1,…, B′

N-1]

y(t)

Fig. 4. In-phase and quadrature-phase modulator

y(t) 900

2cos2πfctLPF

LPF

mI(t)

mQ(t)-1

A/D

A/D

S/PFFT

[A′]

[B′]

[C]

Fig. 5. In-phase and quadrature-phase detector

and mQ(t) are sampled and fed to the S/P con-verter which produces a vector of complex num-bers [D0,D1, ...,DN−1]. The complex numbers

[C0, C1, ..., CN−1] can then be recovered usingFFT operation and thus the transmitted datasequence using the signal de-mapper.

III. Formulation of PAPR Problem

The transmitted OFDM signal may be mod-eled as (from (11)):

y(t) = mI(t) cos 2πfct−mQ(t) sin 2πfct (12)

wheremI(t) andmQ(t) are given by (9) and (10),respectively. The complex modulating signal isgiven by:

x(t) = mI(t) + jmQ(t) =N−1

m=0

Cmej2πm

T t (13)

The transmitted signal in (12) can be written as:

y(t) = m2I(t) +m

2Q(t)

× cos 2πfct+ tan−1 mQ(t)

mI(t)(14)

The envelope of the transmitted signal is thusgiven by

Envelope of y(t) = y(t) = m2I(t) +m

2Q(t)

(15)The PAPR of the envelope of the transmittedsignal can then be written as:

PAPR =max{[y(t)]2}E{[y(t)]2} =

max{|x(t)|2}E{|x(t)|2} (16)

where x(t) denotes the complex modulating sig-nal. It is noted that the instantaneous power ofthe envelope of the transmitted OFDM signal isgiven by:

p(t) = [y(t)]2 = |x(t)|2 (17)

The computation of PAPR in (16) is approxi-mated using discrete-time PAPR, which is ob-tained from the samples of the OFDM signal.The sampling rate is the Nyquist rate or a multi-ple of it (oversampling). An important questionis how large the oversampling factor should bein order for the approximation to be fairly accu-rate. The reasons for discrete-time approach forcomputation of PAPR are: i) most systems areimplemented in discrete-time; ii) computation incontinuous-time is too complex as closed-formexpression is difficult to arrive at; and iii) whenoversampling rate of six times that of Nyquist

rate is used, it closely approximates continuous-time PAPR. Therefore, our approach is to em-ploy discrete-time approximation of continuous-time PAPR by using a sampling rate that is sixtimes the Nyquist rate. The discrete-time com-putation of PAPR is approached by sampling theenvelope of y(t). Which is equivalent to samplingthe complex modulating signal. That is,

y(t = t ) = |x(t = t )| (18)

When Nyquist rate of N samples over the OFDMsymbol interval is taken of y(t), we obtain:

|Dn| = |x(t = nTb)|

= |N−1

m=0

Cmej2πm

N n| (19)

where n = 0, 1, ..., N−1. Using this discrete-timeapproach PAPR in (16) becomes

PAPR =max{|Dn|2, n = 0, 1, ...,N − 1}

1N

N−1

n=0|Dn|2

(20)Using (20) PAPR can be computed easily, since{Dn;n = 0, 1, ..., N − 1} is the output of IFFTblock in response to the information carrying in-put sequence {Cm;m = 0, 1, ..., N − 1}. In arriv-ing at the PAPR using (20) we have used an N-point approximation of the continuous-time en-velope of OFDM signal. It is noted that the N-point approximation may not reveal the real peakpower of the envelope. A better approximation ofPAPR can be obtained by using higher samplingrate than the Nyquist rate N , which is explainednext.Consider using a sampling rate that is two

times the Nyquist rate in which case the inputto the IFFT block would consist of the origi-nal sequence [C] and an additional N zeros ap-pended to it. Denoting this new sequence as[C ] = [C0, ..., CN−1, CN , ..., C2N−1], it is notedthat

Cm =Cm, m = 0, 1, ..., N − 10, m = N,N + 1, ..., 2N − 1

(21)The 2N -point IFFT of [C ] then is given by[D ] = [D0, ...,DN−1,DN , ...,D2N−1], where

Dn =2N−1

m=0

Cmej2π( m2N )n (22)

The sequence of 2N samples [Dn;n =0, 1, ..., 2N − 1] at the output of the IFFT

block, after passing through parallel-to-serial anddigital-to-analog converter can be written as:

x (t) =2N−1

m=0

Cmej2πfmt (23)

wherefm =

m

2N Tb2

(24)

It is noted that by employing a sampling ratethat is twice that of Nyquist rate it is possibleto take into account peaks of the envelope thatlie between the adjacent samples. By increas-ing the sampling rate further the discrete-timePAPR becomes close to that of continuous-timePAPR. The expression of PAPR when the sam-pling rate is L times the Nyquist rate N can bewritten as:

PAPR =max{|Dn|2, n = 0, 1, ..., LN − 1}

1LN

N−1

n=0|Dn|2

(25)

IV. Coding and Weighting

In this Section we briefly describe the codingand weighting functions used in this paper.

A. Coding

In the literature there exist a number of meth-ods for encoding, such as block codes, convolu-tional codes, etc. In this paper we consider aclass of block codes and, in particular, single par-ity check and horizontal and vertical parity checkcodes. Hence we describe these two classes ofcodes and list their properties [24].1) Single Parity Check Codes: In single par-

ity check coding a single bit is appended, called aparity check, to a string of data bits. This paritycheck bit has the value 1 if the number of 1’s inthe bit string is odd, and has the value 0 other-wise. This type of coding is referred to as evensingle parity check coding. While in even single-parity check coding a block of N bits is mappedto N+1 bits with the final bit the modulo 2 sumof the N data bits, in odd single parity checkcoding the final check bit added has a value 0 ifthe number of 1’s in the data bit string is odd,and a value 1 if it is even. The rate of the code isKK+1 , where K is the length of the data sequence.While it is well known that single parity checkbit coding is inadequate from the viewpoint ofreliable detection of errors [25], the objective in

this paper is to investigate this type of codingfrom the viewpoint of PAPR reduction. At thetransmitting end single-parity check coding canbe easily implemented using modulo-2 arithmeticand at the receiving end the decoding consists oftesting whether the modulo-2 sum of the codeword bits yields a zero result (for even parity) orone result (for odd parity). The bit rate at theoutput of a single parity bit encoder is given by:

R =K + 1

KRb bits/ sec (26)

2) Horizontal and Vertical Parity Check Code:Another simple and intuitive approach of encod-ing is to arrange a string of data bits in a twodimensional array with one parity check for eachrow and one for each column. The parity checkin the lower right corner can be viewed as a par-ity check on the row parity checks, on columnparity checks, or on the data array. These hor-izontal and vertical parity checks are also calledrectangular codes and product codes. In generalthe incoming serial data is arranged in the formof a matrix of M rows and N columns and thismatrix is mapped to (M + 1) × (N + 1) matrixthat contains appended check bits. Thus the rateof a rectangular code is given by

k

n=

MN

(M + 1)(N + 1)(27)

The bit rate at the output of a rectangular en-coder is given by:

R =(M + 1)(N + 1)

MNRb bits/ sec (28)

Rectangular code has error correcting capabil-ity besides error detection properties. A majorweakness of this scheme is that it may fail todetect rather short bursts of errors.

B. Weighting

In order to introduce weighting for each sub-carrier in (13), we employ discrete weightingfunction given by wm,m = 0, 1, 2, ..., N − 1, in(13). Thus the weighted complex OFDM modu-lating signal can be written as:

xw(t) =N−1

m=0

Cmwmej2πfmt (29)

Denoting the product Cmwm by Cmw the trans-mitted weighted OFDM signal can be written as

yw(t) = mIw(t) cos 2πfct−mQw(t) sin 2πfct(30)

wheremIw(t) = Re{xw(t)} (31)

mQw(t) = Im{xw(t)} (32)

andxw(t) = mIw(t) + jmQw(t) (33)

The weighted transmitted OFDM signal yw(t)can be written as

yw(t) = m2Iw(t) +m

2Qw(t)

× cos 2πfct+ tan−1 mQw(t)

mIw(t)

(34)

The envelope of the weighted transmitted signalis therefore given by

Envelope of yw(t) = yw(t) = m2Iw(t) +m

2Qw(t)

(35)The PAPR of the envelope is given by:

PAPR =max{[yw(t)]2}E{[yw(t)]2} =

max{|xw(t)|2}E{|xw(t)|2} (36)

where xw(t) denotes the complex modulating sig-nal. The instantaneous power associated withthis envelope is given by :

pw(t) = [yw(t)]2 = |xw(t)|2 (37)

Again we approach the computation of PAPRin (36) using discrete samples of yw(t). WhenNyquist rate is employed, using developmentscarried out for the non-weighted OFDM signalsfrom (12) to (20), the PAPR can be written as

PAPR =max{|Dnw|2;n = 0, 1, ...,N − 1}

1N

N−1

m=0|Dnw|2

(38)where

Dnw =N−1

m=0

Cmwej2π(mN )n (39)

The set of complex number {Dnw, n =0, 1, .., N − 1} is the output of IFFT block in re-sponse to the set of weighted complex numbers{Cnw, n = 0, 1, .., N − 1}. Also, it can be shownthat

|Dnw| = yw(t = nTb) = |xw(t = nTb)|

=N−1

m=0

Cmwej2π(mN )n (40)

where n = 0, 1, ..., N − 1. The PAPR of theweighted OFDM signal given above can be sim-plified by first finding an expression for its in-stantaneous power. That is

pw(t) = |xw(t)|2

=N−1

m=0

N−1

n=0

(Cmwm)(Cnwn)∗ej2π

(m−n)T t

(41)

which can be simplified and written as:

pw(t) =N−1

m=0

|Cmwm|2

+N−1

m=0

N−1

n=0,n=m

(Cmwm)(Cnwn)∗

× ej2π(m−n)

T t (42)

The average power is nothing but the expectationof pw(t) which is given by:

E{pw(t)} =N−1

m=0

|Cm|2|wm|2

+N−1

m=0

N−1

n=0,n=m

E{CmC∗n}wmw∗n

× ej2π(m−n)

T t (43)

since the symbols on different carriers are as-sumed independent

E{CmC∗n} = E{Cm}E{C∗n} (44)

using (44) in (43) we obtain:

E{pw(t)} =N−1

m=0

|wm|2 (45)

for PSK and QPSK. In order to compare theperformance of the OFDM system for variousweighting functions we normalize

N−1

m=0

|wm|2 = 1 (46)

Next, we describe the various weighting func-tions used in the paper:Bartlett: This function has a triangular shapefor 0 ≤ n ≤ N − 1, i.e.,

ω1,n =A .01 + 1− |n−N

2 |N2

; 0 ≤ n ≤ N − 10 otherwise

(47)

Gaussian: Based on the Gaussian function thesefactors are generated, i.e.,

ω2,n =

A exp −(n−M2 )

2

2s2 ; 0 ≤ n ≤ N − 10 otherwise

(48)

where s is the standard deviation of the weight-ing factors around N/2.Shannon: The shape of this weighting functionis the sinc function i.e.

sinc(x) =sin(πx)

πx

and written as

ω3,n =A .01 + sinc 2n−N

N ; 0 ≤ n ≤ N − 10 otherwise

(49)

Half-sine: This weighting function is describedby:

ω4,n =A .01 + sin π n


(50)

Raised Cosine: The shape of this weighting func-tion in the interval [0, N-1] is described by 1-cos(2πn/N) or equivalently by

ω5,n =A .01 + sin2 π n


(51)

Chebyshev: The shape of this weighting functionis exponentially increasing;

ω6,n =

A .01 + sin−1 h nN ;0 ≤ n ≤ N − 1h = .1, .2, ..., .8

0 otherwise(52)

Trapezoid: The shape of this weighting function istrapezoidal form;

ω7,n =

A .01 + n

N4−1 ; 0 ≤ n ≤ N

4− 1

1.01A; N4 ≤ n ≤ 3N

4 − 1A .01 + 4(N−n−1)

N−4 ; 3N4≤ n ≤ N − 1

(53)

The above functions are sketched in Figs. 7 and8. It should be noted that the weighting factorsare discrete. However, for simplicity in Figs. 7

CN-1+kN

x(t)IFFT

C1+kN

C0+kN X

X

X

w0

w1

wN-1

d0+kNMapper

Mapper

Mapper

d1+kN

dN-1+kN

Serialto

ParallelConverter

DataSource Encoder

Fig. 6. Generation of weighted and coded OFDM Signal.

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

n

Wei

ghtin

g fa

ctor

ωi,n

Raised CosineGaussian (s=N/8)Trapezoid

Fig. 7. Different waveforms for the weighting factors ωnof the OFDM signal (N = 256).

0 50 100 150 200 250 3000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

n

Wei

ghtin

g fa

ctor

ωi,n

BartlettChebyshevShannonHalf−sine

Fig. 8. Different waveforms for the weighting factors ωnof the OFDM signal (N = 256).

and 8, the weighting functions are plotted con-tinuous. For performance comparison of OFDMsystems, the amplitude A in (47)-(53) is selectedin such a way that the power of all weightingfactors is constant, i.e.

N−1

n=0

ω2i,n = 1 i = 1, 2, ..., 7 (54)

In (47) - (54) we have used the notation ωi,n todenote the nth weighting coefficient for the ithweighting function. In general when |Cm|2 = 1,m = 0, 1, 2, ...,N − 1, the average power can beset to unity by setting (43) to unity through ap-propriate choice of [wm] for a given [Cm, m =0, 1, 2, ..., N − 1]. Thus far, we have describedweighted OFDM and the various weighting func-tions. Coding can be introduced into these sig-nals prior to serial-to-parallel conversion of theincoming message bits. A block diagram for gen-eration of coded and weighted OFDM signal isshown in Fig.6.

V. Numerical Results and Discussions

In order to illustrate the influence of weight-ing and coding on PAPR of OFDM signals, wecompute the Complementary Cumulative Dis-tribution Function (CCDF). This is a plot ofProb.[PAPR > PAPR0] vs PAPR0. To gen-erate this plot, for a specific OFDM signalingsituation, 50, 000 random OFDM blocks are gen-erated and for each block PAPR is computed andthus CCDF is plotted. In all cases the functionin (13) has been over sampled by a factor of 6to accurately determine PAPR. The number ofsubcarriers N are 8, 16, 32, 64, and 128. It isnoted that PAPR is expressed in dB in all illus-trations. All computations have been carried outusing MATLAB. Next, we illustrate numericalresults for various cases of weighting and codingand offer comments on achievable improvementin PAPR.

A. Odd Parity Coding and Weightings

In Figs. 9 and 10 sample CCDFs have beenplotted. For example, in Fig.9(a) are shownCCDFs for OFDM with BPSK signal mapperwith number of subcarriers equal to 8, odd paritybit coding, and the Bartlett weighting function.Also shown in the same figure are CCDFs forthe following cases: i) CCDF without weight-ing and coding (WOWC); ii) CCDF with onlyweighting (WW); iii) CCDF with only coding(WC); and iv) CCDF with coding and weighting(WCW). The CCDF of WCW can be comparedwith other CCDFs in order to estimate the pos-sible improvements. For easy understanding andcomparison the legends and abbreviations shownin Fig. 9(a) have been maintained throughoutthe paper. Consider the plot of WOWC inFig. 9(a). This refers to OFDM system withoutweighting and coding. This plot has been arrived

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

BPSKN=8TRI

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

QPSKN=8GAU

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

4−QAMN=8TRA

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

16−QAMN=8CHE

(a) (b)

(c) (d)

WOWCWWWCWCW

Fig. 9. CCDF for an 8-subcarriers odd parity codedOFDM system as a function mapper and weightings

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

BPSKN=64SHA

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

QPSKN=64GAU

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

4−QAMN=64HS

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

16−QAMN=64RC

(a) (b)

(c) (d)

WOWCWWWCWCW

Fig. 10. CCDF for an 64-subcarriers odd parity codedOFDM system as a function mapper and weighting

at by examining 50,000 random OFDM symbols.It is noted that the highest value of PAPR inthese OFDM symbols is nearly 9 dB. The proba-bility that the PAPR exceeds or equal to 9 dB isnearly 10−2. Although, CCDF has been shownas a continuous function of PAPR, there areonly finite number of possible values of PAPR.Next, consider the effect of weighting on PAPRby considering the WW plot in the same figure.The highest value of PAPR when Bartlett (TRI)weighting is employed is no greater than 7 dB,which implies an improvement of 2 dB relative tothe case when no weighting/coding is employed.When only odd parity coding is employed thehighest value of PAPR is slightly greater than 5dB, which means a gain of nearly 4dB relative tothe case when no weighting/coding is employed.Finally when both the coding and weighting areemployed the highest value of PAPR turns out tobe nearly 6 dB resulting in a gain of 3 dB relative

to without coding and weighting case. It is notedthat when coding and weighting are jointly em-ployed the relative degradation in PAPR relativeto OFDM system with only coding is less than 1dB.

In Fig. 9(b) CCDFs shown are for OFDM sig-nals with QPSK signal mapper, number of carri-ers equal to 8 and Gaussian weighting function.For QPSK signal mapper odd parity coding doesnot really help in improving the PAPR. The im-provement in PAPR in coding is less than 0.5 dB.However, with weighting a gain of nearly 3.25 dBcan be achieved. The improvement when cod-ing and weighting are simultaneously employedis also nearly 3.25 dB.

In Fig. 9(c) and 9(d) are shown CCDFs for 4-QAM (with Trapezoidal (TRA) weighting) and16-QAM (with Chebyshev (CHE) weighting) re-spectively. While in the former case coding de-teriorates PAPR, the latter maintains the sameperformance relative to the cases without weight-ing and coding. It is noted that in both systemsPAPR can be improved by employing coding andweighting. The achievable improvements in bothOFDM systems is nearly a dB each.

In Figs. 10(a) to (d) are shown CCDFs forBPSK, QPSK, 4-QAM, and 16-QAM OFDMsystems. In the case of QPSK and the number ofsubcarriers equal to 64 when Gaussian weightingis employed, only marginal improvement can beachieved relative to WOWC. However when onlyodd parity coding is employed in this OFDM sys-tem, PAPR deteriorates by less than a dB. In theOFDM systems illustrated in Figs. 9(a), (c), and(d), PAPR improves when coding and weightingsare jointly employed.

All the computations that have been per-formed for odd parity codings have been sum-marized in Tables I to VII. In these tables gainsin PAPR have been tabulated as a function ofthe number of subcarriers and signal mappers.The gains are relative to OFDM systems withoutcoding and weightings. Next, we offer importantobservations based on these tables.

It is observed from Table I that for Bartlettweighting and odd parity coding regardless ofthe signal mapping, in general, as the numberof subcarriers increases in the OFDM system theimprovement in PAPR decreases. The best im-provement is achieved with number of subcar-riers equal to 8. For BPSK signal mapper thebest improvement is achieved with number ofsubcarriers equal to 8. For QPSK, with numberof subcarriers equal to 8 the best improvement

is achieved with either weighting only or withweighting and coding. Coding really does notimprove performance in the case of QPSK. Thus,it is wise to deploy weighting. With 4-QAM and16-QAM OFDM systems one would be requiredto employ weighting to reduce PAPR. In general,when an OFDM system is designed coding couldbe used to combat channel noise, while weightingcould be employed to reduce PAPR.

In Table II, investigation of PAPR for oddparity coding and Gaussian weighting functionis tabulated. For BPSK mapper, it is possibleto employ either coding or Gaussian weightingor both to achieve an improvement in PAPR ofnearly 5 dB. When a 16-QAM signal mapping isused, the best improvement in PAPR is achievedby employing jointly weighting and coding. Thegain achieved is equal to 3.15 dB. With 4-QAMit is possible employ a larger number of subcar-riers and yet achieve improvement in PAPR ofnearly 1.5 dB.

In Tables III to VII achievable gains in PAPRfor odd parity coding as a function of weightingfunctions, number of subcarriers and signal map-pings are tabulated. Chebyshev and raised co-sine weightings are worth considerations, as withthese weightings and odd parity coding it is pos-sible to achieve 5 dB and 3 dB improvementswith BPSK mapping. As the number of sub-carriers increases PAPR deteriorates for 4-QAMand 16-QAM OFDM systems. For large numberof subcarriers it is wise to use only weighting tohave an advantage in PAPR.

TABLE I

Gains as a function of modulations/subcarriers for

Bartlett weighting and odd parity bit coding.

Gain (dB)N BPSK QPSK

WW WC WCW WW WC WCW8 1.96 4.00 3.00 1.70 0.70 1.7016 0.40 2.00 3.00 0.75 -0.60 0.6032 -0.40 1.00 1.40 0.50 -0.60 0.3064 0.25 1.40 1.65 0.00 -2.00 -1.00128 0.09 1.00 1.08 0.30 -0.95 0.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.00 -0.75 1.00 1.00 0.00 1.0016 0.75 0.75 0.75 0.50 0.00 0.5032 0.50 0.25 0.00 0.30 0.00 0,4064 0.60 1.10 1.10 0.60 -0.40 0.60128 0.30 0.20 1.00 0.00 -1.60 1.00

TABLE II


Gaussain weighting and odd parity bit coding.


WW WC WCW WW WC WCW8 5.02 5.02 5.02 3.81 0.75 3.8316 2.10 2.10 4.20 2.60 -0.50 1.8232 0.25 1.00 3.00 1.42 -0.61 1.4264 0.20 1.25 2.20 0.21 -1.82 -0.85128 0.75 1.20 1.25 0.58 -1.21 -0.21

4-QAM 16-QAMN WW WC WCW WW WC WCW8 3.12 -0.75 3.12 2.25 0.00 3.1516 2.75 0.72 2.75 1.51 0.00 1.3532 1.00 0.25 1.00 0.78 0.05 0.9564 1.45 1.00 1.00 1.05 -0.45 1.61128 0.54 0.25 1.45 0.95 -1.55 0.95

TABLE III


Shannon weighting and odd parity bit coding.


WW WC WCW WW WC WCW8 2.15 4.21 3.22 1.62 0.61 1.6216 0.25 2.21 3.22 0.65 -0.55 0.6132 -0.45 1.15 1.48 0.55 -0.58 0.3564 0.15 1.20 1.25 0.91 -0.81 0.91128 0.11 1.21 1.26 0.45 -0.98 0.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.10 -0.75 1.10 1.02 0.00 1.0216 0.63 0.63 0.63 0.28 0.00 0.4832 0.41 0.15 0.00 0.31 0.02 0.4964 0.51 1.05 1.05 0.61 -0.42 0.00128 0.21 0.21 0.62 0.02 -1.62 1.02

B. Even Parity Coding and Weightings

In Figs. 11 and 12 CCDFs are shown as afunction of signal mapping, number of subcarri-ers, weightings and even parity coding. In Fig.11(a), is plotted CCDF for BPSK, number ofsubcarriers equal to 8 and Bartlett weighting.It is noted that even parity coding does notprovide any advantage at all. The maximumPAPR with coding and without coding is nearly9 dB. The probability with which this value

TABLE IV

Gains as a function of modulations/Subcarriers for

half-sine weighting and odd parity bit coding.


WW WC WCW WW WC WCW8 1.12 4.05 3.21 0.65 0.65 0.8516 0.11 2.11 3.15 0.51 -0.52 0.5132 -0.51 1.12 1.45 0.58 -0.60 -0.3264 0.05 1.22 1.25 0.00 -1.98 -1.05128 0.05 1.05 1.21 0.46 -1.10 0.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 0.26 -0.65 0.24 1.00 0.00 1.0016 0.86 0.86 0.86 0.29 0.00 0.2132 0.41 0.35 0.00 0.25 0.05 0.4164 0.48 1.05 1.05 0.18 -0.05 0.21128 0.00 0.25 0.41 0.05 -1.52 -1.15

TABLE V


raised-cosine weighting and odd parity bit coding.


WW WC WCW WW WC WCW8 2.00 4.02 3.10 1.75 0.71 1.9516 0.21 2.10 3.25 1.10 -0.55 0.5832 -0.38 1.20 1.62 0.72 -0.61 0.3864 0.05 1.21 1.55 0.00 -1.95 -1.00128 0.10 1.21 1.21 0.42 -1.25 0.00


occurs is nearly 0.1, which implies that 1 in10 OFDM blocks have PAPR of 9 dB. Hence,even parity coding is not a wise choice. Onthe other hand odd parity coding is a goodchoice (refer Fig. 9(a)) as far as improvement inPAPR is concerned. In the case of QPSK evenparity coding, in fact, deteriorates the PAPRperformance, as can be observed in Fig. 11(b).A similar observation can be made in respect of16-QAM, Fig. 11(d). In Fig. 12(a) CCDFs areshown for Shannon weighting and even paritycoding. It is noted that, BPSK with number

TABLE VI


Chebyshev weighting and odd parity bit coding.


WW WC WCW WW WC WCW8 2.00 4.10 5.00 1.72. 0.71 1.7216 0.31 2.00 3.45 0.51 -0.55 0.5132 -0.15 1.00 2.00 0.55 -0.60 0.3564 0.10 1.20 1.50 0.00 -1.75 1.20128 0.20 1.10 1.10 0.40 -1.00 -1.00


TABLE VII


trapezoidal weighting and odd parity bit coding.


WW WC WCW WW WC WCW8 2.00 4.00 2.00 1.61 0.60 1.6116 0.00 2.00 3.00 0.60 -0.62 -0.3132 -0.75 1.15 1.35 0.41 -0.61 -0.5864 0.00 1.20 1.50 0.00 -1.95 -1.00128 0.00 1.25 1.35 0.05 -1.00 -0.50

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.00 -0.71 1.00 1.00 0.00 1.0016 0.80 0.80 1.00 0.00 0.00 0.1532 0.20 0.25 0.05 0.40 0.00 1.0064 0.40 1.00 1.00 0.00 -0.40 -0.40128 0.00 0.30 0.25 0.00 -1.50 -1.00

of subcarriers equal to 16, even parity codingalone can provide an improvement of nearly 1dB. However, when both coding and weightingare used the gain in PAPR reduces by nearly 0.5dB. For 4-QAM with Gaussian weighting andeven parity coding it is possible to achieve 1.5dB improvement in PAPR.

In Tables VIII-XIV achievable gains in PAPRhave been tabulated, as a function of the num-ber of subcarriers, signal mapping, and weightingfunctions. For an 8 subcarrier BPSK system, the

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

R o]

BPSKN=8TRI

WOWCWWWCWCW

2 4 6 8 10 12 1410

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

R o]

QPSKN=128GAU

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

R o]

4−QAMN=32TRA

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

R o]

16−QAMN=64GAU

(a) (b)

(c) (d)

WOWCWWWCWCW

Fig. 11. CCDF for even parity coded OFDM system asa function mapper, subcarriers, and weightings

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

BPSKN=16SHA

WOWCWWWCWCW

2 4 6 8 10 12 1410

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

QPSKN=32RC

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

4−QAMN=8GAU

WOWCWWWCWCW

2 4 6 8 10 12 1410

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

16−QAMN=128HS

(a) (b)

(c) (d)

WOWCWWWCWCW

Fig. 12. CCDF for even parity coded OFDM system asa function mapper, subcarriers, and weightings

best weighting function is Gaussian and with thisit is possible to achieve nearly 4.2 dB improve-ment in PAPR. The next best weighting func-tion is that of Shannon followed by raised cosine.For an 8 subcarrier BPSK system, the achiev-able improvement in PAPR of 3.75 dB is attainedwhen even parity coding and Gaussian weight-ing is employed. For a QAM signal mapper, theweighting function that provides best improve-ment in PAPR is again the Gaussian function.With Chebyshev weighting and even parity cod-ing maximum achievable improvement in PAPRis 2 dB.

C. Horizontal/Vertical Parity Coding andWeightings

In Figs. 13 and 14 sample CCDFs for BPSK,QPSK and QAM signal mappers with horizon-tal/vertical parity coding are plotted for variousweightings, such as Bartlett, Gaussian, Shannon,

TABLE VIII


Bartlett weighting and even parity bit coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.70 0.71 1.7016 0.40 1.00 1.00 0.71 -0.60 0.5832 -0.50 0.21 0.52 0.58 -0.80 0.4064 0.18 0.95 1.20 0.00 -2.00 -0.85128 0.05 0.85 0.65 0.30 -1.00 0.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.00 0.00 1.00 1.00 0.00 1.0016 1.00 0.85 1.00 0.40 0.00 0.6132 0.52 0.42 0.52 0.38 0.10 0.6264 0.75 0.74 1.15 0.78 -0.50 0.78128 0.38 1.00 1.00 0.10 -1.80 -1.21

TABLE IX


Gaussain weighting and even parity bit coding.


WW WC WCW WW WC WCW8 4.21 0.00 4.21 3.75 0.75 3.7516 2.00 1.00 2.00 2.61 -0.58 1.6232 0.21 0.20 0.00 1.41 -0.80 1.4164 0.20 0.82 1.15 0.20 -1.20 0.61128 0.75 0.81 0.42 0.61 -0.95 0.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 3.00 0.00 3.00 2.32 0.00 3.1016 2.78 0.61 2.78 1.51 0.00 1.5032 1.00 0.35 1.00 1.00 0.15 1.2064 1.51 0.75 2.15 1.20 -0.52 1.85128 0.50 1.05 1.05 0.95 -1.45 0.95

trapezoidal, raised cosine, Chebyshev and half-cycle sinusoid. In Fig. 13(b), QPSK with thenumber of subcarriers equal to 16 and Gaussianweighting is considered. It is observed that withGaussian weighting alone it is possible to achievenearly 1.5 dB improvement in PAPR relativeto OFDM system WOWC. When coding is em-ployed this gain reduces by about 0.25 dB only.BPSK with the number of subcarriers equal to8 with Gaussian weighting provides nearly 4 dBimprovement in PAPR relative to OFDM sys-tem that does not employ coding and weight-

TABLE X


Shannon weighting and even parity bit coding..


WW WC WCW WW WC WCW8 2.10 0.00 2.10 1.72 0.72 1.7216 0.21 1.00 0.51 0.61 -0.60 0.6032 -0.52 0.31 1.10 0.60 -0.82 0.4064 0.15 1.00 1.21 0.00 -2.00 -1.00128 0.08 0.85 0.61 0.50 -1.00 0.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.00 0.00 1.00 1.00 0.00 1.0016 0.60 0.60 0.60 0.42 0.00 0.5132 0.61 0.52 0.21 0.25 0.07 0.5064 0.52 0.60 1.10 0.52 -0.41 0.00128 0.32 1.00 1.00 0.05 -1.80 1.20

TABLE XI


half-sine weighting and even parity bit coding.


WW WC WCW WW WC WCW8 1.00 0.00 1.00 0.60 0.60 1.6016 0.20 1.00 0.40 0.45 -0.45 0.0532 -0.40 0.10 1.00 0.55 -0.78 -0.2164 0.00 0.85 1.00 0.00 -2.00 -1.00128 0.05 0.80 0.65 0.40 -1.10 -3.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 0.26 0.00 0.25 1.00 0.00 1.0016 0.70 0.70 0.70 0.25 0.00 0.1532 0.35 0.35 0.00 0.21 0.10 0.5164 0.40 0.60 1.00 0.00 -0.40 -0.30128 0.00 1.00 1.00 0.00 -1.50 -1.00

ing. This is evident from plots in Fig. 14(a).One of the other weighting functions worth con-sidering is raised cosine which when used withQPSK mapper in a 16 subcarrier system providesa gain of nearly 1.75 dB. In Tables XV to XXIachievable gains relative to OFDM systems with-out coding and weighting are tabulated. Similarobservations have been arrived at for horizontalparity coding. When 16-QAM with the numberof subcarriers equal to 8 is to be employed, itis wise to employ horizontal/vertical coding in-stead of even parity coding, as the former pro-

TABLE XII


raised-cosine weighting and even parity bit coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.70 .70 2.7016 0.20 1.00 1.00 1.00 -0.60 0.5032 -0.35 0.15 0.00 0.65 -0.85 0.3564 0.00 0.91 1.10 0.00 -2.00 -1.00128 0.05 0.90 0.40 0.40 -1.00 0.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.21 0.00 1.21 1.22 0.00 1.2016 1.15 0.75 1.00 1.00 0.00 1.0032 1.00 0.40 1.00 0.25 0.10 1.0064 0.80 0.75 1.00 0.45 -0.40 0.45128 0.40 1.00 1.00 0.05 -1.85 -0.65

TABLE XIII


Chebyshev weighting and even parity bit coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.71 0.71 1.7116 0.30 1.00 1.10 0.51 -0.62 0.5132 -0.10 0.30 0.31 0.61 -0.82 0.3864 0.10 1.00 1.00 1.00 -2.00 1.00128 0.20 0.80 0.80 0.28 -1.10 -1.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.00 0.00 1.00 1.00 0.00 1.0016 0.62 0.62 1.21 1.00 0.00 0.5132 0.21 0.35 0.35 0.45 0.05 1.0064 1.00 0.85 1.00 0.00 -1.00 0.60128 0.00 1.00 1.00 0.05 -1.80 -.95

vides an improvement of 3 dB whereas the latterprovides only 2.3 dB, when Gaussian weighting isemployed. Similarly, 4-QAM , 8-subcarriers andShannon weighting system is superior by 0.75 dBwhen horizontal/vertical coding is employed rel-ative to when even parity is used.

TABLE XIV


trapezoidal weighting and even parity bit coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.71 0.70 1.7116 0.00 1.00 0.60 0.58 -0.60 0.5732 -0.60 0.15 0.45 0.40 -0.80 -0.6264 0.00 0.90 1.00 0.00 -2.00 -1.00128 0.05 0.85 1.00 0.10 -1.00 -0.60

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.00 0.00 1.00 1.00 0.00 1.0016 0.72 0.72 0.72 0.00 0.00 0.2032 0.15 0.40 0.30 0.30 0.15 1.0064 0.40 0.60 1.00 -0.10 -0.40 -0.35128 0.00 1.00 1.00 0.02 -1.80 -1.20

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

BPSKN=8TRI

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

QPSKN=16GAU

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

4−QAMN=32SHA

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

16−QAMN=64TRA

(a) (b)

(c) (d)

WOWCWWWCWCW

Fig. 13. CCDF for horizontal/vertical parity codedOFDM system as a function mapper, subcarriers, andweightings

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

BPSKN=8GAU

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro] QPSK

N=16RC

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

4−QAMN=32CHE

WOWCWWWCWCW

2 4 6 8 10 1210

−3

10−2

10−1

100

PAPRo (dB)

Pr

[PA

PR

> P

AP

Ro]

16−QAMN=128HS

(d)

(a) (b)

(c)

WOWCWWWCWCW

Fig. 14. CCDF for horizontal/vertical parity codedOFDM system as a function mapper, subcarriers, andweightings

TABLE XV


Bartlett weighting and horizontal/vertical parity bit

coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.80 -0.05 1.8016 0.30 0.30 0.80 1.10 -0.05 0.6032 0.40 0.00 0.21 0.20 -0.30 0.0064 0.20 0.00 0.00 0.41 1.10 0.71128 0.15 -0.15 0.65 0.40 0.60 0.80

4-QAM 16-QAMN WW WC WCW WW WC WCW8 2.00 0.10 2.00 1.00 0.00 1.0016 0.70 -0.50 0.60 0.22 0.00 1.0032 0.20 0.00 0.50 1.00 0.00 0.4564 0.10 -0.50 0.20 0.00 -0.52 0.00128 0.15 0.00 0.21 0.42 0.20 0.21

TABLE XVI


Gaussain weighting and horizontal/vertical parity bit

coding.


WW WC WCW WW WC WCW8 4.00 0.00 4.00 3.80 -0.10 3.8016 1.85 0.20 1.85 1.40 -0.10 1.2032 0.60 0.00 0.00 1.00 -0.25 0.9064 0.30 0.00 0.00 1.20 1.00 0.80128 0.60 -0.15 0.25 1.00 0.60 0.60


TABLE XVII


Shannon weighting and horizontal/vertical parity bit

coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.55 0.00 1.5516 0.82 0.82 0.82 0.75 0.00 0.5032 0.20 0.00 0.15 0.20 -0.20 0.0064 0.00 0.00 0.00 0.20 1.00 0.70128 0.00 -0.10 0.60 0.25 0.40 0.75


TABLE XVIII


half-sine weighting and horizontal/vertical parity bit

coding.


WW WC WCW WW WC WCW8 1.00 0.00 1.00 0.65 0.00 0.7516 0.05 0.20 0.05 0.50 0.00 0.5032 0.35 0.00 0.35 0.20 -0.25 0.0064 0.00 0.00 0.00 0.20 1.10 0.85128 -0.10 -0.10 0.60 0.30 0.60 1.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 0.75 0.10 0.75 1.00 0.00 1.0016 0.60 -0.40 0.20 0.00 0.00 0.2532 0.00 0.00 0.20 1.00 0.00 0.2164 0.00 0.10 -0.60 0.00 0.00 -0.60128 0.00 0.00 0.15 0.40 0.22 0.18

TABLE XIX


raised-cosine weighting and horizontal/vertical parity bit

coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.75 0.00 1.8016 0.25 0.25 0.40 1.50 0.00 0.4532 0.50 0.00 0.00 0.30 -0.20 0.2564 0.10 0.00 0.00 0.50 1.00 0.52128 0.15 -0.10 0.50 0.35 0.60 1.00


TABLE XX


Chebyshev weighting and horizontal/vertical parity bit

coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.75 0.00 1.7516 0.30 0.30 0.30 0.50 0.00 0.5032 -0.40 0.00 0.20 0.45 -0.25 0.0064 0.15 0.00 0.20 0.50 1.00 1.00128 -0.10 -0.10 0.00 0.60 0.60 0.95

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.75 0.00 1.75 1.00 0.00 1.0016 0.80 -0.55 0.60 1.00 0.00 0.4132 0.10 0.00 0.00 0.35 0.00 0.4064 0.60 -0.60 0.20 -0.21 -0.45 0.00128 0.41 0.05 0.15 0.61 0.18 0.15

TABLE XXI


trapezoidal weighting and horizontal/vertical parity bit

coding.


WW WC WCW WW WC WCW8 2.00 0.00 2.00 1.75 0.00 1.7516 0.00 0.00 0.20 0.50 0.00 0.0032 0.20 0.00 0.00 -0.40 0.00 0.0064 0.00 0.00 0.00 0.20 1.00 0.60128 -0.15 -0.15 0.51 0.30 0.40 1.00

4-QAM 16-QAMN WW WC WCW WW WC WCW8 1.75 0.00 1.75 1.00 0.00 1.0016 0.40 -0.60 0.20 0.00 0.00 0.0032 0.15 0.00 0.30 1.00 0.00 0.1064 0.05 -0.60 0.05 -0.35 -0.51 0.00128 0.00 0.00 0.00 0.00 0.21 0.10

VI. Conclusions

In this paper we have considered the prob-lem of reduction of PAPR in OFDM systemsby jointly employing coding and weighting. Thecoding schemes that we have considered are sim-ple and easy to implement. Several weightingfunctions have been considered of which Cheby-shev and trapezoidal are new in the context ofPAPR reduction in OFDM systems.Thorough extensive numerical simulations we

have determined the achievable gains in PAPR asa function of number of sub-carriers, signal map-ping, and weighting function. For example, it isobserved that for an 8 subcarrier OFDM-PSKsystem it is possible to achieve nearly 4.2 dBimprovement in PAPR by employing Gaussianweighting. In general Gaussian weighting func-tion provides the best improvements in PAPRrelative to an OFDM system without any cod-ing and weighting. While odd and even par-ity codings have the same distance properties,it is noted that odd parity coding can be gain-fully employed for superior PAPR performance.Chebyshev, trapezoidal and Shannon weightingsprovide PAPR improvements of the order of 2dB. It is noted that with the BPSK signal map-ping, improvement from joint use of coding andweighting can be the order of 3 dB. We concludethat the best way to keep PAPR at a certain leveland at the same time better the error rate per-formance is to jointly use coding and weightingin OFDM systems.It is worthwhile considering coding techniques

such as block codes, convolutional codes, turbocodes etc. jointly with weighting in order to ob-tain a detailed picture of the effects on PAPR.Also the problem is theoretically yet to be mod-elled to obtain analytical bounds on PAPR.

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Documents

PAPR Reduction using Combined Coding, Weighting, and