scfdmaPapr_pimrc06

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The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'06)

1-4244-0330-8/06/$20.00 2006 IEEE

PEAK-TO-AVERAGE POWER RATIO OF SINGLE CARRIER FDMA SIGNALSWITH PULSE SHAPING

Hyung G. Myung Junsung Lim David J. GoodmanDepartment of Electrical and Computer Engineering, Polytechnic University

5 MetroTech Center, Brooklyn, NY 11201USA

{hmyung01; jlim01} @utopia.poly.edu, [email protected]

ABSTRACT

Single carrier frequency division multiple access (SC-FDMA), which utilizes single carrier modulation andfrequency domain equalization is a technique that has similarperformance and essentially the same overall complexity asthose of OFDM, in which high peak-to-average power ratio

(PAPR) is a major drawback. An outstanding advantage ofSC-FDMA is its lower PAPR due to its single carrierstructure. In this paper, we analyze the PAPR of SC-FDMAsignals with pulse shaping. We analytically derive the timedomain SC-FDMA signals and numerically compare PAPRcharacteristics using the complementary cumulativedistribution function (CCDF) of PAPR. The results show thatSC-FDMA signals indeed have lower PAPR compared tothose of OFDMA. Comparing the two forms of SC-FDMA,we find that localized FDMA (LFDMA) has higher PAPRthan interleaved FDMA (IFDMA) but somewhat lower PAPRthan OFDMA. Also noticeable is the fact that pulse shapingincreases PAPR.

I. INTRODUCTIONDemands for media-rich wireless data services have broughtmuch attention to high speed broadband mobile wirelesstechniques in recent years. Orthogonal frequency divisionmultiplexing (OFDM), which is a multicarrier communicationtechnique, has become widely accepted primarily because ofits robustness against frequency selective fading channelswhich are common in broadband mobile wirelesscommunications [1]. Orthogonal frequency division multipleaccess (OFDMA) is a multiple access scheme which is anextension of OFDM to accommodate multiple simultaneoususers.

Despite the benefits of OFDM and OFDMA, they suffer anumber of drawbacks including: high peak-to-average powerratio (PAPR); a need for an adaptive or coded scheme toovercome spectral nulls in the channel; and high sensitivity tofrequency offset [2], [3].

Single carrier frequency division multiple access (SC-FDMA), which utilizes single carrier modulation andfrequency domain equalization is a technique that has similarperformance and essentially the same overall complexity asthose of OFDMA system. One prominent advantage overOFDMA is that the SC-FDMA signal has lower PAPRbecause of its inherent single carrier structure [4]. SC-FDMAhas drawn great attention as an attractive alternative toOFDMA, especially in the uplink communications wherelower PAPR greatly benefits the mobile terminal in terms of

DFT

Serial-to-Parallel

Subcarrier

Mapping

IDFT

Pulse Shaping /

DACChannel

IDFT

Parallel-to-Serial

Subcarrier De-

mapping / *FDE

DFT

ADC

Encoder Decoder

*FDE: Frequency Domain Equalization

Fig. 1 Block diagram of SC-FDMA.

DFT(N-point)

IDFT(M-point)

SubcarrierMapping{ }nx

{ }kX

{ }mx{ }lX

N

T

N

T

M N

NT T

M

>

=

M

T

: number of data symbols,

,

N M

T T : symbol durations

Fig. 2 Generation of SC-FDMA transmit symbols. ThereareMtotal number of carriers, among whichN(


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The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC06)

transmit power efficiency. It is currently a strong candidatefor uplink multiple access scheme in 3G Long TermEvolution of 3GPP [5].

In this paper, we analyze the PAPR of SC-FDMA signalsand compare it with that of OFDMA. We analytically derivethe time domain SC-FDMA signals and numerically comparePAPR characteristics using complementary cumulativedistribution function (CCDF) of PAPR.

The remainder of this paper is organized as follows: SectionII gives an overview of SC-FDMA. Section III derives thetime domain signals for each subcarrier mapping mode of SC-

FDMA and it presents the CCDF of PAPR obtained fromMonte-Carlo simulations.

II.OVERVIEWOFSC-FDMASYSTEMA block diagram of a SC-FDMA system is shown in Fig. 1.SC-FDMA can be regarded as discrete Fourier transform(DFT)-spread OFDMA, where time domain data symbols aretransformed to frequency domain by DFT before goingthrough OFDMA modulation. The orthogonality of the usersstems from the fact that each user occupies differentsubcarriers in the frequency domain, similar to the case ofOFDMA. Because the overall transmit signal is a single

carrier signal, PAPR is inherently low compared to the caseof OFDMA which produces a multicarrier signal.

Fig. 2 details the generation of SC-FDMA transmitsymbols. There are M subcarriers, among which N (< M)subcarriers are occupied by the input data. In the timedomain, the input data symbol has symbol duration of Tseconds and the symbol duration is compressed to

( )NT TM

= after going through SC-FDMA modulation.

There are two methods to choose the subcarriers fortransmission as shown in Fig. 3. In the distributed subcarriermapping mode, DFT outputs of the input data are allocatedover the entire bandwidth with zeros occupying in unused

subcarriers, whereas consecutive subcarriers are occupied by

the DFT outputs of the input data in the localized subcarriermapping mode. We will refer to the localized subcarriermapping mode of SC-FDMA as localized FDMA (LFDMA).

The case of M = QN for the distributed mode with

equidistance between occupied subcarriers is calledInterleaved FDMA (IFDMA) [6]. An example of SC-FDMAtransmit symbols in the frequency domain for N= 4, Q= 4and M= 16 is illustrated in Fig. 4. After subcarrier mapping,the frequency data is transformed back to the time domain byapplying inverse DFT (IDFT).

III.PAPROFSC-FDMASIGNALSIn this section, we analyze the PAPR of the SC-FDMA signalfor each subcarrier mapping mode. For distributed subcarriermapping mode, we will consider the case of IFDMA. In the

subsequent derivations, we will assume M= QNand follow

the notations in Fig. 2.Let { }: 0,1, , 1nx n N= be data symbols to be

modulated. Then, { }: 0,1, , 1kX k N= are frequency

domain samples after DFT of { }: 0,1, , 1nx n N= ,

{ }: 0,1, , 1lX l M= are frequency domain samples after

subcarrier mapping, and { }: 0,1, , 1mx m M= are time

symbols after IDFT of { }: 0,1, , 1lX l M= . The complexpassband transmit signal of SC-FDMAx(t) for a block of datais represented as

1

0

( ) ( )cM

j t

m

m

x t e x r t mT

=

= , (1)

where c is the carrier frequency of the system and r(t) is

the baseband pulse. In our research, we use a raised-cosinepulse, which is a widely used pulse shape in wirelesscommunications, defined as follows in the time domain [7].

cos

( ) sinc2 2

41

2

t

t Tr t

T t

T

=

(2)

0 0 0 0 0 0 0 0 0 0 0 0X0 X1 X2 X3{ }, :l DistributedX

frequency

0 0 0 0 0 0 0 0 0 0 0 0X0 X

1 X

2 X

3{ }, :l LocalizedX

{ }:kX X0 X

1 X

2 X

3

{ } :nx x0 x

1 x

2 x

3

DFT

21

0

, 4N j nk

Nk n

n

X x e N

=

= =

Fig. 4 An example of SC-FDMA transmit symbols in thefrequency domain forN= 4, Q= 4 andM= 16.

,l DistributedX denotes transmit symbols for distributed

subcarrier mapping mode and ,l LocalizedX denotes

transmit symbols for localized subcarrier mapping mode.

x0

x1

x2

x3

x0 x

1 x

2 x

3

{ }nx

{ },m IFDMAx x0 x1 x2 x3 x0 x1 x2 x3 x0 x1 x2 x3

? ? ? ? ? ? ? ? ? ? ? ?x0

x1

x2

x3{ },m LFDMAx

time

Fig. 5 An example of SC-FDMA transmit symbols in the

time domain forN= 4, Q= 4 andM= 16.


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where is the rolloff factor which ranges between 0 and 1.The PAPR is defined as follows for transmit signalx(t).

2

0

2

0

max ( )peak power of ( )

PAPRaverage power of ( ) 1

( )

t MT

MT

x tx t

x tx t dt

MT

= =

(3)

Without pulse shaping, that is, using rectangular pulseshaping, symbol rate sampling will give the same PAPR asthe continuous case since SC-FDMA signal is modulated overa single carrier. Thus, PAPR without pulse shaping withsymbol rate sampling can be expresses as follows.

2

0,1, , 1

12

0

max

PAPR1

mm M

M

m

m

x

xM

=

=

=

(4)

We first examine the PAPR of transmit symbols for each

block without pulse shaping analytically, and then, investigatethe PAPR of transmit symbols with pulse shapingnumerically.

A.Time domain symbols of IFDMAFor IFDMA, the frequency samples after subcarrier mapping

{ }lX can be described as follows.

/, (0 1)

0 , otherwise

l Q

l

X l Q k k NX

= =

(5)

We derive the time symbols

{ }mx which are obtained by

taking inverse DFT of { }lX .Let m N q n= + , where 0 1q Q and 0 1n N .

Then,

( )1 1 22

0 0

1 2

0

1 2

0

1 1 1

1 1

1 1

1

mmM N j kj lNM

m Nq n l k

l k

Nq nN j kN

k

k

nN j kN

k

k

n

x x X e X eM Q N

X eQ N

X eQ N

xQ

+

= =

+

=

=

= = =

=

=

=

. (6)

The resulting time symbols { }mx are simply a repetition of

the original input symbols { }nx in the time domain [6].

Therefore, the PAPR of IFDMA signal is the same as in thecase of conventional single carrier signal. An example of anIFDMA signal is shown in Fig. 5.

B.Time domain symbols of LFDMAFor LFDMA, the frequency samples after subcarrier mapping

{ }l

X can be described as follows.

, 0 1

0 , 1

l

l

X l NX

N l M

=

(7)

Let m Q n q= + , where 0 1n N , and 0 1q Q .

Then,

1 2

0

1 2

0

1

1 1

mM j lM

m Qn q l

l

Qn qN j l

QN

l

l

x x X eM

X eQ N

+

=

+

=

= =

=

. (8)

If q= 0, then,

1 2

0

1 2

0

1 1

1 1

1

QnN j l

QN

m Qn l

l

nN j lN

l

l

n

x x X eQ N

X eQ N

xQ

=

=

= =

=

=

. (9)

If q 0, since

1 2

0

l

pN j lN

p

p

X x e

=

= , then (8) can be

expressed as follows after derivation.

( ){ }

12

20

1 11

1

qNj

pQ

m Qn q n p qj

pN QN

xx x e

Q Ne

+ +

=

= =

(10)

As can be seen from (9) and (10), in the time domain,LFDMA signal has exact copies of input time symbols in theN-multiple sample positions. In-between values are sum of allthe time input symbols in the input block with differentcomplex-weighting, which would increase the PAPR. Anexample of an LFDMA signal is shown in Fig. 5.

C.Numerical resultsCCDF (Complementary Cumulative Distribution Function) ofPAPR, which is the probability that PAPR is higher than acertain PAPR value PAPR0 (Pr{PAPR>PAPR0}), iscalculated by Monte Carlo simulation. CCDFs of PAPR forIFDMA, LFDMA, and OFDMA are evaluated and compared.105 uniformly random data points were generated to acquirethe CCDF of PAPR. Consecutive chunks were used forLFDMA, and in the case of OFDMA, transmission bandwidthof 5 MHz was assumed and 8 times oversampling was usedwhen calculating PAPR [8]. No pulse shaping was applied inthe case OFDMA. In the simulations, the total number ofsubcarriers Mwere set to 256, input data block size Nto 64,

and Qto 4. QPSK, 8-PSK, 16-QAM, 32-QAM, and 64-QAM


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symbol constellations were considered. Raised cosine pulse

r(t) was truncated from -6 T to 6 T time period and it wasoversampled by 8 times [7].

In Fig. 6, plots of CCDF of PAPR for IFDMA, LFDMA,and OFDMA are shown. We compare the PAPR value that is

exceeded with probability less than 0.1% (Pr{PAPR>PAPR0}= 10-3), or 99.9-percentile PAPR. First, in the case of no pulseshaping, IFDMA has lower PAPR than the case of OFDMAby 10.5 dB for QPSK and by 7 dB for 16-QAM, while PAPRof LFDMA is lower than that of OFDMA by 3 dB for QPSKand by 2 dB for 16-QAM but higher than that of IFDMA by7.5 for QPSK and by 5 dB for 16-QAM. With raised-cosinepulse shaping with rolloff factor of 0.5, it can be seen thatPAPR increases significantly for IFDMA whereas PAPR ofLFDMA hardly increases. Full comparison of 99.9-percentilePAPR for different modulation formats is in Table I. We cansee that IFDMA and LFDMA have lower PAPR thanOFDMA consistently.

Fig. 7 shows the impact of rolloff factor on the PAPRwhen using raised cosine pulse shaping, where this impact ismore obvious in IFDMA. As the rolloff factor increases from0 to 1, PAPR reduces significantly for IFDMA. This impliesthat there is a tradeoff between PAPR performance and out-

of-band radiation since out-of-band radiation increases withincreasing rolloff factor.

Table 1. Comparison of 99.9-percentile PAPR

IFDMA LFDMAMod.format No

pulseshaping

Pulseshaping(rolloff

0.5)


0.22)

Nopulse

shaping


0.5)


0.22)

OFDMA

QPSK 0 dB 4.3 dB 6.1 dB 7.5 dB 7.6 dB 7.6 dB 10.7 dB8PSK 0 dB 4.2 dB 5.9 dB 7.4 dB 7.5 dB 7.5 dB 10.6 dB

16QAM 3.5 dB 6.6 dB 7.7 dB 8.4 dB 8.4 dB 8.5 dB 10.5 dB

32QAM 3.4 dB 6.4 dB 7.5 dB 8.2 dB 8.3 dB 8.4 dB 10.6 dB64QAM 4.8 dB 7.1 dB 8.0 dB 8.6 dB 8.7 dB 8.7 dB 10.5 dB

(a)

(b)

Fig. 6 Comparison of CCDF of PAPR for IFDMA,

LFDMA, and OFDMA withM= 256,N= 64, and =0.5. (a) QPSK. (b) 16-QAM.

(a)

(b)

Fig. 7 Comparison of CCDF of PAPR for IFDMA and

LFDMA withM= 256,N= 64, and of 0, 0.2, 0.4, and

0.6, 0.8, and 1. (a) QPSK. (b) 16-QAM.


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IV.CONCLUSIONIn this paper, we analyze the PAPR of SC-FDMA signals andcompare it with the case of OFDMA. Specifically, we derivethe time domain signals of IFDMA and LFDMA, andnumerically compare PAPR characteristics using CCDF ofPAPR. It is shown that SC-FDMA signals indeed have lower

PAPR compared OFDMA. Also, we have shown thatLFDMA incurs higher PAPR compared to IFDMA, butcompared to OFDMA, it is lower, though not significantly.Another noticeable fact is that pulse shaping increases PAPRand that rolloff factor in the case of raised-cosine pulseshaping has a significant impact on PAPR of IFDMA. Apulse shaping filter should be designed carefully in order toreduce the PAPR without degrading the system performance.

In conclusion, to fully exploit the low PAPR advantage ofSC-FDMA, IFDMA is more desirable than LFDMA whenchoosing subcarrier mapping method.

REFERENCES

[1] L. J. Cimini, Jr., Analysis and Simulation of a Digital Mobile ChannelUsing Orthogonal Frequency Division Multipexing, IEEE Trans.Commun., vol. 33, no. 7, July 1985, pp. 665675.

[2] H. Sari, G. Karam and I. Jeanclaude, An Analysis of OrthogonalFrequency-Division Multiplexing for Mobile Radio Applications, Proc.IEEE VTC 94, pp. 16351639, Stockholm, Sweden, June 1994.

[3] H. Sari, G. Karam and I.Jeanclaude, Frequency-Domain Equalization ofMobile Radio and Terrestrial Broadcast Channels, Proc. IEEEGLOBECOM 94, pp. 15, San Francisco, CA, Nov. 1994.

[4] D. Falconer, S. L. Ariyavisitakul, A. Benyamin-Seeyar and B. Eidson,Frequency Domain Equalization for Single-Carrier Broadband WirelessSystems, IEEE Commun. Mag., vol. 40, no. 4, April 2002, pp. 5866.

[5] 3rd Generation Partnership Project (3GPP); Technical SpecificationGroup Radio Access Network; Physical Layer Aspects for Evolved

UTRA, http://www.3gpp.org/ftp/Specs/html-info/25814.htm

[6] U. Sorger, I. De Broeck and M. Schnell, Interleaved FDMA - A NewSpread-Spectrum Multiple-Access Scheme, Proc. IEEE ICC 98, pp.10131017, Atlanta, GA, June 1998.

[7] T. S. Rappaport, Wireless Communications: Principles and Practice,Second Edition. Prentice Hall, 2002

[8] S. H. Han and J. H. Lee, An Overview of Peak-to-Average Power RatioReduction Techniques for Multicarrier Transmission, IEEE WirelessCommun., vol. 12, no. 2, April 2005, pp. 5665.

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