a technique for sidelobe suppression in ofdm systems

A Technique for Sidelobe Suppressionin OFDM Systems

Ivan Cosovic, Sinja Brandes, and Michael SchnellGerman Aerospace Center (DLR)

Institute of Communications and NavigationOberpfaffenhofen, 82234 Wessling, Germany

[email protected] / [email protected] / [email protected]

Abstract— In this paper, a new technique for sidelobe sup-pression in orthogonal frequency-division multiplexing (OFDM)systems is proposed and investigated. This technique can beapplied to both pure OFDM and OFDM based multiple-accesssystems to improve spectral efficiency. Moreover, it can beutilized within OFDM based overlay systems to avoid interferencetowards the legacy systems operating in the same frequency bandas the overlay system.

The proposed sidelobe suppression technique is based onsubcarrier weighting. The real-valued subcarrier weights aredetermined in such a way that the sidelobes of the transmissionsignal are minimized using an optimization algorithm which iscapable to take several optimization constraints into account.Results show that sidelobes can be easily suppressed by more than10 dB with the proposed technique. These results are achievedby allowing only a moderate loss in bit-error rate performance.

I. INTRODUCTION

The ever-growing need for wireless communications whichprovide high data rates entails a substantial demand fornew spectral resources and more flexible and efficient useof existing resources. Over the last several years innovativespectrum management strategies have been suggested whichallow different systems to share the same frequency band [1]–[3]. Furthermore, similar spectrum sharing policies have beenproposed in recent recommendations by the Federal Com-munications Commission (FCC) [4] in the USA and by theInternational Telecommunication Union (ITU) [5] in Europe.

The starting premise of the spectrum sharing concepts isthat, nowadays, at any given time and place, a significant partof spectrum is unused due to the rigid spectrum policies. Afeasible solution to exploit these spectrum gaps is to deploy anorthogonal frequency-division multiplexing (OFDM) system.An important property of OFDM is its flexibility and adjusta-bility to certain spectrum restrictions. With OFDM it is easilypossible to realize transmission systems which do not need acontinuous transmission band. Certain frequency areas can beleft unused by simply turning off the respective subcarriers inthese areas. This makes OFDM especially attractive for theimplementation in spectrum sharing systems. As illustrated inFig.1, a so-called overlay OFDM system can be realized if theexisting systems applied in the considered frequency band donot use the whole frequency band, but leave some frequencygaps.

One of the most challenging problems of spectrum sharingsystems is their successful co-existence in the same frequency

Existing systems (digital and/or analog)

OFDM overlay system

Frequency

Power

Fig. 1. OFDM overlay concept - using the frequency gaps in an existingfrequency bandwidth.

band, i.e., an overlay system should not degrade the perfor-mance of the systems already working in that frequency band.Considering OFDM based overlay systems which utilize thefrequency gaps of existing systems, out-of-band radiation hasto be reduced in order to enable co-existence in the samefrequency band. Hence, the suppression of sidelobes is animportant task in OFDM based overlay systems.

So far, the topic of sidelobe suppression in OFDM systemshas received little attention. An overview of the existing tech-niques can be found in, e.g., [6]. The first method discussedin [6] is based on the deactivation of subcarriers lying atthe borders of the OFDM spectrum, i.e., guard bands areinserted, whereas the second method applies windowing ofthe transmission signal in frequency domain. However, the firstmethod sacrifices bandwidth and reduces system throughput,whereas the latter method expands the signal in time domainand thus, also, results in a reduced system throughput.

In this paper, a different method to significantly suppress theOFDM sidelobes is proposed and analyzed. This technique,referred to as subcarrier weighting, is based on the multipli-cation of the used subcarriers with real-valued factors whichare chosen such that the sidelobes are suppressed and no sideinformation has to be transmitted. In addition, it overcomesthe problems of the techniques presented in [6] as it neitherwastes additional scarce spectral resources nor expands thesignal in time domain.

The paper is organized as follows. In Section II, the sig-nal model and basic notations are described. In Section III,the subcarrier weighting technique is proposed for sidelobesuppression. To highlight the effectiveness of the presentedmethod, numerical results are given in Section IV. Finally, inSection V conclusions are drawn.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE GLOBECOM 2005 proceedings.

0-7803-9415-1/05/$20.00 (C) 2005 IEEE

II. OFDM SIGNAL MODEL

To simplify the analysis, an OFDM system having a singlecontinuous transmission band is considered. As illustrated inFig.1, a real OFDM based overlay system might consist ofseveral transmission sub-bands. Thus, in order to improvesidelobe suppression the proposed algorithm could be appliedto the complete OFDM transmission signal instead of to theseveral continuous transmission sub-bands. However, as weconcentrate on the principle of subcarrier weighting in thiscontribution we leave such investigations for future work.

An OFDM system with a total number of N subcarriersis considered. The block diagram of the OFDM transmitteris illustrated in Fig. 2. The input bits are symbol-mappedapplying unit-energy phase-shift keying (PSK) modulation andN complex-valued data symbols dn, with |dn|2 = 1, n =1, 2, . . . , N , are generated. These symbols are serial-to-parallel(S/P) converted resulting in an N -element data symbol arrayd = (d1, d2, . . . , dN )T, where (.)T denotes transposition.The array d is fed into the sidelobe suppression unit whichoutputs d̄ = (d̄1, d̄2, . . . , d̄N )T. The sidelobe suppression unitperforms the multiplication of each symbol dn with a real-valued weighting factor gn. Hence, the entries of d̄ are givenby

d̄n = gndn, n = 1, 2, . . . , N. (1)

The weighting factors gn, n = 1, 2, . . . , N , are chosen suchthat the sidelobes of the transmission signal are suppressed.Finally, the weighted vector d̄ is modulated on N subcarriersusing the inverse discrete Fourier transform (IDFT). After that,parallel-to-serial (P/S) conversion is performed and a guardinterval that exceeds the delay spread of the multipath channelis added as cyclic prefix. Note that throughout this paper it isassumed that the guard interval is much shorter than the usefulpart of an OFDM symbol.

CY

CL

ICP

RE

FIX

S/P

Ndnd

1d

SID

EL

OB

ES

UP

PR

ES

SIO

N

NN dg

11dg

IDFTSYMBOLMAPPER

Nn ,,, �21�

P/S

inputbits

Fig. 2. Block diagram of the OFDM transmitter with sidelobe suppression.

In the following, we aim at the design of the weightingarray

g = (g1, g2, . . . , gN )T, g ∈ IRN , (2)

according to an optimization criterion by which the sidelobesare suppressed.

III. SIDELOBE SUPPRESSION BY SUBCARRIER WEIGHTING

A single non-weighted subcarrier sn(x), n = 1, 2, . . . , N ,is represented in frequency domain as

sn(x) = dnsin (π(x − xn))

π(x − xn), n = 1, 2, . . . , N. (3)

In (3), x is a normalized frequency given by

x = (f − f0)T0 (4)

where f denotes the frequency, f0 is the center frequency,and T0 is the OFDM symbol duration excluding the guardinterval. In addition, xn is the normalized frequency of thenth subcarrier.

As our goal is to suppress the sidelobes in a certainfrequency range, we consider sn(x) only in that range. We ob-serve M normalized frequency samples ym, m = 1, 2, . . . ,M ,which lie in the frequency range where the optimization of thesidelobes is performed. With that (3) reduces to

sn,m =sn(ym)=dnsin (π(ym − xn))

π(ym − xn), n = 1, 2, . . . , N,

m = 1, 2, . . . ,M.(5)

A simplified frequency domain representation of the non-weighted OFDM signal, the normalized subcarrier frequen-cies xn, n = 1, 2, . . . , N , and the normalized frequenciesym, m = 1, 2, . . . ,M , in the optimization range is shown inFig. 3. For simplicity, a special case is illustrated in whichthe optimization range is divided in two approximately equalparts which start from the first sidelobes outside the OFDMtransmission bandwidth.

optimizationrange

1x 2x Nx

My1y 2y1�My

non-weighted OFDMsignal

� � 12 �My� �2My

Fig. 3. Simplified frequency domain representation of the non-weightedOFDM signal and the optimization range.

Collecting sn,m, m = 1, 2, . . . ,M , into a vector we obtainsn = (sn,1, sn,2, . . . , sn,M )T, n = 1, 2, . . . , N. Finally,stacking the vectors sn, n = 1, 2, . . . , N , into a matrix weget

S = (s1, s2, . . . , sN ). (6)

To minimize the sidelobes of the weighted transmissionsignal d̄, we have to determine the vector g by solving thefollowing optimization problem

ming

‖Sg‖2. (7)

In addition, we include two constraints on the weighting vectorg. The first keeps the transmission power the same as in thecase without weighting, i.e.,

‖d̄‖2 = ‖d‖2. (8)


Exploiting (1) and recalling that the transmission symbols dn

belong to an unit-energy PSK constellation, i.e., |dn|2 = 1, (8)reduces to

‖g‖2 = N. (9)

The second constraint ensures that the elements of g arebetween pre-defined limits, i.e.,

gmin ≤ gn ≤ gmax, n = 1, 2, . . . , N. (10)

Such a constraint guarantees that each subcarrier receives acertain amount of the transmission power which is inherentlycontrolled through the ratio ρ = gmax/gmin. Furthermore, gmin

and gmax can be selected such that a weighted symbol d̄n

remains in the same decision region as the original symbol dn

as in such case no signalling from transmitter to receiver isrequired. For example, in the case of PSK this is satisfied forgmin > 0. Moreover, setting

gmin = 1 − ρ − 1ρ + 1

and gmax = 1 +ρ − 1ρ + 1

(11)

ensures that the upper limit, gmax, and the lower limit, gmin,are equally distant from |dn| = 1.

The optimization problem given in (7), together with theconstraints in (8) and (10), can be solved numerically. Manyeffective and reliable numerical methods to solve this opti-mization problem exist [7].

The principle of the subcarrier weighting technique isillustrated in Fig. 4 for the parameters N = 5, gmax/gmin =√

4, and d = (1, 1, 1, 1, 1)T. The optimization range spanssix sidelobes at each side of the used OFDM bandwidthand it starts at the first sidelobe outside the transmissionbandwidth. To keep the dimensions of the matrix S low onlyone normalized frequency sample per sidelobe is consideredin the optimization range, i.e., M = 12. In addition, thesefrequencies, ym, m = 1, 2, . . . ,M, are chosen such as tosatisfy

sin (π(ym − xn)) = 1, n = 1, 2, . . . , N,

m = 1, 2, . . . ,M, (12)

i.e., they correspond to the frequencies which lie in the middlebetween two zero crossings of the individual subcariers. Notethat considering more samples in the optimization range canlead to a somewhat better sidelobe suppression, but increasesthe dimension of the matrix S. For simplicity, in Fig. 4 only theoptimization range at the right-hand side of the transmissionbandwidth is displayed, whereas the optimization is performedover the complete optimization range. The spectra of theindividual subcarriers as well as of the sum signal of allsubcarriers are shown in frequency domain. From Fig. 4(a)-(b) it can be seen that in the case of subcarrier weightingthe signals of the individual subcarriers are adapted so asto mainly cancel each other in the optimization range. As aconsequence, in the case of subcarrier weighting the sidelobesof the sum signal exhibit significantly lower values in theoptimization range compared to the non-weighted sum signal.Hence, sidelobe suppression is achieved.

spectrum of individual subcarriersspectrum of sum signal

frequency

amplitude

(a)

spectrum of individual subcarriersspectrum of sum signal

gmax

gmin

freq.

amplitude

optimization range

1st sidelobe outside thetransmission bandwidth

(b)

Fig. 4. Illustration of the subcarrier weighting technique: (a) standard OFDMsignal without subcarrier weighting and (b) OFDM signal with subcarrierweighting.

A possible drawback of the subcarrier weighting method is adegradation in bit-error rate (BER) versus signal-to-noise ratio(SNR) performance as, due to the weighting, the subcarriersdo not receive equal amounts of transmission power. Thisdegradation depends on the chosen ratio gmax/gmin and isexpected to grow as gmax/gmin grows. In the next sectionthe BER performance and the corresponding SNR losses areaddressed by numerical simulations assuming different ratiosgmax/gmin.

IV. NUMERICAL RESULTS

In this section, several numerical results are given thatillustrate the effectiveness of the subcarrier weighting methodfor sidelobe suppression.

Binary phase-shift keying (BPSK) modulation is applied andno channel coding is considered. The number of used subcar-riers is set to N = 12. The optimization range consists of 16sidelobes at each side of the spectrum. Unless otherwise stated,the range starts from the first sidelobe outside the OFDMtransmission bandwidth. A single normalized frequency sam-ple is considered per sidelobe in the optimization range, i.e.,M = 32, and the frequency samples ym, m = 1, 2, . . . ,M,are chosen so as to satisfy (12). The ratio gmax/gmin is keptvariable throughout the simulations.


The spectra of the OFDM signals with and without sub-carrier weighting are illustrated in Fig. 5 for the symbolvector d = (1, 1, . . . , 1)T. In the case of subcarrier weightingthe ratio of gmax/gmin =

√4 is used. The benefits of the

subcarrier weighting technique are clearly visible. In compar-ison to OFDM without subcarrier weighting the sidelobes aresuppressed by more than 10 dB in the optimization range.

-20 -10 0 10 20normalized frequency

10-8

norm

aliz

ed p

ower

spe

ctra

l den

sity

in d

B

without subcarrier weightingwith subcarrier weighting

0

-10

-20

-30

-40

-50

-60

-70

-80

optimization rangeoptimization range

Fig. 5. Spectrum of OFDM signal with and without subcarrier weighting;N = 12; M = 32; d = (1, 1, . . . , 1)T; gmax/gmin =

√4.

The sidelobe suppression averaged over all possible datasymbol sequences, i.e., 2N = 212 sequences, for OFDMapplying subcarrier weighting is given in Table I. The sim-ulation results are given for different ratios gmax/gmin. Tocalculate the average sidelobe suppression, standard OFDMwithout subcarrier weighting is taken as a reference. It isnoticeable that already for gmax/gmin =

√4 a remarkable

average suppression of more than 10 dB is achieved. Afurther increase of the ratio gmax/gmin enables even bettersuppression. The reason for this lies in the fact that as this ratiogrows the constraint from (10) becomes looser, thus allowingmore degrees of freedom to find a solution of (7).

TABLE I

AVERAGE SIDELOBE SUPPRESSION FOR OFDM APPLYING SUBCARRIER

WEIGHTING FOR DIFFERENT RATIOS gmax/gmin .

ratio ρ = gmax/gmin 1√

2√

4√

6√

8average sidelobesuppression in dB 0.0 4.9 10.21 13.45 15.76

The OFDM systems applying subcarrier weighting withdifferent ratios gmax/gmin are compared in Fig. 6 on thebasis of BER over SNR curves. SNR is given in Eb/N0

which represents the energy per transmission bit over thenoise spectral density. The underlying mobile radio channelis modelled as a frequency selective Rayleigh fading channel,whereas a maximum ratio combining detector is applied atthe receiver assuming perfect channel knowledge. The Monte-Carlo simulations are performed taking into account all possi-ble data symbol sequences. As a reference the performance

of a standard OFDM system without subcarrier weightingis given. The subcarrier weighting results in an BER loss,since the subcarriers do not receive equal amounts of thetransmission power. From (11) it follows that if gmax/gmin

grows, gmin becomes lower and thus, some subcarriers receivevery small amounts of transmission power and cannot bedecoded properly at the receiver resulting in a performancedegradation. For example, as shown in Fig. 6, the ratio ofgmax/gmin =

√6 causes an SNR degradation of around 3.2

dB.The assessment of Table I and Fig. 6 reveals that there

are two counteracting effects caused by the ratio gmax/gmin.Enlarging this ratio improves sidelobe suppression, but si-multaneously leads to a further loss in SNR performance.Therefore, there is a trade-off between the additional sidelobesuppression obtained by enlarging the ratio gmax/gmin andthe increased loss in SNR performance. Setting gmax/gmin =√

4 seems to be a good compromise. A further increase ofgmax/gmin is not justified as it leads to a relatively high SNRloss with only moderate improvement in sidelobe suppression.

0 2 4 6 8 10 12 14 16 18 20

SNR in dB

10-3

10-2

10-1

100

BE

R

OFDM with g / g = 8OFDM with g / g = 6OFDM with g / g = 4OFDM with g / g = 2OFDM reference, (g / g =1)

max min

max

max

min

min

max min

3.2 dB

max min

√√√√

Fig. 6. BER versus SNR for OFDM applying subcarrier weighting; differentratios gmax/gmin; N = 12; Rayleigh fading channel; BPSK.

To suppress the sidelobes even further the proposed methodcan be easily combined with the standard methods describedin [6]. As an example we give the results obtained by combin-ing the guard band insertion technique [6] and the subcarrierweighting method.

The spectra of the OFDM signals with and without subcar-rier weighting including a guard subcarrier at each side of thetransmission bandwidth are illustrated in Fig. 7 for the symbolvector d = (1, 1, . . . , 1)T. In the case of subcarrier weightingthe ratio of gmax/gmin =

√4 is used. The optimization range

starts with the second sidelobe at each side of the spectrum asone subcarrier is left empty at each side of the spectrum, i.e.,it is used as a guard subcarrier. As the highest sidelobe, i.e.,the first sidelobe, is not attempted to be suppressed with theoptimization algorithm, the sidelobes starting with the second


are suppressed even better than in the case without guardsubcarrier (see Fig. 5).

-20 -10 0 10 20normalized frequency

10-8

norm

aliz

ed p

ower

spe

ctra

l den

sity

in d

B

without subcarrier weightingwith subcarrier weighting

optimization rangeoptimization range

0

-10

-20

-30

-40

-50

-60

-70

-80

subcarrierguard guard

subcarrier

Fig. 7. Spectrum of OFDM signal with and without subcarrier weightingincluding a guard subcarrier at each side of the transmission bandwidth; N =12; M = 32; d = (1, 1, . . . , 1)T; gmax/gmin =

√4.

Finally, the sidelobe suppression averaged over all possibledata symbol sequences for OFDM applying subcarrier weight-ing including a guard subcarrier at each side of the trans-mission band is given in Table II. To obtain these simulationresults different ratios gmax/gmin are considered. Again, theoptimization range starts with the second sidelobe at each sideof the spectrum. As a reference OFDM without subcarrierweighting, but with a guard subcarrier at each side of thetransmission band is used. For example, for gmax/gmin =

√4

the sidelobes are suppressed by more than 13 dB on theaverage. In comparison to the corresponding results from Ta-ble I, the insertion of a guard subcarrier leads to an additionalsuppression of about 3 dB.

TABLE II

AVERAGE SIDELOBE SUPPRESSION FOR OFDM APPLYING SUBCARRIER

WEIGHTING INCLUDING A GUARD SUBCARRIER FOR DIFFERENT RATIOS

gmax/gmin .

ratio ρ = gmax/gmin 1√

2√

4√

6√

8average sidelobesuppression in dB 0.0 6.31 13.22 17.36 20.35

V. CONCLUSIONS

In this paper, we have introduced a new technique tosuppress the sidelobes of OFDM transmission signals. Besidesusing this technique to improve the spectral efficiency ofOFDM based transmission systems, it can be applied toOFDM based overlay systems to avoid interference towardsthe legacy systems sharing the same frequency band. Theproposed sidelobe suppression scheme does not require trans-mission of any side information and is capable of reducingthe sidelobes of OFDM transmission signals by more than10 dB. The price to pay for this achievement is a moderate

loss in BER performance. Moreover, combining the proposedtechnique with well-known suppression methods, like e.g.guard band insertion, the suppression capabilities are furtherincreased by several dB. Currently, the proposed sidelobetransmission technique is extended to applications withinOFDM based overlay systems.

ACKNOWLEDGMENT

This work was supported by the Broadband VHF Aeronau-tical Communications System Based on MC-CDMA (B-VHF)project [8] which is funded by the European Commissionwithin the 6th Framework Programme.

REFERENCES

[1] J. Zander, “Radio resource management in future wireless networks:Requirements and limitations,” in IEEE Communications Magazine, Aug.1997, pp. 30–36.

[2] J. Mitola, “Cognitive radio for flexible mobile multimedia communica-tions,” in Proceedings IEEE International Workshop on Mobile Multime-dia Communications, Nov. 1999, pp. 3–10.

[3] T. Weiss and F. Jondral, “Spectrum pooling - an innovative strategyfor the enhancement of spectrum efficiency,” in IEEE CommunicationsMagazine, Radio Communications Supplement, Mar. 2004, pp. S8–S14.

[4] FCC Spectrum Policy Task Force, Rep. ET Docket no. 02-1935. FederalCommunications Commision (FCC), Nov. 2002.

[5] ITU-R, Framework and Overall Objectives of the Future Development ofIMT-2000 and Systems Beyond IMT-2000. International Telecommuni-cations Union (ITU); Reccomendation M.1645, June 2003.

[6] T. Weiss, J. Hillenbrand, A. Krohn, and F. Jondral, “Mutual interference inOFDM-based spectrum pooling systems,” in Proceedings IEEE VehicularTechnology Conference (VTC’04, Spring), May 2004.

[7] R. Fletcher, Practical Methods of Optimization, 2nd Edition. Wiley,2000.

[8] http://www.b-vhf.org.


a technique for sidelobe suppression in ofdm systems

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