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AnalysisanddesignofOFDM/OQAMsystemsbasedonfilterbanktheory

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1170 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002

Analysis and Design of OFDM/OQAM SystemsBased on Filterbank Theory

Pierre Siohan, Senior Member, IEEE, Cyrille Siclet, and Nicolas Lacaille

Abstract—A discrete-time analysis of the orthogonal frequencydivision multiplex/offset QAM (OFDM/OQAM) multicarriermodulation technique, leading to a modulated transmultiplexer, ispresented. The conditions of discrete orthogonality are establishedwith respect to the polyphase components of the OFDM/OQAMprototype filter, which is assumed to be symmetrical and with arbi-trary length. Fast implementation schemes of the OFDM/OQAMmodulator and demodulator are provided, which are based on theinverse fast Fourier transform. Non-orthogonal prototypes createintersymbol and interchannel interferences (ISI and ICI) that,in the case of a distortion-free transmission, are expressed by aclosed-form expression. A large set of design examples is presentedfor OFDM/OQAM systems with a number of subcarriers goingfrom four up to 2048, which also allows a comparison betweendifferent approaches to get well-localized prototypes.

Index Terms—Filterbanks, multicarrier modulation, polyphaserepresentation, pulse shaping, transmultiplexer.

I. INTRODUCTION

T HE principle of multicarrier modulation (MCM) consistsof splitting up a wideband signal at a high symbol rate into

several lower rate signals, each one occupying a narrower band.A generally recognized advantage of MCM is its robustnessagainst different types of channel distortions, such as multi-path propagation and narrowband interference. Orthogonalfrequency division multiplex (OFDM) is certainly, until now,the most important class of MCM. The OFDM acronym oftenrecovers two different types of modulation. In the first one, asproposed, for instance, in [1], each carrier is modulated usingquadrature amplitude modulation (QAM). In this scheme,which is also called OFDM/QAM [2], QAM symbols areshaped with a rectangular window. In a second category ofOFDM systems, which is also called orthogonally multiplexedQAM in [3] (O-QAM) or OFDM with offset QAM in [2](OFDM/OQAM), the modulation used for each subcarrier is astaggered offset QAM (OQAM). Both the OFDM/QAM andOFDM/OQAM modulation schemes theoretically guaranteeorthogonality and a maximum and identical spectral efficiency.Furthermore, in practice, they can be implemented thanks to

Manuscript received April 27, 2000; revised January 10, 2002. The associateeditor coordinating the review of this paper and approving it for publication wasProf. Gregori Vazquez.

P. Siohan was with France Télécom R&D, DMR/DDH, Cesson-SévignéCedex, France. He is now with IRISA-INRIA, Campus de Beaulieu, RennesCedex, France.

C. Siclet is with the France Télécom R&D, DMR/DDH, Cesson-SévignéCedex, France.

N. Lacaille was with France Télécom R&D, DMR/DDH, Cesson-SévignéCedex, France. He is now with l’Université de Technologie de Belfort-Mont-béliard, Belfort Cedex, France.

Publisher Item Identifier S 1053-587X(02)03151-3.

the discrete Fourier transform (DFT). An important differencecomes from the fact that OFDM/OQAM, unlike OFDM/QAM,allows the introduction of an efficient pulse shaping, whichmakes it less sensitive to the frequency offset due to the trans-mission channel and to the receiver. Therefore, if OFDM/QAMconstitutes the modulation kernel of the famous so-called codedOFDM (COFDM) system, OFDM/OQAM is now presented asbeing a good candidate to get still higher bit rates over wirelesschannels [2].

Based on the earliest works in the field; see, for example,[3]–[5], the orthogonality of OFDM/OQAM has been presentedfor some time, only using continuous pulses of infinite length,as, for instance, the well-known raised-cosine function. The im-portance of getting an orthogonal pulse-shaping satisfying givencriteria has only appeared recently. However, most of the solu-tions already proposed do not take into account the truncationand discretization effects of the pulse shape, i.e., a natural con-sequence of DFT-based implementations. These effects imply aloss of orthogonality, i.e., an intersymbol interference (ISI) andan interchannel interference (ICI), which is implementation de-pendent and arises even for distortion-free channels.

In [6], the authors take into account the truncation effect andchoose to minimize the out-of-band energy of a time-limitedpulse shape. Once again, the discretization step of the resultingpulse shape is ignored, and this only leads to an approximationof the discrete orthogonality conditions. In the case of wirelesstransmission, due to the time and frequency dispersion createdby this type of channel, it seems, however, more appropriate tolook for well-localized pulse shapes [2], [7]. Independently ofthe present work, some authors [8] have recently proposed todiscretize orthogonality conditions given in continuous-time [6]in order to get discrete orthogonality conditions. The evalua-tion in [8] is also based on the time-frequency localization cri-terion. In this paper, our approach is different and more directlyconnected to filterbank formalism, as presented, for instance, intransmultiplexer theory [9]. We start by discretizing the basicdefinition equation of OFDM/OQAM, and afterwards, we de-rive the theoretical properties, the implementation scheme, andthe performances of the system. We represent all signals by theircomplex envelope, and we assume, as usual, that the prototypefilter is real-valued and symmetrical. In this way, we are able toget the following:

• a description of OFDM/OQAM systems being equiva-lent to a modulated transmultiplexer, given in Section II,leading to efficient implementations based on inverse fastFourier transform (IFFT) and presented in Section III;

• the discrete orthogonality conditions with respect to thepolyphase components of a causal finite impulse response

1053-587X/02$17.00 © 2002 IEEE

SIOHAN et al.: ANALYSIS AND DESIGN OF OFDM/OQAM SYSTEMS 1171

(FIR) prototype filter of arbitrary length, which are de-rived in Section III;

• an analytical expression of the overall distortion due tothe ISI and to the interchannel interference ICI created byimperfect filterbanks in distortion-free channels, which isderived in Section IV;

• several design examples presenting a large set of orthog-onal and nearly orthogonal prototype filters with differentpulse shapes. Design charts, tables, and graphical displaysare provided in Section V, allowing different comparisonsin terms of orthogonality and time–frequency localization.

Notations: denote the set of integer and real num-bers, respectively. correspond to the space ofsquare-summable continuous and discrete-time functions,respectively. Vectors and matrices are denoted with bold italicletters, for instance, . Superscript stands for transposition.We denote discrete filters of with lower case letters, for in-stance, , and their -transform with upper case letters, suchas . and designate the real part and imaginarypart of the complex-valued number, respectively. Similarly,for a filter , we get and

. Superscript denotes complexconjugation. For a filter . Thetilde notation denotes paraconjugation: .

is the norm of .

II. OFDM SYSTEMSUSING OFFSET-QAM (OFDM/OQAM)

In this section, we recall the classical continuous-time for-mulation of OFDM/OQAM, and the discrete-time formulationis then derived. We also introduce the notations that allow clas-sical multirate relations of the filterbank theory [10] to be re-covered in discrete time.

A. Continuous-Time Formulation

The principle of OFDM/OQAM is the transmission of offsetQAM symbols rather than QAM symbols. Then, the orthog-onality can be maintained with pulse shapes being differentfrom the rectangular window [4]. The second specificity ofthis scheme is that considering two successive carriers, thetime offset is introduced onto the imaginary part of the QAMsymbols on one of the carriers, whereas it is introduced ontothe real part of the symbols on the other one. The numberof carriers, which is denoted , is also assumed to be even

, and the continuous-time baseband transmittedsignal writes [11]

(1)

where is the signaling interval, the spacing be-tween two successive carriers, and the real and imag-inary parts, respectively, of the QAM complex-valued symbols

we want to transmit, and a symmetrical real-valued

pulse shape that can be different from the rectangular window.At the demodulation side, the symbols are estimated by

(2)

(3)

(4)

(5)

In order to get simplified, easier-to-manipulate expressions, thefollowing notations can be introduced:

(6)

(7)

We also denote . Then, (1) –(5) are rewritten as

(8)

(9)

Another way to formulate the OFDM/OQAM continuous-timetransmitted signal is to write it as an expansion over some basisfunctions with coefficients

(10)

where is defined by

(11)

Then, (9) is rewritten as

(12)Assuming a distortion-free channel, the transmitted symbols areperfectly recovered, i.e., , if and only if the set offunctions constitutes an orthonormal basis of its span, i.e.,

(13)

1172 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002

where is the Kronecker delta. In particular, we get. It is worthwhile mentioning that these conditions are

unchanged if is just defined modulo . Thus, it can beassumed, as, for example, in [2], that .

B. Discrete-Time Formulation

In the previous section, we considered continuous-timesignals in with its classical real-valued inner product

. Now, our goal is to obtain a discrete-time signalin with its classical real-valued inner product ,which is defined by

(14)

Since the duration corresponds to the transmission ofcomplex-valued symbols, the critical sampling period is definedby

(15)

This means that is critically sampled, and therefore, thespectral efficiency is still maximum. In order to get a causal dis-crete-time prototype with length equal to is trun-cated to the interval and is delayed by

time units. Moreover, in order to get a proto-type filter with norm still approximately equal tohas to be normalized with a multiplicative factor equal to

(16)

Thus, based on (8) and using the fact that andthat , the baseband transmitted discrete-timeOFDM/OQAM signal is such that

(17)

with

(18)

At the demodulation stage, we get an estimate of the transmittedreal-valued symbol using the real-valued inner product ofand

(19)

Then, assuming a distortion-free channel, ifand only if the set of functions constitutes an orthonormalbasis of its span, i.e.,

(20)

C. OFDM/OQAM Transmultiplexer

Equation (17) looks like the expression of a signal obtained atthe output of a synthesis bank of a filterbank with sub-bandsand with an expansion factor equal to on each sub-band.Indeed, if and denote the input signal and the filteron the th subband of this synthesis bank, respectively, then theoutput signal is written as [10, p. 117]

(21)

This exactly corresponds to (17) with and

(22)and with

(23)

We can remark that can be defined by (23) since hasjust to be equal to modulo . Similarly, (19) lookslike the expression of the output symbols of an analysis bank ofa filterbank with sub-bands and with a decimation factorequal to on each sub-band. Thus, if denotes the inputsignal of this bank and and denote the filter andthe output signal on the th subband, respectively, then [10, p.117]

(24)

Using the fact that , let us rewrite (19) inorder to illustrate the analogy with (24):

(25)

Let us decompose with two integers and ( and) by

(26)

Then, in order to get a causal system, (25) is rewritten, making

SIOHAN et al.: ANALYSIS AND DESIGN OF OFDM/OQAM SYSTEMS 1173

Fig. 1. Transmultiplexer model of OFDM/OQAM modulations.

the reconstruction delay appear as

(27)

It appears that (27) corresponds to (24) withand

(28)

Then, by defining and , which are the -trans-forms of and , respectively, we get the transmulti-plexer scheme depicted in Fig. 1. In this way, we get a transmul-tiplexer with the following particularities.

• The input and output signals are real-valued symbols.• The number of channels is twice the ex-

pansion and decimation factors , whereas for clas-sical transmultiplexers that are critically decimated [10]or oversampled [12], we necessarily have or

, respectively. Exact reconstruction in this caseis nevertheless made possible by the fact that the inputsymbols are real-valued; therefore, the present transmulti-plexer is equivalent to a critically decimated one withcomplex-valued inputs [cf. (15)].

• Before expansion by a factor , each real-valued input ispremodulated, whereas at the receiver side, the dual oper-ation occurs after decimation, the real part being extractedafterwards (cf. Fig. 1).

• A delay , with depending on the length of the pro-totype filter, has to be included either at the transmitteroutput or at the receiver input.

Compared with previous schemes proposed for orthogonalMCM systems (see, for instance, [11] and [13]), our imple-mentation scheme involves two integer parametersand thatallow us to handle the case of causal prototype filters with arbi-

trary length , which is not considered in theworks cited previously.

III. D ISCRETEORTHOGONALITY FOROFDM/OQAM

The orthogonality conditions of the modulated transmulti-plexer we have just obtained naturally depend on the analysisand synthesis filterbanks, i.e., , respectively.To get a compact representation of this system, we use thepolyphase approach, which leads us to IFFT-based implemen-tations, to the input–output relation and, finally, allows us toget the mathematical orthogonality conditions.

A. Polyphase Approach

The analysis and synthesis filterbanks in Fig. 1 can beexpressed as functions of the prototype filter or of its-transform . Let us decompose as a function of its

polyphase components of order [10]

with

(29)

As is real-valued and symmetrical, using (22) and (28),and can be written as

(30)

where .Wecanthenrewrite andasfunctionsof andderivethepolyphasematricesofthesyn-thesis(modulator)andanalysis(demodulator)parts,whicharede-noted (type 2) and (type 1) [10], respectively:

(31)

(32)

1174 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002

Fig. 2. General polyphase representation of the OFDM/OQAM transmultiplexer.

Fig. 3. OFDM/OQAM modulator realized with an IFFT.

Using these notations and the Noble identities [10, p. 119],the transmultiplexer, as depicted in Fig. 1, is equivalent to itspolyphase representation shown in Fig. 2. Letwith a -transform . Then, denotes the-transform of . Besides, at the receiver side,

denotes the-transform of with .

B. IFFT-Based Implementations

Direct implementation of the systems depicted in Figs. 1 and2 would be very costly. However, as the modulator and demod-ulator are related to a modulated transform, we examine, in thissection, how it is possible to get an efficient IFFT-based imple-mentation.

In order to get the matrices and as functions of, we now introduce the matrices and ,

which are given by

diag

diag (33)

Using these latter notations and (31) and (32), we get

(34)

where is the antidiagonal matrix. Therefore, wecan directly derive from Fig. 2 the IFFT-based modulator anddemodulator schemes depicted in Figs. 3 and 4, respectively.Furthermore, Figs. 3 and 4 only represent two basic schemesthat can be further improved, as shown in Appendix A.

C. Input–Output Relation

Let us now introduce a matrix notation whereand are column vectors with elements

and , respec-tively. is the -transform of . Then, itcan be seen from the scheme depicted in Fig. 2 that we havethe relation

(35)

where denotes the transfer function of the section in-cluding the expanders, delay lines, and decimators. Hence

with

(36)

Thus, from (34) and (36), we get

with

(37)

SIOHAN et al.: ANALYSIS AND DESIGN OF OFDM/OQAM SYSTEMS 1175

Fig. 4. OFDM/OQAM demodulator realized with an IFFT.

After some computation reported in Appendix B, it can beshown that is as in (38), shown at the bottom of the page,where

(39)

D. Mathematical Expression of Orthogonality

Perfect orthogonality is provided when there is no ISI, i.e.,, nor ICI, i.e., .

Using filterbank terminology, we may also say that the orthog-onal transmultiplexer checks the perfect reconstruction (PR)property with the delay if and only if , where

is the identity matrix. In Appendix C, we derivean equivalent formulation of this latter equality, which leadsus to the following theorem showing how the conditions ofperfect orthogonality can be directly related to the polyphasecomponents of the prototype filter.

Theorem 1: Let be a symmetrical real-valued prototypefilter that is assumed to be causal and time-limited with arbi-trary length , and let be its typeI polyphase components. Then, a sub-band OFDM/OQAMsignal, using this prototype, can be transmitted through a dis-tortion-free channel and reconstructed without ISI nor ICI, withthe delay denoting the integer part func-tion by superior value if and only if

(40)

It can be noted that with this latter expression, we recovera PR condition that is identical to the ones already obtained atfirst for PR cosine-modulated filterbanks [10, p. 379], [14] andafterwards, with exception of the normalization constant, for thePR orthogonal MDFT filterbanks [15].1

It is worthwhile mentioning that using the analogy with filter-banks again, each pair of polyphase filterssatisfying (40), for , can be implemented inFigs. 3 and 4 by means of a cascaded lattice that structurally en-sures a perfect orthogonality [10, pp. 380–383], [14].

IV. ISI AND ICI FOR A DISTORTION-FREECHANNEL

In this section, we give, in the case of a distortion-freechannel, explicit expressions of the ISI and ICI terms whenusing OFDM/OQAM with a prototype filter that is not perfectlyorthogonal.

A modulation of each subcarrier using OQAM- , i.e., astates modulation, implies that each real-valued symbol, coded with bits, is such that ,

where is a fixed amplitude, and is an integer in the range.

In practice, most often, prototype functions for communi-cation systems are obtained by truncation and discretizationof continuous-time functions that are no longer orthogonalin discrete time. Then, as (40) is not exactly satisfied, thesenonorthogonal prototypes create ISI, i.e., interference betweensymbols inside a given subband, and ICI, i.e., interferencebetween symbols belonging to different subbands. Using (36)

1In that way, this result is an illustration of the equivalence between these twoaspects of multirate filterbanks.

..... .

......

. . .. . .

. . .

(38)

1176 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002

and (38), it can be seen that each component ofis such that

(41)

where is the integer part function. Therefore, in the timedomain, we get (42), shown at the bottom of the page.

The total distortion is then given by .For a -OQAM, we have , and thus,the maximum distortion is

(43)

Moreover, using (39), we notice thatso that

(44)

V. DESIGN RESULTS

The quality of a given prototype filter can be evaluated bymeans of the resulting ISI and ICI and, generally, by its abilityto satisfy a given criterion. In this section, two situations are con-sidered: a first one where the prototype is obtained by truncationand discretization of orthogonal functions in continuous-timeand is therefore not perfectly orthogonal and a second one wherewe design it in order to exactly satisfy the orthogonality condi-tions. In both cases, the time–frequency localization is used asa criterion either for evaluation, as in Section V-B, or for opti-mization, as in Section V-C.

A. Time–Frequency Localization Measures

It is well known that the wireless channel introduces time andfrequency dispersion. To limit the resulting distortion, a solu-tion proposed in [2] is to use basic signals being localized intime and frequency with the same time–frequency scale as thechannel itself. In this context, time and frequency dimensionsare equally important. Indeed, time–frequency localization isnow often considered as being the right criterion, as it can be

seen in several publications related to OFDM/OQAM over wire-less channels [2], [7], [8], [16]. In this paper, in order to assessour results on a solid basis, we will use two different localiza-tion measures.

Let be a discrete-time signal with finite energy, i.e.,, and with Fourier transform denoted . Its second-

order moment in time, which is denoted , and in frequency,which is denoted , are given by

(45)

where and minimize and , respectively. It can beshown that

(46)

Moreover, for a real-valued signal, mod .It can be deduced from [17] that if , we can

define a localization measure denotedsuch that

(47)

where is the unreachable optimum. Thisinequality related to the uncertainty principle can be applied inparticular when is symmetrical with even length. In order toavoid the restriction mentioned above, we can use, instead of(45), the definition of second-order moments suggested in [18]by Doroslovacki

(48)

(49)

(50)

(51)

where and are the gravity centers in time and frequency,respectively, which minimize and , which are second-

(42)

SIOHAN et al.: ANALYSIS AND DESIGN OF OFDM/OQAM SYSTEMS 1177

order moments in time and frequency, respectively. It can beshown that

mod (52)

(53)

Then, we obtain a second criterion, which is denoted

(54)

The optimum is still .

B. Design Examples Based on Continuous-Time OrthogonalFunctions

By truncation and sampling of a prototype function satis-fying orthogonality conditions in continuous time, it is possibleto get a discrete prototype being nearly orthogonal, as long asits length is sufficient. In our simulations, we have comparedthree types of functions that, initially, were defined in contin-uous-time:

1) the well-known square root raised cosine (SRRC) func-tion (cf. for instance, [19]);

2) a family of pulses introduced by Vahlin and Holte in[20], which we will call optimal finite duration pulses(OFDPs);

3) a new family of functions called extended Gaussian func-tions (EGFs); see [21].

In all cases, the truncation limits the pulse length toand the discretization is carried out with a sampling period

. These three families are interesting fromdifferent points of view. Thus, SRRC functions mainly havea historical interest, and we only consider them in order tomake a comparison with some well-known functions. TheOFDPs have two important properties. The first is that they areorthonormal with a finite duration so that they do not need tobe truncated. The second is that they are optimized in order toget as much of the pulse energy as possible within a frequencyband . Finally, an important property of theEGF is the fact that they can also lead to an orthonormal basisafter discretization and that, furthermore, they can be veryclose to the achievable optimum bound in time–frequencylocalization [2], [22]. Nevertheless, none of these pulses keepstrict orthonormality after both operations of truncation anddiscretization.

First, let us give the continuous-time expressions of SRRCfunctions OFDP and EGF. For a transmission rate equal to,the frequency expression of the SRRC function is as in (55),

shown at the bottom of the page, whereis the roll-off param-eter . Therefore, the continuous-time expression ofSRRC functions is given by

(56)

As for OFDPs, their expression in continuous-time is as fol-lows [6], [20]:

(57)

where is the th prolate spheroidal wave function trun-cated to the interval and where aresome real-valued coefficients obtained thanks to an optimiza-tion procedure [20].

Last, the EGFs are defined by [21]

(58)

where is a real-valued number, are somereal-valued coefficients, and is the Gaussian function

.For OFDM/OQAM signals, we suppose that

, and in order to satisfy (13), we also suppose thatis such that . Given that

, the corresponding filters are multiplied by anormalization factor equal to . The discrete-time versionof the SRRC, OFDP, and EGF prototype filters are then obtainedby truncation to the interval (except forOFDPs, which no longer need to be truncated) and thanks to(16).

Besides, when setting for the EGF, asin [2] and [21], has to be approximately within therange to get the best time–frequency localization measures [22].

In Table I, we have compared, for , the maximum dis-tortion [cf. (44)] obtained for a few EGF and SRRC func-tions. They show that for a given length, except for and

, the EGF clearly outperform the best SRRC func-tion. Some other simulations carried out with and

give the results reported in Figs. 5 and 6. They lead us to theconclusion that for any value of and a given ratio, theEGF can probably provide better results than the SRRC func-tions. They also show that in both cases, the distortion mainlydepends on the ratio.

We can deduce from a result previously obtained for cosine-modulated filterbanks [22] that for OFDM/OQAM systems, the

(55)

1178 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002

TABLE IMAXIMUM DISTORTION (D ) FORSRRC(r) AND EGF(�) WITH

DIFFERENTVALUES OFr AND �,AND FORM = 4

Fig. 5. D values for SRRC prototype functions withr = 1 andM = 4(dashed line),M = 8 (dotted line), andM = 16 (solid line).

Fig. 6. Estimated (lines) and experimental (dots)D values for EGFprototype with� = 0:5 (solid line), � = 1 (dashed line), and� = 1:5(dashdot line) forM = 4 (o);M = 8 (+) andM = 16 (�).

maximum distortion related to discrete EGF prototypes can beestimated by [16]

(59)

Our results reported in Fig. 6 give a good illustration of theaccuracy of this estimate for

and .Concerning the SRRC function, it can be seen in Fig. 5, with

a roll-off value , that an increase in the ratio alsotends to reduce the distortion. However, it is also clear that thisdecrease is not as fast nor as regular as with the EGF.

In Table II, we have reported for different values ofthe maximum distortion related to discrete-time OFDPs withtime duration equal to and , which correspondto and , respectively.These prototype filters have been computed thanks to the values

of the coefficients given in [20]. This table shows that thedistortion of OFDPs slightly increases when the number ofcarriers increases beyond . It also shows that OFDPshave a smaller distortion than SRRCs for the sameratio. Moreover, the distortion of OFDPs is also smaller thanthe distortion of EGFs for and . Finally, for

and , OFDPs are between the EGFsobtained for and .

C. Design Examples Satisfying the Discrete OrthogonalityConditions

From Table I and Figs. 5 and 6 and as illustrated with pre-vious publications (see, for instance, [23]), it also clearly ap-pears that to get low distortion values, high ratios maybe required. Furthermore, the truncation and sampling opera-tions do not allow direct control of the time–frequency localiza-tion. On the other hand, if we choose to directly optimize theprototype filter to get a maximum time–frequency localizationunder the PR constraint (40), we are, at least, ensured to getneither ISI nor ICI distortion. This optimization can be directlyconducted by optimizing the prototype coefficients underthe constraint (40), or it can be also carried out using a set of lat-tice coefficients that corresponds to an efficient implementationof each pair of polyphase components for

, [10, pp. 380–383], [22]. We have chosenthis second approach, which seems preferable, since it leadsto an unconstrained optimization problem with, furthermore, anumber of variables that are approximately reduced by a factorof 2.

Let us refer to the optimization method of the lattice coeffi-cients according to the criterion as the optimized lattice(OL), knowing naturally that more generally, the optimizationof the lattice coefficients can be carried out with different typesof criteria, including, as in [20], the maximization of the in-bandenergy. We can compare the OL method with the truncation anddiscretization method applied to the EGF, OFDP, and SRRCfunctions. We have chosen the parameter value, providing, ingeneral, the best localization measures, i.e., for the EGF[21], [22] and for the SRRC functions. As stated previ-ously, the OL method is the only one that provides a perfectorthogonality, i.e., a distortion theoretically equal to 0but, in practice, due to numerical inaccuracies, aroundor for double precision computers. Furthermore, as ourresults reported in Table III show, it appears that for the twomeasures used (and ), despite the introduction of the PRconstraint, we have the following.

• Compared with the SRRC function, the OL method al-ways provides better localized prototype filters, even for asmaller length than for SRRC filters.

• Compared with the OFDP, OL filters have a better local-ization measure for a given length. This is not surprisingsince OFDPs are not optimized according to the maxi-mization of the time–frequency localization.

• Compared with the EGF, we can see that the OL methodgenerally provides better results. This is particularly truefor the shortest filter lengths ( , i.e., a pulse lengthlimited to the duration of one complex-valued symbol).

SIOHAN et al.: ANALYSIS AND DESIGN OF OFDM/OQAM SYSTEMS 1179

TABLE IIMAXIMUM DISTORTION (D ) FOR THEVAHLIN PULSES

TABLE IIILOCALIZATION MEASURESOBTAINED FOR DISCRETESRRC(r = 1), DISCRETEEGF(� = 1), DISCRETEOFDP,AND FOR THEOL METHOD

Fig. 7. SRRC function withr = 1 for M = 1024 andL = 2048; � =0:0830;D = 7:145e�02.

Some exceptions occur for the highest values offorwhich it may be possible that our optimization procedurefound local optima only.

Nevertheless, if one wants, for simplicity, to use analyticalexpressions, it is also clear that EGF have to be preferred to theOFDP and the SRRC functions to get well-localized prototypefilters.

As for MCM systems, the number of subcarriers is relativelyhigh (generally greater than 64 and sometimes up to 8192 [24]),and as the computational complexity is more often a crucialfactor, the significant improvements we get for prototype filterslimited to one signaling interval are of a particular practicalinterest.

Figs. 7–10 show time and frequency representations obtainedfor OFDM/OQAM systems with . The comparisonfor between the SRRC function, the EGF, and the OLmethod can be made with Figs. 7–9. In Fig. 10, we can comparethe time and frequency responses of the EGF, with , andof the OFDP, for . It is clear that the OFDP that isoptimized to minimize the out-of-band energy yields a betterresult in the frequency domain. On the other hand, the EGF,which has a better localization in time, finally provides, with

Fig. 8. EGF with� = 1 for M = 1024 and L = 2048; � =0:1442;D = 2:846e�02.

Fig. 9. OL for M = 1024 andL = 2048; � = 0:9049;D =7:002e�15.

Fig. 10. EGF (dashed line,� = 1; � = 0:9767;D = 1:476e�03)and OFDP (solid line,� = 0:9328;D = 5:399e�03) for M = 1024andL = 8192.

1180 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002

, a better time–frequency localization measurethan the OFDP with .

Concerning the comparison of the impulse responses for theprototypes of length 2048 (see Figs. 7–9) with the ones withlength 8192 (Fig. 10), one has to take care of the different timescales used for the graphical displays.

VI. CONCLUSION

We have presented an application of the filterbank theory tothe analysis and design of OFDM/OQAM systems. In a firststep, we have derived the transmultiplexer scheme that can beassociated with OFDM/OQAM systems with symmetrical FIRprototypes of arbitrary length. Then, using a polyphase decom-position of this prototype, we have proposed fast implementa-tion schemes of the OFDM/OQAM, modulator, and demodu-lator, which are based on the IFFT, and we have also found thediscrete orthogonality conditions related to the polyphase com-ponents of this prototype. Systems that do not exactly satisfy thediscrete orthogonality conditions create ISI and ICI. For suchnonorthogonal systems, and assuming a distortion-free trans-mission channel, we have derived the analytical expressions ofthe ISI and ICI distortions created by imperfect filterbanks. Fi-nally, we have provided a set of design examples that clearlyshow that the numerical optimization of a discrete orthogonalprototype has, at least, two important advantages compared withtruncation and sampling of prototypes being orthogonal in con-tinuous time. Indeed, the direct design of orthogonal prototypesin discrete time not only avoid ISI and ICI distortions for distor-tion-free channels, but they also allow us to obtain short proto-type filters with very good localization measures. This naturallytends to significantly reduce the computational cost of imple-mentation.

APPENDIX AREDUCTION OF THECOMPLEXITY OF THE IFFT-BASED

IMPLEMENTATIONS

It can be seen from Fig. 3 that for

(60)

Let us define two integers and by

or (61)

Let us also denote

ifif

(62)

Then, two different cases appear.

• If is even, , and for

(63)

Moreover, the FFT in (63) is an FFT with real-valued in-puts; therefore, it can be carried out thanks to an FFT with

complex-valued inputs, instead of real-valued in-puts [25, p. 82].

• If is odd, , and

(64)

Equation (64) corresponds to an odd DFT with real-valuedinputs that can also be realized thanks to an FFT withcomplex-valued inputs instead of real-valued inputs[25, p. 83]. Thus, as in [23], it is possible to use only anFFT of size .

Similarly, at the demodulation part, the IFFT followed by a ro-tation by is also equivalent to an FFTwith a permutation of the inputs . Then, we will onlykeep the real part of the outputs of the FFT so that further re-duction of the complexity is also possible.

APPENDIX BCOMPUTATION OF THE TRANSFERMATRIX

In Section III, we have shown that the transfer matrixcan be written

(65)

where

(66)

Toevaluate , letusfirstnotethatadelay placedbetweenan expansion and a decimation byleads to a transfer function

such that , with if is a mul-tiple of and otherwise. Thus, we obtain

, and from (66), we get. There-

fore, the entries of matrix are

(67)

In order to only retain the nonzero terms in (67), we now define

ifotherwise

(68)

and for , we also introduce (69), shown at the bottom ofthe page. Therefore, the entries in matrixcan be rewritten as

ifif

(69)

SIOHAN et al.: ANALYSIS AND DESIGN OF OFDM/OQAM SYSTEMS 1181

shown in the first equation at the bottom of the page, where

(70)

(71)

Moreover, using (72), shown at the bottom of the page, we ob-tain from (70)–(72) that

and finally

(73)

We can now notice that the argument ofin (73) is real ifand are even; otherwise, it is purely imaginary. Thus,being given, defining as the unique value of , which hasthe same evenness as, we obtain (74), shown at the bottomof the page. Using the decomposition , theexpression of can be simplified as follows whenis even:

(75)

Furthermore, it can be checked by inspection from (29) and (69)that for . Thus,we finally obtain the expression of the transfer matrix given by(38) and (39).

for

for(72)

if is even

otherwise(74)

1182 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 5, MAY 2002

APPENDIX CORTHOGONALITY CONDITIONS

The matrix given by (65) can also be rewritten as

Then, the orthogonality conditions can be derived from theequality or equivalently, from the followingrelation:

This latter expression is also equivalent to

for (76)

Then, using (68) and (69), (76) becomes

(77)

This relationship is always true for . Moreover,and lead to the same relationship; thus

(78)

(79)

(80)

where is the unique value of that has the same even-ness as . Moreover, as in Appendix B, using the fact that

for , we obtain

the necessary and sufficient conditions of orthogonality on thepolyphase components

for(81)

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers forvaluable comments that have led to improvements in this paperand Dr. Vahlin, who provided data files allowing to include itsown prototype function, i.e., the OFDP, in our comparisons.

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Pierre Siohan (M’94–SM’99) was born in Camlez,France, in October 1949. He received the Doctoratdegree from the École Nationale Supérieure desTélécommunications (ENST), Paris, France, in 1989and the Habilitation degree from the University ofRennes, Rennes, France, in 1995.

In 1977, he joined the Centre Commun d’Étudesde Télédiffusion et Télécommunications (CCETT),Rennes, where his activities were first concernedwith the communication theory and its application tothe design of broadcasting systems. Between 1984

and 1997, he was in charge of the Mathematical and Digital Signal ProcessingGroup. From September 1997 to August 2001, he was an Expert Member withthe Terrestrial Broadcasting and Radio Access Networks Laboratory: one ofthe R&D laboratories of France Télécom. He is currently Director of Researchwith the Institut National de Recherche en Informatique et Automatique(INRIA), Rennes. His current research interest are in the areas of filterbankdesign for communication systems and joint source-channel coding.

Cyrille Siclet was born in Champigny-sur-Marne,France, in September 1976. He received the engineerdegree in telecommunications from the ÉcoleNationale Supérieure des Télécommunications deBretagne (ENST Bretagne), Bretagne, France, andthe D.E.A. degree from the University of Rennes,Rennes, France, in 1999. He is pursuing the Ph.D.degree at the University of Rennes.

Since March 1999, he has been working for FranceTélécom R&D, Rennes.

Nicolas Lacaille was born in Paris, France, in1972. He received the engineer degree in electronicengineering from the École Supérieure d’Ingénieursen Electrotechnique et Electronique, Paris, France,in 1995. He received the D.E.A. degree from theUniversity of Rennes, Rennes, France, in 1998.

He joined the École Normale Supérieure deCachan in 1995. Since 1998 he has been withUniversité Technologique de Belfort Montbéliard,Belfort, France, where he is currently a Lecturerwith the Computer Sciences Department, where

he is mainly involved in computer architecture, signal processing, andtelecommunications.