Review Article Filter Bank Multicarrier Modulation: …downloads.hindawi.com/archive/2014/482805.pdfReview Article Filter Bank Multicarrier Modulation: A Waveform Candidate for 5G

Review ArticleFilter Bank Multicarrier ModulationA Waveform Candidate for 5G and Beyond

Behrouz Farhang-Boroujeny

ECE Department University of Utah Salt Lake City UT 84112 USA

Correspondence should be addressed to Behrouz Farhang-Boroujeny farhangeceutahedu

Received 7 August 2014 Accepted 11 November 2014 Published 21 December 2014

Academic Editor Gorazd Stumberger

Copyright copy 2014 Behrouz Farhang-BoroujenyThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Recent discussions on viable technologies for 5G emphasize on the need for waveforms with better spectral containment persubcarrier than the celebrated orthogonal frequency division multiplexing (OFDM) Filter bank multicarrier (FBMC) is analternative technology that can serve this need Subcarrier waveforms are built based on a prototype filter that is designed withthis emphasis in mindThis paper presents a broad review of the research work done in the wireless laboratory of the University ofUtah in the past 15 years It also relates this research to the works done by other researchers The theoretical basis based on whichFBMC waveforms are constructed is discussed Also various methods of designing effective prototype filters are presented Forcompleteness polyphase structures that are used for computationally efficient implementation of FBMC systems are introducedand their complexity is contrastedwith that ofOFDMTheproblems of channel equalization as well as synchronization and trackingmethods in FBMC systems are given a special consideration and a few outstanding research problems are identified Moreover thispaper brings up a number of appealing features of FBMC waveforms that make them an ideal choice in the emerging areas ofmultiuser and massive MIMO networks

1 Introduction

In the past orthogonal frequency division multiplexing(OFDM) has enjoyed its dominance as the most popularsignaling method in broadband wired [1 2] and wireless[3 4] channels OFDM has been adopted in the broad classof DSL standards as well as in the majority of wirelessstandards for example variations of IEEE 80211 and IEEE80216 3GPP-LTE and LTE-Advanced OFDM is known tobe a perfect choice for point-to-point communications forexample from a base station to a mobile node and viceversa It offers a minimum complexity and achieves very highbandwidth efficiency However it has been noted that OFDMhas to face many challenges when considered for adoptionin more complex networks For instance the use of OFDMin the uplink of multiuser networks known as OFDMA(orthogonal frequency division multiple access) requiresfull synchronization of the usersrsquo signals at the base stationinput Such synchronization was found to be very difficult toestablish especially in mobile environments where Doppler

shifts of different users are hard to predicttrack Morelli etal [5] have noted that carrier and timing synchronizationrepresents the most challenging task in OFDMA systems Tocombat the problem some researchers have relaxed on theneed for a close to perfect carrier synchronization amongusers and have proposed multiuser interference cancellationmethods [6ndash9] These methods are generally very complexto implement Their implementation increases the receivercomplexity by orders of magnitude [10] Hence one of themain advantages of OFDM the low complexity will be lost

Another limitation of OFDM appears when attempt ismade to transmit over a set of noncontiguous frequencybands known as carrier aggregation The poor response ofthe subcarrier filters in IFFTFFT filter banks of OFDMintroduces significant out-of-band egress noise to other usersand also picks up significant ingress noise from them Thesame problem appears if one attempts to adopt OFDM forfilling in the spectrum holes in cognitive radios Methodsof reducing OFDM spectral leakage prove to be very lim-ited in performance and may add significant complexity to

Hindawi Publishing CorporationAdvances in Electrical EngineeringVolume 2014 Article ID 482805 25 pageshttpdxdoiorg1011552014482805

2 Advances in Electrical Engineering

the transmitter For instance the side lobe suppressiontechniques like those proposed in [11ndash13] can achieve an out-of-band emission suppression of only 5 to 10 dB while theymay add significant complexity to the transmitter and theywill incur some loss in bandwidth efficiency

Filter bank multicarrier (FBMC) is an alternative trans-mission method that resolves the above problems by usinghigh quality filters that avoid both ingress and egress noisesAlso because of the very low out-of-band emission ofsubcarrier filters application of FBMC in the uplink ofmultiuser networks is trivial [14 15] It can be deployedwithout synchronization of mobile user nodes signals In theapplication of cognitive radios the filter bank that is used formulticarrier data transmission can also be used for spectrumsensing [16ndash20] On the other hand compared to OFDMFBMC falls short in handing multiple-input multiple-output(MIMO) channels although a few solutions to adopt FBMCin MIMO channels have been reported in the literature forexample see [21ndash23] Nevertheless as our recent researchstudy has shown (see Section 8 below) in the emerging areaofmassiveMIMO FBMC is found as powerful as OFDM andin some cases superior to OFDM

In the past many attempts have been made to adoptFBMC in various standards Apparently the earliest pro-posal to use FBMC for multicarrier communications is acontribution from Tzannes et al of AWARE Inc in oneof the asymmetric digital subscriber lines (ADSL) standardmeetings in 1993 [24] The proposed method that was calleddiscrete wavelet multitone (DWMT) was further studied in[25 26] Despite enthusiasm from the research community(see [27] and the cited references therein) DWMT was notadopted in the ADSL standardThis was partly because of theperceived complexity of this method as compared to its rivalDMT (discrete multitone an equivalent name for OFDM inDSL literature) Indeed the DWMT structure proposed byTzannes et al was significantly more complex than that ofDMT The major part of the complexity of DWMT camefrom the equalization method that was adoptedThe detaileddiscussion presented in [26] assumed that one needs anequalizer that combines signals from each subcarrier bandand its adjacent bands Typical equalizer lengths suggestedin [26] were 21 real-valued taps per subcarrier It was laternoted by the author of this paper that if each subcarrier issufficiently narrow such that it can be approximated by aflat gain two real taps per subcarrier would be sufficient forequalization [27] This observation led to further study ofDWMT In power line communications (PLC) communityit has been named wavelet OFDM and was adopted in theIEEE P1901 standard [28] The main motivation for useof DWMT in DSL and its adoption later in PLC was todeal with ingress and egress noises since both DSL andPLC use unshielded copper lines that are subject to strongradio interference Moreover in 1999 an FBMCmethod withnonoverlapping subcarrier bands was proposed as a solutionfor filtering the narrow-band interferences in very high-speedDSL (VDSL) channels [29] The proposed method was calledfiltered multitone (FMT) This proposal that was included asan annex in one of the initial draft documents of VDSL [30]was further developed by a number of researchers [31ndash34]

However to avoid incompatibility with ADSL FMT was notincluded in the final document of the VDSL standard [35]Another unsuccessful story is an attempt by France Telecom[36] to introduce FBMC in the IEEE 80222 a cognitiveradio standard to access TV bands in wireless rural areanetworks (WRAN) Up to now apparently the only standardfor radio transmission that uses FBMC is the TIAs DigitalRadio Technical Standard [37]

Recent discussions on the fifth generation (5G) wirelesscommunications have initiated a much stronger wave ofinterest in deviating from themain stream of OFDM systemsThis shift of interest is clearly due to limitations of OFDM inthe more dynamic and multiuser networks of future A num-ber of proposals have been made to adopt new waveformswith improved spectral containment A good example of suchactivity is the 5GNOW project in Europe which challengesLTE and LTE-Advanced in coping with the dynamic needsof 5G The 5GNOW has identified four alternative choicesof waveforms to better serve 5G needs These waveformsthat are all built based on some sort of filtering may bethought as adoptions of FBMC method to suit differentneeds of various applications We refer interested readersto the documents available at httpwww5gnoweu for thedetails of the proposed waveforms by the 5GNOW groupAnother major activity that has performed a broad study ofFBMC is due to the PYHDYAS project (also in Europe) ThePHYDYAS contributors have published heavily on variousaspects of FBMC including prototype filter design equaliza-tion synchronization and application to MIMO channels Asummary of themajor findings of the PHYDYAS contributorsis presented in [21] and a complete list of their publicationscan be found at httpwwwict-phydyasorg

Themain thrust of this paper is to present a point of viewof FBMC and its future applications as seen by the authorWhile we acknowledge the presence of a large body of workson FBMC the paper details are geared towards the researchoutcomes of the author and his students in past 15 yearsThe paper emphasis is on the recent works of the authorand his students Many shortcomings of OFDM in dealingwith the requirement of the next generation of wirelesssystems are discussed and it is shown how FBMC overcomesthese problems straightforwardly We present a derivation ofFBMC systems that reveals the relationships among differentforms of FBMC A method of designing FBMC systems fora near-optimum performance in doubly dispersive channelsis presented and its superior performance over OFDM isshown The example considered is an underwater acousticchannel Application of FBMC technique to massive MIMOcommunications is introduced and its advantages in thisemerging technology are revealed Last but not the leastthe problems of channel equalization and synchronizationin FBMC systems are also given a special treatment and anumber of outstanding research problems in this field forfuture studies are identified

This paper begins with a historical overview of FBMCmethods in Section 2 In order to keep the presentation in thispaper a complement to thosewe have recently reported in [1938] the rest of this paper is organized as follows A summaryof the theoretical background based on which the various

Advances in Electrical Engineering 3

forms of FBMC are built are presented in Section 3 Methodsof designing prototype filters for FBMC are presented inSection 4 Polyphase structures that are used for efficientimplementation of FBMC systems are reviewed in Section 5A few comments on complexity comparison of FBMC andOFDM are also made in this section Channel equalizationin FBMC systems is discussed in Section 6 Methods ofcarrier and symbol timing acquisition and tracking in FBMCsystems are reviewed in Section 7 Here a few particularfeatures of FBMC systems that need special attention fortheir successful implementation are highlighted Section 8reminds the reader of a number of applications in theliterature where FBMC has been found to be a good fit Thissection also discusses the opportunities offered by FBMC inthe emerging area ofmassiveMIMOThe concluding remarksof the paper are made in Section 9

Notations The presentations in this paper follow a mix ofcontinuous-time and discrete-time formulations as appro-priate While 119909(119905) refers to a continuous function of time 119905119909[119899] is used to refer to its discrete-time version with 119899 denot-ing the time index The notation 119891 is used as frequency vari-able in 119883(119891) the Fourier transform of the continuous-timesignal 119909(119905) and also as normalized frequency in119883(119890

1198952120587119891) the

Fourier transform of the discrete-time signal 119909[119899] We use ⋆

to denote linear convolution and the superscript lowast to denotecomplex conjugate

2 Review of FBMC Methods

FBMCcommunication techniqueswere first developed in themid-1960s Chang [39] presented the conditions required forsignaling a parallel set of PAM symbol sequences througha bank of overlapping filters within a minimum bandwidthTo transmit PAM symbols in a bandwidth-efficient man-ner Chang proposed vestigial sideband (VSB) signaling forsubcarrier sequences Saltzberg [40] extended the idea andshowed howChangrsquosmethod could bemodified for transmis-sion of QAM symbols in a double-sideband- (DSB-) modu-lated format In order to keep the bandwidth efficiency of thismethod similar to that of Changrsquos signaling Saltzberg notedthat the in-phase and quadrature components of each QAMsymbol should be time staggered by half a symbol intervalEfficient digital implementation of Saltzbergrsquos multicarriersystem through polyphase structures was first introducedby Bellanger and Daguet [41] and later studied by Hirosaki[42 43] Another key development appeared in [44] wherethe authors noted that ChangrsquosSaltzbergrsquos method could beadopted to match channel variations in doubly dispersivechannels and hence minimize intersymbol interference (ISI)and intercarrier interference (ICI)

Saltzbergrsquos method has received a broad attention in theliterature and has been given different names Most authorshave used the name offset QAM (OQAM) to reflect the factthat the in-phase and quadrature components are transmittedwith a time offset with respect to each other Moreover toemphasize the multicarrier feature of the method the suffixOFDM has been added hence the name OQAM-OFDM

Others have chosen to call it staggered QAM (SQAM) equiv-alently SQAM-OFDM In [38] we introduced the shortername staggered multitone (SMT)

Changrsquos method [39] on the other hand has receivedvery limited attention Those who have cited [39] have onlyacknowledged its existence without presenting much detailfor example [41 45 46] In particular Hirosaki who hasextensively studied and developed digital structures for theimplementation of Saltzbergrsquos method [43 47] has made abrief reference to Changrsquos method and noted that since ituses VSB modulation and thus its implementation requiresa Hilbert transformation it is more complex to implementthan Saltzbergrsquos method This statement is inaccurate sinceas we have demonstrated in [38] Changrsquos and Saltzbergrsquosmethods are equivalent and thus with aminormodificationan implementation for one can be applied to the other Moreon this is presented in Section 5 Also as noted earlier a vastliterature in digital signal processing has studied a class ofmulticarrier systems that has been referred to as DWMTIt was later noted in [27 48] that DWMT uses the sameanalysis and synthesis filter banks as the cosine modulatedfilter banks (CMFB) [49] CMFB on the other hand maybe viewed as a reinvention of Changrsquos method with a verydifferent application inmind [38] In [38] we also introducedthe shorter name cosine modulated multitone (CMT) to bereplaced for DWMT andor CMFB

One more interesting observation is that another class offilter banks which were called modified DFT (MDFT) filterbank has appeared in the literature [50] Careful study ofMDFT reveals that this although derived independently isin effect a reformulation of Saltzbergrsquos filter bank in discrete-time and with emphasis on compressioncoding The litera-ture onMDFTbegins with the pioneeringworks of Fliege [51]and later has been extended by others for example [52ndash55]

Finally before we proceed with the rest of our presenta-tion it should be reiterated that we identified three types ofFBMC systems (i) CMT built based on the original idea ofChang [39] (ii) SMT built based on the extension made bySaltzberg [40] and (iii) FMT built based on the conventionalmethod frequency division multiplexing (FDM) [34]

3 TheoryThe theory of FBMC particularly those of CMT and SMThas evolved over the past five decades by many researcherswho have studied them from different angles Early studiesby Chang [39] and Saltzberg [40] have presented their findingin terms of continuous-time signals The more recent studieshave presented the formulations and conditions for ISI andICI cancellation in discrete-time for example [46 50] Onthe other hand a couple of recent works [19 38] from theauthor of this paper and his group have revisited the moreclassical approach and presented the theory of CMTand SMTin continuous time It is believed that this formulation greatlysimplifies the essence of the theoretical concepts behind thetheory of CMT and SMT and how these two waveforms arerelated It also facilitates the design of prototype filters thatare used for realization of CMT and SMT systems Thushere also we follow the continuous-time approach of [19 38]


f

T

tminus2Tn = minus2

minusTn = minus1

Tn = 1

2Tn = 2

minus1205872

minus1205872

minus1205872

120587

120587

120587

120587

1205872

1205872

0

0

minus1205872

minus1205872 120587

1205872

1205872

1205872

0

0

0

middot middot middot








F = 12T

f = 74T k = 3

f = 54T k = 2

f = 34T k = 1

f = 14T k = 0

Figure 1 The CMT time-frequency phase-space lattice

t = 0

f14T 34T 54T 74T


(a)

t = T

f14T 34T 54T 74T


(b)

Figure 2 Magnitude responses of the CMT pulse-shaping filters at various subcarriers and time instants 119905 = 0 and 119905 = 119879

Moreover to give a complement presentation to those of[19 38] an attempt is made to discuss the underlyingtheory mostly through the time-frequency phase-space withminimum involvement in mathematical details It is believedthat this presentation also provides a new intuition into therelationship between SMT and CMT Interested readers whowish to see the mathematical details are referred to [20 38]

31 CMT In CMT data symbols are from a pulse amplitudemodulated (PAM) alphabet and hence are real-valuedTo establish a transmission with the maximum bandwidthefficiency PAM symbols are distributed in a time-frequencyphase-space lattice with a density of two symbols per unitarea This is equivalent to one complex symbol per unit areaMoreover because of the reasons that are explained below a90-degree phase shift is introduced to the respective carriersamong the adjacent symbols These concepts are presentedin Figure 1 Vestigial side-band (VSB) modulation is appliedto cope with the carrier spacing 119865 = 12119879 The pulse-shapeused for this purpose at the transmitter as well as for matchedfiltering at the receiver is a square-root Nyquist waveform119901(119905) which has been designed such that 119902(119905) = 119901(119905)⋆119901(minus119905) is aNyquist pulse with regular zero crossings at 2119879 time intervals

Also 119901(119905) by design is a real-valued function of time and119901(119905) = 119901(minus119905) These properties of 119901(119905) as demonstratedbelow are instrumental for correct functionality of CMT

Figure 2 presents a set of magnitude responses of themodulated versions of the pulse-shape 119901(119905) for the datasymbols transmitted at 119905 = 0 and 119905 = 119879 The colors usedfor the plots follow those in Figure 1 to reflect the respectivephase shifts

Let 119899 = minus2 minus1 0 1 2 denote symbol time indexlet 119896 = 0 1 2 denote symbol frequency index let 119904119896[119899]denote the (119899 119896) data symbol in the time-frequency latticeand let 120579119896[119899] = (119896 minus 119899)(1205872) be the phase shift that is addedto the carrier of 119904119896[119899] Accordingly a CMT waveform that isconstructed based on the pulse-shapeprototype filter 119901(119905) isexpressed as

119909 (119905) = sum

119899

sum

119896

119904119896 [119899] 119886119899119896 (119905) (1)

where

119886119899119896 (119905) = 119890119895120579119896[119899]

119901119899119896 (119905)

119901119899119896 (119905) = 119901 (119905 minus 119899119879) 119890119895((2119896+1)1205872119879)119905

(2)


The synthesis of 119909(119905) according to (1) has the followinginterpretations The terms 119886119899119896(119905) may be thought as a set ofbasis functions that are used to modulate the data symbols119904119896[119899] The data symbols 119904119896[119899] can be extracted from 119909(119905)

straightforwardly if 119886119899119896(119905) are a set of orthogonal basisfunctions The orthogonality for a pair of functions V1(119905) andV2(119905) in general is defined as

⟨V1 (119905) V2 (119905)⟩ = int

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (3)

For the case of interest here where the data symbols 119904119896[119899] arereal-valued the orthogonality definition (3) can be replacedby the more relaxed definition

⟨V1 (119905) V2 (119905)⟩119877 = Rint

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (4)

where Rsdot indicates the real part Definition (4) is referredto as real orthogonality

It is not difficult to show that

⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877=

1 119899 = 119898 119896 = 119897

0 otherwise(5)

and hence for any pair of 119899 and 119896

119904119896 [119899] = ⟨119909 (119905) 119886119899119896 (119905)⟩119877 (6)

To develop an in-depth understanding of the CMTsignaling it is instructive to explore a detailed derivation of(5) To this end we begin with the definition

⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895120579119896[119899]

119901119899119896 (119905) 119890minus119895120579119897[119898]

119901lowast119898119897 (119905) 119889119905

(7)

and note that this can be rearranged as

⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895(119898minus119899+119896minus119897)(1205872)

119901 (119905 minus 119899119879)

times 119901 (119905 minus 119898119879) 119890119895((119896minus119897)120587119879)119905

119889119905

(8)

When 119898 = 119899 and 119896 = 119897 after a change of variable 119905 to119905 + 119899119879 (8) reduces to

⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= int

infin

minusinfin

1199012(119905) 119889119905

= 1

(9)

where the second equality follows from the fact that 119901(119905) is areal-valued square-root Nyquist pulse and 119901(119905) = 119901(minus119905)

When 119896 = 119897119898 = 119899 and119898minus 119899 = 2119903 where 119903 is an integer

⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= (minus1)

119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(10)

where the second equality follows since 119901(119905) is a square-rootNyquist pulse designed for a symbol spacing 2119879 On the otherhand when 119896 = 119897 but119898 minus 119899 = 2119903 + 1

⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(11)

Next consider the case where 119896 minus 119897 = 1 and 119898 minus 119899 = 2119903In that case one finds that

⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119890119895(120587119879)119905

119889119905

= minus (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) sin(

120587

119879

119905) 119889119905

= 0

(12)

where the last identity follows by applying the change ofvariable 119905 rarr 119905 + ((119898 + 119899)2)119879 and noting that theexpression under the integral will reduce to an odd functionof 119905 Following similar procedures it can be shown that thereal orthogonality ⟨119886119899119896(119905) 119886119898119897(119905)⟩119877 = 0 also holds when119896 minus 119897 = 1 and 119898 minus 119899 is an odd integer and when 119896 minus

119897 = minus1 and 119898 minus 119899 is either an even or odd integer Finallyfor the cases where |119896 minus 119897| gt 1 the real orthogonality⟨119886119899119896(119905) 119886119898119897(119905)⟩119877 = 0 is trivially confirmed by noting that theunderlying basis functions correspond to filters that have nooverlapping bands The stop-band quality of the frequencyresponse of the prototype filter 119901(119905) determines the accuracyof the equality ⟨119886119899119896(119905) 119886119898119897(119905)⟩119877 = 0 when |119896 minus 119897| gt 1

It is also worth noting that some of the recent derivationsof FBMC that are presented in discrete-time design therespective prototype filter 119901[119899] such that the respective realorthogonality is perfectly satisfied for the cases where |119896 minus

119897| gt 1 as well for example [56] However one mayrealize that in practice the presence of channel destroys theorthogonality of the basis functions The orthogonality ofthe basis functions is commonly recovered at the receiverusing per subcarrier equalizers see Section 6 below Suchequalizers unfortunately will not be able to recover theorthogonality of basis functions that belong to nonadjacentsubcarriers Hence the design of a 119901[119899] that satisfies the realorthogonality for |119896 minus 119897| gt 1may be a waste

To summarize the above derivations revealed that thefollowing settings of the CMT parameters lead to the realorthogonality of the basis functions 119886119899119896(119905) and allow symbolplacement in the time-frequency lattice at the maximumdensity of two real symbols per unit area

(1) The symbol spacing 119879 along the time axis should bematched with the subcarrier spacing 119865 = 12119879 alongthe frequency axis

(2) The pulse-shapeprototype filter 119901(119905) must be a real-valued square-root Nyquist filter for a symbol spacing


s0(t)

s1(t)

sNminus1(t)

SFB

sum

Modulationto RF band

ej2120587f119888t

Channel

eminusj2120587f119888t

Demodulationfrom RF band

AFBs0[n]

s1[n]

sNminus1[n]

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

ej((120587T)t+(1205872))

ej(Nminus1)((120587T)t+(1205872))

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

eminusj((120587T)t+(1205872))

eminusj(Nminus1)((120587T)t+(1205872))

y(t)x(t)

Rmiddot

Rmiddot

Rmiddot

RmiddotRmiddot

Figure 3 CMT transmitter and receiver blocks

2119879 It should be also a linear phase hence whenviewed as a zero phase filter it satisfies the condition119901(119905) = 119901(minus119905) The latter is not a necessary conditionbut is satisfied in most of the designs see [38] for amore relaxed condition

(3) The above constraints are applied to assure orthog-onality of the basis functions that are within thesame subcarrier or adjacent subcarriers only Theorthogonality of basis functions that belong to thenonadjacent subcarriers is guaranteed by virtue ofthe fact that they correspond to filters with nonover-lapping bands A more advanced design presentedin Section 42 allows overlapping of the nonadjacentbands and yet satisfies the orthogonality condition

(4) The phase shifts indicated in Figure 1 can be modifiedto other choices as long as a phase difference plusmn1205872 ispreserved between each pair of adjacent points in thelattice

(5) Although in CMT the position of the lattice pointsis fixed these points can be moved in the time-frequency plane as long as their relative positionand phase differences remain unchanged We usethis point below to arrive at the SMT waveform byapplying a simplemodification to the CMTwaveform(1)

The above equations may be combined to arrive at theCMT transmitter and receiver structures that are presentedin Figure 3 As shown a synthesis filter bank (SFB) is usedto construct the transmit signal and the received signal ispassed through an analysis filter bank (AFB) to separate thedata streams of different subcarrier bands Here it is assumedthat there are119873 subcarrier streams and the data stream of the119896th subcarrier at the input to the SFB is represented by theimpulse train

119904119896 (119905) = sum

119899

119904119896 [119899] 120575 (119905 minus 119899119879) (13)

It is also worth noting that in practice the analyzed signalsat the output of the AFB should be equalized Here to keep

the presentation simple we have not included the equalizersEqualizer details are presented in Section 6

32 SMT SMT may be thought of as an alternative to CMTIts time-frequency phase-space lattice is obtained from thatof CMT (Figure 1) through a frequency shift of the latticepoints down by 14119879 scaling the time axis by a factor 12 andhence the frequency axis by a factor 2 Moreover to remainconsistent with the past literature of SMT (equivalentlyOQAM-OFDM) some adjustments to the carrier phases ofthe lattice points have been made This leads to the time-frequency phase-space lattice that is presented in Figure 4The magnitude responses of the SMT pulse-shaping filters atvarious subcarriers and time instants 119905 = 0 and 119905 = 1198792 arepresented in Figure 5 Note that here the PAM symbols arespaced by 1198792 and subcarriers are spaced by 1119879 In SMTeach pair of adjacent symbols along time in each subcarrieris treated as real and imaginary parts of a QAM symbolThis leads to the transmitter and receiver structures that arepresented as in Figure 6Here each data symbol 119904119896[119899] belongsto a QAM constellation and thus may be written in terms ofits in-phase and quadrature components as 119904119896[119899] = 119904

119868119896[119899] +

119895119904119876

119896[119899] Accordingly the inputs to the SFB in Figure 6 are

119904119868119896 (119905) = sum

119899

119904119868119896 [119899] 120575 (119905 minus 119899119879)

119904119876

119896 (119905) = sum

119899

119904119876

119896 [119899] 120575 (119905 minus 119899119879)

(14)

33 FMT FMT waveforms are synthesized following theconventional method of frequency division multiplexing(FDM) The subcarrier channels have no overlap and thusICI is resolved through use of well-designed filters with highstopband attenuation ISI may be compensated for by adopt-ing the conventional method of square-root Nyquist filteringthat is used in single-carrier communications For doublydispersive channels we have adopted amore advanced design[57] that compensates for both ICI and ISI through aneffective method which is presented in Section 42

For comparison with the CMT and SMT structures inFigures 3 and 6 respectively and also as a basis for further


f

F = 1T

minusT2minusT T

T2

T2

t

n = minus2 n = minus1 n = 1 n = 2

minus1205872

minus1205872120587

1205871205872

12058720

0

minus1205872

120587

1205872

0minus1205872

120587

1205872

0

minus1205872

120587

1205872

0









f = 0 k = 0

f = 3T k = 3

f = 2T k = 2

f = 1T k = 1

Figure 4 The SMT time-frequency phase-space lattice

t = 0

1T 2T 3T f


(a)

t = T2

1T 2T 3T f


(b)

Figure 5 Magnitude responses of the SMT pulse-shaping filters at various subcarriers and time instants 119905 = 0 and 119905 = 1198792

development in later parts of this paper Figure 7 presentsthe structures of transmitter and receiver blocks in an FMTcommunication system Also themagnitude responses of theFMTpulse-shaping filters at various subcarriers are presentedin Figure 8 Note that here the subcarrier spacing is equalto (1 + 120572)119879 where 120572 is the roll-off factor of the pulse-shapingprototype filter Accordingly FMT has a symboldensity of 1(1+120572) complex symbols per unit area in the time-frequency plane Hence FMT is less bandwidth efficient thanCMT and SMT

4 Prototype Filter Design

The results of the previous section suggest that linear phasesquare-root Nyquist filters that have been widely used forsingle carrier data transmission are the most trivial choicefor prototype filter in FBMC systems This indeed remainsan accurate statement as long as the underlying channelis time-invariant or varies slowly However in cases wherethe channel is fast varying a more general class of square-root Nyquist filters that satisfy the Nyquist condition bothalong the time and frequency axis should be adopted In thissection we first review couple of classical Nyquist designs

developed by us [58] and others [59] and then discussan example of more advanced designs that also have beendeveloped in our research group [57] and is more appropriatefor time-varying channels

41 Prototype Filters for Time-Invariant Channels The clas-sical prototype filter for FBMC systems that was suggested in[39 40]was the square-root raised-cosine (SRRC) filterMorespecifically both [39 40] suggested using a SRRC filter withthe roll-off factor 120572 = 1 In practice where FBMC systemsare implemented in discrete-time SRRC response should besampled and truncatedThe truncation of the SRRC responsemay result in a filter with poor frequency response andhence makes SRRC a poor choice Advancements in digitalfilters design have led to very effective methods for designingsquare-root Nyquist (SR-Nyquist) filters for any specifiedfinite length These designs two of which are presented hereaim at balancing the stopband attenuation and the Nyquistproperty of the designs

Martin [59] has proposed a design method whose goal isto satisfy theNyquist criterion approximately while achievinggood attenuation in the stopband A nice property of thismethod is that when the number of samples per symbol


sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot

ej(Nminus1)((2120587T)t+(1205872)) eminusj(Nminus1)((2120587T)t+(1205872))

Rmiddot

Figure 6 SMT transmitter and receiver blocks

s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot

Figure 7 FMT transmitter and receiver blocks

f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T

Figure 8 Magnitude responses of an FMT pulse-shaping filter atvarious subcarriers

interval is 119873 and the filter length is 120573119873 + 1 where 120573 is aninteger greater than 1 there are only120573 parameters that need tobe found for the optimization of the designMore specificallythe prototype filter is constructed as

119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)

where the coefficients 1198960 through 119896120573minus1 are to be optimizedThe optimum choices of the coefficients 1198960 through 119896120573minus1 for

different values of 120573 are tabulated in [59] and also in [60]Beside being a simple and effective method it is important tonote that Martinrsquos design for 120573 = 4 has been adopted by thePHYDYAS project [21]

More recently we have developed an algorithm fordesigning SR-Nyquist filters that can balance between theaccuracy of the Nyquist constraints and the filter stopbandattenuation [58] To select the samples of the zero-phaseimpulse response 119901[119899] = 119901[minus119899] of the SR-Nyquist filter thisalgorithm defines

119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)

and minimizes the cost function

120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)

with respect to the elements 119901[119899] The first two terms on theright-hand side of (17) are to enforce Nyquist property on119902[119899] and the third term is included tominimize the stopbandresponse of 119901[119899] Also the length of 119901[119899] is assumed to be


SRRC4Martin4Farhang4

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency

Figure 9 Magnitude responses of a sampled and truncated SRRCfilter and two discrete-time designed SR-Nyquist filters

equal to 120573119873 + 1 One may note that since 120585 is a forth-orderfunction of the parameters 119901[119899] it is a multimodal functionand hence its minimization may not be straightforwardHowever fortunately if 119901[119899] could be initialized near itsoptimum choice an iterative solution may be applied tosearch for the local minimum of 120585 The search algorithmproposed in [58] follows a similar procedure to the oneoriginally suggested in [61]

Figure 9 presents the magnitude responses of threedesigns based on (i) a sampled and truncated version ofthe impulse response of a SRRC filter SRRC4 (ii) a SR-Nyquist design obtained following [59] Martin4 and (iii)a SR-Nyquist design obtained following [58] Farhang4 Alldesigns are based on the roll-off factor 120572 = 1 and are for anFBMC system with 119873 = 16 subcarriers and the suffix 4 onthe names indicates that for all designs 120573 = 4 that is thedesigned filters have a length 120573119873 + 1 = 4 times 16 + 1 = 65 Inaddition for Farhang4 the parameter 120574 (defined in [58]) isset equal to 01

As seen and one would expect SRRC design performssignificantly inferior to both of the SR-Nyquist designs Fromthe two SR-Nyquist designs the one based on [58] achievesa higher attenuation in the first few sidelobes of its stopbandthan the design based on [59]The first sidelobe of the formerdesign is 10 dB lower Also a study of the time domainresponses of the designs reveals that none of them satisfiesthe Nyquist property perfectly Hence all are near Nyquistdesigns However the design of [59] is the closest to Nyquistfollowed by the design delivered by [58] and then the SRRCdesign Nevertheless all the three designs are very close toNyquist hence the choice of one against the others may bedominantly determined by its superior magnitude response

42 Prototype Filters for Time-Varying Channels When achannel is subject to dispersion both in time (due to multi-path effects) and in frequency (due to variation of the channel

in time) we say the channel is doubly dispersive We also notethat while the dominant effect of time dispersion in a channelis ISI the dominant effect of frequency dispersion is ICIHence in order to design prototype filters that address bothtime and frequency dispersions it has been argued [44] thatone should choose a pulse-shape 119901(119905) with similar behavioralong time and frequency In particular if 119901(119905) is selected tobe a SR-Nyquist along the time axis it should be also madesure that 119875(119891) is a SR-Nyquist along the frequency axis Tothis end the pulse-shape 119901(119905) with the following propertymay be adopted

119875 (119891) = 119901 (120578119891) for a constant scaling factor 120578 (18)

that is a function that has the same form in both time andfrequency domains

The parameter 120578 in (18) is related to the symbol spacingin time 119879 and frequency 119865 and is given by

120578 =

119879

119865

(19)

Also as one may understand intuitively 119879 and 119865 arerespectively chosen proportional to time dispersion Δ120591 andfrequency dispersion Δ] of the channel that is 119879119865 =

Δ120591Δ] Hence the following identity also holds

120578 =

Δ120591

Δ] (20)

The definitions for Δ120591 and Δ] are usually vague The timedispersion Δ120591 may be thought of as a coarse estimate of theduration of the channel impulse response equivalently thespan of themultipaths of the channel Similarly the frequencydispersion Δ] may be thought of as a coarse estimate of thespan of Doppler spread of the channel

The design of the prototype filter for time-varying chan-nels is closely tied to the ambiguity function [62ndash64]

119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2

)119901lowast(119905 minus

120591

2

) 119890minus1198952120587]119905

119889119905 (21)

where 120591 is a time delay and ] is a frequency shiftLet 119901(119905) be a prototype filter and let 119901119899119896(119905) be a time

frequency translated version of it which is defined as

119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)

whereprop indicates proportionate toThe proportionate factoris a phase shift due to the delays of 119898119879 and 119899119879 whose valueis irrelevant to our discussion here Using (23) one may notethat 119901119899119896(119905) of (22) for all choices of 119899 and 119896 will form a set of


complex-valued orthogonal basis functions if 119901(119905) is chosensuch that

119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)

We note that the case where ] = 0 corresponds to thefamiliar (time) correlation function

119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)

which reduces to the Nyquist constraints

119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)

Also one may note that the integral on the right-hand sideof (25) is equal to the convolution of 119901(119905) and its matchedpair 119901(minus119905) evaluated at time 119905 = 120591 Thus 119860119901(119905 0) = 119901(119905) ⋆

119901(minus119905) and hence (26) implies that 119901(119905) is a SR-Nyquist pulseMoreover the ambiguity function119860119901(120591 ])may be thought ofas a generalization of the correlation function119860119901(120591 0)wherecorrelation is found between 119901(119905) and its modulated versionat the frequency 119891 = ] Accordingly we refer to the set ofconstraints (24) as the generalized Nyquist constraints

Another key point that one should consider in theselection of 119901(119905) for time-varying channels is minimizationof its duration Le Floch et al [44] have noted that tomaximize the density of the basis functions 119901119899119896(119905) in thetime-frequency space and hence maximize the bandwidthefficiency of the transmission one should choose a 119901(119905) thathas maximum compactness in the time-frequency spaceMaximum compactness on the other hand is quantifiedby the product 120590119905120590119891 where 120590

2119905 and 120590

2119891 are respectively

the second-order moments of the functions 119901(119905) and 119875(119891)Moreover the Heisenberg-Gabor uncertainty principle statesthat [65]

120590119905120590119891 ge1

4120587

(27)

where the equality holds only when 119901(119905) is the Gaussian pulse119892(119905) = 119890

minus1205871199052 Also the Gaussian pulse 119892(119905) has the interestingproperty that 119866(119891) = 119892(119891) that is it satisfies the desirableproperty (18) with 120578 = 1 However 119901(119905) = 119892(119905) does notsatisfy the orthogonality conditions (24)

Attempts to design filters that satisfy the orthogonal-ity conditions (24) and at the same time approach theHeisenberg-Gabor uncertainty lower bound (27) as close aspossible have been made and design methods have beendeveloped [44 66] The design presented in [44] is calledisotropic orthogonal transform algorithm (IOTA) filter IOTAdesignalgorithm was first introduced in a patent by Alard[67] The designs proposed in [66] on the other hand arereferred to as Hermite pulses reflecting the fact that theirconstruction is based on a linear combination of a set ofHermite functions In the rest of this section we limit ouremphasis to the design of Hermite pulses and emphasize theflexibilities that these designs provide in adopting to doubly

dispersive channels More details on this topic can be foundin [57] (also see [68 69]) Discussion on IOTA design can befound in [19 20 44 70]

The design procedure proposed byHaas and Belfiore [66]constructs an isotropic filter according to the equation

119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)

where ℎ119899(119905) is the set of Hermite functions defined as

ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)

Note that ℎ0(119905) = 119892(119905) and thus it is an isotropic functionwith parameter 120578 = 1 Moreover it can be shown thatthe set of functions ℎ119899(119905) for 119899 = 4119896 119896 = 1 2 arealso isotropic with the same parameter This implies thatthe construction (28) for any set of coefficients 120572119896 leads toan isotropic function In [66] the coefficients 120572119896 have beencalculated to construct a filter 119901(119905) that satisfies the set ofconstraints (24) for parameters 119879 = 119865 = radic2

Haas and Belfiorersquos design [66] allows transmission ofQAM symbols with a density of 1119879119865 = 05 symbol per unitarea in the time-frequency spaceThis design of119901(119905)may alsobe used as the prototype filter in a CMT or SMT structureto increase the density to 2 PAM symbols per unit areaequivalent to one QAM symbol per unit area Alternativelyone can aim for a design with larger density than 05 and staywith transmitting QAM symbols Examples of both designsare presented later In the rest of this section we follow theapproach of [57] to present a broad class of Hermite filterdesigns that includes the design presented in [66] as a specialcase

The basic equations for the design of Hermite pulses areobtained by substituting (28) in (21) This gives

119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[

11986000 (120591 ]) 11986001 (120591 ]) sdot sdot sdot 1198600119871 (120591 ])11986010 (120591 ]) 11986011 (120591 ]) sdot sdot sdot 1198601119871 (120591 ])

d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)


(30) may be rearranged as

119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)

(1) Haas and Belfiore Design In [66] it was proposed that thecoefficients 1205720 through 120572119871 can be determined by substituting(33) into (24) for (119899 119897) = (0 0) and 119871 other significantchoices of (119899 119897) and solving the resulting system of equationsIt was numerically demonstrated that this leads to very gooddesigns To further clarify this procedure and pave theway foradditional developments in the sequel we discuss themethodof [66] in the context of a specific design Figure 10 presentsthe grid of all choices of (119899 119897) Here it is assumed that theconstraints (24) are applied at the origin and the 12 nearestgrid points to it In this figure three different sets of gridpoints are identified

(1) The significant points are the ones at which theconstraints (24) should be imposed There are foursuch points and these are indicated by red disks

(2) Once the desired constraints are imposed at thesignificant points defined in (1) it follows from theeven symmetry and isotropic property of 119901(119905) that thesame constraints will automatically be imposed at therest of the points indicated by green disks

(3) The remaining grid points indicated by white diskswill satisfy the constraints (24) within a good approx-imation since the designed pulse 119901(119905) decays expo-nentiallyfast as |119905| increases

We note that to satisfy the constraints (24) at the originand the 12 nearest grid points to it it is sufficient to applyconstraints to only 3 of the latter points A unique design isthus obtained if the design ismade based on ℎ0(119905) ℎ4(119905) ℎ8(119905)and ℎ12(119905) that is the choice of 119871 = 3 in (30) through (33)

It should be also noted that since here the designed filteris isotropic with the parameter 120578 = 1 the identity 119879 = 119865

should hold Hence given a desired density 119863 = 1119879119865 thedesign must be for the parameters 119879 = 119865 = 1radic119863 Once 119901(119905)is obtained based on these parameters applying a time scalingfactor one can set 119879 to any desired valueThis will set 119865 equalto 1119879119863

(2) Hexagonal LatticeThe spread of data symbols in Figure 10and other lattice grids that have been presented in this paperso far follows an orientation that is referred to as rectangularlattice In [71] it is noted that better designs may be obtainedby adopting the hexagonal orientation The reason behindthis improvement in performance follows the fact that thehexagonal orientation allows maximum separation of pointsfor a given symbol density Interested readers are referred tothe detailed discussions in [19 20 71]

To design prototype filters for an orientation that followsthe hexagonal lattice we choose the constrained pointsaccording to those depicted in Figure 11 It should be notedthat we set119879 = 2119865 and for this choice it is sufficient to enforcedesign constraints at the points indicated by red disks As in

f

2F

F

Tt

2T

minus2F



middot middot middot middot middot middot







minus2T minusT

minusF

Figure 10 Grid of (119899 119896) points at which the constraints (24)should be imposed The red disks indicate the points at which theconstraints (24) are imposed The green disks follow the red onesthanks to symmetric and isotropic property of the design The restof the points (white disks) satisfy the constraints within a goodapproximation since the construction is based on exponentiallydecaying pulses as |119905| increases

the case of rectangular lattice once the constraints are appliedto these points similar constraints will be automaticallyimposed to the points indicated by green disks following thesymmetric and isotropic property of the designsMoreover asin the case of rectangular lattice by design the remaining gridpoints indicated by white disks will satisfy the constraints(24) within a good approximation Finally a time scaling canbe applied to 119901(119905) to set 119879 and accordingly 119865 to any desiredvalue

(3) Robust Design Building on the above findings we notedthat in a practical design the time and frequency dispersionintroduced by a channel smear the null points of the ambi-guity function 119860119901(120591 ]) [57 68 69] Hence as a result of thechannel dispersion the nulls will convert to shallow nulls Itis thus argued that instead of designing 119901(119905) to introduceperfect nulls in 119860119901(120591 ]) a more robust design is obtained byaiming for a design that results in deep (but imperfect) nullareas around the grid points (119899119879 119896119865) To this end a robustHermite pulse is designed by minimizing the cost function

120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)

where A0 is an area around the origin and A1 through A119871are the null areas that are aimed for


f

4F

2F

F

Tt

2T 4T

minus2F









Figure 11 Grid of (119899 119896) points for hexagonal orientation The reddisks indicate the points at which the constraints (24) are imposedThe green disks follow the red ones thanks to symmetric andisotropic property of the design The rest of the points (whitedisks) satisfy the constraints within a good approximation sincethe construction is based on exponentially decaying pulses as |119905|

increases

To develop a numerical method for the minimization of120577 the ambiguity function is sampled along the time axis 120591 andthe frequency axis ] This converts (34) to

120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)

where 119879119904 and 119865119904 are the sampling periods along the time axisand frequency axis respectively and

119906119896 =

1 119896 = 0

0 119896 = 0

(36)

Removing the third term on the right-hand side of (17)one may realize that the cost functions 120585 and 120577 are similarBoth are fourth-order functions of the parameters that weseek to optimize Hence the iterative method developed in[58] can be readily adopted here aswell For this purpose as in[58] an initial guess for the elements of the parameter vector120572 should be made It has been noted in [57] that a properinitial choice for 120572 is the vector whose first element is 1 andthe rest of its elements are 0 That is one should start withthe Gaussian pulse as a first guess and add the higher orderHermite functions in the subsequent iterations

(4) Numerical Examples To conclude this section we presentthe results arising from a few designs of IOTA and Hermiteprototype filters Also we compare the results with those ofthe SR-Nyquist design of [58]

Figure 12 presents the time domain and the magnitude offrequency domain responses of (i) an IOTA design [67] (ii)a Hermite design [66] and (iii) a SR-Nyquist design [58] All

designs have the samefilter length of 4119879 and designed to servea CMTSMT systemwith amaximum of 16 subcarrier bands

Considering the results presented in Figure 12 one maymake the following observations

(1) Both IOTA and Hermite designs have time-domainresponses that are more compact than the time-domain response of the SR-Nyquist filter

(2) In the frequency domain on the other hand the SR-Nyquist design gives a more compact response

(3) The compact responses of IOTA and Hermite designsin time will make them less prone to distortionintroduced by channel variation with time

(4) Thebroader responses of IOTAandHermite design infrequency on the other hand make themmore proneto the channel frequency selectivity

(5) Considering (3) and (4) a balance has to be madebetween the choice of the symbol interval 119879 and thecarrier spacing 119865 Obviously by applying a propertime scaling to the prototype filter 119901(119905) such balancecan be made

(6) When channel is time-invariant or varies slowlythe choice of SR-Nyquist results in the maximumimmunity to the channel frequency selectivity

The robust design approach that was presented abovewas first introduced in [69] and was further studied in [5768] Here to emphasize on the significance of this designapproach we present Figure 13 from [57]This figure presentsa set of results that compare the signal-to-interference ratio(SIR) of the robust isotropic designs (named FMT-dd FMTfor doubly dispersive channels) with OFDM and an FMTdesign according to the SR-Nyquist design of [58] (namedFMC-c conventional FMT) To measure SIR the channelnoise is set equal to zero Three sets of results correspondingto density values119863 = 1119879119865 = 12 23 and 45 are presentedThe robust isotropic designs are obtained for a channelmodelin which dispersion in time and frequency is uniform in theintervals (minus1205751205912 1205751205912) and (minus120575]2 120575]2) respectively andΔ120591 = 02119879 and Δ] = 02119865 The design for each densityis fixed and its performance is examined for varying valuesof Δ120591Δ] in the range of 0 to 01 For FMT-c the prototypefilter is designed in each of the three cases with the aim ofachieving a stopband attenuation of 60 dB or better Alsofollowing the basic principle of FMT-c the roll-off factors ofthe prototype filters for density values 119863 = 12 23 and 45

are set equal to 1 05 and 025 respectively For OFDM thedensity 119863 = 1119879119865 is set by adjusting the ratio of cyclic prefixlength 119879CP over the length of FFT 119879FFT In particular wenote that since in OFDM 119865 = 1119879FFT and 119879 = 119879CP + 119879FFT119863 = 119879FFT(119879CP + 119879FFT) and thus 119879CP119879FFT = 1119863 minus 1

The results presented in Figure 13 clearly show theexpected superior performance of the robust designs overOFDM and the conventional FMT designThemore detailedresults presented in [57] for an underwater acoustic (UWA)communication further confirm these results in an at-seaexperimental setting


035

03

025

02

015

01

005

0

minus005minus2 minus15 minus1 minus05 0 05 1 15 2

tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)

Figure 12 Responses of three designs of a prototype filter (a) The time-domain responses (b) The magnitude of the frequency-domainresponses

50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ

FMT-dd hexagonalFMT-dd rectangular

FMT-cOFDM

D = 12

D = 23

D = 45

Figure 13 The impact of a doubly dispersive design and threedesigns of FMT ldquoFMT-crdquo is the case where a SR-Nyquist design isused ldquoFMT-ddrdquo follows the robust design that was introduced above(the author obtained permission from IEEE to include this figure)

5 Polyphase Structures andComputational Complexity

Polyphase structures are commonly used to implementFBMC transmitter and receiver Direct mimicking of thecontinuous-time structures of Figures 3 and 6 to imple-ment CMT and SMT respectively may lead to structureswhose complexity is not optimized One polyphase structureper each set of real symbols equivalent to two polyphase

structures per each pair of real symbol sets should beimplemented This section highlights the fact that the twopolyphase structures at the transmitter side can be combinedtogether and hence reduce the complexity to one half At thereceiver side special attention has to be paid so that properequalizers can be applied to the analyzed signal componentsThepresentation in this section although results in structureswith the same complexity to those already published arrivesat structures inmore intuitive way and thus hopefully assiststhe reader to have a better grasp of the concepts

Polyphase structures for FMT transmitter and receiveralso need particular attention to take care of the fact thatsampling rate changes in the AFB (at the transmitter side)and in the SFB (at the receiver side) is different from the sizeof IFFTFFT block A method that takes into account thischange of sampling rate is presented in a later part of thissection

51 Polyphase Analysis and Synthesis Filter Banks The basicpolyphase SFB block that may be used to construct theSFB blocks in both CMT and SMT transmitter is presentedin Figure 14 This block which is widely available and welldeveloped in the literature for example see [49 72] has thefollowing characteristics

(1) The inputs 119904119896[119899] 119896 = 0 1 119873 minus 1 are a set of datasequences with the nominal rate of unity

(2) The SFB output has a rate of 119871 or equivalently 119871 timesfaster than the rate at the input

(3) The synthesis is performed based on an IFFT of size119871 gt 119873 with the inputs of index 119873 and greater setequal to zero This is to allocate some guard bandbetween the generated baseband signal and its images


x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

Figure 14 Basic synthesis polyphase filter bank

and hence facilitate additional filtering in the furtherstages of the transmitter

(4) The filtering blocks 1198640(119911) through 119864119871minus1(119911) are thepolyphase components of the prototype filter 119901[119899]

(5) This structure is an efficient implementation of a bankof filters with the respective inputs 1199040[119899] 1199041[119899]

119904119873minus1[119899] and the transfer functions 119875(1198901198952120587119891

)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)

is the Fourier transform of 119901[119899](6) The commutator at the output serializes the outputs of

the polyphase component filters after arrival of eachset of data symbols at the input

Figure 15 presents the basic polyphase AFB that matcheswith the SFB of Figure 14 Here 119877119896(119911) = 119864119871minus119896minus1(119911) In theliterature 119864119896(119911) and 119877119896(119911) are distinguished by referringto them as type I and type II polyphase components [49]When 119909[119899] is fed to its input within the accuracy providedby the prototype filter 119901[119899] the data sequences 119904119896[119899] 119896 =

0 1 119873 minus 1 appear at its first 119873 outputs In other wordif 119909[119899] is passed through an ideal channel (free of distortionand noise) the SFB may be used to recover the transmitteddata symbols 119904119896[119899] Obviously channel introduces distortionand thus the AFB outputs should pass through a bank ofequalizers to recover the data symbols 119904119896[119899]

52 Polyphase Structures for CMT and SMT CMT and SMTsystems as was demonstrated in Section 3 are effectivelythe same modulations For instance a CMT waveformcan be generated using an SMT waveform generator andsubsequently apply a positive spectral shift of one half ofthe subcarrier spacing to the result Alternatively an SMTwaveform may be generated using a CMT waveform gener-ator and subsequently apply a negative spectral shift of onehalf of the subcarrier spacing to the result It turns out thatthe development of structures for SMT waveform is morestraightforward than those of the CMT We thus continuethis section with development of a pair of transmitter andreceiver structures for SMTThe relevant CMT structures will

x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast

Figure 15 Basic analysis polyphase filter bank

sI0(t)

jsI1(t)

jNminus1sINminus1(t)

sQ0 (t)

minusjsQ1 (t)

(minusj)Nminus1sQNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t

ej(Nminus1)(2120587T)t

ej(2120587T)t


j

x[n]

T2delay

sum

sum

Figure 16 A rearrangement of the transmitter of SMT

follow through simple modifications of the SMT structuresfollowing the details provided in Section 3

(1) SMT Transmitter To develop a computationally efficientpolyphase structure for SMT transmitter we begin with rear-ranging the transmitter part of Figure 6 as in Figure 16 Thisclearly is obtained by separating the phase and quadratureparts of the SFB combining the phase shifts at different pointsin the structure and adding the combined results to the realdata symbols 119904119868119896[119899] and 119904

119876

119896[119899] at the inputThe delay of1198792 in

the prototype filters in the quadrature paths has been shiftedto the corresponding filter bank output

The structure that is presented in Figure 16 consists ofa pair for SFBs Clearly each of these filter banks may beimplemented efficiently using the basic polyphase structureof Figure 14 This implementation is presented in Figure 17


sI0[n]

jsI1[n]

jNminus1sINminus1[n]

0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]

(minusj)Nminus1sQNminus1[n] zminusL2

j

x[n]


It involves two separate polyphase structures However acloser look at the inputs to the polyphase structures revealsthat they are modulated versions of two real vectors

s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904

119868119873minus1[119899] 0 sdot sdot sdot 0]

119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)

As demonstrated next one may take advantage of thisproperty to combine the two SFBs into one

Looking back at the synthesis polyphase structure ofFigure 14 itmay be argued that one can start with the real vec-tors s119868[119899] and s119876[119899] at the input to the IFFT block and applythe modulation effect at the output of the IFFT through acircular rotation of the results In that case onemay start withthe computation of IFFT of the real vectors s119868[119899] and s119876[119899]It is well known that this pair of IFFTs can be implementedthrough a single IFFT with a computational complexity of(1198712)log2119871 complex multiplications On the other hand wenote that since the coefficients of the prototype filter 119901[119899]

are real-valued the complexity of implementation of twosets of polyphase component filters in Figure 17 involves 2120573119871real by complex multiplications equivalent to 120573119871 complexmultiplications Adding these results the total complexityof the SMT transmitter is measured as (1198712)log2119871 + 120573119871

complex multiplications per each set of complex outputsymbols Recall that the length of the prototype filter 119901[119899]

was assumed to be equal 120573119871 + 1 Here we have replacedthis by 120573119871 for brevity of the expressions Also we are onlycounting the number of multiplications as the measure ofcomplexity This is because in practice each multiplicationis usually followed by an addition Hence without involvingourselves with details we argue that the number of additionsin each implementation is about the same as the number ofmultiplications

It is also worth noting that a number of other authors alsohave taken note of the same symmetry properties in SMT

y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot

Figure 18 A rearrangement of the receiver of SMT

waveform and accordingly proposed methods of reducingthe complexity [73ndash75] All these works have reached thecomplexity numbers that are similar to those presented here

(2) SMT Receiver In the absence of channel the analysisof an SMT signal at the receiver can be performed withthe same complexity as its synthesis counterpart at thetransmitter [74 75] However the presence of the channeldestroys the symmetry property of SMT waveform Hencethe implementation approach used at the transmitter cannotbe extended to the receiver In particular we note thatalthough the final goal is to extract the real-valued datasymbols the analyzed signals are the preequalized ones andhence are complex-valued

Following the same line of thoughts that led to Figure 17the receiver side of Figure 6 can be rearranged as in Figure 18where the AFB blocks follow the structure presented inFigure 15 This structure yet does not include the channelequalizers that should be added at each output branch Alsoas noted in the previous section the equalizers may be single-tap or multitap Furthermore one may note that since thepair of AFBs in Figure 18 are the same but separated by 1198712

sample delay they may be combined as one AFB Moreoverthe phase rotations (minus119895)119896 at the output side may be absorbedin the equalizers Combining these points we arrive at theSMT receiver structure presented in Figure 19

The receiver structure presented in Figure 19 has a com-plexity of one AFB plus 119873 equalizers for each set of realsymbols 119904119868119896[119899] or 119904

119876

119896[119899] Assuming that each equalizer has

119872 complex tap weights and noting that at each equalizeroutput we need either the real or imaginary part of theresult the total complexity of the receiver structure ofFigure 19 is obtained as (1198712)log2119871 + (119872119873 + 119870119871 + 1)2

complex multiplications per each set of real output symbols


x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

Evenodd(realimaginary)

darr L2

darr L2

darr L2

Figure 19 SMT receiver structureThe equalizers are single-tap or multitap When they are multitap they have a tap spacing of half a symbolinterval

Alternatively we may say the total complexity of the receiverstructure of Figure 19 is 119871log2119871 + 119872119873 + 119870119871 + 1 complexmultiplications per each set of complex output symbols

53 Polyphase Structures for FMT The polyphase SFB that ispresented in Figure 14 implements a bank of synthesis filterswhose center frequencies are at the normalized frequencies0 1119871 2119871 equivalent to the unnormalized frequencies0 1119879 2119879 respectively Also the sampling rate at the SFBoutput is 119891119904 = 119871119879 Equivalently the sampling interval at theSFB output is 119879119904 = 119879119871 The same is true for the AFB that ispresented in Figure 15

In FMT the situation is different While the symbolrate remains equal to 1119879 each subcarrier bandwidth andaccordingly the FMT waveform bandwidth increase by afactor of 1 + 120572 Therefore the sampling frequency of thesynthesizedwaveform should be increased to119891119904 = 119871(1+120572)119879This in turn implies that 119879119904 = 119879(119871(1+120572)) and accordinglyin discrete-time data symbols at the SFB input should be up-sampled119870 = 119871(1+120572) fold (it is assumed that 120572 is chosen suchthat 119870 will be an integer number) Similarly at the receiverside the output of the AFB should be decimated 119870 foldNevertheless it should be noted that the center frequency ofsubcarriers remains unchanged that is it will be at 0 11198712119871

The fact that 119870 = 119871 introduces some irregularityin the polyphase components which needs a special careMost of the work in the literature that have addressed thisissue have looked at the samples of underlying continuoustime and accordingly have discussed how the correspondingpolyphase filter bank should be implementedThis treatmentunfortunately has led to a set of equations that are oftenhard to follow Here we present a solution that directly workswith sampled signalssequences [34] This solution that hasbeen presented for the first time in [72] is believed to beeasier to followHere we quote our original formulation from[72]

s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n

ej(NminusL)(2120587L)n

sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]

Figure 20 FMT transmitter in discrete-time

(1) FMT Transmitter Figure 20 presents a discrete-timeequivalent of the SFB of Figure 7 Following this figure oneobtains

119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)

To proceed we write the time index 119899 as

119899 = 120574119871 + ℓ (39)

where 120574 is the integer part of 119899119871 and ℓ = 0 1 119871minus1 is theremainder of 119899119871 Similarly 119899may be written as

119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]

Figure 21 Polyphase implementation of an FMT transmitter

where 120578 is the integer part of 119899119870 and 120581 = 0 1 119870 minus 1 isthe remainder of 119899119870 Next we use (39) to substitute for 119899 inthe exponential term in (38) to obtain

119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)

where we have noted that 119890119895(2120587119896119899119871)

= 119890119895(2120587119896ℓ119871) We may

also note that 119878ℓ[119898] is the 119871-point IFFT of input symbols1199040[119899] 1199041[119899] 119904119873minus1[119899] for 119899 = 119898 Using (41) and substituting(40) into (38) we obtain

119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)

Equation (42) may be interpreted and accordinglyimplemented as follows For each set of input symbols1199040[119899] 1199041[119899] 119904119873minus1[119899] 119870 samples of 119909[119899] are calculatedThis calculation begins with performing an 119871-point IFFTon the input symbols 1199040[119899] 1199041[119899] 119904119873minus1[119899] The outputsamples from the IFFT are passed through the polyphasecomponents of the prototype filter 119901[119899] where the polyphasecomponents are with respect to the decimation factor119870Thisprocess is depicted in a block diagram form in Figure 21 Thecommutator at the output of Figure 21 takes 119870 steps afterarrival of each set of input symbols 119863(119899)0 (119911) through119863

(119899)119871minus1(119911)

are a set of time-varying filters that are picked up from theset of119870 polyphase components 1198640(119911) through 119864119870minus1(119911) of theprototype filter 119875(119911) = sum119899 119901[119899]119911

minus119899 For a given time 119899 119909[119899] istaken from the output of 119863(119899)

ℓ(119911) and 119863

(119899)ℓ

(119911) = 119864120581(119911) whereℓ and 120581 are the integers defined according to (39) and (40)respectively

Following Figure 21 the complexity of implementation ofFMT transmitter for each set of data symbols is obtained as(1198712)log2119871 + 120573119871 complex multiplications

(2) FMT Receiver Following the same line of arguments thatled to Figure 20 the discrete-time implementation of an FMTreceiver is obtained as in Figure 22 From this figure it followsthat for 119896 = 0 1 119873 minus 1

119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n

eminusj(NminusL)(2120587L)n

y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]

Figure 22 FMT receiver in discrete-time

To proceed we define

119898 = 120574119871 + ℓ (44)

where 120574 is the integer part of119898119871 and ℓ = 0 1 119871minus1 is theremainder part of119898119871 Using (44) (43) may be rearranged as

119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [119899 minus 120574119871 minus ℓ]) 119890minus119895(2120587119896ℓ119871)

(45)

According to Figure 22 the recovered symbol 119904119896[119899] isobtained after 119870-fold decimation of 119910119896[119899] Moreover inpractice a timing recovery loop determines a timing phaseat which 119910119896[119899] should be decimated Equalization is alsonecessary to combat the distortion introduced by the channelA single-tap or a multitap equalizer per subcarrier may beused In the latter case one may use a fractionally spacedequalizer to relax on the sensitivity of the FMT receiver totiming phase In such cases the decimation factor is smallerthan119870

Here we present a general polyphase structure for com-putation of the samples of 119910119896[119899] for an arbitrary value of119899 hence any arbitrary choice of decimation factor is madepossible To this end we write

119899 minus ℓ = 120578119871 + 120581 (46)

where 120578 is the integer part of (119899 minus ℓ)119871 and 120581 = 0 1 119871 minus 1is the remainder of 119899 minus ℓ divided by 119871 Using (46) into (45)we get

119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)

Following (47) one may propose the analysis polyphasestructure that is presented in Figure 23 The commutatortakes the successive samples of the received signal 119910[119899] anddistributes them across the inputs of the time-varying filters


y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast

Figure 23 Polyphase implementation of an FMT receiver

119863(119899)0 (119911) through119863

(119899)119871minus1(119911) The filters119863(119899)0 (119911) through119863

(119899)119871minus1(119911)

are picked up from the set of 119871 polyphase components 1198640(119911)through 119864119871minus1(119911) of the prototype filter 119875(119911) = sum119899 119901[119899]119911

minus119899For a given time 119899 119863(119899)

ℓ(119911) = 119864120581(119911) The reader should

be reminded that while in Figure 21 the time-varying filters119863(119899)0 (119911) through 119863

(119899)119871minus1(119911) are picked up from the polyphase

components of 119875(119911) with respect to the decimation factor 119870in Figure 23 119863

(119899)0 (119911) through119863

(119899)119871minus1(119911) are picked up from the

polyphase components of119875(119911)with respect to the decimationfactor 119871

Following Figure 23 the complexity of implementation ofFMT receiver for computation of each set of output samplesis obtained as (1198712)log2119871 + 120573119871 complex multiplicationsIn FMT unlike CMT and SMT the use of symbol spacedmultitap equalizer is sufficient If an 119872 tap symbol spacedequalizer is used one will find the complexity number(1198712)log2119871 + 120573119871 + 119872119873 complex multiplications per each setof detected data symbols

54 Complexity Comparison with OFDM Being an alterna-tive to the widely adopted OFDM in various standards it isalways of interest and important to compare FBMCmethodsagainst OFDM As noted in Section 1 the much higherquality of the prototype filter in FBMC makes it an idealchoice for many applications where OFDM faces challengesbecause of the poor frequency response of its prototype filterHere we discuss the additional complexity brought to FBMC(compared to OFDM) because of the use of a higher qualityprototype filter

We count the complexity of OFDM as one IFFT(1198712)log2119871 complex multiplications at the transmitter andone FFT and119873 single-tap equalizers (1198712)log2119871+119873 complexmultiplications at the receiverThus recalling the complexitynumbers that were presented above one finds that

120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588

119877119909SMT respectively denote the relative com-

plexity of SMT transmitter and receiver againstOFDMThesenumbers remain about the same in the case of CMT Toget a better feel of these complexity ratios in practical cases

consider a case where 119871 = 1024 119873 = 08119871 (ie 20 ofsubcarriers are left as guard band) 119872 = 1 and 120573 = 4For this case we find 120588

119879119909SMT = 18 and 120588

119877119909SMT = 255 For

119872 = 3 the most likely value of 119872 if a multitap equalizeris used 120588119877119909SMT = 283 Hence one may say CMT and SMThave a complexity that is roughly two to three times that ofthe OFDM

Similarly for FMT we find that

120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)

Comparing (48) and (50) one will find that SMTCMTand FMT transmitters have the same complexity On theother hand comparing (49) and (51) one will find that FMTreceiver has a lower complexity than SMTCMT In any caseit still remains fair to say FBMC systems in general are twoto three times more complex than OFDM Nevertheless itshould be noted that this comparison is with a basicOFDM Itdoes not consider the cases where advanced signal processingmethods may have to be adopted to compensate for otherchannel effects for example carrier frequency offset amongdifferent users in an OFDMA network In such cases oneoften finds that OFDM is more complex than FBMC [10]

6 Equalization

In FBMC receivers equalization is performed at the outputof the analysis filter banks It is often assumed that eachsubcarrier has a small bandwidth hence the channel maybe assumed to be flat over each subcarrier band In thatcase a single-tap equalizer per subcarrier would suffice Incases where the flat gain approximationmay be insufficient amultitap equalizer per subcarrier band may be necessary ForCMT and SMT systems it is necessary to use a fractionallyspaced equalizer A tap-spacing of half symbol interval is themost convenient choice and hence has been suggested in theliterature for example [43] FMT on the other hand allowsthe use of symbol spaced equalizers

Although in many scenarios it may appear that a single-tap equalizer per subcarrier is sufficient in practice wherecarrier and clock mismatch between the transmitter andreceiver is inevitable a multitap equalizer can be instrumen-tal In particular the difference between the clock used attransmitter to pass the signal samples to a digital-to-analogconverter (DAC) and the one used at the receiver to controlthe rate of samples taken by an analog-to-digital converter(ADC) at the receiver introduces a constant drift in thetiming phase of sampled signal This drift as explained inthe next section can significantly deteriorate the receiverperformance The use of a multitap equalizer can resolve thisproblem to a great extent This is demonstrated through anumerical example in the next section

Another point that should be emphasized here is thefollowing Most researchers who have published in the area


of FBMC have limited their results to the use of one single-tap equalizer per subcarrier bandThis has been to emphasizeon the simplicity of equalization in FBMC systems and alsoto stay in par with OFDM Alternatively one may look at theuse of multitap equalizers in FBMC systems as a mechanismthat brings with itself a number of advantages Firstly theuse of multitap equalizers removes any small frequencyselectivity over each subcarrier band and hence facilitatesadoption of a large symbol constellations and this in turnincreases the bandwidth efficiency of transmission Secondlyit allows reducing the number of subcarriers which in turnreduces the system (i) complexity (possibly compensatingfor the added complexity resulting from the use of multitapequalizers) (ii) latency (iii) sensitivity to carrier frequencyoffset and (iv) peak-to-average power ratio (PAPR)

7 Synchronization and Tracking Methods

Synchronization methods are necessary in any receiver tocompensate for any difference between the carrier frequencyof the incoming signal and the local oscillator used to demod-ulate it as well as to compensate for any clock mismatchbetween the transmitter and receiver Pilot aided and blindsynchronization methods have been widely studied in theliterature from the beginning of development of datamodems[76]

In majority of standards including those of OFDMand FBMC the pilot aided approaches have been adoptedFor instance all the OFDM packet formats that have beenproposed for WiFi andWiMAX as well as those suggested inLTE and LTE-Advanced start with a short-training periodicpreamble for coarse carrier acquisitionThis follows by a longpreamble consisting of two or more similar OFDM symbolsthat is used for fine tuning of the carrier timing acquisitionand channel estimation for example see [72 77] In additionto be able to track channel variations during the payloadof each packet it is often proposed to insert distributedpilots in the time-frequency space Interpolationmethods arethen used to obtain the channel gain and accordingly theequalizers gain at the data points [78ndash81]

A few researchers who have looked at the packet designfor FBMC have also suggested a similar packet format tothose of the OFDM [82ndash86] In particular a periodic trainingconsisting of a few repetitions of a pilot FBMC symbol issuggested to be used along with a least squares or amaximumlikelihood estimator to estimate CFO and symbol timingThis is equivalent to long preamble in OFDM Moreover in[82] it is noted that the use of a short preamble similar to thatof OFDM may be useful for coarse estimation of carrier sothat fine tuning of carrier and timing phase based on the longpreamble can be establishedmore robustly Distributed pilotsto track channel variations in payload part of FBMC packetshave also been proposed [87 88] In the rest of this sectionwe make commentssuggestions for future research in threedirections related to synchronization and tracking methodsin FBMC systems

71 Preamble Design Figure 24 presents the packet formatthat has been proposed in [82ndash86] The short preamble

Short preamble Long preamble Datapayload

t1 t2 t10middot middot middot middot middot middot

Figure 24 An FBMC data packet with short and long preambles

consists of a few tones with wider separation than thesubcarrier spacing to allow detection of large CFO valuesIt is used to obtain a coarse estimate of CFO so that theresidual CFO (after removal of the estimated CFO) will bewithin the range of half symbol spacing The long preamblethen may be used for fine tuning of the carrier The workspresented in [83ndash85] do not mention the short preamblebut they make assumption that CFO is within the range ofhalf symbol spacing Naturally to satisfy this assumption amechanism similar to short preamble may be necessary

There is also some difference in the long preambleproposed in [83ndash85] and the one proposed in [82 86] Inthe context of SMT waveform the long preamble proposedin [83ndash85] is constructed based on a set of complex-valuedpilot symbols that occupy all subcarriers The long preambleproposed in [82 86] on the other hand is real-valued andonly occupies the alternate subcarriers the other subcarriersare left empty This modificationsimplification of the longpreamble as demonstrated through numerical results in [86]improves the accuracy of the CFO estimates significantly

Here to make suggestions to further improve the pream-ble design we first note that the preamble is a packetcomponent that constitutes a loss in the bandwidth efficiencyof transmission Clearly an attempt to reduce the lengthof the preamble improves the bandwidth efficiency Theresearch so far in the above cited works has led to thefollowing observations (i) A preamble with a set of pilotsymbols that are well separated in frequency is needed toassure detection and correction of large CFO values (ii)A longer preamble that allows a more accurate estimate ofCFO may be also necessary On the other hand one mayrealize that to obtain the CFO and symbol timing estimatesit is not necessary to send pilots on all the subcarriers Thenumber of parameters that characterize the channel response(determined by the length of the channel impulse response)is usually much smaller than the number of subcarriers Thismeans a fraction of subcarriers whose number is greaterthan or equal to the length of the channel impulse responsesuffice for pilot subcarriers in the preamble This in turnimplies that one may propose to combine the short and longpreambles into one preamble and by taking such approachobviously a shorter preamble will result We note that thepreamble proposed in [82 86] compared to those proposedin [83ndash85] may be viewed as a step forward in this directionWhile the works in [83ndash85] use all subcarriers to transmit thepreamble pilots [82 86] allocate pilots only to alternate (evenor odd) subcarriers

Another consideration that needs onersquos attention indesigning an effective preamble is the choice of the prototypefilter that is used for preamble Assuming that the preambleconsists of only a long preamble the preamble length isdetermined by the length of the prototype filter and the


s[n]SFB DAC Channel ADC AFB

Clock rate = ft Clock rate = fr

Equalizerbank

s[n]

Figure 25 An FBMC transceiver with timing drift and the corresponding compensator block

number of its repetitions Accordingly a shorter prototypefilter leads to a shorter preamble and thus a more bandwidthefficient packet The above suggestion to space out thepilot subcarriers in the preamble can be taken advantageof to shorten the length of the preamble This is becausewider subcarrier allows one to increase the bandwidth ofeach subcarrier and accordingly reduce the symbol periodand thus the length of the corresponding prototype filterNevertheless it should be also noted that a broader band ofeach subcarrier can make it prone to some level of frequencyselectivity that should be taken into account while identifyingthe channel parameters

72 Channel Tracking Tracking of channel variation (includ-ing variations due to CFO and clock timing drift) during thepayload transmission is usually made through distributedpilots that are transmitted along with the data symbols Thistopic as noted above has been well studied for OFDMHowever it has received very limited attention in the FBMCliterature Stitz et al [88] have studied the problem of pilot-aided synchronization and tracking methods in CMTSMT-type systems They have shown that by applying specificpilots designed for CMTSMT the carrier frequency offset(CFO) fractional time delay (FTD) and channel responsecan be accurately estimated The work presented in [89] hastaken a different approach It takes note of the fact thateach signal component at the output of the analysis filterbank before performing equalization and taking its real (orimaginary) part has the form of ℎ119896[119899](119904119896[119899] + 119895120595119896[119899]) (orℎ119896[119899](119895119904119896[119899]+120595119896[119899])) where 120595119896[119899] arises from ICI and ISI Itis thus argued that if the same (known) symbol is transmittedover two instants of times at the same subcarrier or allocatedto the real and imaginary parts of an OQAM symbol in aSMT waveform the analyzed signal samples give a pair ofequations that can be solved to obtain an estimate of ℎ119896[119899]

In [27] (also see [90]) it has been noted that the use ofPAM symbols in CMT (and similarly in SMT if the real andimaginary parts of each OQAM symbol are viewed as a pairof PAM symbols) allows development of a blind equalizationalgorithmThis algorithmmay also be used for channel track-ing Hence in CMT and SMT channel tracking is possiblewithout any use of pilot symbols To elaborate more on thispossibility we note that the convergence behavior of the blindequalizer that is developed in [27] was thoroughly analyzed in[91] (also see [92 93]) It was found that the CMTSMT blindequalizer has a well-behaved performance surface around itsoptimal setting This implies that if the equalizer gain is setnear its optimal value and channel subsequently varies slowlythe blind equalization algorithm of [27] can easily track thechannel variation This observation that yet to be thoroughlystudied will add an additional advantage to CMT and SMTover OFDM in terms of bandwidth efficiency

73 Timing Tracking As noted above any difference betweenthe clock frequency at the transmitter and its counterpart atthe receiver introduces a timing drift The common methodof compensating for this timing drift is to add a timing recov-ery loop to the system In a single carrier receiver the additionof a timing recovery loop as part of the receiver processingis a relatively straightforward task and is well understood[72 94] In an OFDM system with a not very long payloadthe timing driftmay be absorbed by the CP provided that notthe full length of the CP has been consumed by the channeltransient response Drift of timing phase introduces a linearphase across frequency and that can be easily compensatedby the single-tap equalizers at the FFT outputs In FBMCon the other hand the absence of CPguard interval makesan FBMC receiver more sensitive to timing drift Howeverfortunately this problem can be resolved trivially through theuse of a set of multitap fractionally spaced equalizers one persubcarrier

To demonstrate the impact of per subcarrier fractionallyspaced equalizers on resolving the problemof timing drift weconsider a CMTSMT transceiver system which we simulatefollowing the schematic diagram presented in Figure 25 Thesamples of the transmit signal are converted from digital toanalog using a DAC that operated at a clock rate of 119891119905 At thereceiver the received signal is sampled through an ADCwitha clock that has a rate of119891119903The clock frequencies119891119905 and119891119903 aredifferent but typically are close In practice their differencemay be as large as 50 ppm (part permillion)This correspondsto the case where each clock has a precision of 25 ppm

For the purpose of demonstration here we consider thecase where channel is absent In that case the effect of using apair of DAC and ADCwith different clock rates is like resam-pling the generated waveform with a resampling ratio 119891119903119891119905This concept can be easily applied in a MATLAB simulationenvironment that we use using the function ldquoresamplerdquo Theresults of simulating a CMT systemwith 64 active subcarriersand a clock drift of 50 ppm are presented in Figure 26 Theresults show the signal-to-interference ratio (SIR) averagedover all subcarriers as one proceed over successive CMTsymbols Timing phase is set to zero (ie perfect timing) atthe beginning The drift of timing phase introduces some ISIthat cannot be resolved when single-tap equalizers are usedbut can be compensated through multitap equalizers Perfectcompensation may require a long equalizer However a 3-or 5-tap equalizer is sufficient to avoid a significant drop ofSIR We consider an SIR of above 30 dB as acceptable giventhat in typical practice communication systems are oftendesigned for output SINR values 20 dB or less A margin of10 dB should be sufficient to consider ISIICI due to timingdrift as negligible For the numerical example presented herewhile a single-tap equalizer can preserve the SIR of above30 dB for the first 400 CMT symbols the use of a 3-tap and


45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index

5-tap equalizer3-tap equalizer1-tap equalizer

Figure 26 SIR variation as a function of symbol index (equivalentlytiming drift) in a CMT transceiver

5-tap equalizer increases this range to 1200 and 2500 CMTsymbols respectively

Another possible solution that may be adopted to resolvethe difference between the DAC clock at the transmitter andADC clock at the receiver may be proposed by taking note ofthe following observation An FBMCwaveform (being CMTSMT or FMT) is a cyclostationary signal whose statistics arerepeated every 119879119888 second where 119879119888 is an integer multiple ofthe symbol period 119879 In FMT and SMT 119879119888 = 119879 and in CMT119879119888 = 2119879 We conjecture that taking note of this property onecan design a digital resampler circuit that may be added tothe output of ADC at the receiver for tuning the numberof samples per each period of 119879119888 equal to a predefined valueSuch solution is yet to be found through a clever study of theunderlying signals

8 Applications

In [19] OFDMand FBMC are contrastedwith respect to theirsuitability for different applications It is noted that while formany applications OFDM is a perfect choice FBMC maybe a superior choice for other applications The applicationsin which FBMC may be preferred are (i) the uplink ofmultiuser multicarrier systems (ii) cognitive radios (iii)doubly dispersive channels and (iv) digital subscriber linesand power line communications Detailed discussion on whyFBMC is a better choice in these applications is providedin [19] The rest of this section emphasizes on an emergingclass of communication systems massive (multiuser)MIMOwhose significance as an enabling technology for 5G hasrecently been discussed in the literature and other forums

Massive MIMO in essence is a multiuser techniquesomewhat similar to code division multiple access (CDMA)In its simplest form each mobile terminal (MT) has a single

antenna but the base station (BS) has a large number ofantennas The spreading code for each user is then the vectorof channel gains between the respective MT antenna andthe multiple antennas at the BS Accordingly by increasingthe number of antennas at the BS the processing gain ofeach user can be increased to an arbitrarily large value Inthe pioneering work of Marzetta [95] it is noted that in thelimit as the number of BS antennas tends to infinity theprocessing gain of the system tends to infinity and as a resultthe effects of both noise and multiuser interference (MUI)are completely removed Therefore the network capacity (intheory) can be increased unboundedly by increasing thenumber of antennas at the BS [95]

Motivated by Marzettarsquos observations [95] multipleresearch groups in recent past have studied a variety ofimplementation issues related to massive MIMO systems forexample [96ndash100] Also different groups have started thedevelopment of testbeds to confirm the theoretical observa-tion of [95] in practice for example An assumption madeby Marzetta [95] and followed by other researchers is thatorthogonal frequency division multiplexing (OFDM)may beused to convert the frequency selective channels betweeneach MT and the multiple antennas at the BS to a set offlat fading channels Accordingly the flat gains associatedwith the set of channels within each subcarrier constitutethe spreading gain vector that is used for dispreading of therespective data in upstream as well as for precoding of therespective downstream data

In a recent work [101] embarking on the above conceptswe have introduced the application of filter bankmulticarrier(FBMC) to the area of massive MIMO communications Theinteresting finding in this work is that unlike in conventionalMIMO where applicability of FBMC (CMT and SMT inparticular) is found limited FBMC offers a number ofappealing properties in the application of massive MIMOIn particular the linear combining of the signal componentsfrom different antennas at BS (equivalent to dispreading inCDMA) smooths channel distortion across each subcarrierband This property which we call self-equalization allowsone to relax on the requirement of having approximately flatgain for the subcarriersThis in turn allows one to reduce thenumber of subcarriers in an assigned bandwidth Reducingthe number of subcarriers has the following impacts whichall position FBMC as a strong candidate in the application ofmassive MIMO

(1) The complexity of both transmitter and receiver willbe reduced since the underlying polyphase structureswill be based on smaller size IFFTFFT blocks

(2) The system latencywill be reducedThis is because thelength and thus the corresponding group delay of theunderlying prototype filter are reduced

(3) The system sensitivity to CFO will be reduced(4) The PAPR of the transmit signal will be reduced

The preliminary research results that are presented in[101] clearly open up a new research direction for explorationof FBMC techniques in the area of massive MIMO


9 Conclusions

This paper presented a tutorial review of the class of filterbank multicarrier (FBMC) communication systems withemphasis on the recent results of the authors researchlaboratory It was noted that by building FBMC systemsbased on well-designed prototype filters the spectrum ofeach subcarrier can be contained within a limited bandwidthThis in turn allows transmission over noncontiguous bandsa property that makes FBMC an ideal choice for manyapplications including the uplink of multiuser multicarriernetworks and cognitive radios It was also pointed outthat FBMC systems can be adopted to doubly dispersivechannels to achieve a performance far superior to that ofOFDM The polyphase structures that have been proposedfor efficient implementation of FBMC systems were reviewedand their complexities were compared with that of a standardOFDM system It was noted that the complexity of FBMCtransmitters is close to (but less than) twice their OFDMcounterpart and the complexity of FBMC receivers is close to(but less than) three times that of their OFDM counterpartHowever it was also noted that in some applications forexample in the uplink of multiuser systems where syn-chronization cannot be made perfectly hence multiusercancellation techniques should be applied OFDM can befar more complex than FBMC Moreover it was noted thatmore research is needed to better understand the problemsrelated to synchronization equalization and tracking ofchannel variations in FBMC systems Possible applicationof FBMC in the emerging area of massive MIMO was alsohighlighted and a number of advantages that FBMC offers inthis application were identified

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

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[2] J C T Starr and P Silverman Understanding Digital SubscriberLine Technology Prentice Hall Upper Saddle River NJ USA1999

[3] R V Nee and R Prasad OFDM for Wireless MultimediaCommunications Arthec House Boston Mass USA 2000

[4] Y Li and G Stuber Eds Orthogonal Frequency DivisionMultiplexing for Wireless Communications Springer New YorkNY USA 2006

[5] M Morelli C-C J Kuo and M-O Pun ldquoSynchronizationtechniques for orthogonal frequency division multiple access(OFDMA) a tutorial reviewrdquo Proceedings of the IEEE vol 95no 7 pp 1394ndash1427 2007

[6] D Huang and K B Letaief ldquoAn interference-cancellationscheme for carrier frequency offsets correction in OFDMAsystemsrdquo IEEE Transactions on Communications vol 53 no 7pp 1155ndash1165 2005

[7] K Lee and I Lee ldquoCFO compensation for uplink OFDMAsystems with conjugated gradientrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo11) pp 1ndash5Kyoto Japan June 2011

[8] K Lee S-R Lee S-H Moon and I Lee ldquoMMSE-basedCFO compensation for uplink OFDMA systems with conjugategradientrdquo IEEE Transactions on Wireless Communications vol11 no 8 pp 2767ndash2775 2012

[9] A Farhang N Marchetti and L E Doyle ldquoLow complexity LSandMMSE based CFO compensation techniques for the uplinkof OFDMA systemsrdquo in Proceedings of the IEEE InternationalConference on Communications (ICC rsquo13) pp 5748ndash5753 IEEEBudapest Hungary June 2013

[10] H Saeedi-Sourck Y Wu J W M Bergmans S Sadri and BFarhang-Boroujeny ldquoComplexity and performance comparisonof filter bank multicarrier and OFDM in uplink of multicarriermultiple access networksrdquo IEEE Transactions on Signal Process-ing vol 59 no 4 pp 1907ndash1912 2011

[11] S Brandes I Cosovic and M Schnell ldquoSidelobe suppressionin OFDM systems by insertion of cancellation carriersrdquo inProceedings of the 62nd IEEE Vehicular Technology Conference(VTC rsquo05) vol 1 pp 152ndash162 September 2005

[12] Z Yuan S Pagadarai and A M Wyglinski ldquoCancellationcarrier technique using genetic algorithm for OFDM sidelobesuppressionrdquo in Proceedings of the Military CommunicationsConference (MILCOM rsquo08) pp 1ndash5 IEEE 2008

[13] A Selim I Macaluso and L Doyle ldquoEfficient sidelobe suppres-sion for OFDM systems using advanced cancellation carriersrdquoin Proceedings of the IEEE International Conference on Com-munications (ICC rsquo13) pp 4687ndash4692 Budapest Hungary June2013

[14] S N Premnath DWasden S K Kasera B Farhang-Boroujenyand N Patwari ldquoBeyond ofdm best-effort dynamic spectrumaccess using filterbank multicarrierrdquo in Proceedings of the4th International Conference on Communication Systems andNetworks (COMSNETS rsquo12) January 2012

[15] S N Premnath D Wasden S K Kasera N Patwari andB Farhang-Boroujeny ldquoBeyond OFDM best-effort dynamicspectrum access using filterbank multicarrierrdquo IEEEACMTransactions on Networking vol 21 no 3 pp 869ndash882 2013

[16] B Farhang-Boroujeny and R Kempter ldquoMulticarrier commu-nication techniques for spectrum sensing and communicationin cognitive radiosrdquo IEEE Communications Magazine vol 46no 4 pp 80ndash85 2008

[17] B Farhang-Boroujeny ldquoProlate filters for nonadaptive multita-per spectral estimators with high spectral dynamic rangerdquo IEEESignal Processing Letters vol 15 pp 457ndash460 2008

[18] B Farhang-Boroujeny ldquoFilter bank spectrum sensing for cogni-tive radiosrdquo IEEE Transactions on Signal Processing vol 56 no5 pp 1801ndash1811 2008

[19] B Farhang-Boroujeny ldquoOFDM versus filter bank multicarrierrdquoIEEE Signal Processing Magazine vol 28 no 3 pp 92ndash112 2011

[20] A Sahin I Guvenc and H Arslan ldquoA survey on multicar-rier communications prototype filters lattice structures andimplementation aspectsrdquo IEEE Communications Surveys andTutorials vol 16 no 3 pp 1312ndash1338 2013

[21] M Bellanger ldquoFBMC physical layer a primerrdquo Tech Rep062010 2010

[22] M Payaro A Pascual-Iserte and M Najar ldquoPerformancecomparison between FBMC and OFDM in MIMO systemsunder channel uncertaintyrdquo in Proceedings of the EuropeanWireless Conference (EW rsquo10) pp 1023ndash1030 April 2010


[23] T Ihalainen A Ikhlef J Louveaux and M Renfors ldquoChannelequalization for multi-antenna FBMCOQAM receiversrdquo IEEETransactions on Vehicular Technology vol 60 no 5 pp 2070ndash2085 2011

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[25] M Tzannes M Tzannes J Proakis and P Heller ldquoDMTsystemsDWMTsystems and digital filter banksrdquo inProceedingsof the IEEE International Conference on Communications (ICCrsquo94) vol 1 pp 311ndash315 1994

[26] S D Sandberg and M A Tzannes ldquoOverlapped discrete multi-tone modulation for high speed copper wire communicationsrdquoIEEE Journal on Selected Areas in Communications vol 13 no9 pp 1571ndash1585 1995

[27] B Farhang-Boroujeny ldquoMulticarrier modulation with blinddetection capability using cosine modulated filter banksrdquo IEEETransactions on Communications vol 51 no 12 pp 2057ndash20702003

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[30] ldquoVery-high bit-rate digital subscriber lines (vdsl) metallicinterface part 3 technical specification of a multi-carriermodulation transceiverrdquo Working Group R1E14 DocumentT1E142000-013R3 2000

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[32] M Nava and G Okvist ldquoThe zipper prototype a completeandflexibleVDSLmulticarrier solutionrdquo IEEECommunicationsMagazine vol 40 no 12 pp 92ndash105 2002

[33] R Nilsson F Sjoberg M Isaksson J M Cioffi and S KWilson ldquoAutonomous synchronization of a DMT-VDSL systemin unbundled networksrdquo IEEE Journal on Selected Areas inCommunications vol 20 no 5 pp 1055ndash1063 2002

[34] G Cherubini E Eleftheriou and S Olcer ldquoFiltered multitonemodulation for very high-speed digital subscriber linesrdquo IEEEJournal on Selected Areas in Communications vol 20 no 5 pp1016ndash1028 2002

[35] ldquoVery high speed digital subscriber line transceivers 2(VDSL2)rdquo ITU-T Recommendation G9932 2006

[36] M Bellec and P Pirat ldquoOQAM performances and complexityrdquoIEEE P80222 Wireless Regional Area Network 2006 httpwwwieee802org22Meeting documents2006 Janindexhtml

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[38] B Farhang-Boroujeny andC Yuen ldquoCosinemodulated and off-set QAMfilter bankmulticarrier techniques a continuous-timeprospectrdquo EURASIP Journal on Advances in Signal Processingvol 2010 Article ID 165654 16 pages 2010

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[43] B Hirosaki ldquoAn orthogonally multiplexed qam system usingthe discrete fourier transformrdquo IEEE Transactions on Commu-nications vol 29 no 7 pp 982ndash989 1981

[44] B Le Floch M Alard and C Berrou ldquoCoded orthogonalfrequency division multiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995

[45] G Cariolaro and F C Vagliani ldquoOFDM scheme with a halfcomplexityrdquo IEEE Journal on Selected Areas in Communicationsvol 13 no 9 pp 1586ndash1599 1995

[46] P Siohan and C Roche ldquoCosine-modulated filterbanks basedon extended Gaussian functionsrdquo IEEE Transactions on SignalProcessing vol 48 no 11 pp 3052ndash3061 2000

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[48] S Govardhanagiri T Karp P Heller and T Nguyen ldquoPerfor-mance analysis ofmulticarriermodulation systems using cosinemodulated filter banksrdquo in Proceedings of the IEEE InternationalConference on Acoustics Speech and Signal Processing (ICASSPrsquo99) vol 3 pp 1405ndash1408 March 1999

[49] P Vaidyanathan Multirate Systems and Filter Banks PrenticeHall Englewood Cliffs NJ USA 1993

[50] N J Fliege Multirate Digital Signal Processing John Wiley ampSons New York NY USA 1994

[51] N J Fliege ldquoModified DFT polyphase SBC filter banks withalmost perfect reconstructionrdquo in Proceedings of the IEEE Inter-national Conference on Acoustics Speech and Signal Processing(ICASSP rsquo94) vol 3 pp III149ndashIII152 Adelaide AustraliaApril 1994

[52] PNHeller TKarp andTQNguyen ldquoA general formulation ofmodulated filter banksrdquo IEEE Transactions on Signal Processingvol 47 no 4 pp 986ndash1002 1999

[53] T Karp andN J Fliege ldquoModifiedDFT filter banks with perfectreconstructionrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 46 no 11 pp 1404ndash1414 1999

[54] R Bregovic and T Saramaki ldquoA systematic technique fordesigning linear-phase FIR prototype filters for perfect-recon-struction cosine-modulated and modified DFT filterbanksrdquoIEEE Transactions on Signal Processing vol 53 no 8 part 2 pp3193ndash3201 2005

[55] S Salcedo-Sanz F Cruz-Roldan C Heneghan and X YaoldquoEvolutionary design of digital filters with application to sub-band coding and data transmissionrdquo IEEE Transactions onSignal Processing vol 55 no 4 pp 1193ndash1203 2007

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[57] P Amini R-R Chen and B Farhang-Boroujeny ldquoFilterbankmulticarrier communications for underwater acoustic chan-nelsrdquo IEEE Journal of Oceanic Engineering 2014


[58] B Farhang-Boroujeny ldquoA square-root Nyquist (M) filter designfor digital communication systemsrdquo IEEE Transactions onSignal Processing vol 56 no 5 pp 2127ndash2132 2008

[59] K W Martin ldquoSmall side-lobe filter design for multitone data-communication applicationsrdquo IEEE Transactions on Circuitsand Systems II Analog and Digital Signal Processing vol 45 no8 pp 1155ndash1161 1998

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[61] M Rossi J-Y Zhang and W Steenaart ldquoIterative least squaresdesign of perfect reconstruction QMF banksrdquo in Proceedings ofthe Canadian Conference on Electrical and Computer Engineer-ing (CCECE rsquo96) pp 762ndash765 May 1996

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[68] P Amini R-R Chen and B Farhang-Boroujeny ldquoFilterbankmulticarrier for underwater communicationsrdquo in Proceedingsof the 49th Annual Allerton Conference on CommunicationControl and Computing (Allerton rsquo11) pp 639ndash646 September2011

[69] P Amini C H Yuen R-R Chen and B Farhang-BoroujenyldquoIsotropic filter design for MIMO filter bank multicarriercommunicationsrdquo in Proceedings of the IEEE Sensor Array andMultichannel Signal Processing Workshop (SAM rsquo10) pp 89ndash92October 2010

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[72] B Farhang-Boroujeny Signal Processing Techniques for SoftwareRadios Lulu Raleigh NC USA 2nd edition 2009

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[74] G Cariolaro and F C Vagliani ldquoAn OFDM scheme with a halfcomplexityrdquo IEEE Journal on Selected Areas in Communicationsvol 13 no 9 pp 1586ndash1599 1995

[75] L Vangelista and N Laurenti ldquoEfficient implementations andalternative architectures for OFDM-OQAM systemsrdquo IEEETransactions on Communications vol 49 no 4 pp 664ndash6752001

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[78] I Barhumi G Leus andM Moonen ldquoOptimal training designfor MIMO OFDM systems in mobile wireless channelsrdquo IEEETransactions on Signal Processing vol 51 no 6 pp 1615ndash16242003

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[81] C K Ho B Farhang-Boroujeny and F Chin ldquoAdded pilotsemi-blind channel estimation scheme for OFDM in fadingchannelsrdquo in Proceedings of the IEEE Global TelecommunicatinsConference (GLOBECOM rsquo01) vol 5 pp 3075ndash3079 November2001

[82] B Farhang-Boroujeny and P Amini ldquoPacket format design anddecision directed tracking methods for filter bank multicarriersystemsrdquo EURASIP Journal on Advances in Signal Processingvol 2010 Article ID 307983 2010

[83] T Fusco A Petrella and M Tanda ldquoJoint symbol timingand CFO estimation in multiuser OFDMOQAM systemsrdquo inProceedings of the IEEE 10th Workshop on Signal ProcessingAdvances inWireless Communications (SPAWC rsquo09) pp 613ndash617June 2009

[84] T Fusco A Petrella andM Tanda ldquoData-aided symbol timingand CFO synchronization for filter bank multicarrier systemsrdquoIEEE Transactions onWireless Communications vol 8 no 5 pp2705ndash2715 2009

[85] M Tanda T Fusco and A Petrella ldquoJoint symbol timingand CFO estimation for OFDMOQAM systems in multipathchannelsrdquo Eurasip Journal on Advances in Signal Processing vol2010 Article ID 897607 2010

[86] H Saeedi-Sourck S Sadri Y Wu and B Farhang-BoroujenyldquoNear maximum likelihood synchronization for filter bankmulticarrier systemsrdquo IEEE Wireless Communications Lettersvol 2 no 2 pp 235ndash238 2013

[87] T H Stitz A Viholainen T Ihalainen and M Renfors ldquoCFOestimation and correction in a WiMAX-like FBMC systemrdquoin Proceedings of the 10th IEEE Workshop on Signal ProcessingAdvances in Wireless Communications (SPAWC rsquo09) pp 633ndash637 June 2009

[88] T H Stitz T Ihalainen A Viholainen and M RenforsldquoPilot-Based synchronization and equalization in filter bankmulticarrier communicationsrdquo EURASIP Journal on Advancesin Signal Processing vol 2010 article 9 Article ID 741429 2010

[89] M Bellanger and T Dao ldquoMulticarrier digital transmissionsystem using an OQAM transmultiplexerrdquo Granted patentUS7072412 2006

[90] B Farhang-Boroujeny ldquoDiscrete multitone modulation withblind detection capabilityrdquo in Proceedings of the Vehicular


Technology Conference (VTC rsquo02) vol 1 pp 376ndash380 IEEE2002

[91] L Lin and B Farhang-Boroujeny ldquoConvergence analysis ofblind equalizer in a filter-bank-based multicarrier communica-tion systemrdquo IEEE Transactions on Signal Processing vol 54 no10 pp 4061ndash4067 2006

[92] ldquoConvergence an alysis of blind equalizer in a cosine mod-ulated filter bank-based multicarrier communication systemrdquoin Proceedings of the Workshop on Signal Processing Advancesin Wireless Communications (SPAWC rsquo03) pp 368ndash372 IEEE2003

[93] L Lin and B Farhang-Boroujeny ldquoAnalytical study of theperformance surface of blind equalizer in a cosine modulatedmulticarrier communication systemrdquo in Proceedings of theIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo05) pp IV337ndashIV340 March 2005

[94] F J Harris andM Rice ldquoMultirate digital filters for symbol tim-ing synchronization in software defined radiosrdquo IEEE Journal onSelected Areas in Communications vol 19 no 12 pp 2346ndash23572001

[95] T L Marzetta ldquoNoncooperative cellular wireless with unlim-ited numbers of base station antennasrdquo IEEE Transactions onWireless Communications vol 9 no 11 pp 3590ndash3600 2010

[96] J Hoydis S Ten Brink andM Debbah ldquoMassive MIMO in theULDL of cellular networks how many antennas do we needrdquoIEEE Journal on Selected Areas in Communications vol 31 no2 pp 160ndash171 2013

[97] J Hoydis C Hoek T Wild and S Ten Brink ldquoChannelmeasurements for large antenna arraysrdquo in Proceedings of theInternational Symposium on Wireless Communication Systems(ISWCS rsquo12) pp 811ndash815 Paris France August 2012

[98] J Jose A Ashikhmin T L Marzetta and S Vishwanath ldquoPilotcontamination and precoding inmulti-cell TDD systemsrdquo IEEETransactions on Wireless Communications vol 10 no 8 pp2640ndash2651 2011

[99] X Gao O Edfors F Rusek and F Tufvesson ldquoLinear pre-coding performance in measured very-large MIMO channelsrdquoin Proceedings of the IEEE Vehicular Technology Conference(VTC Fall rsquo11) pp 1ndash5 IEEE San Francisco Calif USASeptember 2011

[100] S Payami and F Tufvesson ldquoChannel measurements andanalysis for very large array systems at 26GHzrdquo in Proceedingsof the 6th European Conference on Antennas and Propagation(EuCAP rsquo12) pp 433ndash437 March 2012

[101] A Farhang N Marchetti L Doyle and B Farhang-BoroujenyldquoFilter bank multicarrier for massive MIMOrdquo in Proceedingsof the IEEE International Conference on Vehicular Technology(VTC-Fall rsquo14) Vancouver Canada September 2014

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



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Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



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Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



the transmitter For instance the side lobe suppressiontechniques like those proposed in [11ndash13] can achieve an out-of-band emission suppression of only 5 to 10 dB while theymay add significant complexity to the transmitter and theywill incur some loss in bandwidth efficiency

Filter bank multicarrier (FBMC) is an alternative trans-mission method that resolves the above problems by usinghigh quality filters that avoid both ingress and egress noisesAlso because of the very low out-of-band emission ofsubcarrier filters application of FBMC in the uplink ofmultiuser networks is trivial [14 15] It can be deployedwithout synchronization of mobile user nodes signals In theapplication of cognitive radios the filter bank that is used formulticarrier data transmission can also be used for spectrumsensing [16ndash20] On the other hand compared to OFDMFBMC falls short in handing multiple-input multiple-output(MIMO) channels although a few solutions to adopt FBMCin MIMO channels have been reported in the literature forexample see [21ndash23] Nevertheless as our recent researchstudy has shown (see Section 8 below) in the emerging areaofmassiveMIMO FBMC is found as powerful as OFDM andin some cases superior to OFDM

In the past many attempts have been made to adoptFBMC in various standards Apparently the earliest pro-posal to use FBMC for multicarrier communications is acontribution from Tzannes et al of AWARE Inc in oneof the asymmetric digital subscriber lines (ADSL) standardmeetings in 1993 [24] The proposed method that was calleddiscrete wavelet multitone (DWMT) was further studied in[25 26] Despite enthusiasm from the research community(see [27] and the cited references therein) DWMT was notadopted in the ADSL standardThis was partly because of theperceived complexity of this method as compared to its rivalDMT (discrete multitone an equivalent name for OFDM inDSL literature) Indeed the DWMT structure proposed byTzannes et al was significantly more complex than that ofDMT The major part of the complexity of DWMT camefrom the equalization method that was adoptedThe detaileddiscussion presented in [26] assumed that one needs anequalizer that combines signals from each subcarrier bandand its adjacent bands Typical equalizer lengths suggestedin [26] were 21 real-valued taps per subcarrier It was laternoted by the author of this paper that if each subcarrier issufficiently narrow such that it can be approximated by aflat gain two real taps per subcarrier would be sufficient forequalization [27] This observation led to further study ofDWMT In power line communications (PLC) communityit has been named wavelet OFDM and was adopted in theIEEE P1901 standard [28] The main motivation for useof DWMT in DSL and its adoption later in PLC was todeal with ingress and egress noises since both DSL andPLC use unshielded copper lines that are subject to strongradio interference Moreover in 1999 an FBMCmethod withnonoverlapping subcarrier bands was proposed as a solutionfor filtering the narrow-band interferences in very high-speedDSL (VDSL) channels [29] The proposed method was calledfiltered multitone (FMT) This proposal that was included asan annex in one of the initial draft documents of VDSL [30]was further developed by a number of researchers [31ndash34]

However to avoid incompatibility with ADSL FMT was notincluded in the final document of the VDSL standard [35]Another unsuccessful story is an attempt by France Telecom[36] to introduce FBMC in the IEEE 80222 a cognitiveradio standard to access TV bands in wireless rural areanetworks (WRAN) Up to now apparently the only standardfor radio transmission that uses FBMC is the TIAs DigitalRadio Technical Standard [37]

Recent discussions on the fifth generation (5G) wirelesscommunications have initiated a much stronger wave ofinterest in deviating from themain stream of OFDM systemsThis shift of interest is clearly due to limitations of OFDM inthe more dynamic and multiuser networks of future A num-ber of proposals have been made to adopt new waveformswith improved spectral containment A good example of suchactivity is the 5GNOW project in Europe which challengesLTE and LTE-Advanced in coping with the dynamic needsof 5G The 5GNOW has identified four alternative choicesof waveforms to better serve 5G needs These waveformsthat are all built based on some sort of filtering may bethought as adoptions of FBMC method to suit differentneeds of various applications We refer interested readersto the documents available at httpwww5gnoweu for thedetails of the proposed waveforms by the 5GNOW groupAnother major activity that has performed a broad study ofFBMC is due to the PYHDYAS project (also in Europe) ThePHYDYAS contributors have published heavily on variousaspects of FBMC including prototype filter design equaliza-tion synchronization and application to MIMO channels Asummary of themajor findings of the PHYDYAS contributorsis presented in [21] and a complete list of their publicationscan be found at httpwwwict-phydyasorg

Themain thrust of this paper is to present a point of viewof FBMC and its future applications as seen by the authorWhile we acknowledge the presence of a large body of workson FBMC the paper details are geared towards the researchoutcomes of the author and his students in past 15 yearsThe paper emphasis is on the recent works of the authorand his students Many shortcomings of OFDM in dealingwith the requirement of the next generation of wirelesssystems are discussed and it is shown how FBMC overcomesthese problems straightforwardly We present a derivation ofFBMC systems that reveals the relationships among differentforms of FBMC A method of designing FBMC systems fora near-optimum performance in doubly dispersive channelsis presented and its superior performance over OFDM isshown The example considered is an underwater acousticchannel Application of FBMC technique to massive MIMOcommunications is introduced and its advantages in thisemerging technology are revealed Last but not the leastthe problems of channel equalization and synchronizationin FBMC systems are also given a special treatment and anumber of outstanding research problems in this field forfuture studies are identified

This paper begins with a historical overview of FBMCmethods in Section 2 In order to keep the presentation in thispaper a complement to thosewe have recently reported in [1938] the rest of this paper is organized as follows A summaryof the theoretical background based on which the various




1198952120587119891) the












f

T

tminus2Tn = minus2

minusTn = minus1

Tn = 1

2Tn = 2

minus1205872

minus1205872

minus1205872

120587

120587

120587

120587

1205872

1205872

0

0

minus1205872

minus1205872 120587

1205872

1205872

1205872

0

0

0









F = 12T

f = 74T k = 3

f = 54T k = 2

f = 34T k = 1

f = 14T k = 0


t = 0

f14T 34T 54T 74T


(a)

t = T

f14T 34T 54T 74T


(b)







119909 (119905) = sum

119899

sum

119896

119904119896 [119899] 119886119899119896 (119905) (1)

where

119886119899119896 (119905) = 119890119895120579119896[119899]

119901119899119896 (119905)

119901119899119896 (119905) = 119901 (119905 minus 119899119879) 119890119895((2119896+1)1205872119879)119905

(2)




⟨V1 (119905) V2 (119905)⟩ = int

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (3)


⟨V1 (119905) V2 (119905)⟩119877 = Rint

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (4)



⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877=

1 119899 = 119898 119896 = 119897

0 otherwise(5)


119904119896 [119899] = ⟨119909 (119905) 119886119899119896 (119905)⟩119877 (6)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895120579119896[119899]

119901119899119896 (119905) 119890minus119895120579119897[119898]

119901lowast119898119897 (119905) 119889119905

(7)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895(119898minus119899+119896minus119897)(1205872)

119901 (119905 minus 119899119879)


119889119905

(8)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= int

infin

minusinfin

1199012(119905) 119889119905

= 1

(9)



⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= (minus1)

119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(10)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(11)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119890119895(120587119879)119905

119889119905


infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) sin(

120587

119879

119905) 119889119905

= 0

(12)









s0(t)

s1(t)

sNminus1(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

ej((120587T)t+(1205872))

ej(Nminus1)((120587T)t+(1205872))

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

eminusj((120587T)t+(1205872))


y(t)x(t)

Rmiddot

Rmiddot

Rmiddot

RmiddotRmiddot







119904119896 (119905) = sum

119899

119904119896 [119899] 120575 (119905 minus 119899119879) (13)




119868119896[119899] +

119895119904119876


119904119868119896 (119905) = sum

119899

119904119868119896 [119899] 120575 (119905 minus 119899119879)

119904119876

119896 (119905) = sum

119899

119904119876

119896 [119899] 120575 (119905 minus 119899119879)

(14)




f

F = 1T

minusT2minusT T

T2

T2

t


minus1205872

minus1205872120587

1205871205872

12058720

0

minus1205872

120587

1205872

0minus1205872

120587

1205872

0

minus1205872

120587

1205872

0









f = 0 k = 0

f = 3T k = 3

f = 2T k = 2

f = 1T k = 1


t = 0

1T 2T 3T f


(a)

t = T2

1T 2T 3T f


(b)









sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot


Rmiddot


s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot


f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T



119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)




119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)


120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)




0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1198952120587119891) the












f

T

tminus2Tn = minus2

minusTn = minus1

Tn = 1

2Tn = 2

minus1205872

minus1205872

minus1205872

120587

120587

120587

120587

1205872

1205872

0

0

minus1205872

minus1205872 120587

1205872

1205872

1205872

0

0

0









F = 12T

f = 74T k = 3

f = 54T k = 2

f = 34T k = 1

f = 14T k = 0


t = 0

f14T 34T 54T 74T


(a)

t = T

f14T 34T 54T 74T


(b)







119909 (119905) = sum

119899

sum

119896

119904119896 [119899] 119886119899119896 (119905) (1)

where

119886119899119896 (119905) = 119890119895120579119896[119899]

119901119899119896 (119905)

119901119899119896 (119905) = 119901 (119905 minus 119899119879) 119890119895((2119896+1)1205872119879)119905

(2)




⟨V1 (119905) V2 (119905)⟩ = int

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (3)


⟨V1 (119905) V2 (119905)⟩119877 = Rint

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (4)



⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877=

1 119899 = 119898 119896 = 119897

0 otherwise(5)


119904119896 [119899] = ⟨119909 (119905) 119886119899119896 (119905)⟩119877 (6)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895120579119896[119899]

119901119899119896 (119905) 119890minus119895120579119897[119898]

119901lowast119898119897 (119905) 119889119905

(7)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895(119898minus119899+119896minus119897)(1205872)

119901 (119905 minus 119899119879)


119889119905

(8)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= int

infin

minusinfin

1199012(119905) 119889119905

= 1

(9)



⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= (minus1)

119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(10)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(11)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119890119895(120587119879)119905

119889119905


infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) sin(

120587

119879

119905) 119889119905

= 0

(12)









s0(t)

s1(t)

sNminus1(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

ej((120587T)t+(1205872))

ej(Nminus1)((120587T)t+(1205872))

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

eminusj((120587T)t+(1205872))


y(t)x(t)

Rmiddot

Rmiddot

Rmiddot

RmiddotRmiddot







119904119896 (119905) = sum

119899

119904119896 [119899] 120575 (119905 minus 119899119879) (13)




119868119896[119899] +

119895119904119876


119904119868119896 (119905) = sum

119899

119904119868119896 [119899] 120575 (119905 minus 119899119879)

119904119876

119896 (119905) = sum

119899

119904119876

119896 [119899] 120575 (119905 minus 119899119879)

(14)




f

F = 1T

minusT2minusT T

T2

T2

t


minus1205872

minus1205872120587

1205871205872

12058720

0

minus1205872

120587

1205872

0minus1205872

120587

1205872

0

minus1205872

120587

1205872

0









f = 0 k = 0

f = 3T k = 3

f = 2T k = 2

f = 1T k = 1


t = 0

1T 2T 3T f


(a)

t = T2

1T 2T 3T f


(b)









sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot


Rmiddot


s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot


f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T



119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)




119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)


120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)




0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










f

T

tminus2Tn = minus2

minusTn = minus1

Tn = 1

2Tn = 2

minus1205872

minus1205872

minus1205872

120587

120587

120587

120587

1205872

1205872

0

0

minus1205872

minus1205872 120587

1205872

1205872

1205872

0

0

0









F = 12T

f = 74T k = 3

f = 54T k = 2

f = 34T k = 1

f = 14T k = 0


t = 0

f14T 34T 54T 74T


(a)

t = T

f14T 34T 54T 74T


(b)







119909 (119905) = sum

119899

sum

119896

119904119896 [119899] 119886119899119896 (119905) (1)

where

119886119899119896 (119905) = 119890119895120579119896[119899]

119901119899119896 (119905)

119901119899119896 (119905) = 119901 (119905 minus 119899119879) 119890119895((2119896+1)1205872119879)119905

(2)




⟨V1 (119905) V2 (119905)⟩ = int

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (3)


⟨V1 (119905) V2 (119905)⟩119877 = Rint

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (4)



⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877=

1 119899 = 119898 119896 = 119897

0 otherwise(5)


119904119896 [119899] = ⟨119909 (119905) 119886119899119896 (119905)⟩119877 (6)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895120579119896[119899]

119901119899119896 (119905) 119890minus119895120579119897[119898]

119901lowast119898119897 (119905) 119889119905

(7)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895(119898minus119899+119896minus119897)(1205872)

119901 (119905 minus 119899119879)


119889119905

(8)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= int

infin

minusinfin

1199012(119905) 119889119905

= 1

(9)



⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= (minus1)

119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(10)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(11)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119890119895(120587119879)119905

119889119905


infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) sin(

120587

119879

119905) 119889119905

= 0

(12)









s0(t)

s1(t)

sNminus1(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

ej((120587T)t+(1205872))

ej(Nminus1)((120587T)t+(1205872))

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

eminusj((120587T)t+(1205872))


y(t)x(t)

Rmiddot

Rmiddot

Rmiddot

RmiddotRmiddot







119904119896 (119905) = sum

119899

119904119896 [119899] 120575 (119905 minus 119899119879) (13)




119868119896[119899] +

119895119904119876


119904119868119896 (119905) = sum

119899

119904119868119896 [119899] 120575 (119905 minus 119899119879)

119904119876

119896 (119905) = sum

119899

119904119876

119896 [119899] 120575 (119905 minus 119899119879)

(14)




f

F = 1T

minusT2minusT T

T2

T2

t


minus1205872

minus1205872120587

1205871205872

12058720

0

minus1205872

120587

1205872

0minus1205872

120587

1205872

0

minus1205872

120587

1205872

0









f = 0 k = 0

f = 3T k = 3

f = 2T k = 2

f = 1T k = 1


t = 0

1T 2T 3T f


(a)

t = T2

1T 2T 3T f


(b)









sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot


Rmiddot


s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot


f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T



119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)




119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)


120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)




0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












⟨V1 (119905) V2 (119905)⟩ = int

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (3)


⟨V1 (119905) V2 (119905)⟩119877 = Rint

infin

minusinfin

V1 (119905) Vlowast2 (119905) 119889119905 = 0 (4)



⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877=

1 119899 = 119898 119896 = 119897

0 otherwise(5)


119904119896 [119899] = ⟨119909 (119905) 119886119899119896 (119905)⟩119877 (6)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895120579119896[119899]

119901119899119896 (119905) 119890minus119895120579119897[119898]

119901lowast119898119897 (119905) 119889119905

(7)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= Rint

infin

minusinfin

119890119895(119898minus119899+119896minus119897)(1205872)

119901 (119905 minus 119899119879)


119889119905

(8)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= int

infin

minusinfin

1199012(119905) 119889119905

= 1

(9)



⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877= (minus1)

119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(10)


⟨119886119899119896 (119905) 119886119899119896 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119889119905

= 0

(11)


⟨119886119899119896 (119905) 119886119898119897 (119905)⟩119877

= R119895 (minus1)119903int

infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) 119890119895(120587119879)119905

119889119905


infin

minusinfin

119901 (119905 minus 119899119879) 119901 (119905 minus 119898119879) sin(

120587

119879

119905) 119889119905

= 0

(12)









s0(t)

s1(t)

sNminus1(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

ej((120587T)t+(1205872))

ej(Nminus1)((120587T)t+(1205872))

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

eminusj((120587T)t+(1205872))


y(t)x(t)

Rmiddot

Rmiddot

Rmiddot

RmiddotRmiddot







119904119896 (119905) = sum

119899

119904119896 [119899] 120575 (119905 minus 119899119879) (13)




119868119896[119899] +

119895119904119876


119904119868119896 (119905) = sum

119899

119904119868119896 [119899] 120575 (119905 minus 119899119879)

119904119876

119896 (119905) = sum

119899

119904119876

119896 [119899] 120575 (119905 minus 119899119879)

(14)




f

F = 1T

minusT2minusT T

T2

T2

t


minus1205872

minus1205872120587

1205871205872

12058720

0

minus1205872

120587

1205872

0minus1205872

120587

1205872

0

minus1205872

120587

1205872

0









f = 0 k = 0

f = 3T k = 3

f = 2T k = 2

f = 1T k = 1


t = 0

1T 2T 3T f


(a)

t = T2

1T 2T 3T f


(b)









sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot


Rmiddot


s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot


f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T



119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)




119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)


120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)




0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










s0(t)

s1(t)

sNminus1(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

ej((120587T)t+(1205872))

ej(Nminus1)((120587T)t+(1205872))

p(t)ej(1205872T)t

p(t)ej(1205872T)t

p(t)ej(1205872T)t

eminusj((120587T)t+(1205872))


y(t)x(t)

Rmiddot

Rmiddot

Rmiddot

RmiddotRmiddot







119904119896 (119905) = sum

119899

119904119896 [119899] 120575 (119905 minus 119899119879) (13)




119868119896[119899] +

119895119904119876


119904119868119896 (119905) = sum

119899

119904119868119896 [119899] 120575 (119905 minus 119899119879)

119904119876

119896 (119905) = sum

119899

119904119876

119896 [119899] 120575 (119905 minus 119899119879)

(14)




f

F = 1T

minusT2minusT T

T2

T2

t


minus1205872

minus1205872120587

1205871205872

12058720

0

minus1205872

120587

1205872

0minus1205872

120587

1205872

0

minus1205872

120587

1205872

0









f = 0 k = 0

f = 3T k = 3

f = 2T k = 2

f = 1T k = 1


t = 0

1T 2T 3T f


(a)

t = T2

1T 2T 3T f


(b)









sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot


Rmiddot


s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot


f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T



119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)




119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)


120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)




0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










f

F = 1T

minusT2minusT T

T2

T2

t


minus1205872

minus1205872120587

1205871205872

12058720

0

minus1205872

120587

1205872

0minus1205872

120587

1205872

0

minus1205872

120587

1205872

0









f = 0 k = 0

f = 3T k = 3

f = 2T k = 2

f = 1T k = 1


t = 0

1T 2T 3T f


(a)

t = T2

1T 2T 3T f


(b)









sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot


Rmiddot


s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot


f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T



119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)




119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)


120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)




0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










sI0(t)

jsQ0 (t)

sI1(t)

jsQ1 (t)

sINminus1(t)

jsQNminus1(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

SFB

ej((2120587T)t+(1205872))

sum


ej2120587f119888t

Channel



AFB

eminusj((2120587T)t+(1205872))

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]

T

y(t)x(t)

p(t)

p(t)

p(t)

p(t minus (T2))

p(t minus (T2))

p(t minus (T2))

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot


Rmiddot


s0(t)

s1(t)

sNminus1(t)

p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

SFB

sum


ej2120587f119888t

Channel



AFBs0[n]

s1[n]

sNminus1[n]

ej((2120587(1+120572))T)t

ej(Nminus1)((2120587(1+120572))T)t

eminusj((2120587(1+120572))T)

eminusj(Nminus1)((2120587(1+120572))T)

Rmiddot


f


(1 + 120572)T (2(1 + 120572))T (3(1 + 120572))T



119901 [119899]

=

1

120573119873 + 1

(1198960 + 2

120573minus1

sum

119897=1

119896119897 cos(2120587119897119899

120573119873 + 1

)) 0 le 119899 le 120573119873

0 otherwise(15)




119902 [119899] = 119901 [119899] ⋆ 119901 [minus119899] (16)


120585 = (119902 [0] minus 1)2+

1198702

sum

119898=1

1199022[119898119873] + 120574int

1minus((1+120572)2119873)

(1+120572)2119873

1003816100381610038161003816119875 (119891)

1003816100381610038161003816

2119889119891

(17)




0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

Frequency










120578 =

119879

119865

(19)



120578 =

Δ120591

Δ] (20)



119860119901 (120591 ]) = int

infin

minusinfin

119901(119905 +

120591

2


120591

2

) 119890minus1198952120587]119905

119889119905 (21)



119901119899119896 (119905) = 119901 (119905 minus 119899119879) 1198901198952120587119896119865119905

(22)

We note that

⟨119901119898119896 (119905) 119901119899119897 (119905)⟩

= int

infin

minusinfin

119901 (119905 minus 119898119879) 1198901198952120587119896119865119905

119901 (119905 minus 119899119879) 119890minus1198952120587119897119865119905

119889119905

prop 119860119901 ((119899 minus 119898)119879 (119897 minus 119896) 119865)

(23)




119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119860119901 (119899119879 119897119865) =

1 119899 = 119897 = 0

0 otherwise(24)


119860119901 (120591 0) = int

infin

minusinfin

119901(119905 +

120591

2

)119901(119905 minus

120591

2

) 119889119905 (25)


119860119901 (119899119879 0) =

1 119899 = 0

0 119899 = 0

(26)




2119905 and 120590



120590119905120590119891 ge1

4120587

(27)






119901 (119905) =

119871

sum

119896=0

120572119896ℎ4119896 (119905) (28)


ℎ119899 (119905) =1

(2120587)1198992

1198901205871199052 119889119899

119889119905119899119890minus21205871199052

(29)




119860119901 (120591 ]) = int

infin

minusinfin

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905

=

119871

sum

119899=0

119871

sum

119897=0

120572119899120572119897119860119899119897 (120591 ])

(30)

where

119860119899119897 (120591 ]) = int

infin

minusinfin

ℎ4119899 (119905 +120591

2

) ℎ4119897 (119905 minus120591

2

) 119890minus1198952120587]119905

119889119905 (31)

Defining

120572 = [1205720 1205721 sdot sdot sdot 120572119871]119879

A (120591 ]) =[

[

[

[

[


d

1198601198710 (120591 ]) 1198601198711 (120591 ]) sdot sdot sdot 119860119871119871 (120591 ])

]

]

]

]

]

(32)



119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119860119901 (120591 ]) = 120572119879A (120591 ])120572 (33)










f

2F

F

Tt

2T

minus2F










minus2T minusT

minusF




120577 = 1205740 int

A0

10038161003816100381610038161003816120572119879A(120591 ])120572 minus 1

10038161003816100381610038161003816

2119889120591 119889]

+

119871

sum

119896=1

120574119896 int

A119896

10038161003816100381610038161003816120572119879A(120591 ])120572

10038161003816100381610038161003816

2119889120591 119889]

(34)



f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










f

4F

2F

F

Tt

2T 4T

minus2F










increases


120577 = 119879119904119865119904(

119871

sum

119896=0

sum

(119898119879119904 119899119865119904)isinA119896

120574119896

10038161003816100381610038161003816120572119879A(119898119879119904 119899119865119904)120572 minus 119906119896

10038161003816100381610038161003816

2) (35)


119906119896 =

1 119896 = 0

0 119896 = 0

(36)
















035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










035

03

025

02

015

01

005

0


tT

Am

plitu

de

IOTA4

Hermite4Farhang4

(a)

Frequency

0

minus10

minus20

minus30

minus40

minus50

minus60

minus70

minus80

minus90

minus1000 01 02 03 04 05

Mag

nitu

de (d

B)

IOTA4

Hermite4Farhang4

(b)


50

45

40

35

30

25

20

15

10

50 002 004 006 008 01

SIR

(dB)

Δ120591Δ


FMT-cOFDM

D = 12

D = 23

D = 45











x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x[n]

E0(z)

E1(z)

ELminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0






)

119875(1198901198952120587(119891minus1119871)

) 119875(1198901198952120587(119891minus(119873minus1)119871)

) where 119875(1198901198952120587119891

)






x[n]

R0(z)

R1(z)

RLminus1(z)

L-pointFFT

s0[n]

s1[n]

sNminus1[n]

lowast

lowast


sI0(t)

jsI1(t)


sQ0 (t)

minusjsQ1 (t)


p(t)

p(t)

p(t)

p(t)

p(t)

p(t)

ej(2120587T)t


ej(2120587T)t


j

x[n]

T2delay

sum

sum




119876





sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










sI0[n]

jsI1[n]


0

0

0

0

SFB

SFB

sQ0 [n]

minusjsQ1 [n]


j

x[n]



s119868 [119899] = [1199041198680[119899] 119904

1198681[119899] sdot sdot sdot 119904


119879

s119876 [119899] = [1199041198760 [119899] sdot sdot sdot 119904


119879

(37)







y[n]

zL2

AFB

AFB

minusj

(minusj)Nminus1

lowast

lowast

lowast

lowast

sI0[n]

sI1[n]

sINminus1[n]

sQ0 [n]

sQ1 [n]

sQNminus1[n]

minusj

(minusj)Nminus1

Imiddot

Imiddot

Imiddot

Rmiddot

Rmiddot

Rmiddot







119876





x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x[n minus L + 1]

zminus1

zminus1

zminus1

x[n]

R0(z2)

R1(z2)

RLminus1(z2)

L-point

FFT

Equalizer

Equalizer

Equalizer

lowast

lowast

sI0[n]

sQ0 [n]

sI1[n]

sQ1 [n]

sINminus1[n]

sQNminus1[n]


darr L2

darr L2

darr L2






s0[n]

s1[n]

sNminus1[n]

x[n]

uarr K

uarr K

uarr K

ej(2120587L)n


sum

s0e[n]

s1e[n]

sNminus1e[n]

p[n]

p[n]

p[n]



119909 [119899] =

119873minus1

sum

119896=0

(sum

119898

119904119896 [119898] 119901 [119899 minus 119898119870]) 119890119895(2120587119896119899119871)

= sum

119898

(

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896119899119871)

)119901 [119899 minus 119898119870]

(38)


119899 = 120574119871 + ℓ (39)


119899 = 120578119870 + 120581 (40)


D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-point

IFFT

s0[n]

s1[n]

sNminus1[n]

0

0

x[n]



119878ℓ [119898] =

119873minus1

sum

119896=0

119904119896 [119898] 119890119895(2120587119896ℓ119871)

(41)


= 119890119895(2120587119896ℓ119871) We may


119909 [120578119870 + 120581] = sum

119898

119878ℓ [119898] 119901 [(120578 minus 119898)119870 + 120581]

120581 = 0 1 119870 minus 1

(42)


(119899)119871minus1(119911)



ℓ(119911) and 119863

(119899)ℓ




119910119896 [119899] = (119910 [119899] 119890minus119895(2120587119896119899119871)

) ⋆ 119901 [119899]

= sum

119898

119910 [119898] 119890minus119895(2120587119896119898119871)

119901 [119899 minus 119898]

(43)

y[n]

p[n]

p[n]

p[n]

eminusj(2120587L)n


y0[n]

y1[n]

yNminus1[n]

darr K

darr K

darr K

s0[n]

s1[n]

sNminus1[n]



119898 = 120574119871 + ℓ (44)


119910119896 [119899] = sum

120574

119871minus1

sum

ℓ=0

119910 [120574119871 + ℓ] 119890minus119895(2120587119896ℓ119871)

119901 [119899 minus 120574119871 minus ℓ]

=

119871minus1

sum

ℓ=0

(sum

120574


(45)



119899 minus ℓ = 120578119871 + 120581 (46)


119910119896 [119899] =

119871minus1

sum

ℓ=0

(sum

120574

119910 [120574119871 + ℓ] 119901 [(120578 minus 120574) 119871 + 120581]) 119890minus119895(2120587119896ℓ119871)

(47)



y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










y[n]

D(n)0 (z)

D(n)1 (z)

D(n)Lminus1(z)

L-pointFFT

y0[n]

y1[n]

yNminus1[n]

lowast

lowast


119863(119899)0 (119911) through119863


(119899)119871minus1(119911)







(119899)0 (119911) through119863






120588119879119909SMT =

log2119871 + 2120573

log2119871 (48)

120588119877119909SMT =

119871log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(49)

where 120588119879119909SMT and 120588




119879119909SMT = 18 and 120588

119877119909SMT = 255 For



120588119879119909FMT =

log2119871 + 2120573

log2119871 (50)

120588119877119909FMT =

(1198712) log2119871 + 120573119871 + 119872119873

(1198712) log2119871 + 119873

(51)


6 Equalization





















Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Equalizerbank

s[n]









45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










45

40

35

30

25

20

15

10

50 2000 4000 6000 8000 10000

SIR

(dB)

Symbol index





8 Applications











9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










9 Conclusions




References












































































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






























































































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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