分数域信号与信息处理及其应用 (9).pdf

158IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.56,NO.1,JANUARY 2008Sampling and Sampling Rate Conversionof Band Limited Signals in the FractionalFourier Transform DomainRan Tao,Senior Member,IEEE,Bing Deng,Wei-Qiang Zhang,Student Member,IEEE,and Yue WangAbstractThefractional Fourier transform(FRFT)has becomea very active area in signal processing community in recent years,with many applications in radar,communication,information se-curity,etc.,This study carefully investigates the sampling of a con-tinuous-timebandlimitedsignaltoobtainitsdiscrete-timeversion,as well as sampling rate conversion,for the FRFT.Firstly,basedon product theorem for the FRFT,the sampling theorems and re-construction formulas are derived,which explain how to sample acontinuous-time signal to obtain its discrete-time version for bandlimited signals in the fractional Fourier domain.Secondly,the for-mulas and significance of decimation and interpolation are studiedin the fractional Fourier domain.Using the results,the samplingrate conversion theory for the FRFT with a rational fraction asconversion factor is deduced,which illustrates how to sample thediscrete-time version without aliasing.The theorems proposed inthis study are the generalizations of the conventional versions forthe Fourier transform.Finally,the theory introduced in this paperis validated by simulations.Index TermsFractional Fourier transform(FRFT),samplingrate conversion,sampling theorem.NOMENCLATUREFTFourier transformFRFTFractional Fourier transform.DTFRFTDiscrete-time FRFT.Notation for the FRFT.Transform result offor the FRFT.Notation for the DTFRFT.Transform result offor the DTFRFT.I.INTRODUCTIONTHE fractional Fourier transform(FRFT)has a historydating back to the 1930s 1.It was then employed byNamias to solve some differential and partial differentialManuscript received May 20,2006;revised April 27,2007.The associateeditor coordinating the review of this manuscript and approving it for publi-cation was Dr.Antonia Papandreou-Suppappola.This work was supported inpart by the National Science Foundation of China under Grants 60232010 and60572094,and in part by the National Science Foundation of China for Distin-guished Young Scholars under Grant 60625104.R.Tao and Y.Wang are with the Department of Electronic Engineering,Bei-jing Institute of Technology,Beijing 100081,China(e-mail:).B.Deng is with the Department of Electronic Engineering,Beijing Insti-tute of Technology,Beijing 100081,China,and also with the Department ofElectronic Engineering,Naval Aeronautical Engineering Institute,Yantai City264001,China(e-mail:navy_).W.-Q.Zhang is with the Department of Electronic Engineering,TsinghuaUniversity,Beijing 100084,China(e-mail:).Digital Object Identifier 10.1109/TSP.2007.901666equations in quantum mechanics from classical quadraticHamiltonians 2.The results were later improved by McBrideand Kerr 3.They developed operational calculus to definethe FRFT.The FRFT is a generalization of the conventionalFouriertransform(FT)with potential applications.But,withoutproper physical illumination and fast digital computation algo-rithm,the methodology had remained unknown to the signalprocessing community until the introduction of the efficientdigital computational algorithms of the FRFT and the inter-pretation as rotation in the time-frequency plane 410.TheFRFT processes signals in a unified time-frequency domain.Comparing with the FT,the FRFT is more flexible and suitablefor processing nonstationary signals due to an additional degreeof freedom.Furthermore,the fast algorithm of the discreteFRFT has also been proposed.Therefore,the FRFT has beenwidely applied in radar,communication,information security,etc.518.The FRFT of theorder can be interpreted as a rotation inthe time-frequency plane with an angle,and the time domainand frequencydomain arethe specialcasesof theFRFTdomainwithbeingand,respectively,whereis aninteger 4.Hence,the conventional Shannon sampling theoremfor the FT can also be considered as a special case of the sam-plingtheoremfortheFRFT1923.Basedontherelationshipbetween the FT and FRFT,i.e.,the three decomposition stepsof the FRFT 4,Xia 19,Zayed 20 and Erseghe et al.21,independently generalized the classical Shannon theorem fromthe frequency domain to the FRFT domain.Based on chirp-pe-riodicity Erseghe generalized the FT for continuous-time,peri-odic continuous-time,discrete-time,periodic discrete-time sig-nals to four corresponding versions of the FRFT 21.In 22,Shannonsinterpolation theorem was generalizedfor the FRFT.It was concluded that a signal limited in a certain FRFT domaincan be represented by its samples in any other FRFT domain.In23,Zayedderivedtwosamplingformulastoreconstructabandlimited or time limited signal,which use samples from both thesignalanditsHilberttransformationsampledathalftheNyquistrate.Since signals are always processed within a finite timeinterval and a finite bandwidth in practical engineering appli-cations,signal sampling based on the conventional Shannonsampling theorem can meet the criteria of ideal reconstruction.However,the sampling theorem for the FRFT shows that thesampling method is not always efficient with the possibility ofunnecessary computational cost.In other words,a signal canpotentially be sampled with a rate less than the Nyquist ratewithout aliasing of signals FRFT.1053-587X/$25.00 2007 IEEETAO et al.:SAMPLING AND SAMPLING RATE CONVERSION OF BAND LIMITED SIGNALS159The sampling theorems mentioned earlier explain how tosample a band limited signal without aliasing.The advances indigital signal processing necessitates performing more complexsignal processing operations such as coding,transmitting andstoring.In order to reduce the computational load as wellas saving the storage space,different sampling rates and theconversion between them are typically required in a signalprocessing system.Under these circumstances,the theoryof multirate signal processing was introduced and improved24.This theory explains how to implement the samplingrate conversion from a discrete-time signal to another.Theconventional sampling rate conversion theorem is operatedin the FT domain,which helps to eliminate spectral aliasingdue to decimation and mirror because of interpolation.If asignal is sampled by using the sampling theorem for the FRFT,the conventional sampling rate conversion theorem can notguarantee undistorted sampling rate conversion.Therefore,several questions are raised,such as 1)how to generalize theconventional sampling rate conversion theorem?and 2)how toachieve undistorted sampling rate conversion by eliminatingthe influences on signals FRFT caused by interpolation anddecimation?In this paper,the sampling theorem for band limited signalsin the FRFT domain is deduced from the viewpoint of signalsand systems.Then,some discussions are presented to demon-strate its implementation.Next,we propose the generalizationof conventionalsamplingrateconversion theorem,i.e.,thesam-pling rate conversion theorem with a factor of rational fractionfortheFRFT.Afterthat,simulationsarepresented.Finally,con-clusions are given.II.PRELIMINARIESThe continuous-time FRFT with angleof a signalisdefined as 4(1)whereindicatestherotationangleinthetime-frequencyplane,isthekernelfunction,shownin(2)atthebottomofthepage,where.The discrete-time FRFT(DTFRFT)is defined as 21(3)Its inverse is(4)whereis the sampling period,denotes the integral in-terval with the width.The FRFT has the following four special cases:(5)(6)(7)(8)wheredenotes the FT of.The time domain is theFRFT domain with,while the frequency domain isthe FRFT domain with.Since the FRFT isperiodic with the period of,can be limited in the interval.In this paper,the FRFT withis not takeninto consideration,and if,the following result isobtained:(9)Therefore,it can be assumed that,i.e.,to analyze the fractional power spectrum,i.e.,the squared mod-ulus of the FRFT.In Sections IIIV,this assumption is em-ployed.According to 4 and 25,theth order FRFT can be inter-preted as a rotation in the time frequency plane having an angle.Considering the Wigner distribution,we can write(10)whereandindi-cate the Wigner distributions ofand,respectively,andindicates the operator which rotates a two dimensional functioncounterclockwise by an angle of.Lohmann generalized(10)andobtainedtherelationshipbetweenthefractionalpowerspec-trum and the RadonWigner transform 26 as(11)wheredenotestheRadontransform,i.e.,theoperatorhavingthe integral projection of a two dimensional function onto anaxis making angle ofwith theaxis.Another interpretation(2)160IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.56,NO.1,JANUARY 2008Fig.1.Relationship between the FRFT and the RadonWigner transform of asignal?.of(11)is the marginal integral after rotating the coordinates byan angle,i.e.,(12)The above mentioned relationship expressed by(11)and(12)is shown in Fig.1.The product theorem for the FRFT given below is utilized inSectionIII.Producttheorem27tellsthatif,then(13)III.SAMPLINGTHEOREM OF THEFRFTFOR ABANDLIMITEDSIGNALDefinition of the Band Limited Signal in the FRFT DomainDefinition 1:If the FRFT of signalsatisfies the fol-lowing condition:whenor(14)thenis the band limited signal in the FRFT domain,whosebandwidth is defined as(15)NotethatthebandlimitedsignalreferstotheoneintheFRFTdomain with,i.e.,the FT domain,which is one of thespecial cases of our definition in this paper.A.FRFT of Sampled SignalsThe uniform impulse train is defined as(16)Fig.2.Effect of sampling.(a)Continuous-time signal?.(b)FRFT of?,denoted by?.(c)Sampled signal?.(d)FRFT of?,denoted by?.wheredenotes unit impulse.Then we obtain the sampledsignal(17)Based on the product theorem expressed by(13),we get thefollowing results:(18)Ifissubstituted into(18),after the corresponding manipulations,weobtain(19)Since,we have(20)Equation(20)shows thatreplicates with a period of,along with linear phase modulation depending onthe harmonic order(as shown in Fig.2).WhenTAO et al.:SAMPLING AND SAMPLING RATE CONVERSION OF BAND LIMITED SIGNALS161The expression above shows that the part ofwithmodulates the amplitude ofbybut it does notmodulate the phase.WhenThe equation above shows that the part ofwithmodulates linearly the phase ofin addition to ampli-tude modulation by.Since the main interval isinthe digital frequency domain,the linear phase modulation is de-picted as the polygonal line in Fig.2(d).For general casesTheformulashowsthattheabsolutevalueoftherateoflinearphase modulation increases asincreases,which means thatthe slope of the polygonal line also increases correspondingly.B.Sampling TheoremSincereplicates with a period of,and from the definition of band limited signal inthe FRFT domain,we have,whenor.Then,as illustrated in Fig.3(b),we candetermine an appropriate value for,that satisfies(21a)(21b)whererounds elements to the nearest integers towardszero.Therefore,therewillbenooverlappingbetweentheshiftedversions of,when the sampling rateis determined by(22a)When,which corresponds to the lowpass case,(21b)always holds true.Therefore,from(21a)we have(22b)Fig.3.Reconstruction of a signal.(a)FRFT of the original signal,?.(b)FRFT of the sampled signal,?.(c)Ideal bandpass filter in the FRFTdomain.(d)FRFT of the reconstructed signal,?.The sampling theorem for the band limited signal for the FRFTis expressed by(22b).Let,we have,then thelowpass version of sampling theorem for the FRFT is expressedas follows:(23)Furthermore,assuming,hence,in(22b)and(23),respectively,we obtain the conventional bandpass andlowpass sampling theorems 28.C.Reconstruction FormulasAccording to the sampling formulas mentioned earlier in thispaper,the ideal reconstruction of the signalfrom its sam-pled version can be realized through a bandpass filter in theFRFT domain,as shown in Fig.3.The transfer function of thebandpass filter isotherwise.(24)Thus,the reconstructed signal is derived as follows:(25)162IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.56,NO.1,JANUARY 2008Also(26)Substitute(26)into(25)(27)According to(17),(28)follows,shown at the bottom of thepage.For a lowpass signal,let,and.Then,(28)becomes(29)Theresultisthesameasthatobtainedin19,20.Correspond-ingly,(28)and(29)are changed to the conventional bandpassand lowpass reconstruction formulas when.D.DiscussionThe sampling theorem derived in Section III-C is verified inthis subsection by using a chirp signal as an example.Chirp-like signals are often encountered in signal processing,existingas both real-world and artificial signals.Also,chirp-like sig-nals can be interpreted as the first order approximation of fre-quency varying signals.Therefore,analysis and processing ofchirp-like signals gained considerable importance in signal pro-cessing 29.According to the relationship between the FRFTand time-frequency distribution,we examine the FRFT domainsuperimposed on the time-frequency plane with the 0th FRFTdomain as the time domain and theth FRFT domain as thefrequency domain.It is assumed that the time-frequency distribution of a chirpsignalthat is symmetrical around the origin as shown bythe highlighted line in Fig.4.The symmetry around the originFig.4.Superposition of the FRFT domain on the time-frequency domain ofthe chirp signal?.?and?indicate the transform orders of the FRFT;?is the counterclockwise angle between the Wigner distribution of the signaland the time axis.?and?are the projections from the top of the Wignerdistribution?on?and?axes,respectively.can be achieved by time shift and frequency shift.In this case,the sampling rate does not have to be greater than(where,is the highest frequency of the signalin thefrequencydomain)accordingtotheclassicalsamplingtheorem.On the other hand,the sampling rate should not to be less than(where,is the highest“frequency”of the signalin theth FRFT domain)according to thesampling theorem for the FRFT.The reconstruction filters(thelowpass filters)must be designed in the frequency domain ortheth FRFT domain depending on what sampling theorem isused.Obviously,the sampling rate can be less thanonly ifthe inequality is satisfied asi.e.,(30)From Fig.4,and(31)orand(32)whereequalsthechirprate.Combinethetwoequationsabove(33)(28)TAO et al.:SAMPLING AND SAMPLING RATE CONVERSION OF BAND LIMITED SIGNALS163Substitute(33)into(30)Finally,we obtain(34)The inequality of(34)holds true ifandare in the samequadrant.Thus,it is feasible to sample a chirp signal with thesampling frequency less thanwithout aliasing in thethFRFT domain only ifandare in the same quadrant.Therelative cut-off frequency of the lowpass reconstruction filter intheth FRFT domain is limited in the interval as follows:(35)Although Fig.4 shows the case that the chirp rate is less thanzero,the same conclusion can be drawn for the case that thechirp rate is greater than zero.IV.SAMPLINGRATECONVERSIONBASED ON THEFRACTIONALFOURIERTRANSFORMSection III explains how to sample a continuous-time bandlimited signal to obtain its discrete-time version without over-lapping in its fractional power spectrum.Secti