RELATIONS BETWEEN GABOR TRANSFORMS AND FRACTIONAL FOURIER TRANSFORMS AND THEIR APPLICATIONS FOR SIGNAL PROCESSING


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1 RELATIONS BETWEEN ABOR TRANSFORMS AND FRACTIONAL FOURIER TRANSFORMS AND THEIR APPLICATIONS FOR SINAL PROCESSIN SooChang Pi, JianJiun Ding Dpartmnt o Elctrical Enginring, National Taiwan Univrsity, No., Sc. 4, Roosvlt Rd., 67, Taipi, Taiwan, R.O.C TEL: , Fax: , ABSTRACT Many wondrul rlations btwn th abor transorm and th ractional Fourir transorm (FRFT, which is a gnralization o th Fourir transorm, ar drivd. First, w ind that, as th Wignr distribution unction (WDF, th FRFT is also quivalnt to th rotation opration o th abor transorm. W also driv th shiting, th proction, th powr intgration, and th nrgy sum rlations btwn th abor transorm and th FRFT. Sinc th abor transorm is closly rlatd to th FRFT, w can us it or analyzing th ct o th FRFT. Compard with th WDF, th abor transorm dos not hav th problm o cross trms. It maks th abor transorm a vry powrul assistant tool or ractional sampling and dsigning th iltr in th FRFT domain. Morovr, w show that any combination o th WDF and th abor transorm also has th rotation rlation with th FRFT.. INTRODUCTION Th ractional Fourir transorm(frft is dind as [][]: ( t F u O F u cot ut csc t cot ( t dt cot. ( π It is a gnralization o th Fourir transorm (i.., π/: π / ut FT ( t O ( t /π ( t. ( [ ] ( dt F Th FRFT can xtnd th utilitis o th Fourir transorm (FT. It is usul or iltr dsign, pattrn rcognition, optics analysis, radar systm analysis, communication, tc. Th Wignr distribution unction (WDF is [3] ωτ ω / π ( t τ / ( t τ / dτ * W. (3 I W (t, ω and W F (t, ω ar th WDFs o (t and its FRFT, F (u, thn thy hav th ollowing rlation [4][]: W u, v W u cos v sin, u sin v cos, (4 F That is, W F (u, v is a rotation o W (t, ω. Sinc WDFs and th FRFTs hav such a clos rlation, w otn us WDFs to aid FRFTs or signal procssing applications. For xampl, or iltr dsign in th FRFT domain, WDFs ar hlpul or stimation th optimal ordr []. Cohn s class distributions also hav th rotation rlation with th FRFT []. Howvr, thr is a problm or th WDF and Cohn s class distributions, i.., th cross trm. This can b sn rom Figs. (h, 3(b, and (h. It maks us hard to distinguish th signal part, th nois part, and th crosstrm part. Thus it is hard to us th WDF and Cohn s class distributions to analyz th FRFT charactrs whn th signal consists o multipl timrquncy componnts. In this papr, w driv th rlations btwn th abor transorm and th FRFT. Th abor transorm is [6][7]: ( τ t t ω( τ ( t, ω ( τ dτ. ( π W ind that, as th WDF, th FRFT also corrsponds to th rotation opration or th abor transorm. Thus as th WDF, th abor transorm can also b usd or analyzing th charactrs o signals in th FRFT domain. Comparing with th WDF, th abor transorm has th advantags o ( avoiding th crosstrm problm, ( lss computation tim. Espcially, du to th ability o avoiding th crosstrm, th abor transorm is mor ctiv than th WDF or iltr dsign (s Sction 4. Without th mislading o th crosstrm, th optimal paramtr o th FRFT will b much asir to dtrmin. This problm prplxs th rsarchrs on th ild o th FRFT or many yars. Now w can us th abor transorm to solv it succssully. Morovr, in Sction, w ind that, in addition to th WDF and th abor transorm, i w do arbitrary combination or th abor transorm and th WDF, th rsultant transorm also hav th rotation rlation with th FRFT.. ROTATION RELATION [Rotation Rlation]: I F (u is th FRFT o (t, thn thir abor transorms (dnotd by (t, ω and F (u, v hav th rlation o: F u, v u cos v sin, u sin v cos. (6 That is, th FRFT corrsponds to rotating th abor transorm in th clockwis dirction with angl. ( τ u u v( τ τ d ( τ u u v( τ τ cot τ x csc (Proo: v / π F ( τ F cot π x cot ( x dx dτ
2 cot τ ( cot ( csc [ τ u v x ] dτ. π Thn, applying th act that ( a b τ τ dτ π / a (rom R. [8], w can rwrit (7 as: F π u u v x cot ( u v xcsc b / 4a ( xdx u uv x cot, (7 ( cot ω ( x π π π (8 dx ( u v sin x v x sin v t sin ux sin ( u v sin cos x cot u v sin cos ux cos ω x cos u u v x cot ( x ( xdx ( u cos ω sin x uxcos v xsin u v(cos sin ( u v sin cos x( u sin ω cos ( x [ x( u cos vsin ] u cos v sin ( u sin v cos ( x dx It lads to (6 # W prorm an xprimnt in Fig. to show that th FRFT is quivalnt to th rotation rlation o th abor transorm. Although th WDF also hav th similar proprty, it has th problm o cross trm. I w us th abor transorm instad o th WDF, th cross trm problm can b avoidd. That is, i (t s(t r(t ( and W (t, ω, W s (t, ω and W r (t, ω ar th WDFs o (t, s(t, and n(t, rspctivly, thn W (t, ω W s (t, ω W r (t, ω. ( This is bcaus th ormula o th WDF contains th auto corrlation trm (tτ/*(t τ/, s (. I (t s(t r(t, (tτ/*(t τ/ s(tτ/s*(t τ/ r(tτ/r*(t τ/ s(tτ/r*(t τ/ r(tτ/s*(t τ/. Du to th cross trms s(tτ/r*(t τ/ and r(tτ/s*(t τ/, W (t, ω will not b th sum o W s (t, ω and W r (t, ω. In contrast, whn using th abor transorm, th auto corrlation is avoidd, (s (. I (t, ω, s (t, ω and r (t, ω ar th abor transorms o (t, s(t, and n(t, thn (t, ω s (t, ω r (t, ω, ( and th problm o cross trm can b avoidd. In Fig., w do som xprimnts to show th abor transorms and th WDFs o s(t, r(t, and (t s(tr(t, whr s( t xp( t / 3t or 9 t, s(t othrwis, (3 dx. (9 (a alpha (b alpha pi/6 (c alpha pi/ (dalpha pi/ ( alpha pi/3 ( alpha pi/ Fig. Th abor transorm o F (u whr F (u is th FRFT o a rctangular unction (t, (t or t 3, and (t othrwis. W us gray lvl to show th magnitud o F (u, v. (a T o s(t (b T o r(t (c T o (t (d WDF o s(t ( WDF o r(t ( WDF o (t Fig. Th abor transorm (T and th Wignr distribution unctions (WDF o s(t, r(t, and (t s(tr(t. Not th WDF has th cross trm problm but th T dos not. (a T o xp(t 3 / (b WDF o xp(t 3 / Fig. 3 Th T and th WDF or (t xp(t 3 /. ( t xp( t / 6t xp[ ( t 4 /] r. (4 Although whn (t xp(at bt, th rsolution o th abor transorm may not b as good as th WDT, it has an important advantag o avoiding th crosstrm. Morovr, i (t xp(at k rmaind trms, k 3 and a, ( th abor transorm will hav bttr rsolution than th WDF, as th xampl in Fig. 3.
3 3. PROPERTIES AND IMPLEMENTATION [Rcovry Rlation]: W can rcovr (t rom t (t, ω by: ω t ω dω ( t. (6 π This rlation can b gnralizd into th cas o th FRFT. In (6, i w rplac (t by F (u, thn π π F u v v dv F ( u ( ucos vsin, usin vcos, uv ( dv u. (applying (6. (7 That is, w can obtain F (u rom th abor transorm o (t i w do th scald invrs FT along th dirction o ( sin, cos. (8 [Proction Rlation]: Th rcovry rlation or (t in (6 can b gnralizd into: ( k/ t k ( t ω ω t dω π, ((/ k t. (9 In (9, i w rplac (t by F (u and apply (6, it bcoms k u v ( u v u v dv cos sin, sin cos π ( k/ t (( k / u F ( Whn k /, it bcoms (7. [Powr Intgration Rlation]: I w intgrat th powr o (t, ω along ωaxis, thn ( τ ω dω ( τ t dτ. ( For th cas o th FRFT, w rplac (t in ( by F (u: ( τ u ( u v dv F ( τ τ F, d, ( Thn w apply (6, u cos v sin, u sin v cos dv ( ( τ u F ( τ dτ F. (3 Thror, th powr intgrating along th lin o u(cos, sin v( sin, cos, v (,, (4 will b th local nrgy o F (τ around τ u. [Enrgy Sum Rlation]: From (3, ( τ t t, ω dω dt τ dτ dt, ( ( ( t ω, dω dt π ( τ dτ. ( Thn, rom th act that th nrgy is prsrvd atr rotation and th Parsval s thory o th FRFT [], ( can b gnralizd as ( t d dt F ( u F, ω ω π β du or any, β. [Powr Dcayd Rlation]: I F (u or u > u, thn (6 ( u cos v sin, u sin v cos dv ( u cos vsin, u sin vcos ( uu < dv (7 [Shiting and Modulation Rlations]: I h(t (t t, thn thir abor transorms hav th ollowing rlation: ω t / h ω ( t t, ω, (8 I h(t (txp(ω t, thn ωt / h ω ω ω, (9 nrally, i H (u and F (u ar FRFTs o h(t and (t and H (u F (u u, thn ( t sinω cos u ω ( t u cos, ω u sin h. (3 [Discrtization] Whn doing digital implmntation, in (, w st τ mδ t, t nδ t, and ω sδ ω. To prsrv th rotation proprty and othr proprtis, it is propr to choos δ t δ ω < π / Max(B, whr B is th supporting width o F (u >. (3 x / Morovr, sinc <. whn x > 4.9, rom (, to sav th computational tim, instad o varying m rom to, w can st th rang o m as n 4.9/δ t < m < n 4.9/δ t. (3 4. APPLICATIONS FOR SINAL PROCESSIN IN THE FRFT DOMAIN Sinc th abor transorm is closly rlatd to th FRFT, w can us it as an assistant tool or signal procssing in th FRFT domain. W giv two xampls: (A ractional sampling and (B ractional iltr dsign to show how to us th abor transorm togthr with FRFTs or signal procssing. Whn using th FRFT to do signal sampling [9], w irst try to ind such that th supporting o F (u is minimal supporting Ω : F (u < i u Ω, optimal width(ω is minimal. (33 Whn (t has only on timrquncy (TF componnt, w can us th WDF to stimat width(ω and hnc sarch th optimal. Howvr, whn (t has two or mor TF componnts, as th xampl in Fig. (, using th WDF to stimat may not b propr. In this cas, sinc th orint or ach o th TF componnt is dirnt, it is propr to sparat (t into svral TF componnts and dtrmin th optimal or ach o th componnt. Du to th cross trm, this work is hard to b don by th WDF. Not that, in Fig. (, it is hard to conclud whthr th cntral rgion around (, is th third TF componnt o (t or ust th cross trm o th lt part and th right part. In contrast, whn using th abor transorm, sinc thr is no cross trm, w can asily conclud how may TF componnts (t contains and dcompos (t into th summation o ths componnts. For th xampl in Fig., (a First, w dcompos (t, ω in Fig. (c into th lt part and th right part, which corrsponds to s(t and r(t.
4 dcomposition (t lt part s(t right part r(t Fig. 4 Using th FRFT togthr with th timrquncy componnt dcomposition by th abor transorm to sampl th signal (t (s Fig. in th FRFT domain. (b Thn w us th powr intgration rlation to stimat th optimal ordr (dind by (33 or th lt and th right TF componnts lt .373, right.78. (34 (c Thn, w apply th algorithm in [9] to do ractional sampling or ach o th componnts. Whn th numbr o sampling points is ixd to, th rconstruction rrors ar Using th convntional sampling thory: rr.763% Sampling (t in th FRFT domain:, rr.36%, Sampling (t in th FRFT domain with th aid o th abor transorm: rr.6%. (3 Thn w discuss how to us th abor transorm to dsign th iltr in th FRFT domain. It is known that w can us th FRFT instad o th FT or iltr dsign [9], i.., r( t OF { OF [ h( t ] T ( u } (36 whr h(t and r(t ar th input and th output o th iltr, rspctivly. Although th way or sarching th optimal transr unction T (u has bn dvlopd [][], it lacks an icint way to dtrmin th optimal ordr. It sms that th WDF is hlpul or dtrmining and th cuto lin, howvr, du to th cross trm problm, using th WDF is not suitabl or th cas whr h(t consists o a lot o timrquncy (TF componnts. In this papr, w ind that th abor transorm also has th rotation rlation with th FRFT and it can avoid th cross trm problm. This hints that w can us th abor transorm instad o th WDF or ractional iltr dsign. W giv an xampl in Fig.. Th input signal is s( t cos( t xp( t / (shown in Fig. (a. (37 It is intrrd by th ollowing nois.3t.3t ractional sampling with ractional sampling with.78 n( t. cos(t, (38 and h(t s(t n(t is plottd in Fig. (b. W want to rcovr s(t rom h(t by th iltr in th FRFT domain. Th WDFs o s(t and h(t ar plottd in Fig. (g(h. In Fig. (h, th signal parts, th nois parts, and th cross trm parts ar mixd togthr. It is hard to know how to sparat th signal parts rom th nois parts atr obsrving Fig. (h. In contrast, whn doing th abor transorm, sinc th problm o cross trm can b avoidd, in Fig. (d, th signal parts and th nois parts ar sparatd clarly. W can us th ollowing our lins (in Fig. ( to sparat thm: L: (cos, sin k(sin, cos,.4, L: (cos, sin k(sin, cos L3: 7.(cosβ, sinβ k(sinβ, cosβ, β.3, L4: 7.(cosβ, sinβ k(sinβ, cosβ, (39 (a signal s(t   (c T o s(t (b h(t s(t nois   (d T o h(t ( cuto lin L3 ( Rcovrd signal L L L4   (g WDF o s(t   (h WDF o h(t Fig. Using th FRFT with th abor transorm or iltr dsign. Thror, w can us th ollowing procss to iltr out th nois and rcovr s(t rom h(t. Stp (b coms rom L and L and Stp (d coms rom L3 and L4: (a H ( u OF [ h( t], (4 (b H, (u H (u or u, H, (u or u <, (4 β (c H, β ( u OF [ H, ( t], (4 (d H,β (u H,β (u or u 7., H,β (u or u > 7., (43 β ( s( t OF [ H, β ( u]. (44 Th rcovrd signal is plottd in Fig. (. It is vry clos to th original signal s(t and th rror is only.49%. W giv anothr xampl in Fig. 6. Th input is in Fig. 6(a. It is intrrd by th nois with th 3 rd ordr phas. 3 n( t.9 xp(.t t. (4 Whn using th WDF, th nois and th signal parts ar mixd, s Fig. 6(c. Whn w us th abor transorm, th nois and th signal parts ar obviously sparabl, s Fig 6(d. Sinc th nois and th signal parts can b sparatd by thr lins, w can us thr ractional iltrs to iltr out th nois. Th ordrs o FRFTs or ths ractional iltrs can b dtrmind by th slops o th cuto lins (.6947,.78, With thm, w can rcovr th original signal with a vry small rror, s Fig. 6(b.
5 (a Input signal (b Rcovrd signal rr.946%   (c WDF o signalnois (d abor transorm Fig. 6 Using th FRFT iltr to iltr output th 3 rd ordr phas xponntial unction (a (b Fig. 7 Th WTs o (t whr (t s(tr(t is dind in (3 and (4 and th WTs ar dind in (49 and (.. ABORWINER TRANSFORM W hav known that both th abor transorm and th WDF has th rotation rlation with th FRFT. In act, i w combin th abor transorm with th WDF proprly, th nw timrquncy distribution also has th rotation rlation with th FRFT. [Combination Thorm]: Suppos that p(x, y is any unction with two variabls. I w din a nw tim rquncy C (t, ω (W call it th aborwignr transorm (WT that has th ollowing rlation with th abor transorm (t, ω and th WDF W (t, ω: C ω p( ω, W ω, (46 thn C (t, ω also has th rotation rlation with th FRFT: C F v C ( u cos v sin, u sin v cos, (47 whr C (t, ω and C F (u, v ar th WT o (t and its FRFT, F (u, rspctivly. [Proo]: From (46, CF v p( F v, WF v (48 p( ( u cos v sin, u sin v cos, W ( u cos v sin, u sin v cos C F u cos v sin, u sin v cos. # ( Sinc th WT also has rotation rlation, thus it is possibl to us it instad o th WDF and th abor transorm or signal analysis in th FRFT domain. In act, i p(x, y is chosn proprly, th rsultant WT will combin th advantags o th abor transorm and th WDF. In Fig. 7, w prorm svral xprimnts. For Fig. 7(a, C ω ω W ω, (49 For Fig. 7(b, C ω min( ω, W ω, ( That is, p(x, y xy in (49 and p(x, y min( x, y in (. Compar Fig. 7 with Fig., w ind that, as th abor transorm, whn using th WT, th cross trm problm can also b avoidd. Morovr, its rsolution is obviously bttr than that o th abor transorm. It combins both th advantag o th WDF (highr rsolution and th advantag o th abor transorm (no cross trm. 6. CONCLUSIONS W drivd svral intrsting rlations btwn th FRFT and th abor transorm, including th rotation rlation, th rcovry rlation, and th powr intgration rlation. Sinc th abor transorm can avoid th cross trm, which is a srious problm or th WDF, w could us it instad o th WDF to do signal procss in th FRFT domain, such as ractional sampling and th ractional iltr dsign. 7. REFERENCES [] H. M. Ozaktas, Z. Zalvsky, M. A. Kutay, Th Fractional Fourir Transorm with Applications in Optics and Signal Procssing, Nw York, John Wily & Sons,. [] L. B. Almida, Th ractional Fourir transorm and timrquncy rprsntations, IEEE Trans. Signal Procssing, vol. 4, no., pp , Nov [3] T. A. C. M. Classn and W. F.. Mcklnbraukr, Th Wignr distributiona tool or timrquncy signal analysis; Part I, Philips J. Rs., vol. 3, p. 7, 98. [4] A. W. Lohmann, Imag rotation, Wignr rotation, and th ractional Fourir transorm, J. Opt. Soc. Am. A, vol., no., pp. 886, Oct [] S. C. Pi and J. J. Ding, Rlations btwn th ractional oprations and th Wignr distribution, ambiguity unction, IEEE Trans. Signal Procssing, vol. 49, pp ,. [6] D. abor, Thory o communication, J. Inst. Elc. Eng., vol. 93, pp , Nov [7] M. J. Bastiaans, abor s xpansion o a signal into aussian lmntary signals, Proc. IEEE, vol. 68, pp , 98. [8] M. R. Spigl, Mathmatical Handbook o Formulas and Tabls, McrawHill, 99. [9] X.. Xia, On bandlimitd signals with ractional Fourir transorm, IEEE Signal Procssing Lttrs, vol. 3, no. 3, pp. 774, March 996. [] M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, Optimal iltr in ractional Fourir domains, IEEE Trans. Signal Procssing, v.4, n., pp. 943, 997. [] H. M. Ozaktas, N. Erkaya, and M. A. Kutay, Ect o ractional Fourir transormation on timrquncy distributions blonging to th Cohn class, IEEE Trans. Signal Procssing, vol. 3, no., pp. 44, Fb. 996.
Title. Author(s)Pei, SooChang; Ding, JianJiun. Issue Date Doc URL. Type. Note. File Information. Citationand Conference:
Titl Uncrtainty Principl of th D Affin Gnralizd Author(sPi SooChang; Ding JianJiun Procdings : APSIPA ASC 009 : AsiaPacific Signal Citationand Confrnc: 7 Issu Dat 00900 Doc URL http://hdl.handl.nt/5/39730
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