Title. Author(s)Pei, Soo-Chang; Ding, Jian-Jiun. Issue Date Doc URL. Type. Note. File Information. Citationand Conference:

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1 Titl Uncrtainty Principl of th -D Affin Gnralizd Author(sPi Soo-Chang; Ding Jian-Jiun Procdings : APSIPA ASC 009 : Asia-Pacific Signal Citationand Confrnc: -7 Issu Dat Doc URL Typ procdings Not APSIPA ASC 009: Asia-Pacific Signal and Information Confrnc. -7 Octobr 009. Sapporo Japan. Postr 009. Fil Information TA-P-3.pdf Instructions for us Hokkaido Univrsity Collction of Scholarly and Aca

2 Uncrtainty Principl of th -D Affin Gnralizd Fractional Fourir Transform Soo-Chang Pi * and Jian-Jiun Ding Dpartmnt of Elctrical Enginring National Taiwan Univrsity Taipi Taiwan pi@cc..ntu.du.tw * d@cc..ntu.du.tw TEL: Abstract Th uncrtainty principls of th -D fractional Fourir transform and th -D linar canonical transform hav bn drivd. W xtnd th prvious works and discuss th uncrtainty principl for th two-dimnsional affin gnralizd Fourir transform (-D AGFFT. W find that drivd uncrtainty principl of th -D AGFFT can also b usd for dtrmining th uncrtainty principls of many -D oprations such as th -D fractional Fourir transform th -D linar canonical transform and th -D Frsnl transform. Ths uncrtainty principls ar usful for tim-frquncy analysis and signal analysis. Morovr w find that th rotation and th chirp multiplication of th -D Gaussian function can satisfy th lowr bound of th uncrtainty principl of th -D AGFFT. I. INTRODUCTION Th wll-known Hisnbrg uncrtainty principl stats that if X(ω is th -D Fourir transform (FT of x(t FT: [ (] ω t X ω = FT x t = x( t dt ( and th nd momnts of tim and frquncy ar = ( t t x( t dt / x( t dt x t dt ω ω X( ω dω / X( ω dω whn ( = th following inquality is satisfid [] = (3. ( Thn in [] th uncrtainty principl was gnralizd into th cas of th -D fractional Fourir transform (FRFT [3]: t ω cotα cotα cotα x t dt. (5 u t csc FRFT: Xα ( u = u t α ( If α = (6 u u X ( u du/ X ( u du α sin α t u. (7 Rcntly th uncrtainty principl was gnralizd into th cas of th -D linar canonical transform (LCT [][5]. If d ut a X ( abcd u = x t dt b (8 u t b b b LCT: ( whr b t u (9 u ( abcd / ( abcd = u X u du X u du. (0 Th uncrtainty principl of th -D cas has bn discussd a lot. In this papr w xtnd th prvious works and driv th uncrtainty principl for th two dimnsional affin gnralizd fractional Fourir transform (-D AGFFT. Th drivd uncrtainty principl is shown in Thorm. As Hisnbrg s uncrtainty principl th drivd uncrtainty principl will b usful in signal procssing applications such as tim-frquncy analysis signal synthsis communication sampling thory and filtr dsign. Morovr sinc many -D oprations ar th spcial cass of th -D AGFFT (such as th -D FRFT and th -D Frsnl transform w can us th drivd uncrtainty principl to find th uncrtainty principls for ths oprations. II. TWO-DIMENSIONAL AFFINE GENERALIZED FRACTIONAL FOURIER TRANORM Th two-dimnsional affin gnralizd fractional Fourir transform (-D AGFFT is dfind as [6][7] G( ABCD ( u v = K( ( u v x g( x dxdy ABCD ( whr a a b b c c d d A = a a B = b b C = c c D = d d ( rprsnts th 6 paramtrs of -D AGFFT and ( k u + k u v+ k3 v dt( B K( ABCD u v x y = (( bu+ bv x+ ( bu bv ( p x + p x y+ p3 y dt( B dt( B (3 whr k = d b d b k = ( d b + d b k 3 = d b + d b p = a b a b p = (a b a b p 3 = a b + a b. ( Morovr th following constraints should b satisfid [6][7]: AC=CA BD= DB AD CB= I. (5 Th -D AGFFT is usful for filtr dsign signal analysis data comprssion communication optics and imag procssing [6]. It is a gnralization of many -D oprations. For xampl th -D FT is a spcial cas of th AGFFT whr b = b = c = c = a = a = a = a = 0 b = b = c = c = d = d = d = d = 0. (6 Th -D fractional Fourir transform (-D FRFT [3] is:

3 -D FRFT: ( ( u cotα+ v cot β ( cot α( cot β Gαβ u v = ( x cotα+ y cot β ux ( csc vycsc α + β g( x dxdy. (7 It is a spcial cas of th -D AGFFT whr a =d =cosα b = c =sinα a =d =cosβ b = c =sinβ a = a = b = b = c = c = d = d = 0. (8 Th -D linar canonical transform (LCT is dfind as -D LCT: ( d d ( u + v b b G ( abcda b c d u v = bb ux vy a a ( + ( x + y b b b b g( x dxdy. (9 It is a spcial cas of th -D AGFFT whr a =a a =a b =b b =b c =c c =c d = d d = d a = a = b = b = c = c = d = d = 0. (0 Th -D Frsnl transform is: π kz ( u x + ( v z G( zz u v i λ = g x y dxdy. ( It dscribs th light propagation in th fr spac. If th constant phas is ignord th -D Frsnl transform can b viwd as th spcial cas of th AGFFT whr a = a = d = d = b = b = / ( a = a = b = b = c = c = c = c = d = d = 0. (3 III. UNCERTAINTY PRINCIPLE OF THE -D AGFFT As th -D cas in this papr w always suppos that th signal g(x is normalizd gxy ( dxdy=. ( W will try to find th lowr bound of x y u v whr xy = ( x + y g( x dxdy (5 ( uv= u + v G( ABCD u v dudv. (6 and G (ABCD (u v is th -D AGFFT (dfind in (-(5 of g(x. Th formula of th -D AGFFT is vry complicatd. It has 6 paramtrs. W should us som ways to simplify th drivation of th uncrtainty principl. [Lmma ] First not that if ( p x + p x y+ p3 y dt( B g0 ( xy = g( xy (7 ( k u k u v k3 v dt H ( uv = + + B G( ABCD ( uv (8 sinc g 0 (x = g(x and H(u v = G (ABCD (u v ( xy= x + y g0 x y dxdy (9 uv = ( u + v H ( u v dudv. (30 Not that (( bu+ bv x+ ( bu b v dt( B H ( uv = g0 ( xy dxdy. (3 [Lmma ] Morovr th rotation opration dos not affct th nd ordr momnt. That is if g( x = g0( xcosθ + ysin θ xsinθ + ycosθ (3 H ( u v = H( ucosφ + vsin φ usinφ + vcosφ (33 x + y g ( x dxdy = x + y g ( x dxdy 0 (3 ( u + v H ( u v dudv = ( u + v H( u v dudv. (35 Substituting (3 and (33 into (3 w obtain ( ( u v x ( 3u v H( uv η + η + η + = η g ( xy dxdy (36 whr η η η 3 and η can b calculatd from: b b η η cosθ sinθdt( B dt( B cosφ sinφ η3 η = sinθ cosθ b b sinφ cosφ. (37 dt( B dt( B Not that if η and η 3 ar zro th rlation btwn H (u v and g (x in (36 will b simplifid into th -D scald FT. Th uncrtainty principl of th -D scald FT is asir to find. To mak η = η 3 = 0 θ and φ should satisfy b b dt( B dt( B cosθ sinθ η 0 cosφ sinφ = b b sinθ cosθ 0 η sinφ cosφ dt( B dt( B (38 b + b = ( η + η cos( φ θ b b ( η = η cos( φ+ θ b + b = ( η η sin( φ+ θ b b ( η = + η sin( φ θ. (39 Thrfor ( b + b + ( b b = η + η (0 ( b b + ( b + b = η η. ( Thus w can choos η = /( ( b + b + ( b b + ( b b + ( b + b η = /( ( b + b + ( b b ( b b + ( b + b if (b +b + (b b > (b b + (b +b ( and η = /( ( b + b + ( b b ( b b + ( b + b η = /( ( b+ b + ( b b + ( b b + ( b + b if (b +b + (b b < (b b + (b +b. (3 Thn from (39 φ = (ψ + ψ / θ = (ψ ψ / ( b b b + b whr ψ = cos = sin η η η η b + b b b ψ = cos = sin η + η η + η. (5

4 If w choos η η η 3 η φ and θ as ( (or (3 and ( η = η 3 = 0 and th rlation btwn H (u v and g (x in (36 bcoms th -D scald FT. ηη ( ux v H( uv η + = η g( xy dxdy. (6 [Thorm ] For th -D scald Fourir transform: σσ ( σ fx σ hy G f h = + g( x dxdy. (7 If xy = ( x + y g( x dxdy ( = f h G f h dfdh + (8 ( xy σ + σ. (9 (Proof: Sinc G ( f h = σσ G( σf σh whr G(f h is th FT of g(x if w st f = σ f and h = σ h dfdh = df dh / σ σ and (7 bcoms = pf qh Gfh dfdh +. (50 whr p = /σ and q = /σ. Thn sinc ( p f + q h G( f h ( + ( = + = p f + q h G( f h p f q h G ( f h (5 IFT p f q G f h [ p q ] g( x (5 from Parsval s Thorm of th -D FT: g( xy dxdy G( f h dfdh = (53 if G(f h = FT[g(x ] (5 can b rwrittn as = [ p q ] g( xy [ p q ] + + g ( x dxdy. (5 Furthrmor in (8 x + y gxy ( = x+ ygxy ( x+ yg( x. (55 Thrfor xy = ( x+ g( x [ p + q ] g( x (56 Thn from Cauchy-Schwartz inquality f ( xy gxy ( f( xy gxy ( (57 f( x g( x [ f( x g( x + f ( x g ( x ]/ (56 can b rwrittn as: xy (58 xg p g + yg p g + xg q g + yg q g + p g xg + p g yg + q g xg + q g yg. (59 Not that (56 can also b xprssd as xy = ( x g( x [ p q ] g( x. (60 From th similar procss w obtain xy xg p g yg p g xg q g + yg q g + p g xg p g yg q g xg + q g yg. (6 Adding (6 by (63 and using th fact that a + b + c + d a+ b+ c+ d (6 w obtain xy xg p x g + p x g xg + yg q g + q g yg. (63 Thn xg p g + p g xg = p x g ( x g ( x dxdy ( ( x = = p xg xyg xy g( xyg ( xydx dy x= = p g( x dxdy = p. (6 Similarly yg q g + q g yg = q. (65 Thrfor xy pfqh ( p + q = ( σ + σ. # Lmmas and and Thorm much simplify th drivation of th uncrtainty principl. From (9 (30 (3 (35 (6 and (9 ( xyuv η + η. (66 Morovr from ( and (3 η + η = ( b b b b b b b b max ( + + ( ( + ( +. (67 Thus w obtain: [Thorm ] Uncrtainty Principl of th -D AGFFT: If xy and uv ar th nd ordr momnts of g(x and th -D AGFFT of g(x as in (5 and (6 rspctivly xy uv max ( ( b + b + ( b b b b + b + b. (68

5 IV. THE RELATED PRINCIPLES [Rmark ] Mor gnrally if G ( ( A B C D u v and G (ABCD (u v ar th -D AGFFTs of g(x with paramtrs {A B C D } and {A B C D} rspctivly and ( u v= u + v G( A B C D u v dudv ( uv= u + v G( ABCD u v dudv (69 xy uv max ( ( q + q + ( q q ( q q + ( q + q (70 A B P Q A B q q whr = Q = R S C D C D q q. (7 This can b provn from th fact that G (ABCD (u v is th -D AGFFT of G ( ( A B C D u v with paramtrs {P Q R S}. [Thorm 3] It is known that for th -D FT th -D Gaussian function can satisfy th lowr bound of inquality of th uncrtainty principl []. For th -D AGFFT th chirp multiplication and rotation of th -D Gaussian will satisfy th lowr bound of inquality of th uncrtainty principl. If ( p x + p x y+ p3 y dt( B g x y = π (( cos sin sin cos x θ y θ η + x θ+ y θ η (7 whr p p and p 3 ar dfind in ( η and η ar calculatd from ( or (3 and θ is dtrmind from ( th -D AGFFT of g(x is ( k u + k u v+ k3 v dt( B π (( cos sin sin cos u φ+ v φ η + u φ+ v φ η G( u v = ( ABCD (73 whr φ is calculatd from (. Thn xy= uv= / η + / η / (7 ( xy uv = max ( b + b + ( b b ( b b + ( b + b. (75 Thus th function in (7 satisfis th lowr bound of inquality of th uncrtainty principl for th -D AGFFT. V. SOME IMPORTANT SPECIAL CASES Sinc th -D AGFFT is th gnralization of many oprations w can us (68 to find th uncrtainty principl of ths oprations. [Corollary ] Uncrtainty Principl of th -D FRFT: From (68 and (8 if uv = ( u + v Gαβ ( u v dudv (76 whr G αβ (u v is th -D FRFT of g(x as in (7 xy uv sin α + sin β. (77 Morovr whn ( x cotα + p3 y cot β ( x cscα + y csc β g( x π th quality that = (78 xy uv = sin α + sin β / is satisfid. [Corollary ] Uncrtainty Principl of th -D LCT: From (68 and (0 if ( uv = u + v G( abcda b c d u v dudv (79 xy uv b + b. (80 Mor th quality is satisfid whn a a x + y ( x / b y / b b b + = π (8 g x y [Corollary 3] Uncrtainty Principl of th -D Frsnl Transform: xy λ z uv. (8 π Th quality that = λ π is satisfid whn ( xy uv z / π π ( x + y ( x + y = π. (83 g x y VI. CONCLUSIONS In this papr w drivd th uncrtainty principl of th - D AGFFT (S Thorm. W also showd that th lowr bound can b achivd by th -D Gaussian function with rotation and chirp multiplication (S Thorm 3. Th uncrtainty principl of th -D AGFFT would b vry usful in tim-frquncy analysis dvloping sampling thory in -D cas filtr dsign signal synthsis and optics. VII. REFERENCES [] G. B. Folland A. Sitaram Th uncrtainty principl: a mathmatical survy Th Journal of Fourir Analysis and Applications vol.3 no. 3 pp [] S. Shind and V. M. Gadr An uncrtainty principl for ral signals in th fractional Fourir transform domain IEEE Trans. Signal Procssing vol. 9 n. pp [3] H. M. Ozaktas Z. Zalvsky and M. A. Kutay Th Fractional Fourir Transform with Applications in Optics and Signal Procssing Nw York John Wily & Sons 000. [] K. K. Sharma and S. D. Joshi Uncrtainty principl for ral signals in th linar canonical transform domains IEEE Trans. Signal Procssing vol. 56 pp July 008. [5] A. Strn Uncrtainty principls in linar canonical transform domain and som of thir implications in optics J. Opt. Soc. Am. A. vol. 5 no. 3 pp March 008. [6] S. C. Pi and J. J. Ding Two-dimnsional affin gnralizd fractional Fourir transform IEEE Trans. Signal Procssing vol. 9 no. p Apr. 00. [7] G. B. Folland Harmonic Analysis in Phas Spac th Annals of Math. Studis vol. Princton Univrsity Prss 989. [8] M. Libling T. Blu and M. Unsr Frsnlts: nw multirsolution wavlt bass for digital holography IEEE Trans. Imag Procssing vol. no. pp. 9-3 Jan. 003.