18th European Signal Processing Conference (EUSIPCO-2010) Aalborg, Denmark, August 23-27, 2010

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1 8th Europan Sgnal Procssng Conrnc EUSIPCO- Aalorg Dnmark August 3-7 EIGEFUCTIOS EIGEVALUES AD FRACTIOALIZATIO OF THE QUATERIO AD BIQUATERIO FOURIER TRASFORS Soo-Chang P Jan-Jun Dng and Kuo-W Chang Dpartmnt o Elctrcal Engnrng atonal Tawan Unvrsty o. Sc. 4 Roosvlt Rd. 67 Tap Tawan R.O.C TEL: Fax: Emal: p@cc..ntu.du.tw djj@cc..ntu.du.tw 8993@ntu.du.tw ABSTRACT Th dscrt quatrnon Fourr transorm DQFT s usul or sgnal analyss and mag procssng. In ths papr w drv th gnunctons and gnvalus o th DQFT. W also xtnd our works to th rducd quatrnon cas.. th dscrt rducd quatrnon Fourr transorm DRBQFT. W nd that an vn or odd symmtrc gnvctor o th -D DFT wll also an gnvctor o th DQFT and th DRBQFT. orovr oth th DQFT and th DRBQFT hav 8 gnspacs whch corrspond to th gnvalus o - - j -j k and k. W also us th drvd gnvctors to ractonalz th DQFT and th DRBQFT and dn th dscrt ractonal quatrnon transorm and th dscrt ractonal rducd quatrnon Fourr transorm.. ITRODUCTIO Th quatrnon algra s a gnralzaton o th complx algra []. A numr n th quatrnon ld has thr magnary parts and can xprssd as: q qr + q + q j j + qk k whr j and k satsy th ollowng ruls: j k j k j k k j j k k j k j. Basd on th quatrnon algra th dscrt quatrnon Fourr transorm DQFT [] s dnd as DQFT x m n m n pm xm n j π xp xp. 3 Th DQFT s usul or color mag analyss spctral analyss and ltr dsgn [][3][4]. Thr s anothr algra that also has our lmnts.. th rducd quatrnon algra [5][6]. A numr n th rducd quatrnon ld can xprssd as: q qr + q + q j j + qk k 4 whr k j j j k k k j j k k j. 5 Thr ar two drncs twn th rducd quatrnon and th quatrnon algras. Frst n th rducd quatrnon algra j. Howvr n th quatrnon algra j. orovr th rducd quatrnon algra s always commutatv.. xy yx s always satsd ut n th quatrnon algra j j k k and jk kj. Th dscrt rducd quatrnon Fourr transorm DRBQFT [6] can also dnd as th ollowng orm DRBQFT x m n m n xp pm xp k x m n. 6 As th DQFT th DRBQFT s also usul or color mag analyss and ltr dsgn [5][6]. Th DRBQFT s asr to mplmnt and sutal or multpl channl sgnal analyss. In ths papr w drv th gnvctors and gnvalus o th DQFT and th DRBQFT. W nd that [m n] s an gnvctor o th -D DFT n th complx ld and [m n] ± [m n] or [m n] ± [m n] thn t s also an gnvctor o th DQFT and th DRBQFT. orovr th orgnal DFT has 4 dstnct gnvalus ± and ± ut oth th DQFT and th DRBQFT hav 8 dstnct gnvalus ± ± ±j and ±k. S Sctons and 3. orovr n Scton 4 w us th drvd gnvctors and gnvalus o th DQFT and th DRBQFT to dn th dscrt ractonal quatrnon transorm DFRQFT and th dscrt ractonal rducd quatrnon Fourr transorm DFRRBQFT. Thy ar analogous to th dscrt ractonal Fourr transorm [][] n th complx ld and gnralz th DQFT and th DRBQFT. Furthrmor our rsults can asly xtndd to th contnuous cas.. drvng th gnunctons and gnvalus o th contnuous quatrnon and rducd quatrnon Fourr transorms. S Scton 5. EIGEVECTORS AD EIGEFUCTIOS OF DISCRETE QUATERIO FOURIER TRASFORS Snc th quatrnon algra dos not hav th commutatv rul thr ar two drnt ways to dn th gnvctors and gnvalus o th DQFT: rght-sdd orm m n m n λ 7 lt-sdd orm m n λ m n. 8 W rst dscuss th rght sdd orm gnvctors and gnvalus o th DQFT. W nd that thy can drvd rom thos o th orgnal -D dscrt Fourr transorm -D DFT n th complx ld. [Thorm ] Suppos that [m n] s an gnvctor o th -D DFT: DFT m n m n λ 9 whr π pm π DFT x p q xm. m n EURASIP ISS

2 I [m n] s vn along n.. [m n] [m n] thn s also th gnvctor o th DQFT and th corrspondng gnvalu s also λ. m n m n λ. Proo: Snc [m n] s vn along n m nsn n 3 th DQFT o [m n] s m n π pm cos sn m n π j π m n π pm cos m n π m n π pm cos sn m π π m n π pm m n m DFT m. 4 Thn rom 9 w otan. # [Thorm ] Smlarly [m n] s an gnvctor o th -D DFT that satss 9 and [m n] s odd along n: [m n] [m n] 5 thn t s also an gnvctor o th QDFT ut th gnvalu s changd nto λk. m n m n λk. 6 Proo: Snc m ncos n rom 5 7 m n π pm sn m n π j m n π pm cos sn m n π π j m n DFT m n k m n λk. # [Thorm 3] By contrast [m n] s an gnvctor o th -D DFT that satss 9 and [m n] s vn along m: [m n] [m n] 8 thn [m n] s also an gnvctor o th DQFT ut th gnvalu s changd nto λ q : m n m n λq 9 whr λq λ λ ± λ q ± j λ ±. Proo: From 8 th nnr product o [m n] and snpm/ s zro. Thror DQFT cos m n π pm m n m n j j π m n cos pm jsn pm m n j π j π pm m n DQFT m n m n m n λq. # [Thorm 4] I [m n] satss 9 and s odd along m: [m n] [m n] thn m n m n kλq whr λ q s dnd n. Proo: Snc th nnr product o [m n] and cospm/ s zro w us r [m n] and [m n] to dnot th ral part and th magnary part o [m n] thn m n j π sn π pm m n m n sn j π k j π pm r m n m n + m n j π k r mn mn jsn π pm m n j π j π pm k r mn j mn m n +. 3 From th act that oth r [m n] and [m n] ar th gnvctors o th DFT w otan m n r m n k λq + m n jλq r m n k λq + m n k λq r m n+ m n k λq m n k λq. # [Corollary ] From Thorms -4 w can conclud that [m n] s a -D DFT gnvctor n th complx ld and [m n] ± [m n] or [m n] ± [m n] thn t s also an gnvctor o th DQFT. By contrast [m n] s a -D DFT gnvctor ut non o th symmtry rlatons n 5 8 and s satsd thn [m n] s not an gnvctor o th DQFT. Ths can provn rom th act that th vn part and th odd part o [m n] wll sparatd nto drnt gnspacs o th DQFT. [Corollary ] orovr snc th DFT has our gnvalus: and [7] rom 6 9 and w can conclud that th DQFT has 8 possl gnvalus whch ar j j k and k. Spcally n Thorms -4 w can choos th -D DFT gnvctors as th dscrt Hrmt-Gaussan unctons: m n h a mh n 4 whr h a [m] s th a th dscrt Hrmt-Gaussan uncton o th -D -pont DFT. It can drvd rom th commutng matrx mthod as n [7][8][]. {h a [m]h [n] a } orms a complt and orthogonal gnvctor st o th -D DFT whr s odd s vn 5 and s dnd n th smlar way. Th gnvalu o th -D DFT corrspondng to h a [m]h [n] s : a a DFT h m h n h m h n

3 Tal Th gnvalus o th DQFT corrspondng to th dscrt Hrmt-Gaussan gnvctors h a [m]h [n]. Condtons a 4 4 or a 4 4 a or a a 4 4 or a a 4 4 or a a 4 4 or a 4 4 a 4 4 or a a 4 4 or a a or a 4 4 Snc [7][8][] Egnvalus o th DFT Egnvalus o th DQFT h n h n 7.. [m n] h a [m]h [n] s vn or odd symmtrc along n thus rom Thorms and w hav: [Thorm 5] Th sparal dscrt Hrmt-Gaussan unctons n 4 ar also th gnvctors o th DQFT and a a DQFT h mh n h mh n 8 whn s vn DQFT h a mh n h a mh n k 9 whn s odd. orovr as th cas o th DFT {h a [m]h [n] a.. } also orms a complt and orthogonal gnvctor st or th DQFT. From 8 and 9 th DQFT has 8 dstnct gnvalus whch nclud ± ± ±j and ±k. W lst th rlaton among a and th gnvalus o th DQFT n Tal whr 4 mans th rmandr o a numr atr ng dvdd y 4.g ot that ach gnspac o th orgnal DFT corrsponds to two gnspacs o th DQFT. W can us th smlar way to drv th lt-sdd gnvctors and gnvalus o th DQFT s 8. From th smlar procss as thos n Thorms -4 w otan: [Thorm 6] Suppos that [m n] s an gnvctor o th -D DFT n th complx ld as n 9. Thn a m n λ m n 3 whn [m n] [m n] DQFT mn λk mn 3 whn [m n] [m n] and [m n] s ral c m n λq m n λ q s dnd n 3 whn [m n] [m n] and [m n] s ral or λ ±. d DQFT mn kλq mn 33 whn [m n] [m n] and [m n] s ral or λ ±. k j k j ot that n th rght-sdd cas [m n] s not constrand to a ral uncton. Howvr n th lt-sdd cas whn [m n] [m n] [m n] [m n] and λ ± and [m n] [m n] and λ ± [m n] should a ral uncton. Othrws t s not an gnvctor o th DQFT. [Corollary 3] As th rght-sdd cas w can also prov that th sparal dscrt Hrmt-Gaussan unctons n 4 also orm a complt and orthogonal gnvctor st or th DQFT n th ltsdd cas. orovr th gnvalus lstd n Tal ar also vald or th lt-sdd cas. 3. EIGEVECTORS AD EIGEFUCTIOS OF DISCRETE REDUCED BIQUATERIO FOURIER TRASFORS In th rducd quatrnon algra th dmpotnt lmnts E and E as ollows play vry mportant rols E + j / E j /. 34 Thy satsy th proprts o EE E E... E E 35 n- n n n- E E... E E. 36 A rducd quatrnon numr n 4 can r-xprssd y th dmpotnt lmnt orm as: q qe + qe whr 37 q qr + qj + q + qk q qr qj + q qk. 38 Thror xp k π cos π ksn π 39 π π π π cos sn E+ cos + sn E. Thus th DRBQFT n 6 can rwrttn as DRBQFT x m n m n pm x m n E 4 m n xp pm xp x m n E + xp xp whr x [m n] x r [m n]+x j [m n] + x [m n]+x k [m n] and x [m n] x r [m n]x j [m n] + x [m n]x k [m n]. Thror th DRBQFT also has a clos rlaton wth th -D DFT n th complx ld and w can us th gnvctors and gnvalus o th -D DFT to drv thos o th DRBQFT. [Thorm 7] Suppos that [m n] s th gnvctor o th -D DFT n th complx ld as n 9. I [m n] [m n] 4 thn t satss 3 and th DRBQFT o [m n] s DRBQFT m m n xp pm cos m n E + xp cos pm m n E m n xp π pm π m n E m n E m n m E m E λ m + λ +. 4 That s [m n] s also an gnvctor o th DRBQFT and th corrspondng gnvalu s also λ. 876

4 Tal Th gnvalus o th DRBQFT corrspondng to th dscrt Hrmt-Gaussan gnvctors h a [m]h [n]. Condtons a 4 4 or a 4 4 a or a a 4 4 or a a 4 4 or a a 4 4 or a 4 4 a 4 4 or a a 4 4 or a a or a 4 4 Egnvalus o th DFT Egnvalus o th DRBQFT [Thorm 8] Smlarly [m n] s th gnvctor o th -D DFT and [m n] [m n] 43 thn snc th nnr product o [m n] and snpm/ s zro th DRBQFT o [m n] s DRBQFT [ m n] m n m n λ λ j k pm m n E pm π m n E pm pm m ne m ne cos xp + cos xp π π m n + DFT [ m n] E + IDFT [ m n ] E E+ E m n. 44 That s [m n] s stll th gnvctor o th DRBQFT ut th gnvalus s changd nto λe + λ E. [Thorm 9] Usng th smlar ways w can also prov that [m n] s an gnvctor o th -D DFT that satss 9 and [m n] [m n] thn DRBQFT mn jλ mn. 45 I [m n] satss 9 and [m n] [m n] thn λ λ DRBQFT m n E E m n. 46 Thror rom Thorms 7 8 and 9 [m n] s an gnvctor o th -D DFT n th complx ld and [m n] ± [m n] or [m n] ± [m n] thn t s also an gnvctor o th DRBQFT. Furthrmor rom and 46 th DRBQFT wll hav 8 possl gnvalus ± ± ±j and ±k. [Thorm ] As th quatrnon cas th sparal dscrt Hrmt-Gaussan unctons {h a [m]h [n] a } also orm a complt and orthogonal gnvctor st or th DRBQFT. orovr rom 6 4 and 45 w can conclud that j k DRBQFT h a mh n h a m h n s vn 47 a a DRBQFT h mh n j h mh n s odd. 48 W lst th rlatons among a and th gnvalus o th DRBQFT n Tal. ot that th -D DFT n th complx ld has 4 gnspacs ± and ±. By contrast th DRBQFT wll hav 8 gnspacs ± ± ±j and ±k. 4. DISCRETE FRACTIOAL QUATERIO AD BIQUATERIO FOURIER TRASFORS In [9] Xu t al. drvd th contnuous ractonal quatrnon Fourr transorm asd on gnralzng th ntgral krnl. In ths scton w wll drv ts dscrt countrpart.. th dscrt ractonal quatrnon Fourr transorm DFRQFT and th dscrt ractonal rducd quatrnon Fourr transorm. DFRRBQFT. In [][] th convntonal DFT was gnralzd nto th dscrt ractonal Fourr transorm DFRFT asd on gnvctor dcomposton. Snc th gnvctors and th gnvalus o th DQFT and th DRBQFT hav n drvd n ths papr w can also us th mthod o gnvctor dcomposton to drv th DFRQFT and th DFRRBQFT succssully. As th DFRFT [][] th DFRQFT and th DFRRBQFT wll usul n sgnal and mag procssng. To drv th DFRQFT rst snc th dscrt Hrmt- Gaussan unctons h a [m]h [n] n 4 orm a complt and orthogonal gnvctor st or th DQFT any uncton n th quatrnon ld can dcomposd as a summaton o h a [m]h [n]: τ a a a x mn h mh n 49 whr m n and ar dnd as n 6 τ s vn τ s vn 5 a a m n τ x mnh mh n. 5 Hr w suppos that h a [m]h [n] has n normalzd: a m n h m h n. 5 Atr susttutng 49 nto 3 w otan a τ a a. 53 a DQFT x m n j h m h n Thror w suggst that th DFRQFT can dnd as th ollowng procss: Stp Frst w dcompos th nput x[m n] y 49 and dtrmn th cocnts τ a rom 5. Stp Calculat aφ jθ cos aφ sn aφ cos θ jsn θ. 54 Stp 3 Thn th DFRQFT can dnd as aφ jθ DFRQFTφθ x m n τ a h a m h n. 55 a Thr ar som ntrstng proprts that can notcd. Frst whn φ θ π/ th DFRQFT coms th orgnal DQFT. Whn φ θ π/ t coms th nvrs DQFT. Whn φ π/ θ and φ θ π/ t coms th -D DQFT along th m-axs and th n-axs rspctvly. 877

5 orovr t s no hard to prov that th DFRQFT has th addtvty proprty as ollows: { } x m n DFRQFTφ θ DFRQFTφ θ xm DFRQFTφ + φ θ+ θ. 56 From th addtvty proprty w can conclud that th DFRQFT wth paramtrs φ and θ s th nvrs opraton o th DFRQFT wth paramtrs φ and θ. To dn th DFRRBQFT w can also us th act that th dscrt Hrmt-Gaussan unctons n 4 orm a complt and orthogonal gnvctor st or th DRBQFT. From 4 6 and m n pm a h a m h n h a m h n 57 w otan DRBQFT ha m h n a a a { } h m h n E + E. 58 Thror th DFRRBQFT can dnd as DFRRBQFTφθ x m n { } aφ+ θ aφθ ha m h m E + E. 59 a ot that a a a φ + θ E φ θ E φ + θ E+ θ E cos θ sn θ cos θ sn θ + + aφ E E E E a a k φ + j φ θ. 6 Thus th DFRRBQFT can dnd as th ollowng way Stp Frst dcompos th nput x[m n] y 49 and dtrmn th cocnts τ a rom 5. Stp Calculat aφ kθ cos aφ sn aφ cos θ ksn θ. 6 Stp 3 Thn th DFRRBQFT can dnd as DFRRBQFTφθ xm xp xp τa ha m h m aφ kθ. 6 a As th cas o th DFRFT and th DFRQFT whn φ θ π/ th DFRRBQFT coms th orgnal DRBQFT. orovr th DFRRBQFT also has th addtvty proprty: { } x m n DFRRBQFTφ θ DFRRBQFTφ θ xm DFRRBQFTφ + φ θ+ θ 63 and th DFRRBQFT wth paramtrs φ and θ s th nvrs opraton o th DFRRBQFT wth paramtrs φ and θ. 5. EXTESIO TO THE COTIUOUS CASE In act th rsults n Sctons and 3 can xtndd to th contnuous cas. W can us th smlar way to nd th gnunctons and gnvalus o th contnuous quatrnon Fourr transorm QFT and th contnuous rducd quatrnon Fourr transorm RBQFT. Thorms to can all appld to th contnuous cas xcpt that th -D FT th DQFT and th DRBQFT ar rplacd y thr contnuous countrparts. As th dscrt cas x y s an gnuncton o th contnuous -D FT and x y ± x y or x y ± x y 64 thn x y s also th rght-sdd gnuncton o th contnuous QFT and th gnuncton o th contnuous RBQFT. orovr th contnuous -D Hrmt-Gaussan unctons.. th contnuous countrpart o 4 orm a complt and orthogonal gnuncton st or th QFT and th RBQFT. Furthrmor oth th contnuous QFT and th contnuous RBQFT also hav 8 dstnct gnvalus ± ± ±j and ±k. 6. COCLUSIOS In ths papr w drvd th gnunctons and gnvctors o th DQFT ncludng th rght-sdd and th lt-sdd orms and th DRBQFT. W nd that th -D dscrt Hrmt-Gaussan unctons whch ar th gnvctors o th -D DFT n th complx ld also orm a complt and orthogonal gnvctor st or th DQFT and th DRBQFT. Furthrmor oth th DQFT and th DRBQFT hav 8 gnspacs whch corrspond to th gnvalus o ± ± ±j and ±k. W also us th gnvctors and gnvalus w ound to drv th dscrt ractonal quatrnon and rducd quatrnon Fourr transorms. REFERECES [] W. R. Hamlton Elmnts o Quatrnons Longmans Grn and Co. London 866. [] S. J. Sangwn Th dscrt quatrnon Fourr transorm IEE Con. Pu. vol. pp [3] T. A. Ell and S. J. Sangwn Dcomposton o D hyprcomplx Fourr transorms nto pars o complx Fourr transorms EUSIPCO pp [4] Z. Lu Y. Xu X. Yang L. Song and L. Travrson D quatrnon Fourr transorm: th spctral proprts and ts applcaton n color mag rprsntaton IEEE Intrnatonal Conrnc on ultmda and Expo. pp July 7. [5] V. S. Dmtrov T. V. Cooklv and B. D. Donvsky On th multplcaton o rducd quatrnons and applcatons In. Procss. Ltt. vol. 43 pp [6] S. C. P J. H. Chang and J. J. Dng Commutatv rducd quatrnons and thr Fourr transorm or sgnal and mag procssng IEEE Trans. Sgnal Procssng vol. 5 no. 7 pp. -3 July 4. [7] B. W. Dcknson and K. Stgltz Egnvctors and unctons o th dscrt Fourr transorm IEEE Trans. Acoust Spch Sgnal Procss. vol. 3 pp [8] F. A. Grünaum Th gnvctors o th dscrt Fourr transorm: A vrson o th Hrmt unctons J. ath. Anal. Appl. vol. 88 pp [9] G. Xu X. Wang and X. Xu Fractonal quatrnon Fourr transorm convoluton and corrlaton Sgnal Procssng vol. 88. pp [] O. Arkan. A. Kutay H.. Ozaktas and O. K. Akdmr Th dscrt ractonal Fourr transormaton Procdngs o th IEEE-SP Intrnatonal Symposum on Tm-Frquncy and Tm-Scal Analyss pp [] S. C. P W. L. Hsu and J. J. Dng Dscrt ractonal Fourr transorm asd on nw narly trdagonal commutng matrcs IEEE Trans. Sgnal Procssng vol. 54 no. pp Oct