(7) The Discete Fouie Tasfom The Discete Fouie Tasfom hat is Discete Fouie Tasfom (DFT)? (ote: It s ot DTFT discete-time Fouie tasfom) A liea tasfomatio (mati) Samples of the Fouie tasfom (DTFT) of a apeiodic (with fiite duatio) sequece Etesio of Discete Fouie Seies (DFS) Review: FT, DTFT, FS, DFS Time sigal Tasfom Coeffs. (peiodic/apeiodic) Coeffs. (coti./discete) Aalog apeiodic FT Apeiodic Cotiuous Aalog peiodic FT FS Apeiodic Apeiodic Discete apeiodic DTFT Peiodic Cotiuous Discete peiodic DFS Peiodic Discete Discete fiite-duatio DFT Cotiuous (impulse) Discete CTU EE
(7) The Discete Fouie Tasfom The Discete Fouie Seies Popeties of e jπ, thus e π j -- is peiodic with peiod. (It is essetially cos ad si) : ± ± L -- l, if l m, if l m l m (Pf) (i) If l m, l l (ii) If l m, l l l l -- l Y [] l δ [ l m] m DFS fo peiodic sequeces [ ] [] [ ], peiod + Its DFS epesetatio is defied as follows: Sythesis equatio: j [] [] e [] o Aalysis equatio: [] [] π ote: The tilde i idicates a peiodic sigal. [ ] is peiodic of peiod. CTU EE
(7) The Discete Fouie Tasfom CTU EE 3 Pf) [] [] Pic a ( < ) [] [] [] [] [] [] [] [ ] [] ) ( ) ( + + + + L L That is, [] []. QED Eample: Peiodic Rectagula Pulse Tai [] 4 4 5 si si j K e π π π
(7) The Discete Fouie Tasfom Samplig the Fouie Tasfom Compae two cases: () Peiodic sequece [] [ ] () Fiite duatio sequece [ ] oe peiod of [ ] A apeiodic sequece: jω [] FT ( e ) b? samples jω [] IDFS [] ( e ) π ω Compae: ( t) FT ( jω) samples b? jω [] DTFT ( e ) Eample:, 4, 4 + [] [], othewiase, 5 + 9 + itege CTU EE 4
The Discete Fouie Tasfom CTU EE 5 [] [] ( ) [ ] [ ] [ ] [] [ ] [ ] + + m m m m j m m j j K m m e m e e δ π π ω ω π ω ω ) (Itechage (FT) (Samplig) (IDFS) If [] has fiite legth ad we tae a sufficiet umbe of equally spaced samples of its Fouie Tasfom ( a umbe geate tha o equal to the legth of []), the [] is ecoveable fom [].
The Discete Fouie Tasfom Two ways (equivaletly) to defie DFT: () samples of the DTFT of a fiite duatio sequece [ ] () Mae the peiodic eplica of [ ] [ ] Tae the DFS of [] Pic up oe segmet of [ ] [] DFT peiodic [] DFS [] [ ] oe segmet Popeties of the Discete Fouie Seies -- Simila to those of FT ad z-tasfom Lieaity [] [] [] [] a [] + b [] a [] + b [] Shift m [] [] > [ m] [ ] l [] [ l] Duality Def: [] [] [] [] [] [] [] [ ] () (# ) CTU EE 6
The Discete Fouie Tasfom CTU EE 7 Symmety [] [] [] { } [] [] [ ] ( ) [] { } [] [] [ ] ( ) + j o e Im Re [] [] [ ] ( ) [] { } [] [] [ ] ( ) [] { } j o e Im Re + [] [] [ ] is eal, If. [ ] [ ] [] [ ] [] { } [ ] { } [] { } [ ] { } Im Im Re Re Peiodic Covolutio [] [], ae peiodic sequeces with peiod [ ] [ ] [ ] [ ] [] [] [] [] [ ] 3 l m l l m m
The Discete Fouie Tasfom Discete Fouie Tasfom Defiitio [] : legth, Maig the peiodic eplica: [] [ + ] [] [] [( modulo )] [(( ) ) ] Keep oe segmet (fiite duatio) [], [], othewise This fiite duatio sequece [] Aalysis eq : Sythesis eq : That is, [ ] [(( )) ] is the discete Fouie tasfom (DFT) of [] [] [], [] [], Rema: DFT fomula is the same as DFS fomula. Ideed, may popeties of DFT ae deived fom those of DFS. CTU EE 8
The Discete Fouie Tasfom Popeties of Discete Fouie Tasfom Lieaity [] [] [] [] a legth ma [] + b [] a [] + b [] [, ] Cicula Shift [] [] > m [(( m) ) ] [ ] l [] [(( l) ) ] (Pf) Fom the ight side of the d eq. Km b DFT j m j m [] e [] e [] [(( m) ) ] [ (( m) ) ] [ m] π π IDFS QED Rema: This is cicula shift ot liea shift. (Liea shift of a peiodic sequece cicula shift of a fiite sequece.) CTU EE 9
The Discete Fouie Tasfom Duality [] [] [] [ (( ) ) ], - Symmety Popeties ep [] peiodic cojugate - symmetic e[] { [ (( ) ) ] + [(( ) ) ]}, op [] If ep Re { [] + [ ] }, { [] }, peiodic { [] [ ] }, - - cojugate - atisymmetic - Im{ [] }, [] Re{ [] } [ ] j Im [ ] [] eal, [] [(( )) ], [] [(( ) ) ] { [] } [(( ) ) ] op Re Im { } { [] } Re{ [(( ) ) ]} { [] } Im{ [(( ) ) ]} Re Im { } { [] } [] [(( ) ) ] + [(( ) ) ] ep { } { [] } [] [(( ) ) ] [(( ) ) ] op CTU EE
The Discete Fouie Tasfom Cicula Covolutio Θ 3 [] [] [] m [ m] [ (( m) ) ] [] Θ [] [] [] -poit cicula covolutio Eample: -poit cicula covolutio of two costat sequeces of legth L-poit cicula covolutio of two costat sequeces of legth L CTU EE
The Discete Fouie Tasfom Liea Covolutio Usig DFT hy usig DFT? Thee ae fast DFT algoithms (FFT) How to do it? () Compute the -poit DFT of [ ] ad [ ] [] ad [] () Compute the poduct [] [ ] [ ] 3 (3) Compute the -poit IDFT of 3 [ ] [ ] Poblems: (a) Aliasig (b) Vey log sequece sepaately 3 CTU EE
The Discete Fouie Tasfom Aliasig [], legth L (ozeo values) [], legth P I ode to avoid aliasig, L + P (hat do we mea avoid aliasig? The pecedig pocedue is cicula covolutio but we wat liea covolutio. That is, [ ] 3 equals to the liea covolutio of [] ad [ ] ) CTU EE 3
The Discete Fouie Tasfom [] [] pad with zeos legth pad with zeos legth Itepetatio: (hy call it aliasig?) 3 has a (time domai) badwidth of size L + P [] (That is, the ozeo values of [ ] Theefoe, [ ] 3 ca be at most L + P ) 3 should have at least L + P samples. If the samplig ate is isufficiet, aliasig occus o 3 [ ]. CTU EE 4
The Discete Fouie Tasfom Vey log sequece (FIR filteig) Bloc covolutio Method ovelap ad add Patitio the log sequece ito sectios of shote legth. Fo eample, the filte impulse espose [ ] ealy ifiite. [ + ] [] L, L- Let [ L] whee [], othewise The system (filte) output is a liea covolutio: y h has fiite legth P ad the iput data [ ] [] [] h[] y [ L] whee y [] [] h[] is Rema: The covolutio legth is L + P. That is, the L + P poit DFT is used. y [] has L + P data poits; amog them, (P-) poits should be added to the et sectio. This is called ovelap-add method. (Key: The iput data ae patitioed ito oovelappig sectios the sectio outputs ae ovelapped ad added togethe.) CTU EE 5
The Discete Fouie Tasfom Method ovelap ad save Patitio the log sequece ito ovelappig sectios. Afte computig DFT ad IDFT, thow away some (icoect) outputs. Fo each sectio (legth L, which is also the DFT size), we wat to etai the coect data of legth ( L ( P )) poits Let h [], legth P [], legth L (L>P) The, y [] cotais (P-) icoect poits at the begiig. Theefoe, we divide ito sectios of legth L but each sectio ovelaps the pecedig sectio by (P-) poits. + L P + P, L - [] [ ( ) ( )] This is called ovelap-save method. CTU EE 6