DFT: Discrete Fourier Transform

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1 : Discrt Fourir Trasform Cogruc (Itgr modulo m) I this sctio, all lttrs stad for itgrs. gcd m, = th gratst commo divisor of ad m Lt d = gcd(,m) All th liar combiatios r s m of ad m ar multils of d. a b a is a divisor of b. I a xrssio mod m, m is a strictly ositiv itgr. a b mod m m b If a bmod m th c d mod m a a c b d mod m a c b d mod m a c b d mod m a b mod m with a b mod m m mmod m r for uiqu r that ma m r -oit sigal x[]: -oit sigal Has fiit duratio Duratio itrval [,) Cyclic shift of x[] by ; < ; : = x ; x x ; ; othrwis Dr.Prau Susomog x Examl is aothr -oit sigal

2 = x[] a b c d x 5 d a b c -oit circular covolutio of -oit sigal x [] x [] = x mx m x m x m m m x m x m x m x m x m x m m m m x x x x If x [m] has may s, us mx m that is multilid by x m Examl: = 3 Lt y[] = x [] x [] x x x x x, limiatig ach of x m m y x m x m x x x x x x m x x x x x x x x x x x x x x x x x x y x x x x x 3 x 3 y x x x x x x 3 3 y x x x x x x 3 3 To fid x [] x [] usig circular covolutio rul ˆ X x x x x x A x A 3 3

3 x x x x x A x A 3 3 Th multilicatio ca b asily fid with TI calculator or athcad (slct A, th choos Symbolics > Polyomial Cofficits): x x x x xx 3 4 X ˆ A A A A x x xx x x xx x x x x But 3 4 A, A A ; thrfor, xx xx xx A A xx xx xx. x x xx xx Obviously (actually from ivrs ), Examl: x [] = 3, x [] = y x x xx x x y x x xx x x. y x x xx x x x [] x [] = j mod For v, j 3

4 4 m othrwis j j j j j mod ; if ; if j I I ; if ; if j j I I q q ;,q 4

5 q q q q m q m q ; q q ; q q I -oit of th -oit sigal x[]: ˆ ˆ X X x ; j x x ; x x ˆ X x x ˆ 4 X x x ˆ x X x ˆ ˆ x X X x x x x x ; x X ˆ * x x X ˆ Giv tool for comutig 5

6 Tim rvrs x Aftr -fidr g[q] = ˆ X, q Tim rvrs: g[-q] = ˆ q X ˆ X DTFT x x x x x x x ; < x q ; ; ; othrwis Circular covolutio rul: x [] x [] x m x m x m x m m m m x m m 6

7 To us to comut rgular covolutios of tim-limitd (fiit-duratio) sigal: If x [] has duratio itrval < x [] has duratio itrval < Lt = + -, th x []*x [] = x [] To s this, x [] for < ad = othrwis x []*x [] has fiit duratio at most = + - Thi of x [] ad x [] as -oit sigal whos last valus ar (-addig) Bloc covolutio h[] has duratio itrval < P (imuls rsos of a causal FIR systm) w[] = a (ossibly) ifiit-duratio sigal To comut h[]*w[] Divid w[] ito blocs of som scific lgth L (tyically L >> P) w rl ; L w r duratio itrval < L ; othrwis r w[] = w rl, r y r [] = h[] * w r [] duratio itrval < P + L - Fid y r [] by (P+L )-oit circular covolutio r r h[] * w[] = y rl, h* w h* wr rl h* wr rl r r r y rl ot that h w rl y rl covolutio. * r r r from th tim-ivariac rorty of y [] y [-L] P- L P+L- L P+L- 7

8 Frqucy-samlig aroximatio of h[] To aroximat h[]: Samlig Ĥ at ; < by j h yild Us () -, h ~ Hˆ H ~ˆ j Hˆ Hˆ j Hˆ H ~ˆ r Hˆ ˆ r r h ; < r h ~ h r ˆ y x r Y X ˆ ˆ h H H hm m jm j j m hm m othrwis ; if I j ; if I m j m ; if I m ; if I hm h m m I So, th summatio oly iclud m of th form m r ; r I 8

9 Thus, h h r r ˆ h h r H H r : ˆ j First, ot that ˆ H h h. ˆ H h h r r r h r Substitut r r r r r r r r r Hˆ h h ot that r r x x x. r r r r ˆ H h H. Thus, ˆ Tim-aliasig If h[] has duratio itrval cotaid i <, h ~ = h[] ; < Examl: if h[] is a ()-oit sigal 9

10 j ˆ H h h h h h h h h h h h h h h ; So, h ~ = h[] ; < iff h[] = for (o foldig) Widowig a sigal to gt a aroximatio of Giv x[] - < < x[ ] Loo at y[] = x[] L [] = ; L L ; L Fid Yˆ si L si d (lgth = L- = ) L si L ; L L DTFT ; L si L = d width of s fatur > 4 L If L, lim DTFT, ad Ŷ L L

11 Lots of shar activity i d biggr Giv x[] ; < Ca us -oit s to fid r by lt y[] = x, th Yˆ oly for = = r ; ; r I j x x x r r x x r x Lt x[] b a sigal of duratio. If alrady hav tool for -oit s. Ca fid To do this, Lt q[] = for = +, by costructig y[] so Y j Qˆ Yˆ Q, Wat ˆ x[], th ˆ Q ad From ˆ y[] ˆ. ˆ y x r Y X, so d r r q r othrwis ot that ot all r s ar usd. oly os that satisfy < +r < r < r, but <. So, - < r <

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