Congrunc (Intgr modulo m) : Discrt Fourir Transform In this sction, all lttrs stand for intgrs. gcd ( nm, ) th gratst common divisor of n and m Lt d gcd(n,m) All th linar combinations r n+ s m of n and m ar multils of d. a b a is a divisor of b. In an xrssion mod m, m is a strictly ositiv intgr. a bmodm m b If a bmod m c dmod m thn a a + c b+ d mod m a c bd mod m a c b d mod m n n a b modm with n > n a n b mod m m mmod m + r for uniqu r that mak m+ r < -oint signal x[n]: -oint signal Has finit duration Duration intrval [,) Cyclic shift of x[n] by n ; n < ; n < : xnn ; n n < x n n x n n ; n n + < ; othrwis x n n Examl is anothr -oint signal 5 n - - 3 4 5 6 7 x[n] a b c d
x n 5 d a E F -oint circular convolution of -oint signal x [n] x [n] x [ m] x [ n m ] x [ n m ] x [ m] m m n x n m x[ m] x[ n m] x[ m] + x[ + nm] x[ m] m m m n+ n [ l] [ l] [ l] [ l] x x n + x x n+ l l n+ n If x [m] has many s, us x [ m] x [ n m ] x n m Examl: 3 Lt y[n] x [n] x [n] x l x n l + x l x nl l l n+ m that is multilid by x [ m ], liminating ach of y n x m x n m x x n + x x n + x x n 3 3 3 m x x + x x + x x[ 3] x x + x x + x x + + x x + x x + x x y x x + x x x x 3 + 3 y x x + x x + x x 3 3 y x x x x x x 3 3 To find x [n] x [n] using circular convolution rul ˆ k k X k x + x + x x + x A+ x A 3 3 k k + + + + Xˆ k x x x x x A x A 3 3 Th multilication Xˆ [ k] Xˆ [ k] can b asily find with TI calculator or athcad (slct A, thn choos Symbolics > Polynomial Cofficints):
x x xx xx + 3 4 Xˆ [ k] Xˆ [ k] A A A A xx + xx+ xx xx + xx xx 3 4 But A, A A; thrfor, x x + x x + x x Xˆ [ k] Xˆ [ k] A A xx xx xx + +. xx + xx+ xx y xx + xx + xx Obviously (actually from invrs ), y xx + xx + xx. y x x + x x + x x Examl: x [n] 3, x [n] k k 3k k k 3k ( + + + ) ( + + + ) Xˆ k Xˆ k 3 4 4 4 4 4 4 x [n] x [n] 6 5 8 + 3 + 3 + 8 + 8 + 3 + k k 3k 4k 5k 6k 4 4 4 4 4 4 + 6 + 5 + 8 k k 3k 4 4 4 j ( ) ( ) mod k k For vn n, j π kn k m othrwis ( ) j j
j j j mod k ; if k I j n k ; if I k ; if π π I j kn j nk kn k ; if I ( )( q ) Ψ ;,q q Ψ 4 ( ) ( ) Ψ Ψ Ψ Ψ Ψ Ψ ( l)( q) m( q) ( )( l ) ( l)( q) q l lq l l l m q ; q ( q) ; q q I -oint of th -oint signal x[n]: ˆ ˆ π X k X ω k xn ; k <
jnk Xˆ nk [ k] Xˆ ω k x[ n] x[ n] ; k < Xˆ x x ( ) Xˆ [ ] x x 4 ˆ ( ) X x x Ψ ( ) ( ) ( )( ˆ xn ) X n xn k [ n] [ n] δ δ ˆ ˆ nk ( ) xn X[ k] X [ k] xn ( ) n< k k< x x x x ; Xˆ x n Xˆ Xˆ Xˆ X ˆ [ n] Xˆ Ψ x x Ψ X ˆ Givn tool for comuting * ˆ X k xn Tim rvrs k< k< Aftr -findr g[q] ˆ [ ] X, q Tim rvrs: g[-q] ˆ q X [ ] ˆ X [ ] Xˆ DTFT [ k] x[ n] Xˆ ( ω) kn x nn Xˆ [ k] n< k< q
n x n n xn [ n] + x [ + n n] n [ n n ] n l [ l] ( l n ) x + l [ l] x δ ; n < + k ( l n ) + k l n x [ l] l + n k [ n n] ; n n ; δ < δ n n δ + n n n < n δ n n ; othrwis n< kn δ nn k < Circular convolution rul: Xˆ k Xˆ k x [n] x [n] x[ m] x n m x[ m] x n m m m m mk ( ) ˆ x m X k To us to comut rgular convolutions of tim-limitd (finit-duration) signal: If x [n] has duration intrval n < x [n] has duration intrval n < Lt + -, thn x [n]*x [n] x [n] x [n] for n < and othrwis To s this, x [n]*x [n] has finit duration at most + - Think of x [n] and x [n] as -oint signal whos last valus ar (-adding) Block convolution h[n] has duration intrval n < P (imuls rsons of a causal FIR systm) w[n] a (ossibly) infinit-duration signal To comut h[n]*w[n] Divid w[n] into blocks of som scific lngth L (tyically L >> P) [ + n] w rl ; n < L w r [ n] duration intrval n < L ; othrwis
r w[n] w [ n rl], n r y r [n] h[n] * w r [n] duration intrval n < P + L - Find y r [n] by (P+L )-oint circular convolution r r h[n] * w[n] y [ n rl], n hn * wn hn * wr [ n rl] hn * wr [ n rl] r r r y nrl ot that hn w [ n rl] y [ n rl] of convolution. y [n] * r r r from th tim-invarianc rorty y [n-l] P- L P+L- L P+L- n Frquncy-samling aroximation of h[n] To aroximat h[n]: Samling Ĥ ( ω ) at jk n hn yild ˆ Us () -, [ n] ( Hˆ ( ω ) ω ω ) H ~ˆ ω k ; k < by ˆ H k H ω k ~ˆ nk H k h [ n r ] ; n < k h ~ jnω jnω ( Hˆ ( ω ) ω ω) r [ k] Hˆ ω k h ~ h [ n] [ n r ] ˆ n< r k< r yn ˆ xn r Y k X ω k n < othrwis
ˆ nk hn ˆ π H k H k k k l hm k m jmk j nk j k( n m) hm m k n ; if n I j l n ; if I hm hn m nm I k nk nm jk n m ; if I n m ; if I So, th summation only includ m of th form m n r ; r I Thus, h [ n] hn [ r] r ˆ h n hn r H k H ω k r ˆ jnk π H k hn hn ˆ H[ k] hn h[ n+ r ] r : ˆ Tim-aliasing First, not that. ( hn [ r ] ) + r Substitut l n+ r l r ( l r) k ( ) r+ r+ Hˆ k h l h l r l r r l r ( r+ ) r+ ot that r l r r l r l lk l ˆ. l Thus, ˆ lk ( ) x l x l x l. π H k h H k If h[n] has duration intrval containd in n <, h ~ [ n] h[n] ; n < Examl: if h[n] is a ()-oint signal
jnk ˆ π H k hn hn hn hn hn + ( l+ ) [ l ] l [ l ] l ( hn hn ) hn + h + hn + h + + + ; So, h ~ [ n] hn hn + hn + n < lk k h[n] ; n < iff h[n] for n (no folding) Windowing a signal to gt an aroximation of Xˆ ( ω ) Givn x[n] - < n < Look at y[n] x[n] L [n] π x[ n] ; L < n < L ; n L Find Yˆ ( ω ) Xˆ ( ω µ ) [ n] L π sin L µ µ sin dµ sin L ω ; L < n< L DTFT ; n L ω sin kπ Zros @ L L- @ ω d width of Xˆ ( ω ) s fatur > 4π L (lngth L- ) If L, lim [ n] DTFT δ ( ω), and Yˆ ( ω ) Xˆ ( ω ) L L Lots of shar activity in Xˆ ( ω ) nd biggr Givn x[n] ; n <
Can us -oint s to find [ k ] r by lt y[n] x [ n + ( ) ], thn Xˆ [ k ] Yˆ [ l] Xˆ only for k l rl ; l < ; r I j n nk l Xˆ k xn xn xn r ( + ) r ln r ln xn x n+ x n + Lt x[n] b a signal of duration. If alrady hav tool for -oint s. ln ( n ( ) ) l + Can find Xˆ ( ω ) for ω k ω + k π, by constructing y[n] so Y [ k ] Xˆ ( ω ) To do this, Lt q[n] Qˆ k jω n Xˆ k Want ˆ x[n], thn ˆ( ω ) Xˆ ( ω + ω ) + ω Yˆ k Q k, Q and Xˆ ( ω ) From ˆ y[n] k ˆ. ˆ y n xn r Y k X ω k, so nd r q n r [ + r ] n < othrwis ot that not all r s ar usd. only ons that satisfy < n+r < r < n r < n, but n <. So, - < r < k