,+ -./*/0.0( .01*+1 1/(.1 f ( x ) e iux dx { f ( x ) - - PowerPoint PPT Presentation

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,+ -./*/0.0( .01*+1 1/(.1 * $ ! {f(x)} = F(u) = ∞

−∞

f(x) e−πiux dx * {F(u)} = f(x) = ∞

−∞

F(u) eπiux du * ! {f(x, y)} = F(u, v) = ∞

−∞

∞

−∞

f(x, y) e−πiuxvy dx dy * {F(u, v)} = f(x, y) = ∞

−∞

∞

−∞

F(u, v) eπiuxvy du dv +2 "

• f(x) =

A, x ∈ [0, X] 0, 3

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A f x ( )

x

X

F(u) = ∞

−∞

f(x)e−πiuxdx = # # # = A πu sin(πuX)e−πiuX |F(u)| =

• A

πu

• |sin(πuX)|
• e−πiuX
• = AX
• sin(πuX)

πuX

• abs( ( ))

F u 1 X 2 X

u

− 1

X

− 2

X AX

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f(x) |F(u)| X5 +2 ,+ *(-1++ .01*+1 1/(.1 . 3 3 ! $' 6 7+-.1( ' 8 */*+ 6 f(x)6 8 [0, L]6 6 8 xn ∆x

f x ( ) x0 x1 x2 xN −1 x L

N =

2L

∆ x

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f(x) ˜ f(x) =

∞

• n−∞

dn eπinx/L / L

• eπinx/Le−πikx/Ldx =

0, n = k L, n = k 9 dn = 1 L L

• f(x)e−πinx/Ldx

≈ ∆x L 1 2f(x)e−πinx/L + f(x)e−πinx/L + . . . + 1 2f(xN)e−πinxN/L

• '

≈ ∆x L

N−

• j

f(xj)e−πinxj/L ( f(xjN) = f(xj) ∀ j' ∆x = L N ⇒ ∆x L = 1 N xj = j∆x = jL N ⇒ xj L = j N ) f(xj) = f(j) = fj6 dn = 1 N

N−

• j

fje−πinj/N = Fn

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8 3! Fn = 1 N

N−

• j

fje−πinj/N, n = 0, . . . , N − 1 .1 F(u) = 1 N

N−

• x

f(x) e−πiux/N, u = 0, . . . , N − 1 *! 9 3 eπink/N

N−

• n

!

N−

• n

Fn eπink/N = 1 N

N−

• n

N−

• j

fj e−πinj/N eπink/N = 1 N

N−

• j

fj

N−

• n

eπink−j/N ) r = eπik−j/N6

N−

• n

eπink−j/N =

N−

• n

rn =    rN − 1 r − 1 , r = 1 N, r = 1

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/ * r = 1, j = k * r = 1, j = k (

N−

• n

eπink−j/N =    N, j = k eπik−j − 1 eπik−j/N − 1, j = k / eπik−j = 16

N−

• n

eπink−j/N = N, j = k 0, j = k ⇒

N−

• n

Fn eπink/N = 1 N (fk)N ⇒ fk =

N−

• n

Fn eπink/N

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! Fn = 1 N

N−

• j

fj e−πinj/N, n = 0, . . . , N − 1 *! fj =

N−

• n

Fn eπinj/N, j = 0, . . . , N − 1 / fjN = fj ∀ j ∈

• FnN = Fn ∀ n ∈
• '

( 3 6 '6 3 ! ! F(u, v) = 1 MN

M−

• x

N−

• y

f(x, y) e−πiux/Mvy/N *! f(x, y) =

M−

• u

N−

• v

F(u, v) eπiux/Mvy/N M 9 3 N 9

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f(x + M, y + N) = f(x, y) ∀ x, y ∈

• F(u + M, v + N) = F(u, v) ∀ u, v ∈
• 2

        F F F F # # # FN−         = 1 N         w w w w . . . w w w w w . . . wN− w w w w . . . wN− w w w w . . . wN− # # # # # # # # # # # # . . . # # # w wN− wN− wN− . . . wN−N−                 f f f f # # # fN−        

3 w = e−πi/N6 !

• ! = 1

N Φ * ! = Φ +2 :! - [ 2 3 4 4 ] T     F F F F     = 1 4     w w w w w w w w w w w w w w w w         2 3 4 4    

! "#$ % "#&' ()*+ =$& = 1 4     1 1 1 1 1 −i −1 i 1 −1 1 −1 1 i −1 −i         2 3 4 4     =     3.25 −0.5 + 0.25i −0.25 −0.5 − 0.25i     ⇒ || =     3.25 √ 0.375 0.25 √ 0.375    

>1.>+1*+( . ,+ % $' (% 9 F(u, v) = 1 MN

M−

• x

N−

• y

f(x, y) e−πiux/Mvy/N = 1 M

M−

• x

e−πiux/M 1 N

N−

• y

f(x, y) e−πivy/N = 1 M

M−

• x

e−πiux/M F(x, v)

• F(x, v) = 1

N

N−

• y

f(x, y) e−πivy/N , x = 0, . . . , M − 1

• F(u, v) = 1

M

M−

• x

F(x, v) e−πiux/M , v = 0, . . . , N − 1

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f x y ( , ) F x v ( , ) F u v ( , )

1-D DFT

• f rows

1-D DFT

• f columns

( , ) M − 1 0 ( , ) M N

− −

1 1 ( , ) 0 0

2-D DFT

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'

• f(x, y) eπiux/Mvy/N

= 1 MN

M−

• x

N−

• y
• f(x, y) eπiux/Mvy/N

e−πiux/Mvy/N = 1 MN

M−

• x

N−

• y

f(x, y) e−πiu−ux/Mv−vy/N = F(u − u, v − v)

• f(x, y) eπiux/Mvy/N ⇔ F(u − u, v − v)

( f(x − x, y − y) ⇔ F(u, v) e−πiux/Mvy/N '

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/ 6 u = M 2 v = N 2 f(x, y)(−1)xy ⇔ F

• u − M

2 , v − N 2

• 2 3 )@

/ | {f(x − x, y − y)}| = |F(u, v)| ' 85 4' >( > 19 3 3 3 ! f(x + M, y + N) = f(x, y) F(u + M, v + N) = F(u, v) 3 3 f(x, y) F(u, v)

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(! * $%! |F(u)| = |F(−u)|A * %! |F(u, v)| = |F(−u, −v)|

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"#4 &? 19! 3 6 9

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"' 1! * 3 f(r, θ + θ) ⇔ F(ω, φ + θ) 8 2% ! F(u, v) = ∞

−∞

∞

−∞

f(x, y) e−πiuxvydxdy ) x = r cos θ6 y = r sin θ6 u = ω cos φ6 v = ω sin φ6 F(ω, φ) = π

• ∞
• f(r, θ) e−πirω θ φrω θ φrdrdθ

= π

• ∞
• f(r, θ) e−πirω θ−φrdrdθ
• {f(r, θ + θ)} =

π

• ∞
• f(r, θ + θ) e−πirω θ−φrdrdθ

) θ + θ = θ6 {f(r, θ + θ)} = π

• ∞
• f(r, θ) e−πirω (θ−θφ)rdrdθ

= F(ω, φ + θ)

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:' 9 9 {a f(x, y) + b f(x, y)} = a {f(x, y)} + b {f(x, y)} ( f(ax, by) ⇔ 1 |a b| F u a, v b

• {f(ax, by)} =

∞

−∞

∞

−∞

f(ax, by) e−πiuxvy dx dy ) g = ax h = by ⇒ dx = 1 a dg dy = 1 b dh6 {f(ax, by)} = 1 |a b| ∞

−∞

∞

−∞

f(g, h) e−πi(u

agv bh) dg dh

= 1 |a b| F u a, v b