Properties of the Fourier Transform - Vision with Direction - page 98

Table 7.2. Correspondence table for forward and inverse FT, FC, and DFT

Z

F ( ω )= F ( f )( ω )= P

f ( t )exp( −iωt ) dt (Forward)

t∈D

Z

f ( t )= F − 1 ( F )( t )= F ( ω )exp( itω ) dω (Inverse)

ω∈D

FT FC I FC II DFT

FORWARD
Symbol F F F DFT

Constant P =2 π P =2 π P =1 P = N

exp ` −in 2 T t ´ Ω exp ` −im 2 Ω ω ´

exp( −in 2 N m )

exp exp( −iωt )

R R P m P m

Sum rule

t ∈ [ − 2 , 2 ]

D t ∈ [ −∞,∞ ] m ∈ Z m ∈{ 0 , ··,N − 1 }

t + t Mod( t + t ,T ) m + m Mod( m + m ,N ) Translation

δ Dirac- δ Dirac- δ Kronecker- δ Kronecker- δ

f integrable f FE f FF f FEF f

INVERSE
Symbol F − 1 F − 1 F − 1

IDFT

T exp ` in 2 T

t ´ exp ` im 2 Ω ω ´ exp ` in 2 N m ´

2 π

exp exp( iωt )

R R

P n P n

Sum rule

ω ∈ [ − 2 , 2 ]

D ω ∈ [ −∞,∞ ] n ∈ Z n ∈{ 0 , ··,N − 1 }

ω + ω n + n Mod( ω + ω ,Ω ) Mod( n + n ,N ) Translation

δ Dirac- δ Kronecker- δ Dirac- δ Kronecker- δ

FE : Finite extension functions
FF : Finite frequency functions (band-limited functions)
FEF : Finite extension and frequency functions