3. The Four Settings
Note: for Fourier, bounded periodic.
Infinite continuous domains: f0 (t), t R ... ...
+⇥
ˆ
f0 ( ) = f0 (t)e i t
dt
⇥
Periodic continuous domains: f0 (t), t ⇥ [0, 1] R/Z
1
ˆ
f0 [m] = f0 (t)e 2i mt
dt
0
Infinite discrete domains: f [n], n Z ... ...
ˆ
f( ) = f [n]ei n
n Z
Periodic discrete domains: f [n], n ⇤ {0, . . . , N 1} ⇥ Z/N Z
N 1
ˆ
f [m] = f [n]e
2i
N mn
n=0
4. Sampling and Periodization
f0 (t)
Sampling idealization:
(a)
f [n] = f0 (n/N )
(a)
f [n]
Poisson formula:
ˆ
f (⇥) = ˆ
f0 (N (⇥ + 2k ))
(b)
k
(b) ˆ
f0 ( )
Commutative diagram: (a) 1
1
sampling (a)
f0 (c) f 0
ˆ
f( )
cont. FT (c) discr. FT 0
periodization
ˆ
f0 ˆ
f (b)