This document discusses Fourier processing and covers topics such as continuous and discrete Fourier bases, sampling, 2D Fourier bases, and Fourier approximation. The continuous Fourier basis uses complex exponentials while the discrete Fourier basis uses a finite number of complex exponentials. Sampling and periodization allow transforming between continuous and discrete settings. The 2D Fourier basis is a product of 1D bases. Fourier approximation represents functions as a sum of complex exponentials.
15. 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
16. Fourier Transforms
Periodization f0 (t) ⇥ n f0 (t + n)
f [n] = f0 (n/N )
Sampling
f0 (t), t R f0 (t), t [0, 1] Continuous
f [n], n Z f [n], 0 n<N Discrete
Fourier transform
Isometry f ⇥ f ˆ
Infinite Periodic
f0 (N (⇥ + 2k ))
ˆ
Sampling f0 ( ) ⇥ ˆ
{f0 (k)}k
ˆ
f0 ( ), ˆ
f0 [k], k Z Infinite
Periodization
R
ˆ ˆ
ˆ
f (⇥), ⇥ [0, 2 ] f [k], 0 k<N Periodic
k
f (⇥) =
Continuous Discrete
ˆ
+⇥ 1
ˆ
f0 ( ) = f0 (t)e i t
dt ˆ
f0 [m] = f0 (t)e 2i mt
dt
⇥ 0
N 1
ˆ
f( ) = f [n]e i n ˆ
f [m] = f [n]e
2i
N mn
n Z n=0