Gabriel Peyré
January 01, 2012
1.6k

# Signal Processing Course: Fourier Processing

January 01, 2012

## Transcript

2. ### Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

Basis •Fourier Approximation
3. ### m (x) = em (x) = e2i mx Continuous Fourier

Bases Continuous Fourier basis:
4. ### m (x) = em (x) = e2i mx Continuous Fourier

Bases Continuous Fourier basis:

6. ### Fourier and Convolution x− x+ x 1 2 1 2

f f ∗1[− 1 2 , 1 2 ]
7. ### Fourier and Convolution x− x+ x 1 2 1 2

f f ∗1[− 1 2 , 1 2 ] 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
8. ### Fourier and Convolution x− x+ x 1 2 1 2

f f ∗1[− 1 2 , 1 2 ] 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
9. ### Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

Basis •Fourier Approximation

h[m]

h[m]
14. ### Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

Basis •Fourier Approximation
15. ### Inﬁnite continuous domains: Periodic continuous domains: Inﬁnite discrete domains: Periodic

discrete domains: f0 (t), t R f0 (t), t ⇥ [0, 1] R/Z The Four Settings ˆ f[m] = N 1 n=0 f[n]e 2i N mn ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n Note: for Fourier, bounded periodic. .. . .. . .. . .. . f[n], n Z f[n], n ⇤ {0, . . . , N 1} ⇥ Z/NZ
16. ### ˆ f[m] = N 1 n=0 f[n]e 2i N mn

Fourier Transforms Discrete Inﬁnite Periodic f[n], n Z f[n], 0 n < N Periodization Continuous f0 (t), t R f0 (t), t [0, 1] f0 (t) ⇥ n f0 (t + n) Sampling ˆ f0 ( ) ⇥ { ˆ f0 (k)}k Discrete Inﬁnite Periodic Continuous Sampling ˆ f[k], 0 k < N ˆ f0 ( ), R ˆ f0 [k], k Z Fourier transform Isometry f ⇥ ˆ f ˆ f0 ( ) = +⇥ ⇥ f0 (t)e i tdt ˆ f0 [m] = 1 0 f0 (t)e 2i mtdt ˆ f( ) = n Z f[n]ei n ˆ f(⇥), ⇥ [0, 2 ] Periodization ˆ f(⇥) = k ˆ f0 (N(⇥ + 2k )) f[n] = f0 (n/N)

23. ### Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

Basis •Fourier Approximation
24. ### 2D Fourier Basis em [n] = 1 N e 2i

N0 m1n1+ 2i N0 m2n2 = em1 [n1 ]em2 [n2 ]
25. ### 2D Fourier Basis em [n] = 1 N e 2i

N0 m1n1+ 2i N0 m2n2 = em1 [n1 ]em2 [n2 ]
26. ### Overview •Continuous Fourier Basis •Discrete Fourier Basis •Sampling •2D Fourier

Basis •Fourier Approximation

28. ### 1D Fourier Approximation 0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1