Lecture-DFT

# Lecture-DFT - EE 4780 2D Discrete Fourier Transform(DFT...

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Unformatted text preview: EE 4780 2D Discrete Fourier Transform (DFT) Bahadir K. Gunturk 2 2D Discrete Fourier Transform 2D Fourier Transform 2 ( ) ( , ) [ , ] j um vn m n F u v f m n e π ∞ ∞- + =-∞ =-∞ = ∑ ∑ 1 1 2 1 [ , ] [ , ] k l M N j m n M N m n F k l f m n e MN π --- + ÷ = = = ∑∑ 2D Discrete Fourier Transform (DFT) 2D DFT is a sampled version of 2D FT. Bahadir K. Gunturk 3 2D Discrete Fourier Transform Inverse DFT 1 1 2 1 [ , ] [ , ] k l M N j m n M N m n F k l f m n e MN π --- + ÷ = = = ∑∑ 2D Discrete Fourier Transform (DFT) 1 1 2 [ , ] [ , ] k l M N j m n M N k l f m n F k l e π -- + ÷ = = = ∑∑ where and 0,1,..., 1 k M =- 0,1,..., 1 l N =- Bahadir K. Gunturk 4 2D Discrete Fourier Transform Inverse DFT 1 1 2 1 [ , ] [ , ] k l M N j m n M N m n F k l f m n e MN π --- + ÷ = = = ∑∑ It is also possible to define DFT as follows 1 1 2 1 [ , ] [ , ] k l M N j m n M N k l f m n F k l e MN π -- + ÷ = = = ∑∑ where and 0,1,..., 1 k M =- 0,1,..., 1 l N =- Bahadir K. Gunturk 5 2D Discrete Fourier Transform Inverse DFT 1 1 2 [ , ] [ , ] k l M N j m n M N m n F k l f m n e π --- + ÷ = = = ∑∑ Or, as follows 1 1 2 1 [ , ] [ , ] k l M N j m n M N k l f m n F k l e MN π -- + ÷ = = = ∑∑ where and 0,1,..., 1 k M =- 0,1,..., 1 l N =- Bahadir K. Gunturk 6 2D Discrete Fourier Transform Bahadir K. Gunturk 7 2D Discrete Fourier Transform Bahadir K. Gunturk 8 2D Discrete Fourier Transform Bahadir K. Gunturk 9 2D Discrete Fourier Transform Bahadir K. Gunturk 10 Periodicity 1 1 2 1 [ , ] [ , ] k l M N j m n M N m n F k l f m n e MN π --- + ÷ = = = ∑∑ 1 1 2 1 [ , ] [ , ] k M l N M N j m n M N m n F k M l N f m n e MN π + + --- + ÷ = = + + = ∑∑ 1 1 2 2 1 [ , ] k l M N M N j m n j m n M N M N m n f m n e e MN π π --- +- + ÷ ÷ = = = ∑∑ 1 [M,N] point DFT is periodic with period [M,N] [ , ] F k l = Bahadir K. Gunturk 11 Periodicity 1 1 2 [ , ] [ , ] k l M N j m n M N k l f m n F k l e π -- + ÷ = = =...
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Lecture-DFT - EE 4780 2D Discrete Fourier Transform(DFT...

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