Lecture-08

The2ddftisafourierseriesrepresentation

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Unformatted text preview: n]e j 2 ( m n ) • Recall , thus we can m 0 n 0 write: F [k , l ] F ( • Linearity af [ m, n] bg[m, n] aF [k , l ] bG[k , l ] kl ,) MN • So far: ① The 2‐D DFT is a Fourier Series representation for the periodic extension of ; f [m, n] ② The 2‐D DTF consists of samples of the Fourier Transform; Ghassan AlRegib © Georgia Tech WN e j Ghassan AlRegib © Georgia Tech 10 Symmetry ECE 6258 Fall2012 f [m, n] • If is real, then • Circular Shift mk f [((m m0 )) M , ((n n0 )) N ] WM WNnl F [k , l ] ((q)) N q mod N f [m, n] g[m, n] • When and have the support on the same region M N • Note: If the two images don’t have support on the same region, we perform zero padding 9 Circular Shift ECE 6258 Fall 2012 Properties of the 2‐D DFT ECE 6258 Fall 2012 F [k , l ] F (( M k )) M , (( N l )) N 2 N i.e. If a 2‐D signal is real, its DFT is a 2‐D Hermitian symmetric in a circular sense • Example： Re F [k , l ] Re F (( M k )) M , (( N l )) N Im F [k , l ] Im F (( M k )) M , (( N l )) N Ghassan AlRegib © Georgia Tech 11 Ghassan AlRegib © Georgia Tech 12 3 9/13/2012 Symmetry ECE 6258 Fall 2012 • Similarly, if we have a sequence that has Harmitian symmetric and anti‐symmetric portions, then we can find the following: f s [m, n] • Combining all four equation lead us to: 1 F [k , l ] F [k , l ] Re F[k , l ] 2 1 f a [m, n] F [k , l ] F [k , l ] j Im F [k , l ] 2 f s [m, n] 1 f [m, n] f (( M m)) M , (( N n)) N 2 OR 1 f a [m, n] f [m, n] f (( M m)) M , (( N n)) N 2 Ghassan AlRegib © Georgia Tech Hermitian Symmetric Portion Real Part of DFT Hermitian Anti‐symmetric Portion imaginary part of DFT j 13 Reflection ECE 6258 Fall 2012 Symmetry ECE 6258 Fall 2012 Ghassan AlRegib © Georgia Tech Parseval’s Theorem ECE 6258 Fall 2012 M 1 N 1 • If f [m, n] F [k , l ] • Then f [ m, n ] g [ m , n ] m 0 n 0 _______________ f [n, m] F [l , k ] 14 1 M 1 N 1 F[k , l ]G[k , l ] MN _______________ k 0 l 0 Inner product in sampled frequency domain f (( M m)) M , n F (( M k )) M , l Inner product in spatial domain f m, (( N n)) N F k , (( N l )) N f g • If , then we have two ways to measure the signal energy f (( M m)) M , (( N n)) N F (( M k )) M , (( N l )) N M 1 N 1 m 0 n 0 Ghassan AlRegib © Georgia Tech 15 f [m, n]...
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This note was uploaded on 02/14/2014 for the course ECE 6258 taught by Professor Staff during the Fall '08 term at Georgia Tech.

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