113_1_Week06-print

113_1_Week06-print - EE 113: Digital Signal Processing Week...

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EE 113: Digital Signal Processing Week 6 1. The Fourier domain 2. Discrete-Time Fourier Transform (DTFT) 3. Discrete Fourier Transform (DFT) 4. Convolution with the DFT
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Fourier Transforms Discrete finite/pdc X [ k ] Discrete finite/pdc x [ n ] Discrete FT (DFT) Continuous periodic X ( e j ω ) Discrete infinite x [ n ] Discrete-Time FT (DTFT) Continuous infinite X ( ) Continuous infinite x ( t ) Fourier Transform (FT) Discrete infinite c k Continuous periodic x ( t ) Fourier Series (FS) Frequency Time ~ ~
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Preliminaries
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Discrete Time FT (DTFT) ± FT defined for discrete sequences: ± Summation (not integral) ± Discrete (normalized) frequency variable ω ± Argument is e j ω , not plain ω DTFT X ( e j ω ) = x [ n ] e j n n =−∞
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Discrete Time FT (DTFT) ± Two ways to compute it ± Using the z-transform X (z) and replace z with e j ω ± Compute the sum X ( e j ω ) = x [ n ] e j n n =−∞ Examples!
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± x [ n ] = δ [ n ] ± i.e. x [ n ] X ( e j ω ) [ n ] 1 DTFTs of simple sequences (for all ) n -2 2 3 1 -1 -3 - ππ x [ n ] X ( e j ) () [ ] 1 j jn n j Xe x n e e =−∞ = ==
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DTFT example ± e.g. x [ n ] = α n · µ [ n ], | | < 1 n -1 1234567 X ( e j ω ) = n [ n ] e j n n =−∞ = e j () n n = 0 = 1 1 −α e j
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Periodicity of X ( e j ω ) ± X ( e j ω )
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This note was uploaded on 11/07/2009 for the course EE 113 taught by Professor Walker during the Spring '08 term at UCLA.

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113_1_Week06-print - EE 113: Digital Signal Processing Week...

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