Lec2 - IEG 4190 Lecture Notes 2 Lecturer Jianzhuang Liu Outline 1 Continuous-time Fourier Transform 2 Discrete-time Fourier Transform 3 2-D

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1 IEG 4190 Lecture Notes 2; Lecturer: Jianzhuang Liu Outline : 1. Continuous-time Fourier Transform 2. Discrete-time Fourier Transform 3. 2-D Discrete Space Fourier Transform 4. Signal Synthesis and Reconstruction from Phase 5. The Projection-Slice Theorem Computed Tomography (CT)
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2 Continuous-time Fourier Transform Let ) ( ~ t x be a periodic signal of period 0 T , define = other T t T t x t x , 0 2 2 ), ( ~ ) ( 0 0 Fourier series of ) ( ~ t x , 0 () jk t k k xt ae ω +∞ =−∞ = ± 0 0 0 /2 0 1 T jk t k T ax t e d t T = ± 0 2 T 0 2 T 0 2 T 0 2 T ±
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3 CTFT, cont’d 0 00 0 /2 11 () T jk t jk t k T ax t e d t x t e d t TT ωω +∞ −− == ∫∫ ± Let ( ) jt Xx t e d t ω +∞ −∞ = Coefficients k a can be expressed as 12 ( ), k aX k π 0 0 ( ) ( ) 2 jk t jk t kk xt Xk e e T + ∞+ =−∞ ∑∑ ±
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4 CTFT, cont’d As 0 T , 0 0 d ω →→ , ) ( ) ( ~ t x t x 0 00 () ( ) jk t jt k Xk e X e d ωω +∞ −∞ =−∞ So we have 1 ( ) 2 xt X π +∞ −∞ = ( ) Xx t e d t +∞ −∞ = Note: both and X are continuous, aperiodic.
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5 Discrete-time Fourier Transform Consider a finite duration signal 0 ) ( = n x , for 1 N n > and periodic signal ) ( ~ n x constructed to equal ) ( n x over one period N .
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6 DTFT, cont’d Fourier series of ) ( ~ n x : > =< = N k n N jk k e a n x ) / 2 ( ) ( ~ π > =< = N n n N jk k e n x N a ) / 2 ( ) ( ~ 1 Since ) ( ~ ) ( n x n x = over one period, then ) ( ~ n x can be replaced by () xn , therefore: + −∞ = = = = n n N jk N N n n N jk k e n x N e n x N a ) / 2 ( ) / 2 ( ) ( 1 ) ( 1 1 1
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7 DTFT, cont’d Defining a function () X ω as ( ) jn n Xx n e +∞ =−∞ = 00 12 , k aX k NN π ωω == Then 0 0 1 ( ) jk n kN xn Xk e N =< > = ±
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8 DTFT, cont’d Since 0 1/ /2 N ω π = , then 0 00 1 () ( ) 2 jk n kN xn Xk e w =< > = ± As ) ( ~ , n x N equals ) ( n x for any finite value of n and 0 0 d →→ , 2 1 ( ) 2 jn X e dw = ( ) n Xx n e +∞ =−∞ = Note: is discrete & aperiodic, but X is continuous & periodic ( 2) XX ωπ =+ .
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9 Comparison among CTFS, DTFS, CTFT, and DTFT CTFS: DTFS: k a continuous, period = T discrete, aperiodic discrete, period = N discrete, period = N () xn k a xt
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10 Comparison among CTFS, DTFS, CTFT, and DTFT, cont’d CTFT: DTFT: continuous, aperiodic continuous, aperiodic discrete, aperiodic continuous, period = 2 π () X ω xt xn X
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11 2-D Discrete Space Fourier Transform Eigenfunction of LSI Systems: a sequence ) , ( 2 1 n n x is said to be an eigenfunction of a system T if ) , ( )] , ( [ 2 1 2 1 n n kx n n x T = , 11 2 2 jn jn ee ω can be
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This note was uploaded on 10/13/2009 for the course CS 5150 taught by Professor Xulei during the Spring '09 term at University of Central Arkansas.

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Lec2 - IEG 4190 Lecture Notes 2 Lecturer Jianzhuang Liu Outline 1 Continuous-time Fourier Transform 2 Discrete-time Fourier Transform 3 2-D

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