2025-L19su09 - ECE2025 Summer 2009 Lecture 19 Fourier...

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1 7/14/2009 1 ECE2025 Summer 2009 Lecture 19 Fourier Transforms 15 Jul 09 7/14/2009 3 ANNOUNCEMENTS ± HW #8 due July 14 (in recitation)-15 (in lab) ± Do Lab #9 on July 15-16 ± Do Lab #10 on July 22-23. All in-lab. 7/14/2009 4 THIRD QUIZ ± Average : 72.5 ± Median: 69.0 ± +8 added as adjustment. ± Standard deviation: 18.5 ± Correlation coefficients: 0.647, 0.670 7/14/2009 5 Fourier Transforms ± For non-periodic signals Fourier Synthesis Fourier Analysis = dt e t x j X t j ω ) ( ) ( = π d e j X t x t j ) ( ) ( 2 1
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2 7/14/2009 6 Table of Fourier Transforms x ( t ) = e at u ( t ) X ( j ω ) = 1 a + j x ( t ) = 1 t < T /2 0 t > T X ( j ) = sin( T /2) () x ( t ) = sin( 0 t ) π t X ( j ) = 1 < 0 0 > 0 x ( t ) = δ ( t t 0 ) X ( j ) = e j t 0 x ( t ) = e j 0 t X ( j ) = 2 πδ ( 0 ) 7/14/2009 7 Fourier Transform of a General Periodic Signal ± If x(t) is periodic with period T 0 , −∞ = = = 0 0 0 0 0 ) ( 1 ) ( T t jk k k t jk k dt e t x T a e a t x ) ( 2 since Therefore, 0 0 k e t jk −∞ = = k k k a j X ) ( 2 ) ( 0 7/14/2009 8 Square Wave Signal x ( t ) = x ( t + T 0 ) T 0 2 T 0 T 0 2 T 0 0 t a k = e j 0 kt j 0 kT 0 0 T 0 e j 0 kt j 0 kT 0 T 0 /2 T 0 = 1 e j k j k a k = 1 T 0 (1) e j 0 kt dt + 1 T 0 ( 1) e j 0 kt dt T 0 T 0 0 T 0 7/14/2009 9 Square Wave Fourier Transform X ( j ) = 2 a k ( k 0 ) k =−∞ x ( t ) = x ( t + T 0 ) T 0 2 T 0 T 0 2 T 0 0 t
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3 7/14/2009 10 Strategy for using the FT ± Develop a set of known Fourier transform pairs.
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This note was uploaded on 01/24/2010 for the course ECE 2025 taught by Professor Juang during the Spring '08 term at Georgia Institute of Technology.

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2025-L19su09 - ECE2025 Summer 2009 Lecture 19 Fourier...

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