2025-L19su10

2025-L19su10 - ECE2025 Summer 2010 Lecture 19 Fourier...

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1 7/9/2010 1 ECE2025 Summer 2010 Lecture 19 Fourier Transforms (Part II) 19 Jul 10 Guest Lecturer: Justin Romberg 7/9/2010 3 ANNOUNCEMENTS ± HW #8 due July 20 (in recitation)-21 (in lab L01, L02 only) ± Do Lab #9 on July 21-22 ± Do Lab #10 on July 28-29. All in-lab. 7/9/2010 4 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 7/9/2010 5 Table of Fourier Transforms x ( t ) = e at u ( t ) X ( j ) = 1 a + j x ( t ) = 1 t < T /2 0 t > T X ( j ) = sin( T /2) () x ( t ) = sin( 0 t ) 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 )
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2 7/9/2010 6 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 ω ) ( 2 since Therefore, 0 0 πδ k e t jk −∞ = = k k k a j X ) ( 2 ) ( 0 δ π 7/9/2010 7 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 /2 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 7/9/2010 8 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 7/9/2010 9 Strategy for using the FT ± Develop a set of known Fourier transform pairs.
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This note was uploaded on 10/21/2010 for the course ECE 2025 taught by Professor Juang during the Summer '08 term at Georgia Tech.

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2025-L19su10 - ECE2025 Summer 2010 Lecture 19 Fourier...

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