2025-L18su09 - ECE2025 Summer 2009 Lecture 18 Frequency...

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1 7/9/2009 1 ECE2025 Summer 2009 Lecture 18 Frequency Response of Continuous-Time Systems 10 Jul 09 7/9/2009 3 ANNOUNCEMENTS ± HW #8 due July 14 (in recitation)-15 (in lab) ± Do Lab #9 on July 15-16: ± Quiz 3: Monday June 13 7/9/2009 4 THIRD QUIZ ± Quiz #3, in lecture, Monday, July 13 ± 10% of final grade ± You can use the full 1:20-2:30 time ± Review session Sunday July 12, 8 pm. ± Emphasis: ± HWs some of #5 through all of #7 ± Lectures #11 through #15 ± No Fourier Transforms ± Emphasis on DT frequency response and z-transforms. ± Closed book, closed notes, except: ± One 8.5”X11” crib sheet allowed, handwritten , OK to write on both sides 7/9/2009 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 7/9/2009 6 Example 1: x ( t ) = e at u ( t ) X ( j ω ) = 1 a + j X ( j ) = e at 0 e j t dt = 0 e ( a + j ) t dt X ( j ) =− e at e j t a + j 0 = 1 a + j a > 0 7/9/2009 7 Frequency Response ± Fourier Transform of h(t) is the Frequency Response j j H t u e t h t + = = 1 1 ) ( ) ( ) ( ) ( ) ( t u e t h t = 7/9/2009 8 Magnitude and Phase Plots j a j H + = 1 ) ( = a j H 1 tan ) ( 2 2 1 1 + = + a j a ) ( ) ( j H j H = 7/9/2009 9 X ( j ) = sin( T / 2) /2 () Example 2: x ( t ) = 1 t < T / 2 0 t > T X ( j ) = e j t j T T = e j T e j T j X ( j ) = (1) e j t dt T T = e j t dt T T
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3 7/9/2009 10 x ( t ) = 1 t < T / 2 0 t > T /2 X ( j ω ) = sin( T /2) () 7/9/2009 11 Example 3: > < = b b j X 0 1 ) ( t t t x b π ) sin( ) ( = = = b b d e d e j X t x t j t j 1 2 1 ) ( 2 1 ) ( jt e e jt e t x t j t j t j b b b b = = 2 1 2 1 ) ( 7/9/2009 12 > < = = b b b j X t t t x 0 1 ) ( ) sin( ) ( 7/9/2009 13 Example 4: X ( j ) = δ ( t ) e j t dt −∞ = 1 Shifting Property of the Impulse ) ( ) ( 0 t t t x = δ 0 ) ( ) ( 0 t j t j e dt e t t j X = =
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4 7/9/2009 14 x ( t ) = δ ( t ) X ( j ω ) = 1 7/9/2009 15 Example 5: X ( j ) = 2 πδ ( 0 ) x ( t ) = 1 2 π 2 ( 0 ) e j t d −∞
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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-L18su09 - ECE2025 Summer 2009 Lecture 18 Frequency...

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