Lesson+31

# Lesson+31 - 1 Lesson 31 Lesson 31 Challenge 30...

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Lesson 31 1

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Challenge 30 Continuous-Time Fourier Transform Chap 17 Challenge 31 Lesson 31 2 Lesson 31
2 2 2 / 2 / 2 2 / 2 / ) 2 / sin( 2 ) 2 / cos( 2 1 2 1 2 2 2 2        A j j e j e A e A dt e t A F j j t j t j Lesson 31 3 Challenge 30 What is the Fourier transform of the signal x(t) shown below. - /2 /2 A Integration challenging Justifies use of tables. Old school

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Lesson 31 4 - /2 /2 A Note: F(0)=0/0 (indeterminate), what do you do now? L’Hopital’s rule: F(0)=0 or simply guess it (i.e., 0) 2 2 2 2 2 ω ωτ ωτ ωτ τ ω ) / sin( ) / cos( A j F
Lesson 31 5 =10 = 3.16 Suppose h(t)=te -at u(t) is the impulse response of a lowpass filter. 2 2 2 2 2 ω ωτ ωτ ωτ τ ω ) / sin( ) / cos( A j F f(t) and F( ) are shown below. - /2 /2 A a=10

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Lesson 31 6 =10 = 3.16 a=10 Then the filter’s output response can be found using machine calculations: Y( ) = F( ) G( Virtually impossible to solve manually.
Lesson 31 7 dt e t x t j ) ( ) X( Bilateral Fourier transform Analysis equation d e t x t j ) X( 2 1 ) ( Synthesis equation Unilateral Fourier transform 0 ) ( ) X( dt e t x t j Analysis equation 0 ) X( 2 1 ) ( d e t x t j Synthesis equation

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Using Tables Lesson 31 8 What is the Fourier transform of a non-causal cosine wave? What is the Fourier transform of a causal cosine wave? x(t)=cos( 0 t) [ ( - 0 ) + ( + 0 )] x(t)=cos( 0 t)u(t) ( /2)[ ( - 0 ) + ( + 0 ) + j /( 0 2 - 2 )] - 0 0 0 - 0 0 0
(rect(t/ ) ) / ( ) / ( ) ( / / / / 2 sinc 1 2 2 2 2 ωτ ω τ ω ωτ ωτ τ τ ω ω j j t j t j e e j dt e dt e t rect X Lesson 31 9 Old School Fourier Transforms - /2 0 /2 In this manner a library of Fourier transforms have been constructed over time.

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Fourier Transforms Lesson 31 10 Tables provide a foundation for articulating the CTFT of primitive signals and CTFT properties (see back cover of the class text). Some properties will be explored in this Lesson. There is also a CTFT uncertainty principle that should be kept in mind. 1 . In order to have absolute knowledge of x(t) at a point in time, X( ) must be known for all frequencies (  [- , ]). 2 . In order to have absolute knowledge of X( ) at a specific frequency, x(t) must be known for all time (t [- , ]).
CTFT vs CTFS What is the relationship between the Fourier transform and Fourier series?

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