HW10 Solutions(1).pdf - ESE 351-01 Fall 2017 Homework Set#10 due Nov 14 Read this Most problems below do not require a lot of calculation(The 1st two

HW10 Solutions(1).pdf - ESE 351-01 Fall 2017 Homework...

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ESE 351-01, Fall 2017 Homework Set #10, due Nov. 14 Read this : Most problems below do not require a lot of calculation. (The 1 st two require just a little.) The problems do require some insight and some use of complex numbers. (You’ll probably want to refer to the first part of Lecture 14 as needed.) The problems are also in sequence. You may use (in fact I expect you to) the results of previous problems in this set as you solve subsequent ones. 1 . Find the Fourier transforms of the functions below. Use the definition of the Fourier Transform, not a table. a ( ) f t ( ) = e at 1 t ( ) a > 0 b ( ) f t ( ) = e a t a > 0 2 . Find the Inverse Fourier transform of the “pulse” in frequency (height of 1 from – ω 0 to + ω 0 , and 0 elsewhere). Use the definition of the Inverse Fourier Transform, not a table. ( ) ( ) ( ) 0 0 1 1 ω ω ω ω ω + = F 3 . Given that F ( ω ) is the Fourier Transform of f ( t ), find the Fourier Transforms of the functions in (a) and (b) below in terms of F ( ω ). Then do part (c). (a) f ( t - τ ). (b) f ( at ), a > 0. (c) Find the Fourier Transform of the function sketched below. (See part (a), and the formula for a pulse centered at t = 0 from Lecture 18.) 4 . Prove each of the following statements. (Each of these is a short proof, maybe only one line. But you must be familiar with complex numbers. See the 1 st part of Lecture 14.) (a) If f ( t ) is real, then F ω ( ) = F ω ( ) . (b) If f ( t ) is real, then F ω ( ) F ω ( ) = F ω ( ) 2 . (Use result of (a).) (c) If f ( t ) is even, meaning f (- t ) = f ( t ), then so is F ( ω ). (d) If f ( t ) is odd, meaning f (- t ) = - f ( t ), then so is F ( ω ).
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ESE 351-01, Fall 2017 Homework Set #10, due Nov. 14 5 . (a) Show that if f ( t ) is real and even, then F ( ω ) is real (no imaginary component). (b) Show that if f ( t ) is real and odd, then F ( ω ) is purely imaginary (no real component). (c) Determine the angle, F ω ( ) , where F ( ω ) is the Fourier Transform of the function below 6 . Determine the imaginary part of the Fourier Transform of the function below. (See previous problem and problem 3. Also, linearity of the FT and Euler’s formula.) 7 . In class we derived the following relationship: e i ω t dt −∞ = 2 πδ ω ( ) . (a) Use that fact to find the Fourier Transform of e i ω 0 t . (b) Use the result from (a), linearity and Euler’s formula, to find the Fourier Transforms of cos ω 0 t and sin ω 0 t . 8 . Given that f ( t ) and F ( ω ) form a Fourier Transform pair, (a) Find the inverse Fourier Transform of a frequency shifted transform, F ω ω 0 ( ) .
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  • Spring '14
  • Fuhrmann
  • π ω, = F, iφ ω

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