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...

• Homework Help
• 8

This preview shows page 1 - 3 out of 8 pages.

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 ).

Subscribe to view the full document.

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 ( ) .
• Spring '14
• Fuhrmann
• π ω, = F, iφ ω

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes