ps8 - EE 350 PROBLEM SET 8 DUE: 12 November 2007 Reading...

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EE 350 PROBLEM SET 8 DUE: 12 November 2007 Reading assignment: Lathi Chapter 4, Sections 4.5 through 4.7 and Chapter 5, Section 5.1 Recitation sections will meet during the week of November 5. Exam II is scheduled for Thursday, November 15 from 8:15 pm to 10:15 pm in 121 Sparks (all sections). The third exam covers material from Problem Sets 6 through 8, and Laboratory #3. The date and location of a review session will be announced on the EE 350 web page. Problem 38: (17 points) 1. (2 points) Consider two functions f ( t ) and g ( t ) that have Fourier transforms F ( ω ) and G ( ω ), respectively. For arbitrary constants a and b , ±nd an expression for F{ af ( t )+ bg ( t ) } in terms of F ( ω ) and G ( ω ). 2. (3 points) Consider a real-valued signal f ( t ) with Fourier transform F ( ω )= ± -∞ f ( t ) e - ωt dt. The Fourier transform F ( ω ) is a complex-valued function of the real variable ω . Said diﬀerently, when evaluated for a given value ω ,the Fourier transform F ( ω ) will be a complex number. Because complex numbers can be expressed in either polar or rectangular form, we can also express F ( ω ) in either polar or rectangular form. Show that F ( ω ) can be expressed in rectangular form as F ( ω R ( ω I ( ω ) , where R ( ω ) and I ( ω ) are de±ned as R ( ω ± -∞ f ( t )cos( ωt ) dt I ( ω ± -∞ - f ( t )sin( ) dt. The function R ( ω ) is the real part of F ( ω ) and the function I ( ω ) is the imaginary part of F ( ω ). 3. (3 points) The polar form of the Fourier transform F ( ω ) is given by F ( ω | F ( ω ) | e ± F ( ω ) . Express the magnitude | F ( ω ) | and angle ± F ( ω ) in terms of R ( ω ) and I ( ω ). 4. (3 points) If f ( t ) is real-valued, show that (a) F ( - ω F * ( ω ). (b) The magnitude | F ( ω ) | is an even function of ω . (c) The phase angle ± F ( ω ) is an of odd function of ω . 5. (3 points) If f ( t ) is real-valued and an even function of t , show that F ( ω R ( ω )=2 ± 0 f ( t ) dt is real and an even function of ω .

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ps8 - EE 350 PROBLEM SET 8 DUE: 12 November 2007 Reading...

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