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HW9 - h t = x t and z t = h t x t Find out the Fourier...

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ECE 301, Homework #9, due date: 10/28/2010 http://cobweb.ecn.purdue.edu/ chihw/10ECE301F/10ECE301F.html Question 1: [Basic] For a continuous time x ( t ) = 2 -| t | , find out the Fourier transformation of x ( t ). (Hint: Example 4.2) Question 2: [Basic] For a continuous time x ( t ) = U ( t +2) -U ( t - 2), find out the Fourier transformation of x ( t ). (Hint: Example 4.4) Question 3: [Basic] For a continuous time x ( t ) = cos(2 πt ) + sin(4 t ), find out the Fourier transformation of x ( t ). [There is one more question on the back.] Question 4: [Basic] For a continuous time signal with X ( ) = U ( ω +3) -U ( ω - 3), find out the inverse Fourier transformation of X ( ). Question 5: [Basic] What is the “time-shifting” property of the Fourier transformation. Please describe it carefully. From Question 4, we have found x ( t ) when knowing X ( ) = U ( ω + 3) - U ( ω - 3). Suppose we know y ( t ) = x ( t ) e j 3 t . Find out and plot the Fourier transform of y ( t ). Question 6: [Basic] Continue from the previous questions Suppose we know

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Unformatted text preview: h ( t ) = x ( t ), and z ( t ) = h ( t ) * x ( t ). Find out the Fourier transform of z ( t ). Question 7: [Basic] Suppose x 1 ( t ) = δ ( t-t ) and x 2 ( t ) = U ( t + 2)- U ( t-2). Find out and plot x 3 ( t ) = x 1 ( t ) * x 2 ( t ) when t = 1. If t changes from 1 to 5, how will your x 3 ( t ) change? Question 8: p. 338, Problem 4.21(b,g,i). Question 9: p. 338, Problem 4.22(b,c,d). Question 10: p. 339, Problem 4.23. Question 11: p. 341, Problem 4.25. (a,b,c) Question 12: p. 341, Problem 4.25. (e) • Do (e) by the Parseval’s relationship. • Evaluate 1 2 π R ∞-∞ X ( ω ) e j 2 ω dω . Hint: View it as 1 2 π R ∞-∞ X ( ω ) e jωt dω with t = 2, which is the inverse Fourier transform of X ( ω ) evaluated at t = 2....
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