HW8 - [There is one more question on the back Question 10...

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ECE 301, Homework #8, due date: 3/10/2010 chihw/10ECE301S/10ECE301S.html Question 1: [Basic] What is the “time-shifting” property of the Fourier transformation. Please describe it carefully. Question 2: [Basic] From HW7, 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 x ( t ). Question 3: [Basic] Continue from the previous questions Suppose we know h ( t ) = x ( t ), and z ( t ) = h ( t ) * x ( t ). Find out the Fourier transform of z ( t ). Question 4: [Basic] Suppose x 1 ( t ) = δ ( t - t 0 ) 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 0 = 1. If t 0 changes from 1 to 5, how will your x 3 ( t ) change? Question 5: p. 338, Problem 4.21(b,g,i). Question 6: p. 338, Problem 4.22(b,c,d). Question 7: p. 339, Problem 4.23. Question 8: p. 341, Problem 4.25. (a,b,c) Question 9: p. 341, Problem 4.25. (e) Do (e) by the Parseval’s relationship. Evaluate 1 2 π R -∞ X ( ω ) e j 2 ω . Hint: View it as 1 2 π R -∞ X ( ω ) e jωt with t = 2, which is the inverse Fourier transform of X ( ω ) evaluated at t
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Unformatted text preview: [There is one more question on the back.] Question 10: We will solve p. 344, Problem 4.30 step by step. • Consider a test signal x ( t ), whose Fourier transform X ( ω ) is as specified in Fig-ure P4.28(a). Let y ( t ) = x ( t )cos( t ). Find Y ( ω ). Hint 1: You should know the following equations very well: cos( t ) = 1 2 ( e jt + e-jt ) and F{ cos( t ) } = 1 2 (2 πδ ( ω-1) + 2 πδ ( ω + 1)). Hint 2: Use the property that multiplication in time equals the convolution in frequency. Compute the 1 2 π X ( ω ) * F{ cos( t ) } , which should be the Fourier transform of x ( t )cos( t ). • Use the previous sub-question, “guess” what type/shape of X ( ω ) can generate the given G ( jω ) based on the formula 1 2 π X ( ω ) * F{ cos( t ) } . • Once the correct X ( ω ) is “guessed,” then do the inverse Fourier transform to obtain x ( t )....
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