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HW8 - Texas A&M University Mechanical Measurements...

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Texas A&M University Mechanical Measurements Assignment #8 Due: Monday November 8, 2010 Problem 1) Complex Numbers a) Convert the complex number z = 1+1 j to the form φ j Ae , where A and φ are the magnitude and phase angle of complex number, respectively. b) Convert the complex number z = (1+1 j )/(-1+5 j ) to the form φ j Ae . c) Determine expressions for the magnitude and phase of the following complex function: ( ) ωτ ωτ ω j j j G + = 1 . Assuming that 10 = τ , plot the magnitude and phase vs. frequency. You should use a log-scale for the frequency. Problem 2) Fourier Series a) Find Fourier series of the wave function in Figure (a). b) Find Fourier series of the wave function in Figure (b). c) What is the wave function that is combined by above two waves? Plot the result wave function. Problem 3) Fourier Transform Definition Using the definition of the Fourier Transform, ( ) ( ) j t Y y t e dt ω ω −∞ = , integrate to obtain: a) the Fourier Transform for the function T t y 1 ) ( = ( T t 0 ). b) the Fourier transform of the non-periodic pulse signal represented by 1 for ( ) 2 2 0 otherwise t x t τ τ = c) the Fourier transform of the decaying exponential signal t e t u t x α = ) ( ) ( where α is a positive constant and 1 for 0 ( ) 0 for 0 t u t t = < is known as the unit-step function.
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Problem 4) Fourier Transforms by Tables Using the Fourier Transform Pairs Table in below: a) Compute the (generalized) Fourier transform of the signal ( ) 0.5 ( ) x t u t = − + b) Compute the (generalized) Fourier transform of the signal ( ) 5sin(2 ) 2cos(5 ) x t t t = + .
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