20111ee102_1_homework8

20111ee102_1_homework8 - y t when the input is x t = 1...

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WINTER 2011: Put Discussion section number in the corner →→→ (* Otherwise Your HW will be LOST) Name(Print) (LAST, Middle, First): ————————————— Student ID : ——————————————————– HW: # 8 NO LATE HOMEWORK POLICY! Posted: March 4 Hand In: March 10 IN CLASS Attach This Sheet To Your HW 1. Consider a linear system with impulse response h ( t ) = e - t (cos t - sin t ) U ( t ) . a) Find the transfer function H ( s ). b) Compute the input x ( t ) which generates the output y ( t ) = e - t cos( t - 1) U ( t - 1) . c) Find H ( ). Is H ( ) = H ( s ) | s = ? Why? d) The input x 2 ( t ) = 100 + cos(2 t ) is now applied to the system. What is the average (DC) value of the output? [Think before you start writing the equations.] e) For the input x 2 ( t ) of the previous question, what is the mean square error of the output when it is approximated up to the third harmonic? [Again, think.] 1
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h) What is the output y 3 ( t ) when the input is x 3 ( t ) = cos( 2 t ) + U ( t )? 2. The impulse response of a LTI system is h ( t ) = 2 sin(2 t ) πt [cos(6 t )] 2 a) Find H ( ). b) Find the output
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Unformatted text preview: y ( t ), when the input is x ( t ) = 1 + sin(2 . 5 t ) + cos(5 t ). 3. Consider the two LTI systems whose frequency functions are given by: H 1 ( jω ) = F { h 1 ( t ) } = e-jω/ 2 rect p ω 2 π P H 2 ( jω ) = F { h 2 ( t ) } =-j sgn ( ω ) rect p ω 2 π P where rect ( ω/ω ) = U ( ω + ω / 2)-U ( ω-ω / 2) and sgn ( ω ) = 2 U ( ω )-1. a) Sketch the amplitude and phase spectra of H 1 ( jω ) and H 2 ( jω ). b) Compute h 1 ( t ) and h 2 ( t ). c) Compute the output y ( t ) of the cascade x ( t ) → H 1 ( jω ) → H 2 ( jω ) → y ( t ) when the input is x ( t ) = cos( πt/ 2) + 10 sin(2 πt ). 4. Compute the signal f ( t ) whose Fourier transform F ( jω ) is given in terms of its amplitude | F ( jω ) | = 1 for-2 ≤ ω < 0, | F ( jω ) | = ω for 0 ≤ ω < 2 and | F ( jω ) | = 0 otherwise . and its phase: Θ( ω ) = π 2 for ω < 0 and Θ( ω ) =-π 2 for ω ≥ . 2...
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20111ee102_1_homework8 - y t when the input is x t = 1...

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