midterm_fall05

midterm_fall05 - j 2 πf of the signal x t = u t 2-2 u t u...

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UCSB Fall 2005 ECE 130A: Midterm Examination Problems INSTRUCTIONS: Problems are weighted as shown. Show your work. No credit without proper justification, even if your answers are correct. The exam is closed book, closed notes, except for the sheet of formulas provided separately. Problem 1 (20 points) This question has four unrelated parts. 1(a) 4 points A system with input x ( t ) has output y ( t ) = sin( π t ) e j | x ( t ) | Is the system stable? Is it time-invariant? 1(b) 4 points Evaluate the integral 1 - 1 δ ( t + 2) e t + δ ( t - 1 2 ) sin( π t ) dt 1(c) 7 points Sketch the following three signals as a function of time, carefully labeling the important points. x ( t ) = (1 - t )( u ( t ) - u ( t - 1)) x 1 ( t ) = x ( t - 2) + x (2 - t ) x 2 ( t ) = x 1 (5 + 2 t ) 1(d) 5 points Evaluate the integral -∞ sin 2 5 t t 2 dt Problem 2 (10 points) Let x ( t ) = e - 2 t u ( t ). Find and sketch the output y ( t ) = ( x * h )( t ) when x ( t ) is passed through a filter with impulse response h ( t ) = x ( - t ). Problem 3 (10 points) Find the Fourier transform X ( j ω ) or X
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Unformatted text preview: j 2 πf ) of the signal x ( t ) = u ( t + 2)-2 u ( t ) + u ( t-1) Evaluate the magnitude | X ( jω ) | and the phase arg( X ( jω )) at ω = π . Problem 4 (25 points) A periodic waveform x ( t ) has period 4. We know that the values taken by x ( t ) over the interval 100 ≤ t ≤ 104 are as follows: x ( t ) = ´ 1 100 ≤ t < 103-1 103 ≤ t < 104 (a) 5 points Sketch x ( t ) over the interval-4 ≤ t ≤ 4, labeling both axes carefully. (b) 10 points Find the complex exponential Fourier series { a k } for x ( t ) and specify the value of the fundamental frequency ω in radians/sec. Simplify your expression for the Fourier series coe±cients as much as possible. (c) 10 points Find the output y ( t ) when x ( t ) is passed through a filter with impulse response h ( t ) = sinc( t 6 )....
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