midterm_fall06

# midterm_fall06 - h ( t ) = I [0 , 1] ( t ). Problem 4 (20...

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UCSB Fall 2006 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 and your two sides of handwritten notes. Problem 1 (10 points) A system with input x ( t ) has output y ( t ) = w t -∞ τx ( τ ) 1(a) 3 points Is the system causal? 1(b) 3 points Is the system stable? 1(c) 4 points Find and sketch y ( t ) when the input x ( t ) = δ ( t - 1). Problem 2 (10 points) Sketch the following signals as a function of time, carefully labeling represen- tative points on the axes. x ( t ) = sin( πt ) I [0 , 1] ( t ) x 1 ( t ) = x (2 - t ) - x ( t ) x 2 ( t ) = x (1 - t 2 ) Problem 3 (10 points) Let x ( t ) = 3 I [ - 1 , 1] ( t ) - 2 I [0 , 2] ( t ). Find and sketch the output y ( t ) when x ( t ) is passed through an LTI system with impulse response
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Unformatted text preview: h ( t ) = I [0 , 1] ( t ). Problem 4 (20 points) Let x ( t ) denote a periodic function with fundamental period 6, specied as follows: x ( t ) = W 2 , &lt; t &lt; 2-1 , 2 &lt; t &lt; 6 4(a) 10 points Find the complex exponential Fourier series { a k } for x ( t ) and specify the value of the fundamental frequency in radians/sec. Simplify your expression for the Fourier series coecients as much as possible. 4(b) 5 points Set x 1 ( t ) = x (2 t + 4). What is the fundamental period T 1 of x 1 ( t )? Sketch x 1 ( t ) over the interval [-T 1 , T 1 ], carefully labeling the axes. 4(c) 5 points Find the complex exponential Fourier series for x 1 ( t ), specifying the fundamental fre-quency 1 ....
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## This note was uploaded on 08/06/2008 for the course ECE 130A taught by Professor Madhow during the Fall '07 term at UCSB.

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