102_1_midterm1

# 102_1_midterm1 - Systems and Signals EE102 Lee Fall 2011-12...

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Systems and Signals Lee, Fall 2011-12 EE102 Midterm Exam 1 NAME: You have 2 hours for 4 questions. Show enough (neat) work in the clear spaces on this exam to convince us that you derived, not guessed, your answers. Put your ﬁnal answers in the boxes at the bottom of the page. Closed notes, closed book, 1 letter sized sheet allowed. Problem Score 1 2 3 4 Total 1

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Problem 1. (25 Points) A LINEAR system is shown below. Suppose there are three test signals ( x 1 [ n ], x 2 [ n ], x 3 [ n ]), for which the system outputs are y 1 [ n ], y 2 [ n ], y 3 [ n ], respectively. 0 1 -1 2 1 2 0 1 -1 2 1 -1 0 1 -1 2 2 1 0 1 2 3 1 0 1 2 3 1 0 1 2 3 1 input output x 1 [n] x 2 [n] x 3 [n] y 1 [n] y 2 [n] y 3 [n] 2
1. If the output of the system ( y 4 [ n ]) when x 4 [ n ] is the input as shown below, ﬁnd the input signal x 4 [ n ] as a linear combination of x 1 [ n ], x 2 [ n ], and x 3 [ n ]. Also, plot x 4 [ n ]. 0 1 -1 2 3 1 -1 y 4 [n] Solution: y 4 [ n ] = - y 1 [ n ] - y 2 [ n ] + y 3 [ n ]. Therefore, x 4 [ n ] = - x 1 [ n ] - x 2 [ n ] + x 3 [ n ] . 0 1 -1 2 -5 -2 -2 3

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2. What are the outputs of the system when the inputs are δ [ n - 1] and δ [ n - 2], respec- tively? Also, plot the two outputs seperately. Solution: δ [ n - 1] = x 2 [ n ] - x 1 [ n ] - x 3 [ n ] therefore, the output is y 2 [ n ] - y 1 [ n ] - y 3 [ n ] 0 1 2 3 -1 -1 δ [ n - 2] = x 2 [ n ] - x 1 [ n ] therefore, the output is y 2 [ n ] - y 1 [ n ] 0 1 2 3 1 4
3. Based on what you have found from the last question, do you think this system is LTI? Why? Please explain your reason in detail. Solution: NO. Because when the input signal δ [ n - 1] gets delayed by 1 to δ [ n - 2], the output is not the output of input δ [ n - 1] delayed by 1. 5

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Problem 2. (25 Points) Determine the Linearity and Time Invariance of the following systems with the given input x ( t ) and output y ( t ) relationship. 1.
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102_1_midterm1 - Systems and Signals EE102 Lee Fall 2011-12...

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