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# HW4 - Question 6[Basic Consider a linear time-invariant...

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ECE 301, Homework #4, due date: 9/22/2010 http://cobweb.ecn.purdue.edu/ chihw/10ECE301F/10ECE301F.html Question 1: [Basic] Sketch the following discrete-time signals from n = - 2 to n = 3. x [ n ] = U [ n - 4] - 2 U [ n ] + U [ n + 1] x [ n ] = ( n + 1) U [ n ] - 2 U [ n - 3] - ( n - 1) U [ n + 1] x [ n ] = δ [ n ] - δ [ n - 2] + U [ n - 2] - U [ n + 1] Question 2: [Basic] For any discrete time signals, x [ n ] can be decomposed as an infinite sum of δ [ n - k ] as follows: x [ n ] = X k = -∞ α k δ [ n - k ] . Find the coefficients α k . [Advanced] x [ n ] can be decomposed as an infinite sum of U [ n - k ] as follows: x [ n ] = X k = -∞ β k U [ n - k ] . Find the coefficients β k . Hint: Use the equality: δ [ n - k ] = U [ n - k ] - U [ n - k - 1]. [Basic] For any continuous time signals, x ( t ) can be decomposed as an integral of δ ( t - s ) as follows: x ( t ) = Z s = -∞ α s δ ( t - s ) ds. Find the coefficients α s . Question 3: [Basic] Problem 1.27 (a,b,c) Question 4: [Basic] p. 61, Problem 1.28 (a,b,c).

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Unformatted text preview: Question 6: [Basic] Consider a linear time-invariant system. Suppose we know that when the input is x [ n ] = δ [ n ], the output y [ n ] = 2 n e-jn U [ n-1]. Solve the following questions in order. 1. If the input is x [ n ] = δ [ n-1], what is the output y [ n ]? (Hint: Use the time-invariance property.) 2. If the the input is x [ n ] = U [ n ]-U [ n-2], what is the output y [ n ]? (Hint: First plot the signal, and see what x [ n ] looks like. Then use the linearity of the system and the result of the previous sub-question.) Question 7: [Advanced] p. 62, Problem 1.30(a,b,c,e). Question 8: [Advanced] p. 62, Problem 1.30(f,g,j,n). Question 9: [Basic] p. 62, Problem 1.31....
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