20102ee113_1_HW 1 Solution Spring 2010

20102ee113_1_HW 1 Solution Spring 2010 - EE113: Digital...

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EE113: Digital Signal Processing Spring 2010 Prof. Mihaela van der Schaar Homework #1 Solutions Prepared by Nick Mastronarde, Yu Zhang, and Shaolei Ren 2.5. (a) The sequence x ( n ) can be written as x ( n ) = 1 3 δ ( n - 1) + " ± 1 3 n + ± 1 2 n - 1 # u ( n - 2) Then the energy of x ( n ) is given by: ² x = X n = -∞ | x ( n ) | 2 (1) = 1 9 + X n =2 ± 1 3 n + ± 1 2 n - 1 2 (2) = 1 9 + X n =2 " ± 1 3 2 n + 4 ± 1 6 n + 4 ± 1 2 2 n # (3) = 1 9 + X n =2 •± 1 9 n + 4 ± 1 6 n + 4 ± 1 4 n (4) = 1 9 + X n =0 1 81 ± 1 9 n + 1 9 ± 1 6 n + 1 4 ± 1 4 n (5) = 1 9 + 1 81 1 1 - 1 9 + 1 9 1 1 - 1 6 + 1 4 1 1 - 1 4 = 71 120 (6) (b) By inspecting the sequence x ( n ), we find that x ( n ) = 0 for n 0. Then y ( n ) = 0 when x ( - 2 n - 3) = 0, which is equivalent to - 2 n - 3 0 n ≥ - 1 where n must be an integer. Then y ( n ) = 0 for n ≥ - 1. 2.8. Procedure (b) is the correct one. By following (a), we will actually get y (2 n ) = x (2 n - 1). By following (c) we actually get w ( n - 2) = x (2( n - 2)) = x (2 n - 4). However, if we do the time-shift and plot w ( n - 1), we get w ( n - 1) = x (2( n - 1)) = x (2 n - 2). 2.10. Answer true or false. (a) False. u ( n ) is not an energy sequence. However, it is a power sequence.
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(b) True. An energy sequence
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20102ee113_1_HW 1 Solution Spring 2010 - EE113: Digital...

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