probset-one - EE533 Problem Set 1 1 Consider x(n = cos(0 n...

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EE533 Problem Set 1 1. Consider x ( n ) = cos( ω 0 n ) for various values of ω 0 in the set { 0 , 0 . 2 π, 0 . 6 π, 0 . 9 π, π, 1 . 1 π, 31 . 2 π, 2 } . What is the fundamental period of x ( · ) in each case? 2. Consider the following systems whose input-output relationships are given below. Deter- mine which of the properties (a) linearity (b) time-invariance (c) causality (d) stability hold for each system. In each case, x ( · ) is the input sequence and y ( · ) denotes the system output. (a) y ( n ) = x ( n ) n 1 0 n = 0 x ( n + 1) n ≤ - 1 (b) y ( n ) = x ( n 2 ) (c) y ( n ) = ( x ( n ) ) 5 (d) y ( n ) = k = n - 1 x ( k ) (e) y ( n ) = x 2 ( n ) - x ( n - 1) x ( n + 1) 3. Determine the output of an LTI system if the impulse response h ( · ) and the input x ( · ) are as follows: (a) x ( n ) = u ( n ) and h ( n ) = a n u ( - n - 1), with a > 1. (b) h ( n ) = 2 n u ( - n - 1) and x ( n ) = u ( n ) - u ( n - 10). Use linearity and time-invariance to minimize the work in part(b). 4. Consider the following three sequences: x 1 ( n ) = A (a constant); x 2 ( n ) = u ( n ); x 3 ( · ) = { (1) , - 1 } . Determine (a) x 2 ( x 3 x 1 ), and (b) ( x 2 x 3 ) x 1 . Are the two results the same?
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