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HW1_2005

# Digital Signal Processing: A Practical Approach (2nd Edition)

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Homework #Set 1 Signals, Systems, and Fourier Transforms 1. For each of the following systems, determine whether the system is (1) stable, (2) causal, (3) linear (4) time invariant, and (5) memoryless . (a) y [ n ]= x [2 n ]; (b) h [ n ] = min{ x [ n 1], x [ n ], x [ n -1]}; (c) y [n] = 2 y [ n 1] + 3 x [ n ] with y [ 1] = 0; (d) y [ n ] = nx [ n 2] + 3 x [ n + 2] + 5; (e) h [ n ] = 2 n u [ n 1]; (f) y [ n ] = 3 x [ n ] + 2x [ n 1] + n ; (g) h [ n ] = 3 ( n 2) δ [ n ]+ 2 δ [ n 1]; (h) y [ n ] = odd n even n n x = = if if 0 ] 2 / [ 2. The impulse responses of two linear time-invariant system are shown below. Determine and sketch the responses of these two systems when they use the same input as x [ n ] = δ [ n ] + 2 δ [ n 1] δ [ n 2] + 2 δ [ n 3]. 3 . Let x [ n ] and X ( e j ω ) represent a sequence and its Fourier transform, respectively. Determine, in terms of X (e j ω ) the Fourier transforms of y s [ n ], y d [ n ], y e [ n ]. In each case, sketch their Fourier transforms if X (e j ω ) is shown below. X (e j ω ) 1 -2 π - π - π/3 π/3 π 2 π ω (a) y s [ n ] = k n k n n x 2 2 , 0 ], [ = ; (b) y d [ n ] = x [2 n ]; (c) y e [ n ] = k n k n n x 2 2 , 0 ], 2 / [ = 4. If a complex sequence
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