EE 3054: Signals, Systems, and Transforms Fall 2011 Test 1: Discrete-time Signals and Systems • No notes, closed book • Show your work • Simplify your answers 1. Accurately sketch of each of the following discrete-time signals. (a) x ( n ) = ∞ X k = -∞ ( - 1) k δ ( n - 5 k ) (b) x ( n ) = cos π 3 n u ( n ) (c) x ( n ) = u ( n ) + u ( n - 4) - 2 u ( n - 8) 2. A discrete-time system is described by the following rule y ( n ) = ( - 1) n x ( n ) + 2 x ( n - 1) where x is the input signal, and y the output signal. Accurately sketch the output signal, y ( n ), produced by the input signal x ( n ) illustrated below. 1 2 3 1 -2 -1 0 1 2 3 4 5 6 n x ( n ) 3. Classify the system in Problem 2 as: (a) causal/non-causal (b) linear/nonlinear (c) time-invariant/time-varying 4. For the discrete-time signal x ( n ) illustrated in Problem 2 . . . (a) What is the Z -transform, X ( z ), of the signal, x ( n )? (b) Sketch the signal, v ( n ), defined by V ( z ) = z - 3 X ( z ). (c) Sketch the signal, y ( n ), defined by Y ( z ) = X ( - z ). (d) Sketch the signal, g ( n ), defined by G ( z ) = X (1 /z ). 5. Accurately sketch the convolution of the discrete-time signal x ( n ) in Problem 2 with each of the following signals. (a) f ( n ) = δ ( n ) (b) f ( n ) = u ( n ) (c) f ( n ) = 1 (d) f ( n ) = 2 δ ( n - 2) 6. Sketch the convolution of the discrete-time signal x ( n ) in Problem 2 with itself. That is, find f ( n ) = x ( n ) * x ( n ). 1
7. A causal discrete-time LTI system is implemented using the difference equa- tion y ( n ) = x ( n ) - √ 2 x ( n - 1) + x ( n - 2) - 0 . 5 y ( n - 2) where x is the input signal, and y the output signal.
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