test1sol - ECEN 314 Test#1 Georghiades Given Instructions...

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ECEN 314 Test #1 Georghiades Given: October 20, 2009. Instructions : Exam is open-book, open notes. No calculators are allowed. Please give your solution in the allocated space. If more space is needed, use the back of the page. All problems carry equal weight. 1. A system, S , is defined by the following equation relating the input x ( t ) to the output y ( t ): y ( t ) = S [ x ( t )] = sin(2 t ) x ( t - 1) . Is the system: (a) Causal? (b) Memoryless? (c) Invertible? (d) Stable? (e) Linear? (f) Time-invariant? Explain your answers briefly (a yes or no answer will not suffice). Solution (a) The system is causal because the output depends only on past inputs; the output at time t depends only on the past input at time t - 1. (b) The system is not memoryless because the output at time t is a function of the input at another time besides t . (c) The system is not invertible since at times for m and integer the output is 0 and there is no way to determine the input from it. (d) Assume | x ( t ) | ≤ A < . Then | y ( t ) | = | sin(2 t ) x ( t - 1) | = | sin(2 t ) || x ( t - 1) | ≤ A | sin(2 t ) | ≤ A < . Thus, the system is stable. (e) Let y 1 ( t ) = S [ x 1 ( t )] = sin(2 t ) x 1 ( t - 1) and y 2 ( t ) = S [ x 2 ( t )] = sin(2 t ) x 2 ( t - 1). Now let the input be c 1 x 1 ( t ) + c 2 x 2 ( t ). Then the output is sin(2 t )[ c 1 x 1 ( t - 1)+ c 2 x 2 ( t - 1)] = c 1 sin(2 t ) x 1 ( t - 1)+ c 2 sin(2 t ) x 2 ( t - 1) = c 1 y 1 ( t )+ c 2
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