96下信號與系çµ&plusmn

96下信號與系統

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Signals and Systems, Midterm Exam Solutions Spring 2008, Edited by bypeng 1. [10] Consider a system H to be tested as being memoryless , causal , linear , time invariant , and invertible . Three signals 1 ( ) x t , 2 ( ) x t , and 3 ( ) x t are sent to the system, and the corresponding output signals 1 ( ) y t , 2 ( ) y t , and 3 ( ) y t are obtained as shown in Figure 1. Based on the three input-output pairs, is it possible to determine each of the five properties for system H ? If yes, what is it? If no, why? Justify your answer. Figure 1 Solution: i. NEITHER memoryless NOR causal. Observe that 1 3 ( ) ( ) x t x t = for 2 t < , but 1 3 ( ) ( ) y t y t for 0 2 t < < . ii. NOT linear. Observe that 3 1 2 ( ) ( ) ( ) x t x t x t = + , but 3 1 2 ( ) ( ) ( ) y t y t y t + . iii. NOT time invariant. Observe that 2 1 ( ) ( 1) x t x t = , but 2 1 ( ) ( 1) y t y t . iv. NOT invertible. Observe that 1 2 ( ) ( ) x t x t , but 1 2 ( ) ( ) y t y t = . 2. Consider a system as shown in Figure 2, where ( ) h t is the impulse response of the LTI sub-system in the block, and 2D is the operation of time delay for 2 units. H H H 1 2 3 4 5 1 2 3 0 4 x 1 ( t ) 1 2 3 4 5 1 2 3 0 4 x 2 ( t ) 1 2 3 4 5 1 2 3 0 4 x 3 ( t ) 1 2 3 4 5 1 2 3 0 4 y 1 ( t ) 1 2 3 4 5 1 2 3 0 4 y 2 ( t ) 1 2 3 4 5 1 2 3 0 4 y 3 ( t ) h ( t ) 2D + x ( t ) y ( t )
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