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Unformatted text preview: Review Quiz Solution 1. Consider the system whose input-output relationship is given by y [ n ] = e-| x [ n ] | . (a) Linear: No. Let x 1 [ n ] = 2 x [ n ] . Then y 1 [ n ] = e-| x 1 [ n ] | = e-| 2 x [ n ] | = ( y [ n ]) 2 6 = 2 y [ n ] . (b) Causal: Yes. Output depends on present value of input and hence it is causal. (c) Time-invariant: Yes. Let x 1 [ n ] = x [ n- n ] . Then y 1 [ n ] = e-| x 1 [ n ] | = e-| x [ n- n ] | = y [ n- n ] (d) Stable: Yes: For | x [ n ] | < M, we have | y [ n ] | ≤ 1 . The output is always bounded for a bounded input. 2. For Parseval’s theorem, see text, section 4.2.5, pages 254-255 3. (a) The system has a pole at z = 1 2 and z = 4 5 . Since the system is causal, the region of convergence must be exterior to a circle and cannot include any pole, so the ROC is | z | > 4 5 . Since the region of convergence includes the unit circle ( | z | = 1), the system is stable....
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- Spring '09
- 3 sec, 4 sec, Partial fractions in complex analysis, π π, 2 3 4 5 k, 5 3 1 2 k