MIT16_30F10_rec03

MIT16_30F10_rec03 - 16.30/31, Fall 2010 Recitation # 3 1 On...

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Unformatted text preview: 16.30/31, Fall 2010 Recitation # 3 1 On Linearity and Time Invariance Consider the systems described by the following input/output relationships: y ( t ) = u ( t ) + 1 , (1) and y ( t ) = 2 t 3 u ( t ) . (2) Are these systems linear or nonlinear? Are they time-invariant or time-varying? Linearity A system is linear if any linear combination of two arbitrary inputs results in an output that is the linear combination of the outputs corresponding to the original inputs. Let us consider first system ( 1 ). We have that: u 1 ( t ) = 1 y 1 ( t ) = 2 , and u ( t ) = 0 y ( t ) = 1 . Clearly, u ( t ) = 0 u 1 ( t ), but y ( t ) = 0 y 1 ( t ). This counterexample shows that the system is not linear. Now consider system ( 2 ). Considering two generic inputs u 1 and u 2 , we get y 1 ( t ) = 2 t 3 u 1 ( t ) , y 2 ( t ) = 2 t 3 u 2 ( t ) . Now consider the input u 3 ( t ) = u 1 ( t )+ u 2 ( t ). The corresponding output can be computed as y 3 ( t ) = 2 t 3 ( u 1 ( t ) + u 2 ( t )) = 2 t 3 u 1 ( t ) + 2 t 3 u 2 ( t ) , that is, y 3 ( t ) = y 1 ( t ) + y 2 ( t ). Hence, the...
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MIT16_30F10_rec03 - 16.30/31, Fall 2010 Recitation # 3 1 On...

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