{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MIT16_30F10_rec03

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

This preview shows pages 1–2. Sign up to view the full content.

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 0 ( t ) = 0 y 0 ( t ) = 1 . Clearly, u 0 ( t ) = 0 u 1 ( t ), but y 0 ( 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 ) = αu 1 ( t )+ βu 2 ( t ). The corresponding output can be computed as y 3 ( t ) = 2 t 3 ( αu 1 ( t ) + βu 2 ( t )) = α 2 t 3 u 1 ( t ) + β 2 t 3 u 2 ( t ) , · · that is, y 3 ( t ) = αy 1 ( t ) + βy 2 ( t ). Hence, the system is linear. Time invariance A system is time-invariant if it commutes with a time delay, i.e., if the output of the system does not depend on the “origin” of the

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online