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Definition

A time invariant system is a system whose response to the shifted input $ x(t-t_0) $ is the shifted output $ y(t-t_0) $ for any input x(t) and any time $ t_0 \varepsilon \real $


Time Invariant Example

First take the equation: $ y(t)=17x(t) $

Consider an input: $ y_1(t)=17x_1(t) $

Now, shift that input in time: $ x_2 (t)= x_1(t-t_0) $

$ y_2(t)=17x_2(t)= 17x_1(t-t_0) $

$ y_1(t-t_0)=17x_1(t-t_0) $

Note that $ y_2(t) = y_1(t-t_0) $, this means that the system is time invariant


Time Variant Example

First take the equation: $ y(t)=17tx(t) $

Consider an input: $ y_1(t)=17tx_1(t) $

Now, shift that input in time: $ x_2 (t)= x_1(t-t_0) $

$ y_2(t)=17tx_2(t)= 17tx_1(t-t_0) $

$ y_1(t-t_0)=17(t-t_0)x_1(t-t_0) $

Note that $ y_2(t) \ne y_1(t-t_0) $, this means that the system is time variant

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva