Difference between revisions of "HW2-D Seung Seob Lee ECE301Fall2008mboutin" - Rhea

Latest revision as of 16:22, 12 September 2008

TIME INVARIANCE

Time invariance, in my definition, is such a system that does not stretch or shrink the input function and does not change time shift of input is called "time invariance."

Example of Time invariant system and its proof

$ \,y(t)=e^{x(t)}\, $

Proof:

$ x(t) \to System \to y(t)=e^{x(t)} \to Time Shift(t0) \to z(t)=y(t-t0) $

$ \, =e^{x(t-t0)}\, $

$ x(t) \to Time Shift(t0) \to y(t)=x(t-t0) \to System \to z(t)=e^{y(t)} $

$ \, =e^{x(t-t0)}\, $

Both cascades yielded the same outputs, thus $ \,y(t)=e^{x(t)}\, $ is time invariant.

Example of Time variant system and its proof

$ \,y(t)=x(2t)\, $

Proof:

$ x(t) \to System \to y(t)=x(2t) \to Time Shift(t0) \to z(t)=y(t-t0) $

$ \, =x(2t-2t0)\, $

$ x(t) \to Time Shift(t0) \to y(t)=x(t-t0) \to System \to z(t)=y(2t) $

$ \, =x(2t-t0)\, $

They yielded different outputs, thus $ \,y(t)=x(2t)\, $ is time-variant.