Part D. Time Invariance
A system is called time invariant if a time shift has no affect on the shape of the output. For example, a time shift in the input produces an output with that same time shift, but that has the same shape. Suppose x(t)=y(t), then for a time shift, x(t-t0)=y(t-t0).
Example of a Time Invariant System
According to one of the definitions Professor Boutin gave in class, if the cascade of the input followed by the time delay followed by the system and the cascade of the input followed by the system followed by the time delay both yield the same result, then the system can be called time invariant. We can use this definition to prove whether a system is time invariant or not.
Suppose there is a system y(t)=exp(j*pi*x(t)).
x(t) -> time delay -> x(t-t0) -> system -> exp(j*pi*x(t-t0))
x(t) -> system -> exp(j*pi*x(t)) -> time delay -> exp(j*pi*x(t-t0))
Since both cascades produce the same output signal, this is an example of a time invariant system.
Example of a Non-Time Invariant System
Suppose there is a system y(t)=exp(j*pi*t*x(t)).
x(t) -> time delay -> x(t-t0) -> system -> exp(j*pi*t*x(t-t0))
x(t) -> system -> exp(j*pi*t*x(t)) -> time delay -> exp(j*pi*(t-t0)*x(t-t0))
Since both cascades produce different outputs, this is an example of a NON-time invariant system.
