Proof of time-invariance of continuous-time system

1. How do you prove a time invariance?
2. How do you know if a discrete time system is time invariant?
3. How do you prove shift invariant?

How do you prove a time invariance?

A system is time-invariant if its output signal does not depend on the absolute time. In other words, if for some input signal x(t) the output signal is y1(t)=Trx(t), then a time-shift of the input signal creates a time-shift on the output signal, i.e. y2(t)=Trx(t−t0)=y1(t−t0).

How do you know if a discrete time system is time invariant?

◦ The system is time invariant. This means that if the input signal x(t) generates the output signal y(t), then, for each real number s, the time shifted input signal ˜x(t) = x(t − s) generates the time shifted output signal ˜y(t) = y(t − s).

How do you prove shift invariant?

If g(x + x0) = H[f(x + x0)] then the system is shift invariant, otherwise it is not.