Chapitre 5

Time-invariant (or stationary) systems reproduce their behavior identically over time.

We say that the system is stationary if and only if the output is \(y\left(t-\tau\right)\) when the input, denoted as \(x\left(t-\tau\right)\), is a shifted version of the input \(x\left(t\right)\).

For example,

\(y(t)=ax\left(t\right)\)

Consider the case of an ideal amplifier, where the output signal \(y(t)=ax\left(t\right)\) (with a > 1) is associated with the input signal \(x\left(t\right)\).

In this scenario, the system is clearly time-invariant (stationary).

For example (Counter-example),

Consider the system described by :

\(y(t)=x\left(t\right)sin(t)\)

When the input is be \(x\left(t\right)\) , the output is \(\ y(t)=x\left(t\right)sin(t)\) . If the input is shifted \(x\left(t-\tau\right)\) the output becomes \(x\left(t-\tau\right)\sin(t)\), which is not equal to :

\({y(t)|}_{t-t}=x\left(t-t\right)sin(t-\tau)\)

Therefore, this system is not stationary.