# Time-invariant system

A time-invariant (TIV) system is one whose output does not depend explicitly on time.

If the input signal $x(t)$ produces an output $y(t)$ then any time shifted input, $x(t + \delta)$, results in a time-shifted output $y(t + \delta)$

This property can be satisfied if the transfer function of the system is not a function of time except expressed by the input and output. This property can also be stated in another way in terms of a schematic

If a system is time-invariant then the system block is commutative with an arbitrary delay.

## Simple example

To demonstrate how to determine if a system is time-invariant then consider the two systems:

• System A: $y(t) = t\, x(t)$
• System B: $\,\!b(t) = 10 x(t)$

Since system A explicitly depends on t outside of $x(t)$ and $y(t)$, it is not time-invariant. System B, however, does not depend explicitly on t so it is time-invariant.

## Formal example

A more formal proof of why system A & B from above differ is now presented. To perform this proof, the second definition will be used.

System A:

Start with a delay of the input $x_d(t) = \,\!x(t + \delta)$
$y(t) = t\, x(t)$
$y_1(t) = t\, x_d(t) = t\, x(t + \delta)$
Now delay the output by $\delta$
$y(t) = t\, x(t)$
$y_2(t) = \,\!y(t + \delta) = (t + \delta) x(t + \delta)$
Clearly $y_1(t) \,\!\ne y_2(t)$, therefore the system is not time-invariant.

System B:

Start with a delay of the input $x_d(t) = \,\!x(t + \delta)$
$y(t) = 10 \, x(t)$
$y_1(t) = 10 \,x_d(t) = 10 \,x(t + \delta)$
Now delay the output by $\,\!\delta$
$y(t) = 10 \,x(t)$
$y_2(t) = y(t + \delta) = 10 \,x(t + \delta)$
Clearly $y_1(t) = \,\!y_2(t)$, therefore the system is time-invariant. Although there are many other proofs, this is the easiest.

## Abstract example

We can denote the shift operator by $\mathbb{T}_r$ where $r$ is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system

$x(t+1) = \,\!\delta(t+1) * x(t)$

can be represented in this abstract notation by

$\tilde{x}_1 = \mathbb{T}_1 \, \tilde{x}$

where $\tilde{x}$ is a function given by

$\tilde{x} = x(t) \, \forall \, t \in \mathbb{R}$

with the system yielding the shifted output

$\tilde{x}_1 = x(t + 1) \, \forall \, t \in \mathbb{R}$

So $\mathbb{T}_1$ is an operator that advances the input vector by 1.

Suppose we represent a system by an operator $\mathbb{H}$. This system is time-invariant if it commutes with the shift operator, i.e.,

$\mathbb{T}_r \, \mathbb{H} = \mathbb{H} \, \mathbb{T}_r \,\, \forall \, r$

If our system equation is given by

$\tilde{y} = \mathbb{H} \, \tilde{x}$

then it is time-invariant if we can apply the system operator $\mathbb{H}$ on $\tilde{x}$ followed by the shift operator $\mathbb{T}_r$, or we can apply the shift operator $\mathbb{T}_r$ followed by the system operator $\mathbb{H}$, with the two computations yielding equivalent results.

Applying the system operator first gives

$\mathbb{T}_r \, \mathbb{H} \, \tilde{x} = \mathbb{T}_r \, \tilde{y} = \tilde{y}_r$

Applying the shift operator first gives

$\mathbb{H} \, \mathbb{T}_r \, \tilde{x} = \mathbb{H} \, \tilde{x}_r$

If the system is time-invariant, then

$\mathbb{H} \, \tilde{x}_r = \tilde{y}_r$