# Initial rest condition applied on $x(t)$ vs $h(t)$

Define the LTI system $$\mathcal{H} : x\mapsto y$$

Define the convolution for continuous-time system : $$(x*h)(t)=\int_{-\infty}^{\infty}x(\tau)h(t-\tau)\;\text{d}\tau$$ The initial rest condition states that :

An LTI system with an input signal $$x(t)$$ is causal if and only if $$x(t)=0$$, $$\forall t<0$$

Now I have noticed that in some textbooks, the definition of convolution would have $$h(\tau)$$ instead of $$x(\tau)$$ and $$x(t-\tau)$$ instead of $$h(t-\tau)$$ (since $$x*h=h*x$$) but the initial rest condition now becomes :

An LTI system with an impulse response $$h(t)$$ is causal if and only if $$h(t)=0$$, $$\forall t<0$$

My question is :

Why does commutativity allow that for a LTI system to be causal then in one definition $$x(t)=0$$, $$\forall t<0$$ while in another definition $$h(t)=0$$, $$\forall t<0$$ knowing that $$x$$ and $$h$$ are two different terms?

It should be clear that a property of a system, such as causality, cannot be determined by looking at its input signals. For a linear time-invariant system, it is its impulse response $$h(t)$$ from which properties such as causality or stability can be determined.

Only the second definition in the question is correct: a causal LTI system has an impulse response $$h(t)$$ which equals zero for $$t<0$$. This implies that the system's output at time $$t_0$$ only depends on values of the input signal $$x(t)$$ for $$t\le t_0$$, and not on future values $$x(t)$$, $$t>t_0$$.

The output signal of a causal LTI system can be written as

$$y(t)=\int_{0}^{\infty}h(\tau)x(t-\tau)d\tau=\int_{-\infty}^tx(\tau)h(t-\tau)d\tau$$

An initial-rest condition just means that if the input $$x(t)$$ is zero for $$t then the output $$y(t)$$ must also be zero for $$t.

• So if we say something like "For any $t_{0}$ and any input $x(t)$ such that $x(t)=0$ for $t<t_{0},$ the corresponding output $y(t)$ must also be zero for $t<t_{0}"$ is only true if we know that the system is LTI causal? Feb 28 at 17:12
• Alright, got it thank you! Feb 28 at 17:15
• @xXACEXx: I correct my previous comment: the condition given in your comment is valid for causal linear systems. There are non-linear systems that satisfy this requirement but are not causal, and there are causal non-linear systems that do not satisfy this condition. Mar 1 at 14:23
• So if we prove that $h(t)=0$ when $t<0$, then we would be proving causality for LTI systems so then the statement in my previous comment would be valid. right? Because I am confusing between the necessary condition and the sufficient condition for the causality of LTI systems and how they are related. Mar 1 at 14:30
• @xXACEXx: Yes, sure, as written. Mar 6 at 18:18