# Why the impulse response needs to be right-sided for a system to be causal?

I have been revisiting my school notes about the definition of Causality for signal and systems. I already understand that for a system to be causal, the output at any time can only depend on past and present inputs. Wikipedia says

Suppose $$h(t)$$ is the impulse response of any system $$H$$ described by a linear constant coefficient differential equation. The system $$H$$ is causal if and only if $$\begin{equation*} h(t) = 0, \forall t < 0 \end{equation*}$$

https://en.wikipedia.org/wiki/Causal_system

I could not understand why this is true. For example, for a system whose impulse response is $$h(t) = 1, t \in R$$, the output at for example $$h(-1) = 1$$ clearly doesn't depend on future inputs, so it should be a causal system. Does this mean the definition on Wikipedia is wrong?

• Check the definition of "impulse response". Feb 27, 2021 at 4:01
• Please elaborate. I can't seem to find anything more specific than 'the output of a system when the input is the dirac delta function'. Feb 27, 2021 at 4:15
• Differential equations also come with boundary conditions. Implicit in this definition of a Causal system is the boundary condition $h(-\infty) = 0$. I.e., there is no pre-existing source of value for $h$. It really should have been made explicit, so in that sense, this definition in Wikipedia is not perfect. Feb 27, 2021 at 16:09
• Thank you for your answer! What about a system whose impulse response $g(t)$ is a shifted unit step $g(t) = u(t + 1)$. Here $g(t) \vert _{t=-\infty} = 0$ and it doesn't depend on future inputs, so it should be causal. It still doesn't match the Wikipedia definition Feb 27, 2021 at 19:27

I think I found out. The definition on Wikipedia is derived from the response of Linear Time Invariant systems whose output is the result of a convolution. The reason for a causal system $$h(t) = 0$$ $$\forall t < 0$$ must hold true is:
For linear time invariant system $$H$$ whose impulse response is represented by $$h(t), t \in \mathbb{R}$$, the output of the system $$y(t)$$ in response to an input signal $$x(t)$$ at time $$t$$ is the result of convolution $$$$y(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau$$$$
For the system to be causal, it must not depend on future inputs, which means at time $$t$$, $$x(\tau)$$ in the above convolution must not be de-referenced for all $$\tau > t$$. This can only hold true when $$h(t-\tau) = 0$$ $$\forall \tau > t$$. Hence, \begin{align} h(t) = 0 && \forall t < 0 \end{align}