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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?

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  • $\begingroup$ Check the definition of "impulse response". $\endgroup$ Feb 27, 2021 at 4:01
  • $\begingroup$ 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'. $\endgroup$
    – John Doe
    Feb 27, 2021 at 4:15
  • $\begingroup$ 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. $\endgroup$ Feb 27, 2021 at 16:09
  • $\begingroup$ 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 $\endgroup$
    – John Doe
    Feb 27, 2021 at 19:27

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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 \begin{equation} y(t) = \int_{-\infty}^{\infty} x(\tau) h(t-\tau) d\tau \end{equation}

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}

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