Consider a system $y(t) = \dot{x}(t)$ where $y$ is the output and $x$ is the input. Given an initial condition $x_0$ and two inputs $x_1$ and $x_2$ such that $$x_1(t)=x_2(t) , 0 \le t < t_0$$ the outputs $y_1,y_2$ corresponding to $x_1,x_2$ respectively should satisfy $$y_1(t)=y_2(t) , 0 \le t < t_0$$ since $$x_1(t)=x_2(t) , 0 \le t < t_0 \implies \dot{x}_1(t)=\dot{x}_2(t) , 0 < t < t_0$$ How is this system non causal then?

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    $\begingroup$ What is the question? $\endgroup$ – Cherny Oct 29 '18 at 7:15
  • $\begingroup$ Added the question. $\endgroup$ – sixtyTonneAngel Oct 29 '18 at 11:00
  • $\begingroup$ (1) Why do you think the system is non-causal? (2) How is the analysis you did related to causality? (double check the definition of causal system) $\endgroup$ – MBaz Oct 29 '18 at 13:49

I am not sure why what you wrote is relevant for the causality. Anyways, the derivative operator is depends on the future signal. Actually this almost causal, in the sense that you need an infinitesimal environment around t to define the derivative.

Just for example, we can take 2 cases: $$ x_1(t) = -t , x_2=|t| \Rightarrow \dot x_1(t)=-1, \dot x_2(t) =undefined$$ $$\Rightarrow y_1(t)\neq y_2(t)$$ despite $\forall t\leq 0 : x_1(t)=x_2(t)$

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