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I want to understand the meaning of unit impulse response when time flows. An example is given below:

enter image description here

In the example the left signal depicts a $\delta[n]$ function value of which is $1$ only at $n=0$. The LHS signal is response of the system to the unit impulse.

  • I want to understand how to interpret this when time flows, i.e. at $n=-4,\ n=-3$, what is response of the system?
  • Or at $n=0,\ n=5$, etc. what is response of the system?
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In the figure that you have posted, the right plot shows the impulse response $h[n]$ according to the convention that only its nonzero samples are displayed. And therefore, it's understood that all the remaning samples of $h[n]$ are zero unless otherwise stated explicitly.

This means that $h[0]=0$, $h[1]=0$,...,$h[12]=0$. Which can be stated in the following:

$$ h[n] = \begin{cases} 0 &,& \text{ for } n \le 4 \\ \alpha_n &,& \text{ for } 5 \le n \le 11 \\ 0 &,& \text{ for } n \ge 12 \\ \end{cases} $$

In this setting the domain of support for the finite length sequence $h[n]$ is stated as $n \in [5,11]$ , which means that $h[n]$ is certainly zero for $n$ out of this interval, while it can have nonzero as well as zero values inside this interval.

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