I have $x(n)$ = {$4,-1,-3,1$} and $h(n)$ = {$2,1,3,5$}.
I would like to know how can I check whether
$h(n)$ is stable filter or not and $h(n)$ is causal filter or not?
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1$\begingroup$ hey! Your notation isn't quite right, and that makes it impossible for you to answer this yourself :) $h(n)$ is a single value, the value of the signal (or impulse response) $h$ at time $n$. Now, $\{2,1,3,5\}$ is a set of values, and that's not what you wanted. So, first step, figure out for which $n$ the value of $h$ becomes $2$, for which it becomes $1$ and so on. Then check the definition of "causal"! $\endgroup$– Marcus MüllerApr 19, 2020 at 9:12
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$\begingroup$ let say h(n = 0) = 2, h(n = 1) = 1, h(n = 2) = 3, h(n = 3) = 5. $\endgroup$– user1761275Apr 19, 2020 at 9:17
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2$\begingroup$ if you say so, congratulations, you've just answered your own causality question. $\endgroup$– Marcus MüllerApr 19, 2020 at 9:23
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Firstly, you do not need to specify an input $x(n)$ to know whether the system is stable or causal, these are properties of the system itslef, I will interpret $h(n)$ you mention as a array describing the system response.
For causality in time domain perspectivefor an LTI system check if $h(n) =0 $ for all $n < 0$, that is the $h(n)$ is essentially a right sided sequence.
For causality in Z domain perspective equivalently, the Region of Convergence is the right side of the outermost pole (outermost in terms of the highest magnitude pole in z domain)
For stability in time domain perspective, the system response $h(n)$ should be absolutely summable, since you have only finite coefficients, this system is stable, any FIR filter is always stable.
For stability from Z domain perspective, the region of Convergence must contain the unit circle.
For stable and causal system all these conditions hold at the same time.
