I know is that for a system to be invertible it should be mapping one-one, but what happens in this signal?, I feel like I need to know the value of $x[n]$ before.
$y[n]= \mathrm{median}\{x[n-1],x[n],x[n+1]\}$
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Consider a signal $x$ such that $x(n) = x_0$ for almost every $n$, but for every 10$^{th}$ sample $x(n) = 100 x_0$. Then a median filter will return $y(n) = x_0$ always.
You can change the value of that "every tenth sample" in any way you want: let it equal $x_0 - 100$, $\pi$, $42$, whatever, and the result is the same.
Because the results are all the same when you have different inputs, you cannot reconstruct the input from the output -- this is the essence of the filter being non-invertible.
answered Dec 19, 2021 at 20:37
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$\begingroup$ rackin' up that rep, Tim. You'll be whizzing past me in no time, flat. $\endgroup$ Commented Dec 20, 2021 at 0:46
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$\begingroup$ Regardless of what StackExchange may think, I don't consider it a contest. I just answer interesting questions as best as I can. $\endgroup$ Commented Dec 20, 2021 at 4:10
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1$\begingroup$ I'm kind of amused by the fact that I spend a lot more work hours doing DSP than circuit design, yet I have way more points over on electronics.stackexchange.com than I do here. $\endgroup$ Commented Dec 20, 2021 at 4:11
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2$\begingroup$ I am afraid that if an input value is 42, the filter should spit out this value, always $\endgroup$ Commented Dec 20, 2021 at 7:10
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1$\begingroup$ i don't consider this to be a contest either. but i notice when someone i know from the comp.dsp days shows up and immediately is useful. i sorta peetered out on both forums. $\endgroup$ Commented Dec 21, 2021 at 4:09