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Hi guys i'm studying signals and systems, and my professor ask us if this signal is linear or not

$$y(t) = \big[x(t − 1)x(t + 1)\big]^{\frac 12}$$

the fact that is in the form of $x\cdot x$ told me that that's not linear but the square root confuses me can you explain to me if this is linear or not?

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    $\begingroup$ Have you applied the Standard Linearity Test ? $\endgroup$ – Fat32 Dec 9 '20 at 22:09
  • $\begingroup$ yes but i can't undestand it $\endgroup$ – Giovanni Cerciello Dec 9 '20 at 22:11
  • $\begingroup$ which square root? the + one or the - one? $\endgroup$ – robert bristow-johnson Dec 9 '20 at 22:12
  • $\begingroup$ idk the exercise is written exactly this way $\endgroup$ – Giovanni Cerciello Dec 9 '20 at 22:14
  • $\begingroup$ write down what you have applied, and lets which part you don't understand. $\endgroup$ – Fat32 Dec 9 '20 at 22:15
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Let $x_1(t) = t^2$ and $x_2(t) = 1$. We have $$y_1(t) =\sqrt{(t-1)^2(t+1)^2}$$ $$y_2(t) = 1$$ Now let $x_3(t) = x_1(t) + x_2(t) = t^2 + 1$. Then we have $$y_3(t) = \sqrt{((t-1)^2 +1)((t+1)^2 +1)}$$ Obviously $$y_3(t) \not = y_1(t) + y_2(t)=\sqrt{(t-1)^2(t+1)^2} +1$$Therefore the system isn't linear.

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