# Linear systems: Square root of input product

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?

• Have you applied the Standard Linearity Test ? – Fat32 Dec 9 '20 at 22:09
• yes but i can't undestand it – Giovanni Cerciello Dec 9 '20 at 22:11
• which square root? the + one or the - one? – robert bristow-johnson Dec 9 '20 at 22:12
• idk the exercise is written exactly this way – Giovanni Cerciello Dec 9 '20 at 22:14
• write down what you have applied, and lets which part you don't understand. – Fat32 Dec 9 '20 at 22:15

## 1 Answer

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.