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Given a signal

$$y(n)=x(n)\cos(wn)$$

How to determine if the given signal is linear or non linear?

I am getting so confused reading the theory. If someone could help step by step to determine this. It would be great thankyou

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  • $\begingroup$ If you double $x(n)$ (or multiply by any other factor), will $y(n)$ scale accordingly? $\endgroup$ – user32113 Nov 21 '17 at 10:00
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    $\begingroup$ @user32113: This is the check for homogeneity. In general one must also check for additivity to make sure that the system is linear. $\endgroup$ – Matt L. Nov 21 '17 at 12:20
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As Matt L. says you'll need to check for homogeneity and, possibly, additivity.

Homogeneity

That test says that if: $$ y[n] = f(x[n]) $$ then $$ A \cdot y[n] = f(A \cdot x[n]) $$ for all scalar $A$.

Additivity

This test says that if $$ y_1[n] = f(x_1[n]) $$ and $$ y_2[n] = f(x_2[n]) $$ then $$ y_{\rm tot}[n] = f(x_1[n] + x_2[n]) = y_1[n] + y_2[n] $$

You just need to apply these for your system.


I say "possibly" because if the system fails one of these tests, then it will not be linear --- so there is little point testing the second condition.

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