# How to determine if the system is linear or nonlinear

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

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

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 $A$.

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