# Linearity/Non-Linearity of $y(t) = x(t) +\cos(w_0 t)$

I have the following System with (input x(t), output(y(t)), that I have to check for linearity and time-invariance.

$$y(t) = x(t) + \cos(w_o t )$$

I am able to show that it is time-variant. For linearity I am not that sure. Here is what I have done so far.

$$y_1(t)= x_1(t) +\cos(w_0t)\\ y_2(t)= x_2(t) +\cos(w_0t)$$

Adding these give me $$y_1(t) + y_2(t) = x_1(t) + \cos(w_0t) +x_2(t) +\cos(w_0t)$$

Now $$x_1(t) + x_2(t) \rightarrow$$ system $$\rightarrow x_1(t) + x_2(t) +\cos(w_0t)$$.

Is this correct?

• Looks correct to me!
– MBaz
Apr 26 at 18:36

$$2x(t) \rightarrow \text{system} \rightarrow 2x(t) + \cos(\omega_0 t) \neq 2 y(t)$$
Not linear. You've also shown it: need $$2\cos(\omega_0t)$$ to satisfy $$x_1(t) + x_2(t) \Leftrightarrow y_1(t) + y_2(t)$$.