Basics of Linearity

Question

A linear system's response to $ e^{j2t} \ $ is $ te^{-j2t} \ $ and its response to $ e^{-2jt}\ $ is $ t e^{2jt}\ $.

What is the system's response to $ cos(2t)\ $?

Answer

Accord to Euler's formula, $ cos(t) = \frac{e^{jt}+e^{-jt}}{2} $

Hence, the response to $ cos(2t)\ $ would be $ \frac{e^{j2t}+e^{-j2t}}{2}\ $.

With that in mind, the response to $ e^{j2t}\ $ is $ t e^{-j2t}\ $ and its response to $ e^{-j2t}\ $ is $ t e^{j2t}\ $.

Thus the output of the system will be:

$ =t\frac{e^{j2t}+e^{-j2t}}{2}\, $ $ =t\cos(2t)\, $