Discrete Time Fourier Analysis
The sequences $e^{j\omega_0n}$ are eigensequences of discrete-time LTI systems, i.e., the response to such a sequence is the same sequence scaled by a complex constant (the eigenvalue). This can be shown as follows:

$$\begin{align}y[n]&amp;=(h\star x)[n]\\&amp;=\sum_{k=-\infty}^{\infty}h[k]x[n-k]\\&amp;=\sum_{k=-\infty}^{\infty}h[k]e^{j\omega_0(n-k)}\\&amp;=e^{j\omega_0n}\sum_{k=-\infty}^{\infty}h[k]e^{-j\omega_0k}\\&amp;=e^{j\omega_0n}H(\omega_0)\end{align}$$

where $h[n]$ is the system's impulse response, and $H(\omega)$ is its frequency response. A completely analogous proof can be given for continuous-time systems.