Real Sinewave Analysis

As a special case of this example, obtained by taking the real part, the response of an LTI filter to

$$x(n) = A \cos(\omega_0 nT + \phi)$$

is

$$y(n) = AG(\omega_0) \cos[\omega_0 nT + \phi + \Theta(\omega_0)].$$

Thus, for a real sinusoidal input at radian frequency $\omega_0$, the amplitude is multiplied by $G(\omega_0)$, and the phase is additively changed by $\Theta(\omega_0)$.