Complex Sinewave Analysis Revisited

Consider again the sine-wave analysis method used in Chapter 1 (§1.3.1) for the simplest lowpass filter. We let the input signal be a complex sinusoid

$$x(n) = A e^{j\omega_0 n T+ \phi}$$

having amplitude $A$ and phase $\phi$. Recall that for complex sinusoidal inputs, the amplitude response is measurable, after any transient response has settled down, as the instantaneous output amplitude divided by the instantaneous input amplitude, or $G(\omega_0) = \left\vert y(n)\right\vert/\left\vert x(n)\right\vert$. (This simple formula for $G(\omega_0)$ holds only for complex sinusoidal inputs, since a constant-amplitude signal is required.) Secondly, the phase response is given by the phase of the output sinusoid minus the phase of the input sinusoid, or $\Theta(\omega) = \angle y(n) - \angle x(n)$. Thus, the output $y(n)$ must be

$$\begin{eqnarray*} y(n) &=& [G(\omega_0)A ] e^{j[\omega_0 n T + \phi + \Theta(\o... ...T}) A e^{j[\omega_0 n T + \phi]}\\ &=& H(e^{j\omega_0T}) x(n). \end{eqnarray*}$$

This shows that the output of an LTI filter in response to a complex sinusoid at frequency $\omega_0$ is obtained by (1) scaling by $G(\omega_0)$ and phase-shifting by $\Theta(\omega_0)$, or, equivalently, (2) multiplying the input complex sinusoid by the (complex) frequency response at frequency $\omega_0$.