From Sinewaves to Arbitrary Signals

By superposition, we may readily generalize complex sinewave analysis to the case in which $x(n)$ is an arbitrary superposition of input sinusoids:

$$x(n) \isdef \int_{-\pi/T}^{\pi/T}A(\omega) e^{j[\omega n T + \phi(\omega)]} d\omega$$

(This is formally $2\pi$ times the inverse DTFT of $X(e^{j\omega T})\isdef A(\omega) e^{j\omega n T + \phi(\omega)}$.) Passing this superposition of sinusoids through any LTI filter ${\cal L}_n$ having transfer function $H(z)$ yields, by linearity,

$$\begin{eqnarray*} y(n) &=& {\cal L}_n\left\{\int_{-\pi/T}^{\pi/T} X(e^{j\omega ... ... \int_{-\pi/T}^{\pi/T} H(e^{j\omega T}) X(e^{j\omega T}) d\omega \end{eqnarray*}$$

where $G(\omega)$ is the amplitude response, and $\Theta(\omega)$ the phase response of the LTI filter. We have thus shown by superposition that, given any input signal $x$, the output spectrum $Y(e^{j\omega T})$ is equal to the input spectrum $X(e^{j\omega T})$ multiplied by the frequency response $H(e^{j\omega T})$, where the frequency response can be measured one frequency at a time using a sinusoidal input signal.

In contrast to the polar representation of frequency response $H(e^{j\omega T})= G(\omega)e^{j\Theta(\omega)}$, the real and imaginary parts do not have such intuitively appealing individual interpretations. Consequently, the polar form is usually preferred for expressing filter responses as a function of frequency.