Now we may isolate the filter phase response by taking a ratio of the and in Eq. (1.5):
Substituting the expansions of and yields
Thus, the phase response of the simple lowpass filter is
produces the output
Thus, the gain versus frequency is and the change in phase at each frequency is given by radians. These functions are shown in Fig.1.7. With these functions at our disposal, we can predict the filter output for any sinusoidal input. Since, by Fourier theory , every signal can be represented as a sum of sinusoids, we've also solved the more general problem of predicting the output given any input signal.