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

We have completely solved for the frequency response of the simplest low-pass filter given in Eq.(1.1) using only trigonometric identities. We found that an input sinusoid of the form

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 [84], every signal can be represented as a sum of sinusoids, we've also solved the more general problem of predicting the output given

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