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

(2.7) 
We have completely solved for the
frequency response of the simplest
lowpass 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
any input signal.
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