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

having amplitude and phase . 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
. (This simple formula for
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
.
Thus, the output must be

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