An important property of sinusoids at a particular frequency is that they
are closed with respect to addition. In other words, if you take a
sinusoid, make many copies of it, scale them all by different gains,
delay them all by different time intervals, and add them up, you always get a
sinusoid at the same original frequency. This is a nontrivial property.
It obviously holds for any constant signal (which we may regard as
a sinusoid at frequency ), but it is not obvious for (see
Fig.4.2 and think about the sum of the two waveforms shown
being precisely a sinusoid).

Since every linear, time-invariant (LTI^{4.2}) system (filter) operates by copying, scaling,
delaying, and summing its input signal(s) to create its output
signal(s), it follows that when a sinusoid at a particular frequency
is input to an LTI system, a sinusoid at that same frequency always
appears at the output. Only the amplitude and phase can be changed by
the system. We say that sinusoids are eigenfunctions of LTI
systems. Conversely, if the system is nonlinear or time-varying, new
frequencies are created at the system output.

To prove this important invariance property of sinusoids, we may
simply express all scaled and delayed sinusoids in the ``mix'' in
terms of their in-phase and quadrature components and then add them
up. Here are the details in the case of adding two sinusoids having
the same frequency. Let be a general sinusoid at frequency
:

Now form as the sum of two copies of with arbitrary
amplitudes and phase offsets:

Focusing on the first term, we have

We similarly compute

and add to obtain

This result, consisting of one in-phase and one quadrature signal
component, can now be converted to a single sinusoid at some amplitude and
phase (and frequency ), as discussed above.