### Sinusoids at the Same Frequency

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 :

Focusing on the first term, we have

We similarly compute

**Next Section:**

Constructive and Destructive Interference

**Previous Section:**

In-Phase & Quadrature Sinusoidal Components