### 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 :

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