A Sum of Sinusoids at the
Same Frequency is Another
Sinusoid at that Frequency

It is an important and fundamental fact that a sum of sinusoids at the same frequency, but different phase and amplitude, can always be expressed as a single sinusoid at that frequency with some resultant phase and amplitude. An important implication, for example, is that

$\parbox{0.8\textwidth}{sinusoids are eigenfunctions of linear time-invariant (LTI) systems.}$

That is, if a sinusoid is input to an LTI system, the output will be a sinusoid at the same frequency, but possibly altered in amplitude and phase. This follows because the output of every LTI system can be expressed as a linear combination of delayed copies of the input signal. In this section, we derive this important result for the general case of $N$ sinusoids at the same frequency.