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

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 appendix, we derive this important result for the
general case of sinusoids at the same frequency.