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 section, we derive this important result for the general case of sinusoids at the same frequency.
We want to show it is always possible to solve
for and , given for . For each component sinusoid, we can write
Applying this expansion to Eq.(A.2) yields
Equating coefficients gives
where and are known. We now have two equations in two unknowns which are readily solved by (1) squaring and adding both sides to eliminate , and (2) forming a ratio of both sides of Eq.(A.4) to eliminate . The results are
which has a unique solution for any values of and .
Thus, equality holds when we define
Since is just the polar representation of a complex number, there is always some value of and such that equals whatever complex number results on the right-hand side of Eq.(A.5).
As is often the case, we see that the use of Euler's identity and complex analysis gives a simplified algebraic proof which replaces a proof based on trigonometric identities.
Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier
The heart of the preceding proof was the algebraic manipulation
For an arbitrary sinusoid having amplitude , phase , and radian frequency , we have
Elementary Filter Sections
Complex and Trigonometric Identities