You may remember the following trigonometric identity for sinusoids:
(4)
In this way, a sinusoidal signal with arbitrary phase offset can be written:
(5)
(6)
where and are constants proportional to the amplitude and the phase offset .
Thus, a sinusoid with arbitrary phase offset can be “separated” into sine and cosine terms. Said another way, a sinusoid with arbitrary phase offset can be completely reconstructed from properly scaled sine and cosine terms, each with zero phase.
This result suggests a solution to the dot-product problem discussed above. That is, in order to fully detect a sinusoidal component in a given signal, we need to compute the dot-product with both a cosine and a sine term for each frequency (which produces results that correspond to the constants and above and from which we can also calculate the amplitude and the phase offset ).