Fixing the Phase Problem

• You may remember the following trigonometric identity for sinusoids:
    $$\cos( \alpha \pm \beta ) = \cos( \alpha ) \cos( \beta ) \mp \sin( \alpha ) \sin( \beta )$$ (4)

• In this way, a sinusoidal signal with arbitrary phase offset can be written:
  

  $$x(t) = A \cos( \omega t + \phi ) = A \left[ \cos( \omega t ) \cos( \phi ) - \sin( \omega t ) \sin( \phi ) \right]$$ (5)

  $$= B \cos( \omega t ) + C \sin( \omega t ),$$ (6)

  where $B$ and $C$ are constants proportional to the amplitude $A$ and the phase offset $\phi$.

• 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 $B$ and $C$ above and from which we can also calculate the amplitude $A$ and the phase offset $\phi$).