In-Phase & Quadrature Sinusoidal Components

From the trig identity $\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ , we have

$$\begin{eqnarray*} x(t) &\isdef & A \sin(\omega t + \phi) = A \sin(\phi + \omega t) \\ &=& \left[A \sin(\phi)\right] \cos(\omega t) + \left[A \cos(\phi)\right] \sin(\omega t) \\ &\isdef & A_1 \cos(\omega t) + A_2 \sin(\omega t). \end{eqnarray*}$$

From this we may conclude that every sinusoid can be expressed as the sum of a sine function (phase zero) and a cosine function (phase $\pi/2$ ). If the sine part is called the ``in-phase'' component, the cosine part can be called the ``phase-quadrature'' component. In general, ``phase quadrature'' means ``90 degrees out of phase,'' i.e., a relative phase shift of $\pm\pi/2$ .

It is also the case that every sum of an in-phase and quadrature component can be expressed as a single sinusoid at some amplitude and phase. The proof is obtained by working the previous derivation backwards.

Figure 4.2 illustrates in-phase and quadrature components overlaid. Note that they only differ by a relative $90$ degree phase shift.

Figure 4.2: In-phase and quadrature sinusoidal components.