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In-Phase & Quadrature Sinusoidal Components

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

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).

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.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2021-02-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University