**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
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Fourier Transform (DFT), with Audio Applications --- Second
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Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## In-Phase and Quadrature Sinusoidal Components

From the trig identity , we have

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 ). 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 .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 5.2 illustrates in-phase and quadrature components overlaid. Note that they only differ by a relative degree phase shift. (See §5.4 for the Mathematica code for this figure.)