# In-Phase and Quadrature Sinusoidal Components

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. In-Phase and Quadrature Sinusoidal Components

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 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.)

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