In-Phase and Quadrature Sinusoidal Components

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). In-Phase and Quadrature Sinusoidal Components

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In-Phase and Quadrature Sinusoidal Components

From the trig identity $\, we have

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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 $90$ degree phase shift. (See §5.4 for the Mathematica code for this figure.)

Figure 5.2:In-phase and quadrature sinusoidal components.
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