Quadrature phase

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Communication signals often have the form:

$\begin{align} A(t)\cdot \sin[2\pi ft + \phi(t)] &= I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \cos(2\pi ft) \\ &= I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \sin(2\pi ft + \begin{matrix} \frac{\pi}{2}) \end{matrix} \end{align}$ ^[1]

where $f\,$ represents a carrier frequency, and

$I(t)\ \stackrel{\mathrm{def}}{=}\ A(t)\cdot \cos[\phi(t)], \,$

$Q(t)\ \stackrel{\mathrm{def}}{=}\ A(t)\cdot \sin[\phi(t)].\,$

$A(t)\,$ and $\phi (t)\,$ represent possible modulation of a pure carrier wave: $\sin(2\pi f t).\,$ The modulation alters the original $\sin\,$ component of the carrier, and creates a (new) $\cos\,$ component, as shown above. The component that is in phase with the original carrier is referred to as the in-phase component. The other component, which is always 90° ( $\begin{matrix} \frac{\pi}{2} \end{matrix}$ radians) out of phase, is referred to as the quadrature component.