Quadrature phase

Two periodic waveforms whose phase difference is $\tfrac{1}{4}$ of their output period are said to have a quadrature phase relationship. The term is also used in communication systems to describe one of the components of orthogonal decomposition.

A composite signal described by its envelope-and-phase form $A(t)\cdot \sin[2\pi ft + \phi(t)]$ can be decomposed to an equivalent quadrature-carrier(IQ) form as:

$I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \cos(2\pi ft) = I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \sin(2\pi ft + \begin{matrix}\frac{\pi}{2}\end{matrix})$

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 direct or 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.

See also

• In-phase and quadrature (I&Q) components
• Quadrature
• Quadrature amplitude modulation
• Constellation diagram