 Quadrature detection

Quadrature detection measures a signal against two reference oscillations separated by a quarter cycle. The resulting in-phase and quadrature components retain both amplitude and relative phase; a single component can vanish merely because the signal and reference are out of phase. A two-channel lock-in amplifier is a concrete implementation.^[ref]

For a signal $s(t)=A\cos (\omega t+\varphi )$ with constant $A$ and $\varphi$, choose the normalized references $2\cos (\omega t)$ and $-2\sin (\omega t)$. Multiplication followed by a low-pass filter that removes terms at $2\omega$ givesThus $A=\sqrt{I^2+Q^2}$, while the pair $(I,Q)$ specifies phase including its quadrant. For $\varphi =\pi /2$, the in-phase reading is zero but the quadrature reading is $A$: the signal has not disappeared. The sign of $Q$ depends on the chosen reference convention.

In an actual detector, finite filter bandwidth admits nearby frequencies and noise. Synchronization does not reject an unwanted signal at exactly the same reference frequency. If the reference is detuned, the output components rotate at the difference frequency rather than remaining constant.^[ref]