Quadrature samples are essentially complex numbers. Complex numbers can be represented as two real numbers in two equivalent ways:
- Cartesian form: real (here called I or in-phase) and imaginary (here called Q or quadrature).
- Polar form: magnitude (or absolute value) and phase (or angle, or argument).
When you have I and Q signals or samples, those are in Cartesian form. However, when you want to demodulate, or even describe mathematically, a complex signal, the magnitude/phase form is more relevant; this is because the received phase is arbitrary (unless the transmitter and receiver have perfectly synchronized clocks and mixers and the path length never changes), which means that the signal has an arbitrary phase shift and therefore has no specific relationship with your I and Q "axes".
To help understand this, visualize an unmodulated complex (analytic) signal as a helix in 3D space: the axes are I, Q, and time. Unlike a real-valued signal, there are no zero crossings; the sample values follow a circle about the origin over time, and never meet it except when the amplitude is 0.
Furthermore, if the signal is baseband (after a receiver's mixer or before a transmitter's), then the rotation rate is 0 by definition: your samples have a constant value, except for the effects of the modulation. And this is the condition under which modulation and demodulation are typically done!
You ask for analog demodulation examples:
Demodulating AM from complex samples consists of taking the magnitude of the samples (and then subtracting the carrier amplitude from that, or equivalently using a high-pass filter), because that is exactly the amplitude of the original signal.
Demodulating FM from complex samples consists of taking the difference between the phase of successive samples, because that difference is the instantaneous frequency; if the signal is at baseband, then the instantaneous frequency is exactly the modulating signal!