Amplitude and phase: what are they really?

For sinusoidal signals, the idea of amplitude and phase are pretty clear. The amplitude is one half of the peak to valley distance of the signal and phase coincides with the position of the signal at a point in time or space. By signal, I mean any physical quantity that is undergoing simple harmonic motion, e.g. a mass on a spring. In optics, the amplitude and phase of plane waves are well-defined as well: the amplitude is related to the irradiance and the phase typically refers to the position of the wavefronts with respect to some reference.

What's often taken for granted by new physics students (and even experienced ones) is that plane waves and sinusoidal signals are not real. Any real signal must be of finite duration; a sinusoid extends to infinity both forwards and backwards in time. As a result, the amplitude and phase of real signals become somewhat ambiguous.

The two concepts are retained in practice, however, because they simplify the interpretation of many physical situations, such as the effects of a filter on a signal or the output of an optical interferometer or two-slit experiment. A real signal s(t) has an analytic representation z(t) that consists of s(t) plus the signal's Hilbert transform times the imaginary number, j (I was trained as an engineer, but consider it the same as i used by physicists and mathematicians). Two things should be noted here: 1) the analytic signal is unique, and 2) the analytic signal is complex; it has a real and imaginary part. Like all complex numbers, it can also be represented by a magnitude and phase angle. The magnitude is often denoted the amplitude and the phase angle is thought of as the phase of the signal.

Unfortunately, the amplitude and phase of the analytic signal can not always be directly correlated to physically meaningful quantities. For frequency modulated signals, the time-varying amplitude and phase are independent only so long as their individual spectra are non-overlapping. Furthermore, even if the time-varying amplitude and phase are spectrally separate, it is very counter-intuitive (at least to me) to be able to represent one function of time (the real signal) with two functions of time. This last point I believe is related to the Kramers-Kronig relationship between the real and imaginary parts of a signal. So, while a time-varying amplitude and phase can represent a real modulated signal and simplify interpretation, the only thing about the signal with any physical significance is what can be measured, such as current or voltage.

Most of this post comes from thoughts I've had recently concerning how casually we in optical sciences use the terms amplitude and phase. Once an optical signal becomes incoherent, either in space or time, its amplitude and phase are said to vary randomly. This statement only really makes since once one considers that amplitude and phase are purely theoretical constructs for describing real data and that one must not place too much physical significance in their meaning.

I also learned a lot and reproduced some information from this very good article by Boashash in the Proceedings of the IEEE: http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=135376&tag=1.