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The usual answers when asking "why use complex numbers to model measurements?" is usually: (a) Because you are measuring a phenomenon like electricity and magnetism. (b) Because it is super useful for analysis- Fourier, convolutions, differentiation: these are easier on the complex plane.

When discussing seismology, acoustics, and other real-valued measurements, authors still use complex numbers, even if analysis in the time-domain, and no convolutions involved. The usual argument is that the complex numbers represent the complex envelope of the real valued measurement. But why do that? Is this a sociological/communal habit, or is there a practical advantage in working with the convex envelope of a signal, and not its real version?

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'Imaginary' is a misnomer; they 'exist' as much as reals do. As for the envelope: it provides a meaningful representation of phase and amplitude over time, not otherwise doable with reals alone.

To illustrate, consider a real $x(t) = a(t)\cos(\phi(t))$, where $a(t)$ is amplitude and $\phi(t)$ is phase at time $t$. Suppose we observe $x(t)$, but don't know its function (i.e. righthandside of above). For any $t=t_0$, how do we tell its amplitude or phase? At $t_0$, all we have is $x(t_0)$, which can be at any point in the cycle:

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Enter the analytic representation:

enter image description here

with amplitude & phase computed as:

$$ a_a(t) = \sqrt{\Re^2 + \Im^2},\ \phi_a(t) = \tan^{-1}(\Im / \Re), \\ \Re = \text{real}(x_a(t)),\ \Im = \text{imag}(x_a(t)) $$

Example with trickier $\phi(t)$:

enter image description here

But I seek $x$'s amplitude & phase, not $x_a$'s: that's the thing, they're the same: $x_a(t)$ is constructed precisely so to keep $x(t)$'s $a(t)$ and $\phi(t)$. Not only can we now compute these quantities, but also instantaneous frequency, and with some luck, defy Heisenberg's limitations. Particularly once entering the discrete realm however, the method isn't flawless, but works well enough in general.


Why is this appropriate for measured data? Note that we're not transforming the data in any fundamental way, but rather expressing it in an equivalent but more useful form. So we don't say "H2O is liquid", which assumes certain temperatures and pressures, but more like "H2O has two hydrogen and one oxygen atoms", i.e. exact same thing but reformulated.

Whether the results are meaningful, however, does use assumptions: we assume the data is mainly composed of (1) periodic (repeating over time), and (2) sinusoidal processes. For acoustics, both hold very well; air compresses and expands periodically and sinusoidally over a local time segment. Then it's meaningful to ask: what's $a(t)$ and $\phi(t)$ of this physical sinusoidal process?

Assumption (3) is equally crucial: the data has a single intrinsic mode or 'major frequency component'. For example, if we have two A.M. sines, $f=2\text{Hz}, 4\text{Hz}$ out of phase, then we need two amplitudes and phases, $a_1(t), a_2(t)$ and $\phi_1(t), \phi_2(t)$, so a simple analytic transform won't suffice. For this we use methods like STFT, CWT, and synchrosqueezing to decompose the signal into its components.

Lastly, assumption (2) is flexible: if signal is, say, triangular waves instead of sinusoidal, then we simply re-interpret resulting $a(t)$ and $\phi(t)$.

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  • $\begingroup$ Thank you for a great answer. From which I learn that the complex envelope decomposes a signal into the instantaneous phase and amplitude. But this is an analysis/processing method. I still wonder why to assume this is the measurement model? Say in acoustics? Or seismology? $\endgroup$ – JohnRos Jan 4 at 6:36
  • $\begingroup$ @JohnRos Added a section; let me know if something's unclear, I may have misunderstood the question. $\endgroup$ – OverLordGoldDragon Jan 4 at 7:06
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    $\begingroup$ Thank you for an excellent answer and clarification! $\endgroup$ – JohnRos Jan 4 at 7:08

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