Can a 1-dimensional signal be in a domain other than the time or frequency domain?

Question

In signal processing I have only heard of signals being in the "Time" domain or in the "Frequency" domain, but are there situations where a signal is not in either of these?

If so, what are these other domains called? How could you translate a time or frequency domain signal into one of these other domains? Are there any practical use cases for doing so?

This question is referring specifically to 1-dimensional signals (for example a signal read from an audio file, or a simple sine wave).

Answer 1

If you have a one dimensional signal $s(\cdot)$, it somehow belongs to a combination of two different domains:

• the sampling domain: where samples are considered or taken, or how the signal is sampled. When samples are taken in some order, this is often called: the ordinal variable. This is generally related to some "physically-sound" unit, like time (you sample every second), space (you sample every meter, e.g. with aligned sensors), or else. It can be temperature (chromatography), mass (mass spectrometry), age, weight, whatever related to acquisition devices.
• the data domain: is the cardinal variable data binary, discrete, continuous, multi-valued (like RBG images), etc.?

When time is the ordinal variable, frequency is the corresponding (Fourier) dual frequency version. When space is the ordinal variable, the corresponding quantity is the wave-number, etc. Being "dual", as it "preserves properties" (like energy, scalar-products), is important to:

• mathematicians who need to prove stuff,
• algorithm people who needs to make stuff work,
• practitioners, who need to understand things.

Where I dare to differ from Marcus's answer is on: "The math doesn't care whether you're transforming amplitude over time...". The inherent structure of either the primal space (ordinal) or dual space (cardinal) may convey how you can convert one into the other, and back (to about or limit information loss). Whether the ordinal is uniform, whether the cardinal is discrete, provides constraints on what numerical tools you can build, and how precise the interpretation can be.

To wrap to up: to convert one signal into another domain roots on how the other domain helps you understand in a meaningful way how your signal behaves. And you should be careful on units to study behavior.

Answer 2

Really, there's nothing special about time and frequency domain.

The math doesn't care whether you're transforming amplitude over time, gravel over mountain height, or smell intensity over fridge temperature. To the math, you map elements from a function space to elements from a function space when you do the Fourier transform. Full stop.

Any scalar physical entity can be used as basis for a 1D signal. Spatial signals are the most obvious: what's the height profile of something over distance, or the temperature over distance, or color over distance, whatever over distance. In some contexts, the resulting entities are called "spatial frequencies", but sometimes something else. Names are just names.

Or what is the temperature of something over pressure? That's another 1D signal. And it makes a lot of sense to Fourier transform it, if you're dealing with compressible media in a container, things will oscillate, and not necessarily over time. Do I know whether that has a name? No. Did some PhD give it a name in his dissertation about pressure in containers? Pretty likely, if you ask me. Does that name matter to anything but the notation in that dissertation? Maybe. Maybe not.

Very commonly observed: impulse / state space its Fourier "dual" location space in solid state physics.

We can go on with this all day, because, as said anything can be a 1D signal, and some have Fourier transforms with special names, other don't. It's by pure coincidence that you've heard of the time/frequency pair first. If you were a math student, you might have heard about probability densities and characteristic functions first – yet another 1D signal/FT pair.

So, really, to cite someone I think Tukey,

One can Fourier transform anything – often meaningfully.

Your question is just asking for an arbitrary list of names. Names don't actually matter – they just make it easier to communicate about something. Whether I call something a "time signal" or a "frequency signal", however, makes little difference. For all we care, I could be calling it "original space" and "image space" (under the Fourier transform), because that's what "time" and "frequency" are. It's just convenient that these two have a name.