[zeehio]

Fully agree. Switching from the time domain to the frequency domain is just like switching coordinate systems.

If you have a signal with a single narrow peak in the time domain you can represent it in a very compact or sparse way just using a delta at the position of a peak. If you try to represent it in the frequency domain you will not find such compact representation. Similarly if you have a sinusoidal signal in the time domain, you won't get a compact representation there, but you will get it in the frequency domain where you just have a couple of deltas.

Time and frequency are two ways of representing the same thing. Sometimes it's easier to represent something in one domain, sometimes it's easier in the other domain.

It can be proven that anything bounded in the time domain will be unbounded in the frequency domain and viceversa. So compact stuff in one domain always spreads when represented in the other domain.

Beautifully, quantum mechanics tell us that position and momentum are conjugated variables (like time and frequency in the example above) and therefore if something has a bounded position (we know where it is) its momentum will be unbounded (we won't know its speed), and viceversa.

That's the main idea of Heisenberg's uncertainty principle.