Why do Discrete Time Signals Produce Repeating Frequency Spectra?

A discrete time signal is like a jump rope that only moves at certain moments, not all the time, and when you look at its frequency spectrum, it's like seeing how many different ways the rope can swing, but only some of them repeat.

Imagine you're jumping rope. If someone times your jumps to only happen every second, you might be doing a slow, steady rhythm, but if they time it so that you jump every half-second, or every third-second, the pattern changes.

Now imagine you're watching this from above and drawing how many times you jumped in each second, that's like looking at the frequency of your jumps. But since the rope only moves at certain moments, instead of showing all possible jumping speeds, it shows only those that fit into the timing, and these speeds repeat every so often.

A music box can only play certain tunes because of how its gears work. Similarly, a discrete time signal is like a limited music box, it can only show frequencies that match up with the timing it uses. That's why you see repeating frequency patterns when you look at the spectrum, they're just the tunes this special music box knows how to play!