Thinking of signals as components that are mixed together, such as sine waves and transients.

The very simplest model for this would be a bank of filters where we can break a signal up into different bands, and then sum them back together to regain our original signal

Splitting a signal into archetypes such as sinusoidal components and transients is trickier, but can be very effective when your signal fits the assumptions of the algorithm well (and can produce interesting artefacts some of the rest of the time)

More on specific techniques: