This topic uses linear system theory to motivate looking at methods for signal decompositions. It starts by observing that linear systems can be described by ordinary differential equations (ODEs), which have harmonic steady-state solutions in response to a harmonic input. This conceptually simple solution enables the frequency response of the system to be easily determined, and therefore the system response to an arbitrary input can be found, if the input signal can be represented in the frequency domain. Fourier analysis enables this, and this Topic provides the narrative journey to explain this.