# Motivation

We have studied the modeling of systems as functions that map function spaces into function spaces. We now start to look at how to specify and analyze such functions.A class of systems that yield to sophisticated analysis techniques is
the class of *linear time-invariant* (LTI) systems. LTI systems have
a key property: given a sinusoidal input, the output is a sinusoidal signal
with the same frequency (but possibly different amplitude and phase).

We can justify describing audio signals as sums of sinusoids on purely psychoacoustic grounds. However, because of this property of LTI systems, it is often convenient to describe any signal as sums of sinusoids, regardless of whether there is a psychoacoustic justification.

The real value in this mathematical trick is that by using the theory of LTI systems, I can design systems that operate more-or-less independently on the sinusoidal components of a signal. For example, note that abrupt changes in the signal value require more of the signal to be at higher frequencies. Thus, I can enhance or suppress these abrupt changes by enhancing or suppressing the higher frequency components.