Summary: Linear time-invariant (LTI) systems are the most important class of systems in communications. They ensure that a system acting on a signal can be modeled by the system acting individually on the component parts of the signal and summed.

A signal's complexity is not related to how wiggly it is. Rather, a signal
expert looks for ways of decomposing a given signal into a *sum of
simpler signals*, which we term the signal decomposition.
Though we will never compute a signal's complexity, it essentially equals the
number of terms in its decomposition. In writing a signal as a sum of component
signals, we can change the component signal's gain by multiplying it by a constant
and by delaying it. More complicated decompositions could contain derivatives
or integrals of simple signals. In short, signal decomposition amounts to
thinking of the signal as the output of a linear system having simple signals as
its inputs. We would build such a system, but envisioning the signal's components
helps understand the signal's structure. Furthermore, you can readily compute a
linear system's output to an input decomposed as a superposition of simple signals.

As an example of signal complexity, we can express the pulse

Express a square wave having period

Because the sinusoid is a superposition of two complex exponentials, the
sinusoid is more complex. We could not prevent ourselves from the pun in
this statement. Clearly, the word "complex" is used in two different ways
here. The complex exponential can also be written (using Euler's relation) as a sum of a sine and a
cosine. We will discover that virtually every signal can be decomposed into a
sum of complex exponentials, and that this decomposition is *very
*useful. Thus, the complex exponential is more fundamental, and
Euler's relation does not adequately reveal its complexity.