2.2 Signal decomposition

A signal's complexity is not related to how wiggly itis. 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 equalsthe number of terms in its decomposition. In writing a signal as a sum of component signals, we can change the component signal'sgain by multiplying it by a constant and by delaying it. More complicated decompositions could contain derivatives orintegrals of simple signals.

Because the sinusoid is a superposition of two complex exponentials, the sinusoid is more complex. We could not preventourselves from the pun in this statement. Clearly, the word "complex" is used in two different ways here. The complexexponential can also be written (using Euler's relation ) as a sum of a sine and a cosine. We will discover that virtuallyevery 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 adequatelyreveal its complexity.