Perhaps the most common real-valued signal is the sinusoid.

Summary: Complex signals can be built from elemental signals, including the complex exponential, unit step, pulse, etc. This module presents the elemental signals in brief.

*Elemental signals are the building blocks with which
we build complicated signals.* By definition,
elemental signals have a simple structure. Exactly what we
mean by the "structure of a signal" will unfold in this
section of the course. Signals are nothing more than
functions defined with respect to some independent variable,
which we take to be time for the most part. Very interesting
signals are not functions solely of time; one great example of
which is an image. For it, the independent variables are

Perhaps the most common real-valued signal is the sinusoid.

The most important signal is complex-valued, the complex exponential.

The complex exponential defines the notion of frequency: it is
the *only* signal that contains only one
frequency component. The sinusoid consists of two frequency
components: one at the frequency

This decomposition of the sinusoid can be traced to Euler's
relation.

The complex exponential signal can thus be written in terms of
its real and imaginary parts using Euler's relation. Thus,
sinusoidal signals can be expressed as either the real or the
imaginary part of a complex exponential signal, the choice
depending on whether cosine or sine phase is needed, or as the
sum of two complex exponentials. These two decompositions are
mathematically equivalent to each other.

Using the complex plane, we can envision the complex
exponential's temporal variations as seen in the above figure
(Figure 1). The magnitude of
the complex exponential is

As opposed to complex exponentials which oscillate, real exponentials decay.

The quantity *
A decaying complex exponential is the product of a real and
a complex exponential.
*

The unit step function
is denoted by

This signal is discontinuous at the origin. Its value at the
origin need not be defined, and doesn't matter in signal
theory.

The unit pulse
describes turning a unit-amplitude signal on for a duration of

The square wave

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