Complex signals can be built from elemental signals, including the complex
exponential, unit step, pulse, etc. This module presents the elemental signalsin brief.
Elemental signals are the building blocks with which
we build complicated signals. By definition,
elemental signals have a simple structure. Exactly what wemean by the "structure of a signal" will unfold in this
section of the course. Signals are nothing more thanfunctions defined with respect to some independent variable,
which we take to be time for the most part. Very interestingsignals are not functions solely of time; one great example of
which is an image. For it, the independent variables are
$x$ and
$y$ (two-dimensional space).
Video signals are functions of three variables: two spatialdimensions and time. Fortunately, most of the ideas underlying modern
signal theory can be exemplified with one-dimensional signals.
Sinusoids
Perhaps the most common real-valued signal is the sinusoid.
$s(t)=A\cos (2\pi {f}_{0}t+\phi )$
For this signal,
$A$ is its amplitude,
${f}_{0}$ its frequency, and
$\phi $ its phase.
Complex exponentials
The most important signal is complex-valued, the complex
exponential.
Here,
$i$ denotes
$\sqrt{-1}$ .
$Ae^{i\phi}$ is known as the signal's
complex amplitude .
Considering the complex amplitude as a complex numberin polar form, its magnitude is the amplitude
$A$ and its angle the signal
phase. The complex amplitude is also known as a
phasor . The complex exponential cannot be further
decomposed into more elemental signals, and is the
most important signal in electrical
engineering ! Mathematical manipulations at first
appear to be more difficult because complex-valued numbers areintroduced. In fact, early in the twentieth century,
mathematicians thought engineers would not be sufficientlysophisticated to handle complex exponentials even though they
greatly simplified solving circuit problems.
Steinmetz introduced complex exponentials to electrical engineering, and
demonstrated that "mere" engineers could use them to goodeffect and even obtain right answers! See
Complex Numbers for a
review of complex numbers and complex arithmetic.
The complex exponential defines the notion of frequency: it is
the
only signal that contains only one
frequency component. The sinusoid consists of two frequencycomponents: one at the frequency
${f}_{0}$ and the other at
$-{f}_{0}$ .
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 choicedepending on whether cosine or sine phase is needed, or as the
sum of two complex exponentials. These two decompositions aremathematically equivalent to each other.
Using the complex plane, we can envision the complex
exponential's temporal variations as seen in the above figure(
[link] ). The magnitude of
the complex exponential is
$A$ ,
and the initial value of the complex exponential at
$t=0$ has an angle of
$\phi $ .
As time increases, the locus of points traced by the complexexponential is a circle (it has constant magnitude of
$A$ ). The number of times per
second we go around the circle equals the frequency
$f$ . The time taken for the
complex exponential to go around the circle once is known asits
period$T$ , and
equals
$\frac{1}{f}$ . The projections onto the real and imaginary axes
of the rotating vector representing the complex exponentialsignal are the cosine and sine signal of Euler's relation
(
[link] ).
Real exponentials
As opposed to complex exponentials which oscillate,
real exponentials decay.
$s(t)=e^{-\left(\frac{t}{\tau}\right)}$
The quantity
$\tau $ is known as
the exponential's
time constant , and corresponds
to the time required for the exponential to decrease by afactor of
$\frac{1}{e}$ , which approximately equals
$0.368$ .
A decaying complex exponential is the product of a real and
a complex exponential.
In the complex plane, this signal corresponds to an
exponential spiral. For such signals, we can define
complex frequency as the quantity multiplying
$t$ .
This signal is discontinuous at the origin. Its value at the
origin need not be defined, and doesn't matter in signaltheory.
This kind of signal is used to describe
signals that "turn on" suddenly. For example, tomathematically represent turning on an oscillator, we can
write it as the product of a sinusoid and a step:
$s(t)=A\sin (2\pi ft)u(t)$ .
Pulse
The
unit pulse describes turning a unit-amplitude signal on for a duration of
$\Delta $ seconds, then turning it
off.
We will find that this is the second most important signal in
communications.
Square wave
The
square wave$\mathrm{sq}(t)$ is a periodic signal like the sinusoid. It too has an
amplitude and a period, which must be specified tocharacterize the signal. We find subsequently that the sine
wave is a simpler signal than the square wave.
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