Along with the classification of signals below, it is also
important to understand the Classification of Systems.
As the names suggest, this classification is determined by
whether or not the time axis (x-axis) is
discrete (countable) or continuous
(Figure 1). A continuous-time signal will
contain a value for all real numbers along the time axis.
In contrast to this, a discrete-time signal is often created by
using the sampling
theorem to sample a continuous signal, so it will
only have values at equally spaced intervals along the time
axis.
The difference between analog and
digital is similar to the difference between
continuous-time and discrete-time. In this case, however,
the difference is with respect to the value of the function
(y-axis) (Figure 2). Analog corresponds to a
continuous y-axis, while digital corresponds to a discrete
y-axis. An easy example of a digital signal is a binary
sequence, where the values of the function can only be one
or zero.
Periodic signals
repeat with some period
TT, while aperiodic, or
nonperiodic, signals do not (Figure 3). We can
define a periodic function through the following
mathematical expression, where t
t can be any number and T
T is a positive constant:
ft=fT+t
f
t
f
T
t
(1)
The
fundamental period of our function,
ft
f
t
, is the smallest value of
T
T that the still allows
Equation 1 to be true.
Causal signals are signals that are zero for
all negative time, while anticausal are signals
that are zero for all positive time. Noncausal
signals are signals that have nonzero values in both
positive and negative time (Figure 4).
An even signal is any signal
ff such that
ft=f-t
f
t
f
t
. Even signals can be easily spotted as they are
symmetric around the vertical axis. An
odd signal, on the other hand, is a signal
ff such that
ft=-f-t
f
t
f
t
(Figure 5).
Using the definitions of even and odd signals, we can show
that any signal can be written as a combination of an even and
odd signal. That is, every signal has an odd-even
decomposition. To demonstrate this, we have to look no
further than a single equation.
ft=12ft+f-t+12ft-f-t
f
t
1
2
f
t
f
t
1
2
f
t
f
t
(2)
By multiplying and adding this expression out, it can be shown
to be true. Also, it can be shown that
ft+f-t
f
t
f
t
fulfills the requirement of an even function, while
ft-f-t
f
t
f
t
fulfills the requirement of an odd function (
Figure 6).
A deterministic signal is a signal in which
each value of the signal is fixed and can be determined by a
mathematical expression, rule, or table. Because of this
the future values of the signal can be calculated from past
values with complete confidence. On the other hand, a
random
signal has a lot of uncertainty about its
behavior. The future values of a random signal cannot be
accurately predicted and can usually only be guessed based
on the averages
of sets of signals (Figure 7).
A right-handed signal and
left-handed signal are those signals whose
value is zero between a given variable and positive or
negative infinity. Mathematically speaking, a right-handed
signal is defined as any signal where
ft=0
f
t
0
for
t<
t
1
<∞
t
t
1
, and a left-handed signal is defined as any signal where
ft=0
f
t
0
for
t>
t
1
>-∞
t
t
1
. See (Figure 8) for an example. Both
figures "begin" at
t
1
t
1
and then extends to positive or negative infinity
with mainly nonzero values.
As the name applies, signals can be characterized as to
whether they have a finite or infinite length set of
values. Most finite length signals are used when dealing
with discrete-time signals or a given sequence of values.
Mathematically speaking,
ft
f
t
is a finite-length signal if it is
nonzero over a finite interval
t
1
<ft<
t
2
t
1
f
t
t
2
where
t
1
>-∞
t
1
and
t
2
<∞
t
2
. An example can be seen in Figure 9.
Similarly, an infinite-length signal,
ft
f
t
, is defined as nonzero over all real numbers:
∞≤ft≤-∞
f
t
Useful Signals
System Classification and Properties
Sampling Theorem
