EEL 3135:
Dr. Fred J. Taylor, Professor
Lesson Title: Continuous-Time Signal and
Systems
Lesson Number: 25 (Section 9-1 to 9-4)
Background:
Continuous-time
signals and systems are known to the general public as
analog
signals and
systems.
They have always been a part of man’s natural world in which our analog sensory
systems are presumed to process analog signals. A bridge between discrete-time signal and
system domains, studied to this point (Chapters 1-8), has been constructed using the Sampling
Theorem, z-transforms, and similar tools.
They are, however, differences between continuous-
and discrete-time signal and system models that should be understood.
The studies presented
in Chapter 9 develop these ideas and builds a framework in which the study of continuous-time
signal and systems can take place.
Continuous-Time Signals:
Continuous-time signals x(t) persist over an interval of time called the signal’s
support
. This
interval of time can be
finite
(t
∈
[t
1
,t
2
]),
one-side
(t
∈
[0,
∞
], also called “right sided”), or
two-sided
(t
∈
[-
∞
,
∞
]).
Over the signal’s support, the continuous-time signal is continuously resolved in time
and amplitude (meaning that the time and amplitude values of x(t) are known with infinite
precision).
Over their support, continuous-time signals maybe
deterministic
(e.g., sinewave),
random
(e.g., noise) or
arbitrary
.
They maybe 1-
dimensional
(e.g., x(t)=speech), 2-
dimensional
(e.g., x(k
1
,k
2
)=2-D image), or multi-
dimensional
(e.g., x(k
1
, … , k
n
) = mechanical
system with n-degrees of freedom).
Signals may also be
real
(e.g., A cos(
ϖ
0
t)) or
complex
(e.g., Ae
j
ϖ
0
t
).
Two-Sided Signals:
Two-sided are assumed to exist for all time.
For example, a common sinusoidal test signal:
x
1
(t)=cos(
ϖ
0
t), t
∈
[-
∞
,
∞
]
1.
can be used to mathematically study the dynamic behavior of continuous-time systems. The
signal model assumes that the signal has always been present and will always be present.
This
raises questions of physical realizably which are often ignored.
Two-sided signals can also be
periodic
(i.e., x(t)=x(t+kT
0
), a property that a one-side or finite support signal can not
mathematically possess.
One-Sided Signals:
One-sided signals are generally assume to be
right sided
((t
∈
[0,
∞
]).
For example, a common
test signal shown below:
1