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System Classifications and Properties
Summary: Describes various classifications of systems.
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Example links
- [4] Simple Systems
Supplemental links
Introduction
In this module some of the basic classifications of systems
will be briefly introduced and the most important properties
of these systems are explained. As can be seen, the
properties of a system provide an easy way to separate one
system from another. Understanding these basic difference's
between systems, and their properties, will be a fundamental
concept used in all signal and system courses, such as digital
signal processing (DSP). Once a set of systems can be
identified as sharing particular properties, one no longer has
to deal with proving a certain characteristic of a system each
time, but it can simply be accepted do the the systems
classification. Also remember that this classification
presented here is neither exclusive (systems can belong to
several different classifications) nor is it unique (there are
other methods of classification
). Examples of simple systems can be found here.
Classification of Systems
Along with the classification of systems below, it is also
important to understand other Classification of Signals.
Continuous vs. Discrete
This may be the simplest classification to understand as the
idea of discrete-time and continuous-time is one of the most
fundamental properties to all of signals and system. A
system where the input and output signals are continuous is a
continuous system, and one where the input and
output signals are discrete is a discrete system.
Linear vs. Nonlinear
A linear system is any system that obeys the
properties of scaling (homogeneity) and superposition
(additivity), while a nonlinear system is any
system that does not obey at least one of these.
To show that a system H H
H k f t = k H f t
H
k
f
t
k
H
f
t
 Figure 1: A block diagram demonstrating the scaling property of
linearity |
To demonstrate that a system H H
H
f
1
t +
f
2
t = H
f
1
t + H
f
2
t
H
f
1
t
f
2
t
H
f
1
t
H
f
2
t
 Figure 2: A block
diagram demonstrating the superposition property of
linearity |
It is possible to check a system for linearity in a single
(though larger) step. To do this, simply combine the first
two steps to get
H
k
1
f
1
t +
k
2
f
2
t =
k
2
H
f
1
t +
k
2
H
f
2
t
H
k
1
f
1
t
k
2
f
2
t
k
2
H
f
1
t
k
2
H
f
2
t
Time Invariant vs. Time Variant
A time invariant system is one that does not
depend on when it occurs: the shape of the output does not
change with a delay of the input. That is to say that for a
system H H
H f t = y t
H
f
t
y
t
H
H T T
H f t - T = y t - T
H
f
t
T
y
t
T
 Figure 3:
This block diagram shows what the condition for time
invariance. The output is the same whether the delay is put
on the input or the output.
|
When this property does not hold for a system, then it is said
to be time variant, or time-varying.
Causal vs. Noncausal
A causal system is one that is
nonanticipative; that is, the output may depend
on current and past inputs, but not future inputs. All
"realtime" systems must be causal, since they can not have
future inputs available to them.
One may think the idea of future inputs does not seem to
make much physical sense; however, we have only been
dealing with time as our dependent variable so far, which is
not always the case. Imagine rather that we wanted to do
image processing. Then the dependent variable might represent
pixels to the left and right (the "future") of the current
position on the image, and we would have a
noncausal system.
 Subfigure 4.1: For a typical system to be causal...  Subfigure 4.2: ...the output at time
Figure 4 |
Stable vs. Unstable
A stable system is one where the output does
not diverge as long as the input does not diverge. There
are many ways to say that a signal "diverges"; for example
it could have infinite energy. One particularly useful
definition of divergence relates to whether the signal is
bounded or not. Then a system is referred to as
bounded input-bounded output (BIBO) stable if
every possible bounded input produces a
bounded output.
Representing this in a mathematical way, a stable system
must have the following property, where
x t x t y t y t
| y t | ≤
M
y
< ∞
y
t
M
y
| x t | ≤
M
x
< ∞
x
t
M
x
M
x
M
x
M
y
M
y
t
t
If these conditions are not met, i.e. a
system's output grows without limit (diverges) from a
bounded input, then the system is unstable.
Note that the BIBO stability of a linear time-invariant
system (LTI) is neatly described in terms of whether or not
its impulse response is absolutely integrable.
More about this content: Metadata | Version History | Cite This Content
This work is licensed by Melissa Selik and Richard Baraniuk. See the Creative Commons License about permission to reuse this material.
Last edited by Charlet Reedstrom on Jun 13, 2005 7:52 pm GMT-5.
"My introduction to signal processing course at Rice University."