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Properties of Systems
Summary: Properties of different types of systems
Linear Systems
If a system is linear, this means that when an input to a given system is scaled
by a value, the output of the system is scaled by the same amount.
Linear Scaling
Figure 1 |
In subfigure 1.1 above, an input x x L L y y x x α α α α
A linear system also obeys the principle of superposition. This
means that if two inputs are added together and passed through a linear system,
the output will be the sum of the individual inputs' outputs.
Figure 2 |
Superposition Principle |
That is, if Figure 2 is true, then
Figure 3 is also true for a linear system. The scaling property mentioned
above still holds in conjunction with the superposition principle. Therefore,
if the inputs x and y are scaled by factors α and β, respectively, then the
sum of these scaled inputs will give the sum of the individual scaled outputs:
Figure 4 |
Time-Invariant Systems
A time-invariant system has the property that a certain input will
always give the same output, without regard to when the input was applied to the system.
Time-Invariant Systems
Figure 6: subfigure 6.1 shows an input at time |
In this figure,
x t x t
x t - t 0 x t t 0 x t x t
x t - t 0 x t t 0
x t - t 0 x t t 0 t 0 t 0
Whether a system is time-invariant or time-varying can be seen
in the differential equation (or difference equation) describing it.
Time-invariant systems are modeled with constant coefficient equations.
A constant coefficient differential (or difference) equation means that the parameters of the
system are not changing over time and an input now will give the same
result as the same input later.
Linear Time-Invariant (LTI) Systems
Certain systems are both linear and time-invariant, and are thus referred
to as LTI systems.
Linear Time-Invariant Systems
Figure 7: This is a combination of the two cases above. Since the input to
subfigure 7.2 is a scaled, time-shifted
version of the input in subfigure 7.1, so is the output. |
As LTI systems are a subset of linear systems, they obey the principle of
superposition. In the figure below, we see the effect of applying time-invariance
to the superposition definition in the linear
systems section above.
Figure 8 |
Superposition in Linear Time-Invariant Systems Figure 9: The principle of superposition applied to LTI systems |
LTI Systems in Series
If two or more LTI systems are in series with each other,
their order can be interchanged without affecting the overall output of the system.
Systems in series are also called cascaded systems.
Cascaded LTI Systems Subfigure 10.1  Subfigure 10.2 Figure 10: The order of cascaded LTI systems can be interchanged without
changing the overall effect. |
LTI Systems in Parallel
If two or more LTI systems are in parallel with one another,
an equivalent system is one that is defined as the sum of these individual systems.
Parallel LTI Systems
Figure 11: Parallel systems can be condensed into the sum of systems. |
Causality
A system is causal if it does not depend on future values of
the input to determine the output. This means that if the first input
to a system comes at time
t 0 t 0
Non-causal System Figure 12: In this non-causal system, an output is produced due to an input
that occurs later in time. |
A causal system is also characterized by an
impulse response h t h t t < 0 t 0
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This work is licensed by Thanos Antoulas and JP Slavinsky. See the Creative Commons License about permission to reuse this material.
Last edited by Erin Maloney on Nov 24, 2003 1:51 pm US/Central.
"My introduction to signal processing course at Rice University."