Biomedical Engineering Reference
In-Depth Information
11.6 LINEAR SYSTEMS
A system is a process, machine, or device that takes a signal as an
input
and manipulates
it to produce an
output
that is related to, but is distinctly different from,
its input.
Figure 11.15 shows a system block diagram.
Biological systems and organs are very often modeled as systems. The heart, for instance,
is a large-scale system that takes oxygen-deficient blood from the veins (the
input
) and
pumps it through the lungs. This produces a blood
via the main arteries of the heart
that is rich in oxygen content. Neurons in the brain can also be thought of as a simple
microscopic system that takes electrical nerve impulses from various neurons as the input
and sums these impulses to produce action potentials: the output. Linear systems are a spe-
cial class of systems with a unique set of properties that make them easy to analyze.
output
11.6.1 Linear System Properties
While biological systems are not per se linear, very often they can be approximated by a
linear system model. This is desired because it makes their analysis and the subsequent
interpretation more tractable.
All linear systems are characterized by the principles of
superposition
(or
additivity
)
and
. The superposition property states that the sum of two independent inputs
produces an output that is the sum or superposition of the outputs for each individual
input. The scaling property tells us that a change in the size of the input produces a com-
parable change at the output. Mathematically, if we know the outputs for two separate
inputs—that is,
scaling
Input
Output
x
(
t
)
y
(
t
)
f
{
x
(
t
)}
=
f
{.
(a)
Input
Output
x
(
t
)
y
(
t
)
x
(
t
)
*
h
(
t
)
h
(
t
)
=
(b)
Output
Input
X
(
ω)
Y
(
ω
)
= X
(
ω
)
H
(
ω
)
H
(
ω)
(c)
FIGURE 11.15
(a) Block diagram representation of a system. The input signal,
x
(
t
), passes through the system
transformation
). (b) Time domain representation of a linear system. The output of the
linear system is represented by the convolution of the input and impulse response. (c) Frequency domain repre-
sentation of a linear system. The output corresponds to the product of the input and the system transfer function.
f
fg
to produce an output,
y
(
t
