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IV. Linear systems analysis
Linear systems analysis refers to a set of mathematical techniques that can be used to
analyze and describe input/output systems that satisfy certain linearity assumptions.
The
inputs and outputs of linear systems are represented by functions.
These functions can be
of any number of variables, but for present purposes there will only be one variable, time
(t)
or space
(x)
, or two variables, spacespace
(x,y)
, or spacetime
(x,t)
.
For example the
input functions might represent the variation in light level on a single receptor over time,
the spatial distribution of light entering the eye, or the spatial distribution of neural
activity transmitted from the eye to the brain along the optic nerve.
The output functions
might represent the electrical response of the receptor over time, the spatial distribution
of light falling on the retina, or the spatial distribution of neural activity of a group of
cells receiving inputs from the optic nerve.
The letter
L
will be used to represent a linear system;
i(t)
,
i(x,y)
, etc. will represent
input functions; and
o(t)
,
o(x,y)
, etc. will represent output functions.
Thus, we can write
in the onedimensional case,
o
(
t
)
=
L i
(
t
)
{
}
(4.1)
or in the twodimensional case,
o
(
x
,
y
)
=
L i
(
x
,
y
)
{
}
(4.2)
The curly brackets are used as a reminder that a linear system takes a
whole function as
input and gives a whole function as output.
For example, the optical system of the eye
can be regarded as a linear system that transforms the twodimensional light distribution
in an object plane,
i(x,y)
, into a twodimensional light distribution in the image plane,
o(x,
y)
.
Figure 4.1 illustrates graphically a onedimensional input function, a linear system,
and a one dimensional output function.
To make the figure concrete the reader might
imagine that the left side represents a temporal signal (rectangular pulse) turned on
shortly after time 0, while the right side represents the response of the linear system to
this signal.
A. Constraints defining linear systems
In order for a onedimensional system to be linear, it must satisfy the following
constraint,
L ai
1
(
t
)
+
bi
2
(
t
)
{
}
=
aL i
1
(
t
)
{
}
+
bL i
2
(
t
)
{
}
(4.3)
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where
i
1
(t)
and
i
2
(t)
are two arbitrary input functions and
a
and
b
are arbitrary constants
(scalars).
In words, the constraint is that the response
of a linear system to the sum of two input function equals the sum of the responses to
each input taken alone, and also scaling an input function by some amount scales the
output function by exactly the same amount. In Figure 4.1B, another pulse has been
added to the one in Figure 4.1A.
If the system is linear then the response must be the
sum of the individual responses (as shown). In Figure 4.1C, the input pulse in Figure
4.1A has been doubled in amplitude.
If the system is linear the response should double in
amplitude (as shown).
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 Fall '07
 GEISLER
 LTI system theory

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