Convolution and Linear Systems
Convolution and Linear Systems
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Convolution and Linear Systems
Introduction
Analyzing Systems
Goal: analyze a device that turns one signal into another.
Notation:
f
(
t
)
→
g
(
t
)
f
(
t
)
is the input signal
g
(
t
)
is the output signal
(We’ll write this for 1dimensional signals, but all of the theory
applies to higherdimensional signals as well.)
Convolution and Linear Systems
Introduction
Linearity (Revisited)
An operation
T
is
linear
if and only if
T
(
ax
+
by
) =
aT
(
x
) +
bT
(
y
)
Implications:
T
(
ax
) =
aT
(
x
)
T
(
x
+
y
) =
T
(
x
) +
T
(
y
)
Convolution and Linear Systems
Introduction
Multiplying an Input to a Linear Operation
If the operation is linear:
f
(
t
)
→
g
(
t
)
a f
(
t
)
→
a g
(
t
)
Applying a linear operation to an signal multiplied by a constant
is the same as applying the operation and then multiplying by
that constant.
This is called the
scaling property
of linear operations.
Convolution and Linear Systems
Introduction
Adding Inputs to a Linear Transformation
If the operation is linear:
f
1
(
t
)
→
g
1
(
t
)
f
2
(
t
)
→
g
2
(
t
)
f
1
(
t
) +
f
2
(
t
)
→
g
1
(
t
) +
g
2
(
t
)
Applying a linear operation to the sum of two signals is the
same as applying it to each separately and adding the results.
This is called the
superposition property
of linear operations.
Convolution and Linear Systems
Introduction
Shift Invariance
Shift invariance
means an operation is invariant to translation.
Implication:
If you shift the input, you get the same (shifted) output.
In other words,
f
(
t
)
→
g
(
t
)
implies
f
(
t
+
T
)
→
g
(
t
+
T
)
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Convolution and Linear Systems
Introduction
Systems
Linearity and shift invariance are nice properties for a
signalprocessing operation to have:
input devices
output devices
processing
A transformation that is both
linear
and
shift invariant
is called a
system
.
Convolution and Linear Systems
Introduction
The Little White Lie
No physical device is really a linear system:
Linearity is limited by saturation.
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 Fall '09
 Image processing, Digital Signal Processing, Linear Systems, LTI system theory, Impulse response, Dirac delta function, Convolution and Linear Systems

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