6.003: Signals and Systems
Lecture 10
October 15, 2009
1
6.003: Signals and Systems
Convolution
October 15, 2009
Multiple Representations of CT and DT Systems
Verbal descriptions:
preserve the rationale.
Diﬀerence/diﬀerential equations:
mathematically compact.
y
[
n
] =
x
[
n
] +
z
0
y
[
n
−
1]
˙
y
(
t
) =
x
(
t
) +
s
0
y
(
t
)
Block diagrams:
illustrate signal ﬂow paths.
+
R
z
0
X
Y
+
A
s
0
X
Y
Operator representations:
analyze systems as polynomials.
Y
X
=
1
1
−
z
0
R
Y
X
=
A
1
−
s
0
A
Transforms:
representing diﬀ. equations with algebraic equations.
H
(
z
) =
z
z
−
z
0
H
(
s
) =
1
s
−
s
0
Convolution
Representing a system by a single signal.
Responses to arbitrary signals
Although we have focused on responses to simple signals (
δ
[
n
]
,δ
(
t
)
)
we are generally interested in responses to more complicated signals.
How do we compute responses to a more complicated input signals?
No problem for diﬀerence equations / block diagrams.
→
use stepbystep analysis.
Check Yourself
What is
y
[3]
?
+
+
R
R
X
Y
x
[
n
]
n
y
[
n
]
n
Superposition
Break input into additive parts and sum the responses to the parts.
n
x
[
n
]
y
[
n
]
n
=
+
+
+
+
=
n
−
1 0 1 2 3
4
5
n
n
n
n
−
1 0 1 2 3
4
5
n
Superposition works if the system is
linear
.
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Lecture 10
October 15, 2009
2
Linearity
A system is linear if its response to a weighted sum of inputs is equal
to the weighted sum of its responses to each of the inputs.
Given
system
x
1
[
n
]
y
1
[
n
]
and
system
x
2
[
n
]
y
2
[
n
]
the system is linear if
system
αx
1
[
n
] +
βx
2
[
n
]
αy
1
[
n
] +
βy
2
[
n
]
is true for all
α
and
β
.
Superposition
Break input into additive parts and sum the responses to the parts.
n
x
[
n
]
y
[
n
]
n
=
+
+
+
+
=
n
−
1 0 1 2 3
4
5
n
n
n
n
−
1 0 1 2 3
4
5
n
Reponses to parts are easy to compute if system is
timeinvariant
.
TimeInvariance
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 Spring '11
 DennisM.Freeman
 Digital Signal Processing, Superposition, LTI system theory, Impulse response, Hubble Space Telescope, Corrective Optics Space Telescope Axial Replacement

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