6.003: Signals and Systems
Lecture 8
October 6, 2009
1
6.003: Signals and Systems
Operator Representations for ContinuousTime Systems
October 6, 2009
Midterm Examination #1
Tomorrow, October 7, 7:309:30pm, Walker Memorial.
No recitations tomorrow.
Coverage:
DT Signals and Systems
Lectures 1–5
Homeworks 1–4
Homework 4 includes practice problems for midterm 1.
It will not collected or graded. Solutions are posted.
Closed book: 1 page of notes (
8
1
2
×
11
inches; front and back).
Designed as 1hour exam; two hours to complete.
Analyzing CT Systems
We have used differential equations to represent CT systems.
r
0
(
t
)
r
1
(
t
)
h
1
(
t
)
τ
˙
r
1
(
t
) =
r
0
(
t
)
−
r
1
(
t
)
Methods to solve differential equations:
•
homogeneous and particular equations
•
singularity matching
•
Laplace transform
Today: new methods based on
block diagrams
and
operators
.
Block Diagrams
Block diagrams illustrate signal flow paths.
DT:
adders, scalers, and delays – represent systems described by
linear difference equations with constant coefficents.
+
Delay
p
x
[
n
]
y
[
n
]
CT:
adders, scalers, and integrators – represent systems described
by a linear differential equations with constant coefficients.
+
t
−∞
(
·
)
dt
p
x
(
t
)
y
(
t
)
Operator Representation
Block diagrams are concisely represented with
operators
.
We will define the
A
operator
for functional analysis of CT systems.
Applying
A
to a CT signal generates a new signal that is equal to
the integral of the first signal at all points in time.
Y
=
A
X
is equivalent to
y
(
t
) =
t
−∞
x
(
τ
)
dτ
for
all
time
t
.
Evaluating Operator Expressions
As with
R
,
A
expressions can be manipulated as polynomials.
+
+
A
A
X
Y
W
w
(
t
) =
x
(
t
) +
t
−∞
x
(
τ
)
dτ
y
(
t
) =
w
(
t
) +
t
−∞
w
(
τ
)
dτ
y
(
t
) =
x
(
t
) +
t
−∞
x
(
τ
)
dτ
+
t
−∞
x
(
τ
)
dτ
+
t
−∞
τ
2
−∞
x
(
τ
1
)
dτ
1
dτ
2
W
= (1 +
A
)
X
Y
= (1 +
A
)
W
= (1 +
A
)(1 +
A
)
X
= (1 + 2
A
+
A
2
)
X
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6.003: Signals and Systems
Lecture 8
October 6, 2009
2
Evaluating Operator Expressions
Expressions in
A
can be manipulated using rules for polynomials.
•
Commutativity:
A
(1
− A
)
X
= (1
− A
)
A
X
•
Distributivity:
A
(1
− A
)
X
= (
A − A
2
)
X
•
Associativity:
(1
− A
)
A
(2
− A
)
X
= (1
− A
)
A
(2
− A
)
X
ContinuousTime Feedback
What is the response of a CT system with feedback?
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 Spring '11
 DennisM.Freeman
 Trigraph, Complex number, Impulse response, CT Systems

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