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6.003: Signals and Systems
Lecture 6
September 29, 2009
1
6.003: Signals and Systems
ContinuousTime Systems
September 29, 2009
Midterm Examination #1
Wednesday, October 7, 7:309:30pm, Walker Memorial.
No recitations on the day of the exam.
Coverage:
DT Signals and Systems
Lectures 1–5
Homeworks 1–4
Homework 4 will include practice problems for midterm 1.
However, it will not collected or graded. Solutions will be posted.
Closed book: 1 page of notes (
8
1
2
×
11
inches; front and back).
Designed as 1hour exam; two hours to complete.
Review sessions during open oﬃce hours.
Conﬂict? Contact freeman@mit.edu before Friday, October 2, 5pm.
Multiple Representations of DiscreteTime Systems
We have developed several useful representations for DT systems.
Verbal descriptions:
preserve the rationale.
“To reduce the number of bits needed to store a sequence of
large numbers that are nearly equal, record the ﬁrst number,
and then record successive diﬀerences.”
Diﬀerence equations:
mathematically compact.
y
[
n
] =
x
[
n
]
−
x
[
n
−
1]
Block diagrams:
illustrate signal ﬂow paths.
−
1
Delay
+
x
[
n
]
y
[
n
]
Operator representations:
analyze systems as polynomials.
Y
= (1
−R
)
X
Multiple Representations of ContinuousTime Systems
Today we start to develop similarly useful representations for CT.
Representing ContinuousTime Systems
As with DT systems, we begin with verbal descriptions.
Example: Leaky Tank
r
0
(
t
)
r
1
(
t
)
h
1
(
t
)
Verbal descriptions
of continuoustime systems typically specify
relations among signals (e.g., inputs and outputs) and their rates of
change.
“Water ﬂows into a tank at rate
r
0
(
t
)
and ﬂows out at a rate
proportional to the depth of water in the tank. Determine .
..”
Representing ContinuousTime Systems
Translate the verbal description into
diﬀerential equations.
r
0
(
t
)
r
1
(
t
)
h
1
(
t
)
Assume
r
1
(
t
)
∝
hydraulic pressure at the bottom of the tank:
r
1
(
t
)
∝
ρgh
1
(
t
)
Height
h
1
(
t
)
will grow in proportion to
r
0
(
t
)
−
r
1
(
t
)
:
˙
h
1
(
t
)
∝
r
0
(
t
)
−
r
1
(
t
)
Combining ﬁrst equation with second:
˙
r
1
(
t
)
∝
˙
h
1
(
t
)
∝
r
0
(
t
)
−
r
1
(
t
)
→
˙
r
1
(
t
) =
α
(
r
0
(
t
)
−
r
1
(
t
)
)
Derivatives play key role in CT, analogous to time shifts in DT.
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Lecture 6
September 29, 2009
2
Check Yourself
˙
r
1
(
t
) =
α
(
r
0
(
t
)
−
r
1
(
t
)
)
r
0
(
t
)
r
1
(
t
)
h
1
(
t
)
What are the dimensions of
α
?
Charactering Responses of CT Systems
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 Spring '11
 DennisM.Freeman

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