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6.003: Signals and Systems
ContinuousTime Systems
September 29, 2009
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View Full Document 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 [email protected] 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
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View Full Document 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 .
..”
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View Full Document 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.
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
?
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˙
r
1
(
t
) =
α
(
r
0
(
t
)
−
r
1
(
t
))
Flow rates
r
0
(
t
)
and
r
1
(
t
)
have dimensions
r
0
(
t
)
≈
r
1
(
t
)
≈
±
quantity of water
time
²
Quantity of water could be mass or volume, but must be consistent.
The derivative of
r
1
(
t
)
has dimensions
˙
r
1
(
t
)
≈
quantity of water
time
time
Therefore,
α
has dimensions
α
≈
±
1
time
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
?
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This note was uploaded on 08/24/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.
 Spring '11
 DennisM.Freeman

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