6.003: Signals and Systems
Lecture 3
September 17, 2009
1
6.003: Signals and Systems
Feedback, Poles, and Fundamental Modes
September 17, 2009
Last Time: Multiple Representations of 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
Last Time: Feedback, Cyclic Signal Paths, and Modes
Systems with signals that depend on previous values of the same
signal are said to have
feedback
.
Example: The accumulator system has feedback.
Delay
+
X
Y
By contrast, the difference machine does not have feedback.
−
1
Delay
+
X
Y
Last Time: Feedback, Cyclic Signal Paths, and Modes
The
effect
of
feedback
can
be
visualized
by
tracing
each
cycle
through the cyclic signal paths.
Delay
+
p
0
X
Y
−
1 0
1
2
3
4
n
x
[
n
] =
δ
[
n
]
−
1 0
1
2
3
4
n
y
[
n
]
Each cycle creates another sample in the output.
The response will persist even though the input is transient.
Geometric Growth: Poles
These system responses can be characterized by a single number —
the
pole
— which is the base of the geometric sequence.
Delay
+
p
0
X
Y
y
[
n
] =
p
n
0
,
if
n >
= 0
;
0
,
otherwise.
−
1 0
1
2
3
4
n
y
[
n
]
−
1 0
1
2
3
4
n
y
[
n
]
−
1 0
1
2
3
4
n
y
[
n
]
p
0
= 0
.
5
p
0
= 1
p
0
= 1
.
2
Check Yourself
What value of
p
0
represents the signal below?
y
[
n
]
n
1.
p
0
= 0
.
8
2.
p
0
=
−
0
.
8
3.
p
0
= 0
.
64
interspersed with
p
0
=
−
0
.
64
4.
none of the above
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6.003: Signals and Systems
Lecture 3
September 17, 2009
2
Geometric Growth
The value of
p
0
determines the rate of growth.
y
[
n
]
y
[
n
]
y
[
n
]
y
[
n
]
−
1
0
1
z
p
0
<
−
1
:
magnitude diverges, alternating sign
−
1
< p
0
<
0
:
magnitude converges, alternating sign
0
< p
0
<
1
:
magnitude converges monotonically
p
0
>
1
:
magnitude diverges monotonically
SecondOrder Systems
The unitsample responses of more complicated cyclic systems are
more complicated.
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 Spring '11
 DennisM.Freeman
 Complex number, Geometric progression, Fibonacci number, fundamental modes

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