2.160 System Identification, Estimation, and Learning
Lecture Notes No.
9
March 8, 2006
Part 2 Representation and Learning
We now move on to the second part of the course, Representation and Learning. You will
learn various forms of system representation, including linear and nonlinear systems. We
will use “Prediction” form as a generic representation of diverse systems. You will also
learn data compression and learning, which are closely related to system representation
and prediction.
5 Prediction Modeling of Linear Systems
5.1 Impulse Response and Transfer Operator (Review)
Linear TimeInvariant system
u(t)
)
1
(
g
y(t)
)
2
(
g
)
(
g(
τ
)
y(t)
y(t)
t
g
Impulse response:
a complete characterization
t
t
t
Continuous time
impulse response
Discrete time impulse response
Any linear timeinvariant system can be characterized completely with Impulse Response:
(
( )
t
g
) . For an arbitrary input,
{
s
s
u
≤
t
}
, the output
y
(
t
) is given by the convolution of the
input and the impulse response given by
∞
Continuous Time Convolution
t
y
)
=
g
(
τ
)
t
u
−
τ
τ
(1)
(
(
)
d
∫
0
For discretetime systems, an impulse response is given by an infinite time series:
1
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g
(
),
0
g
( ),
1
g
(
),
2
"
, or
{
g
(
τ
τ
=
0
…
∞
}
)
Output
{
s
s
y
≤
t
}
{
s
s
u
≤
t
}
( )
( )
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 Spring '06
 HarryAsada
 Signal Processing, LTI system theory, Impulse response

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