1
LINEAR CIRCUITS AND SIGNALS
Some Basic System Theoretic Concepts
Department of Electrical and Computer Engineering
University of Miami
I. P
RELIMINARIES
Recall that
u
(
t
)
↔
U
(
ω
)
→
h
(
t
)
↔
H
(
ω
)
→
y
(
t
) = (
h
*
u
)(
t
)
↔
Y
(
ω
) =
H
(
ω
)
U
(
ω
)
.
(1)
The zerostate response
y
(
t
)
of an LTI system with frequency response
H
(
ω
)
to an arbitrary
input
u
(
t
)
is given by the convolution formula
y
(
t
) = (
h
*
u
)(
t
)
. Here
h
(
t
)
↔
H
(
ω
)
.
As we demonstrate later, this convolution formula turns out to be more general than the FT
formula
Y
(
ω
) =
H
(
ω
)
U
(
ω
)
. Indeed, while the FT
H
(
ω
)
of
h
(
t
)
may not exist, the convolution
formula is always valid for a LTI system. We will also discuss the notions of
causality
and
stability
and how they are related to the impulse response of a LTI system.
II. I
MPULSE
R
ESPONSE AND THE
Z
ERO
S
TATE
R
ESPONSE
A. Measuring the Impulse Response
Two methods of determining the impulse response of a system in a laboratory setting:
(a) Note that
(
δ
*
h
)(
t
) =
h
(
t
)
⇐⇒
lim
→
0
(
p
*
h
)(
t
) =
h
(
t
)
,
where
p
(
t
) =
1
rect
t
.
(2)
So, application of
p
with decreasing
should result in an output that converges to
h
(
t
)
.
If convergence is not forthcoming (usually because of the presence of an impulsive
signal in the impulse response), then the next method can be used instead.
(b) Take the unit step response and use the following property applicable to LTI systems:
1(
t
)
→
LTI
→
(
h
*
1)(
t
) =
⇒
d
dt
1(
t
)
→
LTI
→
(
h
*
δ
)(
t
) =
h
(
t
)
.
(3)
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2
Example
Suppose the unit step response of a LTI system is
y
(
t
) =
e

t
1(
t
)
.
To determine the system’s impulse response, let us use approach (b) described above.
h
(
t
) =
y
(1)
(
t
) =
e

t
δ
(
t
)

e

t
1(
t
) =
δ
(
t
)

e

t
1(
t
)
.
Can we use approach (a) instead? Note that
(
p
*
h
)(
t
) = [
p
*
(
δ
(
t
)

e

t
1(
t
))](
t
) =
p
(
t
)

p
*
e

t
1(
t
)
.
So
lim
→
0
(
p
*
h
)(
t
) = lim
→
0
p
(
t
)

(
δ
(
t
)
*
e

t
1(
t
)) = lim
→
0
p
(
t
)

e

t
1(
t
)
.
In a laboratory setting, it will be impossible to determine the area corresponding to
p
(
t
)
as
→
0
, and therefore approach (a) will not be of much use in this scenario.
B. Response of an LTI System to Arbitrary Inputs
We now demonstrate the convolution formula related to the zerostate response of an LTI
system without recourse to frequency domain notions.
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 Spring '10
 KAMALPREMERATNE
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, LTI System

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