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Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
D.
Total Response of LTI Systems
The convolution integral provided a viable tool to
derive the output
()
yt
of a Linear TimeInvariant (LTI)
system, which has an impulse response
( )
ht
,
in
response to an input
( )
x t
:
() ()
( ) (
)
xt ht
x ht
d
τ
ττ
∞
−∞
=∗=
−
∫
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
However, it is well known that:
¾
A system can generate output in response to
internal energy stored within the system, even if the
(external) input to the system is zero
.
¾
Internal energies stored within a system are
sometimes referred to as
initial conditions
. These
initial conditions impact the response of the system
independent of the nature of the external input
x t
.
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
The convolution integral does not provide the
response of the system to the internal energy stored
within the system; and hence we need another tool to
identify the system response
when the input is zero
0
xt
=
, and when the system generates an output in
response to some internal energy. This type of
response is known as the
ZeroInput Response
.
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
Definition
Zeroinput Response (ZIR)
ZIR is the response of the system when no input is
applied, i.e.
() 0
x t
=
.
The response of the system is then the result of internal
energy storages, and it is independent of any external
inputs.
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Copyright © 20052009 – Hayder Radha
Definition
Zerostate Response (ZSR)
The response of the system when the system is in zero
state, meaning the absence of all internal energy storages;
i.e.
all initial conditions are zero
. In other words, the ZSR
is the response of the system to (external) input signals.
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
The response provided by the
convolution
integral
assumes that the system has zero energy, or as
commonly known,
the system is in
zero state
.
In other
words, convolution provides the
ZeroState Response
of an LTI system:
() ()
()(
)
x
th
t
xh
t
d
τ
ττ
∞
−∞
=∗=
−
∫
Zero State Response
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
Since an LTI system adheres to the superposition
principle, the total response of the system is the
summation of the zerostate response and the zeroinput
response:
Total Response
Input
=+
Total Response
Zero State Response
Zero
Response
Hence,
x t
h t
Input
=∗+
TotalResponse
Zero
Response
Introduction to Signal Processing
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Copyright © 20052009 – Hayder Radha
Therefore, a key question is how to derive the zero
input response of an LTI system. To answer this
question, we focus on a class of LTI systems that are
known as
linear differential systems
. These are LTI
systems that satisfy linear differential equations. To
motivate the study of linear differential systems, we
look at a familiar application from linear circuits.
Introduction to Signal Processing
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This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.
 Spring '08
 STAFF
 Signal Processing

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