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Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
ECSE 304 Signals and Systems II
Lecture 7: The zTransform and Linear
Time Invariant Systems
Reading: O and W, Section 10.7 and 10.8
Richard Rose
McGill University
Dept. of Electrical and Computer Engineering
2
Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
• Analysis of DLTI Systems using zTransforms
– Causality
– Stability
• LTI difference systems and rational transfer
functions
• Relating System Behavior to Transfer Functions
• Block Diagram Representations
Outline
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Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
• Match the impulse
responses with the
correct polezero
plots
• See “Lecture 7
Example Problems”
for Solutions
a ___
b ___
c ___
d ___
e ___
a)
b)
c)
d)
e)
Review: Inverse zTransforms
(j)
(k)
(l)
(m)
(n)
Time
sequence
indices
used in
online
notes
4
Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
• For a DLTI system with impulse response
, the
convolution property of zTransforms gives:
•
is the
system function
or
transfer function
• Properties of DLTI systems
–
causality
and
stability
• … are associated with the characteristics of
–
Poles
,
zeroes
, and
ROC
Analysis of DLTI Systems Using zTransforms
[]
x n
y n
hn
()
Hz
X z
Yz
HzXz ROC R
R
xh
() ()
,
=
⊃∩
H z
H z
5
Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
• Establish necessary and sufficient
conditions on
to
guarantee the
causality
of a DLTI system
• For a system to be causal,
–
must be rightsided and
for
• Therefore:
– The ROC of
is the exterior of a circle in the z plane
– The ROC of
must include
Why does causality imply that
?
For
, as
,
for
This is not true for
•
A DLTI system is causal iff the ROC of its system
function is the exterior of a circle including infinity
Causality of DLTI Systems
()
H z
hn
[]
=
0
n
<
0
H z
H z
z
=∞
z
→∞
0
n
n
H
zh
n
z
∞
−
=
= ∑
0
n
z
−
→
0
n
>
0
n
<
h
zR
O
C
=∞⊂
6
Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
Region of Convergence for the zTransform
Sequence Type
Region of Convergence
Right Sided
Sequence
Left Sided
Sequence
Two Sided
Sequence
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Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
•
A DLTI System represented by
rational transfer function,
is causal iff:
a)
The ROC is the exterior of a circle with radius equal to the
magnitude of the outermost pole, and
b)
With
expressed as the ratio of polynomials in z, the
order of the numerator is less than or equal to the order of
the denominator
Causality of DLTI Systems
()
H z
8
Lecture 7 – The zTransform and LTI Systems
ECSE304 Signals and Systems II
•
Example:
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This note was uploaded on 08/17/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.
 Spring '11
 ChenandBacsy

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