1
Lecture 2: Discrete-Time Systems and
z
-Transform
•
Discrete-time signals v.s. continuous-time signals
•
Discrete-time systems v.s. continuous-time systems
•
z
-Transform
•
Inverse
z
-Transform
•
z
-Transform for solving linear difference equations
Continuous-Time Signals
•
A signal that changes continuously in time:
•
Defined at all times
t
(no gap)
•
Signal can take arbitrary values
• Example: temperature, position, velocity,…
e
(
t
)
;
¡1
< t <
1
;
or
t
¸
0
e
(
t
)
t

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2
•
A signal (sequence, or series) whose values are
defined only at discrete times:
•
Signal defined only at integer times
k
•
Signal can still take arbitrary values
Discrete-Time Signals
4
¢¢¢
¡
1
k
0
1
3
¢¢¢
e
(
k
)
2
e
(
k
)
;
k
=
:::;
¡
1
;
0
;
1
;:::
or
k
= 0
;
1
;
2
;:::
Origin of Discrete-Time Signals
•
Some come naturally
–
Population of a species in different generations
–
Annual growth percentage of GDP
–
Results of a numerical algorithm in different rounds of iteration
•
Some arise by sampling continuous-time signals at regular
time intervals, say, every
T
seconds
e
(
t
)
t
e
(
kT
)
0
T
2
T
3
T
4
T
¡
T
¢¢¢
¢¢¢
e
(
t
)
;
¡1
< t <
1
)
e
(
kT
)
;
k
=
:::;
¡
1
;
0
;
1
;:::
)
e
(
k
)
;
k
=
¢ ¢ ¢
;
¡
1
;
0
;
1
;:::;
(if
T
is known from the context)