14. 14
Sum of two sequences
Product of two sequences
Multiplication of a sequence by a numberα
Delay (shift) of a sequence
Basic Sequence Operations
]
[
]
[ n
y
n
x
integer
:
]
[
]
[ 0
0 n
n
n
x
n
y
14
]
[
]
[ n
y
n
x
]
[n
x
16. 16
Basic sequences
Unit step sequence
0
0
0
1
]
[
n
n
n
u
7/6/2023
16
n
k
k
n
u
]
[
0
]
[
]
2
[
]
1
[
]
[
]
[
k
k
n
n
n
n
n
u
]
1
[
]
[
]
[
n
u
n
u
n
First backward difference
0, 0 ,
1, 0
0 0
1 0
since
n
k
when n
k
when n
k
k
k
17. 17
Basic Sequences
Exponential sequences
n
A
n
x
]
[
7/6/2023
17
A and α are real: x[n] is real
A is positive and 0<α<1, x[n] is positive and
decrease with increasing n
-1<α<0, x[n] alternate in sign, but decrease
in magnitude with increasing n
: x[n] grows in magnitude as n increases
1
18. 18
EX. 1 Combining Basic sequences
0
0
0
]
[
n
n
A
n
x
n
7/6/2023
18
If we want an exponential sequences that is
zero for n <0, then
]
[
]
[ n
u
A
n
x n
Cumbersome
simpler
19. 19
Periodic Sequences
A periodic sequence with integer period N
n
all
for
N
n
x
n
x ]
[
]
[
N
w
n
w
A
n
w
A 0
0
0 cos
cos
7/6/2023
19
integer
,
2
0 is
k
where
k
N
w
0
2 / , integer
N k w where k is
21. 21
Discrete-Time System
Discrete-Time System is a trasformation
or operator that maps input sequence
x[n] into a unique y[n]
y[n]=T{x[n]}, x[n], y[n]: discrete-time
signal
7/6/2023
21
T{‧}
x[n] y[n]
Discrete-Time System
22. 22
EX. 5 The Ideal Delay System
n
n
n
x
n
y d ],
[
]
[
7/6/2023
22
If is a positive integer: the delay of the
system. Shift the input sequence to the
right by samples to form the output .
d
n
d
n
If is a negative integer: the system will
shift the input to the left by samples,
corresponding to a time advance.
d
n
d
n
24. 24
Properties of Discrete-time systems
Linear Systems
7/6/2023
24
If
n
y1
T{‧}
n
x1
n
y2
n
x2
T{‧}
n
ay
n
ax T{‧}
n
bx
n
ax
n
x 2
1
3
n
by
n
ay
n
y 2
1
3
T{‧}
n
y
n
y 2
1
n
x
n
x 2
1 T{‧} additivity property
homogeneity or scaling
property
principle of superposition
and only If:
25. 25
Example Nonlinear Systems
7/6/2023
25
Method: find one counterexample
2
2
2
1
1
1
1
counterexample
2
]
[n
x
n
y
For
]
[
log10 n
x
n
y
1
10
log
1
log
10 10
10
counterexample
For
26. 26
Properties of Discrete-time systems
Time-Invariant Systems
Shift-Invariant Systems
7/6/2023
26
0
1
2 n
n
x
n
x
0
1
2 n
n
y
n
y
n
y1
T{‧}
n
x1
T{‧}
29. 29
Properties of Discrete-time systems
Causality
A system is causal if, for every choice
of , the output sequence value at
the index depends only on the
input sequence value for
0
n
0
n
n
7/6/2023
29
0
n
n
30. 30
Ex:9 Example for Causal System
Forward difference system is not Causal
Backward difference system is Causal
n
x
n
x
n
y
1
1
n
x
n
x
n
y
7/6/2023
30
31.
32. 32
Properties of Discrete-time system
Stability
Bounded-Input Bounded-Output (BIBO)
Stability: every bounded input sequence
produces a bounded output sequence.
n
all
for
B
n
x x ,
n
all
for
B
n
y y ,
7/6/2023
32
if
then
33. 33
Ex:10 Test for Stability or Instability
2
]
[n
x
n
y
n
all
for
B
n
x x ,
n
all
for
B
B
n
y x
y ,
2
7/6/2023
33
if
then
is stable
34. 34
Accumulator system
n
k
k
x
n
y
bounded
n
n
n
u
n
x :
0
1
0
0
Ex:11 Test for Stability or Instability
7/6/2023
34
bounded
not
n
n
n
k
x
k
x
n
y
n
k
n
k
:
0
1
0
0
Accumulator system is not stable
36. 36
LTI Systems: Convolution
k
k
n
k
x
n
x
k
k
k
n
h
n
x
k
n
h
k
x
k
n
T
k
x
k
n
k
x
T
n
y
7/6/2023
36
Representation of general sequence as a
linear combination of delayed impulse
principle of superposition
An Illustration Example（interpretation 1）
37.
38. 38
Properties of LTI Systems
Convolution is commutative
n
x
n
h
n
h
n
x
7/6/2023
38
h[n]
x[n] y[n]
x[n]
h[n] y[n]
n
h
n
x
n
h
n
x
n
h
n
h
n
x 2
1
2
1
Convolution is distributed over addition
39. 39
Cascade connection of systems
n
h
n
h
n
h 2
1
7/6/2023
39
x [n] h1[n] h2[n] y [n]
x [n] h2[n] h1[n] y [n]
x [n] h1[n] ]h2[n] y [n]
41. 41
Stability of LTI Systems
LTI system is stable if the impulse response
is absolutely summable .
k
k
h
S
41
Causality of LTI systems 0
,
0
n
n
h
HW: proof, Problem 2.62
