2. 2 10/17/2021
2
Chapter 2 Discrete-Time Signals and Systems
2.0 Introduction
2.1 Discrete-Time Signals: Sequences
2.2 Discrete-Time Systems
2.3 Linear Time-Invariant (LTI) Systems
2.4 Properties of LTI Systems
2.5 Linear Constant-Coefficient
Difference Equations
3. 3 10/17/2021
3
Chapter 2 Discrete-Time Signals and Systems
2.6 Frequency-Domain Representation
of Discrete-Time Signals and systems
4. 4 10/17/2021
4
2.0 Introduction
Signal: something conveys information
Signals are represented mathematically as
functions of one or more independent variables.
Continuous-time (analog) signals, discrete-
time signals, digital signals
Signal-processing systems are classified along the
same lines as signals: Continuous-time (analog)
systems, discrete-time systems, digital systems
Discrete-time signal
Sampling a continuous-time signal
Generated directly by some discrete-time process
5. 5 10/17/2021
5
2.1 Discrete-Time Signals: Sequences
Discrete-Time signals are represented as
In sampling,
1/T (reciprocal of T) : sampling frequency
integer
:
,
, n
n
n
x
x
period
sampling
T
nT
x
n
x a :
,
Cumbersome, so just use
x n
6. 6 10/17/2021
6
Figure 2.1 Graphical representation
of a discrete-time signal
Abscissa: continuous line
: is defined only at discrete instants
x n
7. 7 Figure 2.2
EXAMPLE Sampling the analog waveform
)
(
|
)
(
]
[ n T
x
t
x
n
x a
n T
t
a
8. 8 10/17/2021
8
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
]
[
]
[ n
y
n
x
]
[n
x
11. 11 10/17/2021
11
Basic sequences
Unit step sequence
0
0
0
1
]
[
n
n
n
u
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
12. 12 10/17/2021
12
Basic Sequences
Exponential sequences
n
A
n
x
]
[
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
13. 13 10/17/2021
13
EX. 2.1 Combining Basic sequences
0
0
0
]
[
n
n
A
n
x
n
If we want an exponential sequences that is
zero for n <0, then
]
[
]
[ n
u
A
n
x n
Cumbersome
simpler
15. 15 10/17/2021
15
Exponential Sequences
0
jw
e
j
e
A
A
n
w
A
j
n
w
A
e
A
e
e
A
A
n
x
n
n
n
w
j
n
n
jw
n
j
n
0
0 sin
cos
]
[ 0
0
1
1
1
Complex Exponential Sequences
Exponentially weighted sinusoids
Exponentially growing envelope
Exponentially decreasing envelope
0
[ ] jw n
x n Ae
is refered to
16. 16 10/17/2021
16
Frequency difference between
continuous-time and discrete-time
complex exponentials or sinusoids
n
jw
n
j
n
jw
n
w
j
Ae
e
Ae
Ae
n
x 0
0
0 2
2
]
[
: frequency of the complex sinusoid
or complex exponential
: phase
0
w
0 0
[ ] cos 2 cos
x n A w r n A w n
17. 17 10/17/2021
17
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
integer
,
2
0 is
k
where
k
N
w
0
2 / , integer
N k w where k is
18. 18 10/17/2021
18
EX. 2.2 Examples of Periodic Sequences
Suppose it is periodic sequence with period N
]
[
]
[ 1
1 N
n
x
n
x
4
/
cos
]
[
1 n
n
x
4
/
cos
4
/
cos N
n
n
integer
:
,
4
/
4
/
2
4
/ k
N
n
k
n
0
1, 8 2 /
k N w
2 / ( / 4) 8
N k k
19. 19 10/17/2021
19
Suppose it is periodic sequence with period N
]
[
]
[ 1
1 N
n
x
n
x
8
/
3
cos
8
/
3
cos N
n
n
integer
:
,
8
/
3
8
/
3
2
8
/
3 k
N
n
k
n
16
,
3
N
k
EX. 2.2 Examples of Periodic Sequences
8
/
3
cos
]
[
1 n
n
x
8
3
8
2
0
2 / 2 / (3 / 8)
N k w k
0 0
2 3/ 2 / ( continuous signal)
N w w for
20. 20 10/17/2021
20
EX. 2.2 Non-Periodic Sequences
Suppose it is periodic sequence with period N
]
[
]
[ 2
2 N
n
x
n
x
n
n
x cos
]
[
2
)
cos(
cos N
n
n
2 , :integer,
integer
for n k n N k
there is no N
21. 21 10/17/2021
21
High and Low Frequencies in Discrete-time signal
0
[ ] cos( )
x n A w n
(a) w0 = 0 or 2
(b) w0 = /8 or 15/8
(c) w0 = /4 or 7/4
(d) w0 =
22. 22 10/17/2021
22
2.2 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
T{‧}
x[n] y[n]
Discrete-Time System
23. 23 10/17/2021
23
EX. 2.3 The Ideal Delay System
n
n
n
x
n
y d ],
[
]
[
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 10/17/2021
24
x[m
]
m
n
n-5
dummy index
m
EX. 2.4 Moving Average
2
1
1 2
1 1 2
1 2
1
1
1
1 ... 1 ...
1
M
k M
y n x n k
M M
x n M x n M x n x n x n M
M M
for n=7, M1=0, M2=5
25. 25
Effect of a moving average filter. (Sample values are connected
by straight lines to enable easier viewing of stock exchange
trends)
26. 26 10/17/2021
26
Properties of Discrete-time systems
2.2.1 Memoryless (memory) system
Memoryless systems:
the output y[n] at every value of n depends
only on the input x[n] at the same value of n
2
]
[n
x
n
y
27. 27 10/17/2021
27
Properties of Discrete-time systems
2.2.2 Linear Systems
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:
28. 28 10/17/2021
28
Example of Linear System
Ex. 2.6 Accumulator system
n
k
k
x
n
y
n
by
n
ay
k
x
b
k
x
a
k
bx
k
ax
k
x
n
y
n
k
n
k
n
k
n
k
2
1
2
1
2
1
3
3
n
k
k
x
n
y 1
1
n
k
k
x
n
y 2
2
n
x
and
n
x 2
1
for arbitrary
n
bx
n
ax
n
x 2
1
3
when
29. 29 10/17/2021
29
Example 2.7 Nonlinear Systems
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
30. 30 10/17/2021
30
Properties of Discrete-time systems
2.2.3 Time-Invariant Systems
Shift-Invariant Systems
0
1
2 n
n
x
n
x
0
1
2 n
n
y
n
y
n
y1
T{‧}
n
x1
T{‧}
31. 31 10/17/2021
31
Example of Time-Invariant System
Ex. 2.8 Accumulator system
n
k
k
x
n
y
0
1 ]
[ n
n
x
n
x
0
1
0
1
1
0
1
n
n
y
k
x
n
k
x
k
x
n
y
n
n
k
n
k
n
k
32. 32 10/17/2021
32
Properties of Discrete-time systems
2.2.4 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
0
n
n
33. 33 10/17/2021
33
Ex. 2.10 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
34. 34 10/17/2021
34
Properties of Discrete-time systems
2.2.5 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 ,
if
then
35. 35 10/17/2021
35
Ex. 2.11 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
if
then
is stable
36. 36 10/17/2021
36
Accumulator system
n
k
k
x
n
y
bounded
n
n
n
u
n
x :
0
1
0
0
bounded
not
n
n
n
k
x
k
x
n
y
n
k
n
k
:
0
1
0
0
Ex. 2.11 Test for Stability or Instability
Accumulator system is not stable
37. 37 10/17/2021
37
Properties of Discrete-time systems (Repeat)
2.2.2 Linear Systems
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:
38. 38 10/17/2021
38
2.3 Linear Time-Invariant (LTI)
Systems
Impulse response
0
n
n
n
h
n
0
n
n
h
T{‧}
T{‧}
39. 39 10/17/2021
39
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
Representation of general sequence as a
linear combination of delayed impulse
principle of superposition
An Illustration Example(interpretation 1)
41. 41 10/17/2021
41
Computation of the Convolution
reflecting h[k] about the origion to obtain h[-k]
Shifting the origin of the reflected sequence to
k=n
(interpretation 2)
k
k
n
h
k
x
n
y
n
k
h
k
n
h
k
h
k
h
43. 43
Convolution can be realized by
–Reflecting h[k] about the origin to obtain h[-k].
–Shifting the origin of the reflected sequences to k=n.
–Computing the weighted moving average of x[k] by
using the weights given by h[n-k].
44. 44 10/17/2021
44
Ex. 2.13 Analytical Evaluation
of the Convolution
otherwise
N
n
N
n
u
n
u
n
h
0
1
0
1
For system with impulse response
h(k)
0
n
u
a
n
x n
input
Find the output at index n
45. 45
45
0
0
y n n
otherwise
N
n
n
h
0
1
0
1
n
u
a
n
x n
h(k)
0
0
h(n-k) x(k)
h(-k)
0
46. 46 10/17/2021
46
1 0
0, 0 1
n n N n N
a
a
a
n
k
h
k
x
n
y
n
n
k
k
n
k
1
1 1
0
0
h(-k)
0
h(k)
0
47. 47 10/17/2021
47
a
a
a
a
a
a
a
n
k
h
k
x
n
y
N
N
n
n
N
n
n
N
n
k
k
n
N
n
k
1
1
1
1
1
1
1
1
h(-k)
0
h(k)
0
1 0 1
n N n N
49. 49 10/17/2021
49
2.4 Properties of LTI Systems
Convolution is commutative
n
x
n
h
n
h
n
x
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
50. 50 10/17/2021
50
Cascade connection of systems
n
h
n
h
n
h 2
1
x [n] h1[n] h2[n] y [n]
x [n] h2[n] h1[n] y [n]
x [n] h1[n] ]h2[n] y [n]
52. 52
52
Stability of LTI Systems
LTI system is stable if the impulse response
is absolutely summable .
k
k
h
S
k
k
k
n
x
k
h
k
n
x
k
h
n
y
x
B
n
x
x
k
y n B h k
Causality of LTI systems 0
,
0
n
n
h
HW: proof, Problem 2.62
53. 53 10/17/2021
53
Impulse response of LTI systems
Impulse response of Ideal Delay systems
,
d d
h n n n n a positive fixed integer
Impulse response of Accumulator
n
u
n
n
k
n
h
n
k 0
,
0
0
,
1
54. 54 10/17/2021
54
Impulse response of Moving
Average systems
otherwise
,
M
n
M
,
M
M
k
n
M
M
n
h
M
M
k
0
1
1
1
1
2
1
2
1
2
1
2
1
55. 55
Impulse response of Forward Difference
n
n
n
h
1
1
n
n
n
h
Impulse response of Backward Difference
56. 56
Finite-duration impulse
response (FIR) systems
The impulse response of the system has
only a finite number of nonzero samples.
The FIR systems always are stable.
otherwise
,
M
n
M
,
M
M
k
n
M
M
n
h
M
M
k
0
1
1
1
1
2
1
2
1
2
1
2
1
n
S h n
such as:
57. 57
Infinite-duration impulse
response (IIR)
The impulse response of the system is
infinite in duration.
n
u
n
n
k
n
h
n
k 0
,
0
0
,
1
n
u
a
n
h n
n
S h n
Stable IIR System:
1
a
58. 58
Equivalent systems
1 1
h n n n n
1 1 1
n n n n n
59. 59
Inverse system
n
n
u
n
u
n
n
n
u
n
h
1
1
n
n
h
n
h
n
h
n
h i
i
60. 60
2.5 Linear Constant-Coefficient
Difference Equations
M
m
m
N
k
k m
n
x
b
k
n
y
a
0
0
An important subclass of linear time-
invariant systems consist of those
system for which the input x[n] and
output y[n] satisfy an Nth-order linear
constant-coefficient difference equation.
61. 61
Ex. 2.14 Difference Equation
Representation of the Accumulator
,
n
k
y n x k
1
1
k
n
y n x k
1
1
n
y
n
x
k
x
n
x
n
y
n
k
n
x
n
y
n
y
1
62. 62
Block diagram of a recursive
difference equation representing an
accumulator
1
y n y n x n
63. 63
Difference Equation
Representation of the System
An unlimited number of distinct
difference equations can be used to
represent a given linear time-invariant
input-output relation.
64. 64
Solving the difference equation
Without additional constraints or
information, a linear constant-
coefficient difference equation for
discrete-time systems does not provide
a unique specification of the output for
a given input.
M
m
m
N
k
k m
n
x
b
k
n
y
a
0
0
65. 65
Solving the difference equation
Output:
n
y
n
y
n
y h
p
Particular solution: one output sequence
for the given input
n
yp
n
xp
Homogenous solution: solution for
the homogenous equation( ):
n
yh
0
0
N
k
h
k k
n
y
a
N
m
n
m
m
h z
A
n
y
1
where is the roots of
m
z 0
0
N
k
k
k z
a
0
x n
M
m
m
N
k
k m
n
x
b
k
n
y
a
0
0
66. 66
Example 2.16 Recursive Computation of
Difference Equation
1 , , 1
y n ay n x n x n K n y c
K
ac
y
0
aK
c
a
K
ac
a
ay
y
2
0
0
1
K
a
c
a
aK
c
a
a
ay
y 2
3
2
0
1
2
K
a
c
a
K
a
c
a
a
ay
y 3
4
2
3
0
2
3
1
0
n n
y n a c a fo
K r n
67. 67
Example 2.16 Recursive Computation of
Difference Equation
n
x
n
y
a
n
y
1
1
c
a
x
y
a
y 1
1
1
1
2
c
a
c
a
a
x
y
a
y 2
1
1
1
2
2
3
1
1
n
y n a c for n
c
a
c
a
a
x
y
a
y 3
2
1
1
3
3
4
1
n n
y n a c Ka for l
n l n
u a
1
for n
c
y
n
K
n
x
n
x
n
ay
n
y
1
1
68. 68
Periodic Frequency Response
The frequency response of discrete-time
LTI systems is always a periodic function of
the frequency variable with period
w 2
2 2
j w j w n
n
H e h n e
2 2
j w jwn j n jwn
e e e e
2
j w jw
H e H e
2
,
j w r jw
H e H e for r an integer
Signal Processing - Dr. Arif Wahla
10/17/2021
69. 69
Periodic Frequency Response
The “low frequencies” are frequencies
close to zero
The “high frequencies” are frequencies
close to
More generally, modify the frequency with
, r is integer.
2 r
jw
e
H
or w
0 2
w
We need only specify over
Signal Processing - Dr. Arif Wahla
10/17/2021
74. 74
Example 2.20 Frequency Response of
the Moving-Average System
otherwise
,
M
n
M
,
M
M
n
h
0
1
1
2
1
2
1
1
1 2
1 2
2
1
1
2
1
1
1
1 1
jw
M
n M
jw M
jwn
H e
M M
jwM
jw
M M
e
e e
e
75. 75
Impulse response and
Frequency response
The frequency response of a LTI
system is the Fourier transform of the
impulse response.
jw jwn
n
H e h n e
dw
e
e
H
n
h jwn
jw
2
1