This document provides an overview of discrete-time signals and systems in signal processing. It begins with an introduction to discrete-time signals as sequences and sampling of continuous-time signals. Common basic discrete-time sequences like impulses, steps, and exponentials are described. Properties of linear, time-invariant discrete-time systems are introduced. The frequency domain representation of discrete-time signals using complex exponentials is also covered. Examples of periodic and non-periodic discrete-time sequences are provided to illustrate key concepts.

1. Signal processing
Chapter 2 Discrete-Time Signals and
Systems
1

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
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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
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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



9. 9 10/17/2021
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Basic sequences
Unit sample sequence
(discrete-time impulse,
impulse)
 






0
1
0
0
n
n
n


10. 10 10/17/2021
10
Basic sequences






k
k
n
k
x
n
x ]
[
]
[
]
[ 
arbitrary
sequence
         
7
2
1
3 7
2
1
3 






  n
a
n
a
n
a
n
a
n
p 



A sum of scaled, delayed impulses

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
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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
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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

14. 14 10/17/2021
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Basic sequences
Sinusoidal sequence
  n
all
for
n
w
A
n
x 

 0
cos
]
[

15. 15 10/17/2021
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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
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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
  
    
 
 

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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
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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

  

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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 = 

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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
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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
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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
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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
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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
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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
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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

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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
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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）

40. 40 10/17/2021
40

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

42. 42 10/17/2021
42
Ex. 2.12

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
     

48. 48 10/17/2021
48
 































n
N
a
a
a
N
n
a
a
n
n
y
N
N
n
n
1
,
1
1
1
0
,
1
1
0
,
0
1
1

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]

51. 51 10/17/2021
51
Parallel connection of systems
     
n
h
n
h
n
h 2
1 


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

70. 70
Example 2.19 Ideal
Frequency-Selective Filters
Frequency Response of Ideal Low-pass Filter
Signal Processing - Dr. Arif Wahla
10/17/2021

71. 71
Frequency Response of Ideal
High-pass Filter

72. 72
Frequency Response of Ideal
Band-stop Filter

73. 73
Frequency Response of Ideal
Band-pass Filter

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

76. 76
76
Chapter 2 HW
2.1, 2.4, 2.5, 2.7,
2.11, 2.12,2.15, 2.20