The document discusses discrete-time signals and systems. It defines discrete-time signals as sequences represented by x[n] and discusses important sequences like the unit sample, unit step, and periodic sequences. It then defines discrete-time systems as devices that take a discrete-time signal x(n) as input and produce another discrete-time signal y(n) as output. The document classifies systems as static vs. dynamic, time-invariant vs. time-varying, linear vs. nonlinear, and causal vs. noncausal. It provides examples to illustrate each classification.

1. ‫ر‬َ‫ـد‬ْ‫ق‬‫ِـ‬‫ن‬،،،‫لما‬‫اننا‬ ‫نصدق‬ْْ‫ق‬ِ‫ن‬‫ر‬َ‫د‬
LECTURE (3)
Discrete-Time Signals and Systems
Amr E. Mohamed
Faculty of Engineering - Helwan University

2. Agenda
 Discrete-time signals: sequences
 Discrete-time system
 Linear Time-Invariant Causal Systems (LTIC System)
 Linear Constant-Coefficient Difference Equations
2

3. Discrete-Time Signals---Sequences
3

4. The Taxonomy of Signals
 Signal: A function that conveys information
4
Time
Amplitude
analog signals
continuous-time
signals
discrete-time
signals
digital signals
Continuous
Continuous
Discrete
Discrete

5. Discrete-Time Signals: Sequences
 Discrete-time signals are represented by sequence of numbers
 The nth number in the sequence is represented with 𝑥[𝑛]
 Often times sequences are obtained by sampling of continuous-time signals
 In this case 𝑥[𝑛] is value of the analog signal at 𝑥 𝑐(𝑛𝑇)
 Where T is the sampling period
0 20 40 60 80 100
-10
0
10
t (ms)
0 10 20 30 40 50
-10
0
10
n (samples)
5

6. Representation by a Sequence
 Discrete-time system theory
 Concerned with processing signals that are represented by sequences.
6
 nnxx )},({
1 2
3 4 5 6 7
8 9 10-1-2-3-4-5-6-7-8
n
x(n)

7. Important Sequences
 Unit-sample sequence (𝑛)
 Sometime call
 a discrete-time impulse; or
 an impulse
7






00
01
)(
n
n
n
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n
(𝒏)
)()()()(: mnmxmnnxNote  

8. Important Sequences
 Unit-step sequence 𝑢(𝑛)
8






00
01
)(
n
n
nu
 Fact:
)1()()(  nunun
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
𝒏
𝒖(𝒏)




0
)()(
m
mnnu 

9. Important Sequences
 Real exponential sequence
9
n
anx )(
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
𝒏
𝒙(𝒏)
. . .
. . .
𝟎 < 𝒂 < 𝟏

10. Important Sequences
 Sinusoidal sequence
10
)cos()( 0  nAnx
n
x(n)

11. Important Sequences
 Complex exponential sequence
11
njnnjnnj
eaeeenx 000 )(
)( 
 

12. Periodic Sequences
 A sequence x(n) is defined to be periodic with period N if
12
NNnxnx allfor)()( 
 Example: consider
nj
enx 0
)( 

)()( 0000 )(
Nnxeeeenx njNjNnjnj
 
 kN 20 0
2



k
N
0
2

 must be a rational
number

13. Examples of Periodic Sequences
 Suppose it is periodic sequence with period N
][][ 11 Nnxnx 
 4/cos][1 nnx 
    4/cos4/cos Nnn  
integer:,4/4/24/ kNnkn  
01, 8 2 /k N w   
2 / ( / 4) 8N k k  
13

14. Examples of Periodic Sequences
 Suppose it is periodic sequence with period N
][][ 11 Nnxnx 
    8/3cos8/3cos Nnn  
integer:,8/38/328/3 kNnkn  
16,3  Nk
 8/3cos][1 nnx 
8
3
8
2 

02 / 2 / (3 /8)N k w k   
14

15. Example of Non-Periodic Sequences
 Suppose it is periodic sequence with period N
][][ 22 Nnxnx 
 nnx cos][2 
  )cos(cos Nnn 
2 , :integer,
integer
for n k n N k
there is no N
  
15

16. Operations on Sequences
 Sum
 Product
 Multiplication
 Shift
 Reflection
16
)}()({ nynxyx 
)}()({ nynxyx 
)}({ nxx 
)()( 0nnxny 
)()( nxny 

17. Sequence Representation Using delay unit




k
knkxnx )()()( 
1
2
3 4 5 6
7
8 9 10-1-2-3-4-5-6-7-8
𝒏
𝒙(𝒏)
a1
a2 a7
a-3
)7()3()1()3()( 7213   nananananx
17

18. Energy of a Sequence
 Energy of a sequence is defined by
|)(| 2




n
n
nxE
18

19. DISCRETE-TIME SYSTEMS
19

20. Signal Process Systems
analog
system
signal outputContinuous-time
Signal
Continuous-time
Signal
discrete-
time
system
signal outputDiscrete-time
Signal
Discrete-time
Signal
digital
system
signal outputDigital Signal Digital Signal
20

21. Discrete-Time Systems
 Definition: A discrete-time system is a device or algorithm that
operates on a discrete-time signal called the input or excitation (e.g.
𝑥(𝑛)), according to some rule (e.g. 𝑇[. ]) to produce another discrete-
time signal called the output or response (e.g. 𝑦(𝑛)).
 This expression denotes also the transformation 𝑇[. ] , (also called
operator or mapping) or processing performed by the system on x(n) to
produce y(n).
21
𝑇 [ ]𝑥(𝑛) 𝑦(𝑛) = 𝑇[𝑥(𝑛)]
output signal or
response
input signal or
excitation

22. Discrete–Time Systems
 Discrete-Time System Analysis
 It is the process of determining the response, y(n) of that system described by an
operator, transformation T[.] or its impulse response, h(n) to a given excitation,
x(n).
 This process could done also in z- or frequency domains.
 Discrete-Time System Design
 It is the process of synthesizing the system parameters that satisfy the input
output specification.
 This process could done also in time, z- or frequency domains
 Digital Filter
 It is a digital system that can be used to filter discrete –time signal.
 Filtering is a process by which the frequency spectrum of the signal could be
modified or manipulated according to some desired specifications.
22

23. Classification of Discrete-Time Systems
 Static (Memoryless) Vs Dynamic system (Memory)
 Time varying Vs Time Invariant system
 Linear Vs Nonlinear System
 Causal Vs Noncausal System
 Stable vs. Unstable System
23

24. Static Vs Dynamic System
 Definition: A discrete-time system is called static or memoryless if its
output at any time instant n depends on the input sample at the same
time, but not on the past or future samples of the input.
 In the other case, the system is said to be dynamic or to have memory.
If the output of a system at time n is completely determined by the
input samples in the interval from n-N to n (𝑁 ≥ 0), the system is said
to have memory of duration N.
 If 𝑁 ≥ 0, the system is static or memoryless.
 If 0 < 𝑁 < ∞ , the system is said to have finite memory.
 If 𝑁 → ∞, the system is said to have infinite memory.
24

25. Examples:
 The static (memoryless) systems:
 The dynamic systems with finite memory:
 The dynamic system with infinite memory:
3
( ) ( ) ( )y n nx n bx n 
0
( ) ( ) ( )
N
k
y n h k x n k

 
0
( ) ( ) ( )
k
y n h k x n k


 
25

26. Time Varying Vs Time Invariant System
 Definition: A discrete-time system is called time-invariant if its input-
output characteristics do not change with time. In the other case, the
system is called time-variable.
 Time-Invariant (shift-invariant) Systems
 A time shift at the input causes corresponding time-shift at output
𝑇 [ ]
𝑥(𝑛) 𝑦(𝑛) = 𝑇[𝑥(𝑛)]
𝑥(𝑛 − 𝑛 𝑜) 𝑦(𝑛 − 𝑛 𝑜) = 𝑇[𝑥(𝑛 − 𝑛 𝑜)]
26

27. Examples:
 The time-invariant systems:
 The time-variable systems:
3
( ) ( ) ( )y n x n bx n 
0
( ) ( ) ( )
N
k
y n h k x n k

 
3
( ) ( ) ( 1)y n nx n bx n  
0
( ) ( ) ( )
N
N n
k
y n h k x n k

 
27

28. Example of Time-Invariant System
 Accumulator system 𝑦 𝑛 = 𝑘=−∞
𝑛
𝑥 𝑘
 Shift the system by (𝑚)
 then the output: y 𝑛 − m = 𝑘=−∞
𝑛−m
𝑥1(𝑘)
 Let the input: 𝑥(𝑛 − 𝑚),
 then the output: 𝑦1 𝑛 = 𝑘=−∞
𝑛
𝑥(𝑘 − 𝑚)
 Let: 𝐾 = 𝑘 − 𝑚, then 𝑦1 𝑛 = 𝑘=−∞
𝑛−𝑚
𝑥(𝑘) = y 𝑛 − m
 Hence , the system is Time-Invariant System
28

29. Example of Time-Varying System
 Accumulator system 𝑦 𝑛 = 𝑛𝑥(𝑛)
 Shift the system by (𝑚)
 then the output: y 𝑛 − m = (𝑛 − 𝑚)𝑥(𝑛 − 𝑚)
 Let the input: 𝑥(𝑛 − 𝑚),
 then the output: 𝑦1 𝑛 = 𝑛𝑥(𝑛 − 𝑚) ≠ y 𝑛 − m
 Hence , the system is Time-Varying System
29

30. Linear vs. Non-linear Systems
 Definition: A discrete-time system is called linear if only if it satisfies
the linear superposition principle. In the other case, the system is
called non-linear.
)]([)]([)]()([ 2121 nxbTnxaTnbxnaxT 
𝑇 [ ]𝑥(𝑛) 𝑦(𝑛) = 𝑇[𝑥(𝑛)]
30

31. Examples:
 The linear systems:
 The non-linear systems:
0
( ) ( ) ( )
N
k
y n h k x n k

 
2
( ) ( ) ( )y n x n bx n k  
3
( ) ( ) ( 1)y n nx n bx n  
0
( ) ( ) ( ) ( 1)
N
k
y n h k x n k x n k

   
31

32. Example of Linear System
 Accumulator system 𝑦 𝑛 = 𝑘=−∞
𝑛
𝑥 𝑘
 For arbitrary input: 𝑥1(𝑛),
 then the output: 𝑦1 𝑛 = 𝑘=−∞
𝑛
𝑥1(𝑘)
 For arbitrary input: 𝑥2(𝑛),
 then the output: 𝑦2 𝑛 = 𝑘=−∞
𝑛
𝑥2(𝑘)
 Let the input: 𝑥 𝑛 = 𝑎𝑥1 𝑛 + 𝑏𝑥2(𝑛),
 then the output:
• 𝑦 𝑛 = 𝑘=−∞
𝑛
𝑎𝑥1 𝑘 + 𝑏𝑥2(𝑘)
• 𝑦 𝑛 = 𝑎 𝑘=−∞
𝑛
𝑥1 𝑘 + 𝑏 𝑘=−∞
𝑛
𝑥2 𝑘
• Then, 𝑦 𝑛 = 𝑎𝑦1 𝑛 + 𝑏𝑦2 𝑛
 Hence , the system is Linear System
32

33. Example of Nonlinear Systems
 Accumulator system 𝑦 𝑛 = 𝑥2(n)
 For arbitrary input: 𝑥1(𝑛),
 then the output: 𝑦1 𝑛 = 𝑥1
2
(n)
 For arbitrary input: 𝑥2(𝑛),
 then the output: 𝑦2 𝑛 = 𝑥2
2
(n)
 Let the input: 𝑥 𝑛 = 𝑎𝑥1 𝑛 + 𝑏𝑥2(𝑛),
 then the output:
• 𝑦 𝑛 = 𝑎𝑥1 𝑛 + 𝑏𝑥2(𝑛) 2
• 𝑦 𝑛 = 𝑎2 𝑥1
2
n + ab𝑥1 𝑛 𝑥2(𝑛) + 𝑏2 𝑥2
2
(n)
• Then, 𝑦 𝑛 ≠ 𝑎𝑦1 𝑛 + 𝑏𝑦2 𝑛
 Hence , the system is Nonlinear System
33

34. Causal vs. Non-causal Systems
 Definition: A system is said to be causal if the output of the system at
any time n (i.e., y(n)) depends only on present and past inputs (i.e.,
x(n), x(n-1), x(n-2), … ). In mathematical terms, the output of a causal
system satisfies an equation of the form
 where is some arbitrary function. If a system does not satisfy this
definition, it is called non-causal.
[.]F
 ( ) ( ), ( 1), ( 2),y n F x n x n x n   L
34

35. Examples:
 The causal system:
 The non-causal system:
0
( ) ( ) ( )
N
k
y n h k x n k

 
2
( ) ( ) ( )y n x n bx n k  
3
( ) ( 1) ( 1)y n nx n bx n   
10
10
( ) ( ) ( )
k
y n h k x n k

 
35

36. Example :
 Check the discrete-time system for causality if its response is of the form:
a) y(n) = R[x(n)] = 3 x(n − 2) + 3 x(n + 2)
b) y(n) = R[x(n)] = 3 x(n − 1) − 3 x(n − 2)
 Solution:
a) Let x1(n) and x2(n) be distinct excitations that satisfy Eq. (2.4b) and assume that x1(n )  x2(n )
for n > k
For n = k
R[x1(n)]|n=k = 3x1(k − 2) + 3x1(k + 2)
R[x2(n)]|n=k = 3x2(k − 2) + 3x2(k + 2)
and since we have assumed that x1(n)  x2(n ) for n > k, it follows that:
x1(k + 2)  x2(k + 2) and thus 3 x1(k + 2)  3 x2(k + 2)
Therefore, R[x1(n)] = Rx2(n) for n = k
that is, the system is noncausal.
b) For this case
R[x1(n)]|n=k = 3x1(k − 1) + 3x1(k - 2)
R[x2(n)]|n=k = 3x2(k − 1) + 3x2(k - 2)
If n ≤ k,
x1(k - 1) = x2(k – 1) and x1(k - 2) = x2(k - 2)
for n ≤ k or
R[x1(n)] = Rx2(n) for n ≤ k
that is, the system is causal. 36

37. Stable vs. Unstable of Systems
 Definition: An arbitrary relaxed system is said to be bounded input -
bounded output (BIBO) stable if and only if every bounded input
produces the bounded output. It means, that there exist some finite
numbers say and , such that
 for all n. If for some bounded input sequence x(n) , the output y(n) is
unbounded (infinite), the system is classified as unstable.
xM yM
( ) ( )x yx n M y n M      
37

38. Testing for Stability or Instability
   2
][nxny 
  nallforBnx x ,
  nallforBBny xy ,2

if
then
is stable
38

39. Testing for Stability or Instability
 Accumulator system
 Accumulator system is not stable
   

n
k
kxny
    bounded
n
n
nunx :
01
00






     





  
boundednot
nn
n
kxkxny
n
k
n
k
:
01
00
39

40. Characterization of Discrete Time System
 Any Discrete-time can be characterized by one of the following
representations:
1) Difference Equation
2) Impulse Response
3) Transfer Function
4) Frequency Response
40

41. Difference Equation
 Continuous-time systems are characterized in terms of differential equations.
Discrete-time systems, on the other hand, are characterized in terms of
difference equations.
 A LTI system can be described by a linear constant coefficient difference
equation of the form:
or
 This equation describes a recursive approach for computing the current
output, y(n) given the input values, x(n) and previously computed output
values y(n-i). b0 =1, M and N are called the order of the system, ai and bi are
constant coefficients . Two types of discrete-time systems can be identified:
nonrecursive and recursive. 41
   inxainyb
M
i
N
i
ii  
 0 0
      
 

N
i
M
i
ii inybinxany
0 1

42. Recursive vs. Non-recursive Systems
 Definition: A system whose output y(n) at time n depends on any
number of the past outputs values ( e.g. y(n-1), y(n-2), … ), is called a
recursive system. Then, the output of a causal recursive system can be
expressed in general as
𝑦 𝑛 = 𝐹 𝑦 𝑛 − 1 , 𝑦 𝑛 − 2 , … , 𝑦 𝑛 − 𝑁 , 𝑥 𝑛 , 𝑥 𝑛 − 1 , … , 𝑥(𝑛 − 𝑚)
 where F[.] is some arbitrary function.
 In contrast, if y(n) at time n depends only on the present and past inputs
then such a system is called Nonrecursive.
𝑦 𝑛 = 𝐹 𝑥 𝑛 , 𝑥 𝑛 − 1 , … , 𝑥(𝑛 − 𝑚)
42

43. Examples:
 The Non-recursive system:
 The recursive system:
0
( ) ( ) ( )
N
k
y n h k x n k

 
0 1
( ) ( ) ( ) ( ) ( )
N N
k k
y n b k x n k a k y n k
 
    
43

44. Solving the difference equation
 Example: compute impulse response of LTICS that described by the following
Difference Equation.
𝑦 𝑛 − 𝑎𝑦 𝑛 − 1 = 𝑥(𝑛)
 Solution:
 Since the system is Causal-System, then 𝑦 𝑛 = 0 𝑓𝑜𝑟 𝑛 < 0.
 The system input is 𝑥 𝑛 = 𝛿(𝑛)
 Then the output can be calculated as follows:
𝑦 𝑛 = 𝑎𝑦 𝑛 − 1 + 𝛿(𝑛)
 At n=0: 𝑦 0 = 𝑎𝑦 −1 + 𝛿 0 = 1 = 𝑎0
 At n=1: 𝑦 1 = 𝑎𝑦 0 + 𝛿 1 = 𝑎 = 𝑎1
 At n=2: 𝑦 2 = 𝑎𝑦 1 + 𝛿 2 = 𝑎2
 At n=3: 𝑦 3 = 𝑎𝑦 2 + 𝛿 3 = 𝑎3
𝑇ℎ𝑒𝑛 𝑡ℎ𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑓𝑜𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑢𝑡𝑝𝑢𝑡 𝑦 𝑛 = 𝑎 𝑛
𝑓𝑜𝑟 𝑛 > 0
44

45. Discrete-Time system Networks
 The basic elements of discrete-time
systems are the adder, the
multiplier, and the unit delay.
 Ideally, the adder produces the sum
of its inputs and the multiplier
multiplies its input by a constant
instantaneously.
 The unit delay, on the other hand,
is a memory element that can store
just one number. At instant (nT), in
response to a synchronizing clock
pulse, it delivers its content to the
output and then updates its content
with the present input.
 The device freezes in this state
until the next clock pulse. In effect,
on the clock pulse, the unit delay
delivers its previous input to the
output.
45

46. Example
 For the discrete-time system shown in the Figure, if the system is initially
relaxed, that is, y(n) = 0 for n < 0, and p is a real constant, do the following:
a) Derive the difference equation.
b) Derive the impulse response, h(n).
c) Check the system stability.
d) Find the unit-step response of this system
46

47. Solution
a) From the Figure, the signals at node A is y(nT − T) and at node B is [p y(nT − T)],
respectively. Thus, the difference equation has the form:
𝑦(𝑛𝑇) = 𝑥(𝑛𝑇) + 𝑝𝑦(𝑛𝑇 − 𝑇) 𝑜𝑟 𝑠𝑖𝑚𝑝𝑙𝑦 𝑦(𝑛) = 𝑥(𝑛) + 𝑝𝑦(𝑛 − 1)
b) The impulse response, h(n) is the response of the system, y(n) when it is excited
by input 𝑥(𝑛) = 𝛿(𝑛). With 𝑥(𝑛𝑇) = 𝛿(𝑛𝑇), we can write:
𝑦(𝑛) = 𝑥(𝑛) + 𝑝 𝑦(𝑛 − 1)
𝑦(𝑛) = ℎ(𝑛) = 𝛿(𝑛) + 𝑝 𝑦(𝑛 − 1)
𝑦(0) = ℎ(0) = 1 + 𝑝 𝑦(−1) = 1 + 0 = 1
𝑦(1) = ℎ(1) = 0 + 𝑝 𝑦(0) = 𝑝
𝑦(2) = ℎ(2) = 0 + 𝑝 𝑦(𝑇 ) = 𝑝2
· · · · · · · · · · · · · · · · · · · · · · · · ·
ℎ(𝑛) = 𝑝 𝑛
and since 𝑦(𝑛) = 0 for 𝑛 ≤ 0, we have → ℎ(𝑛) = 𝑝 𝑛 𝑢(𝑛)
47
 






00
01
n
n
n

48. Solution (Cont.)
 The impulse response, ℎ(𝑛) of this system is illustrated in the Figure.
48
(a) Stable system (b) Unstable system
(c) Unstable system

49. Solution (Cont.)
c) check the stability:
For p< 1 and n   the p(n+1) )  0 and
In this case this system is stable one.
d) With x(n) = u(n), we get: y(n) = x(n) + py(n − 1)
𝑦(0) = 1 + 𝑝 𝑦(−1) = 1
𝑦 1 = 1 + 𝑝 𝑦(0) = 1 + 𝑝
𝑦 2 = 1 + 𝑝 𝑦(1) = 1 + 𝑝 + 𝑝2
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
49
  


 


n
k
n
k
n
k p
p
pkh
0
)1(
0 1
1
  





 00 1
1
k
k
k p
pkh
p
p
nupnuny
nn
k
k





 1
1
)()()(
)1(
0

50. Solution (Cont.)
 The unit-step response for the three values of p is illustrated in the
Figure. Evidently, the response converges if p < 1 and diverges if p ≥ 1.
50
(a) P < 1 (b) P > 1
(c) P = 1

51. LINEAR TIME-INVARIANT CAUSAL SYSTEMS
(LTIC SYSTEM)
51

52. Signal Process Systems
 A important class of systems
 In particular, we’ll discuss
52
Linear Shift-Invariant Systems.
Linear Shift-Invariant Discrete-Time Systems.

53. Linear Shift-Invariant Systems
)()()( knkxnx
k
 


 





 


)()()( knkxTny
k

)]([)()( knTkxny
k
 


 )(*)()()( nhnxknhkx
k
 


Time k impulse
The output value at time n
Only with the time
difference
𝑇 [ ]𝑥 𝑛 = 𝛿(𝑛) 𝑦 𝑛 = 𝑇 𝑥 𝑛 = ℎ(𝑛)
impulse responseunit impulse
 LTI system description by convolution (convolution sum):
53

54. Impulse Response
0 0
00
𝑇 [ ]𝑥(𝑛) 𝑦(𝑛) = 𝑇[𝑥(𝑛)]
54

55. Convolution Sum
)(*)()()()( nhnxknhkxny
k
 


Convolution
A linear shift-invariant system is completely
characterized by its impulse response.
𝑇 [ ]𝑥(𝑛)
𝑦 𝑛 = 𝑇 𝑥 𝑛
𝑦(𝑛) = 𝑥(𝑛) ∗ ℎ(𝑛)
55

56. Properties of Convolution (Cumulative)
)(*)()()()( nhnxknhkxny
k
 


)(*)()()()( nxnhknxkhny
k
 


)(*)()(*)( nxnhnhnx 
56

57. Properties of Convolution (Associative):
Cascaded Connection
ℎ1(𝑛)𝑥(𝑛) ℎ2(𝑛) 𝑦 𝑛 = 𝑥 𝑛 ∗ ℎ1 𝑛 ∗ ℎ2 𝑛
ℎ2(𝑛)𝑥(𝑛) ℎ1(𝑛) 𝑦 𝑛 = 𝑥 𝑛 ∗ ℎ2 𝑛 ∗ ℎ1 𝑛
ℎ1(𝑛) ∗ ℎ2(𝑛)𝑥(𝑛) 𝑦 𝑛 = 𝑥 𝑛 ∗ ℎ1 𝑛 ∗ ℎ2 𝑛
These systems are identical.
57

58. Properties of Convolution (Distributive):
Parallel Connection
ℎ1(𝑛) + ℎ2(𝑛)𝑥(𝑛) 𝑦 𝑛 = 𝑥 𝑛 ∗ ℎ1 𝑛 + ℎ2 𝑛
These two systems are identical.
ℎ1(𝑛)
𝑥(𝑛)
ℎ2(𝑛)
𝑦 𝑛 = 𝑥 𝑛 ∗ ℎ1 𝑛 + 𝑥 𝑛 ∗ ℎ2 𝑛+
58

59. Example: Parallel and Cascade
ℎ1(𝑛)
𝑥(𝑛)
ℎ 𝑒𝑞 𝑛 = [ℎ1 𝑛 ∗ ℎ3 𝑛 + ℎ2 𝑛 ]
+
ℎ3(𝑛)
ℎ2(𝑛)
𝑦 𝑛
ℎ 𝑒𝑞(𝑛)
𝑦 𝑛 = 𝑥 𝑛 ∗ [ℎ1 𝑛 ∗ ℎ3 𝑛 + ℎ2 𝑛 ]
𝑦 𝑛𝑥(𝑛)
59

60. Example
0 1 2 3 4 5 6
)()()( Nnununx 






00
0
)(
n
na
nh
n
𝑦(𝑛) =?0 1 2 3 4 5 6
60

61. Example
)()()(*)()( knhkxnhnxny
k
 


0 1 2 3 4 5 6
𝑘
𝑥(𝑘)
0 1 2 3 4 5 6
𝑘ℎ(𝑘)
0 1 2 3 4 5 6
𝑘ℎ(0𝑘)
61

62. Example
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
compute y(0)
compute y(1)
How to computer y(n)?
𝑘
𝑥(𝑘)
𝑘ℎ(0𝑘)
𝑘ℎ(1𝑘)
)()()(*)()( knhkxnhnxny
k
 


62

63. Example Two conditions have to be considered.
n<N and nN.
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0 1 2 3 4 5 6
compute y(0)
compute y(1)
How to computer y(n)?
)()()(*)()( knhkxnhnxny
k
 


𝑘
𝑥(𝑘)
𝑘ℎ(0𝑘)
𝑘ℎ(1𝑘)
63

64. Ways to find D.T. Convolution
 Three ways to perform digital convolution
 Graphical method
 Table method
 Analytical method
64

65. 1- Graphical Method
 Example: Find the convolution of the two sequences x[n] and h[n] given
by 𝒙[𝒏] = [𝟑, 𝟏, 𝟐] and 𝒉[𝒏] = [𝟑, 𝟐, 𝟏]
 Solution: On the Board
65

66. 2- Table Method
 Example: Find the convolution of the two sequences x[n] and h[n] given
by 𝒙[𝒏] = [𝟑, 𝟏, 𝟐] and 𝒉[𝒏] = [𝟑, 𝟐, 𝟏]
K -2 -1 0 1 2 3 4 5
x[k]: 3 1 2 y[n<0]: 0
h[0-k]: 1 2 3 y[0]: 9
h[1-k]: 1 2 3 y[1]: 9
h[2-k]: 1 2 3 y[2]: 11
h[3-k]: 1 2 3 y[3]: 5
h[4-k]: 1 2 3 y[4]: 2
h[5-k]: 1 2 3 y[n≥5]: 0
66

67. Example:
 Find the convolution of the two sequences x[n] and h[n] represented by,
𝑥[𝑛] = [2, 1, −2, 3, −4] and ℎ[𝑛] = [3, 1, 2, 1]
 Solution: On the Board
67

68. 3- Analytical Method (Example)
1
1
1
)1(
00 11
1
)( 












  a
aa
a
a
aaaany
nn
n
n
k
kn
n
k
kn
For n < N:
For n  N: 11
1
0
1
0 11
1
)( 













  a
aa
a
a
aaaany
NnnN
n
N
k
kn
N
k
kn
)()()(*)()( knhkxnhnxny
k
 


)()()( Nnununx 






00
0
)(
n
na
nh
n
𝑦(𝑛) =?
68

69. 3- Analytical Method (Example) Cont…
0
1
2
3
4
5
0 5 10 15 20 25 30 35 40 45 50
1
1
1
)1(
00 11
1
)( 












  a
aa
a
a
aaaany
nn
n
n
k
kn
n
k
kn
For n < N:
For n  N: 11
1
0
1
0 11
1
)( 













  a
aa
a
a
aaaany
NnnN
n
N
k
kn
N
k
kn
)()()(*)()( knhkxnhnxny
k
 


)()()( Nnununx 






00
0
)(
n
na
nh
n
𝑦(𝑛) =?
69

70. Example:
 Find the output 𝑦[𝑛] of a Linear, Time-Invariant system having an
impulse response ℎ[𝑛], when an input signal 𝑥[𝑛] is applied to it ℎ 𝑛
= 𝑎 𝑛
𝑢 𝑛 𝑎𝑛𝑑 𝑥 𝑛 = 𝑢 𝑛 , where 𝑎 < 1.
 Solution:
 By definition of Convolution sum, the output y[n] is given as
)()()(*)()( knhkxnhnxny
k
 








0
)(.)().()(
k
k
k
k
knuaknukuany
a
a
any
nn
k
k





 1
1
)(
)1(
0
70

71. Causal LTI Systems
 A relaxed LTI system is causal if and only if its impulse response is zero
for negative values of n , i.e.
 Then, the two equivalent forms of the convolution formula can be
obtained for the causal LTI system:
0
( ) ( ) ( ) ( ) ( )
n
k k
y n h k x n k x k h n k

 
    
0for0)(  nnh
71

72. Stable LTI Systems
 A LTI system is stable if its impulse response is absolutely summable,
[i.e. every bounded input produce a bounded output (BIBO)]
 



k
khS |)(|
72

73. Example:
 Show that the linear shift-invariant system with impulse response
ℎ 𝑛 = 𝑎 𝑛
𝑢(𝑛) where |𝑎| < 1 is stable.
73


 



 1
1
|)(|
00 k
k
k a
akhS

74. Impulse Response of the Ideal Delay System
Ideal Delay System: )()( dnnxny 
)()( dnnnh 
By letting 𝑥(𝑛) = 𝛿(𝑛) and 𝑦(𝑛) = ℎ(𝑛),
(n nd)
0 1 2 3 4 5 6 nd
74

75. Impulse Response of the Ideal Delay System
You must know:
(n nd)
0 1 2 3 4 5 6 nd
)()(*)( dd nnxnnnx 
(𝒏 𝒏 𝒅) plays the following functions:
• Shift; or
• Copy
75

76. Impulse Response of the Moving Average
Moving Average: 





Mk
Mk
knx
MM
ny
1
)(
1
1
)(
21




M
Mk
kn
MM
nh
1
)(
1
1
)(
21






otherwise
MnM
MMnh
0
1
1
)( 21
21
M1  0  M2
. . . . . .
Can you explain with (n k)?
M1  0  M2
. . . . . .
Can you explain with u(n)?
76

77. Forward Difference Vs Backward Difference
 Impulse response of Forward Difference
 Impulse response of Backward Difference
     nnnh   1
     1 nnnh 
77

78. Impulse Response of the Accumulator
)(
0,0
0,1
)()( nu
n
n
knh
n
k 





 

Can you explain?
Accumulator: 

n
k
kxny )()(
0
. . .
Unstable system 


  n
S u n
78

79. Inverse system
        
     nnunu
nnnunh




1
1
         nnhnhnhnh ii 
x[n] h[n] hi[n] y[n]= x[n]*]= x[n]
δ[n]
79

80. Discrete-Time Frequency
Response (DTFT)
FREQUENCY-DOMAIN REPRESENTATION OF DISCRETE-TIME
SIGNALS AND SYSTEMS
80

81. Frequency Response
 DTFT, Discrete-Time Fourier Transform of ℎ(𝑛) is
 IDTFT, Inverse Discrete-Time Fourier Transform is
    ,jw jwn
n
H e h n e



 





deeHnh njj

 )(
2
1
)(
ℎ(𝑛)
H(𝑒 𝑗𝑤
)
𝑥(𝑛) 𝑦 𝑛 = 𝑥 𝑛 ∗ ℎ(𝑛)
𝑋(𝑒 𝑗𝑤) Y 𝑒 𝑗𝑤 = 𝑋 𝑒 𝑗𝑤 . 𝐻(𝑒 𝑗𝑤)
81

82. Example: DTFT of The Ideal Delay System
)()( dnnnh  








k
nj
d
k
njj
enkekheHSince 
 )()()(,
1|)(| j
eH
d
j
neH  
)(
The Magnitude:
The phase:
dnjj
eeHthen  
)(,
82

83. Ideal Frequency-Selective Filters

|)(| j
eH
 cc
1

|)(| j
eH
 aa
1
bb

|)(| j
eH
 aa
1
bb
Lowpass Filter
Bandstop Filter
Bandpass Filter

|)(| j
eH
 aa
1
bb
Highpass Filter
83

84. Example: IDTFT of Ideal Lowpass Filter


cc
)( j
eH






c
cj
eH
0
||1
)(


 



 deeHnh njjc
c
)(
2
1
)(
 

c
c
c
c
njde
nj
de njnj









)(
2
1
2
1
n
n
e
nj
cnj
c
c





 sin
2
1


84

85. Example
 Find the response of the ideal delay system ℎ 𝑛 = 𝛿(𝑛 − 𝑛 𝑑), if the input is
𝑥 𝑛 = 𝛿 𝑛 + 2𝛿(𝑛 − 1).
 Solution:
 By DTFT of ℎ 𝑛 :
𝐻 𝑒 𝑗𝜔
= 𝑛=−∞
∞
ℎ(𝑛) 𝑒−𝑗𝜔𝑛
= 𝑛=−∞
∞
𝛿(𝑛 − 𝑛 𝑑) 𝑒−𝑗𝜔𝑛
= 𝑒−𝑗𝜔𝑛 𝑑
 By DTFT of 𝑥 𝑛 :
𝑋 𝑒 𝑗𝜔 = 𝑛=−∞
∞ 𝑥(𝑛) 𝑒−𝑗𝜔𝑛 = 𝑛=−∞
∞ 𝛿 𝑛 + 2𝛿(𝑛 − 1) 𝑒−𝑗𝜔𝑛
𝑋 𝑒 𝑗𝜔
= 1 + 2𝑒−𝑗𝜔
 The system output in frequency domain:
𝑌 𝑒 𝑗𝜔
= 𝑋 𝑒 𝑗𝜔
. 𝐻 𝑒 𝑗𝜔
= 𝑒−𝑗𝜔𝑛 𝑑 + 2𝑒−𝑗𝜔(𝑛 𝑑+1)
85

86. Example (Cont…..)
 By the IDTFT, The system output in Time domain:
𝑦 𝑛 =
1
2𝜋 −𝜋
𝜋
𝑌 𝑒 𝑗𝜔 . 𝑒 𝑗𝜔𝑛 . 𝑑𝜔
𝑦(𝑛) =
1
2𝜋 −𝜋
𝜋
𝑒−𝑗𝜔𝑛 𝑑 + 2𝑒−𝑗𝜔(𝑛 𝑑+1)
. 𝑒 𝑗𝜔𝑛
. 𝑑𝜔
𝑦(𝑛) =
1
2𝜋 −𝜋
𝜋
𝑒 𝑗𝜔(𝑛−𝑛 𝑑) + 2𝑒 𝑗𝜔(𝑛−𝑛 𝑑−1) . 𝑑𝜔
 Then, y(n) is
𝒚(𝒏) = 𝜹(𝒏 − 𝒏 𝒅) + 𝟐𝜹(𝒏 − 𝒏 𝒅−𝟏)
86

87. 87