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
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
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
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
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
63. Example Two conditions have to be considered.
n<N and nN.
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(0𝑘)
𝑘ℎ(1𝑘)
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
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