Outline
DiscreteTime Signals
Dr. Bokamoso Basutli
Lecturer and Group Leader (SPNS)
Department of Electrical, Computer and Telecommunications
Engineering
Digital Signal Processing
Lecture 2: DiscreteTime Signals
and Systems
Outline
DiscreteTime Signals
Outline
DiscreteTime Signals
Outline
DiscreteTime Signals
Elementary DiscreteTime Signals
(1)
Unit sample sequence (a.k.a. Kronecker delta function):
δ
[
n
] =
0
,
n
6
= 0,
1
,
n
= 0.
(2)
Unit Step Signal:
u
[
n
] =
1
,
n
≥
0,
0
,
n
<
0.
(3)
Unit Ramp Signal:
u
r
[
n
] =
n
,
n
≥
0,
0
,
n
<
0.
NOTE:
δ
[
n
]
=
u
[
n
]

u
[
n

1] =
u
r
[
n
+ 1]

2
u
r
[
n
] +
u
r
[
n

1]
=
u
r
[
n
+ 1]

u
r
[
n
]
Outline
DiscreteTime Signals
Unit Step Manipulations
•
A unit step sequence,
u
[
n
]
is given as;
u
[
n
] =
1
n
≥
0,
0
n
<
0.
or in terms of
δ
[
n
]
u
[
n
]
=
n
X
k
=
∞
δ
[
k
]
=
∞
X
k
=0
δ
[
n

k
]
which is analogous to the
continoustime unit step
being written as
Figure 1:
Rectangular pulses using unit
step sequence.
u
(
t
) =
Z
t
∞
δ
(
λ
)
d
λ
=
Z
∞
0
δ
(
t

λ
)
d
λ
Outline
DiscreteTime Signals
Signal Symmetry
Even Signal:
x
[

n
] =
x
[
n
]
Odd Signal:
x
[

n
] =

x
[
n
]
Outline
DiscreteTime Signals
Signal Symmetry
Even Signal component:
x
e
[
n
] =
1
2
[
x
[
n
] +
x
[

n
]]
Odd Signal component:
x
o
[
n
] =
1
2
[
x
[
n
] +

x
[

n
]]
Note:
x
[
n
] =
x
e
[
n
] +
x
o
[
n
]
Outline
DiscreteTime Signals
Simple Manipulation of DiscreteTime Signals
•
Transformation of independent variable:
•
time shift:
n
→
n

k
,
k
∈
Z
.
•
What if
k
/
∈
Z
•
time scale:
n
→
α
n
, α
∈
Z
.
•
What if
α /
∈
Z
•
Additional, multiplication and scaling:
•
amplitude scaling:
y
[
n
] =
Ax
[
n
]
,
∞
<
n
<
∞
•
sum:
y
[
n
] =
x
1
[
n
] +
x
2
[
n
]
,
∞
<
n
<
∞
•
product:
y
[
n
] =
x
1
[
n
]
x
2
[
n
]
,
∞
<
n
<
∞
Outline
DiscreteTime Signals
Simple Manipulation of DiscreteTime Signals
•
Sequence shift or delay
is obtained by letting
n
→
n

k
which defines a new signal
y
[
n
] =
x
[
n

k
]
.
•
If
k
>
0 then
y
[
n
] is delayed and if
k
<
0 then
y
[
n
] is
advanced, with respect to
x
[
n
].
•
Unit sample sequence or impulse,
δ
[
n
]
is the discretetime
equivalent to
δ
[
n
] =
0
n
6
= 0,
1
n
= 0.
Any sequence can be written as a linear combintaion of
δ
[
n
],
e.g.
x
[
n
] =
∞
X
k
=
∞
x
[
k
]
δ
[
n

k
]
Outline
DiscreteTime Signals
Tutorial: Simple Manipulation of DiscreteTime Signals
•
Consider the sequence
x
[
n
] given in the figure below.
•
Determine
x
[
n
] =
x
[
n
+ 1]
Outline
DiscreteTime Signals
Tutorial: Simple Manipulation of DiscreteTime Signals
•
Determine
x
[
3
2
n
+ 1]
Outline
DiscreteTime Signals
Tutorial: Simple Manipulation of DiscreteTime Signals
•
Graph of
x
[
3
2
n
+ 1]
Outline
DiscreteTime Signals
InputOutput Description
•
Inputoutput description (exact structure of system is
unknown or ignored):
y
[
n
] =
T {
x
[
n
]
}
•
“black box” representation:
x
[
n
]
T
→
y
[
n
]
Outline
DiscreteTime Signals
DiscreteTime Systems
DiscreteTime System
•
A discretetime system is an operator that maps an input
sequence into an output sequence.
y
[
n
] =
T {
x
[
n
]
}
•
Note that
T {·}
is an operator that maps the input sequence
to the output sequence.
Figure 2:
Discretetime system block diagram definition.
Outline
DiscreteTime Signals
Classifiying DiscreteTime Systems
•
Discretetime system can be classified based on their
properties, such as:
•
static
or
dynamic (
memory
or
memoryless,)
•
linearity
or
nonlinearity,
•
timeinvariant
or
timevarying,
•
causal
or
noncausal,
•
boundedinput boundedoutput (BIBO stability)
or
unbounded.
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 Winter '20
 Digital Signal Processing, Signal Processing, N0, Classifiying DiscreteTime Systems, exact structure of system