(v) The set of all distinct values that a discrete time sinusoidal sequence can have
occurs for values of
ranging from
or simply
]
,
[
Because of these last two properties, any sinusoid having the frequency
is
called an
alias.
(f)
Complex exponential sequence
n
j
e
n
x
)
(
)
(
An arbitrary sequence can be expressed as a sum of scaled and delayed unit
samples as:
)
7
(
)
2
(
)
1
(
)
3
(
)
(
7
2
1
3
n
a
n
a
n
a
n
a
n
p
3
a
0
1
8
7
6
5
4
3
2
4
3
2
1
)
(
n
p
n
7
a
2
a
1
a
Figure 2.3: An arbitrary sequence
This can be written in general terms as
k
k
n
k
x
n
x
)
(
)
(
)
(
1.6
LINEAR SHIFT INVARIANT SYSTEM
A discrete system may be thought of as a unique transformation or operator that
maps some input sequence
)
(
n
x
to some output sequence
)
(
n
y
through some
transformation
]
[
T
such that
)]
(
[
)
(
n
x
T
n
y
]
[
T
)
(
n
x
)
(
n
y

ECE 524E
–
DIGITAL SIGNAL PROCESSING
(Mr. Chemweno)
12
Figure 2.4 : Representation of a system
A linear system obeys the principle of superposition which states:
If
)]
(
[
)
(
1
1
n
x
T
n
y
and
)]
(
[
)
(
2
2
n
x
T
n
y
, then a system is linear if
)
(
)
(
)]
(
[
)]
(
[
)]
(
)
(
[
2
1
2
1
2
1
n
y
n
y
n
x
T
n
x
T
n
x
n
x
T
,
are constants
Example 2.1
: Determine the linearity of a 3-sample average given by
)]
(
[
)]
1
(
)
(
)
1
(
[
3
1
)
(
n
x
T
n
x
n
x
n
x
n
y
Solution:
)]
(
)
(
[
2
1
n
x
n
x
T
)
1
(
)
1
(
)
(
)
(
)
1
(
)
1
(
[
3
1
2
1
2
1
2
1
n
x
n
x
n
x
n
x
n
x
n
x
)]
1
(
)
(
)
1
(
[
3
)]
1
(
)
(
)
1
(
[
3
2
2
2
1
1
1
n
x
n
x
n
x
n
x
n
x
n
x
)
(
)
(
2
1
n
y
n
y
Hence linear.
Example 2.2
: Determine the linearity of the system
)
(
)]
(
[
)
(
2
n
x
n
x
T
n
y
Solution:
)]
(
)
(
[
2
1
n
x
n
x
T
2
2
1
)]
(
)
(
[
n
x
n
x
)
(
)
(
2
)
(
)
(
2
1
2
2
2
2
1
2
n
x
n
x
n
x
n
x
)
(
)
(
2
2
2
2
1
2
n
x
n
x
Hence non linear
A discrete time system is shift invariant for all
n
and
o
n
)]
(
[
)
(
o
o
n
n
x
T
n
n
y
,
o
n
is the number of delay samples.
Example 2.3
:
A system is described by
)
(
)]
(
[
)
(
n
nx
n
x
T
n
y
Determine whether the system is linear and whether it is time varying
Solution

ECE 524E
–
DIGITAL SIGNAL PROCESSING
(Mr. Chemweno)
13
Linearity:
)]
(
)
(
[
2
1
n
x
n
x
T
2
2
1
)]
(
)
(
[
n
nx
n
nx
)]
(
[
)]
(
[
2
1
n
nx
n
nx
)
(
)
(
2
1
n
y
n
y
Hence lnear
Shift invariance
)
(
)]
(
[
)
(
o
o
o
o
n
n
x
n
n
n
n
x
T
n
n
y
Since the coefficient is time varying, then the system is not shift invariant.
A system that satisfies both the above two conditions is known as Linear Shift
Invariant system.
1.7
UNIT SAMPLE RESPONSE
A Linear shift invariant system can be characterized by its unit sample response
)
(
n
h
.

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