Digital Signal Processing Reference
In-Depth Information
+
+
m
,
K
0
to denote a set of positive integer
values. Based on the above discussion, we make the following proposition.
∈
Z
, where we use the notation
Z
Proposition 1.1
An arbitrary DT sinusoidal sequence x
[
k
]
=
A
sin(
Ω
0
k
+ θ
)
is
periodic iff
Ω
0
/
2
π
is a rational number.
The term
rational number
used in Proposition 1.1 is defined as a fraction of
two integers. Given that the DT sinusoidal sequence
x
[
k
]
=
A
sin(
Ω
0
k
+ θ
)is
periodic, its fundamental period is evaluated from the relationship
=
m
K
0
Ω
0
2
π
(1.7)
as
=
2
π
Ω
0
K
0
m
.
(1.8)
Proposition 1.1 can be extended to include DT complex exponential signals.
Collectively, we state the following.
(1) The fundamental period of a sinusoidal signal that satisfies Proposition 1.1
is calculated from Eq. (1.8) with
m
set to the smallest integer that results
in an integer value for
K
0
.
(2) A complex exponential
x
[
k
]
=
A
exp[j(
Ω
0
k
+ θ
)] must also satisfy Propo-
sition 1.1 to be periodic. The fundamental period of a complex exponential
is also given by Eq. (1.8).
Example 1.4
Determine if the sinusoidal DT sequences (i)-(iv) are periodic:
(i)
f
[
k
]
=
sin(
π
k
/
12
+ π/
4);
(ii)
g
[
k
]
=
cos(3
π
k
/
10
+ θ
);
(iii)
h
[
k
]
=
cos(0
.
5
k
+ φ
);
(iv)
p
[
k
]
=
e
j(7
π
k
/
8
+θ
)
.
Solution
(i) The value of
0
in
f
[
k
]is
π/
12. Since
Ω
0
/
2
π =
1
/
24 is a rational number,
the DT sequence
f
[
k
] is periodic. Using Eq. (1.8), the fundamental period of
f
[
k
]isgivenby
=
2
π
Ω
0
K
0
m
=
24
m
.
Setting
m
=
1 yields the fundamental period
K
0
=
24.
To demonstrate that
f
[
k
] is indeed a periodic signal, consider the following:
f
[
k
+
K
0
]
=
sin(
π
[
k
+
K
0
]
/
12
+ π/
4)
.
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