Digital Signal Processing Reference
In-Depth Information
Im
e
j
0
n
n
2
•
•
•
n
1
2
0
n
0
0
1
e
Figure 9.15
Representation of the
complex exponential.
If we make the change of variable in (9.37), we have the complex
exponential signal of (9.35), and all the preceding conclusions apply directly. The
variable has the units of radians; we refer to as
normalized discrete frequency,
or simply,
frequency
. For sampled signals, real frequency
Æ
0
= v
0
T
Æ
Æ
v
and discrete frequency
Æ
are related by
We now consider (9.37) in a different manner. The complex exponential signal
of (9.37) is periodic, provided that
vT =Æ.
x[n] = e
jÆ
0
n
= x[n + N] = e
j(Æ
0
n+Æ
0
N)
= e
j(Æ
0
n+ 2pk)
,
(9.38)
where
k
is an integer. Thus, periodicity requires that
k
N
2p,
Æ
0
N = 2pk Q Æ
0
=
(9.39)
so that
Æ
0
must be expressible as
2p
multiplied by a rational number. For example,
x[n] = cos
(2n)
is
not
periodic, since
Æ
0
= 2.
The signal
x[n] = cos(0.1pn)
is peri-
odic, since For this case, , and satisfies (9.39).
As a final point, from (9.39), the complex exponential signal is periodic
with
N
samples per period, provided that the integer
N
satisfies the equation
Æ
0
= 0.1p.
k = 1
N = 20
e
jÆ
0
n
2pk
Æ
0
N =
.
(9.40)
In this equation,
k
is the smallest positive integer that satisfies this equation,
such that
N
is an integer greater than unity. For example, for the signal
the number of samples per period is
x[n] = cos(0.1pn),
2pk
0.1p
= 20k = 20,
N =
k = 1.
(9.41)
