Digital Signal Processing Reference
In-Depth Information
We now consider periodic discrete-time signals. By definition, a discrete-time signal
x
[
n
] is
periodic
with period
N
if
x[n + N] = x[n].
(9.32)
Of course, both
n
and
N
are integers.
We first consider the case that the signal
x
[
n
] is obtained by sampling a sinu-
soidal signal
x(t) = cos
(v
0
t)
every
T
seconds; that is,
x[n] = x(nT).
[
T
is the sam-
ple period and not the period of
x(t).
] For
x
[
n
] to be periodic, from (9.32),
x[n] = cos
(nv
0
T) = x[n + N] = cos
[(n + N)v
0
T]
= cos
(nv
0
T + Nv
0
T).
Hence,
Nv
0
T
must be equal to
2pk,
where
k
is an integer, because
cos(u
+2pk)=
cos u.
Therefore,
2pk = Nv
0
T = N
2p
T
0
k
N
=
T
T
0
T Q
,
(9.33)
where is the fundamental period of the continuous-time sinusoid. Thus,
the ratio of the sample period
T
to the period of the sinusoid
T
0
= 2p/v
0
T
0
must be a ratio of
integers; that is, must be
rational
.
The result in (9.33) can also be expressed as
T/T
0
NT = kT
0
.
(9.34)
This relation states that there must be exactly
N
samples in
k
periods of the signal
This statement applies to the sampling of
any
periodic continuous-time
cos(v
0
t).
signal
In summary, the sampled signal is periodic if exactly
N
sam-
ples are taken in exactly every
k
periods, where
N
and
k
are integers. Note the sur-
prising conclusion that the sampling of a periodic continuous-time signal does not
necessarily result in a periodic discrete-time signal. We now give an example.
x(t).
x[n] = cos(nv
0
T)
Sampling of a sinusoid
EXAMPLE 9.6
In this example, we will consider sampling the periodic signal
x(t) = sin pt,
which has the pe-
riod
T
0
= 2p/v
0
= 2 s.
First, we sample with the period
T = 0.5 s.
There are exactly four
samples for each period of sinusoid; in (9.34),
kT
0
= (1)
(2) = 2 s
and
NT = 4(0.5) = 2 s.
The signals are illustrated in Figure 9.14(a).
Next, we sample with the period
3
8
T
0
= 0.75 s.
T =
In this case, we have exactly eight
samples in every three periods
(8T = 3T
0
),
or in every 6 s. These signals are illustrated in
Figure 9.14(b).
As a final example, we sample a triangular wave that is periodic with a period of
This signal is sampled with sample period as shown in Figure 9.14(c).
In this case, there is less than one sample per period of the triangular wave; however,
T=
4
T
0
=
2.5 s,
T
0
= 2 s.
