Digital Signal Processing Reference
In-Depth Information
Fig. 1.5. Analog signal with its
digital approximation. The
waveform for the analog signal
is shown with a line plot; the
quantized digital approximation
is shown with a stem plot.
1.125
0.875
0.625
0.375
0.125
−0.125
−0.375
−0.625
−0.875
−1.125
0
sampling time
t
=
kT
samples per second. The sampling interval
T
is given by 1
/
44 100, or 22.68
microseconds (
µ
s). Each sample is then quantized with a 16-bit uniform quan-
tizer. In other words, a sample of the recorded music signal is approximated
from a set of uniformly distributed values that can be represented by a 16-bit
binary number. The total number of values in the discretized set is therefore
limited to 2
16
entries.
Digital signals may also occur naturally. For example, the price of a com-
modity is a multiple of the lowest denomination of a currency. The grades of
students on a course are also discrete, e.g. 8 out of 10, or 3.6 out of 4 on a 4-point
grade point average (GPA). The number of employees in an organization is a
non-negative integer and is also digital by nature.
1.1.3 Periodic and aperiodic signals
A CT signal
x
(
t
) is said to be
periodic
if it satisfies the following property:
x
(
t
)
=
x
(
t
+
T
0
)
,
(1.2)
at all time
t
and for some positive constant
T
0
. The smallest positive value
of
T
0
that satisfies the periodicity condition, Eq. (1.3), is referred to as the
fundamental period
of
x
(
t
).
Likewise, a DT signal
x
[
k
] is said to be
periodic
if it satisfies
x
[
k
]
=
x
[
k
+
K
0
]
(1.3)
at all time
k
and for some positive constant
K
0
. The smallest positive value of
K
0
that satisfies the periodicity condition, Eq. (1.4), is referred to as the fun-
damental period of
x
[
k
]. A signal that is not periodic is called an
aperiodic
or
non-periodi
c signal. Figure 1.6 shows examples of both periodic and aperiodic
Search WWH ::
Custom Search