Digital Signal Processing Reference
In-Depth Information
step is to output the digital value of the signal and hold it for the duration of the
sample period. The second step is to pass that signal through a lowpass filter. In
order to understand the reasons why these steps are necessary for analog and
digital conversion, we need to study the frequency spectrum of a sampled signal
and the requirements placed on the sampling rate.
Figure 5.1
Comparison of analog and digital signals.
5.1.1 Frequency Spectrum and Sampling Rate
When an analog signal
x
(
t
) is sampled, as shown in Figure 5.1, the samples are
usually taken at equal intervals of time. This sampling period
T
s
is the inverse of
the sampling frequency
f
s
. The resulting digitized waveform
x
s
(
nT
) can be
specified with an argument indicating the sampling period
T
s
, as shown in (5.1):
x
(
nT
)
=
x
(
t
)
(5.1)
s
t
=
nT
s
For example, if we had an analog signal
x
(
t
) as specified in (5.2), the sampled
version of the signal
x
s
(
nT
s
) as shown in (5.3) would result:
−
100
⋅
t
x
(
t
)
=
50
⋅
e
⋅
cos(
200
⋅
t
)
(5.2)
−
100
⋅
nT
x
(
nT
)
=
50
⋅
e
⋅
cos(
200
⋅
nT
)
s
s
(5.3)
If we assume that the analog signal is sampled at a frequency of 1,000
samples per second, then we can calculate the values of
x
s
(
nT
s
) with
T
s
= 0.001
second. The values that result can be stored as a sequence of numbers, as shown
