Digital Signal Processing Reference
In-Depth Information
below. This is an important point: once an analog signal has been digitized, it is
nothing more than a sequence of numbers that can be stored, manipulated,
transmitted, or processed in any way we see fit.
x
(
nT
)
=
{
50.0,
44.3,
37.7,
30.6,
23.4,
16.4,
...
},
n
=
0
1
…
s
(5.4)
Since the sampling period seldom changes, discrete-time equations usually
drop the sampling period
T
s
from the expressions to produce an expression such as
(5.5). The sampling period will not be needed again until the digitized waveform
is converted back to analog form.
−
10
⋅
n
x
(
n
)
=
25
⋅
e
⋅
cos(
100
⋅
n
)
(5.5)
Although it doesn't appear that much has changed in the representation of the
signal in the time domain, a great deal has changed in the frequency domain. The
frequency spectrum of a signal is shown in Figure 5.2(a) before sampling. If this
signal were sampled at a frequency of
f
s
, the spectrum of the sampled signal would
be as shown in Figure 5.2(b). The original analog spectrum is replicated
throughout the spectrum at intervals of
f
s
(although only one instance of that is
shown). Because of this replication, it is feasible that there will be corruption of
the frequency components of the original signal by components of the replicated
signals. This corruption is referred to as
aliasing
, and the offending frequencies
are
alias
frequencies. (Further details of aliasing and the upcoming Nyquist
criteria can be found in most of the digital filter design references in Appendix A.)
Figure 5.2
Spectrum of signal (a) before and (b) after sampling.
