103
6.5 Digital Signal Processing
Summary of the practical consequences of the sampling theorem in
frequency space
Recall the general formula for calculating the frequency of an aliased signal that has been sampled at
some rate less than
f
ny
, the Nyquist frequency,
obs
s
f
ff
=
−
,
and the graphical demonstration of the same:
f (Hz)
E
80
90
100
fj
fny
fs
180
f
The following are important general principles of dealing with aliasing in discrete signals. Let the true
frequency be denoted
f,
f
obs
is the observed frequency,
f
s
is the sampling frequency, and
f
ny
=
f
s
/2 is the
Nyquist frequency.
1.
Frequencies
f
≤
f
ny
are recovered correctly.
2.
When
f
>
f
ny
,
the information does not disappear, it reappears as phantom peaks, contaminating the
signal at
f
obs
=
f
s
–
f
.
3.
In any single realization, there is no way to distinguish phantoms from the real thing, unless you
have some other information about likely values.
4.
If the apparent signal is
a phantom, then
f
obs
will change when
f
s
changes. Thus real and phantom
peaks can be distinguished, if
f
s
can be modified.
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 '07
 Pottebaum
 Digital Signal Processing, Fny

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