WOLFRAM|DEMONSTRATIONS PROJECT

Sampling Theorem

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frequency
f [Hz]
1000
sampling rate
f
s
8 kHz
16 kHz
48 kHz
The top figure shows a sine wave of frequency
f
and its samples. The bottom figure shows the Fourier spectrum
S(f)
of the sampled signal. It consists of two spectral lines at
±f
, repeated periodically at integer multiples of the sampling rate
f
s
.
According to the sampling theorem, for
f≤
f
s
/2
, the samples uniquely represent the sine wave of frequency
f
. For
f>
f
s
/2
, aliasing occurs, because the replicated spectra begin to overlap. In the range
0≤f≤
f
s
/2
, a spectral line appears at the frequency
|f-
f
s
|
. In the upper figure the sine wave with the corresponding frequency and color appears.
Note that for
f
s
=8kHz
and
f>6kHz
, additional lines at
-f+2
f
s
and
f-2
f
s
appear in the spectrum.