As I remember it, the sampling theorem says that the sampling rate used to
sample a signal must be at least twice the highest frequency being sampled in
order to get a faithful reproduction when the samples are turned back into a
(continuous) output signal. In practice, because it is necessary to band limit
most signals to prevent aliasing artifacts, the sampling rate usually needs to
be about 2.2 times the highest frequency being sampled, since it is impossible
in practice to create low pass filters that are extremely steep, steep enough
to allow a sampling rate of only 2 times the highest frequency involved to
prevent aliasing.
Take the standard 8000 Hz sampling rate for telephone toll quality voice used
with mu-law and a-law voice codecs for long distance digital transmission. The
specified upper frequency is about 3400 Hz, as I recall. 8000 Hz is more than
2 times 3400 Hz, it's close to 2.2 times.
I think you can have frequencies changing amplitude and jumping in and out
subject to the constraints given above. Obviously, telephone voice signal will
have different frequencies at different times depending on the speaker and the
words being spoken. Music has different frequencies and amplitudes throughout a
particular performance.
I'm not sure what you mean by discontinuous. When people speak on the
telephone, there are often periods of silence between periods of speech signal.
Signals that have sharp rising edges like the unit step function and the
infinite impulse (Dirac Delta) function will obviously be band limited and
their shape and frequency content changed by the anti-aliasing low pass filter
used before sampling takes take place.
I don't think you get an exact reconstruction of the signal, but with proper
filtering after being converted by a digital to analog converter, you can get a
signal that sounds (or looks like, on an oscilloscope) a lot like the original
signal, such that it is recognizable and intelligible.
Music is band limited to 20 KHz and sampled at at least 44,100 Hz for recording
on CDs.
What is your application?
From: Doug Houghton doug_hough...@sympatico.ca
To: A discussion list for music-related DSP music-dsp@music.columbia.edu
Sent: Wednesday, March 26, 2014 10:42 PM
Subject: [music-dsp] Nyquist–Shannon sampling theorem
I can't seem to get to the bottom of this with the usual internet pages.
Is the test signal, while possibly containing any number of wave compenents at
various frequencies, required to be continous ansd uniform?
By this I mean you can't have frequencies jumping in and out, changing in
amplitude etc...
I'm guessing this somehow scratches at the surface of what I've read about no
signal being properly band limited unless it's infinit.
I fail to see how a readable proof is possible to explain exact reconstruction
of any real recording sound, whether it's music or crickets chirping.
I sort of see maybe how an infinit signal could solve some of these issues,
meaning any amplitude/frequency complexities over infinity may simply resolve
to something that can be bandlimited and described as a frequency of a steady
signal, something like that.
Curouis, I am starting to suspect there is a lot of typical misconceptions
about what the math really proves, I can't read the equations I'm turning to
this list.
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