Klarinet Archive - Posting 000729.txt from 1997/11

From: Ian Dilley <imd@-----.uk>
Subj: RE: Nyquist, web page ok
Date: Thu, 20 Nov 1997 11:18:23 -0500

You now seem to agree that Nyquist is correct and that the definition on
the standards page is correct but have shifted your attention to the
resolution of the sampling as opposed to the sampling rate.

The problem you describe is entirely due to the resolution of your
samples. You only allow 10 sample levels. In a CD recording there are
65535 sample levels (not 32767). This is many more levels of volume
than the human ear can resolve so it seems reasonable to me that from
this sampled data you can reconstruct a waveform which is
indistinguishable from the original to the human ear. Surely this is
all that is necessary and not an exact reproduction of the original.

Ian Dilley

-----Original Message-----
From: Jerry Korten [SMTP:jerryk@-----.com]
Sent: Thursday, November 20, 1997 4:03 PM
To: klarinet@-----.us
Subject: Nyquist, web page ok

Well, I understand now why I am in disagreement with the web
page. In the
case of a sampled waveform, in a non A/D world, you can in fact
represent
the waveform provided you use an interpolating function
sin(2*pi*t/T/(pi*t/T)) to filter (interpolate) the sampled
signal in order
to reconstruct the sampled signal. I guess the standards page is
correct
;^)

This issue remains however, that a signal sampled by an A/D
cannot be
"exactly" reconstructed from its samples. I thought this over
last night
and have decided that an example is the best way to describe
this:

An A/D is a discrete quantizing device. It takes a varying
analog signal (a
voltage) and converts it into one of several levels. For a 16
bit A/D it is
32767 levels. But for the sake of example I would like to refer
to an
imaginary A/D converter that converts our music signal into 10
levels. Say
we have things set up so that a voltage varying from 0 to 10
volts are
digitized such that anything between 0 and .99 volts gets a
value of 0,
anything from 1 to 1.99 gets a value of 1, a voltage from 2 to
2.99 a value
of 2 and so on up until 9 A/D units (0 - 9 or 10 different
possibilities).
The A/D converter will "look" at this voltage once every
hundredth of a
second. This is called the sample period.

Now we can visualize the problem with the sampling theorem when
applied to
A/D signals for the following reason:

Suppose the signal was ramping up so that when we first sample
it the level
was 1 volt, this would correspond to a 1. The next period the
waveform is
at 2.9 volts corresponding to a reading of 2. The next reading
is 3
corresponding to an A/D reading of 3 and so on. Our sampled
sequence is 1,
2, 3

Now we consider a voltage that starts at 1.1 volts (A/D value =
1), goes to
2.0 volts (A/D value @-----. Our sample
sequence is
now also 1, 2, 3.

The interpolation requried to reconstruct the sampled signal
when applied
to these two equal data streams will produce the same "replica"
of a
waveform. And only one waveform is possible from the
interpolation. But we
clearly saw two different waveforms being measured.

This is why something that quantizes to discreet values (an A/D
converter)
cannot behave the same as a continuously sampled system which
exists only
in textbooks and minds.

The effect of this quantizing affects both magnitude and phase
of a signal.

I'm sorry for the laborious sequence above, but we musicians
(whether by
avocation or vocation) are now in the digital age and it is
incumbent upon
us to understand the shortcomings of this medium.

Jerry Korten
NYC