Klarinet Archive - Posting 000715.txt from 1997/11

From: "Jerry Korten" <jerryk@-----.com>
Subj: Nyquist, web page ok
Date: Thu, 20 Nov 1997 11:02:50 -0500

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