[Dick Pierce[_2_]]

Edmund wrote:
On Thu, 14 Feb 2013 22:14:07 +0000, Dick Pierce wrote:


Scott wrote:

On Wednesday, February 13, 2013 7:41:41 PM UTC-8, Dick Pierce wrote:

Scott wrote:

"The Nyquist theorem (which is mathematically proven) says
that the exact waveform can be reproduced if the original signal is
frequency limited to less than half the sampling frequency."

The quote you supplied does NOT say that "digital is perfect."

In effect it does.

To you. It does not to me. It simply, to me, states that when the
nyquist criteria is met, and that means the signal must be limited to
less than half the sampling rate, samplig does NOT lose any information
needed to reproduce an exact replica of the signal meeting the
criterion.


So if I offer you an analog signal, limited to 40kHz you can sample that
at 80 kHz and 4 bit, you can recreate the input signal exactly?
Well, would you like to prove that?

You just went and changed the conditions. To this point, all
the discussion had to do with sampling. One other poster is
clearly confused about the difference between sampling in the time
domain and quantization in the amplitude domain, and it appears
that you have similarly bundled the two together as well.

Let's examine your statement, for it contains several errors:

"So if I offer you an analog signal, limited to 40kHz
you can sample that at 80 kHz

First error, the bandwidth must be limited to LESS THAN HALF
the sample rate, NOT half the sample rate.

"and 4 bit"

Second error, just like in the sampling error above, you made
an assumption about the signal which may, in fact, not be true.
Now IF the dynamic range is limited to 24 dB, then with proper
dithering, yes, all of the information present in the original
signal WILL be available at the output of the system.

Now, the separation between sampling and quantization is NOT
some clever symantic trick, rather it is at the very basis of
the process. If it helps, you can think of the sampling
process as quantization in the time domain, and what many here
term "quantization" as quantization in the amplitude domain.
Assuming the wo are inextricably tied together is the root
of much confusion, as exhibited by your question as one example.

Let's rephrase your question slightly, relating it to more
practical terms: what is the capability of a sampling system
with a sample rate of 88.2 kHz (twice Redbook CD rate) and a
signed 16-bit linear integer quantization (redbook CD spec),
using adequate dithering (or noise shaping)?

Well, IF the input signal is limited to less than half the
sample rate, let's say 40 kHz, and the dynamic range of the
input signal does not exceed 96 dB, ALL the information in
that signal will be captured and be available in the output
of the system. NO information in either the time domain or
the amplitude domain will be lost.

Now, recall where I said that the resolution of the system is
defined by the product of the bandwidth and quantization level?
That allows us to trade off one form the other, for example,
I could double the sample rate (to 176.4 kHz) and, in doing
so, gain an extra 3 dB of resolution, and, assuming my signal
is STILL limited to 40 kHz, I can now use a 15-bit encoding
instead of 16 bit and still achieve that 40 kHz, 96 dB
resolution with no information loss. Double it again
(352.8 kHz), and I get to throw away another bit and still
achieve the same base-band resolution (40 kHz, 96 dB).

And, by the way, that 96 dB I stated is the BROADBAND dynamic
range: it DOES NOT represent a hard floor like so many assume,
it is simply the braodband noise floor of the system: valid
signal information is still encoded BELOW that noise floor, so
that narrow-band avergaing systems, like spectrum analyzers
or human hearing, are QUITE capable of hearing the real signal
BELOW that noise floor.

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+--------------------------------+
+ Dick Pierce |
+ Professional Audio Development |
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