On 7/22/15 10:16 PM, Peter S wrote:
You have your signal S. When you digitize that signal, you add the
noise floor of the ADC (among other noises), let's call it N1. When
you reconstruct the signal, you add the noise floor of the DAC (among
other noises), let's call that N2. So you have
S + N1 + N2
okay, since there is no processing, just passing the signal from A/D to
D/A converter, there is only one quantization operation, at the A/D. if
it's an old-fashioned "conventional" A/D, the quantization operation is,
essentially, a non-linear staircase operation and the error is a
measurable and then predictable function of the input. if it's "properly
dithered", the error will be well described as "noise" that's white with
DC=0 and AC power decoupled from the input value. with noise-shaping,
the noise wouldn't have to be white, but the AC power level would
increase. sigma-delta (Σ-Δ or Δ-Σ) converters might have a similar
signal+noise model. that's N1.
there *is* error at the D/A, due to non-linearity, but i wouldn't call
that noise. and it's a function of the input. if it's a "1-bit" D/A
(a.k.a. delta-sigma) the effective mathematical non-linearity sorta goes
away, even if the physical non-linearity remains stark (a binary step
function). that's N2.
Then you subtract the original signal (let's disregard the delay) and
measure what you have:
S + N1 + N2 - S
Since S - S = 0, what you're measuring is:
N1 + N2
In other words, you're measuring the combined noise floor from your
ADC and DAC (among other noises). And you're trying to "fix" that
noise.
Here's why it won't work - the noise floor of the ADC and DAC are
pretty much close to uniform distribution noise, without any pattern.
The lowest bits are just random.
you need to read about Δ-Σ converters a bit. one of the earliest papers
on the concept of (from and audio POV) is "Resolution Below the Least
Significant Bit in Digital Systems with Dither"
http://www.aes.org/e-lib/browse.cfm?elib=4523 . there is information,
even in those lowest bits that look pretty noisy. if the signal was DC
3/4 between to adjacent LSB levels, the noisy bits would have a mean
that is 3/4 one level and 1/4 the adjacent level. the lowest bits are
not *only* random, but they have both a deterministic and a stochastic
component.
In order to be able to "fix" that
noise, you'd need to be able to "predict" that noise.
or make use of the statistics of that noise (known in advance).
But since you cannot "predict" noise,
uniform p.d.f. white noise cannot be predicted at all (except that it's
within the rails of the uniform p.d.f.). that's what you get when you
hook up a really good random number generator to a D/A. but if the noise
is colored (and you know about the spectrum) and if the noise has a
p.d.f. that is not uniform, you can make guesses about where the next
sample is that are better than wild-assed guesses.
Peter, if you can take or sit-in some grad courses, i might recommend a
course in Statistical Communications or at least a course in Random or
Stochastic Processes. there is ostensibly some stuff you're missing here.
there's pretty much nothing you
can do to "fix" that noise... Once you reach the noise threshold of
your ADC/DAC, there's nothing you can do to improve your
reconstruction (other than, buying another soundcard that costs a few
thousand dollars more, and has a lower noise floor).
depends on what we have available for sample rates. essentially we are
only limited by the laws in Information Theory. if i have a 192 kHz
system and i only need to measure a 30 Hz waveform, there is a lot i can
do to mitigate noise.
If your soundcard is not so high-end, and has a slight roll-off at
high frequencies, then the best you may do is correct that with a
filter, and improve your readings slightly. About that's all you can
do, you cannot "fix" noise.
If noise were "fixable", then sound card manufactures would already be
doing it.
they're doing it. it's called "noise shaping" or "error shaping". it's
how we mitigate the huge quantization error of a 1-bit converter and
make it sound like a 20-bit converter, and a linear one at that.
How much you can "fix" noise, correlates with how many
thousand dollars you want to spend on your sound card, and even that
works only up to some limit, above which, there's nothing you can do -
you have noise, whatever you do.
that's just not always true. it depends on what you know about the
signal you're after. if it's not broad-banded, there *are* things you
can do about the noise getting added to it.
--
r b-j r...@audioimagination.com
"Imagination is more important than knowledge."
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