In article , Ian Bell
wrote:
Jim Lesurf wrote:
Once you apply dither correctly then the level at which signals can be
resolved is limited by the duration of the signal and/or its
coherence/predictability. If the noise is white then this would be the
same across the band. If the noise is shaped it may depend upon the
relative spectra of the noise and the coherent signal.
No problem so far but the resolution that can be obtained below the ls
bit depends on the number of samples taken per cycle.
The question here is, why are you assuming we have to impose a duration in
each case which is limited to just one cycle? Also a secondary question
about your unstated definition of "resolution". Please see below...
For a 1KHz tone this is about 44 or equivalent to another 5 or so bits.
At 20KHz it's 2 samples equivalent to about 1 extra bit.
Yes *if* you are assuming that dither, etc, only works over one cycle.
However this isn't the case.
Lets consider two different situations/arguments which I think will show
what I mean.
Case one is where we have a low level sinewave whose level is fixed, but
whose frequency can be either 1 kHz or 20kHz and we sample this, with
'random' dither for one second. Here we assume the dither pattern also has
a white power spectral density ('white noise').
No matter which signal frequency we have chosen, we now have 44,1000
samples over which we can perform detection processes (e.g. correlation or
Fourier Transformation, etc). It is correct that each individual 20kHz
cycle is only 1/20th of the length of each 1kHz cycle - however this is
balanced by having 20 times as many of them in the total duration.
We have to avoid using the 'same' added random dither in each cycle. i.e.
we must have dither which shows no patterns across our total duration of 1
second (44,100 samples). If we do so, and correlate/ detect/ filter/
Fourier/ whatever we end up with the same level of recovery from the
'noise' of the dither irrespective of the choice of the signal frequency
buried in the 'noise'. This follows from the usual correlation gain
arguments against the white noise.
In terms of normal spectral arguments we have the same noise power density,
uniformly cut up into 1 Hz bins by having a 1 seond total duration for our
process. Thus in both the 1 kHz bin and in the 20 kHz bin we have only have
1/20,000'th of the noise power from the dither above which our signal can
now poke.
Case two is a different argument based upon the properties of human
perception. Here we allow for a specific detection process where hearing
essentially divides the incoming signal into frequency bins whose bandwidth
varies with frequency. If you use Bob Stewart's arguments these are of the
order of about tenth of an octave (but depend upon the signal level, which
I'll ignore here).
As the bandwidth scales with the center frequency for each hearing sensor
'bin' we find that the background noise hitting each sensor rises with
frequency if we have white noise. Thus in this case we would need a bigger
signal in the first place at higher frequency to be noticable against a
white background. This can be argued to be one of the mechanisms that lead
to our hearing sensivity degrading as we go above a few kHz.
However the key point I wish to make is that "case two" is not really
anything specifically to do with CD or digital. It relates to having a
quite specific detection system (hearing) that has a specific property.
In fact we can now add:
Case three. Here the dither noise does not have a white spectrum. This is
because of a different property of human hearing, namely that the actual
sensitivity falls rapidly as we move above a few kHz. (In some ways this is
a sensible adaptation given the above as otherwise the higher noise power
in the wider bins would tend to become an annoyance. Having lower
sensitivity drops this back down again.)
This means it makes sense to fiddle the spectrum of the dither so that most
of the dither power is at the frequencies where our perception is least
sensitive. It still works OK as dither. It still gives the same correlation
gain, so it improves the detection SNR by the same amount at 1kHz and at
20kHz. However now we have more noise at 20kHz, Hence we may find that
although the improvement in SNR is the same in both cases, for the 1kHz
case the signal may now poke above the noise in the 1kHz bin, but the 20kHz
signal may not against the higher noise spectral density at 20kHz.
However this result is a cosequence of choosing to have dither with a
non-white spectrum.
Since the noise is spectrally shaped to be as inaudible as possible, if we
can't hear the noise, then the 20kHz would be too low in level to be
audible as well. Hence in a sense this does not matter, although this is a
different point to the one I am making above.
For the above reasons your statements may be at cross purposes with mine as
you are making different assumptions about what we are discussing. But it
leads me back to repeating my initial statement at the top of this message
as being correct for CD or any other general signal system. The
implications depend upon what you are applying the system for, though,
hence cases two and three. :-)
Slainte,
Jim
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