Digital BW, The Print

> If you use dB values they are universal from
> end to end in the audio system.

For any system you use dB for...not just audio.

> Now we define "dynamic range".  Again its just a dB measure
> of the power ratio:  power of largest signal / power of the
> smallest discernable signal.  "discernable" is important in
> that we have to have some sound signal detectable above
> the noise level -- otherwise you might be tempted to
> consider a totally zero signal

Not really, as noise will have some power level to it (it will never be 0 or
below 0), and therefore will be above zero.  As I've said, as noise
approaches zero, your dynamic range approaches infinity.

> If we come up with a 60dB number for DyR,
> it means that the loudness sound is 60dB louder than the
> quietest sound.

OK

> The kicker is that since all the dB's
> are perceived equivalently we can say StepSize = .5dB,
> Range = 60dB and voila: number of steps = 60/.5  = 120
> different loudness levels.

That works for me as far as I can tell...

> With a fixed step size, the
> dynamic range specifies both the Height and NumSteps.
> Only one independent variable.

Well, I'm not quite with you here.  Dynamic range, in and of it self, only
specifies the "number of relative steps"...it has no reference to the
height, in and of it self, as dB doesn't relate back to what you are
determining the dB of.  In other words, you measured voltage of, say 6V, and
a "stepsize" of .01V, you would have a dynamic range of X dB = 20 (for
audio) log10(6/.01) = ~55dB.  But, there is no way to get back to 6V or .01V
just by saying 55dB.

> Moving more quickly now to the digital signal processing
> world, I don't know who designed DSP concepts and all their
> motivations.  But if you are designing a digital system to
> mimic all the audio systems I describe above, it sure makes
> sense to spec it such that you handle down to human
> discernable levels and no further.

That's debatable...as there are nth order harmonics that DO effect the
audio...but we can ignore that for this discussion.

> You'd use number of
> bits that fit the computer (16 usually) thus determining
> number of steps, the step size would be the human reception
> level (.5dB) and as the Higgin's shows on the very same
> page we calculate the dynamic range. Really its just the
> same as the analog with the addition 2^NumBits = NumSteps
>
> I think all this agrees with your statements about
> dynamic range.

I'd say you might want to 2x that just to be above the Nyquist limit, as I
do believe it applies to any sensory system...

> -----------------------
>
> Now let's look at it in the image world.
>
> To start: let's take a print with two grays that are very
> close but just discernable.  For the sake of argument let's
> say they differ in density by 0.03 log density units.  This
> is the minimum discernable density step.  This print came
> from a negative that has densities on it with in the printing
> process produce the density step on the print.  What is the
> density difference measured on the megative??  You can't
> tell.

Correct, there is never any DIRECT backwards relationship from the print to
the negative, except for some sense of relativity.  In other words, you can'
t jump tones around...

> The print may have been made on a grade 1, a grade 3
> or a grade 5 paper.  It the different cases you could come
> up what the negative difference must have been.  Maybe
> 0.021 or 0.024 or 0.027.  The negative by itself cannot
> tell you if the two tones on the print will be discernable.

Correct.

> This is very different than audio: a 3dB difference is
> always a 3dB difference no matter what stage you look at.

Correct.

> On prints there is no notion of dB tonal difference.

Oh, I disagree with that.  I can measure the tones, and since dB is a
relative scale (as is density BTW ;-) that gives me dB.

> Its
> not a matter of it hasn't been defined yet. As shown in the
> example its a matter of being impossible for the concept
> to exist.

Well, I disagree with that, and can't see where you made that "leap".

> The discernability changes from stage to stage
> because of the very nature of how we do imaging.

It matters not that it changes from stage to stage, since the dynamic range
of each stage is entirely separate.  No one said it had to follow through
the process!

>  (In fact
> I think its even more intractable than that:  As Martin
> observed a minimum discernable density on a single print
> varies from the dark portions to the light portions.
> Giving rise to the concept of Gamma.)

Yes it does, but that doesn't mean it doesn't have dynamic range!  Dynamic
range, as I've said in other posts, does NOT mean the steps HAVE to be the
same.

> So for Imaging its just impossible to state one property
> "dynamic range" and have it specify both the max /min
> and the number of step (or number of bits).  My argument
> basically goes back to the staircase model, with imaging
> its a two independent variable system.  Whereas with
> audio it was possible simplify the analysis to a one
> independent variable system.

Ever hear of integration?  I brought exactly this variability up in a
previous post.  I take it you missed that?

> ---------------------
>
> In our multitude of discussions, I never considered assuming
> all the steps would be the same.

Neither had I ;-)

> And I think your
> extensive audio experience lead you to assume there
> was "some" fixed step size.  Actually, more accurately, all
> the formulas that you brought over had the fixed step
> size builtin long, long ago -- no need to ever worry about
> variable size steps in audio.

Correct, but those were simply for a concept...as I said above, and said in
other posts, that isn't difficult to deal with using calculus.

BTW, typically, in a system such as this, the "smallest discernable signal"
can be taken from the worst case, if you want a single value.  That's a
simple and dirty way to do it, but it, as you can well imagine, isn't the
best way to characterize a system.

> I think this is the root of all our collective confusion.
> Maybe if we can agree on why we disagree, we can regroup
> and agree to agree on a new view.

Well, I believe you have flaws in what you believed I believed, and that you
are also trying to carry "dynamic range" from step to step.  The dynamic
range of the film is entirely different than the dynamic range of the
scanner and then of the print.  They all have their own dynamic range in and
of themselves, and none of them are directly related to each other.


Regards,

Austin