Digital BW, The Print

--- In DigitalBlackandWhiteThePrint@y..., "Austin Franklin" <darkroom@i...> wrote:
> 
> > 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.
> 

Well, well, well, Austin, its good to see these two paragraphs together.
The fallacy of your logic, your feeling of the whole dynamic range concept
is well illustrated here.  I'm sure you've been able to do your job for years.
Plugging in numbers is NOT the same understanding and feeling the concept.

Let's also add the formula:   DynRange = 10 * log10 (maxpower/minpower)
 and yes, maxpower = power of maximum signal 
 and  minpower i= power of minimum discernable signal

The basic fallacy is the mixing of linear scales and exponential (or geometric)
scales --  especially as it relates to "What is a step?".  In the real world just
about every ordinary human experience "feels" linear.  There's no difficulty
thinking of a 1 foot ruler and saying there's a 1 inch step so 12/1 = 12 steps
or 12 inches -- its so basic it seems absurd to even say it.   You are
looking at the DynRange formula and seeing the term (maxpower/minpower)
and "feeling"  that its the same thing as the ruler.  Its like minpower is the
step size and when you divide it into maxpower you get "how many steps".
This is exactly what you've said in the second paragraph above.
This is completely wrong as a concept.  Why?? Because power is an
exponential scale from a perceptual point of view.  I.E.  To get a "linear" feel
of loudness in the audio system, the power output has to go up exponentially.

Here's a short audio example:
We have an audio amplifier -- puts out 100 watts at the max end, and measuring
the low end (smallest discernable signal) we get 1 watt.    Let's throw that
into the DynRange formula:  the inside term is (100/1), you are thinking that
this is 100 steps.  Not TRUE!!
This would mean the steps would be:
1,2,3,4,5,6,7,8 ...95, 96, 97, 98, 99, 100  watts
but these steps make no sense, the intervals in the beginning are huge and
the ones at the end are miniscule.

The steps have to be like:
1, 2, 4, 8, 16, 32, 64
or
1, 3, 9, 27, 81
or
1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16 ....

Each of these sequencies produce and "even" set of intervals or steps.   Also
notice they all  have a different number of steps --- the original statement
100 watt -- 1 watt spec says nothing whatsoever about number of steps.
Just like a staircase going from floor 1 to 2, can be built with any number of
steps -- you just have to pick an appropriate step size.

Bottom line: In a exponential scale, a step is always a ratio, a range is
always a ratio.  StepRatio ^ NumSteps = RangeRatio

Yeah, but ...  What about the first paragraph?  Here I said:
Range = 60dB, Stepsize = 0.5dB  and then I calculated (60dB/.5dB)
getting 120 levels or steps.  Isn't this what I just argued against??
NO, not at all.  The reason is dB's are a linear scale so the ruler analogy
works just fine.  dB's are the log of an exponential scale.  Taking the log
converts an exponetial scale to a linear scale.  We as human beings use
this conversion all the time because we are so much more comfortable
having an intuitive feel for a linear scale.

This is pure and simple mathematical transformation.   There's no hand
waving arguments --- logs are defined very precisely.

Here's probably the most basic of log transformations:
log (A*B) = log(A) + log(B)
log (A/B)  = log(A) - log(B)

Look at these two terms:

log10 (max/min)    <<<<NOT EQUAL>>>>     log10(max)/log10(min)

The left side is the dynamic range term.  The right side is the number
of steps example and calculation.  Austin, although you've always punched
in the calcs like the left side, but you are thinking (feeling) like the right side.
It's produced a desire on your part to keep trying to get the same
"feeling" in Imaging but that has distorted your transformation from 
the audio world to the imaging world.

Taking the DynRange formula and transforming it a little:

DyR = 10 * log10 (maxpower/minpower)
DyR = 10 * (log10(maxpower) - log10(minpower))
DyR = ( 10*log10(maxpower)  - 10*log10(minpower))

Imagine measuring maxpower and stating it as a db relative to a standard,
let's call it maxdb.  Similarly, minpower as a db relative to the same 
standard, calling it mindb.  If   this isn't clear, check:
         http://www.phys.unsw.edu.au/~jw/dB.html

See now have:
DyR  = maxdb - mindb

Example:   connect the dbmeter on amp at full power -- reads 40dB
 with amp at min power -- read -20dB

So, DyR = 40dB - (-20db)  = 60dB

QED

All things called "ranges" are just a top end and a bottom end.  No more,
no "internal" information whatsoever.  "Range" can apply to a linear
scale and the size of the range = max - min.   "Range" can also apply
to an exponential scale, then the size of the range = max / min.
An exponential range can be converted to a linear range by taking the log.
A linear range can be converted to an exponential range by putting it in
the exponent.

Dynamic range is a range just like any other range,  the word "dynamic"
is just a qualifier that says "perceptually interesting part of".

In photography, we can "dynamic range" various different parts.  Some
of them are exponential scales like exposure, some have been linearized
by taking the log like density.   The concept of "stops" is also a linearized
form of the exposure range.

-------

Roy