If this appears twice, it is because I thought it was rejected first time
due attachment. Here it is again, without attachment.
Austin - maybe we are zeroing in on something here, I hope so.
As I read through your replies, light bulbs keep flashing 'aha!!' above my
head as to what is happening here, so let me have a go at some of
them...hopefully we might come to a resolution.
Anybody else reading this - my apologies for the length but it seems there
is no other way. If you have any remaining interest in this topic at
all(!) please at least skim all of this. The meat is in the middle.
My biggest aha! - more like an ahaaaaaaaa - was at the following:
>It is VERY easy to show your interpretation of terms simply makes NO sense:
>Largest signal - 100V
>Smallest signal - 98V
>Overall range 2V
>By your interpretation of terms, the dynamic range would be 100/98 or 1.02.
>Largest signal - 3V
>Smallest signal - 1V
>Overall range 2V
>Again, by your interpretation of terms, the dynamic range would be 3/1 or 3
>How can that be? They both have the SAME overall range of 2...one is simply
>offset by 97V.... They SHOULD have the exact same dynamic range, why not?
>Again, they are BOTH of the same overall range, merely offset...and offset
>should not effect dynamic range, as it is merely an offset...and if you
>believe offset should, then please explain of what use that would be?
You are talking about *DC signals* and are therefore concerned about
offsets!! Aha!! *DC signals*!!!. BIG LIGHT BULB. Signal theory and all
the definitions of dynamic range that I know of are discussing AC signals -
or DC levels by pre-supposing any offsets are removed. Of *course* the
effect of any offset will have to be removed to make any sense of anything,
because - as you point out - an offset is entirely arbitrary. This also
explains maybe why you were having a problem with RMS
In your example above, *IF* your 100v and 98 v etc were AC signals (and so
the 98V is the noise level), then the dynamic ranges are exactly as you
disparagingly calculate them, 1.02 in the first case and 3 in the second
case. That is correct, and that is what the definition formula says. Do
you have any trouble with that?
But I now realise you were actually talking about a 98VDC minimum, and a
100V DC max signal, hence your above quoted thoughts, and hence a lot of
the confusion in our discussions.
Now if I change my way of thinking to DC signals, then I can see why you
say what you are saying, but you then have made life extremely complex for
the more usual situation and ruined a perfectly good definition, and then
had to make up your own. I have no problem with any of this, except that
you HAVE made up your own definition to get rid of offsets, and this is not
only irrelevant to signal theory, it is slightly inaccurate. The
definitions in the books are AC signal based or assume no offset, which
seems to me to be self-evident.
In the case of your above example, I would have just subtracted the 98v
from the first example before applying the book definition. I mean that is
trivial and preserves the normal definition of dynamic range. The REASON
that signal theory assumes AC signals is precisely because it is concerned
with noise, and noise is by definition, a varying signal and therefore
AC. This is also what makes it so easy to remove offsets, because it is so
easy to distinguish between the non-varying part of a "minimum signal" (the
offset) and the varying part (the noise).
> > >Julian: This example system for some reason has a noise of 1V,
> > a smallest
> > >discernable signal of 2V and a largest signal of 10V.
> > You often tell me that noise and smallest discernable signal are not
> > necessarily the same thing and I agree with you.
>No, that is a misinterpretation of what I said. I have said that the
>smallest signal level and noise are not the same thing. The smallest
>discernable signal and noise ARE the same thing. What you mean is "smallest
>signal" or "smallest signal level" or whatever you want to call it, is 2V,
>NOT "smallest discernable signal", as you have already defined noise as
>being 1V, so you are using two terms that mean the same thing and assigning
>them two different numbers.
??? You have made up an entire lexicon to support an unusual view of
dynamic range! I THINK I see what you are doing in all this, I just don't
know why, unless it is ENTIRELY to do with DC signals in which case as I
said, it is unnecessary. IF you want to allow DC offsets within your
calculations, then you need these definitions I agree. But it is much
easier to just remove any DC offset first, then use the book definition of
dynamic range. And it gives you the right answer, not a slightly incorrect
EG1) Use your "DC" example of 98 min signal to 100 V max signal. Say the
noise is 100mV.
CASE 1 - doing it your way.
First: what is your smallest discernable signal, and what is the smallest
signal level? I'm damned if I know. If 98 is the minimum POSSIBLE level,
then the minimum SIGNAL level must be 98+noise = 98.1V. Then putting the
correct values into your (incorrect) formula gives(100-98.1)/0.1 =
19. Unfortunately for my ease of discussion, putting in the INCORRECT
value into the (incorrect) formula in this case gives the right answer
thus: DR = (100-98) / O.1 = 20.
CASE 2 - Use normal definition
Remove the offset by subtracting the offset DC component of 98V from the
upper and lower limits. Then the max signal is 2V, the min signal is 100mV
(i.e. noise) and DR = max/min = 20.
In this example, as I said, your formula is incorrect but by using an
incorrect value as well, you get the right answer. This becomes more clear
in the next example.
EG2) Example using AC signals:
Say noise = 100mV = "min" (= minimum signal = minimum discernable signal),
and Max signal = 2V = "max". (Note that under AC conditions minimum signal
and minimum discernable signal are the SAME, always. Minimum discernable
signal and noise are the same, NEARLY always).
CASE 1: DR by your definition = (max-min) / noise = 1.9V / 100mV =
19. This is unfortunately just plain wrong again and is the reason I
originally got into this discussion. Not just because of the (usually
small) difference in the numbers, but because it was confusing to the
discussion in terms of definitions and what things meant.
CASE 2: DR by book definition = max/min = 2V/100mV = 20.
Now I KNOW you will object here and say that "max" should be"max absolute
signal" or something, but it just isn't true. There is no range involved
in that figure of "max", it is just a level. There is no justification for
using this difference, and it is not supported or mentioned in tests. It
also gives you the wrong answer!!! Heck, please take a moment to think
about this.Take a look at the books, the numerator in this equation is NOT
a range, it is a straightforward level. The dynamic range is the ratio
between two levels, that's all it is. It is not a ratio between a range
and a level. The DR in this example is definitely 20, and not 19.
I don't know how to get you to accept that your calculation of DR is
incorrect, but here's three attempts.
I'd say please read the book definitions critically. Look at the
definition of DR and see that they are NOT saying that the numerator is
something minus something else. They say, exactly, that the numerator is
the "largest signal". This is NOT the largest signal minus the smallest
signal, it is just the largest signal. Using the second of the above
examples (2V, 100mV, AC), if you had a meter, you'd stick it on the output
and read 2V when the thing was limiting. That's what you'd read, and
that's what you'd put as the numerator of the equation. If you removed the
signal source so you were looking at noise on the output, you'd read 100mV,
and that's what you'd put as the denominator. It is really that simple.
Elsewhere you say:
>Absolutely not, as it is wrong. You are confusing the absolute largest
>signal (as in //, like absolute numbers) with highest signal level. The
>"largest" signal (as in absolute) is (max measured signal - minimum
>measured) signal or 10-2 or 8.
No!!! This "absolute largest signal" is TOTALLY your creation.
Here again is the formula (YOUR QUOTE) you are referring to:
DR (dB) = 10log10(largest signal/smallest discernable signal)
I am not confusing highest signal level with "absolute largest signal",
simply because I'm not even MENTIONING "absolute largest signal"! I don't
even consider "absolute largest signal" and neither does the
formula. There is no need to put it anywhere, to write it down, think
about it, or to consider it in any way. This "absolute largest signal" is
your creation which you require to get rid of offsets, but you do NOT need
to use it even for DC offset situations, and you CERTAINLY don't need it in
AC situations where it is not just wrong, it is meaningless - and
confusing. What exactly IS the difference between two AC signals of
different phase and frequency? It is a complex matter, and is not
meaningfully represented by subtracting the two amplitude values. V1 - V2
is totally meaningless unless you know the exact frequencies and phases
involved, it is a vectorial thing. The "largest signal" in the definition
is just that - the largest signal, or what you called "max measured
signal". I know that this does not agree with your thinking because this
interpretation does not give the same answer as your own personal formula
does. None-the-less, it is correct and I wish you'd see that and let me
get on doing important things like eating. The problem is with your own
As a cross check - another reference of **YOURS** defines DR in the
following terms: DR = "maximum level RMS" / "noise level RMS". (see the
actual formula at
this case too, the maximum level is just what it says, it is NOT a
difference between two levels. It is just and simply what you would
measure on your meter, oscilloscope or whatever. Why do you insist on
interpreting straightforward things in a complex way? The only answer is -
to make the book definition of DR fit with your personal definition.
Please show us a reference to YOUR definition. You say it is not your
personal definition, so please show the reference. I have used the
definition that YOU quoted throughout from a book, and this accords with my
understanding. Please show your OTHER definition that gives you the answer
of 19 in the above second example. That is, I guess, where the book talks
about the numerator as being the difference between two levels.
You cannot go on saying your two definitions are equivalent because you
choose to change the meaning of one term in the book definition! The fact
is that when numbers are applied by anyone other than you to the two
equations, they come out with two different answers!
Use your definition on the following example and give me the answer in log
The perfectly possible situation of 10V max signal and 6V noise.
The book definition has no trouble -
DR (dB) = 10 log (10/6) = 2.2dB
Your definition takes us into the negative logs. Hmmm.
I can demonstrate the relationship between YOUR equation and the BOOK
equation if you like, as follows.
This example has max = 1000mV, min = 10mV.
First calculate the dynamic range...
Dynamic Range = 1000mV / 10mV = 100. That *IS* the dynamic
range! Truly, it is.
THEN, and only then, I'd say that the number of STEPS you need in an A/D
converter to accurately represent that range is given by your equation:
Number of steps = (range in terms of a difference) / (resolution of
[I think you'd agree with this equation. The numerator is what you would
call the "absolute range".
The resolution is as you keep saying and I think we all agree, the noise.]
Num of steps = (1000 - 10) / 10 = 99 steps.
Eq(2) is your equation on the right hand side, and it gives the answer that
is NEARLY the same as the dynamic range.
The Dynamic range, expressed as a ratio to one, is in fact (in the normal
case) exactly the same as the number of LEVELS that your A/D should resolve.
Number of LEVELS required in this A/D is one more than the number of STEPS.
Number of LEVELS = number of STEPS + 1 = 99+1 = 100. Eq(3)
[To illustrate why you add 1 to get the number of levels, and my def of
steps and levels, see
As I keep saying, what you are calculating with your formula is the number
of steps, not the dynamic range. I grant you that it is close to the same
number in the normal case where min signal is the same as noise, and I
grant you that using your equation you get NEARLY the right answer, and I
grant you that your method will get rid of DC offsets, but the point is
that what you are doing in the name of removing offsets is confusing, it
changes the definition of dynamic range, and it gets the wrong answer.
Going back to the relationship between your formula and the book formula
for DR. The reason the calculations for number of A/D levels and the
formula for dynamic range give the same answer when min signal = noise is
because the two formulas collapse to be exactly the same... as follows ...
NB noise = min...
Dynamic range = max / min ....(1)
Austin's formula = (max - min) / min = num of A/D steps...(2)
Num of A/D levels = Austin's formula plus 1
= (max-min) / min +1 ...(3)
= max/min - min/min + 1
= max/min - 1 + 1
= dynamic range formula!!!
So the relationship in AC terms between your formula and the DR formula is
that the DR gives the number of resolvable levels, and your formula gives
one less than that, or the number of resolution steps.
My point is:
a) that your way just makes it complicated in that you have to keep track
of strange and otherwise uninteresting things called minimum signals as
distinct from minimum discernable signals for no reason. You then have to
make up your own definition of DR using these terms to deal with it.
b) that when you apply your definition it gives you the wrong answer.
As a PS I need to say that I only introduced min discernable signals that
were different from noise in yesterday's post to accommodate what I thought
you were saying. I agree that almost always min discernable signal and
noise are the same thing and everything I have said today assumes min disc
signal = noise. It *IS* in fact perfectly possible for a device to have a
different noise from it's smallest discernable signal, but only I think if
it is non-linear. And in such a circumstance it is only the book
definition that will give you the correct dynamic range. Your formula
would be even more incorrect, because as I said yesterday, the dynamic
range is dependent ONLY on max signal and min signal. When the min signal
is NOT determined by noise, then your formula has a problem because it
still has the noise term in it.
If you want me to give you an example of a system with min discernable
signal not equal to noise, say so and I'll do it separately.
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