Why is electric pulse attenuation measured in dB using natural logarithms?

[Efthimios]

Hallo to all,

I have been reading a paper regarding electric pulse attenuation and
the values are quoted in dB. The conversion to dB was done by
multiplying the value with (20log10e) where "e" is the base of the
natural logarithm.

I see this conversion to dB fist time. I can not find any information
in my math books or the web.

Does any of you have any information regarding this that I can read so
I can understand it better?

 

[Rich L.]

On Dec 6, 12:37 pm, Efthimios <eangelopou...@gmail.com> wrote:
> Hallo to all,
>
> I have been reading a paper regarding electric pulse attenuation and
> the values are quoted in dB. The conversion to dB was done by
> multiplying the value with (20log10e) where "e" is the base of the
> natural logarithm.
>
> I see this conversion to dB fist time. I can not find any information
> in my math books or the web.
>
> Does any of you have any information regarding this that I can read so
> I can understand it better?

"dB" is short for "decibel". It is a logarithmic power scale relative
to some reference power. The most general formula is:

dB = 10 Log10( P1/P0)

Where P1 is the power being described and P0 is the reference power
level. On this scale zero dB means the power is equal to the
reference power level. 10 dB means the power is 10 times the
reference level. 20dB means it is 100x the reference level. etc.

When used often the reference level is not stated explicitely. Really
it ought to be. In low power RF (radio frequency) electronics they
often use dBm meaning decibels wrt 1 milliwatt. You also sometimes
see dBuW (wrt to 1 micro Watt) and dBW (wrt 1 Watt). As electrical
signals are often measured in volts, you often see decibels defined
as:

dB = 20 Log10 (V1/V0)

This is correct assuming that the impedance of the circuit with V1 and
V0 are the same, and that the power is related to V^2/R. If the
impedances are not the same (e.g. between the input and output of an
amplifier) then this formula is not correct. Decibels are always a
power ratio expressed as a log.

There is more on this in the Wikipedia article on
"decibel" (www.wikipedia.org)

Rich L.

 

[Efthimios]

On Dec 7, 4:01 am, "Rich L." <ralivings...@sbcglobal.net> wrote:
> On Dec 6, 12:37 pm, Efthimios <eangelopou...@gmail.com> wrote:
>
> > Hallo to all,
>
> > I have been reading a paper regarding electric pulse attenuation and
> > the values are quoted in dB. The conversion to dB was done by
> > multiplying the value with (20log10e) where "e" is the base of the
> > natural logarithm.
>
> > I see this conversion to dB fist time. I can not find any information
> > in my math books or the web.
>
> > Does any of you have any information regarding this that I can read so
> > I can understand it better?
>
> "dB" is short for "decibel". It is a logarithmic power scale relative
> to some reference power. The most general formula is:
>
> dB = 10 Log10( P1/P0)
>
> Where P1 is the power being described and P0 is the reference power
> level. On this scale zero dB means the power is equal to the
> reference power level. 10 dB means the power is 10 times the
> reference level. 20dB means it is 100x the reference level. etc.
>
> When used often the reference level is not stated explicitely. Really
> it ought to be. In low power RF (radio frequency) electronics they
> often use dBm meaning decibels wrt 1 milliwatt. You also sometimes
> see dBuW (wrt to 1 micro Watt) and dBW (wrt 1 Watt). As electrical
> signals are often measured in volts, you often see decibels defined
> as:
>
> dB = 20 Log10 (V1/V0)
>
> This is correct assuming that the impedance of the circuit with V1 and
> V0 are the same, and that the power is related to V^2/R. If the
> impedances are not the same (e.g. between the input and output of an
> amplifier) then this formula is not correct. Decibels are always a
> power ratio expressed as a log.
>
> There is more on this in the Wikipedia article on
> "decibel" (www.wikipedia.org)
>
> Rich L.

Thanks Reich, but have you seen the expression (20log10e) where "e" is
the base of the natural logarithm.?
This Log is multiplied with the value of the attenuation to convert it
to the logarithmic dB.

I am trying to see where this particular strange (20log10e) expression
of comes from. This does not look like the normal dB expression that
you mentioned above.

Brgds

Efthimios

 

[William R. Frensley]

Rich L. wrote:
> On Dec 6, 12:37 pm, Efthimios <eangelopou...@gmail.com> wrote:
(on the subject of the decibel)
>
Let me add one thing to Rich's excellent summary: While the formula
10\log_{10}(r) looks completely arbitrary, ratios of 1 dB \approx 1.26
turn up in a surprising number of places, always in some way associated
with human perception. The association of dB with audio technology is
well-known, and the quantization of modern digital volume controls is
almost certainly 1 dB steps.

A number of years ago, I needed to design an optical resolution test
pattern, consisting of different sets of lines with varying pitch.
After a bit of drawing I concluded that stepping the pitch such that
it doubled every three patterns was just right. In other words, the
ratio between two steps was cube root of 2, \aprox 1.26.

The most surprising occurrence of a dB scale was found when a
colleague and I attempted to estimate the salary scale at the company
for which we worked. The annual stockholder's meeting report gave us
enough information to estimate the salary of the Vice President at the
top of our reporting chain, and a simple calculation gave us a factor of
about 1.25 per level of management.

- Bill Frensley

 

[Rich L.]

On Dec 7, 10:50 am, "William R. Frensley" <frens...@utdallas.edu>
wrote:
> Rich L. wrote:
> > On Dec 6, 12:37 pm, Efthimios <eangelopou...@gmail.com> wrote:
>
> (on the subject of the decibel)
>
> Let me add one thing to Rich's excellent summary: While the formula
> 10\log_{10}(r) looks completely arbitrary, ratios of 1 dB \approx 1.26
> turn up in a surprising number of places, always in some way associated
> with human perception. The association of dB with audio technology is
> well-known, and the quantization of modern digital volume controls is
> almost certainly 1 dB steps.
>
> A number of years ago, I needed to design an optical resolution test
> pattern, consisting of different sets of lines with varying pitch.
> After a bit of drawing I concluded that stepping the pitch such that
> it doubled every three patterns was just right. In other words, the
> ratio between two steps was cube root of 2, \aprox 1.26.
>
> The most surprising occurrence of a dB scale was found when a
> colleague and I attempted to estimate the salary scale at the company
> for which we worked. The annual stockholder's meeting report gave us
> enough information to estimate the salary of the Vice President at the
> top of our reporting chain, and a simple calculation gave us a factor of
> about 1.25 per level of management.
>
> - Bill Frensley

I recall an article in Scientific American in the 70's or early 80's
making a similar observation about such random things as the first
digit of street addresses, the area of lakes (irrespective of the
units used), etc. It appears that any measurement that spans a
population of more than a few factors of 10 shows a similar
logarithmic frequency of the first digit. Not quite your observation,
but possibly related.

As for Efthimios' question: I was assuming that the "e" in that
expression was the voltage (sometimes represented by "e" or "E" for
electric field. If he is using the natural logarithim, it seems like
he is doing some conversion from a base 10 log to a base e log, but
that doesn't make sense for decibels. By definition dB is base 10
log.

One point that I didn't cover that might be key to understanding
this. What makes dB so handy in electronics is that amplifier gains
and attenuator losses are multiplicative properties. That is the gain
is 2x or the loss is 0.75x, and to get the output level you multiply
the input by the gain/attenuation. Since dB is a log of the signal
level, when working with dB signal levels you add/subtract the gain/
attenuation to get the output level in dB.

What you are describing does not make sense to me. If you would like
me to look at the paper I'd be happy to try to figure out what they
are doing. That is assuming it is in English, however. I'm an
American and thus monolingual...

Rich L.

 

[Murray Arnow]

Efthimios wrote:
>Hallo to all,
>
>I have been reading a paper regarding electric pulse attenuation and
>the values are quoted in dB. The conversion to dB was done by
>multiplying the value with (20log10e) where "e" is the base of the
>natural logarithm.
>
>I see this conversion to dB fist time. I can not find any information
>in my math books or the web.
>
>Does any of you have any information regarding this that I can read so
>I can understand it better?
>

It looks like is using Nepers, which is the natural log of the power
ratio. See

http://www.sengpielaudio.com/calculator-neper.htm

for details about converting Np to dB.

 

[Jim Black]

On Fri, 7 Dec 2007 16:50:31 +0000 (UTC), Efthimios wrote:

> On Dec 7, 4:01 am, "Rich L." <ralivings...@sbcglobal.net> wrote:
>> On Dec 6, 12:37 pm, Efthimios <eangelopou...@gmail.com> wrote:
>>
>>> Hallo to all,
>>
>>> I have been reading a paper regarding electric pulse attenuation and
>>> the values are quoted in dB. The conversion to dB was done by
>>> multiplying the value with (20log10e) where "e" is the base of the
>>> natural logarithm.
>>
>>> I see this conversion to dB fist time. I can not find any information
>>> in my math books or the web.
>>
>>> Does any of you have any information regarding this that I can read so
>>> I can understand it better?
>>
>> "dB" is short for "decibel". It is a logarithmic power scale relative
>> to some reference power. The most general formula is:
>>
>> dB = 10 Log10( P1/P0)
>>
>> Where P1 is the power being described and P0 is the reference power
>> level. On this scale zero dB means the power is equal to the
>> reference power level. 10 dB means the power is 10 times the
>> reference level. 20dB means it is 100x the reference level. etc.
>>
>> When used often the reference level is not stated explicitely. Really
>> it ought to be. In low power RF (radio frequency) electronics they
>> often use dBm meaning decibels wrt 1 milliwatt. You also sometimes
>> see dBuW (wrt to 1 micro Watt) and dBW (wrt 1 Watt). As electrical
>> signals are often measured in volts, you often see decibels defined
>> as:
>>
>> dB = 20 Log10 (V1/V0)
>>
>> This is correct assuming that the impedance of the circuit with V1 and
>> V0 are the same, and that the power is related to V^2/R. If the
>> impedances are not the same (e.g. between the input and output of an
>> amplifier) then this formula is not correct. Decibels are always a
>> power ratio expressed as a log.
>>
>> There is more on this in the Wikipedia article on
>> "decibel" (www.wikipedia.org)
>>
>> Rich L.
>
> Thanks Reich, but have you seen the expression (20log10e) where "e" is
> the base of the natural logarithm.?
> This Log is multiplied with the value of the attenuation to convert it
> to the logarithmic dB.
>
> I am trying to see where this particular strange (20log10e) expression
> of comes from. This does not look like the normal dB expression that
> you mentioned above.
>
> Brgds
>
> Efthimios

Most likely it comes from the fact that

20 log10(V1/V2) = 20 log10(e) * ln(V1/V2).

Note that if V1 is very close to V2, ln(V1/V2) is approximately (V1-V2)/V1.

--
Jim E. Black

 

[Dirk Bruere at NeoPax]

Efthimios wrote:
> Hallo to all,
>
> I have been reading a paper regarding electric pulse attenuation and
> the values are quoted in dB. The conversion to dB was done by
> multiplying the value with (20log10e) where "e" is the base of the
> natural logarithm.

Not heard of it being done with base e

> I see this conversion to dB fist time. I can not find any information
> in my math books or the web.
>
> Does any of you have any information regarding this that I can read so
> I can understand it better?
>

http://en.wikipedia.org/wiki/Decibel

--
Dirk

http://www.transcendence.me.uk/ - Transcendence UK
Remote Viewing classes in London

 

[Dirk Bruere at NeoPax]

Efthimios wrote:
> Hallo to all,
>
> I have been reading a paper regarding electric pulse attenuation and
> the values are quoted in dB. The conversion to dB was done by
> multiplying the value with (20log10e) where "e" is the base of the
> natural logarithm.

Not heard of it being done with base e

> I see this conversion to dB fist time. I can not find any information
> in my math books or the web.
>
> Does any of you have any information regarding this that I can read so
> I can understand it better?
>

http://en.wikipedia.org/wiki/Decibel

--
Dirk

http://www.transcendence.me.uk/ - Transcendence UK
Remote Viewing classes in London

 

[Jonathan Thornburg [remove -animal to reply]]

distribution of leading digits of numbers (was: Re: Electric pulseattenuation)

Rich L. <ralivingston@sbcglobal.net> wrote:
> I recall an article in Scientific American in the 70's or early 80's
> making a similar observation about such random things as the first
> digit of street addresses, the area of lakes (irrespective of the
> units used), etc. It appears that any measurement that spans a
> population of more than a few factors of 10 shows a similar
> logarithmic frequency of the first digit. Not quite your observation,
> but possibly related.

Moving distinctly off-topic [[note to moderators: this is arguably
outside the s.p.r. charter, but does at least supplement an ambiguous
reference with a solid primary source for anyone wanting more info]]...

For a detailed discussion (including further references) of the
logarithmic distribution of leading digits of numbers written in
scientific notation, see section 4.2.4 of Donald E. Knuth's
"The Art of Computer Programming, volume 2: Seminumerical Algorithms"
(3rd Edition, Addison-Wesley 1997 hardcover, ISBN 0-201-89684-2).

ciao,

--
-- Jonathan Thornburg (remove -animal to reply) <J.Thornburg@soton.ac-zebra.uk>
School of Mathematics, U of Southampton, England
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

 

[Jonathan Thornburg [remove -animal to reply]]

distribution of leading digits of numbers (was: Re: Electric pulseattenuation)

Rich L. <ralivingston@sbcglobal.net> wrote:
> I recall an article in Scientific American in the 70's or early 80's
> making a similar observation about such random things as the first
> digit of street addresses, the area of lakes (irrespective of the
> units used), etc. It appears that any measurement that spans a
> population of more than a few factors of 10 shows a similar
> logarithmic frequency of the first digit. Not quite your observation,
> but possibly related.

Moving distinctly off-topic [[note to moderators: this is arguably
outside the s.p.r. charter, but does at least supplement an ambiguous
reference with a solid primary source for anyone wanting more info]]...

For a detailed discussion (including further references) of the
logarithmic distribution of leading digits of numbers written in
scientific notation, see section 4.2.4 of Donald E. Knuth's
"The Art of Computer Programming, volume 2: Seminumerical Algorithms"
(3rd Edition, Addison-Wesley 1997 hardcover, ISBN 0-201-89684-2).

ciao,

--
-- Jonathan Thornburg (remove -animal to reply) <J.Thornburg@soton.ac-zebra.uk>
School of Mathematics, U of Southampton, England
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

 

[J. J. Lodder]

William R. Frensley <frensley@utdallas.edu> wrote:

> The most surprising occurrence of a dB scale was found when a
> colleague and I attempted to estimate the salary scale at the company
> for which we worked. The annual stockholder's meeting report gave us
> enough information to estimate the salary of the Vice President at the
> top of our reporting chain, and a simple calculation gave us a factor of
> about 1.25 per level of management.

In general the income distributions are often
to a good approximation lognormal.
(The logarithms of the incomes are distributed normally)
At the high end this reduces to a power law distribution.
(Known since the 19th century as Pareto's law)
The Pareto exponent is a measure of the inequality
of the distribution.
Jan

 

[jwill@BasicISP.net]

Hi.

The 20 factor is when the quantity is an amplitude, not a power.

Because dB always is log power, to get the dB ratio for one
amplitude A vs a standard amplitude S, one has to square
the ratio. Thus 20log(A/S) == 10log[(A/S)^2].

 

[Rich L.]

On Dec 8, 2:01 pm, Jim Black <fmla...@organization.edu> wrote:
> On Fri, 7 Dec 2007 16:50:31 +0000 (UTC), Efthimios wrote:
> > On Dec 7, 4:01 am, "Rich L." <ralivings...@sbcglobal.net> wrote:
> >> On Dec 6, 12:37 pm, Efthimios <eangelopou...@gmail.com> wrote:
>
> >>> Hallo to all,
>
> >>> I have been reading a paper regarding electric pulse attenuation and
> >>> the values are quoted in dB. The conversion to dB was done by
> >>> multiplying the value with (20log10e) where "e" is the base of the
> >>> natural logarithm.
>
> >>> I see this conversion to dB fist time. I can not find any information
> >>> in my math books or the web.
>
> >>> Does any of you have any information regarding this that I can read so
> >>> I can understand it better?
>
> >> "dB" is short for "decibel". It is a logarithmic power scale relative
> >> to some reference power. The most general formula is:
>
> >> dB = 10 Log10( P1/P0)
>
> >> Where P1 is the power being described and P0 is the reference power
> >> level. On this scale zero dB means the power is equal to the
> >> reference power level. 10 dB means the power is 10 times the
> >> reference level. 20dB means it is 100x the reference level. etc.
>
> >> When used often the reference level is not stated explicitely. Really
> >> it ought to be. In low power RF (radio frequency) electronics they
> >> often use dBm meaning decibels wrt 1 milliwatt. You also sometimes
> >> see dBuW (wrt to 1 micro Watt) and dBW (wrt 1 Watt). As electrical
> >> signals are often measured in volts, you often see decibels defined
> >> as:
>
> >> dB = 20 Log10 (V1/V0)
>
> >> This is correct assuming that the impedance of the circuit with V1 and
> >> V0 are the same, and that the power is related to V^2/R. If the
> >> impedances are not the same (e.g. between the input and output of an
> >> amplifier) then this formula is not correct. Decibels are always a
> >> power ratio expressed as a log.
>
> >> There is more on this in the Wikipedia article on
> >> "decibel" (www.wikipedia.org)
>
> >> Rich L.
>
> > Thanks Reich, but have you seen the expression (20log10e) where "e" is
> > the base of the natural logarithm.?
> > This Log is multiplied with the value of the attenuation to convert it
> > to the logarithmic dB.
>
> > I am trying to see where this particular strange (20log10e) expression
> > of comes from. This does not look like the normal dB expression that
> > you mentioned above.
>
> > Brgds
>
> > Efthimios
>
> Most likely it comes from the fact that
>
> 20 log10(V1/V2) = 20 log10(e) * ln(V1/V2).
>
> Note that if V1 is very close to V2, ln(V1/V2) is approximately (V1-V2)/V1.
>
> --
> Jim E. Black- Hide quoted text -
>
> - Show quoted text -

Efthimios,

I got the paper you sent, and Jim Black has the correct answer. The
parameter they are working with is the propagation constant, gamma =
alpha +i*beta. The voltage attenuation they are calculating is:

V = V0*e^(alpha*x)

So the voltage ratio is:

V/V0 = e^alpha

To calculate the dB attenuation per unit length:

dB = 20*log10(V/V0) = 20*Log10(e^alpha) = 20*Log10(e)*alpha

Which is what is shown in the paper. The factor Log10(e) comes about
because dB is defined with Log10, but alpha in the paper was defined
for an exponential of the natural base (e).

Rich L.