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A Decibel Primer

Studio Overviews

Posted by CapnZilog - Oct 27, 2006
@ 09:00:51 PM

A short but deep exploration of
recording's most confusing subject

(for
a more readable and properly formatted version, please visit:
http://dallashodgson.info/files/DecibelPrimer.pdf)

A
Decibel Primer

Dallas
Hodgson
10/23/06
v1.2

In
all the world of audio, there is no measurement that is so
key, yet so difficult to understand, than the decibel (dB.)
They�re so commonly used both in live sound and recording that
we often take them for granted without understanding they
really mean or worse, understanding them incorrectly. And it�s
very easy to understand them incorrectly, because there are
many flavors of dB (which we�ll get to in a moment) and
without the proper frame of reference, are as meaningless as
they are convenient. But audio convenience is at the heart of
all this, because the decibel scale would not exist were it
not for Alexander Graham Bell, inventor of the telephone
(which it was named after.)

Part of the reason for the
decibel scale, which is derived from logarithms, is to
compress the extreme range of loudness perceivable by the
human ear into something more compact.

(threshold of
hearing/pain chart derived from
http://ohioline.osu.edu/cd-fact/0190.html, Dave Moulton and
myself)

Safe Range (dB SPL) Sound Pressure Ratio over
0dB (times)

Threshold of Hearing (.0002 dynes/cm^2) 0
dB 1 x
(convenient reference) 6.02 dB 2 x
(convenient
reference) 10 dB 3.16 x
Stream flow, rustling leaves 15 dB
5.62 x
Watch ticking 20 dB 10 x
Soft whisper 30 dB 32
x
Quiet street noises 40 dB 100 x
Conversation, quiet
room 45 dB 178 x
Conversation, crowded room 60 dB 1,000 x

Normal city or freeway traffic 70 dB 3,162 x
Vacuum
cleaner 75 dB 5,623 x
Hair dryer 80 dB 10,000
x
Motorcycle, electric shaver 85 dB 17,783 x
Lawn mower,
heavy equipment 90 dB 31,623 x
Garbage truck 100 dB 100,000
x
Screaming baby 115 dB 562,341 x

Injury Range (dB
SPL) Sound Pressure Ratio over 0dB (times)

Jack hammer
3 feet /
Jet airplane takeoff from 120 feet /
Threshold
of discomfort 120 dB 1,000,000 x
Air begins to distort 125
dB 1,778,279 x
Race car, loud thunder, rock band 120-130 dB
1,000,000 x - 3,162,278 x
Sensation of eardrums moving
(�tickle�) 130 dB 3,162,278 x
�Horrific� Pain 140 dB
10,000,000 x
Rocket launch from 150 feet 180 dB
1,000,000,000 x
Theoretical maximum at 1 atmosphere 194 dB
5,011,872,336 x

Some interesting things from this table
� you can see that the human ear has a very wide dynamic
range, with the loudest common sounds (120dB) being one
million times louder in intensity (pressure) than the quietest
sound we can hear. In fact, it is generally considered that
for something to sound twice as loud to our ears its SPL must
increase 20dB, a sound pressure increase of ten times. You can
begin to see why faders on mixing boards are calibrated in dB,
not sound pressure or volts; it would take a huge fader to go
all the way from 0 to a million, and at low volumes you�d be
moving the fader in impossibly small steps whereas at high
volumes the hand motions required would be huge.

Some
Interesting Sound Facts:

1) The ear is so sensitive,
that if the Threshold of Hearing were much lower we�d be
hearing the Brownian noise of air-molecules as they randomly
bump into each other all day long.
2) I�ve also read the
Threshold Of Hearing being described as .0002 microbars, 20
micropascals (�Pa = 1�10^-6 Pa), 10^-12 W/m^2 (watts per
square meter), and roughly the sound of a mosquito flying 3
metres away!
3) Ever wonder why orchestras are so large?
Because they need to be! It takes ten violinists to sound
twice as loud as one (an increase of 20dB SPL.)
4) Few
sounds in Nature (like lightning) are so loud that that the
air itself distorts like a guitar amp. When a speaker moves
forward, it creates air pressure; and when it moves backward,
a partial vacuum. Now imagine a sound so large that the
partial vacuum actually becomes total, and simply can�t get
any larger � that�s what happens at 194dB, and you�ll probably
be dead - earplugs or not.
5) SPL stands for �Sound
Pressure Level� which is relative to the threshold of hearing.
There are many other possible thresholds, such as dbFS, dbM,
dbU, dbV and dbW which we�ll get into later but for now, just
remember that they�re all different beasts and often can�t be
compared directly, like apples and oranges.
6) A 1 dB
change in gain is about the smallest we can notice in
isolation. In the context of a mix though, .3dB-.5dB changes
in gain are common for me. Professional recording engineer
Roger Nichols has been known to make tweaks as small as
.1dB!

The fact that the ear gets less sensitive to
sounds as they get louder is by design. Internally, a physical
mechanism actually �turns the volume down� on the ear like a
compressor, but unfortunately this mechanism is not
instantaneous. One of the worst things you can do to your ears
is to expose them to the sound of brief, impulsive noises such
as explosions or gunfire. It may only last a fraction of a
second, but an unexpected blast of 140dB SPL could render you
deaf permanently. How about rock concerts? Well, at least our
ears are somewhat prepared for that; but as OSHA�s Table G-16
(below) shows, we can�t listen to loud sounds forever without
sustaining some damage: (from
http://www.osha.gov/pls/oshaweb/owadisp.show_document?p_table=STANDARDS&p_id=9736)

A-weighted
SPL (slow response) Max Recommended Exposure

80dB 32
hrs
85dB 16 hrs
90dB 8 hrs
95dB 4 hrs
100dB 2
hrs
105dB 1 hrs
110dB 30 min
115dB 15 min
120dB 8
min
125dB 3.78 min
130dB 1.8 min

(Another
Factoid: �A-weighting� refers to one of two common methods of
measuring sound with a sound level meter, the other being
�C�-weighting. �A�-weighting is less sensitive to low
frequencies, while the more accurate �C� weighting is fairly
flat. For reasons that are unclear and probably have to do
with lobbyists, �A�-weighting is the preferred weighting in
laws regarding noise pollution.)

Note that 8 min is not
a very long time to spend at a 120dB rock concert! When you
leave the show, if you ears are ringing you have received some
(possibly permanent) nerve damage which may degrade your
ability to mix. So what level should one mix at? 85dB is most
often cited as the ideal level. For one thing, your ears can
tolerate a full day of it and secondly, the ear�s
Fletcher-Munson frequency response (see
http://en.wikipedia.org/wiki/Fletcher-Munson_contours)
flattens out nicely giving you a more accurate read on your
mix. Much louder than that? Wear earplugs! There are some good
ones out there aimed at audiophiles and musicians that sound
much better than what you�re probably used to, and they�re a
very cheap investment for protecting your ears: See
http://www.etymotic.com/ephp/er20.aspx

This Is Where It
Gets Tricky

So what makes  understanding dB�s so
confusing? Part of it is that it�s not intuitive to think in
terms of exponentials and logarithms. For example, when I say
that in order to sound twice as loud it takes 10 times the
sound pressure, expressed as an increase of 20dB, that�s just
weird. So what do you get if you merely double the sound
pressure? Only a 6.02dB increase, as it turns out. 6.02dB is
one of those magic numbers that�s worth memorizing (along with
20dB) because it comes up again and again in digital audio.
Try this test:

1) Open up a project in SONAR, find a
guitar track somewhere, and solo it. What�s the highest peak
reported by the output bus meter? Let�s say it�s �10dB.
2)
Clone that guitar track, solo it, and play both tracks at the
same time. What�s the highest peak reported by the output bus
meter now?

I don�t even have to look to know � it�s
around �4dB, which is ~6dB louder than what you got by playing
the single track alone. Why? Because you�ve just doubled the
sound intensity by playing two identical guitars at the same
time. OK, now you should be able to figure this out � what if
you wanted to make a single guitar sound twice as loud? You�d
make 8 more copies (for a total of 10) and play�em all at once
� this yields a 20dB increase, but be careful at this point
because your output meters will be clipping severely! Why?
Because �10dB + 20dB = 10dB, which is 10dB over the digital
mixing maximum of 0dBFS. (We�ll explain the �dBFS� part in a
moment, but at this point it�s worthwhile to note that to
avoid the clipping problem at this point you should either be
reducing the output bus level by 0dB, or reducing all of the
10 tracks feeding the Output Bus by 10dB apiece, which has the
same effect.)

Ramifications For Digital
Audio

Remember how I said that cloning an instrument
(thereby doubling its intensity, or sound pressure) raises
things by 6.02dB? Well, we like to double things in the
digital world; it happens every time we add an extra bit of
precision. How many times have you bought a CD and read the
little liner note thingy explaining how great CD�s are? (�The
high resolution of this Compact disc may reveal limitations in
the master tape, including noise and other distortions.�)
Well, that�s because being 16-bit, Compact discs have a
dynamic range of 16*6.02dB = 96.32dBFS which is well beyond
the 60 dB or so of dynamic range that LP records were capable
of. Dynamic Range is the difference between the loudest sound
playable to the softest, and for anybody who caught my
�Preserve Your Bits� article on Guitar School that�s 65,536
times louder. So is this enough? Most certainly not; the ear
has a good 120dB dynamic range (up to the �threshold of pain�)
so while the dynamic range of the compact disc is not perfect,
CD�s are much better than any consumer format we had
prior.

So what kind of wordlength should CD�s have been
given in order to supply to 120dB� worth of dynamic range? The
answer is  120dB / 6.02dB, or 19.93 bits. Looks like
those 20-bit ADAT decks from the 90�s were good enough! That�s
not something that consumers ever saw, however the 24-bit
consumer �audiophile� standard today is better still (and more
than good enough.) In the old days those �limitations in the
master tape and other distortions� were always there � we just
couldn�t hear them, because they were lost among all the noise
and turntable rumble!

�SPL� and �dBFS� Explained, Along
With All The Rest

Up until this point, we�ve avoided
giving a formal definition of what a decibel is. This is where
I�m going to step aside and let Ken Pohlmann�s classic
�Principles Of Digital Audio (5th ed.) do it for
me:

Specifically, the decibel is defined to be 10 times
the logarithm of a power ratio:

Intensity level = 10
log(P1/P2) dB

where P1 and P2 are values of acoustical
or electrical power.

If the denominator is set to a
reference value, standard measurements can be made. In
acoustic measurements, an intensity level (IL) can be measured
in decibels by setting the reference intensity to the
threshold of hearing, which is 10^-12 W/m^2 (watts per square
meter). Thus the intensity level of a rock band producing
sound power of 10W/m^2 can be calculated:

Intensity
level = 10 log(P1/P2) dB = 10 log(10^1/10^-12) = 130dB
SPL

Going in the reverse (antilog) direction is
trivial:

Sound Pressure Ratio = antilog(130dB SPL/10) =
10^(130/10) = 10,000,000,000,000 (x 0dB SPL)

Note that
when ratios of currents, voltages or sound pressures are used
(quantities whose square is proportional to power), the above
decibel formulas must be multiplied by 2.

(DJ�s Math
Refresher: The (base-10) log of a number is simply the number
you�d have to raise 10 to the power of in order to get the
desired number. In other words, the log of 1000 (i.e., 10^3)
is 3. The Antilog of 3 is 10^3, or 1000. It helps to open up
the Windows Calculator in Scientific Mode and play around with
these formulas a bit until they feel comfortable.)

I
know I�ve probably lost everybody at this point, but what Ken
is trying to say is that dB�s don�t exist in a vacuum; they�re
always relative to something and that relationship is always
expressed as a ratio of powers. What is power? In physics,
Power is the ability to do Work. And what is work? Imagine a
river that feeds a hydroelectric plant, like Niagra Falls.
What kind of river can generate the most electricity (work)?
One that has a lot of water (current) moving at a high rate of
speed (pressure). Multiplying these two figures gives you the
river�s power, which might be expressed in terms of something
like gallons per minute (GPM). In electricity, the analogy is
the same, only Power is expressed in Watts, and Watts =
current (there�s that word again) * voltage
(pressure).

When the Bel scale was invented it was
originally used for measuring signal power loss over telephone
lines, hence its standardization around �power� measurements.
Much of the audio world, however, involves measuring pressure
or intensity, not power. However, power is proportional to
intensity, and you can perform �pressure to power� conversions
on the fly simply by squaring the pressure. (To understand
this fully, consider that in Ohm�s Law, in a circuit with a
resistance of 1 ohm voltage and current are always identical �
and therefore, the formula of Watts (power) = Volts * Amps is
equivalent to Volts or Amps squared.) Incidentally, a deciBel
is simply a Bel measurement multiplied by 10, which figures
into the formulas above and below. This gives us an extra
digit of precision when quoting measurements, as saying things
like �the human ear has 12.0 Bel of dynamic range� seemed a
little too coarse back when the scale was invented.

So
when computing dB values based on pressure or intensity,
(which would include almost everything that wasn�t a Watt or
pre-squared in some other fashion) the decibel formula needs
to be written like so:

Intensity level = 20
log(p1/p2)dB

where p1 and p2 are values of acoustical
of electrical pressure. (Changing the �10� to �20� before the
log is equivalent to squaring p1 and p2 in the formula above,
only easier � such is the power of logarithms.)

In this
example, we�ll compute the maximum dBFS (FS=Full Scale) of a
16-bit word as used in a compact disc. The maximum value would
be 65536, and the minimum value would be one. The dynamic
range of a compact disc is, therefore:

Intensity level
= 20 log(p1/p2) dB = 20 log(65536/1) = ~96.32
dBFS

Going in the reverse (antilog) direction is even
easier:

16-Bits Full Scale = antilog(96.32 dbFS/20) =
10^(96.32/20) = ~65,536 (x 1)

Doing Decibels In
Your Head (taken from Dave Moulton�s �Total
Recording�)

Doing all this in your head is fairly easy,
believe it or not. Keep in mind the following basic
equivalents:

FOR POWER FOR PRESSURE
3 dB = 2 x power
6 dB = 2 x pressure
-3dB = .5 x power -6 dB = .5 x
pressure
6 dB = 4 x power 10 dB = 3.16 x pressure
-6 dB
= .25 x power -10dB = .316 x pressure
7 dB = 5 x power 12
dB = 4 x pressure
9 dB = 8 x power -12 dB = .25 x
pressure
10 dB = 10 x power 14 dB = 5 x pressure
-20 dB
= .01 x power 18 dB = 8 x pressure
13 dB = 20 x power 20 dB
= 10 x pressure
17 dB = 50 x power -40dB = .01 x
pressure
20 dB = 100 x power 26 dB = 40 x pressure
30 dB
= 1,000 x power, etc. 34 dB = 50 x pressure
40 dB = 100 x
pressure
60 dB = 1,000 x pressure, etc.

Using these,
you can work out just about anything you want. For instance,
33dB is a power ratio of 2000 (3 dB over 1000), or a pressure
ratio of a little less than 50. A little practice and you�ll
have it in no time.

Decibel References (also taken from
Dave Moulton�s �Total Recording�)

As I�ve mentioned,
decibels are simply alternative numbers that represent ratios
of power (and, indirectly, pressure). Naturally, there are
many occasions when we want to know the ratio, expressed in
decibels, between some value and a known, standardized
reference. As a result of this need, an array of decibel
references have evolved over the years. Below are several such
references that are in common use today for audio, along with
some discussion of their applications.

0 dB SPL = .0002
microbars = .0002 dynes/cm^2 = .00002 Pascals
This
reference is for the threshold of hearing. It is an acoustic
expression of pressure (all calculations should use the
pressure multiplier, 20) that represents the pressure level of
the softest 1KHz sound that average humans can detect. Any
expression of acoustical amplitude using dB is usually derived
from this value.

0 dbW = 1 Watt
This reference is a
power reference tied to Watts, generally used with power
amplifiers.

0dBM = 1 milliWatt
This reference has
become a little ambiguous. In its original form, it
unequivocally referred to 1 milliWatt of power, for use in
telephone and other audio transmission lines.

At that
time, power management was a critical issue, and so
transmission systems used matched impedance systems for the
most efficient transfer of power between elements in a system.
In telephone practice, the standard source and input impedance
of all devices was 600 ohms. Given that impedance, the voltage
level needed to generate 1 milliWatt of power is .775 volts.
As a result, 0 dBM has, over the years, come to refer
informally to .775 volts in common practice, regardless of the
load. (If the load is not 600 ohms, then .775 volts will not
generate 1 milliWatt of power.)

So, the literal usage
of 0dbM is as a power reference of 1 milliWatt, while the
sloppier common-practice usage is as a voltage reference of
.775 volts. When you run across the dBM reference in specs and
advertising for modern audio gear, it almost always is the
common-practive reference. We simply aren�t concerned about
the power issue at line levels anymore.

For the more
literal-minded and anal-retentive among us, there is a more
formal voltage reference to use:

0 dBu = .775
Volts
This reference is unequivocal. It is the voltage
reference that is equivalent to 0dBM when a 600-ohm load is
present. It is a standard audio reference. (Dallas� note:
+4dbU (1.23V) is the standard signal level for balanced
professional audio gear)

0dBV = 1 Volt
This
reference is an obvious and convenient one. It is unfortunate
that it is only 2.2 dB from 0dBu,which causes the two to get
tangled up a lot. The confusion is magnified by the casual
practice of some engineers to refer to 1 Volt with a capital
�V� and to .775 volts with a lower-case �v�. So, the bad news
is that this particular specification is a bit of a minefield,
and you�ve got to proceed with a little caution. The good news
is that if you get it wrong, you�re only off by 2dB!

If
I were king, I�d abolish the .775 volt reference now, and have
a single line-level voltage of 0 dBV = 1 Volt. (Dallas�s note:
-10dbV (3.16V) is the standard signal level for unbalanced
consumer audio gear)

0 dBFS = Digital Audio Level at
the top of the  Most Significant Bit
dBFS (Full Scale)
is a digital audio reference. It equals the maximum level on a
meter used to measure a digital audio sine wave signal, and
typically in digital audio, it refers to the level of a sine
wave at the top of all bits, or the maximum capacity of the
digital word used to represent the audio information. The
practical consequence of this is that 0 dBFS represents the
peak level the system can handle without
overloading.

There are numerous other decibel
references. The principle is the same: a symbol following the
term dB refers to some standard value that will be 0dB for
that reference system. All you�ve got to do is find out what
the reference is!

Actually, you can even have fun with
this and make up your own decibel references. For instance, as
a teaching example I created the Nugent, or dBN, where 0dbN =
the sound pressure level of Ted Nugent playing a power chord
at full level. Question: How many Ted Nugents playing a single
power chord all at once does it take to generate a sound
pressure level of 40 dbN? Answer, 10,000! It�s enough to make
you think carefully about this business!

Finally, as
you work with decibels, you�ve got to stay aware of exactly
when you�re working with decibels as ratios and when as
decibel references. Any decibel value tied to a reference
(say, -21 dBu) is a fixed value, while 21dB is just a ratio.
If, for example, you say, �I�d like a 3dB pay increase,� you
are saying you�d like to double your salary! This is quite a
bit different than saying �I�d like to earn 3
dB$.�

(dB$ is an obscure capitalist reference, where
0dB$ = $1. In the example, you are asking for
$2!)

Dallas� Wrap Up

That�s it for this
tutorial, and believe me it was a very difficult one to write.
Decibels were on my mind this week and there�s no better way
to test one�s degree of understanding in a concept then by
trying to explain it to someone else. For more info, (a lot
more) check out David Moulton�s �Total Recording� and Ken
Pohlmann�s �Principles Of Digital Audio�, now in its fifth
edition. This little write-up represents what I find most
useful in thinking about dB�s, and as wordy as it is I felt
like I couldn�t make it any shorter with leaving something
important out!

Happy recording,

-djh

4 comment(s) | viewed 97 time(s)
| |

Your Comments

Commented by CapnZilog - Oct 29, 2006 @
04:28:05 AM

OK, now that the School's PHP is
working again, let's try a second time posting this one. I
gave up trying to make the HTML formatting work properly here
though, as there's too much HTML munging going on behind the
scenes. Maybe in the future we'll be allowed to post PDF's
instead?

Commented by CapnZilog - Dec 20, 2006 @
06:17:00 PM

Version 1.3 up at
http://dallashodgson.info/files/DecibelPrimer.pdf, correcting
a bug in the "130dB rock band" SPL formula
example

Commented by CIRCLEjerk - Jan 03, 2007 @
08:57:52 PM

that was extremely interesting...i
really liked the part on the ramifications of a compact
disk...i need to brush up on my learnin' as far as that stuff
goes.

Commented by GtrZan - Jan 31, 2007 @ 11:11:02
AM

That ws great Dallas!

Being
an RF engineer I LIVE in the world of dB (dBm, to be more
precise). This presents the entire concept including the
history of the scale very
succinctly.

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