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Decibels

Introduction

 

 

Decibel scales are used in many branches of science and engineering as a way to measure derived physical quantities like force and power.

In the field of acoustics, derived quantities include the pressure, power, and intensity of sound waves.

In the field of electronics, derived quantities include electrical voltage and power.

Here's a quick review of sound power, intensity and pressure :

 

  1. SOUND POWER :

    Sound power measures the acoustic energy radiated per second by a sound source.

    Sound power is measured in watts ( W ) and is a property of the sound source, independent of the surrounding space.

    One watt equals one joule of energy per second.


  2. SOUND INTENSITY :

    Sound intensity measures the power passing through a unit of perpendicular area at a particular location in a sound field, distant from the sound source.

    Sound Intensity is measured in watts per square meter ( W / m 2 ).


  3. SOUND PRESSURE :

    Sound pressure measures the force of a sound wave on a unit of perpendicular area at a particular location in a sound field, distant from the sound source.

    The old, CGS (centimeter-gram-second) unit of pressure is the dyne per square centimeter ( dyn / cm 2 ) which also equals one microbar ( μbar ).

    The standard international ( SI ) unit of pressure is the pascal ( Pa ).  One pascal equals one newton of force per square meter of area :

    1 Pa = 1 N / m 2

           = 10 μbar

 

 

 

Sound Pressure

 

 

We're starting with pressure because “Sound Pressure Level” ( SPL ) may very well be the most widely familiar measurement expressed in decibels.

Sound pressure, of course, is what vibrates your eardrums.  The strength of the vibration depends on the amplitude of the pressure wave that we call sound.

“Amplitude” measures the breadth of the back-and-forth displacement of the air particles that transmit the pressure wave.

Sound pressure is a deviation from the surrounding atmospheric pressure and its amplitude decreases with its distance from the sound source.

The standard unit of pressure is the pascal, abbreviated ‘ Pa ’ in honor of Blaise Pascal.

 



  Pascal ( 1623 - 1662 ) was a French mathematician, physicist, inventor, philosopher, and Catholic writer.  He laid the foundation for the modern theory of probabilities and also invented the syringe and the hydraulic press.


 

 

 

Sound Speed

 

 

In the air, audio-frequency pressure waves (sounds) are measured using microphones.  Underwater, specially constructed hydrophones are used to pick up audio-frequency pressure waves.

Sound waves travel faster in water than they do in the air:  1500 meters / sec instead of 340 meters / sec.  The faster travel is due to the higher density of water.

The speed ( c ) of a wave equals its frequency ( f ) times its wavelength ( λ ):

 


Wave Speed formula [ 1 ]

 

So its wavelength equals its speed divided by its frequency :

 


Wave length formula [ 2 ]

 

Equation [2] shows us that when the speed (c) of a wave increases, so does its wavelength (λ).

For example, when a 20 hertz sound wave moves from air to water, it's wavelength increases from 17 meters ( 340 / 20 ) to 75 meters ( 1500 / 20 ).

 

 


  Heinrich Hertz ( 1857 - 1894 ) was a German physicist and experimentalist who first conclusively proved the existence of electromagnetic waves.  The unit of frequency, one "cycle per second", is called a hertz, abbreviated Hz.


 

 

 

Airwaves

 

 

Sound pressure is transmitted by air particles mechanically vibrating forward and back, propagating waves of air compression and rarefaction.

Humans can perceive sound waves having frequencies between about 20 Hz and 20,000 Hz (20 kHz).

Sound waves are longitudinal—that is, each air particle's displacement is parallel to the wave's propagation.  This is unlike transverse ocean waves, where water particles move vertically and waves propagate horizontally.

The following animations show how air particles simply move back and forth about their equilibrium positions while a wave of compression travels from left to right.

 

Longitudinal Wave Particles
Click animation to enlarge it
Longitudinal Waves
Animations courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State

 

 

 

The Absolute Threshold
of Hearing

 

 

The very faintest 1 kHz tone that our ears can perceive is often used as a reference level when expressing a sound pressure, intensity or power.  This reference level is called the “Absolute Threshold of Hearing” ( abbreviated ATH ).

A person's actual threshold of hearing varies with the person's age and other factors but the following standard values have been adopted for a 1 kHz sound wave :

 

ATH Power :

W 0  =  10 -12 watts  =  1 picowatt  ( pW )

ATH Intensity :

I 0  =  1 picowatt per square meter  ( pW / m 2 ).

ATH Pressure :

p 0  =  20 micropascals  ( μ Pa )

      =  0.0002  microbar ( μ bar ) or dyn / cm 2

 



  The ATH sound pressure is about ten one-billionths of the standard atmospheric pressure ( 1 atm = 101,325 Pa ).

  The ATH pressure roughly compares to that of a mosquito flying 10 feet ( 3 m ) away, with air particle displacements on the order of a tenth of an atomic diameter.


 

 

 

Exponential Sensitivity

 

 

At the LOUD extreme of our hearing lies the threshold of pain.  Its pressure has been standardized to 20 pascals at a frequency of 1 kHz.  That's one million ( 10 6 ) times greater than the ATH pressure of 20 μPa.

In terms of power and intensity, the threshold of pain is roughly a trillion ( 1012  =  1,000,000,000,000 ) times greater than the ATH levels.

This huge numeric range speaks to the exponential sensitivity of our hearing.

A convenient way to record both huge and tiny numbers on a single, compact scale is to use logarithmic ratios.

Ratios make the numbers dimensionless and thus applicable to a wide variety of physical quantities and reference levels.

Logarithms squeeze more and more numbers into each scale division by compressing them into powers of ten :

 

 

Log Scale

A Logarithmic Scale

 

 

Logarithms also simplify the chaining of ratios by making them additive, not multiplicative.  Moreover, logarithmic scales roughly align with the psychoacoustics of our hearing.

 

 

 

The Bel

 

 

A.G. Bell

 

 

The convenient, logarithmic ratio we need is named the ‘bel’ and abbreviated capital B for Alexander Graham Bell (1847-1922), a Scottish-born Canadian-American inventor and engineer remembered for his work in sound technology and education for the deaf.


 

A more commonly used value is the decibel, abbreviated dB.  One decibel is one tenth of a bel :

 

10 decibels  =  1 bel

 

 

 

Logarithms

 

 

Since decibels use logarithms, let's quickly review what the logarithm ( or log ) of a number is.

The base-10 log of a number is simply the power ( or exponent ) of 10 that produces that number.

For example, if the number is 100 then the power of 10 is 2 since :


100 = 10 2

So,


log 10 (100) = 2

 

If the number is 2, then the power of 10 is almost exactly 0.30103 since :


2 ≅ 10 0.30103

 

So, to a very close approximation, we can say that


log 10 (2) = 0.3

 

 


  If you're wondering how 10 can be multiplied by itself a non-whole number of times, it can't.  But graphs, tables and slide rules have long been used to interpolate the values of non-whole powers of ten.

  Today, you can type a number into a pocket calculator or cell phone and press the  [ 10 x ]  button.  Calculator algorithms give excellent results for any power of 10 and also for the [ LOG ] of any number, for example ‘2’ .


 

 

Here are a few clear-cut examples of whole-number logarithms :

 


Whole Number Logs [ 3 ]

 

 

   Notice that “log10 (2)” must fall somewhere between  0 and  1.  A good first guess would be 0.3.

 

 

 

Why Does 10 0 = 1 ?

 

Raising a number to the zeroth power means multiplying that number by itself zero times.  But that means not multiplying anything, which is the same as multiplying by ‘1’, the multiplicative identity.  ‘1’ times anything doesn't change anything.

 

 

 

Why is  log10 ( 0 ) Undefined ?

 

You can never get a result of zero by raising any number (except zero) to any power whatsoever.

 

 

 

 

Defining the Bel & Decibel

 

 

Bels and decibels are dimensionless measures of the ratio of two values.  As such, they communicate a comparison of values or a gain / loss.

A bel ( abbreviated B) is defined as the logarithm of the ratio of two power quantities, W2 and W1 :

 


dB equation [ 4 ]

Notice that there are 10 decibels [dB] in one bel [B].

 

   A 10-fold gain in power [W2 = 10 W1] equals 1 bel or 10 decibels since log10 (10) = 1.

   A 100-fold gain is 20 decibels, a thousand-fold gain is 30 decibels and a million-fold gain is 60 decibels.

   A doubling of power is about 3 dB since log 2 ≅ 0.30103.

   If W2 = W1, the gain is 0 dB since log 1 = 0.

 

Remember that decibels are logarithms, or exponents, and that exponents (of any base) are additive not multiplicative.  So, an amplifier with a gain of 2 dB turns a 3 dB signal into a 5 dB signal :

 

103 × 102 = 105

     1000 × 100 = 100,000

 

 


   Notice that we often write the log operator without using the subscript ‘10’.  That's because logs to the base 10, also called "common" logs, are commonly called "logs".

The only other log base regularly used is ‘e’, an irrational and transcendental number used in higher math.  Logs to the base e, also called "natural" logs, are commonly denoted by the symbol ln, not by log e .


 

 

 

Sound Power & Intensity

 

 

Although decibels express a ratio of two power levels, they can also communicate a specific or absolute power level if the ratio is made to a specific reference power.

A commonly used reference power is that of a sound at the Absolute Threshold of Hearing (W0 =  1.0 picowatt).

Using this reference, the "Sound Power Level" (abbreviated LW or SWL) of a wattage ‘W’ can be expressed in decibels as :

 


SWL Definition [ 5 ]
Where:

W    =  an absolute sound power in watts
W0   =  the ATH reference sound power ( = 1 pW )
L =  SWL =  the absolute sound power in decibels


The 10 multiplier appears because there are 10 decibels in a bel.

 

 



 

Likewise, an absolute sound intensity ‘I’ can be expressed in decibels by comparing it to the ATH reference intensity (I 0 =  1.0 picowatt per square meter)

This "Sound Intensity Level" is abbreviated LI or SIL :

 


SIL Definition [ 6 ]
Where:

I   =  an absolute sound intensity in watts per square meter
I 0  =  the ATH reference sound intensity ( = 1 p W / m 2 )
L I  =  S I L  =  the absolute sound intensity in decibels


The 10 multiplier appears because there are 10 decibels in a bel.

 

 

 

Root-power Quantities

 

 

Measurement quantities fall into two categories:  power quantities and root-power quantities.

A power quantity is directly proportional to power or energy.

A root-power quantity must be squared for it to be proportional to power or energy.  These quantities are proportional to the square root of a power quantity.

The following table lists various power and root-power quantities :

 

 

 

POWER QUANTITIES
( Directly proportional to energy or power )
Power (Energy per second)
Power Density (Power per volume)
Energy Density (Energy per volume)
Acoustic Intensity (Power per area)
Luminous Intensity (Power per steradian)

ROOT-POWER QUANTITIES
( Their squares are
proportional to power or energy )
EXAMPLES
Sound Pressure  p Acoustic Intensity = p2 / Z
Voltage  V Power = V2 / R
Current  I Power = I2 × R
Speed  v, c Kinetic Energy = 1⁄2 m v2      E = m c2
Electric Field Strength  E Power density = E2 ÷ 377
Magnetic Field Strength  H Power density = H2 × 377

 

 

The above table shows that sound pressure and electric voltage are root-power quantities.  Decibels, however, are defined in terms of power quantities.

So, in order to conform to the decibel scale, pressures and voltages must always be squared when calculating decibels.

 

 

 

Decibels for
Pressure & Voltage

 

 

Since sound pressure (p) and electric voltage (V) are root-power quantities, their values must be squared when calculating bels (B) and decibels (dB).  For example :

 


Root-power decibels [ 7 ]

 

   A 10-fold gain in voltage or pressure [p 2 = 10 p1] equals 2 bels (or 20 decibels) since log(10) = 1.

   A 100-fold gain is 40 decibels, a thousand-fold gain is 60 decibels and a million-fold gain is 120 decibels.

 

 

 

Log Arithmetic

 

 

   Equations [7] take advantage of the following identity :


x n / y n = ( x / y )n

   For example, suppose  x  =  10y  =  5 and n  =  2 :


 100 / 25 = ( 10 / 5 )2


4 = 4

 

 

   They also take advantage of the following identity :


log x n  = n × log x

     For example, suppose  x  =  10  and n  =  2 :


log ( 100 ) = 2 × log ( 10 )


 2 = 2 × 1

 

 

 

SPL — Sound Pressure Level

 

 

We now have the wherewithal to communicate a specific Sound Pressure Level (abbreviated SPL or Lp) in decibels rather than pascals.

Just replace p1 in equation [ 7 ] with the ATH reference pressure (p 0  =  20 micropascals) and replace p2 with p:



SPL Definition [ 8 ]
Where:

p     =  an absolute sound pressure in pascals
p 0  =  the ATH reference sound pressure  ( = 20 μPa )
L p  =  SPL  =  the absolute pressure level in decibels

 

 


 

Sound Pressure Levels are commonly used to rank sound levels in the environment.  Shown below is a list of Sound Pressure Levels, in decibel order, along with some sound sources that produce that pressure level.

 

 


S o u n d   P r e s s u r e   L e v e l s    ( Lp  ref  20 μ Pa )
dB SPL Sound Source Examples
0 "THRESHOLD OF HEARING"
Mosquito flying 3 m away
10 Rustling leaves in the distance • Light wind
Calm breathing • Ticking watch
20 Unoccupied TV studio • Whisper 1 m away
30 Woods • Quiet bedroom at night
40 Quiet library • Light rain • Computer hum
50 Average suburban home • Quiet office
60 Conversational speech 1 m away
70 Busy business road 15 m away • Vacuum cleaner 1 m away
80 Busy business road 5 m away • Noisy restaurant
Garbage disposal • Freight train 30 m away
90 Heavy truck 10 m away • Inside a school bus
100 Loud "disco" music 1 m from speaker
Alongside a mainline railway • A nearby motorcycle
110 Jet engine 100 m away • Jackhammer 10 m away
Chainsaw 1 m away • Rock music venue (front row)
120 "THRESHOLD OF DISCOMFORT"
Large jet taking off 300 m away • Loud car horn 1 m away
Ship's engine room • Hard rock concert
130 "THRESHOLD OF PAIN"
Trumpet 0.5 m away • Pneumatic drill nearby
Loudest human voice 1 inch away
140 Large jet aircraft taking off 50 m away • Colt 45 pistol 8 m away
150 Jet engine 1 m away • M-80 firecracker close up
160 Fighter jet taking off nearby • Shotgun close up
170 Stun grenade • Space shuttle launch
310 Pressure wave from the Krakatoa eruption in 1883

 

 

 

130 dB SPL
130 dB SPL      

 

 

 

 

dB Suffixes

 

 

 

Next, we'll go over a few of the dB suffixes that are commonly used in audio and electronic measurement.

 

 

 

The dBm

 

 

The ‘ dBm ’ is a logarithmic power ratio relative to one milliwatt :


0 dBm = 1 mW

By substituting 1 mW for W1 in the decibel definition [ EQ 4 ], you can see that 10 mW = 10 dBm, 100 mW = 20 dBm, 1000 mW = 30 dBm, etc.

SIM settings

 

 

The signal strength being received by a cell phone is often given in dBm in the phone's SIM card status settings [see photo].

Note:  “asu” is an “arbitrary strength unit” whose scale varies with the network generation.

The dBm is also used to report Wi-Fi connection strengths.

Since the power of cellular and Wi-Fi signals is less than one milliwatt, their dBm values are negative numbers ( refer to  EQ 3 ).


 

Excellent phone signals can range from −50 dBm to −80 dBm.  Poor signals can register as low as −120 dBm or less.  The actual values depend on the phone's sensitivity and the cellular network's generation.

To get a feeling for these dBm values, let's use [ EQ 4 ] to calculate what the above −100 dBm signal strength is equal to in watts (W).  Remember, the dBm reference level is 1 mW or 10−3 watts :

 


-100dBm wattage conversion [ 9 ]

 

So, a signal strength of −100 dBm is 10−13 watts, 10−10 milliwatts or 0.1 picowatts (1 pW = 10−12 watts).

 

  The dBm is also used to express signal power levels in mixing consoles, tape recorders, and other audio gear.

 

 

 

 

The dBu

 

 

In The Beginning…

 

The ‘dBu’, formerly known as ‘dBv’, has its roots in the time of Alexander Graham Bell and the telephone industry that standardized the 600-ohm telephone line that's still in use today.

 

telephone pole
"It Came From Outer Space"

 

In order to measure power losses in telephone lines, vintage voltmeters often had a decibel scale marked “ dBm ” even though the dBm is a power measurement not a voltage measurement.

The dBm scale that's printed onto vintage voltmeters acts to convert a voltage into its power equivalent but only if the tested voltage is across an impedance of 600 Ω.  Only then will the meter's scale graduations correctly convert the voltage.

Today, audio gear has input impedances that are greatly higher than 600 Ω, making those dBm scales useless.  And besides, at high impedances the main concern is voltage, not power, since very little power is needed to pass a voltage signal.

 

 

 

Enter The dBu

 

 

So, eventually, a purely voltage-referenced decibel measurement – the ‘dBu’ – came along.

This new measurement is unloaded in the sense that it doesn't presume any particular load impedance but it does remain referenced to the same voltage as that implied by the ‘ 1 mW @ 600 Ω ’ telephony standard.

So, just for fun, let's calculate the voltage (V) that corresponds to 1 mW of power (P) in a 600 Ω load (R).

We already know that electric power  P = V × I  and that electric current  I = V / R.  So let's substitute "V/R" for "I" in the first equation to get :

 

P = V × ( V / R ) = V 2 / R

 

Solving for "V", we find that

 

V = √ P × R

 

So, to find the dBu reference voltage, we plug 1 mW and 600 Ω into the last equation :

 


dBu reference voltage [ 10 ]

   So, the dBu reference voltage is 0.775 volts.

 

 

 

The Professional Audio Standard

 

 

The dBu is the standard voltage quantifier for interfacing pro-audio gear.

The pro-audio interface specifies a balanced line-level signal of +4 dBu connected by a balanced 3-wire cable using XLR connectors.

 

XLR cables

The ‘XLR’ connector was first introduced in the 1950s by the Canon Electric Co. ( now ITT Canon ) as the "XL Series" audio connector.

The name "XLR" stands for "eXternal Line Return".


We can calculate the voltage level that corresponds to +4 dBu by using equation [ 7 ], the decibel definition for root-power quantities.

In the definition, we simply set  V2 = V ,  V1 = 0.775 ( the dBu reference voltage )  and  dB = +4.  That gives us :


Calculating +4dBu [ 11 ]

 

   So, the +4 dBu pro-audio line level is 1.23 volts.

 

The dBu was previously called the dBv (lower case v) but to avoid confusion with the dBV, which uses a 1.0 V reference, the lower case ‘u’ was adopted.  Occasionally, however, you may find the ‘dBv’ notation still being used to indicate a 0.775 reference voltage.

 

 

 

 

The dBV

 

 

The ‘dBV’ is a logarithmic voltage ratio relative to one volt.  Like the dBu, it presumes no particular load impedance.

So, 1 V = 0 dBV.  And since voltage is a root-power quantity, we use equation [ 7 ] to find that 10 V = 20 dBV,  100 V = 40 dBV,  1000 V = 60 dBV,  etc.

 

 

 

The Consumer Audio Standard

 

 

The dBV is the standard voltage quantifier for interfacing consumer and semi-pro audio gear.

The consumer interface specifies an unbalanced line-level signal of −10 dbV connected by an unbalanced 2-wire cable using RCA connectors ( ‘phono plugs’ ).

RCA cables


This connector was first used by RCA to connect phonograph tonearms to amplifiers, hence the label ‘phono’ plug.

The plug was so common in RCA equipment that it became widely known as the ‘RCA’ plug.

 

We can calculate the voltage that corresponds to a −10 dBV line-level signal by using equation [ 7 ], the decibel definition for root-power quantities.

In the definition, we simply set  V2 = V ,  V1 = 1.0 ( the dBV reference voltage )  and  dB = −10.  That gives us :


0.316 volts [ 12 ]

 

   So, the −10 dBV consumer line level is 316 mV.

 

 

 

Pro/Consumer Level Difference

 

 

It may be obvious that the difference between the +4 dBu professional line level and the −10 dBV consumer line level is not 14 dB.  That's because the two decibel values reference different voltages:  0.775 V for the dBu and 1.0 V for the dBV.

To find the true difference in decibels, we must compare the absolute voltages to which +4 dBu and −10 dBV refer.

Calculation [11] showed us that +4 dBu = 1.23 volts.  Calculation [12] showed us that −10 dBV = 0.316 volts.

 

So, using the decibel definition for voltages [ 7 ] , we find that 1.23 volts is about 12 dB above 0.316 volts :


 

20 log (1.23/0.316) = 20 log (3.892)

= 20 × 0.590 = 11.8 dB

 

And, not surprisingly, we find that 0.316 volts is about 12 dB below 1.23 volts :


 

20 log (0.316/1.23) = 20 log (0.257)

= 20 × (− 0.590) = −11.8 dB

 

 

   So, the difference between the Professional and the Consumer line-level voltage standards is roughly 12 dB.

 

 

 

 

VU Meters

 

 

The VU (Volume Unit) meter is a tool for visually gauging the "loudness" of an audio signal (more about loudness in a moment).

The VU meter's electromechanical ballistics act to average audio levels over short periods of time, which simulates how we perceive loudness.

The original VU meters were fitted with a 200 μA d'Arsonval (moving-coil) DC ammeter fed by a full-wave, copper-oxide rectifier.

D'Arsonval

 

 

 

  Jacques-Arsène d'Arsonval (1851-1940) was a French physician, physicist and inventor of the moving-coil galvanometer and the thermocouple ammeter.

He was also a contributor to the emerging field of electrophysiology.


The needle of an analog VU meter has a mass that slows down the meter's response time.  Short-duration signal peaks and troughs are smoothed out, better depicting the perceived loudness of an audio program.

The needle's rise time (the time it takes to reach the level of the sound) and its fall time (the time it takes to drop to a lower level) are each about 300 milliseconds.

VU Scale

 

The VU scale ranges from −20 VU to +3 VU, with −2 VU near the middle.

0 VU (100%) equals a signal level of +4 dBu (1.23 V) – the same level as the pro-audio line-level standard.

0 VU is sometimes referred to as 0 dBVU or simply 0 dB.

 

The VU meter isn't designed to show you peak volumes but rather to help you guide the average volume to a 100% target.

Electronic and digital VU meters must carefully emulate the VU meter's original ballistics in order to match the d'Arsonval movement.

 

 

 

Loudness

 

 

The intensity of a sound in watts / meter 2 doesn't correlate to its perceived "loudness".

One reason for this is that our hearing sensitivity depends on a sound's wavelength.  This dependency is largely due to our roughly 2.5 cm long, curved ear canals.

Another reason is that, as the next chart will show, two tones played at the same SPL (sound pressure level) may not sound equally loud if they differ in frequency.  In fact, the difference in perceived loudness could be quite large.

In the end, loudness is a subjective measure obtained statistically by surveying the judgments of a large number of listeners having normal hearing.  In this way, a ‘ typical listener ’ is established.

 

 

 

Phons

 

 

Loudness surveys use a 1 kHz tone at 60 dB SPL as a reference loudness, which is defined as 60 phons.

 

60 phons = 60 dB SPL @ 1 kHz

 

The loudness scale, in phons, logarithmically tracks the SPL of the 1 kHz reference tone both up and down from 60 phons.

 

Headphones

So, to judge the loudness of a random test tone, the SPL of the reference tone is adjusted up or down until the two tones sound equally loud.

At that point, the perceived loudness of the test tone, in phons, exactly matches the SPL of the reference tone.

 

 

 

Equal Loudness Curves

 

Harvey Fletcher

 

In the 1930s, equal-loudness curves were researched by Harvey Fletcher, the “father of stereophonic sound” (left) and Wilden Munson.

As technology has improved, the original curves have been updated.


 

These so-called "Fletcher-Munson" curves show the relationship of a tone's perceived loudness to its frequency and pressure:



Equal Loudness Curves from ISO 226:2003

Equal Loudness Curves

 

 

Look at the blue, 60-phon loudness curve.  It shows that a 1000 Hz tone at 60 dB SPL sounds 60 phons loud.  That's the definition of a phon.

But when you follow the blue curve to the left, you see that a low 20 Hz bass tone must be 110 dB SPL to reach that same 60-phon loudness curve!  Clearly, our hearing is less sensitive to low-frequency sound waves.

 

 LOW-FREQUENCY NOTE:  Notice that the louder phon curves rise less steeply in the low frequency range than do the quieter curves.  In other words, the low-end sensitivity of our hearing gets flatter with increasing loudness.

 

 

 

Decibel Weighting

 

 

Since some audio frequencies are more important to perceived loudness than are others, it makes sense to overweight them when measuring sound levels.

Our ears are most sensitive to frequencies between about 600 Hz and 7 kHz (especially around 3 kHz).  At lower and higher frequencies our ears are less sensitive, as depicted below :

 

Hearing Sensitivity Chart

Hearing Sensitivity
This photo by Unknown Author is licensed under CC-BY-SA

 

 

 

Equalization Curves

 

 

Sound level meters often use "frequency weighting" to skew their sensitivity toward those frequencies to which our own hearing is most sensitive.

The meters employ equalization (EQ) curves that control the degree of emphasis and de-emphasis given to various frequency ranges.

Three common EQ curves have been named 'A', 'B' and 'C'.  Whenever an EQ curve is applied to a measurement, its letter designation is appended to the dB label – for example, dBA, dB(A) or LpA.

Since our hearing is less sensitive to low frequencies, all three of these weighting curves de-emphasize the lows :

 

 

dB Weighting Curves

Decibel Weighting

 

 

 

dB(A)

 

 

The ‘A’ weighting curve is the most widely used EQ curve.  Its aggressive exclusion of low frequencies and its slight emphasis of the 3 kHz range make it useful for defining noise regulations

Sound Level Meter

Though the ‘A’ curve may have some deficiencies, like not falling off quickly enough above 10 kHz, the dB(A) is mandated worldwide for hearing damage risk.

The ‘A’ curve is also used to measure the sound pressure level (SPL) of musical performances.  However, it works best for music having a moderate volume, without loud low-end frequencies.

 

 

dB(C) and dB(B)

 

 

Notice how steeply the low frequencies are rolled off by the A-weighted EQ curve.

Loudspeaker

By contrast, the C-weighted curve has a moderate low-end cut which may render a better measure of music having loud bass tones.

That's because, as mentioned previously, the low-end sensitivity of our ears gets flatter with increasing loudness.

As a result, ‘C’ weighting can be helpful when monitoring high-SPL music at shows or in the studio.  The ‘C’ curve is practically linear over many musical octaves.

 


 

The ‘B’ curve has a low-end slope that's midway between the severe cut of the ‘A’ curve and the minimal cut of the ‘C’ curve.

B-weighting may be best for overall music listening levels but it has been phased out of many sound level meters and is seldom used.

 

 

 

Average Sound Levels

 

 

It turns out that our perception of the loudness of a piece of music depends, not only on which sound frequencies and pressure levels are present, but also on its average sound level rather than peak levels.

Sound levels can peak in a fraction of a second but they won't sound “loud” if the level isn't sustained.

A short burst of a high-frequency tone sounds much lower in level to us than it actually is because it takes time for our auditory processing system (our hearing) to become aware of and attach meaning to sounds.

The Ear

 

 

RMS Meters

 

 

In the early days of sound technology, mechanical VU meters approximated what's called “RMS averaging” to provide an acceptable indication of program loudness.

But toward the end of the 20th century, the mechanical meter's limited decibel range (half of its scale displays only 6 dB) made it less suited to advanced recording technology.

In the 1980s, electronic RMS meters were developed, offering scale ranges of 50 dB or more.  These meters calculate RMS averages over a roughly 300 millisecond time span.

RMS averaging smoothes out any sharp peaks in the audio so that the meter readings more closely agree with how we perceive loudness.

Since neither RMS nor VU meters register sudden peaks in audio level, they're often used in conjunction with Peak-Level meters or indicators.

 

 

 

What's RMS Averaging?

 

 

Recall that the amplitude of the back-and-forth displacement of air particles determines the pressure of the longitudinal acoustic wave propagated through the air.

Likewise, the amplitude of the back-and-forth displacement of charged particles (like electrons) determines the voltage of the longitudinal electric wave propagated through wires and components.

In both waves, particle displacements are constantly increasing, decreasing and reversing direction, like the displacement of a swinging pendulum bob.

 

 

Sine Wave

— AMPLITUDES —
Peak (PK),  Peak-to-Peak (PP),  and  RMS

 

 

To calculate the effective power (energy / time) of vibrating particles, we need to use an average, steady amplitude which delivers the same power as the fluctuating amplitude.

The average value we need is called the root-mean-square or RMS.  It's defined as the square root of the mean (average) of a series of squared amplitudes over time.  In short, it's the root of the mean squared amplitude :

 


RMS equation [ 13 ]
Where:

x rms   =   the root mean square
  n       =   the number of measurement values
  x i     =   each value

 

The RMS amplitude of a sine wave is equal to its peak amplitude ( V PK ) multiplied by about 0.707 ( 1 ⁄ √2 ).

Inversely, its peak amplitude is equal to its RMS amplitude multiplied by about 1.414 ( √2 ).  For example, the peak voltage of a 120 VAC wall outlet is about 170 volts (120 × 1.414).

Decibel ratios are very often calculated from RMS averages.

Averaging is especially important when dealing with complex, non-repeating signals or noise.

 

 

 

Digital Audio

 

 

Digital audio and analog audio exist in markedly different realms.

In the analog realm, an audio signal is a smooth wave of physical force, like acoustic pressure or electric voltage.

In the digital realm, an audio signal is a stream of numbers (or "samples") each representing one instantaneous level of force.

Each number in the stream has been quantized onto one step of a staircase of fixed levels having a top step – a highest level – called the clipping point.

For example, a 16-bit digital audio system has 216 = 65,536 (64 K) numeric steps.  A signal voltage higher than the top step can't be represented by the system, so it gets clipped down to the top step.

 

 

 

dBFS (Full Scale)

 

 

Another name for the digital clipping point is Full Scale.  At full scale, a program's dynamic range is using the entire bit depth of the fixed-point digital system.

‘dBFS’ is a logarithmic ratio referenced to Full Scale.  It measures how many decibels below full scale (0 dBFS) an audio signal level is.

So, while analog signal levels are measured upward from the noise floor, digital signal levels are measured downward from full scale.

 

Signal voltages higher than full scale are clipped down to the highest numeric step.  In a 16-bit system, a one-step error is a relative distortion of 1 ⁄ 65,536.  In decibels, that's

 

20 × log (1 ⁄ 65,536) = −96 dBFS.

 

In other words, the maximum, unclipped dynamic range of a 16-bit system is 96 dB.

A 24-bit system supports a dynamic range of 144 dB and a 32-bit fixed-point system supports a range of 1680 dB.

 

 

 

dBTP (True Peak)

 

 

The digital music that's released for streaming, download, digital radio or CD is a stream of numbers that are converted into a smooth wave by Digital-to-Analog Converters (DACs) in our phones, cars, laptops, Bluetooth speakers, CD players, etc.

The conversion of digital steps into a smooth wave can result in changes to the audio level.


True Peak

For example, the DAC may convert two adjacent, full-scale samples into a wave with a true peak higher than either sample peak (see figure).

These inter-sample peaks can result in audible clipping if the DAC lacks sufficient analog headroom.  Otherwise, they'll play back cleanly.


 

‘dBTP’ measures the True Peak level of a signal.  Like dBFS, it's a logarithmic ratio referenced to the system's full-scale clipping point.  So,

 

0 dBTP = 0 dBFS.

 

But while dBFS values are negative, dBTP values are positive (above full scale).

Some peak meters detect only sample peaks whereas a true-peak meter can predict inter-sample peaks by using oversampling and software algorithms.

Peak metering can help you gauge the dynamic range, digital clipping and headroom of an audio program, but it isn't a good measure of loudness.

Average sound levels are a better measure of loudness than are instantaneous peak levels.

 

 

 

LUFS & LKFS

 

 

‘LUFS’ and ‘LKFS’ are two different abbreviations for the exact same method of measuring perceived loudness.  There's no difference between the two and both are measured in decibels relative to full-scale.

In this article, we'll mostly use the abbreviation ‘LUFS’.

 



  The International Telecommunications Union ( ITU ) first standardized this loudness measurement as LKFS, an abbreviation of "Loudness K-weighted relative to Full Scale".  Later, the European Broadcast Union ( EBU ) standardized the same measurement as LUFS, an abbreviation of "Loudness Units relative to Full Scale".


 

 

LUFS meters give us a more accurate measure of perceived loudness than do either RMS or peak meters.

They do this by taking into account two important characteristics of our hearing :

 

  1. its differing sensitivity to different frequencies and pressure levels, and

  2. its dependence on average sound levels.

 

 

 

Both dBFS and LUFS are measured in negative decibels relative to full scale.  However, dBFS is a measure of signal strength while LUFS is a measure of perceived loudness.

 

 

0 LUFS = 0 dBFS

 

 

As LUFS readings move more negative, the sound is perceived as quieter.

 

 

 

Loudness Units (LU)

 

 

LUFS loudness is an absolute measure even though it's relative to full scale, just like the height of an airplane is an absolute measure even though its relative to Earth's surface.

The Loudness Unit ( LU ) was created as a relative loudness (in decibels) that's equal to the difference between two LUFS levels.

For example, if a music streaming service has a loudness target of −16 LUFS, then that target can be considered as 0 LU, much like 0 VU is considered a volume target in the analog realm.

By using −16 LUFS as a reference loudness :

   −19 LUFS  becomes  −3 LU

   −14 LUFS  becomes  +2 LU

 

 

 

K-Weighting

 

 

One improvement baked into the LUFS method is a frequency weighting filter called K-weighting.  K-weighting is the first step of the LUFS method.

The following graph shows how K-weighting compares to the traditional A- and C-weightings.

 

 

K Weighting Curve

‘K’ Weighting Curve compared to ‘A’ and ‘C’

 

 

LUFS meters K-weight audio signals so they more closely match the way we perceive loudness—with more sensitivity to mid and high frequency signals and less sensitivity to low ones.

So, if a program has a lot of low bass content, LUFS meters deemphasize it, lessening its contribution to the loudness readings.

 

  K-weighting is actually a combination of a Pre-filter that applies a 4 dB high shelf above 2000 Hz followed by a Revised Low-frequency B-curve or RLB filter—a simple high-pass filter that applies a 2nd order rolloff below about 100 Hz.

Together, the two filters leave the frequencies between 100 Hz and 1000 Hz unchanged, at 0 dB.

  In a multi-channel program, K-weighting is applied separately to each audio channel (Left, Right, Center, Left-Surround and Right-Surround).

 

 

 

Mean Squares and Channel Summing

 

 

After K-weighting the audio, the LUFS method computes the "mean-square" voltage of each channel.

A mean square voltage is the average (mean) of the squares of a succession of voltage measurements over a period of time.  It's basically a root mean square (RMS), without taking the square root.

After the channels are mean-squared, they're summed together.  1.5 dB of gain is added to the two surround channels to simulate a position nearer to each side of the listener than are the three front channels.

Mean-square voltages are used to compute the RMS power of the audio over various durations.

 


 

At this point, take a look at the following Flowchart to see how far down the LUFS recipe we've gotten—we're just past the channel summation.

 

 

LKFS / LUFS Loudness Measurement Flowchart

LUFS Flow Chart

 

 

 

 

Momentary Loudness

 

 

After summing together all the audio channels, LUFS meters compute the RMS average power contained within 400-millisecond blocks of time.

One 400 ms time block is averaged every 100 ms.  This means that every new time block overlaps the previous block by 75%, ensuring that Momentary Loudness readings transition smoothly :

 

LUFS Loudness Blocks

LUFS Momentary Loudness Readings

 



  Recall that the d'Arsonval VU Meter ballistics have an effective averaging time of about 300 ms.  So, Momentary LUFS is somewhat similar to a traditional RMS meter, but with the addition of K weighting.


 

 

The final step to a LUFS "Momentary Loudness" is taking the log10 of the current time block's frequency-weighted, RMS-averaged power, and then multiplying by 10 to convert bels to decibels.

 

  Momentary Loudness is the foundation for two longer duration LUFS measurements:  Short-Term Loudness and Integrated Loudness.

 

 

 

Short-Term Loudness

 

 

A LUFS Short-Term loudness reading is an average of the last three seconds of Momentary loudness readings.

Since momentary values are calculated 10 times per second, each 3-second-long short-term value is the average of a pool of 30 momentary values.

Every 100 milliseconds, the newest momentary value is added to the pool of 30 values and the oldest value is removed.  Then, a new Short-Term value is computed :

 

 

Short-Term Average Diagram

LUFS Short-Term Loudness Readings

 

 

This method, called a simple moving average, is also used to chart financial data like stock prices and unit sales.  The method smoothes out momentary spikes in the data, helping to reveal trends that otherwise would be jittery.

The advantage of Short-Term LUFS readings is that they aren't overly affected by a few extra-loud, momentary peaks.

The final step to a LUFS "Short-Term Loudness" is taking the log10 of the most recent 3-second power average, and then multiplying by 10 to convert bels to decibels.

 



  Since Short-Term LUFS react more slowly than Momentary LUFS, True-Peak metering is a good complement to Short-Term LUFS.


 

 

 

Integrated Loudness

 

 

LUFS Integrated Loudness is a gated average of all the Momentary Loudness blocks within a long-duration time interval.

It measures the loudness of an entire audio program, be it a song; an album; a video; a podcast; a TV/radio spot or episode; or a feature film.

There are two gates that determine whether a particular 400 ms momentary time block is included in the gated average or ignored for being too quiet :

  1. Low-Level Absolute Gate

    The first gate is a low-level, absolute gate having a fixed threshold of −70 LUFS.  This gate keeps very quiet background content from biasing the overall program loudness.

    All the momentary blocks that measure below this threshold are ignored.  They will not contribute to the integrated loudness.

  2. Foreground Relative Gate

    The second gate has a relative threshold that floats higher or lower, so as to focus on the foreground portion of the program.

    The floating threshold is determined by averaging all the momentary blocks that have successfully passed through the low-level gate, and then decreasing the result by 10 LU ( loudness units ).

    Any momentary blocks that measure below this floating threshold are ignored.  They, too, will not pass on to the final integrator.

 

 

Final Integrator and Timer

 

 

The Final Integrator arithmetically averages all the momentary time blocks that have successfully passed through both the absolute and the relative gates.

The starting time for this average is always "0" and the ending time is some chosen duration.

Many LUFS meters run in real time, continually updating the integrated loudness reading as new momentary blocks pass through the gates.  At the ending time, the timer is reset to zero.

LUFS Loudness Meter

 

 


The final step to displaying a LUFS "Integrated Loudness" is taking the log10 of the most recent integrated power level, and then multiplying by 10 to convert bels to decibels.


 

 

 

Loudness Normalization

 

 

LUFS Integrated Loudness has become the standard for maintaining loudness consistency between individual pieces of audio, like songs, podcast episodes or spots in a newscast.

Volume Control


A consistent loudness level frees listeners from having to keep readjusting the playback volume on their devices.

Many media outlets use Integrated LUFS as their primary metric for defining the loudness targets that perform best on their platforms.


Music streaming services will apply negative gain to songs that exceed the service's target loudness and positive gain to songs that fail to reach the target.

They may also use peak program levels to help determine the amount of applied gain.

Matching songs to a target loudness is called "Loudness Normalization"

Loudness normalization helps to ensure that

 

Consistency

 

 

 

 

So… What's a Decibel ?

 

 

Okay, here are three Merriam-Webster™ style definitions of the word ‘decibel’ for those occasions when a short answer is needed to the question, "What's a decibel?" :

 

 

  1. A decibel is a unit for expressing sound pressure level (SPL) on a scale from zero for the average least perceptible sound to about 130 for the average pain level.

  2. A decibel (symbol dB) is a unit for expressing the ratio of two amounts of acoustic or electric signal power.  One decibel equals ten times the common logarithm of this ratio.

  3. A decibel (dB) is a unit for expressing the ratio of two amounts of acoustic pressure or electric voltage.  One decibel equals twenty times the common logarithm of this ratio.

 

 

 


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