# decibel

Alexander Graham Bell

For examples of noise levels in dB, see noise.

• The meaning now
• Original definition: the transmission unit
• A refresher on logarithms
• Extending it to voltage and currents
• Extending it to sound
• Power ratios for electrical units
• dBm
• dBm0
• dBW
• dBi
• dBd
• Voltage ratios for electric signals
• dBV
• dbv
• dBu
• dBFS
• Sound intensity levels
• db
• dBA, dBB, dBC
• Sound pressure levels
• dB SPL
• Radar signal back-scattered from clouds
• dBZ

A unit used in electrical engineering and acoustics to express the ratio between two values with the same dimensions. The quantities compared may be two power levels, two voltages, two sound pressure levels, and so on. Since the quantities in the ratio always have the same dimensions, the dimensions cancel out; the decibel itself is dimensionless. Some examples of other dimensionless units are the radian and ppm.¹ Symbol, dB, but see below.

A measurement in decibels can express an absolute magnitude, provided one side of the ratio is a reference level explicitly or implicitly specified. A large number of reference levels, measuring several different properties, have been used. What level of which property is often shown by adding a suffix to the dB symbol, for example, “dBm”. Sometimes the portion after “dB” is printed as a subscript, or follows as a separate word, for example, “dB SPL”. In several cases the suffixes have themselves become part of the names of units. Recording engineers, for example, refer to “dee-bee-you”, the dBu.

The bel is named for Alexander Graham Bell (1847-1922).

The decibel is not an SI unit, though the use of the decibel with SI units is sanctioned by the CIPM.² The symbol for the decibel was originally “db”, but over time the influence of the SI usage rules has been so strong that they are now applied to the decibel symbol  : the first letter that represents a person's name in a symbol is capitalized, while the name of the unit is not. Compare watt, W.; pascal, Pa; volt, V; hertz, Hz, etc. The lowercase “d” comes from the symbol for the metric prefix “deci-”. Hence, dB.

The Bell System has adopted the name “decibel” for the “transmission unit,” based on a power ratio of 10.1. This is in accordance with the terminology for the decimal unit, the prefix “deci” being the usual one for indicating a one-tenth relation.3

The one-tenth in the definition of the decibel occurs in 10 to the one-tenth power. That is not what “deci-” means in the metric system. (If you'd like a quick refresher on , check this out.)

The decibel began as the transmission unit, defined by researchers at AT&T to replace of the “mile of standard cable”, a unit of power ratio used in telephone engineering.¹ One of the shortcomings of the mile of standard cable was that it was frequency-dependent. The transmission unit was not; it was purely a unit of power ratio.

The transmission unit was soon renamed the bel (for Bell)3, but the bel is inconveniently large, and the decibel is also.

### Extending the decibel to voltage and current

In time, the decibel began to be used to express ratios between two current levels, or two voltage levels. In other words, it was at first used to describe a ratio between measurements in watts, but became extended to volts and amperes. In the telephone system, impedences were standardized at 600 ohms, and under those circumstances power is proportional to the square root of the voltage or amperage. In logarithm, square roots are , so became 2 × 10 times log V/V; or log .

Similarly for electric currents,

Later, as the decibel spread beyond the confines of Bell Labs, the difference between 10 and 20 would become a source of confusion.

0.467 milliwatt into 600 ohms/1 milliwatt into 600 ohms = 0.467  (note, dimensionless, no units!)

The common (base 10) log of 0.467 is −0.331

Multiply by 10 (we're dealing with power) 10 × −0.331 = −3.3 dBm

That the dBm is, in fact, a unit is demonstrated by the fact that the calculation process can be run in reverse to get milliwatts (no question that that is a unit!) from a measurement expressed in dBm. Notice that, without the "m", that would be impossible. -3.3 dB also describes 467 kilowatts and a reference level 1000 kilowatts.

## The Use grows

The First International Acoustical Conference (Paris, July 1937) adopted the decibel as an international unit at scales of energy and pressure levels.²

Important documents which have been written and reviewed by men eminent in their profession have contained statements which are demonstrable contradictions of basic physical laws. Measurements of power magnitudes have been found to yield results which have differed by several orders of magnitude from estimates based on computation. These errors can be attributed directly to the practice of expressing current ratios, or acoustic pressure ratios, in decibels when these ratios are not the square roots of corresponding power ratios.

Horton (1954) page 551.

The writer, as chairman of the ASA C42 Subcommittee on Definitions of Communication Terms, has been bombarded with letters divided about equally between pleas for extension of the db [sic] to new kinds of ratios and pleas for strict limitation of the meaning.

E. I. Green.
IRE Transactions, April 1954, page 43.

The use of the term “decibel” in connection with a quantity other than power is a violation of the original definition of the decibel. It has, however, become so general that it may have to be lived with, at least in the foreseeable future.

Hartley 1955

A variety of other names have been proposed for the decibel, including logit36, decilit37, decilog, decomlog38 and decilu. 39

The decibel was originally primarily used in Britain and the United States; in continental Europe the neper played the same role. In the early 21st century, the CCU considered recommending adding the neper to the list of SI units as a coherent unit of logarithmic decay, leaving the bel outside SI. To date nothing has come of this effort.3

1. Rather than call these units dimensionless, some experts prefer to treat them as having the dimension 1. Dimensionlessness is a very controversial subject among metrological philosphers. Horton (1954), cited below, is a good example of willful incomprehension of the subject.

2. See NIST Special Publication 330 (2008 edition), page 35 and footnotes h and i on page 36.

2. W. H. Martin.
The transmission unit and telephone transmission reference systems.
AIEE Transactions, vol. 42 (June 1924), pages 797-801.

R. V. L. Hartley.
The transmission unit.
Electrical Communication, (July 1924), pages 34-42.

3. W. H. Martin.
Decibel–The name for the transmission Unit.
Bell System Technical Journal, January 1929.
Available online at www3.alcatel-lucent.com/bstj/vol08-1929/bstj-vol08-issue01.html

4. The First International Acoustical Conference.
Nature, volume 140, page 370 (August 28, 1937).

V. V. L. Rao.
The Decibel Notation.
New York: Chemical Publishing Co., 1946.

The first edition was published in Madras, India.

36. J. W. Horton.
The bewildering decibel.
Electrical Engineering, vol. 73, issue 6 (June 1954).
doi:10.1109/EE.1954.6438830

37. V. V. L. Rao and S. Lakshminarayanan.
The Decilit: A New Name for the Logarithmic Unit of Relative Magnitudes.
Journal of the Acoustical Society of America, vol. 27, issue 2, page 376 (1955).
doi:10.1121/1.1907863

38. John B. Moore.
Letter to the editor: The decilog.
Electrical Engineering, vol. 73, issue 10 (1954) page 960.
doi:10.1109/EE.1954.6439050

39. Attributed in Green's 1954 article to an unpublished memo by M. W. Baldwin and R. E. Graham.

## Power ratios for electric signals

In audio and broadcast engineering, for power of electric signals, the decibel is 10 times the common logarithm of the ratio of the power of the signal being described to the power of the reference level. When the meter reads "0 dB", the signal being measured equals the reference level.

### dBm

The reference level is 1 milliwatt across an impedance of 600 ohms. The “m” stands for "milliwatt". The 600 ohms value came from standards in the telephone industry, the high-tech of the early 20th century, in which maximizing power transfer by matching output and input impedances was an important consideration. Note that a 0 dBm signal in a circuit with an impedance of 600 ohms corresponds to 0.775 volt rms. A signal change of −3 dBm is roughly a halving of the power.

### dBm0

decibels above 1 milliwatt at the zero transmisson level point.

page 789.

### dBW

The reference level is 1 watt.

### dBi

Used in engineering antennas. The "i" stands for "isotropic". imaginary isotropic antenna, radiating equally in all directiobs, withe the  The strength of the radiation is measured in microwatts per square meter.

## Voltage ratios for electric signals

As tubes (valves) gave way to transistors, matching impedances became less important. Audio engineers went from using decibels based on power to those based on voltage. For voltages, the decibel is 20 times the common logarithm of the ratio of voltage being measured to the reference voltage.

### dBV

The reference level is 1 volt rms across any impedance. To convert dBV to dBu, add 2.2 dB. Consumer audio gear is designed for an normal input level of −10 dBV, which corresponds to 0.316 volt rms. This voltage level arose because it was the optimal maximum level for a signal fed directly to an electron tube (valve to Brits), which was a practice in consumer equipment.

### dBv

The reference level is 0.775 volt rms across any impedance. The 0.775 volt value comes from the definition of dBm, since it is the voltage when a 0 dBm sine wave is fed into 600 ohms. This symbol was too easily confused with dBV, and so was renamed dBu.

Some writers do not observe the distinction between the upper and lower case V and treat both dBV and dBv as referenced to 1 volt rms.

### dBu

The reference level is 0.775 volt rms across any impedance. See dBm above for the origin of the value. The “u” stands for unterminated.  Professional audio equipment is designed for a normal input level (“line level”) of +4 dBu, which corresponds to 1.23 volts rms. A signal change of −6 dBu is about a halving of the voltage.

### dBFS

“FS” stands for “full scale.” A unit used in monitoring signal levels in digital signal processing, such as digital audio. To record audio (or any analog signal) digitally, the signal level is measured at equal, small intervals and the value recorded as a number. For audio this requires recording as many as 192,000 numbers each second, so it is necessary to choose a way of representing numbers that a computer can process very quickly and efficiently. All such systems for representing numbers have a biggest number that can be represented (and also a smallest negative number, what follows applies to negative numbers as well). This is the full scale point. If the level of the analog signal is greater than the level assigned to the biggest possible number, there is no way of recording its actual value. Every level greater than the one assigned to the largest possible number is usually simply recorded as the largest possible number. The signal is "clipped." The dBFS scale provides a way of indicating how close one is coming to this undesirable situation.

In such a system, the maximum level before clipping of a sine wave is -3 dBFS.

The relevant standard is IEC 6106-3 ed 1.0 (2008-10). The corresponding Audio Engineering Society standard is AES17.

The AES Information Document for Digital audio engineering

– Guidelines for the use of the AES3 interface,

AES-2id-2006 defines Full-Scale Amplitude as: “the rms voltage that corresponds with a sine wave whose positive peak value reaches the maximum positive digital value and whose negative peak reaches one LSB greater than the minimum negative digital value.” This means a full scale sine wave input would read +3 dBFS, and so would a full-scale square wave, and its rms value would equal +3 dBFS.

Further, AES Information Document for Digital audio engineering – Personal computer audio quality measurements, AES-6id-2006 defines Decibels, Fullscale (dB FS): “Digital signal rms amplitude expressed as a level in decibels relative to full-scale amplitude (20 times the common logarithm of the amplitude over the full-scale amplitude [defined as the ‘rms amplitude of a 997 Hz sine wave in the digital domain whose positive peak value reaches the positive digital full scale, leaving the negative maximum code unused.’]). Note that dB FS expresses a signal level of a digital signal and should not be used to express the signal level of an analog signal.”

For a good discussion of the use and misuse of the dBFS scale, see the papers by Nielsen and Lund at www.tcelectronic.com/TechLibrary#LoudnessControl%20and%20Mastering

### dBr

A reference level is specified in the immediate context.

## Sound intensity levels

In acoustics, the decibels used to express sound intensity levels are 10 times the common logarithm of the ratio between the measured intensity and a reference intensity.

### dB

The reference level is usually 1 picowatt per square meter.

### dBA, dBB, dBC

The measurement is frequency-weighted according to the "A" curve. B and C curves also exist.

## Sound pressure levels

In acoustics, sound pressure levels in dB are 20 times the common logarithm of the ratio between the measured pressure level and a reference pressure level.

### dB SPL

The reference level is usually 20 micropascals, the level of the faintest sound humans can be hear.

The sound pressure level Lp is defined as 20 log10 (p/po) in decibel (dB), where p is the root mean square value of the measured sound pressure, and po is the root mean square value of a reference pressure. For air: po = 2 × 10⁻⁵ N m⁻² μbar is generally used.

K. Diem and C. Lentner.
Scientific Tables. 7th edition.
Ardsley, NY: Geigy Pharmaceuticals, 1970.
Page 224.

## Reflected radar signal level

### dBZ

In meteorology, a measure of the strength of a radar signal reflected off a distant object, such as clouds and falling rain or snow, compared to the strength of the emitted signal. The reference level is 1 millimeter⁶/ meter³, which equals 1 cubic micrometer. See VIP levels.

dBZ Rainfall
40 heavy
24 – 39 moderate
8 – 23 light
0 – 8 barely anything

NOAA.
Glossary: D's.
www.srh.noaa.gov/jetstream/append/glossary_d.htm
Retrieved 1 January 2010.