# Decibeles y nepers

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THE DECIBEL AND THE NEPER
Admiralty Handbook of Wireless Telegraphy, B.R.230 (Vol. II). HMSO London, 1938., Appendix "A". Crown Copyright 1938 (Now public domain). Updated, corrected, and references added by D W Knight. (Version 1.01, 19th Oct. 2010) (Units are changed to use camelcase, e.g., db becomes dB, etc.).

1. Historical. - The "deciBel" is the 1/10th part of a "Bel" (afterAlexander Graham Bell, inventor of the telephone sounder), a unit in which one may express power ratios, and gain or loss ratios of related quantities such as current and voltage. It originated in line telephony in 1923, when the American Telephone and Telegraph Company introduced a new unit, then called a "transmission unit"; this was to replace an older conception based on a ratio comparison betweenthe decrease in signal strength produced by the given telephone line, and that produced by a "mile of standard cable" 1 2. In 1924, an international advisory committee on long distant telephony in Europe, together with the representatives of the Bell system, agreed to recommend their countries to adopt as standards EITHER the "Bel", a unit based on logarithms to the base 10, and equal to 10 of theAmerican company's "transmission units", OR the "Neper" (after Napier), a unit based on Naperian logarithms to the base e. The growth in popularity of the deciBel, since 1929, has been so great that it is now almost a household word throughout all branches of Electrical Engineering and Acoustics. 2. Definitions

THE DECIBEL :- Two powers P1 and P2 are said to differ by N "Bels" when P1 = 10N P2i.e. N = Log10 P2 P1 . . . . . . . in Bels

Or in words - "The logarithm to the base 10 of the ratio of the powers, gives the gain or loss in Bels." If P1 = P2 , then N = 0. In practice, a unit of one Bel is found to be inconveniently large, and the l/10th part of it - the deciBel - is more often used. Using the smaller unit we have P1 = 100.1N P2 or N = 10 Log10 P2 P1 . . . . . . . in deciBelsThe basic power ratio is 100.1 , that is, 1.259.

1 "The Transmission Unit and Telephone Transmission Reference Systems", W H Martin. Bell System Technical Journal. 3(3), July 1924. p400-408. Available from http://bstj.bell-labs.com/ . 2 "Decibel - The Name for the Transmission Unit", W H Martin. Bell Sys. Tech. Journal. 8(1), Jan. 1929. p1-2.

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THE NEPER : - Two powers P1 and P2 aresaid to differ by N "Nepers" when P1 = (e2)N P2 Therefore (P1/P2)½ = eN i.e. N = ½ Loge(P1/P2) . . . . . . . in Nepers

Any ratio in dB may be readily converted to Nepers, for example 60 dB We have Log10(P1/P2) = 6 Bels Therefore ½ Loge(P1/P2) = 2.3026 × 6 × ½ = 6.907 Nepers The Neper is used in some European countries, but is less commonly encountered than the deciBel.

3. Cables, Amplifiersand Attenuators. - With these units, if the signal strength of a cable signal is 1/10th of that at the transmitting end, the loss is 1 Bel. With two similar cables in series, the received signal would be l/100th of the transmitted one, and the loss would be two Bels. In any amplifier, if the output power is 100 times the input, the "gain" is two Bels or 20 dB ; with two such amplifiers used inseries, the gain in power ratio would be 10 000 : 1, or 40 dB. One of the many advantages of the deciBel is that the enormous ranges of power involved in communication work can be expressed in figures conveniently small, instead of astronomically big ones. In the above example, it will be noted that the nett power ratio of two amplifiers in series, involves the product of their individual powerratios, but only the sum of their deciBel equivalents. This is due to the logarithmic nature of the unit employed. An attenuator may be regarded as the converse of an amplifier, and its power loss may be described in similar units. If two stages of amplification, one of p and the other of q dB is followed by a line having an attenuation of s dB, the nett level at the output of the system will be (p +...