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With a foundation in mathematics and physics, audio engineers can accurately describe sound and manipulate its effects with the decibel. The decibel, however, has worked its way into common usage. In the beginning, it was among several units of measurement.
Prior to 1923 "gains" and "losses" of telephone circuits were labeled in miles of standard cable (MSC). A "standard" cable was a 19 gauge open wire cable with a resistance of 88 Ω/mi and a capacitance of 0.054 µF/mi.
The standard carbon transmitter's electrical output level with normal speech as an input was considered as "zero level" (at that time 0.006 W). The loss that occurred over one mile of this "standard" cable very nearly equaled what Harvey Fletcher was measuring in the Bell Telephone Laboratories as the smallest increment easily detected by a normal listener and which he labeled the sensation unit (SU).
In 1923 W. H. Martin wrote in the Bell System Technical Journal an article introducing the transmission unit (TU) devised to replace both the MSC and the SU while reconciling both uses to the telephone system as a whole. The TU was defined as:
(N)TU = 10log(P1/P2) (2-1)
TU was the transmission unit,
P1 was the power measured,
P2 was the power used as a reference,
N is the numerical value to be labeled as (N)TU.
There were 0.947 MSC to 1.0 TU or 1.056 TU to 1.0 MSC.
2.1 The Decibel
In 1929 W. H. Martin wrote again in the Bell System Technical Journal an article entitled "Decibel The Name for the Transmission Unit." As a result of Bell Telephone's participation as invited attendees, the European International Advisory Committee recommended to the various European telephone administrations that they adapt either the decibel or the Naperian unit and designate them the "Bel" and the "neper," respectively.
The Bell system adopted the Bel, converting it into the deci-Bel (one tenth of a Bel) for the convenience of higher resolution of the measured level. Over the 20 years in which they had used the MSC they found that they could meaningfully resolve 0.1 mi. The decibel was defined as the logarithmic form of that power ratio having a value of 100.1. Two amounts of power differ by 1 dB where they are in the ratio of 100.1 and any two amounts of power differ by (N) dB when they are in the ratio of 10N(0.1):
P1/P2 = 10N(0.1)
log10(P1/P2) = log1010 • N • 0.1
log1010 = 1
log10(P1/P2)/0.1 = N
10log10(P1/P2) = N dB
N units are labeled dB.
Hereafter it is assumed that log is to the base 10 in this book unless otherwise stated.
2.2 The Neper
The neper (Np) is defined as:
ln(E1/E2) = N(nepers). (2-3)
One neper is a voltage ratio such that:
(E1/E2) = e1 = 2.718.
N nepers is a voltage ratio such that:
E1/E2 = eN = 2.718...N.
ln(E1/E2) = lne • N
lne = 1
ln(E1/E2) = N (nepers)
The question arises as to what is the relationship between the neper and the decibel. Take the voltage ratio corresponding to one neper and square it to put it into the form of a power ratio and manipulate as follows:
(P1/P2) = (E1/E2)2 = (e2) (2-4)
10log(P1/P2) = 10loge2 = 8.686 dB/neper.
The conclusion is that one neper is equivalent to 1/8.686 dB which is 0.115 Np/dB. Alternately you can use the decineper dNp (1 dB = 1.15 dNp).