Neper unit: From Napier to Telephone Signals
How much signal disappears along a telephone cable? The Neper unit belongs to that mathematical world of ratios: one neper equals about 8.685 890 decibels. Its symbol is Np, and its name honors John Napier, the Scottish mathematician who developed logarithms. The unit carries his name into comparisons of power and amplitude, where a logarithm turns a multiplicative relationship between quantities into a difference between levels. Its history joins personal names, mathematical scales and the practical demands of communication.
Napier lived from 1550-1617, and the spelling of his surname was far from fixed. He often signed himself Jhone Neper and used Ioanne Napero in Latin writings. Neper thus preserves a form of the mathematician's own name. The mathematics behind the Neper unit uses the natural logarithm, written ln. For two power quantities whose ratio is r, the level difference is half of ln r nepers. A power ratio of about 7.389056 therefore corresponds to one neper. For a root-power quantity, such as amplitude, one neper instead corresponds to a change by a factor of e.
Telephone engineers initially expressed transmission losses through a physical length. Until the mid-1920s, miles of standard cable served as their measure. One such mile represented the power lost along a mile of standard telephone cable at a specified frequency, and the loss closely matched the smallest attenuation a listener could detect. The cable itself had defined electrical properties, including resistance and capacitance. This made a length of cable a practical benchmark for a loss of signal. As telephone work developed, however, engineers moved toward a unit defined directly through the relationship between measured power and reference power.
Bell Telephone Laboratories replaced miles of standard cable with the Transmission Unit. The new unit was ten times the base-ten logarithm of a power ratio. Its size stayed close to the older benchmark: one Transmission Unit equaled 1.056 miles of standard cable. In 1928, the Bell system renamed it the decibel. The bel honored telecommunications pioneer Alexander Graham Bell, while the decibel represented a tenth of that larger unit. The smaller measure became the working unit; the bel itself remained comparatively rare.
The Neper unit and the decibel express the same relationships through different logarithms. For n nepers, the equivalent number of decibels is 20n divided by ln 10. In power measurements, the decibel value is ten times the base-ten logarithm of the ratio. With root-power quantities, the corresponding multiplier is twenty. Electrical voltage illustrates why the two formulas differ: when impedance is constant, dissipated power is proportional to voltage squared. A change in voltage and its associated change in power can therefore produce matching level differences, even though the underlying ratios have different numerical values. Current has a similar relationship under the corresponding conditions.
Logarithmic scales also changed the arithmetic of telephone transmission. Gains and losses through successive parts of a circuit could be added or subtracted as level differences, rather than multiplied as power ratios. Positive levels indicate a measured quantity greater than its reference, negative levels indicate a smaller quantity, and equal quantities give a level of zero. The same approach spread into electronics, where amplifier gain, signal attenuation and signal-to-noise ratios are frequently expressed in decibels. These measurements describe relationships between quantities, with a reference establishing the level whenever an absolute value is expressed through the scale.
Sound engineering brought this vocabulary into rooms and other enclosed spaces. In 1962, Kosten and van Os introduced a noise rating system whose numerical rating equaled the sound level in decibels at 1000 hertz. At higher frequencies, where noise was more objectionable, the system required lower levels. A room's acceptability therefore depended on the frequency of its noise as well as its level. Beranek's noise criterion system, introduced in 1957, focused on preserving speech communication. It required lower levels at both high and low frequencies. An update in 1971 produced the preferred noise criterion, known as PNC.
The Neper unit also belongs to an international vocabulary whose boundaries have been debated. In 2002, the International Committee for Weights and Measures considered adding the decibel to the International System of Units and decided against the proposal. Other bodies, including the International Electrotechnical Commission and the International Organization for Standardization, recognized its use. The IEC permitted decibels for root-power quantities as well as power quantities, a practice followed by national standards bodies such as NIST for voltage ratios. Meanwhile, engineers continued using the logarithmic measures across acoustics, electronics and control theory, from sound power to the attenuation of electrical signals.