# Decibels

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Decibels

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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# Decibels

A decibel (abbreviated dB) is defined as one tenth of abel. The bel4.2 is an amplitude unit defined for sound as the log (base 10) of the intensityrelative to some reference intensity,4.3 i.e.,

The choice of reference intensity (or power) defines the particular choice of dB scale. Signal intensity, power, and energy are always proportional to the square of the signalamplitude. Thus, we can always translate these energy-related measures into squared amplitude:

Since there are 10 decibels to a bel, we also have

A just-noticeable difference (JND) in amplitude level is on the order of a quarter dB. In the early days of telephony, one dB was considered a reasonable ''smallest step'' in amplitude, but in reality, a series of half-dB amplitude steps does not sound very smooth, while quarter-dB steps do sound pretty smooth. A typical professional audio filter-design specification for ''ripple in the passband'' is 0.1 dB.

Exercise: Try synthesizing a sawtooth waveform which increases by 1/2 dB a few times per second, and again using 1/4 dB increments. See if you agree that quarter-dB increments are ''smooth'' enough for you.

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