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Properties of DB Scales

In every kind of dB, a factor of 10 in amplitude increase corresponds to a 20 dB boost (increase by 20 dB):

$\displaystyle 20\log_{10}\left(\frac{10 \cdot A}{A_{\mbox{\small ref}}}\right)
= \underbrace{20\log_{10}(10)}_{\mbox{$20$\ dB}} + 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)
$

and $ 20\log_{10}(10) = 20$ , of course. A function $ f(x)$ which is proportional to $ 1/x$ is said to ``fall off'' (or ``roll off'') at the rate of $ 20$ dB per decade. That is, for every factor of $ 10$ in $ x$ (every ``decade''), the amplitude drops $ 20$ dB.

Similarly, a factor of 2 in amplitude gain corresponds to a 6 dB boost:

$\displaystyle 20\log_{10}\left(\frac{2 \cdot A}{A_{\mbox{\small ref}}}\right)
= \underbrace{20\log_{10}(2)}_{\mbox{$6$\ dB}}
+ 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)
$

and

$\displaystyle 20\log_{10}(2) = 6.0205999\ldots \approx 6 \;$   dB$\displaystyle . \protect$

A function $ f(x)$ which is proportional to $ 1/x$ is said to fall off $ 6$ dB per octave. That is, for every factor of $ 2$ in $ x$ (every ``octave''), the amplitude drops close to $ 6$ dB. Thus, 6 dB per octave is the same thing as 20 dB per decade.

A doubling of power corresponds to a 3 dB boost:

$\displaystyle 10\log_{10}\left(\frac{2 \cdot A^2}{A^2_{\mbox{\small ref}}}\right)
= \underbrace{10\log_{10}(2)}_{\mbox{$3$\ dB}}
+ 10\log_{10}\left(\frac{A^2}{A^2_{\mbox{\small ref}}}\right)
$

and

$\displaystyle 10\log_{10}(2) = 3.010\ldots \approx 3\;$dB$\displaystyle . \protect$

Finally, note that the choice of reference merely determines a vertical offset in the dB scale:

$\displaystyle 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)
= 20\log_{10}(A) - \underbrace{20\log_{10}(A_{\mbox{\small ref}})}_{\mbox{constant offset}}
$


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``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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA