Logarithms of Negative and Imaginary Numbers

Logarithms and Decibels

Decibels

A *decibel* (abbreviated dB) is defined as one tenth of a
*bel*. The bel^{F.1} is an amplitude unit
defined for sound as the log (base 10) of the *intensity*
relative to some *reference intensity*,^{F.2}
*i.e.*,

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.

Properties of DB Scales

Logarithms of Negative and Imaginary Numbers

Logarithms and Decibels

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.

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