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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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Sound Pressure Level (SPL) is defined using a reference which is approximately the intensity of 1000 Hz sinusoid that is just barely audible (0 ''phons''). In pressure units:


In intensity units:

which corresponds to a root-mean-square (rms) pressure amplitude of $20.4$$\Pa, or about $20$ $\Pa, as listed above. The wave impedanceof air plays the role of ''resistor'' in relating the pressure- and intensity-based references exactly analogous to the dBm case discussed above.

Since sound is created by a time-varying pressure, we compute sound levels in dB-SPL by using the average intensity (averaged over at least one period of the lowest frequency contained in the sound).

Table 4.1 gives a list list of common sound levels and their dBequivalents4.5 [9]:

Table 4.1:Ballpark figures for the dB-SPL level of common sounds.
Sound dB-SPL
Jet engine at 3m 140
Threshold of pain 130
Rock concert 120
Accelerating motorcycle at 5m 110
Pneumatic hammer at 2m 100
Noisy factory 90
Vacuum cleaner 80
Busy traffic 70
Quiet restaurant 50
Residential area at night 40
Empty movie house 30
Rustling of leaves 20
Human breathing (at 3m) 10
Threshold of hearing (good ears) 0

In my experience, the ''threshold of pain'' is most often defined as 120 dB.

The relationship between sound amplitude and actual loudness is complex [10]. Loudness is a perceptual dimension while sound amplitude is physical. Since loudness sensitivity is closer to logarithmic than linear in amplitude (especially at moderate to high loudnesses), we typically use decibels to represent sound amplitude, especially in spectral displays.

The sone amplitude scale is defined in terms of actual loudness perception experiments [10]. At 1kHz and above, loudness perception is approximately logarithmic above 50 dB SPL or so. Below that, it tends toward being more linear.

The phon amplitude scale is simply the dB scale at 1kHz [10, p. 111]. At other frequencies, the amplitude in phons is defined by following the equal-loudness curve over to 1 kHz and reading off the level there in dB SPL. In other words, all pure tones have the same loudness at the same phon level, and 1 kHz is used to set the reference in dB SPL. Just remember that one phon is one dB-SPL at 1 kHz. Looking at the Fletcher-Munson equal-loudness curves [10, p. 124], loudness in phons can be read off along the vertical line at 1 kHz.

Classically, the intensity level of a sound wave is its dB SPL level, measuring the peak time-domain pressure-wave amplitude relative to$10^{-16}$ watts per centimeter squared (i.e., there is no consideration of the frequency domain here at all).

Another classical term still encountered is the sensation level of pure tones: The sensation level is the number of dB SPL above thehearing threshold at that frequency [10, p. 110].

For further information on ''doing it right,'' see, for example,

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