# dBFS

Clipping of a digital waveform. The red lines indicate full scale, and the waveform is shown before and after hard clipping (grey and black outlines respectively).

Decibels relative to full scale (dBFS or dB FS) is a unit of measurement for amplitude levels in digital systems, such as pulse-code modulation (PCM), which have a defined maximum peak level. The unit is similar to the units dBov and decibels relative to overload (dBO).[1]

The level of 0 dBFS is assigned to the maximum possible digital level.[2] For example, a signal that reaches 50% of the maximum level has a level of −6 dBFS, which is 6 dB below full scale. Conventions differ for root mean square (RMS) measurements, but all peak measurements smaller than the maximum are negative levels.

A digital signal that does not contain any samples at 0 dBFS can still clip when converted to analog form due to the signal reconstruction process interpolating between samples.[3] This can be prevented by careful digital-to-analog converter circuit design.[4] Measurements of the true inter-sample peak levels are notated as dBTP or dB TP ("decibels true peak").[5][6]

## RMS levels

Since a peak measurement is not useful for qualifying the noise performance of a system,[7] or measuring the loudness of an audio recording, for instance, RMS measurements are often used instead.

A potential for ambiguity exists when assigning a level on the dBFS scale to a waveform rather than to a specific amplitude, because some engineers follow the mathematical definition of RMS, which is −3 dB below the peak value, while others choose the reference level so that RMS and peak measurements of a sine wave produce the same result.[8][9][10][11][12]

The unit dB FS or dBFS is defined in AES Standard AES17-1998,[13] IEC 61606,[14] and ITU-T Recs. P.381[15] and P.382,[16] such that the RMS value of a full-scale sine wave is designated 0 dB FS. This means a full-scale square wave would have an RMS value of +3 dB FS.[17][18] This convention is used in Wolfson[19] and Cirrus Logic[20] digital microphone specs, etc.

The unit dBov is defined in the ITU-T G.100.1 telephony standard such that the RMS value of a full-scale square wave is designated 0 dBov.[21][22] All possible dBov measurements are negative numbers, and a sine wave cannot exist at a larger RMS value than −3 dBov without clipping.[21] This unit can be applied to both analog and digital systems.[21] This convention is the basis for the ITU's LUFS loudness unit,[23] and is also used in Sound Forge[10] and Euphonix meters,[24] and Analog Devices digital microphone specs[25] (though referred to as "dBFS").

## Dynamic range

The measured dynamic range of a digital system is the ratio of the full scale signal level to the RMS noise floor. The theoretical minimum noise floor is caused by quantization noise. This is usually modeled as a uniform random fluctuation between −12 LSB and +12 LSB. (Only certain signals produce uniform random fluctuations, so this model is typically, but not always, accurate.)[26]

As the dynamic range is measured relative to the RMS level of a full scale sine wave, the dynamic range and the level of this quantization noise in dBFS can both be estimated with the same formula (though with reversed sign):

${\displaystyle \mathrm {DR} =\mathrm {SNR} =20\log _{10}{\left(2^{n}{\sqrt {\tfrac {3}{2}}}\right)}\approx 6.0206n+1.761}$

The value of n equals the resolution of the system in bits or the resolution of the system minus 1 bit (the measure error). For example, a 16-bit system has a theoretical minimum noise floor of −98.09 dBFS relative to a full-scale sine wave:

${\displaystyle \mathrm {DR} =\mathrm {SNR} =20\log _{10}{\left(2^{16}{\sqrt {\tfrac {3}{2}}}\right)}\approx 6.0206\cdot 16+1.761\approx 98.09\,}$

In any real converter, dither is added to the signal before sampling. This removes the effects of non-uniform quantization error, but increases the minimum noise floor.

## History

The phrase "dB below full scale" has appeared in print since the 1950s,[27][28][29] and the term "dBFS" has been used since 1977.[30]

Although the decibel (dB) is permitted for use alongside units of the International System of Units (SI), the dBFS is not.[31]

## Analog levels

dBFS is not defined for analog levels, according to standard AES-6id-2006. No single standard converts between digital and analog levels, mostly due to the differing capabilities of different equipment. The amount of oversampling also affects the conversion with values that are too low having significant error. The conversion level is chosen as the best compromise for the typical headroom and signal-to-noise levels of the equipment in question. Examples:[32][33][34]

• EBU R68 is used in most European countries, specifying +18 dBu at 0 dBFS.
• In Europe, the EBU recommend that −18 dBFS equates to the alignment level.
• UK broadcasters, alignment level is taken as 0 dBu (PPM 4 or −4 VU)
• The American SMPTE standard defines −20 dBFS as the alignment level.
• European and UK calibration for Post & Film[clarification needed] is −18 dBFS = 0 VU.
• US installations use +24 dBu for 0 dBFS.
• American and Australian Post: −20 dBFS = 0 VU = +4 dBu.
• In Japan, France, and some other countries, converters may be calibrated for +22 dBu at 0 dBFS.
• BBC specification: −18 dBFS = PPM "4" = 0 dBu
• German ARD and studio, PPM +6 dBu = −10 (−9) dBFS. +16 (+15) dBu = 0 dBFS. No VU.
• Belgium VRT: 0 dB (VRT ref.) = +6 dBu; −9 dBFS = 0 dB (VRT ref.); 0 dBFS = +15 dBu.

## References

1. ^ M. Thurston, A; H. Pearce, T; D. Higman, M; Hawksford, Malcolm (1993-01-01), Bandpass Sigma Delta A-D Conversion, pp. 259–281, ISBN 9781441951311, retrieved 2018-07-27, In all cases the reference power level used for measurements will be the overload point of the converter in question, and figures will be quoted in dBO.
2. ^ Price, Jim. "Understanding dB". Professional Audio. Retrieved 2007-03-13.
3. ^ Nielsen, Søren H.; Lund, Thomas. "0dBFS+ Levels in Digital Mastering" (PDF). TC Electronic A/S. Denmark. inter-sample peaks may be considerably higher than 0dBFS.
4. ^ Aldrich, Nika (July 2003). "Digital Distortion in CD's and DVD's: The Consequences of Traditional Digital Peak Meters" (PDF). Trillium Lane Labs. Archived from the original (PDF) on 2010-08-16. Retrieved 20 November 2010.
5. ^ "BS.1770-4 (10/2015): Algorithms to measure audio programme loudness and true-peak audio level". International Telecommunication Union. Retrieved 2018-07-27. Meters that ... use an oversampled sampling rate of at least 192 kHz, should indicate the result in the units of dB TP [which] signifies decibels relative to 100% full scale, true-peak measurement.
6. ^ "Terminology for Loudness and Level dBTP, LU and all that - BBC R&D". BBC. Retrieved 2018-07-27. As can be seen from the figure, if the peak sample values are 0 dBFS, the true peak will be higher than 0 dBTP.
7. ^ Davidson, J. (1961). "Average vs RMS meters for measuring noise". IRE Transactions on Audio. AU-9 (4): 108–111. doi:10.1109/tau.1961.1166333. ISSN 0096-1981. the conclusion is reached that the meaningful quantities are found by rms measurements. ... At this point it may be objected that other measurements, peak, or peak-to-peak voltage measurements in particular, are also significant. This is true, but not from the standpoint adopted here. Such measurements are applicable only to the field of nonlinear response, such as dielectric breakdown, etc.
8. ^ "RMS Settings" (PDF). Adobe Audition – User Guide for Windows. Adobe. 2003. Archived from the original (PDF) on 2007-01-27. Retrieved 2007-03-16. - Allows "0dB = FS Sine Wave" or "0dB = FS Square Wave"
9. ^ "0 Db Reference". Active Voice / Noise Level Monitor User's Guide. GL Communications, Inc. Retrieved 2007-03-16. - "0 Db" reference can be either "FS Sine Wave" or "FS Square1 1Wave"
10. ^ a b Katz, Robert (2000-10-28). "Zero dBFS Defined". Digital Domain. Retrieved 2017-06-11. This method yields a result of -3dB for a full scale sine wave and 0dB for a full scale square wave. Sound Forge uses this method.
11. ^ "A/85 – Techniques for Establishing and Maintaining Audio Loudness for Digital Television — ATSC". ATSC. Retrieved 2018-07-27. many software programs indicate level on virtual meters by means of a conventional RMS calculation, leading to a full scale sine wave reading -3.01 dB FS, which is incorrect in the context of this document.
12. ^ "LUFS, dBFS, RMS... WTF ?!? How to read the new loudness meters - Production Advice". Production Advice. 2013-01-08. Retrieved 2018-07-27.
13. ^ "AES Standard » AES17-2015: AES standard method for digital audio engineering - Measurement of digital audio equipment". www.aes.org. Retrieved 2016-04-29. Because the definition of full scale is based on a sine wave, it will be possible with square-wave test signals to read as much as + 3,01 dB FS.
14. ^ "IEC 61606-3:2008 Audio and audiovisual equipment - Digital audio parts - Basic measurement methods of audio characteristics - Part 3: Professional use". International Electrotechnical Commission. 2008. Retrieved 2018-07-27. the r.m.s. amplitude of a ... 997 Hz sinusoid whose peak positive sample just reaches positive digital full-scale ... is defined as 0 dB FS
15. ^ "P.381 (03/17): Technical requirements and test methods for the universal wired headset or headphone interface of digital mobile terminals". International Telecommunication Union. 2017. Retrieved 2018-07-27. 0 dBFS represents the root mean square (RMS) level of a full-scale sinusoidal
16. ^ "P.382 (07/16): Technical requirements and test methods for multi-microphone wired headset or headphone interfaces of digital wireless terminals". International Telecommunication Union. Retrieved 2018-07-27. 0 dBFS represents the root mean square (RMS) level of a full-scale sinusoidal signal
17. ^ Digital and Analog Measurement Units for Digital CMOS Microphone Preamplifier ASICs (Analog Devices) - "The definition of 0 dBFS as a full-scale sine wave is used by several audio analyzers, and the rms and peak values in the digital domain for a sine wave are equal for these analyzers. … Thus, a square wave whose top and bottom are at the maximum digital codes has an rms value of 1.414 FFS or 3.01 dBFS"
18. ^ "10 Audio Recording". Tonmeister. 23 October 2011. Retrieved 30 January 2016.
19. ^ "WM7216E datasheet" (PDF). May 2016. Note that, because the definition of FSR is based on a sine wave, it is possible to support a square wave test signal output whose level is +3dBFS.
20. ^ "CS7250B datasheet" (PDF). Note that, because the definition of FSR is based on a sine wave, it is possible to support a square wave test signal output whose level is +3 dBFS.
21. ^ a b c "G.100.1 (06/15): The use of the decibel and of relative levels in speechband telecommunications". International Telecommunication Union. Retrieved 2018-07-27. The level of a tone with a digital amplitude (peak value) of xover is therefore L= –3.01 dBov.
22. ^ Zopf, Robert. "Real-time Transport Protocol (RTP) Payload for Comfort Noise (CN)". tools.ietf.org. Retrieved 2016-04-30. For example, in the case of a u-law system, the reference would be a square wave ... and this ... represents 0dBov
23. ^ "BS.1770-4 (10/2015): Algorithms to measure audio programme loudness and true-peak audio level". International Telecommunication Union. 2015. Retrieved 2018-07-27. If a 0 dB FS, 1 kHz ... sine wave is applied to the ... input, the indicated loudness will equal −3.01 LKFS.
24. ^ "Application Note 1: System 5 Metering: Peak vs. Average" (PDF). January 2002. On a logarithmic dB scale, the difference between a sine wave's peak and RMS average level is 3 dB. Euphonix bases its metering on the Audio Precision measurement system, which adheres to the RMS average technique.
25. ^ "Understanding Microphone Sensitivity". Analog. Retrieved 30 January 2016. so a digital microphone’s output must be scaled from peak to rms by lowering the dBFS value. For a sinusoidal input, the rms level is 3 dB (the logarithmic measure of (FS√2) below the peak level ... A 94 dB SPL sinusoidal input signal will give a –26 dBFS peak output level, or a –29 dBFS rms level.
26. ^ Watkinson, John (2001). The Art of Digital Audio 3rd Edition. Focal Press. ISBN 978-0-240-51587-8.
27. ^ Automatic Control. Reinhold Publishing Corporation. 1957-01-01.
28. ^ Hewlett-Packard Journal. Hewlett-Packard Company. 1962-01-01.
30. ^ Robert, Talambiras (1977-05-01). "Some Considerations in the Design of Wide-Dynamic-Range Audio Digitizing Systems". It is convenient when working with A/D converters to define a 0 dB reference for a full-scale-to-full-scale sine wave. ... The quantizing noise in the Nyquist bandwidth for a 16 bit converter would be -98.08dbFS Cite journal requires |journal= (help)