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dB or not dB
- 1. Alexis Baskind dB or not dB Why do we need the decibel ? How to calculate it ? Alexis Baskind, https://alexisbaskind.net
- 2. Alexis Baskind dB or not dB Course series Fundamentals of acoustics for sound engineers and music producers Level undergraduate (Bachelor) Language English Revision January 2020 To cite this course Alexis Baskind, dB or not dB, course material, license: Creative Commons BY-NC-SA. Full interactive version of this course with sound and video material, as well as more courses and material on https://alexisbaskind.net/teaching. dB or not dB Except where otherwise noted, content of this course material is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License.
- 3. Alexis Baskind Why do we need the decibel ? • The decibel (dB) is a generic logarithmic scale that is used to measure levels as ratios relatives to a reference level. • It is widely used in many domains, especially: – Acoustics: dB SPL, dB(A), dB(B), dB(C)… – Electronics and Electricity: dBu, dBV, dBm – Optics: dBm – Video, digital imaging – Telecommunications… dB or not dB
- 4. Alexis Baskind What is the logarithm ? • The common logarithm (“log10”), the inverse function of the exponentiation, is a mathematical function used to describe quantities with a very wide dynamic range: dB or not dB Log10(100)=2 Log10(10)=1 Log10(1)=0
- 5. Alexis Baskind What is the logarithm ? • The logarithm is defined through its base. In audio, the base 10 is the most important: • The logarithm transforms products into sums, and ratios into differences: dB or not dB log10 x´ y( )= log10 x( )+ log10 y( ) log10 x/ y( )= log10 x( )- log10 y( ) y= log10 x( )Û x =10y
- 6. Alexis Baskind Why do we need the Logarithm (1)? • In many engineering sciences like communication engineering and sound engineering, signal levels fluctuates usually over several orders of magnitude. Example 1: the voltage amplitude at the output of a condenser microphone can evolve between fractions of microvolts and several tens of millivolts. Example 2: hearing can perceive sound pressure amplitudes between ca. 0,00002 Pa and ca. 20 Pa => Thanks to logarithm, the range of numerical values reduces drastically and is therefore easier to handle. dB or not dB
- 7. Alexis Baskind Why do we need the Logarithm (2)? • Calculations relative to amplifications and attenuations are much easier to handle with the decibel, since they simply correspond to additions (respectively subtractions), without the need for multiplications (respectively divisions) dB or not dB
- 8. Alexis Baskind Why do we need the decibel ? • Weber-Fechner Law: Human perception is generally more sensitive to ratios than to differences. Example: dB or not dB x2 x2 1 kg 2 kg 4 kg There is perceptually as much difference between 1 kg and 2 kg… …as between 2 kg and 4 kg • Thus in fact, the relation between a stimulus and its perception is (roughly) logarithmic
- 9. Alexis Baskind Why do we need the decibel ? dB or not dB x10 There is perceptually as much difference between 0,02 Pa and 0,2 Pa… …as between 0,2 Pa and 2 Pa therefore, the perception of loudness is roughly proportional to the dB SPL scale pRMS = 0,2Pa pRMS = 2Pa LSPL = 60dBSPL pRMS = 0,02Pa LSPL = 80dBSPL LSPL =100dBSPL x10 +20dB +20dB
- 10. Alexis Baskind How to define the dB ? • The Relations in relative dB units is defined thanks to the logarithm of the ratio between two values: RelationdB = 20.log10 Value1 Value2 æ è ç ö ø ÷ • For intensity and power, a factor of 10 instead of 20 is used (this is also true for absolute dB scales)
- 11. Alexis Baskind How to define the dB ? • Absolute dB scales, generally speaking, are related to the logarithmic ratio of the measured value to a reference value : • If the value equals the reference value, it corresponds to 0 dB • This is true for all kinds of decibels (see next page). The only difference between different dB scales (dB SPL, dBu, dBV, dBFS…) is the unit (pressure, voltage, etc…) and the reference value dB or not dB ValuedBX = 20.log10 ValueLinear referenceValue æ è ç ö ø ÷
- 12. Alexis Baskind Different dB scales dB or not dB Scale Physical Quantity Unit Reference Value (corresponding to 0 dB) dB SPL Sound pressure Pascal 0,00002 Pa dBu Voltage Volt 0,775 V dBV Voltage Volt 1V dBFS Digital amplitude none 1.0 (floating point) or 2Nbits (integers)
- 13. Alexis BaskinddB or not dB Appendix – The dB – useful values (not valid for power and intensity) • Multiplying a signal (amplitude, RMS,…) by 2 is equivalent to adding 6 dB • Dividing a signal by 2 is equivalent to subtracting 6 dB • Multiplying a signal by 10 is equivalent to adding 20 dB • Dividing a signal by 10 is equivalent to subtracting 20 dB • Multiplying a signal by is equivalent to adding 3 dB • Dividing a signal by is equivalent to subtracting 3 dB 2 2