A Brief Evolution of the dB (decibel)
dB (deciBel): The logarithmic Unit of Level
dB Notation References
dB Examples
The dB decibel is ubiquitous in the field of audio technology, recording, broadcast, etc. However, a need for a better notation has arisen to avoid confusion on the dB usage for certain measurements. This article explains this new and preferred dB notation with selected examples. In the early 1920’s telephone engineers needed a way to measure the loss in a standard one-mile long cable (1msc). This would allow the engineers to see if the telephone signal (voice) would need an amplifier or not to regenerate the signal strength. The development of the TU (Transmission Unit) in 1924 was created and used. The TU was defined by the following mathematical formula: $$TU = \log_{10}\left( P_o \over P_i\right) $$ Where \(P_o\) is the power output in watts and \(P_i\) is the power input in watts. A pair of telephones with a one standard mile long cable (1msc) to measure the loss. This was done bi-directionally. This TU worked out very well for several years. However as the telephone industry grew exponentially, so did the use of the TU for measurements. The TU was then decided to become the de facto standard unit at this time and then was renamed the Bel commemorating Alexander Graham Bell the inventor of the telephone. See formula: $$Bel = \log_{10}\left( P_o \over P_i\right) $$ The Bel was considered too large of a unit to use with upcoming telephone circuits and systems. In 1928 the Bel unit was replaced with the smaller unit called the decibel dB, meaning one tenth (1/10) of a Bel. This was now the birth of the dB (decibel) which is still being used in many technical areas of sound, light, electricity, fiber optics, audio/video and other technologies. In the Journal of the Audio Engineering Society, Dr. R.W. Young explained the Unit of Level in 1971. The dB unit had acquired some various appendages or suffixes. The idea of these appendages was to develop reliable measurements that would compare the measured quantity to a known reference quantity. In this way the readings of the levels would be more accurate and reliable with less confusion. Using this in the field of audio technology the following table will show the concept of Levels in dB (decibels). Based on the following definition the capital letter L stands for level in dB. By adding a subscript to the letter L will represent the type of level. The below levels are considered to be electrical levels. <tbody> <tr> <td width="20%">L</td> <td>Level in dB.</td> </tr> <tr> <td>Lp</td> <td>Power level in dBm; where dBm is referenced to 1mW of power.</td> </tr> <tr> <td>Lv</td> <td>Voltage level in dBu or dBV; where dBu is referenced to .775VRMS and dBV is referenced to 1VRMS.</td> </tr> <tr> <td> </td> <td> </td> </tr> </tbody> The below Lsp is considered to be an acoustic level. <tbody> <tr> <td width="20%">Lsp</td> <td>Sound pressure level in \(dB_{Pa}\) or \(dB_{µBar}\); where \(dB_{Pa}\) is referenced to a sound pressure of .00002Pa and \(dB_{µbar}\) is referenced to a sound pressure of .0002µBar.</td> </tr> </tbody> We can use the following general formulas for any particular technical field. The unit shall be in \(dB_x\), where x is the appendage letter of the reference value chosen. <tbody> <tr> <td width="20%">\(L_x = 10 * log_{10}R_P\)</td> <td>\(dB_x\)</td> </tr> <tr> <td width="20%">\(L_x = 20 * log_{10}R_V\)</td> <td>\(dB_x\)</td> </tr> <tr> <td width="20%">\(L_x = 20 * log_{10}R_{SP}\)</td> <td>\(dB_x\)</td> </tr> </tbody> On the other hand, level gains typically used for amplifiers/pre-amplifiers are in absolute dB (no appendage letter) which designates no reference. \(R_P\) = power ratio in watts, \(R_P = P_O / P_R\), where \(P_O\) is the power under test and \(P_R\) is the power reference. \(R_V\) = voltage ratio in volts, \(R_V = V_O/V_R\), where \(V_O\) is the voltage under test and \(V_R\) is the voltage reference. \(R_{SP}\) = sound pressure ratio in Pascals(Pa) or micro Bars(µBar), \(R_{SP} = sp/sp_r\), where \(sp\) is the sound pressure under test and \(sp_r\) is the sound pressure reference. Here is a list of adapted audio formulas using the above notations: Power 1. Power Level: \(L_P = 10 * log \left(P\over 1mW\right)\) dBm 2. Power in watts: \(P = .001(10)^{L_P\over 10}\) W 3. Power Gain Level: \(L_{PG} = 10 * logG_P\) dB 4. Power Gain Level: \(L_{PG} = L_{P_o} - L_{P_i}\) dB 5. Power Gain: \(G_P = P_O / P_I\) 6. Power Gain: \(G_P = (10)^{L_{PG}\over 10}\) Voltage 7. Voltage Level: \(L_V = 20 * log\left(V\over.775V\right)\) dBu 8. Voltage Level: \(L_V = 20 * log\left(V\over 1V\right)\) dBV 9. Voltage in Volts: \(V = V_R(10)^{L_V\over 20}\) V \(V_R\) is the voltage reference equal to .775V or 1V 10. Voltage Gain Level: \(L_{VG} = 20 * logG_V\) dB 11. Voltage Gain Level: \(L_{VG} = L_{V_o} - L_{V_i}\) dB 12. Voltage Gain: \(G_V = V_O/V_I\) 13. Voltage Gain: \(G_V = (10)^{L_{VG}\over20}\) Sound Pressure (Acoustic) 14. Sound Pressure Level: \(L_{SP} = 20 * log\left(sp\over sp_r\right)\) dB 15. Sound Pressure: \(sp = sp_r(10)^{L_{sp}\over 20}\) Pa or µBar Sound Pressure References: spr = .00002 Pa ; spr = .0002 µBar Voltage Gain Level with Impedance dependance 16. \(L_{VG} = 20 * log\left(V_o\over V_i\right) + 10 * log\left(Z_i\over Z_o\right)\) dB where \(V_i\) is the input voltage \(V_o\) is the output voltage \(Z_i\) is the input impedence \(Z_o\) is the output impedence Power Gain Level with Impedance dependance 17. Sound Pressure Level: \(L_{PG} = 10 * log\left(P_o\over P_i\right) + 10 * log\left(Z_i\over Z_o\right)\) dB where \(P_i\) is the input power \(P_o\) is the output power \(Z_i\) is the input impedence \(Z_o\) is the output impedence Note: voltage is in volts (V), power is in watts (W), impedance is in ohms (Ω). Let me now show you a few examples on how we use these preferred formulas: 1. A voltage of 1.23 V is given at the input of the console. What is the voltage level \(L_V\) in dBu? Using equation number 7: \(L_V = 20 * log\left(V\over.775V\right)\) dBu; you get \(L_V = 20 * log\left(1.23\over.775V\right) = 4.012 dBu\) 2. Given a headphone with a maximum power of 100mW, what is the power level \(L_P\) in dBm? Using equation number 1: \(L_P = 10 * log \left(P\over 1mW\right)\) dBm; you get \(L_P = 10 * log \left(100mW\over 1mW\right)\) = 20 dBm 3. Given a condenser microphone with a voltage level \(L_V\) = -34dBV, what is the voltage V? Using equation number 9: \(V = V_R(10)^{L_V\over 20}\) V; you get \(V = 1(10)^{-34\over 20}\) 19.953mV 4. Given a sound pressure \(sp = 4Pa\), what is the sound pressure level \(L_{SP}\) in \(dB_{PA}\)? Using equation number 14: \(L_{SP} = 20 * log\left(sp\over sp_r\right)\) \(dB_{PA}\); you get \(L_{SP} = 20 * log\left(4\over .00002\right)\) = 106.021 \(dB_{PA}\) This article was written in dedication to the late Albert B. Grundy, the founder of the Institute of Audio Research in New York City and a past president of the AES Audio Engineering Society. I had the pleasure to work for Al for many years in which he became a good friend and mentor on anything related to audio. The information in this article is now being used by my students which allows for less confusion on understanding the use of the dB. By Ronald G. Ajemian, Adjunct, Institute of Audio Research, New York, NY 10003 AES Audio Engineering Society — Member References: 1. R.V.L. Hartley, The Transmission Unit, Electrical Communications, New York, NY, pp. 34-42, (July 1924). 2. W.H. Martin: Decibel — The Name for the Transmission Unit, Bell System Technical Journal, New York, NY, pp. 1-2 (January 1929). 3. R.W. Young: Decibel, A Unit of Level, JAES Vol. 19 No. 6, 1971, pp. 512-516 (1971). 4. AES project report: For articles on professional audio and for equipment specifications and Notations for expressing levels refer to: # AES-R2-2004 Standards preprint, also IEC 60027-3 and IEC 60268-2 documents. 5. Stop Using the Ambiguous dBm! by Herman A.O. Wilms, Paper Number: M01, AES Convention: 2ce (March 1972). 6. dB Formulas for Pro-Audio, by R.G. Ajemian, (March 2005).