Using the Decibel - Part 3: Combining decibels and using log charts

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[Part 1 introduces the decibel and examines concepts underlying its use in sound systems. Part 2 looks at expressing power as an audio level, the decibel in acoustics, the inverse square law, directivity factor and Ohm's law.]

2.15 Combining Decibels

Adding Decibel Levels
The sum of two or more levels expressed in dB maybe found as follows:

L_T = 10log(10^L₁/10 + 10^L₂/10 + ... + 10^L_N/10.        (2-32)

If, for example, we have a noisy piece of machinery with an L_P = 90 dB, and wish to turn on a second machine with an L_P = 90 dB, we need to know the combined L_P. Since both measured levels are the result of the power being applied to the machine, with some percentage being converted into acoustic power, we can determine L_T by using Eq. 2-32. Therefore:

L_T = 10log(10^90/10 + 10^90/10)

      = 10log(10⁹ + 10⁹)
      = 10log(2 x 10⁹)
      = 93 dB.

Doubling the acoustic power results in a 3 dB increase.

An alternative dB addition technique is given through the courtesy of Gary Berner.

L_T = 10log(10^{-(diff in dB)/10} + 1) + smallest number        (2-33)

Example
If we wish to add 90 dB to 96 dB, using Eq. 2-33, take the difference in dB (6 dB) and put it in the equation:

L_T = 10log(10^6/10 +1) + 90
      = 96.97 dB.

Input signals to a mixing network also combine in this same manner, but the insertion loss of the network must be subtracted. Two exactly phase-coherent sinewave signals of equal amplitude will combine to give a level 6 dB higher than either sine-wave.

The general case equation for adding either sound pressure, voltages, or currents is:

Combined L_P =

20log√((10^E₁/20)² + (10^E₂/20)² + 2(10^E₁/20)(10^E₂/20)(cos[a₁ - a₂]))        (2-34)

Table 2-9 shows the effects of adding two equal amplitude signals with different phase together using Eq. 2-34.

Table 2-9. Combining Pure Tones of the Same Frequency but Differing Phase Angles

Subtracting Decibels
The difference of two levels expressed in dB may be found as follows:

L_diff = 10log(10^{Total Level/10 - Level with one source off/10}).        (2-35)

Page 2: Combining Levels of Uncorrelated Noise Signals
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