Decibel: Encyclopedia II - Decibel - Reckoning

Decibel - Reckoning

Decibels are handy for mental calculation, because adding them is easier than multiplying ratios. First, however, one has to be able to convert easily between ratios and decibels. The most obvious way is to memorize the logs of small primes, but there are a few other tricks that can help.

Decibel - Round numbers

The values of coins and banknotes are round numbers. The rules are:

1. One is a round number
2. Twice a round number is a round number: 2, 4, 8, 16, 32, 64
3. Ten times a round number is a round number: 10, 100
4. Half a round number is a round number: 50, 25, 12.5, 6.25
5. The tenth of a round number is a round number: 5, 2.5, 1.25, 1.6, 3.2, 6.4

Now 6.25 and 6.4 are approximately equal to 6.3, so we don't care. Thus the round numbers between 1 and 10 are these:

Ratio  1    1.25 1.6  2    2.5  3.2  4    5    6.3  8   10
dB     0    1    2    3    4    5    6    7    8    9   10

This useful approximate table of logarithms is easily reconstructed or memorized.

Decibel - The 4 → 6 energy rule

To one decimal place of precision, 4.x is 6.x in dB (energy).

Examples:

• 4.0 → 6.0 dB
• 4.3 → 6.3 dB
• 4.7 → 6.7 dB

Decibel - The 789 rule

To one decimal place of precision, x → (½ x + 5.0 dB) for 7.0 ≤ x ≤ 10.

Examples:

• 7.0 → ½ 7.0 + 5.0 dB = 3.5 + 5.0 dB = 8.5 dB
• 7.5 → ½ 7.5 + 5.0 dB = 3.75 + 5.0 dB = 8.75 dB
• 8.2 → ½ 8.2 + 5.0 dB = 4.1 + 5.0 dB = 9.1 dB
• 9.9 → ½ 9.9 + 5.0 dB = 4.95 + 5.0 dB = 9.95 dB
• 10.0 → ½ 10.0 + 5.0 dB = 5.0 + 5.0 dB = 10 dB

Decibel - −3 dB ≈ ½ power

A level difference of ±3 dB is roughly double/half power (equal to a ratio of 1.995). That is why it is commonly used as a marking on sound equipment and the like.

Another common sequence is 1, 2, 5, 10, 20, 50 ... . These preferred numbers are very close to being equally spaced in terms of their logarithms. The actual values would be 1, 2.15, 4.64, 10 ... .

The conversion for decibels is often simplified to: "+3 dB means two times the power and 1.414 times the voltage", and "+6 dB means four times the power and two times the voltage ".

While this is accurate for many situations, it is not exact. As stated above, decibels are defined so that +10 dB means "ten times the power". From this, we calculate that +3 dB actually multiplies the power by 10^3/10. This is a power ratio of 1.9953 or about 0.25% different from the "times 2" power ratio that is sometimes assumed. A level difference of +6 dB is 3.9811, about 0.5% different from 4.

To contrive a more serious example, consider converting a large decibel figure into its linear ratio, for example 120 dB. The power ratio is correctly calculated as a ratio of 10¹² or one trillion. But if we use the assumption that 3 dB means "times 2", we would calculate a power ratio of 2^120/3 = 2⁴⁰ = 1.0995 × 10¹², giving a 10% error.

Decibel - 6 dB per bit

In digital audio, each bit offered by the system doubles the (voltage) resolution, corresponding to a 6 dB ratio. For instance, a 16-bit (linear) audio format offers an approximate theoretical maximum of (16 × 6) = 96 dB, meaning that the maximum signal (see 0 dBFS, above) is 96 dB above the quantization noise.

Decibel - dB cheat sheet

As is clear from the above description, the dB level is a logarithmic way of expressing ratios. The following tables are cheat-sheets that provide values for various dB levels.

Adapted from the Wikipedia article "Reckoning", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki