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## Tricks for dB scale conversion

The use of the decibel scale is ubiquitous in electronic systems. For example, the dynamic range through a system is measured by the signal-to-noise ratio in decibels, or dB. When working in the lab, or viewing the output of any measurement device, it is useful know how to make quick mental conversions between the linear and dB scales.

The amplitude or power of a signal is typically quantified relative to a fixed reference. A full-scale signal has a ratio of 1:1, and is expressed as 20.log(1/1) = 0 dB. Remember that multiplication (division) in the linear domain is equivalent to addition (subtraction) in the log domain. One needs to only remember a few values in order to compute most conversions:

1. A ten-fold (10x) increase or decrease in linear signal amplitude results in a +20 dB or -20 dB change on the decibel scale, respectively.
2. A doubling or halving (2x or ½x) of linear signal amplitude results in [approximately] a +6 dB or -6 dB change on the decibel scale, respectively.
3. A tripling (3x) of linear signal amplitude can be approximated by using 3 ≈ √10. The square-root is equivalent to a power of half, and in the log domain, this simply halves the dB value. This results in ½ x 20 dB = 10 dB.

Using these basic rules, it is easy to quickly compute the linear ratios corresponding the dB value. Note that in the list above, the 3x ↔ 10 dB conversion is the greatest source of error in the final approximation.

Some examples:

• Convert 54 dB to the linear scale: note that 54 dB = (60 – 6) dB, which is equivalent to 1000 x ½ = 500 in the linear domain (this is a good approximation to the actual value of 501.2)
• Convert 7x to the dB scale: note that 7 = √49 ≈ √50 = √(100 x ½), which is equivalent to ½ x (40 – 6) = 17 dB (the answer should be 16.9)
• Convert 30 dB to the linear scale: note that 30 dB = (20 + 10) dB, which is equivalent to (10 x 3) = 30 in the linear domain (the answer should be 31.6). Alternatively, we can use 30 dB = (40 – 10) dB, which converts to (100 ÷ 3) = 33.3 (the magnitude of the approximation error is about the same as that of the first answer)

In the computerized world of today (or when no one brings a calculator to the lab), these mental shortcuts can be very useful as a quick sanity check.