### dB vs. dBW vs. dBm Tutorial

 Alexander Graham Bell, after which the bel, and therefore the decibel, is named.

## dB

The decibel (dB) is a dimensionless unit used for quantifying the power ratio between two values. The unit is useful for describing the gain or loss in a system. It is used when characterizing amplifiers, attenuators, mixers, or RF chains as a whole. It is based on the bel and is defined as follows:

A power ratio is defined as:

power ratio = ouput_power / input_power

A bel is defined as:

power in bels = log10(output_power / input_power)

A decibel is 1/10 as large as a bel so ten decibels make up one bel:

1 bel = 10 decibels

And therefore power in decibels is defined as:

power in decibels = 10 * log10(output_power / input_power)

The table below shows the loss/gain as a power ratio compared to the loss/gain in decibels for several different values:

The table below shows a amplitude ratio, power ratio (which is just the amplitude ratio squared), and associated dB value. Note that the dB range for corresponding power ratios is much smaller, i.e, -100 dB to +100 dB corresponds to 0.0000000001 to 10000000000 power range. That makes the use of dBs desirable when dealing with large power ranges. An example would be a satellite communications system where transmit antenna signal power is +14 dBW and receiver antenna signal power is -160 dBW.

As you can be seen from the table above, 1 dB is approximately 26% power gain, 3 dB is approximately 2× power gain, and 10 dB is 10× power gain. This is useful to know for "back of the envelope" power calculations.

## dBm

dBm, also written as dBmW, is an abbreviation for the power ratio in decibels (dB) of the measured power referenced to one milliwatt (mW):

power in dBm = 10 * log10((output power in watts)/1mW) = 10 * log10((output power in watts) / 0.001)

For 1mW power, you get 0 dBm.

For 2mW power, you get about 3 dBm.

For 1W power, you get 30 dBm.

So doubling the power is equivalent to about a 3 dBm increase.

## dBm to dBW Conversion

We know the following:

power in dBW = 10 * log10(output power in watts/ 1W) = 10 * log10(output power in watts)

and

power in dBm = 10 * log10(output power in watts/ 1mw) =

10 * log10(output power in watts / 0.001) =
10 * log10(output power in watts * 1/1000)

So we need to divide the dBW power ratio by 1000 to equate it to the dBm power ratio. But we know that 10 * log10(1/1000) is -30 dB.

And since we know:

log10(X * Y * Z) = log10(X) + log10(Y) + log10(Z)

we can just subtract 30 dB from the dBm value instead of dividing the power ratio by 1000 to arrive at the dBW value.

So for example, 50 dBm - 30 dB = 20 dBW.

Both dBm and dBW are used for measuring absolute power.

Below is a table showing the relationships between dBm, W, mW, and dBW.

## References

https://en.wikipedia.org/wiki/DBm
https://en.wikipedia.org/wiki/Decibel_watt
https://en.wikipedia.org/wiki/Decibel