Calculating with dBs

By far the most important technique when calculating power is calculating with decibels (dB). There is no new physics hidden in this -it is just a convenient method which makes calculations a lot simpler.

The decibel is a dimensionless unit1, that is, it defines a relationship between two measurements of power. It is defined by:

dB = 10 * Log (P1/P0)

where P1 and P0 can be whatever two values you want to compare. Typically, in our case, this will be some amount of power.

Why are decibels so handy to use? Many phenomena in nature happen to behave in a way we call exponential. For example, the human ear senses a sound to be twice as loud as another one if it has ten times the physical signal.

Another example, quite close to our field of interest, is absorption. Suppose a wall is in the path of our wireless link, and each meter of wall takes away half of the available signal. The result would be:

0 meters = 1 (full signal)

1 meter = 1/2

2 meters = 1/4

3 meters = 1/8

4 meters = 1/16

n meters = 1/2n = 2-n

This is exponential behaviour.

But once we have used the trick of applying the logarithm (log), things become a lot easier: instead of taking a value to the n-th power, we just multiply by n. Instead of multiplying values, we just add.

Here are some commonly used values that are important to remember:

+3 dB = double power

-3 dB = half the power

+10 dB = order of magnitude (10 times power)

-10 dB = one tenth power

In addition to dimensionless dBs, there are a number of relative definitions that are based on a certain base value P0. The most relevant ones for us are:

dBm relative to P0 = 1 mW

dBi relative to an ideal isotropic antenna

An isotropic antenna is a hypothetical antenna that evenly distributes power in all directions. It is approximated by a dipole, but a perfect isotropic antenna cannot be built in reality. The isotropic model is useful for describing the relative power gain of a real world antenna.

Another common (although less convenient) convention for expressing power is in milliwatts. Here are equivalent power levels expressed in milliwatts and dBm:

1 mW = 0 dBm

2 mW = 3 dBm

100 mW = 20 dB

m1 W = 30 dBm

Another example of a dimensionless unit is the percent (%) which can also be used in all kinds of quantities or numbers. While measurements like feet and grams are fixed, dimensionless units represent a relationship.