## The Basics

A decibel (dB) is a logarithmic comparison of two power measurements.

Let P1 and P2 be two measurements of power.

A Bel (as in Alexander Graham Bell) is defined as

`   B = log10 (P1 / P2)`

A Bel is a huge difference, so customarily a tenth of a bel, or deciBel is used.

`  dB = 10 * log10 (P1 / P2)`

(Decibels have never made any sense to me -- John Gilmore -- but Bels make
perfect sense. So just divide by ten and think in Bels, which are just
logarithms base ten. Thus 20dB is 2 Bels which is a factor of 100. Why the radio
world complicated the simplicity of logs by multiplying them all by ten baffles me -- it's like calculating in furlongs per fortnight.)

## dB's vis-a-vis Volts

In the electrical domain, it's sometimes more convenient to measure voltage than power.

Volts can be related to power like this: P = V*V / R, where R is a resistance.
That is, power goes as the square of voltage.

For two voltages V1 and V2, measured across two resistances (impedances) R1 and R2 a little algebra gives you:

```         P1 = V1^2 / R1

P2 = V2^2 / R2

dB = 10 * log10 ((V1^2 / R1) / (V2^2 / R2))

If R1 == R2, this simplifies to

dB = 10 * log10 ((V1/V2)^2)

db = 10 * 2 * log10 (V1/V2)

db = 20 * log10 (V1/V2)
```

## dBm's

If you thought that was simple, they've thought of more stuff to complicate
your life. A dBm is a dB value calculated against a 1-milliwatt value.
Yep, not one watt, but a thousandth of a watt. Again the bogons have inserted
random factors into what ought to be simple.

Thus a 5dBm signal is a 0.5 Bel signal times 0.001 watt. 0.5 Bel is 10**0.5
or about 3.16. Thus 5 dBm is an inconvenient way to write 0.00316 watts. Didn't you always want to know
that?

## A few handy numbers

Power

Conversion from Power to dB's

```         3 dB is a factor of 2   [10 * log10 (2) = 3.01]

6 dB is a factor of 4

10 dB is a factor of 10

20 dB is a factor of 100

-3 dB is a factor of 1/2

-6 dB is a factor of 1/4

-10 dB is a factor of 1/10

-20 dB is a factor of 1/100
```

Voltage

Conversion from voltage to dB's

```

3 dB is a factor of 1.414

6 dB is a factor of 2

12 dB is a factor of 4

20 dB is a factor of 10

40 dB is a factor of 100

-3 dB is a factor of 0.707

-6 dB is a factor of 1/2

-12 dB is a factor of 1/4

-20 dB is a factor of 1/10

-40 dB is a factor of 1/100
```