Decibel (dB, dBm)

The decibel is a logarithmic unit for expressing the ratio between two power levels:

$$\text{Ratio [dB]} \;=\; 10 \log_{10}\!\left( \frac{P_1}{P_2} \right).$$

For voltage or field amplitude (where power scales as the square of amplitude):

$$\text{Ratio [dB]} \;=\; 20 \log_{10}\!\left( \frac{V_1}{V_2} \right).$$

The factor of 10 vs 20 is the most common source of dB-related errors. Optical power meters report dB (factor of 10); RF voltage measurements report dB (factor of 20).

dBm is power referenced to 1 milliwatt:

$$P_\text{dBm} \;=\; 10 \log_{10}\!\left( \frac{P}{1 \text{ mW}} \right).$$

The "m" denotes milliwatt reference. Conversion table:

Power	dBm
1 mW	0 dBm
10 mW	+10 dBm
100 mW	+20 dBm
1 W	+30 dBm
100 μW	$-10$ dBm
1 μW	$-30$ dBm
1 nW	$-60$ dBm
1 pW	$-90$ dBm
1 fW	$-120$ dBm

Arithmetic. dB makes multiplication into addition and division into subtraction:

• Cascade gain: $G_\text{total} = G_1 + G_2 + G_3$ (in dB)
• Path loss: $\text{out [dBm]} = \text{in [dBm]} - \text{loss [dB]}$
• Doubling power: $+3.01$ dB
• Halving power: $-3.01$ dB
• 10× power: $+10$ dB
• 100× power: $+20$ dB

Common related units:

Unit	Reference
dBm	1 mW
dBW	1 W (so 0 dBW = +30 dBm)
dBμ	1 μW (so 0 dBμ = $-30$ dBm)
dBV	1 V (voltage)
dBi	Isotropic radiator (antenna gain)
dBc	Carrier reference (for sidebands)

Nepers and dB. Nepers use the natural logarithm:

$$\text{Loss [nepers]} \;=\; \frac{1}{2} \ln(P_1/P_2).$$

The conversion: 1 neper = 8.686 dB. Mathematical optical propagation in $e^{-\alpha L}$ form uses $\alpha$ in nepers/length (or 1/length); engineering loss in dB/length is 4.343 times higher in numerical value.

The dB is unitless (a ratio); dBm has units (power referenced to 1 mW). Mixing them ("the loss is 3 dBm") is a common error — the loss is 3 dB, while the power level might be $-3$ dBm.