# Decibel

## Logarithmic unit used to describe ratios of signal levels - like power or intensity - to a reference level.

The **decibel** is a logarithmic unit used to describe the ratio of a signal level - like power or intensity - to a reference level.

- the decibel express the level of a value relative to a reference value

### Decibel Definition

The decibel level of a signal can be expressed as

L_{ }= 10 log (S / S_{ref})(1)

where

L = signal level (decibel, dB)

S= signal - intensity or power level (signal unit)

S_{ref}= reference signal- intensity or power level(signal unit)

Decibel is a dimensionless value of relative ratios. The signal units depends on the nature of the signal - can be *W* for power.

A decibel is one-tenth of a *Bel *- named after Alexander Graham Bell, the inventor of the telephone.

**Note!** - the decibel value of a signal increases with *3 dB* if the signal is doubled *(L = 10 log (2) = 3)*.

If the decibel value and reference level are known the absolute signal level can be calculated by transforming *(1)* to

S = S_{ref}10^{(L / 10)}(2)

### Decibel Calculator

*S - signal - intensity or power level (signal unit)*

S_{ref} - **reference** signal *- intensity or power level* (signal unit)

### Example - Lowest Hearable Sound Power

*10 ^{-12} W* is normally the lowest sound power possible to hear and this value is normally used as the reference power in sound power calculations.

The sound power in decibel from a source with the lowest sound hearable can be calculated as

*L = 10 log ((10^{-12} W) / (10^{-12} W))*

* = 0 dB*

### Example - Highest Hearable Sound Power

*100 W* is almost the highest sound power possible to hear. The sound power in decibel from a source with the highest possible to hear sound power can be calculated as

*L = 10 log ((100 W) / (10^{-12} W))*

* = 140 dB*

### Example - Sound Power from a Fan

A fan creates *1 W* of sound power. The noise level from the fan in decibel can be calculated as

*L = 10 log ((1 W) / (10^{-12} W))*

* = 120 dB*

### Example - Sound Intensity and Decibel

The difference in decibel between sound intensity *10 ^{-8} W/m^{2} *and sound intensity

*10*(

^{-4}W/m^{2}*10000 units*) can be calculated as

ΔL= 10 log ((10^{-4}W/m^{2}) / (10^{-12}W/m^{2})) - 10 log ((10^{-8}W/m^{2}) / (10^{-12}W/m^{2}))

= 40 dB

Increasing the sound intensity by a factor of

*10 raises its level by 10 dB**100 raises its level by 20 dB**1000 raises its level by 30 dB**10000 raises its level by 40 dB and so on*