File:Plot of decibel and inverse.pdf

(Click on the plot to enlarge) The plot represents the decibel scaling function dB and its inverse, defined by

${\displaystyle x\,\mathrm {dB} =10^{x/10}}$ for real-valued x;

${\displaystyle x\,[\mathrm {dB} ]=10\,\mathrm {lg} \,x}$ for strictly positive real-valued x. Note: ${\displaystyle [\mathrm {dB} ]}$ is read " x in dB or " x expressed in dB".

Examples

(1) Illustrating the functions: ${\displaystyle 3\,\mathrm {dB} =2}$ and ${\displaystyle 30\,\mathrm {dB} =1000}$; conversely: ${\displaystyle 2\,[\mathrm {dB} ]=3}$ and ${\displaystyle 1000\,[\mathrm {dB} ]=30}$.

(2) A representative application: let input power ${\displaystyle P}$, output power ${\displaystyle P'}$ and gain ${\displaystyle G}$ be related by ${\displaystyle G\,P=P'}$. Then one can write rigorously the following

(a) ${\displaystyle G\,[\mathrm {dB} ]+P\,[\mathrm {dBW} ]=P'\,[\mathrm {dBW} ]}$

(b) instantiating ${\displaystyle G\,P=P'}$ with ${\displaystyle G=g\,\mathrm {dB} }$ and ${\displaystyle P=p\,\mathrm {dB} }$ and ${\displaystyle P'=p'\,\mathrm {dB} }$ yields ${\displaystyle g+p=p'}$

(c) numerical example: ${\displaystyle 10\times 100\,\mathrm {W} =1000\,\mathrm {W} }$ is equivalent to ${\displaystyle 10\,\mathrm {dB} \times 20\,\mathrm {dBW} =30\,\mathrm {dBW} }$

(3) Conversion to scientific notation: ${\displaystyle x\,\mathrm {dB} =10^{x\,\mathrm {div} \,10}\times (x\,\mathrm {mod} \,10)\,\mathrm {dB} }$, for instance ${\displaystyle 23\,\mathrm {dB} =10^{2}\times 3\,\mathrm {dB} =200}$.

The scientific notation example shows that the graph for the interval from 0 to 10 suffices for calculating quick approximations for any argument value. Boute (talk) 05:25, 29 March 2012 (UTC)

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