Properties of DB Scales
In every kind of dB, a factor of 10 in amplitude increase corresponds to a 20 dB boost (increase by 20 dB):
![$\displaystyle 20\log_{10}\left(\frac{10 \cdot A}{A_{\mbox{\small ref}}}\right)
...
...)}_{\mbox{$20$\ dB}} + 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)
$](http://www.dsprelated.com/josimages_new/mdft/img1936.png)
![$ 20\log_{10}(10) = 20$](http://www.dsprelated.com/josimages_new/mdft/img1937.png)
![$ f(x)$](http://www.dsprelated.com/josimages_new/mdft/img268.png)
![$ 1/x$](http://www.dsprelated.com/josimages_new/mdft/img308.png)
![$ 20$](http://www.dsprelated.com/josimages_new/mdft/img273.png)
![$ 10$](http://www.dsprelated.com/josimages_new/mdft/img1908.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ 20$](http://www.dsprelated.com/josimages_new/mdft/img273.png)
Similarly, a factor of 2 in amplitude gain corresponds to a 6 dB boost:
![$\displaystyle 20\log_{10}\left(\frac{2 \cdot A}{A_{\mbox{\small ref}}}\right)
=...
...2)}_{\mbox{$6$\ dB}}
+ 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)
$](http://www.dsprelated.com/josimages_new/mdft/img1938.png)
![$\displaystyle 20\log_{10}(2) = 6.0205999\ldots \approx 6 \;$](http://www.dsprelated.com/josimages_new/mdft/img1939.png)
![$\displaystyle . \protect$](http://www.dsprelated.com/josimages_new/mdft/img860.png)
![$ f(x)$](http://www.dsprelated.com/josimages_new/mdft/img268.png)
![$ 1/x$](http://www.dsprelated.com/josimages_new/mdft/img308.png)
![$ 6$](http://www.dsprelated.com/josimages_new/mdft/img1940.png)
![$ 2$](http://www.dsprelated.com/josimages_new/mdft/img109.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ 6$](http://www.dsprelated.com/josimages_new/mdft/img1940.png)
A doubling of power corresponds to a 3 dB boost:
![$\displaystyle 10\log_{10}\left(\frac{2 \cdot A^2}{A^2_{\mbox{\small ref}}}\righ...
...{\mbox{$3$\ dB}}
+ 10\log_{10}\left(\frac{A^2}{A^2_{\mbox{\small ref}}}\right)
$](http://www.dsprelated.com/josimages_new/mdft/img1941.png)
![$\displaystyle 10\log_{10}(2) = 3.010\ldots \approx 3\;$](http://www.dsprelated.com/josimages_new/mdft/img1942.png)
![$\displaystyle . \protect$](http://www.dsprelated.com/josimages_new/mdft/img860.png)
Finally, note that the choice of reference merely determines a vertical offset in the dB scale:
![$\displaystyle 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)
= 20\log_...
...(A) - \underbrace{20\log_{10}(A_{\mbox{\small ref}})}_{\mbox{constant offset}}
$](http://www.dsprelated.com/josimages_new/mdft/img1943.png)
Next Section:
Specific DB Scales
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Logarithms of Negative and Imaginary Numbers