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):

$$20\log_{10}\left(\frac{10 \cdot A}{A_{\mbox{\small ref}}}\right) = \underbrace{20\log_{10}(10)}_{\mbox{$20$\ dB}} + 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)$$

and $20\log_{10}(10) = 20$ , of course. A function $f(x)$ which is proportional to $1/x$ is said to ``fall off'' (or ``roll off'') at the rate of $20$ dB per decade. That is, for every factor of $10$ in $x$ (every ``decade''), the amplitude drops $20$ dB.

Similarly, a factor of 2 in amplitude gain corresponds to a 6 dB boost:

$$20\log_{10}\left(\frac{2 \cdot A}{A_{\mbox{\small ref}}}\right) = \underbrace{20\log_{10}(2)}_{\mbox{$6$\ dB}} + 20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right)$$

and

$$20\log_{10}(2) = 6.0205999\ldots \approx 6 \;$$   dB$$. \protect$$

A function $f(x)$ which is proportional to $1/x$ is said to fall off $6$ dB per octave. That is, for every factor of $2$ in $x$ (every ``octave''), the amplitude drops close to $6$ dB. Thus, 6 dB per octave is the same thing as 20 dB per decade.

A doubling of power corresponds to a 3 dB boost:

$$10\log_{10}\left(\frac{2 \cdot A^2}{A^2_{\mbox{\small ref}}}\right) = \underbrace{10\log_{10}(2)}_{\mbox{$3$\ dB}} + 10\log_{10}\left(\frac{A^2}{A^2_{\mbox{\small ref}}}\right)$$

and

$$10\log_{10}(2) = 3.010\ldots \approx 3\;$$dB$$. \protect$$

Finally, note that the choice of reference merely determines a vertical offset in the dB scale:

$$20\log_{10}\left(\frac{A}{A_{\mbox{\small ref}}}\right) = 20\log_{10}(A) - \underbrace{20\log_{10}(A_{\mbox{\small ref}})}_{\mbox{constant offset}}$$