Bel: Difference between revisions

Revision as of 17:31, 10 October 2006

dB

The decibel (abbreviated to dB) is a logarithmical unit used to express relative difference in intensity or power between two signals, usually acoustic.

It is equal to ten times the common logarithm of the ratio of the two levels.

  ${diff}=10\cdot log_{10}\left({\frac {P_{2}}{P_{1}}}\right)$

The 10 factor comes from the fact the Bel unit is too large for everyday use; decibel is just more convenient.

There is a second definition of the decibel. In acoustics, it is used to express the ratio in sound pressure, because power is roughly proportional to the square of effective sound pressure.

Let 'p' be pressure, 'P' power.

  ${diff}=10\cdot log_{10}\left({\frac {P_{2}}{P_{1}}}\right)$ 
 
  ${diff}=10\cdot log_{10}\left({\frac {p_{2}^{2}}{p_{1}^{2}}}\right)$ 
 
  ${diff}=20\cdot log_{10}\left({\frac {p_{2}}{p_{1}}}\right)$  because  $log_{10}(a^{2})=2\cdot log_{10}(a)$

In this scale, it means an increase of 20 dB is equal to a ten-fold ratio, or 6 dB a two-fold ratio ( $20\cdot log_{10}(2)$ is approximately equal to 6).

@@ Line 4: / Line 4: @@
 It is equal to ten times the common logarithm of the ratio of the two levels.
- diff = 10 × log<sub>10</sub>(P<sub>2</sub> / P<sub>1</sub>)
+  <math>{diff} = 10 \cdot log_{10}\left(\frac{P_2}{P_1}\right )</math>
 The ''10'' factor comes from the fact the Bel unit is too large for everyday use; decibel is just more convenient.
@@ Line 11: / Line 12: @@
 Let 'p' be pressure, 'P' power.
- diff = 10 × log<sub>10</sub>(P<sub>2</sub> / P<sub>1</sub>)
+  <math>{diff} = 10 \cdot log_{10}\left(\frac{P_2}{P_1}\right )</math>
- diff = 10 × log<sub>10</sub>(p<sub>2</sub><sup>2</sup> / p<sub>1</sub><sup>2</sup>)
+  <math>{diff} = 10 \cdot log_{10}\left(\frac{p_2^2}{p_1^2}\right )</math>
- diff = 20 × log<sub>10</sub>(p<sub>2</sub> / p<sub>1</sub>)  because log<sub>10</sub>(a<sup>2</sup>) = 2 × log<sub>10</sub>(a)
+  <math>{diff} = 20 \cdot log_{10}\left(\frac{p_2}{p_1}\right )</math> because <math>log_{10}(a^2) = 2 \cdot log_{10}(a)</math>
-In this scale, it means an increase of 20 dB is equal to a ten-fold ratio, or 6 dB a two-fold ratio (<tt>20 × log<sub>10</sub>(2)</tt> ''is approximately equal to 6'').
+In this scale, it means an increase of 20 dB is equal to a ten-fold ratio, or 6 dB a two-fold ratio (<math>20 \cdot log_{10}(2)</math> ''is approximately equal to 6'').
 [[Category:Technical]]