The bel (B) is a unit of measurement for the ratio of a physical quantity
(power, intensity, magnitude, etc.) and a specified or implied reference
level in base-10 logarithm. As it is a ratio of two quantities with the same
unit, it is dimensionless.
For example, consider a power amplifier with input signal power
and output signal power
, then the power gain of the amplifier is
, which can be more concisely expressed in base-10 log scale:
 |
(484) |
The unit bel (B) was first used in early 1920's in honor of
Alexander Bell (1847 – 1922),
a telecommunication pioneer and founder of the Bell System (the Bell Labs).
As bel (B) is often too big a unit (a gain of 100 is only 2 B), a smaller
unit of decibel (dB), 1/10 of the unit bel (B), is more widely used instead.
Now the power gain above can be expressed as:
or |
(485) |
Similarly a 1,000 fold power gain is expressed as:
or |
(486) |
Given the input power
and the power gain in decibel, e.g.,
, the output power can be obtained as:
i.e. |
(487) |
As another example, the sound level is measured in decibel, in terms of the ratio
of the sound intensity (power per area, e.g.,
) and the threshold of human
hearing (
) as the reference. The human hearing has a large range
from 0 dB (threshold) to 140 dB (military jet takeoff,
times the threshold,
i.e.,
). 160 dB sound level will cause instant membrane/eardrum
perforation.
In general, power (and energy) is always proportional to the amplitude of certain
quantity squared (e.g.,
,
,
). Therefore a
different definition is used for ratios between two amplitudes, for example, the
output and input voltages
and
of a voltage amplifier, we have:
 |
(488) |
If the input to a voltage amplifier is 10 mV and the output voltage is 1 V, then the
voltage gain in terms of decibel is:
 |
(489) |
If the output voltage is 10 V, then
 |
(490) |
We see that the difference of one order of magnitude in the gain corresponds to 20 dB.
Given the input voltage
and the voltage gain in decibel, e.g.,
,
the output voltage can be obtained as:
i.e., |
(491) |
A related issue is the half power point. Recall that for a second order system, when
is small (e.g.,
), the magnitude
of the frequency
response function has a peak at
. The bandwidth of
the peak is defined as the difference between two cut-off frequencies
and
(
) at which
i.e., |
(492) |
The ratio between the half-power point and the peak in decibel is
 |
(493) |