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1022 ⏐⏐⏐ DECIBELS, FILTERS, AND BODE PLOTS
(bels) (23.8)
However, to provide a unit of measure of less magnitude, a decibel
is defined, where
(23.9)
The result is the following important equation, which compares
power levels P 2 and P 1 in decibels:
P2
dB � 10 log10 �� P
(decibels, dB) (23.10)
If the power levels are equal (P 2 � P 1), there is no change in power
level, and dB � 0. If there is an increase in power level (P 2 > P 1), the
resulting decibel level is positive. If there is a decrease in power level
(P 2 < P 1), the resulting decibel level will be negative.
For the special case of P 2 � 2P 1, the gain in decibels is
dB � 10 log 10
P2
B � log10 �� P
1 bel � 10 decibels (dB)
P 2
� P1
� 10 log 10 2 � 3 dB
Therefore, for a speaker system, a 3-dB increase in output would require
that the power level be doubled. In the audio industry, it is a
generally accepted rule that an increase in sound level is accomplished
with 3-dB increments in the output level. In other words, a
1-dB increase is barely detectable, and a 2-dB increase just discernible.
A 3-dB increase normally results in a readily detectable
increase in sound level. An additional increase in the sound level is
normally accomplished by simply increasing the output level another
3 dB. If an 8-W system were in use, a 3-dB increase would require
a 16-W output, whereas an additional increase of 3 dB (a total of
6 dB) would require a 32-W system, as demonstrated by the calculations
below:
P2 16
dB � 10 log10 � � 10 log10 � � 10 log10 2 � 3 dB
P1
8
P2 32
dB � 10 log10 � � 10 log10 � � 10 log10 4 � 6 dB
P1
8
For P2 � 10P1, P2 dB � 10 log10 � � 10 log10 10 � 10(1) � 10 dB
P1
resulting in the unique situation where the power gain has the same
magnitude as the decibel level.
For some applications, a reference level is established to permit a
comparison of decibel levels from one situation to another. For communication
systems a commonly applied reference level is
P ref � 1 mW (across a 600-� load)
Equation (23.10) is then typically written as
1
1
dB