Copyright © 2014 by Wayne Stegall
Created November 21, 2014. See Document History at end for
details.
Decibels
The
calculation and use of decibels in audio
Introduction
Because human senses – including hearing – perceive signal levels
exponentially (or logarithmically depending on point of view), it was
deemed useful to represent the levels of these signals by a logarithm
of their magnitude. As a result the Bell(B) – named after
Alexander Graham Bell – was designated to be the unit representing the
base ten logarithm of signal power ratio.
(1)

power_{Bell} = log_{10}(powerratio)

These units are rather large for many uses so the smaller unit of
decibel (dB) became more commonly used. In this system, level
differences of 1dB are just noticeable, 3dB more so, and 10dB is a
significant change.
(2)

power_{dB} = 10 ×
log_{10}(powerratio)

Because power is proportional to the square of voltage (or current)
decibels can represent voltage or current ratios as well.
(3)

voltage_{dB} = 20
×
log_{10}(voltageratio)

Used alone dB can represent voltage or power ratios and which is
inferred must be determined by context. Additionally, dB also is
used to represent absolute values against some fixed reference.
Examples used in electronics and audio are
Figure 1: Absolute decibel units
Unit

Quantity represented

Reference value

Calculation





dBV

Voltage

1V

20
×
log_{10}(voltage) 
dBW

Power

1W

10 ×
log_{10}(power) 
dB(SPL)

Sound pressure level

20µPa

20 ×
log_{10}(soundpressure / 2×10^{–5}Pa ) 
dB(SWL)

Sound power or intensity

1pW (10^{–12}W)

10 ×
log_{10}(soundpower /
10^{–12}W) 
Reference levels for dB(SPL) and dB(SWL) were chosen so that they
produced the same decibel values for the same sound level. The
following table relates decibel sound levels to their measurements and
to some common sound experiences.
Figure 2: Sound pressure level
related data^{1}
Decibels (dB(SPL) or dB(SWL)) 
Sound pressure level (Pa or N/m^{2}) 
Intensity (W/m^{2})

Type of sound 




130 
63.2456

10 
artillery fire at close proximity
(threshold of pain) 
120 
20 
1 
amplified rock music; near jet
engine 
110 
6.32456

10^{–1} 
loud orchestral music, in audience 
100 
2 
10^{–2} 
electric saw 
90 
6.32456×10^{–1} 
10^{–3} 
bus or truck interior 
80 
2×10^{–1} 
10^{–4} 
automobile interior 
70 
6.32456×10^{–2} 
10^{–5} 
average street noise; loud
telephone bell 
60 
2×10^{–2} 
10^{–6} 
normal conversation; business
office 
50 
6.32456×10^{–3} 
10^{–7} 
restaurant; private office 
40 
2×10^{–3} 
10^{–8} 
quiet room in home 
30 
6.32456×10^{–4} 
10^{–9} 
quiet lecture hall; bedroom 
20 
2×10^{–4} 
10^{–10} 
radio, television, or recording
studio 
10 
6.32456×10^{–5} 
10^{–11} 
soundproof room 
0 
2×10^{–5} 
10^{–12} 
absolute silence (threshold of
hearing) 
Relating speaker sensitivity and power to resulting sound level
Calculations relating speaker sensitivity, power, and sound level
seem the most useful for audio hobbyists. Speaker sensitivity
often has to be recalculated because it is not power sensitivity as
thought. Because sensitivity specifications are given
specifically for 8Ω it instead represents voltage sensitivity referred
to the voltage necessary to drive 1W into 8Ω, that is 2.82843V.
Before power decibels can be added to speaker sensitivity to give sound
level, sensitivity has to be in power decibels.
To convert speaker sensitivity to power decibels.
(4)

sensitivity_{power} =
sensitivity_{voltage} + dB


speaker impedance
8Ω 

Then sound level can be calculated.
(5)

volume_{dB(SPL)} =
sensitivity_{power}
+ power_{dBW} 
Example:
A speaker of 90dB/1W sensitivity will produce a sound level of 117dBSPL
when driven with 17dB (50W) of power.
Head math
Scientific calculators are not always handy when you want to do decibel
calculations. However because decibels are logarithms, the laws
of logarithms simplify calculations.
Figure 3: Some basic laws of
logarithms in decibel form
dB(x × y)

=

dB(x) + dB(y)

dB(x / y)

=

dB(x) – dB(y)

dB(1 / x)

=

–dB(x) 
dB(x^{y})

=

dB(x) × y 
From a few known decibel relations, the remainder can be calculated in
your head or on paper without undue difficultly, so long as
calculations are short enough not to accumulate error approaching 1dB.
Figure 4: A table of decibels and now
they would have been derived from head math
Ratio


dBW

dBV

Calculating decibels from
estimated ratio (you can infer how to calculate in the other direction)






1


0

0

A given

1.25893


1

2

dB(1.25) = dB(10/2^{3}) =
dB(10)
–
(dB(2)
× 3) = 10dB – (3dB × 3) = 1dB 
1.58489


2

4

dB(3.16228 / 2) = dB(3.16228) –
dB(2) = 5dB – 3dB = 2dB 
1.99526


3

6

Round to 2. This is a given
you would expect to know and begin with.

2.51189


4

8

dB(2.5) = dB(10/(2
× 2)) = dB(10) – dB(2) – dB(2) = 10dB – 3dB – 3dB = 4dB 
3.16228


5

10

dB(

10

) = dB(10) / 2 = 10dB/2 =
5dB (It is helpful to remember this one too.)


3.98107


6

12

dB(4) = dB(2 × 2) = dB(2) +
dB(2) = 3dB + 3dB = 6dB 
5.01187


7

14

dB(5) = dB(10 / 2) = dB(10) –
dB(2) = 10dB – 3dB = 7dB 
6.30957


8

16

dB(3.16228 × 2) = dB(3.16228)
+ dB(2) = 5dB + 3dB = 8dB 
7.94328


9

18

dB(2^{3}) = dB(2)
× 3 = 3dB × 3 = 9dB 
10


10

20

A given

100


20

40

A given

1000


30

60

A given

Example Calculations
Example 1: Determine the sound pressure level to which a 30W amp
will drive a 6Ω speaker of 90dB sensitivity.
From
equation 4 calculate
speaker sensitivity.
(6)

sensitivity_{power} =
90dB
+ dB


6Ω
8Ω 

= 90dB + dB


3
4 

= 90dB + dB 

3
2 × 2


(7)

sensitivity_{power} ≈
90dB
+ dB(3.16228) – (dB(2) + dB(2)) = 90dB + 5dB – (3dB + 3dB) 
(8)

sensitivity_{power} ≈ 89dB/W (88.7506dB by calculator) 
Calculate power using head math and table of decibels in
figure 4.
(9)

power_{dBW} = dB(30W) =
dB(10W × 3) = dB(10) + dB(3.16228) = 10dB + 5dB 
Finally calculate sound pressure level using
equation 5.
(11)

volume_{dB(SPL)} = 89dB
+ 15dBW = 104dB(SPL) 
Example 2: Determine power required to drive a 4Ω speaker of 88dB
sensitivity to 110dB.
From
equation 4 calculate
speaker sensitivity.
(12)

sensitivity_{power} =
88dB
+ dB


4Ω
8Ω 

= 88dB + dB


1
2 

(13)

sensitivity_{power} =
88dB – dB(2) = 88dB – 3dB = 85dB/W

Calculate power required by reversing
equation
5.
(14)

power_{dBW} =
volume_{dB(SPL)} – sensitivity_{power} = 110dB – 85dB =
25dBW 
Convert power in watts to decibels.
(15)

power = dB^{1}(25dBW)
=
dB^{1}(20dBW + 5dBW) = dB^{1}(20dBW) × dB^{1}(5dBW)

(16)

power = (100 x
3.16)W
= 316W 
^{1}Table partially based on
Encyclopedia Britannica, "Table 1: Sound Levels for Nonlinear (Decibel)
and Linear (Intensity) Scales," 2002.
Document History
November 21, 2014 Created.