
Decibels

The decibel, so loved by engineers around the world, is often misunderstood  not because of
the maths of decibels, more to do with the nomeclature. This page is my attempt to try and
clear up some of the confusion.


A Brief History
The Bel  a unit of the logarithm of the ratio between two power levels 
was named after Alexander Graham Bell, the inventor of the telephone. It
is used predominantly in the field of telecommunications, mainly because, I
think, the human senses are predominantly logarithmic in nature, and so the
phone people devised this logarithmic scale to simplify their work.
Generally, the Bel is too large a unit for practical use, so the deciBel
(one tenth of a Bel) is used almost exclusively.
How It Works
The deciBel (dB) is defined as the base10 logarithm of the ratio of two power levels:
N (dB) = 10 log_{10} ( P_{1} / P_{2} )
From the relation P = V^{2}/R, we can derive the formula for ratios of voltages:
N (dB) = 20 log_{10} ( V_{1} / V_{2} )
Two things to note here:
 The impedances of both top and bottom are assumed to be equal (R_{1} = R_{2}).
 The factor of 2 comes from the squaring of the volts.
A large table below gives a number of dB figures and the power and voltage ratios they represent.
Nomenclature  What Do The Letters Mean?
A figure quoted in basic "dB"s is fine if you talk about a ratio between
two values. In practice, though, we usually specifiy the
deciBels and some standard reference, indicated by a suffix appended to
"dB".
Suffix 
Description 
Reference Level 
Comments 
c  Signal relative to the carrier signal  
Mostly used in communications systems (signal generators, receivers, etc)
where signals (typically unwanted ones) are related to some carrier signal. 
FS  Full Scale  Fullscale input (or output) of A/D or D/A converter. 
Used in digital audio systems.
See below. 
m  Power level relative to 1 mW  1 mW 
Usually quoted with reference to a source or
load impedance. For example, 600 Ohms for audio,
50 or 75 Ohms for RF. 
u  Unterminated or Unloaded (Volts)  0.775V 
Derived from 1 mW across 600 Ohms. Used
where the circuit impedances are significantly
different from 600 Ohms, typical of audio
circuits (high input impedance, low output
impedance). 
V  Volts  1 V  An easy reference! 
W 
Power relative to 1W 
1 W 
Popular in audio amplifier design, because the log scale is more representative of
loudness differences between amplifiers than linear Watts. 
Some examples to illustrate the above:
 +5dBm  a signal 5dB greater than 1mW, i.e. 3.16mW.
 4dBu  a signal 4dB less than 0.775 Volts, i.e. 0.49 Volts.
For a thoroughly readable and concise reference for all of this I refer interested readers to
ITU Recommendation ITUR V.5744.
Easy Lookup Table
(To three significant figures)
dB  Power Ratio 
Voltage Ratio 

dB  Power Ratio 
Voltage Ratio 
0  1.0  1.0   0  1.0  1.0 
0.1  1.02  1.01   0.1  0.977  0.988 
0.2  1.05  1.02   0.2  0.954  0.977 
0.3  1.07  1.03   0.3  0.933  0.966 
0.4  1.10  1.05   0.4  0.912  0.954 
0.5  1.12  1.06   0.5  0.891  0.944 
0.6  1.15  1.07   0.6  0.871  0.933 
0.7  1.17  1.08   0.7  0.851  0.923 
0.8  1.20  1.10   0.8  0.832  0.912 
0.9  1.23  1.11   0.9  0.813  0.902 
1  1.26  1.12   1  0.794  0.893 
2  1.58  1.26   2  0.633  0.794 
3  2.00  1.41   3  0.500  0.707 
4  2.51  1.58   4  0.398  0.633 
5  3.16  1.77   5  0.316  0.565 
6  4.00  2.00   6  0.250  0.500 
7  5.01  2.24   7  0.200  0.446 
8  6.31  2.51   8  0.158  0.398 
9  7.94  2.82   9  0.126  0.354 
10  10  3.16   10  0.1  0.316 
20  100  10   20  0.01  0.1 
30  1,000  31.6   30  0.001  0.0316 
40  10,000  100   40  0.0001  0.01 
50  100,000  316   50  0.00001  0.00316 
60  1,000,000  1,000   60  0.000001  0.001 
80  1 x 10^{8}  1 x 10^{4}   80  1 x 10^{ 8}  1 x 10^{4} 
100  1 x 10^{10}  1 x 10^{5}   100  1 x 10^{10}  1 x 10^{5} 
120  1 x 10^{12}  1 x 10^{6}   120  1 x 10^{12}  1 x 10^{6} 
140  1 x 10^{14}  1 x 10^{7}   140  1 x 10^{14}  1 x 10^{7} 
160  1 x 10^{16}  1 x 10^{8}   160  1 x 10^{16}  1 x 10^{8} 
In lowfrequency electronic circuits, such as audio filters and amplifiers,
we work with volts (the righthand column). For the rest of this page
unless otherwise stated we talk about voltage or current rather than power.
From highschool maths, positive log values multiply and negative log values divide. For example, a
value of "+10dB" is a gain of 3.16, and a value of "10dB" is an attenuation of 3.16.
Because a dB value can specify an increase
or a decrease in signal level it is possible to say confusing but correct statements. For example,
a filter might be described as having a "gain" of 10dB, meaning the output is 0.316 times the input. Or
a resistive pad might be specified as having an attenuation of 10dB, again meaning the output is 0.316 times
the input. Both are correct, if a little confusing!
To calculate dB values not shown in the table, work out which values you need to sum to the dB value you
have, and then multiply or divide the factors in the appropriate column to calculate the final ratio.
For example, suppose we have a lowpass filter with a slope of 24dB per octave.
From the table, 24 = 20 + 4. We're dealing with voltages, so look up the right column
to get factors of 10 and 1.58 respectively.
Multiplied together, we get 15.8. So for every octave step in the
stop band, the output level is attenuated by a factor of 15.8.
Put another way, 24 = 20 + 4.
From the table above we see that 20dB is x0.1
and 4dB is x0.633. Multiplying together gives a gain of 0.0633, which is 1/15.8.
Filter cutoff frequencies are specified by the 3dB point, when the power output of the filter is
half that of the input. So a 10Hz lowpass filter has a power attenuation of 3dB at 10Hz.
However, we typically refer to voltages when dealing with filters and the like (e.g., crosstalk specifications
for mixers).
So when a filter is at its 3dB cutoff frequency the output voltage is 0.707x that of its
input (e.g., input = 1V, output = 0.707V).
Standard Units
In the audio world there are two standard references for signal levels: +4dBu and 10dBV.
The former is widely used in professional audio. The latter is predominantly
used for domestic or home hifi use. Some equipment can be configured to work at
either of the two signal levels.
The table below shows how they relate to each other.
dBV  dBu  Volts 
10  7.8  0.316 
2.2  0  0.775 
0  +2.2  1.0 
+1.8  +4  1.23 
Conversion Between Units
Units are very useful, but they can also lead to some confusion. For example, suppose we have an output at +4dBu that we wish to connect to a 10dBV input. Clearly, we need an attenuator to match the two signal levels. But how much attentuation is needed? At first, you might think 14dB. WRONG!
Applying a 14dB attenuator to a +4dBu signal will produce a 10dBu signal, not a 10dBV signal. Looking at the above table, the correct jump is from +4dBu to 7.8dBu (which is equivalent to 10dBV), an attenuation of 11.8dB.
Digital audio systems specify their own dB scale: the dBFS.
Here, the FS refers to the fullscale range of the A/D or D/A converter, as these do not like their
inputs exceeding full scaleif they do, then they hardlimit, which sounds awful!
To avoid this in the recording chain the mixer output is operated at a level comfortably
below the fullscale level to allow headroom for peaks that help maintain the audio's
dynamic range. Like most things in life, there are several `standard' levels, described in the
table below.
Scale  Percent of Full Scale  Where used 
12dBFS  25%  Japanese prosumer equipment^{(1)} 
14dBFS  20%  
18dBFS  12.5%  AES Standard 
20dBFS  10%  Lucid 8824 converter factory setting 
Notes:
 Fostex D824 quotes THD at 1kHz, 12dB. In other words, this is the sweetspot.
The whole purpose here is not to exceed 0dBFS or else the audio signal will be
severely distorted.
To convert these figures into realworld volts we also need to know the reference level
of the input or output. This is typically +4dBu or 10dBV. Then apply the headroom
number above to determine the actual peak signal signal level before clipping.
For example, a prosumer
hard disk recorder with 10dBV inputs and operated at 12dBFS
means that the input signal can peak up to ±1.8V (12dB x 316mV x 1.414) without clipping,
where 1.414 is the peak/rms ratio of a sine wave.
I hope this page is of use. If you find any errors, or have any ideas for additions, please email
me@njohnson.co.uk

