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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 base-10 logarithm of the ratio of two power levels:

N (dB) = 10 log10 ( P1 / P2 )

From the relation P = V2/R, we can derive the formula for ratios of voltages:

N (dB) = 20 log10 ( V1 / V2 )

Two things to note here:

1. The impedances of both top and bottom are assumed to be equal (R1 = R2).
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
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.
FSFull ScaleFull-scale 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.
(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 ITU-R V.574-4.

## Easy Lookup Table

(To three significant figures)

dB Power Ratio Voltage Ratio   dB Power Ratio Voltage Ratio
0 1.0 1.0 01.01.0
0.1 1.02 1.01 -0.10.9770.988
0.2 1.05 1.02 -0.20.9540.977
0.3 1.07 1.03 -0.30.9330.966
0.4 1.10 1.05 -0.40.9120.954
0.5 1.12 1.06 -0.50.8910.944
0.6 1.15 1.07 -0.60.8710.933
0.7 1.17 1.08 -0.70.8510.923
0.8 1.20 1.10 -0.80.8320.912
0.9 1.23 1.11 -0.90.8130.902
1 1.26 1.12 -10.7940.893
2 1.58 1.26 -20.6330.794
3 2.00 1.41 -30.5000.707
4 2.51 1.58 -40.3980.633
5 3.16 1.77 -50.3160.565
6 4.00 2.00 -60.2500.500
7 5.01 2.24 -70.2000.446
8 6.31 2.51 -80.1580.398
9 7.94 2.82 -90.1260.354
10 10 3.16 -100.10.316
20 100 10 -200.010.1
30 1,000 31.6 -300.0010.0316
40 10,000 100 -400.00010.01
50 100,000 316 -500.000010.00316
60 1,000,000 1,000 -600.0000010.001
80 1 x 108 1 x 104  -801 x 10 -81 x 10-4
100 1 x 1010 1 x 105 -1001 x 10-101 x 10-5
120 1 x 1012 1 x 106 -1201 x 10-121 x 10-6
140 1 x 1014 1 x 107 -1401 x 10-141 x 10-7
160 1 x 1016 1 x 108 -1601 x 10-161 x 10-8

In low-frequency electronic circuits, such as audio filters and amplifiers, we work with volts (the right-hand column). For the rest of this page unless otherwise stated we talk about voltage or current rather than power.

From high-school 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 low-pass 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 cut-off frequencies are specified by the -3dB point, when the power output of the filter is half that of the input. So a 10Hz low-pass filter has a power attenuation of 3dB at 10Hz. However, we typically refer to voltages when dealing with filters and the like (e.g., cross-talk 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 hi-fi 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.

## The Digital Age

Digital audio systems specify their own dB scale: the dBFS. Here, the FS refers to the full-scale range of the A/D or D/A converter, as these do not like their inputs exceeding full scale---if they do, then they hard-limit, which sounds awful!

To avoid this in the recording chain the mixer output is operated at a level comfortably below the full-scale 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.

ScalePercent of Full ScaleWhere used
-12dBFS 25% Japanese pro-sumer equipment(1)
-14dBFS 20%
-18dBFS 12.5% EBU R68 Standard (PDF)
-20dBFS 10% Lucid 8824 converter factory setting

Notes:

1. 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 real-world 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.