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Decibels

Decibels are a fundamental way for us to measure various electrical and acoustical quantities that are associated with sound. However, decibels are frequently misunderstood because they aren't particularly intuitive, and the underlying math behind decibels is not just everyday arithmetic. This article aims to describe decibels in an easy-to-understand way, and help to avoid common misunderstandings surrounding decibels.

History

While decibels are most commonly thought of as describing sound volume, one might be surprised to learn that the origins of the decibel trace back to measuring power loss over long-distance transmission cables. In the late 19th and early 20th centuries, telegraph and telephone lines were quickly circling the globe. Predicting and measuring the performance and efficiency of these long-distance transmission lines involved complex calculations, especially when summing the power loss over multiple segments of long-distance transmission cables. 

To simplify these calculations, Bell Telephone Labs invented the decibel for describing the magnitude of power loss. This decibel unit allowed engineers to eliminate the previously-required complex calculations, and instead use simple addition. The unit they invented was initially called the Transmission Unit (TU) but was quickly renamed to the decibel (dB), in honor of telecommunications pioneer Alexander Graham Bell. The official unit is the Bel, which is an order of magnitude or a power of ten, a single unit in the logarithmic number system. It is almost never used, as the deci-bel (1/10 of a Bel, since "deci" is the SI prefix meaning "one-tenth") is far more convenient.

The decibel is not a unit of measurement

The first thing to understand about decibels is that they are not units of measurement. Standard units of measurement (like meters, gallons, degrees Celsius, seconds) always measure physical quantities against a constant reference. In contrast, decibels are simply a way of comparing two arbitrary values on a logarithmic scale.

Even though decibels are commonly used to measure sound levels, they can be used to compare any two values. The following statements are perfectly valid:

  • "The distance from New York to Tokyo is 5.9dB more than the distance from Chicago to Los Angeles."
  • "It takes Bob 3dB more time to run a mile than it takes Jill."
  • "Bill Gates has 59dB more money than I do."

Of course, these comparisons are somewhat nonsensical, and you'd never hear them used in a real scenario. But, this reinforces that decibels do not implicitly measure any physical quantity, they only compare two values on a logarithmic scale. We're used to comparing values on a linear scale, which is much more intuitive. The linear equivalent of the decibel unit is the "x", or "times" unit. For example, "My car is 2 times faster than Jim's car", or "my computer takes 3x longer to boot up than yours." Note that the "x" or "times" unit is also not a unit of measurement; it only compares two arbitrary values on a linear scale.

Linear vs. logarithmic scales

Measuring quantities on a linear scale makes sense for things that combine additively. For example, if the distance from point A to point B is 10 meters, and the distance from point B to point C is 100 meters, it makes sense to say that "the total distance from A to B to C is 110 meters." It also makes sense to say that "the distance from B to C is 10 times further than A to B".

Linear quantities like distance increase arithmetically (1, 2, 3, 4, 5...). In other words, the difference between 1 meter and 2 meters is the same as the difference between 100 meters and 101 meters. Most quantities that we're used to dealing with on a day-to-day basis increase arithmetically and combine additively, and therefore linear comparisons make sense for them. However, not all quantities behave nicely like this. Some quantities increase geometrically (1, 2, 4, 8, 16...) and combine by multiplication. In other words, the difference between 1 pascal (a unit of pressure) and 2 pascals is perceived as the same as the difference between 2 pascals and 4 pascals.

The image below shows a linear scale on top and a logarithmic scale on the bottom:

logscale.png

Note that on the linear scale, the distance from 0 to 10 is the same as the distance from 10 to 20. On the logarithmic scale, the distance between 1 and 10 is the same as the distance between 10 and 100. This is because linear scales compare additively, whereas logarithmic scales compare by multiplication. This is clear on the scales above: on the linear scale, to get from one of the large hash marks to the next, you need to add or subtract by 10. On the logarithmic scale, to get from one of the large hash marks to the next, you need to multiply or divide by 10.

Examples of logarithmic scales

For example, humans perceive the pitch of a musical note on a logarithmic scale. The difference between a 100Hz frequency and a 200Hz frequency is perceived to be the same as the difference between a 1000Hz frequency and a 2000Hz frequency; both intervals are considered to be one octave. The Hz unit is linear, but other units like octaves and steps and logarithmic. Like sound pressure levels, frequency is linear, but the human perception of it is logarithmic; the change is relative to the whole. To find the difference between two quantities that are perceived logarithmically (like frequency), I would divide them by each other. To find the difference between two quantities that are percieved linearly (like distance), I would subtract them from each other.

Sound is caused by variations in air pressure, and the perceived volume of a sound is governed by how far its pressure level deviates from normal atmospheric pressure. The standard unit for pressure is pascals. However, sound pressure levels are also not perceived linearly by humans. For instance, the difference in volume between two sounds played with a pressure level of 1 pascal and 2 pascals is not perceived the same as the difference in volume between two sounds played at 100 pascals and 101 pascals. In fact, to get the same perceived difference in volume, you'd need to go from 100 pascals to 200 pascals. Similarly, you'd get the same difference in volume between two sounds played at 0.001 pascals and 0.002 pascals.

Logarithms

In order to fully understand decibels, you must understand logarithms. Logarithms are simply the inverse operation of exponentiation, just like subtraction is the inverse operation of addition, and division is the inverse operation of multiplication:

  Operation Inverse Operation
Addition/Subtraction 7 + 3 = 10 10 - 7 = 3
Multiplication/Division 5 * 4 = 20 20 ÷ 5 = 4
Exponent/Logarithm 103 = 1000 log10(1000) = 3

A base b raised to the power of the exponent y gives a number x. The logarithm reverses that operation: the logarithm to the base b of a number gives the exponent y

by = x        logb(x) = y

Logarithms can be performed on any base. However, when dealing with decibels, the base is always 10. Therefore, it's common to see log10(x) abbreviated as simply log(x) in this context.

Decibel definition

The Bel is defined as the logarithm (to base 10) of the ratio of two values. In other words:

Bel = log10(Value1 ÷ Value2)

Since a decibel is 1/10th of a bel, you'll always have 10 times as many decibels as compared to Bels. Therefore, the definition of a decibel is:

decibel = 10 * log10(Value1 ÷ Value2)

The decibel is used much more commonly than the Bel, because it offers a broader range of values to work with.

Decibel calculation examples

To drive this point home, let's do some examples of decibel calculations on quantities that have nothing to do with sound at all. If you'd like to try these yourself, many calculators can calculate logarithms, or you can also use Google's built-in calculator.

If I can run a mile in 16 minutes, and you can run a mile in 8 minutes, how many decibels slower am I?

10 * log(16 ÷ 8) = 10 * log(2) = 3.01029995664 ≈ 3dB

If I'm 300 meters away from an object, and you're 150 meters away from the same object, how many decibels further am I from that object than you?

10 * log(300 ÷ 150) = 10 * log(2) = 3.01029995664 ≈ 3dB

Already, we can see that the difference between 16 and 8 is the same as the difference between 300 and 150: they're both about 3 decibels apart. As you'll soon see, 3dB generally represents the doubling of a value. Here's a different example:

If I have a half liter bottle of water and you have a 2-liter bottle of water, how much less water do I have than you?

10 * log(0.5 ÷ 2) = 10 * log(0.25) = -6.02059991328 ≈ -6dB

As you can see, when the first value is less than the second value (i.e. when a value is reduced instead of increased), the decibel value becomes negative. One more example:

If Jeff Bezos has a net worth of $114 billion, and your net worth is $5700, how many decibels richer is Jeff Bezos than you?

10 * log($114,000,000,000 ÷ $5700) = 10 * log(20000000) = 73.0102999566 ≈ 73dB

Here, we can also see that another great property of decibels is that they can reduce large numbers down to manageable levels. Someone that has 20 million times more money than you is only 73dB richer.

Decibel shortcuts

As we've seen in some of the above examples, doubling a value is the same as increasing it by (about) 3dB. Correspondingly, halving a value is the same as decreasing it by (about) 3dB. This is an easy shortcut to remember. If something has quadrupled, that's the same as doubling twice, so it has increased by 3dB twice, which is 6dB. If something has increased by 8x, that's the same as doubling it, then doubling it, then doubling it again; which is the same as 3dB + 3dB + 3dB = 9dB.

Another shortcut is that increasing something by 10x is the same as increasing it by 10dB. The tables below illustrate both of these shortcuts:

Factor Decibel
1/64x -18dB
1/32x -15dB
1/16x -12dB
1/8x -9dB
1/4x -6db
1/2x -3dB
1x 0dB
2x 3dB
4x 6dB
8x 9dB
16x 12dB
32x 15dB
64x 18dB
Factor Decibel
1/1,000,000x -60dB
1/100,000x -50dB
1/10,000x -40dB
1/1,000x -30dB
1/100x -20dB
1/10x -10dB
1x 0dB
10x 10dB
100x 20dB
1,000x 30dB
10,000x 40dB
100,000x 50dB
1,000,000x 60dB

Putting theory into practice

Now that we've learned the theory behind decibels, let's understand how they're used in the real world. First, we'll review why decibels are useful:

Why do we need decibels?

As we've seen in the above examples, decibels provide two primary advantages:

  1. They allow for easy manipulation of quantities that behave or are perceived logarithmically.
  2. They can reduce very large ranges of numbers down to more manageable numbers.

How do these advantages help us when we're measuring more relevant quantities, like electrical power or Sound Pressure Level (SPL)? In the case of SPL, the units for pressure are pascals. The human ear can perceive a startlingly large range of pressure levels. The lowest sound that the average human can detect is 0.00002 pascals or 20 micropascals (also called the threshold of hearing), about the sound of a mosquito flying 10 feet away you. The loudest sounds become painful when they reach around 200 pascals (also called the threshold of pain). This may not seem like much, but it represents a ten-million-fold difference between the pressure difference at the threshold of hearing and the threshold of pain. Luckily, we don't hear it as being ten million times louder, because our sense of hearing is logarithmic too.

Additionally, humans perceive Sound Pressure Level on a logarithmic scale (meaning that we perceive every doubling of pressure to be the same difference; 0.0001 to 0.0002 pascals is the same difference as 50 to 100 pascals). In other words, if you looked at it on a logarithmic graph, the "distance" from 0.0001 to 0.0002 pascals would be the same as the distance from 50 to 100 pascals. Therefore, comparing SPL values using a logarithmic scale makes a lot more sense than using a linear scale. After all, without this scale, it would be difficult to intuitively understand the difference in loudness for sounds at, say, 0.00355 pascals, 0.714 pascals and 71.5 pascals (45 dBSPL, 91 dBSPL, and 131 dBSPL, respectively).

The loudest undistorted sound possible in Earth's atmosphere is 101,325 pascals, which is about 5 billion times higher pressure than the human threshold of hearing. Decibels help us to deal more easily with such large ranges of ratios. The dBSPL difference between the threshold of hearing and the loudest undistorted sound possible on Earth is around 194dB, despite the latter being 5 billions times higher pressure than the former.

Now let's break the rules

The sections below deal with aspects of how decibels are used in real-world scenarios. And, as you'll soon see, some of the rules that we learned about decibels are broken in practice. In particular:

Rule: Decibels are not units of measurement.
Reality: While this rule is technically true, there are ways to transform decibels into actual units of measurement. We'll discuss this in the next section on decibel units.

Rule: 3dB always represents a doubling of the underlying values.
Reality: While this is true for pure decibels, things get a little more complicated when we start introducing decibel units. We'll learn in the sections below that 3dB still represents a doubling for some types of units, but for other types of units, 6dB represents a doubling.

Decibel units

Above, we learned that decibels are not units of measurement; they don't measure any particular physical quantity, they are just a logarithmic comparison of two numbers. This is true for raw, pure decibels. However, many "decibel units of measurement" have been created for various purposes. These are decibel scales that compare values to an agreed-upon reference value. In these cases, decibels can actually become units of measurement, and can do more than just compare two numbers.

When using pure decibels, the abbrevation "dB" is used. When using decibel units, one or more letters are appended to the end of the dB unit. For instance, dBu, dBV, dBW, dBFS, dBSPL, etc. Each of these decibel units is intended to measure a real physical quantity, compared to a reference value for that quantity. Therefore, the new equation for decibel units becomes:

DecibelUnits = 10 * log10(Value1 ÷ ReferenceValue)

The ReferenceValue always remains the same for a given decibel unit, so we're always comparing values to a known, stable reference value.

Below are just a few examples of common decibel units, but be aware that there are dozens if not hundreds of different decibel units that have been defined for various purposes.

dBSPL

dBSPL is a good example of a decibel unit. dBSPL allows us to compare Sound Pressure Levels (SPL) in pascals using a decibel scale. The reference value for dBSPL is 0.00002 pascals, which is the threshold of hearing. This means that when using dBSPL, we're always calculating how much a given sound is louder than the threshold of hearing, in decibels. The equation for dBSPL would be:

DecibelUnits = 20 * log10(SPLValue ÷ 0.00002)

You may have noticed the dBSPL uses a 20*log equation instead of the normal 10*log equation. Hold that thought for now, we'll learn more about that soon.

dBu

dBu is a decibel unit that allows us to measure electrical voltages. The reference value for dBu is 0.775 volts. That may seem like a strange value, but it was chosen for a good reason (which is beyond the scope of this article). Therefore, dBu allows us to measure RMS voltages compared to 0.775 volts, on a decibel scale.

dBm

dBm is a decibel unit that allows us to measure electrical power. The reference value for dBm is 1 milliwatt, or 0.001 watts. Therefore, dBm allows us to measure electrical power values compared to 0.001 watts, on a decibel scale.

dB(A), dB(B), dB(C)

dB(A), dB(B), and dB(C) are a bit different than the above decibel units. All of these values actually still measure dBSPL; that is, they're measuring pressure in pascals compared to 0.00002 pascals, on a decibel scale. However, each of these units apply a different "weighting" on the SPL values based on their frequency. So, we essentially get the pure dBSPL value, apply the weighting to it to modify the values, and then calculate the new weighted dBSPL value. To denote which weighting scheme was used, we use dB(A), dB(B), dB(C) (sometimes written as dBA, dBB, dBC), or several other available weighting schemes.

Weighting is used to modify the frequency response of a measurement for various purposes. For instance, the human ear does not perceive all frequencies at the same loudness. If you play a 50 Hz tone and a 2000Hz tone at the same dBSPL value, the 2000 Hz tone will be perceived much louder. Therefore, A-weighting attempts to apply weights to the levels of different frequencies such that it approximates the level that the average human would actually perceive. This can be useful when measuring environmental noise levels. If there is noise present at a frequency that humans don't perceive very well, then that noise isn't as important and shouldn't affect the overall dBA value as much as a noise at a frequency that the human ear perceives well.

dBFS

dBFS is also another decibel abbreviation that is a bit different than the rest. In fact, dBFS is actually not a decibel unit, because it is not referenced to a stable value. The "FS" in dBFS stands for "full scale", which is another way to describe the clipping point of a digital audio device. Therefore, dBFS is referenced to the clipping point of a digital audio device, which means that 0dBFS is equal to the clipping point of that device.

Since dBFS values are usually measuring how far a signal is below the clipping point of a device (since it's generally not as useful to know how far above the clipping point a signal is), dBFS values are usually negative. -10dBFS means 10dB below the clipping point. Since the clipping point of different devices can often be different, dBFS is not referenced to a stable value and is therefore not a unit of measurement.

10log vs. 20log, 3dB or 6dB per doubling?

You may have noticed above that the formula for dBSPL is 20 * log10(Value ÷ RefValue) instead of the normal 10 * log10(Value ÷ RefValue). And, if you've worked with decibels before, you might be aware that sometimes the doubling of a value is represented by a 3dB increase, but other times a doubling is represented by a 6dB increase. This section will explain this phenomenon of decibels.

Decibels as a ratio of two powers

Back when Bell Telephone Labs came up with the decibel, they were primarily concerned with measuring the loss of power over long-distance transmission lines. Therefore, they started using a convention where decibels are always formulated to describe the change in power, or more specifically, the logarithmic ratio of two powers. This convention has continued to this day. This means that when we're using decibels to measure a power quantity, nothing changes. The formula is still:

10 * log10(Power1 ÷ Power2)

However, when we're measuring something different than power (like voltage or current), things need to change a bit to ensure that we're always ultimately talking about the change in power. To do this, we must use 20 * log10(Value1 ÷ Value2) to calculate the decibel values of these quantities. This is because power is proportional to the square of voltage and current, as evidenced by Ohm's Law:

P = V2 ÷ R
P = I2 * R

The derivation for a decibel voltage value is shown below. Notice that even though we're measuring voltage, we start with an equation that uses a ratio of powers, and manipulate it to get to voltage.

dB(Power) = 10 * log10(Power1 ÷ Power2)
dB(Power) = 10 * log10((V12 ÷ R) ÷ (V22 ÷ R))        since Power = V2 ÷ R
dB(Power) = 10 * log10(V12 ÷ V22)
dB(Power) = 10 * log10((V1 ÷ V2)2)
dB(Power) = 10 * (2 * log10(V1 ÷ V2))                      since log(x2) = 2 * log(x)
dB(Power) = 20 * log10(V1 ÷ V2)

This is a potentially confusing convention that has long been used with decibels, but in some cases this convention can make thing easier. For instance, when the engineers at Bell Laboratories wanted to measure the power loss over a long-distance transmission line, they couldn't directly measure the power on the line. Instead, they measured the voltage loss over the cable with a volt meter, and used the voltage loss and resistance to calculate the power loss. Using this convention, they could easily measure the voltage loss in dB and immediately know the power loss in dB without any additional calculations required.

Similarly, when describing audio devices like amplifiers, this convention allows us to ignore the differences between types of units when specifying levels. For example, one can describe an amplifier as having "40dB" of gain, without having to specify whether it's voltage gain or power gain. This convention ensures that 40dB of voltage gain (measured at 20*log) always results in 40dB of power gain (measured at 10*log).

Put another way, this convention is what gives us the luxury to simply say "that signal is 6dB louder", without having to specify whether we're talking about power, or voltage, or current, or pressure. All units are equalized so that a 6dB increase is a 6dB increase, regardless of the underlying unit of measurement being used. Without this convention, we'd have to always be careful to say, "that signal has 3dB more voltage", or "the pressure of that signal is 10dB lower".

Which units are 10log and which are 20log?

When we're talking about electrical units like voltage, current, and power, it's pretty easy to remember which units use the 10log equation and which use 20log: any units related to power use 10log and the rest use 20log. However, with other units, like sound pressure level, it's not quite as obvious. Below is a list of common units and whether they use 10log or 20log when using decibel units. Also note that any units that use 10log are increased 3dB per doubling, whereas any units that use 20log are increased 6dB per doubling.

10log units - 3dB per doubling

Any units that are directly proportional to power use the 10log equation. They are often referred to as "power quantities".

  • Power, in watts (dBm)
  • Sound intensity level, in watts per m2 (dBSIL)

20log units - 6dB per doubling

Any units that are directly proportional to the square root of power use the 20log equation. They are often referred to as "root-power quantities".

  • Voltage, in volts (dBu, dBV)
  • Sound pressure level, in pascals (dBSPL, dB(A), dB(B), dB(C))

Psychoacoustics

To complicate things even further, humans generally don't perceive a 3dB increase as a doubling of volume. In fact, a 3dB increase can be barely perceptible under some conditions. It's generally accepted that most people perceive a doubling of volume to be around a 10dB difference. So, if someone wants to hear a signal that "sounds" twice as loud as another, that signal would need to be around 10dB louder than the other signal.

Try it out for yourself below. Choose a decibel value and click the play button to get a feel for different decibel levels. At what value can you start to hear a difference in volume between the two tones? At what value does the second tone sound about twice as loud as the first tone?

Play two tones that are apart in level.

 

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