Technical Report: What is dB (Decibel)?

Decibels (dB) are used in a variety of fields, including electricity, communications, optics, acoustics, and vibration. This section summarizes the basic definition of decibels (dB), their significance, their usefulness, and how they are used in various fields.

The definition of a decibel (dB) uses logarithms, and understanding the properties of logarithms is essential to understanding decibels, so let's start with an explanation of logarithms.

【Note】
This chapter is intended for humanities students, so science students should skip it.

When representing large numbers, the number of digits increases, so we often use exponentiation. For example, the number 1,000,000 is:

1 million = 1,000,000 = 10 ⁶

Expressing it as a power of 10 results in fewer digits and is simpler. In general, when a number N is expressed as a power of any positive number a:

This is called the exponential representation of N, where a is the base and m is the exponent. A more generalized version of this, where m can be a real number as well as an integer, is the exponential function.

Important formulas related to indices;

Next, we define logarithms. In equation (2-1), we consider what power of a N is, and we call the part m that represents this power the logarithm of N, and express it using the following formula.

Here, m is called the logarithm, a is the base, and N is the anti-logarithm. It is important to note that even though the value of m is the same, the name differs between equations (2-1) and (2-4) due to the difference in how the equations are expressed. An exponential function is expressed with N as the center, and a logarithmic function is expressed with m as the center; they are inverse functions of each other. In terms of English grammar, this is like the relationship between the active and passive voice.

Substituting m in equation (2-4) into equation (2-1) gives:

This yields a relationship. Furthermore, similar to the exponential formulas, important formulas concerning logarithms can be obtained.

Logarithms are said to have been discovered in the 16th century by John Napier and others to simplify multiplication and exponentiation calculations. In an era without calculators or computers, they were apparently an extremely useful method of calculation.

What specific value would be best for the base 'a' in equation (2-4)?

The following logarithms are used depending on the base:

Table 1: Various logarithms with different bases

Since the decibel (dB) unit, which is the main focus of this discussion, uses the common logarithm, from now on, when we refer to logarithms, we will be referring to the common logarithm.
Also, the base a = 10 in equation (2-4) is omitted.

[example]
The logarithm of 1 million is 6 ⇒ log (1,000,000) = log (10^ ⁶) = 6

Table 3 in the lower right shows the logarithms of numbers from 1 to 10. These can now be easily calculated with a calculator, but as shown in Table 2 below, some can be calculated from other values.

Figure 1 shows an example graph of the common logarithm y = log(x), and Table 3 shows numerical examples when the argument x is an integer. Depending on the value of the argument x:

When x < 1, y < 0
When x = 1, y = 0
When x > 1, y > 0

This means that the decibel values, which will be explained next, can also take negative values.

Table 3: Examples of Logarithmic Numbers

The decibel was initially used in electrical systems to express the degree (i.e., ratio) of power transmission attenuation. Now, if we take the common logarithm of the ratio of the powers at two points _P1 and _P2, and let it be x;

This x is called a bell (B). This is because Alexander Graham Bell of the United States first used it to express the attenuation of power transmission in telephones. Also, since the value of a bell (B) itself is too large, decibels (deci Bel = dB), which are one-tenth of a bell, are usually used.

【Note】
The statement "Bel (B) itself is too large" means that the quantitative value of 1 Bel is large, but numerically it can be small. For example, 7B is the same quantity as 70 dB. This is clear if you compare the length of 1 m with the length of 1 mm.

A decibel L is defined as 10 times the logarithm of the power ratio between two points (_P1, _P2).

Thus, while the definition of a decibel is fundamentally a power ratio, it is also frequently used as a voltage ratio (or current ratio).

Since power is proportional to the square of the voltage (or current);

Therefore, the decibel [L] in terms of voltage ratio is defined as 20 times the logarithm of the voltage ratio between two points (_V1, _V2).

Thus, the decibel value is the same regardless of whether the ratio is power or voltage (effective value or linear value).

【Note】

Since it can be converted as such, 1Np corresponds to 8.686 dB. Nepa is taken from Napier, who invented logarithms, as discussed in Chapter 2 above.

Table 4 shows the relationship between commonly used decibel values and their corresponding true values (power ratio and voltage ratio). Using this table, you can easily determine the approximate decibel value for a given voltage multiplier.

Table 4. Commonly Used Decibel Values and Their Conversion Values

[example]
What about 5 times?

【Note】
As shown in Table 4 above, when the true value is greater than 1, the dB value is positive, and when the true value is less than 1, the dB value is negative. This is the same property of logarithms mentioned earlier. In other words, even if the decibel value is negative, the true value itself is not a negative number; it simply means that the power ratio (or voltage ratio) is less than 1.

As defined, a decibel represents the ratio of two quantities and is a relative level value. If a reference value (the denominator of the ratio) is defined as a constant physical quantity, the resulting decibel value can be easily converted to the absolute value of that physical quantity and can therefore be considered an absolute level value.

Generally, in fields such as electricity and sound (vibration), quantities expressed in decibels are called "levels" (unit: dB). From this point forward, level and decibel will be treated as almost synonymous.

Here, let's consider an example of representing voltage values. If we define the reference value for the voltage ratio as 1 V, any voltage value can be represented by substituting it with a decibel value. The unit in this case is dBV. For example, if x (V) is y (dBV):

Therefore, the baseline value of 1 can be omitted during calculations.

【Note】
In the future, if the absolute reference value is 1, the definition formula for the absolute level value will omit the 1. That is, you can use the table by treating the voltage ratio values in Table 4 above as voltage values.

[example]
2 V ⇒ 6 dBV
3.16 V ⇒ 10 dBV

Table 5 Voltage Range and Decibels

In FFT analyzers, this absolute level value is used to represent the voltage range of the input (see Table 5).

The same applies to other physical quantities besides voltage. For example, consider vibration acceleration. If we define the reference value as 1 m/ ^s², then 5 m/ ^s² is approximately 14 dBm/ ^s².

In the previous examples, the baseline value was 1 for a simple calculation, but the baseline value defined by sound pressure level, which is commonly used in the field of acoustics, is not 1. However, even in this case, the decibel value is expressed in the same way as the absolute value (sound pressure value).

This will be explained later. Although the decibel itself is not an SI unit, it is used in the field of acoustics as a quantity equivalent to a unit.

In summary, absolute level values represent the physical quantity itself, treating dB as the unit. Therefore, you will often see notations like "dB ○○" rather than simply dB.

As explained above, there are many advantages to using decibels. Below are some of the main reasons.

Since both decibels and percentages (%) represent ratios, it is possible to convert between them. In this case, decibels are usually considered as voltage ratios. For example, 10% is -20 dB, as shown by 20 log (10/100) = -20.

When expressing tolerances for resistance values or the frequency response range of a sensor, they are sometimes expressed as a ratio from a reference value, such as ±5%. This value can also be converted to a decibel value. For example, +10% is (1 + 10/100) times, so 20 log (1.1) = 0.83, or equivalent to 0.83 dB. Similarly, -10% is 20 log (0.9) = -0.92, or equivalent to -0.92 dB.

It's important to note here that percentages are linear, while dB are logarithmic.

+1 dB ⇒ +12.2 %
−1 dB ⇒ −10.9 %

Therefore, if the intervals are equal geometrically (logarithmically), the linear interval will be wider on the positive side. To put it very roughly, ±1 dB is approximately ±10%.