Technical Report: What is dB (Decibel) 1

Decibels (dB) are used in various fields such as electricity, communication, optics, acoustics, and vibration. Here, we summarize the basic definition of decibels (dB), their significance, convenience, and how they are used in different fields.

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

【Note】
This chapter is aimed at humanities students, so those in the sciences should skip it.

When representing large numbers, exponential notation is often used because the number of digits increases. For example, the number one million is represented as:

1 million = 1,000,000 = 10 ⁶

When expressed as a power of 10, it becomes simpler and has fewer digits. 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 referred to as the base and 'm' as the exponent. Generalizing this further, when 'm' is extended from integers to real numbers, it becomes the exponential function.

Important formulas regarding indices.

Next, we define the logarithm. In equation (2-1), we consider what power of a gives N, and we call that exponent m the logarithm of N, which is expressed in the following formula.

Here, we refer to m as the logarithm, a as the base, and N as the anti-logarithm. It is important to note that even with the same m, the names differ due to the different ways of expressing the equations in (2-1) and (2-4). The exponential function is centered around N, while the logarithmic function is centered around m, making them inverse functions of each other. In terms of English grammar, this relationship is akin to that of active and passive voice.

Substituting m from equation (2-4) into equation (2-1);

A relationship formula can be obtained. Furthermore, important formulas regarding logarithms can be obtained, similarly to exponential formulas.

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

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

Due to the difference in bases, the following logarithms are used.

Table 1: Various Logarithms Based on Differences in Bases

The main purpose of the decibel (dB) used here is based on the common logarithm, so from now on, when we refer to logarithm, we will mean the common logarithm.
Also, the base a = 10 in equation (2-4) will be omitted in the notation.

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

Table 3 in the lower right shows the logarithmic values for numbers from 1 to 10, which can now be easily calculated using calculators, but as shown in Table 2, some can be computed from other values.

Figure 1 is an example of the 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 x;

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

This means that the decibel values explained below can also take on negative values.

Table 3: Examples of Logarithmic Values

Decibels were first used to represent the degree of power transmission attenuation (i.e., ratio) in electrical systems. Now, let the power at two points be P ₁ and P ₂, and taking the common logarithm of their ratio gives us x;

This x is called a Bell (B). This is named after Alexander Graham Bell, who first used it to express the attenuation of power transmission in telephones. Additionally, since the value of Bell (B) itself is too large, the decibel (deci Bel = dB), which is one-tenth of a Bell, is commonly used.

【Note】
The phrase "the bell (B) itself is too large" means that the quantitative value of 1 bell is large, but numerically it becomes small. For example, 7B is equivalent to 70 dB. This will be clear when comparing a length of 1 m to a length of 1 mm.

The decibel L is defined as ten times the logarithm of the power ratio between two points (P ₁, P ₂).

In this way, the definition of decibels is fundamentally a power ratio, but it is also commonly used in terms of voltage ratio (or current ratio).

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

Therefore, the decibel [L] for the voltage ratio is defined as 20 times the logarithm of the voltage ratio between two points (V ₁, V ₂).

In this way, the decibel value will be the same regardless of whether the ratio is of power or voltage (effective value or linear value).

Notice

Since it can be converted, 1Np corresponds to 8.686 dB. The term 'neper' is derived from Napier, who invented logarithms as mentioned in the previous Chapter 2.

Table 4 shows the relationship between commonly used decibel values and their corresponding absolute values (power ratio and voltage ratio). This table can be used to easily determine the decibel value (approximate value) corresponding to the voltage ratio.

Table 4: Commonly Used Decibel Values and Their Conversion Values

[Example]
What about 5 times?

[Note]
As shown in Table 4, 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 very nature 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.

Decibels, as defined, represent the ratio of two quantities and signify a relative level value. If a reference value (the denominator of the ratio) is defined as a constant physical quantity, the decibel value in that case can be easily converted to the absolute value of the physical quantity, allowing it to be regarded as an absolute level value.

Generally, the magnitude of physical quantities in fields such as electricity and sound (vibration) expressed in decibels is referred to as 'level (unit dB)'. From this point onward, 'level' and 'decibel' are explained as nearly synonymous.

Here, let's consider an example of expressing voltage values. If we define the reference voltage ratio as 1 V, any voltage value can be represented in decibels. The unit at this time will be dBV. For example, if x (V) is y (dBV);

You can omit the reference value 1 during calculations.

【Note】
If the absolute reference value is 1 in the future, the definition formula for the absolute level value will omit 1. In other words, the values of the voltage ratio in the previous Table 4 can be considered as voltage values and used in the table.

[Example]
2 V ⇒ 6 dBV
3.16 V ⇒ 10 dBV

Table 5: Voltage Range and Decibel

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

It is exactly the same for other physical quantities, not just voltage values. For example, let's consider the case of vibration acceleration. If we define the reference value as 1 m/s ², then 5 m/s ² means approximately 14 dBm/s ².

In previous examples, the reference value was 1 for simplicity, but in the field of acoustics, the reference value defined for sound pressure levels is not 1. However, even in this case, the decibel value similarly represents an absolute value (sound pressure value).

This will be discussed later. The decibel itself is not an SI unit, but it is used as a quantity corresponding to a unit in the field of acoustics.

In summary, absolute level values can be seen as representing physical quantities themselves, treated as units like dB. Therefore, it is often the case that instead of simply using dB, expressions like 'dB ○○' are used.

As mentioned in the previous explanation, there are many advantages to using decibels. Below, I will outline some representative reasons.

Decibels and percentages both represent ratios, so it is possible to convert between the two. In this case, decibels are usually considered as voltage ratios. For example, 10% corresponds to -20 dB, calculated as 20 log (10/100) = -20.

When expressing the tolerance of resistance values or the allowable deviation of a sensor's frequency characteristic range, it may be represented as a ratio from a reference value, such as ±5%. This value can also be converted into decibels. For example, +10% means (1 + 10/100) times, so 20 log(1.1) = 0.83, which corresponds to 0.83 dB. Similarly, -10% corresponds to 20 log(0.9) = -0.92, which is -0.92 dB.

Here, it is important to note that % is linear, while dB is logarithmic.

+1 dB ⇒ +12.2 %
−1 dB ⇒ −10.9 %

In addition, being equidistant in a geometric (logarithmic) sense means that the positive side of the linear scale becomes wider. To put it very roughly, ±1 dB corresponds to about ±10%.