The Principle of Arithmetic Operations in Decibels(dB)
Decibels, or dB, are widely used in acoustics, vibration, electronics, and signal processing. However, arithmetic operations in decibels can be confusing because dB values are expressed on a logarithmic scale, not a linear scale. Therefore, ordinary addition, subtraction, multiplication, and division cannot always be applied directly to dB values.
The most important principle is decibels represent ratios on a logarithmic scale. Therefore, physical quantities must often be converted back to the linear scale before performing arithmetic operations.
What Does a Decibel Represent?
A decibel expresses the ratio between a measured quantity and a reference quantity.
For power-like quantities, the decibel value is defined as
where P is the measured power and P0 is the reference power.
For amplitude-like quantities such as pressure, voltage, velocity, or acceleration, the decibel value is usually written as
This is because power is proportional to the square of amplitude.
For example, sound pressure level is defined as
where p0 = 20 μPa in air.
Why dB Values Cannot Be Added Directly
Suppose there are two independent sound sources: L1 = 60dB, L2 = 60dB
It is incorrect to say 60dB + 60dB = 120dB
This is wrong because dB values are logarithmic representations of sound energy. To add two sound levels, each dB value must first be converted back to a linear energy ratio.
The correct formula is
For two 60 dB sources, Ltotal = 10log10 (10 60/10 + 10 60/10) = 10log10 (10 60/10 + 10 60/10)  = 10log10 (2×106) ≈ 63dB
Therefore, 60dB + 60dB ≠ 120dB
Instead, 60dB + 60dB ≈ 63dB
This is one of the most important rules in dB arithmetic.
1. Addition of Decibel Levels
When adding independent power, energy, or mean-square quantities expressed in dB, the general formula is
This formula is commonly used for
- sound pressure level summation
- vibration level summation
- octave-band or one-third-octave-band level summation
- power spectral density band summation
- contribution analysis
For example, if three independent sources have levels of 70 dB, 65 dB, and 60 dB, the total level is
Ltotal = 10log10 (10 70/10 + 10 65/10 + 10 60/10)
The largest level usually dominates the total value. Smaller levels contribute less because the scale is logarithmic.
2. Subtraction of Decibel Levels
Subtraction in decibels requires special care because the meaning of “subtraction” depends on the physical situation. There are two fundamentally different cases
- Subtracting two dB levels to obtain a ratio or level difference
- Subtracting one physical energy contribution from another
These two cases look similar mathematically, but they are physically different.
Subtraction as a Level Difference or Ratio
In many engineering applications, subtracting two dB values is completely valid. This is the case when the purpose is to express a ratio between two physical quantities.
For example, insertion loss is commonly defined as
If the sound level before treatment is 80dB and the sound level after treatment is 65dB, then IL = 80 − 65 = 15dB
This direct subtraction is correct because insertion loss represents the ratio between the acoustic energy before and after inserting a noise-control treatment.
For power-like quantities, Lbefore = 10log10 (Pbefore / P0), Lafter = 10log10 (Pafter / P0)
Therefore, Lbefore − Lafter = 10log10 (Pbefore / Pafter)
So, the subtraction of two dB levels is equivalent to taking the logarithm of a linear ratio.
This type of subtraction is used for
| Quantity | Typical expression | Meaning |
|---|
| Insertion Loss | IL = Lbefore − Lafter | Reduction due to inserting a treatment or device |
| Transmission Loss | TL = Lincident − Ltransmitted | Ratio of incident to transmitted acoustic power |
| Noise Reduction | NR = L1 − L2 | Level difference between two positions or rooms |
| Gain | G = Lout − Lin | Output-to-input ratio |
| Attenuation | A = Lin − Lout | Input-to-output reduction |
In these cases, direct dB subtraction is valid because the calculation is not removing one energy component from another. It is calculating a ratio expressed in dB.
Noise reduction before and after improvement measures
Subtraction as Removal of an Energy Contribution
A different situation occurs when one wants to remove a physical contribution from a measured total.
For example, suppose the measured total noise level is Ltotal = 80dB
and the background noise level is Lbackground = 75dB
It is incorrect to calculate the source-only level as 80 − 75 = 5dB
This is wrong because the total level and background level represent logarithmic forms of energy-like quantities. The correct operation is to convert them back to the linear scale, subtract the physical energy contribution, and then convert the result back to dB.
The correct formula is
For the example above, Lsource = 10log10 (10 80/10 − 10 75/10) ≈ 78.35dB
So, 80dB − 75dB ≠ 5dB
Instead, the corrected source level is approximately 78.35dB
This type of subtraction is used for
| Situation | Correct method |
|---|
| Background noise correction | Convert dB to linear, subtract, then covert back |
| Removing one source contribution from total noise | Convert dB to linear, subtract, then covert back |
| Seperating target signal from total measured level | Convert dB to linear, subtract, then covert back |
| Contribution analysis | Perform subtraction on the linear energy scale |
3. Multiplication and Division in Decibels
Multiplication and division behave differently on a logarithmic scale.
If two linear ratios are multiplied A × B
their dB values are added LA×B = LA + LB
This is because 10log10 (AB) = 10log1 0 (A) + 10log10 (B)
Similarly, if two linear ratios are divided A / B
their dB values are subtracted LA/B = LA − LB
This is why gain and attenuation are conveniently expressed in decibels.
For example, if an amplifier has a gain of 20 dB and a filter has an attenuation of 6 dB, the total gain is 20dB − 6dB = 14dB
In this case, ordinary dB addition and subtraction are valid because they represent multiplication and division of linear ratios.
The Key Distinction
The most important point is to distinguish between two different situations.
| Operation meaning | Correct dB operation |
|---|
| Adding independent energies or powers | Convert dB to linear, add, then covert back |
| Removing one contribution from total energy | Convert dB to linear, subtract, then convert back |
| Multiplying gains or ratios | Add dB values directly |
| Dividing gains or ratios | Subtract dB values directly |
This distinction explains why dB arithmetic sometimes seems inconsistent. It is not actually inconsistent; the correct operation depends on the physical meaning of the calculation.
Quick dB Addition
| dB difference (L1 - L2) | Level increase |
|---|
| 0dB | +3.0dB |
| 1dB | +2.5dB |
| 2dB | +2.1dB |
| 3dB | +1.8dB |
| 10dB | +0.4dB |
| > 10dB | almost 0dB |
80dB + 80dB = 80 + 3dB = 83dB
80dB + 77dB = 80dB + 1.8dB = 81.8dB (refer to MALMIJAL example below)
80dB + 70dB = 80dB + 0.4dB = 80.4dB
Set 80 dB using a Step signal, and set 77 dB in the same way
Use Arithmetic Operation → “dB Sum” to calculate the energy sum of 80 dB and 77 dB
Right-click the graph and select “Data Tip” from the popup menu to view the numerical dB Sum result
Conclusion
Arithmetic operations in decibels are governed by the logarithmic nature of the dB scale. When dB values represent independent energy or power quantities, they cannot be added or subtracted directly. They must first be converted to the linear scale, combined, and then converted back to dB.
On the other hand, when dB values represent gains, attenuations, or ratios in a signal chain, direct addition and subtraction are appropriate because multiplication and division on the linear scale become addition and subtraction on the logarithmic scale.
In short, the secret of dB arithmetic is simple.
Add or subtract dB values directly only when the original linear quantities are being multiplied or divided. When the original physical quantities are being added or subtracted, convert the dB values back to linear scale first.
Suggested Further Reading
##You may also find these topics helpful:
The Principle of Arithmetic Operations in Decibels(dB)
Decibels, or dB, are widely used in acoustics, vibration, electronics, and signal processing. However, arithmetic operations in decibels can be confusing because dB values are expressed on a logarithmic scale, not a linear scale. Therefore, ordinary addition, subtraction, multiplication, and division cannot always be applied directly to dB values.
The most important principle is decibels represent ratios on a logarithmic scale. Therefore, physical quantities must often be converted back to the linear scale before performing arithmetic operations.
What Does a Decibel Represent?
A decibel expresses the ratio between a measured quantity and a reference quantity.
For power-like quantities, the decibel value is defined as
where P is the measured power and P0 is the reference power.
For amplitude-like quantities such as pressure, voltage, velocity, or acceleration, the decibel value is usually written as
This is because power is proportional to the square of amplitude.
For example, sound pressure level is defined as
where p0 = 20 μPa in air.
Why dB Values Cannot Be Added Directly
Suppose there are two independent sound sources: L1 = 60dB, L2 = 60dB
It is incorrect to say 60dB + 60dB = 120dB
This is wrong because dB values are logarithmic representations of sound energy. To add two sound levels, each dB value must first be converted back to a linear energy ratio.
The correct formula is
For two 60 dB sources, Ltotal = 10log10 (10 60/10 + 10 60/10) = 10log10 (10 60/10 + 10 60/10)  = 10log10 (2×106) ≈ 63dB
Therefore, 60dB + 60dB ≠ 120dB
Instead, 60dB + 60dB ≈ 63dB
This is one of the most important rules in dB arithmetic.
1. Addition of Decibel Levels
When adding independent power, energy, or mean-square quantities expressed in dB, the general formula is
This formula is commonly used for
For example, if three independent sources have levels of 70 dB, 65 dB, and 60 dB, the total level is
Ltotal = 10log10 (10 70/10 + 10 65/10 + 10 60/10)
The largest level usually dominates the total value. Smaller levels contribute less because the scale is logarithmic.
2. Subtraction of Decibel Levels
Subtraction in decibels requires special care because the meaning of “subtraction” depends on the physical situation. There are two fundamentally different cases
These two cases look similar mathematically, but they are physically different.
Subtraction as a Level Difference or Ratio
In many engineering applications, subtracting two dB values is completely valid. This is the case when the purpose is to express a ratio between two physical quantities.
For example, insertion loss is commonly defined as
If the sound level before treatment is 80dB and the sound level after treatment is 65dB, then IL = 80 − 65 = 15dB
This direct subtraction is correct because insertion loss represents the ratio between the acoustic energy before and after inserting a noise-control treatment.
For power-like quantities, Lbefore = 10log10 (Pbefore / P0), Lafter = 10log10 (Pafter / P0)
Therefore, Lbefore − Lafter = 10log10 (Pbefore / Pafter)
So, the subtraction of two dB levels is equivalent to taking the logarithm of a linear ratio.
This type of subtraction is used for
In these cases, direct dB subtraction is valid because the calculation is not removing one energy component from another. It is calculating a ratio expressed in dB.
Subtraction as Removal of an Energy Contribution
A different situation occurs when one wants to remove a physical contribution from a measured total.
For example, suppose the measured total noise level is Ltotal = 80dB
and the background noise level is Lbackground = 75dB
It is incorrect to calculate the source-only level as 80 − 75 = 5dB
This is wrong because the total level and background level represent logarithmic forms of energy-like quantities. The correct operation is to convert them back to the linear scale, subtract the physical energy contribution, and then convert the result back to dB.
The correct formula is
For the example above, Lsource = 10log10 (10 80/10 − 10 75/10) ≈ 78.35dB
So, 80dB − 75dB ≠ 5dB
Instead, the corrected source level is approximately 78.35dB
This type of subtraction is used for
3. Multiplication and Division in Decibels
Multiplication and division behave differently on a logarithmic scale.
If two linear ratios are multiplied A × B
their dB values are added LA×B = LA + LB
This is because 10log10 (AB) = 10log1 0 (A) + 10log10 (B)
Similarly, if two linear ratios are divided A / B
their dB values are subtracted LA/B = LA − LB
This is why gain and attenuation are conveniently expressed in decibels.
For example, if an amplifier has a gain of 20 dB and a filter has an attenuation of 6 dB, the total gain is 20dB − 6dB = 14dB
In this case, ordinary dB addition and subtraction are valid because they represent multiplication and division of linear ratios.
The Key Distinction
The most important point is to distinguish between two different situations.
This distinction explains why dB arithmetic sometimes seems inconsistent. It is not actually inconsistent; the correct operation depends on the physical meaning of the calculation.
Quick dB Addition
80dB + 80dB = 80 + 3dB = 83dB
80dB + 77dB = 80dB + 1.8dB = 81.8dB (refer to MALMIJAL example below)
80dB + 70dB = 80dB + 0.4dB = 80.4dB
Conclusion
Arithmetic operations in decibels are governed by the logarithmic nature of the dB scale. When dB values represent independent energy or power quantities, they cannot be added or subtracted directly. They must first be converted to the linear scale, combined, and then converted back to dB.
On the other hand, when dB values represent gains, attenuations, or ratios in a signal chain, direct addition and subtraction are appropriate because multiplication and division on the linear scale become addition and subtraction on the logarithmic scale.
In short, the secret of dB arithmetic is simple.
Add or subtract dB values directly only when the original linear quantities are being multiplied or divided. When the original physical quantities are being added or subtracted, convert the dB values back to linear scale first.
Suggested Further Reading
##You may also find these topics helpful: