Decibels

Logarithmic scales, ratios, and reference levels

Lorenz Schwarz
Karlsruhe University of Arts and Design (HfG)

Winter Semester 2024/25
Course info

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Human hearing range

The human ear perceives sound pressures from 20 µPa (threshold of hearing) to 20 Pa (pain threshold), a ratio of 1 : 1,000,000.

The pain threshold is about a million times louder than the weakest audible sound.

Scaling human hearing

The human hearing is too large for a linear scale, so the logarithmic decibel scale is used, matching human hearing.

Decibel

A decibel is one-tenth of a Bel, a unit named after Alexander Graham Bell, expressing power ratios logarithmically.

• Expresses relative change (ratio between two physical quantities)
• A logarithmic unit (compresses large ranges into manageable numbers)

Logarithm

The logarithm is the inverse function of exponentiation:

Example:

Note:

Properties of logarithmic scales

Human perception of intensity follows an approximately logarithmic relationship.

Benefits:

• Matches nonlinear human perception
• Represent ratios (×2, ×10, etc.) consistently
• Display data spanning many orders of magnitude on one scale
• Reveal details at both low and high ends

→ Used for frequency and magnitude displays

With a logarithmic scale, the values of the tick marks increase by the same factor over equal distances (e.g., a base value of 10 raised to the powers 0, 1, 2, 3, etc.)

With a linear scale, the values of the tick marks increase by the same amount over equal distances.

Audio demonstration: Decibel changes

Pink noise demonstrating relative dB changes (each compared to reference):

▶ Play demonstration

Three A/B comparisons:

1. Reference → +3 dB (2× power, clearly noticeable)
2. Reference → +6 dB (4× power, 2× amplitude, noticeably louder)
3. Reference → +10 dB (10× power, about 2× as loud)

Absolute and relative dB

• With suffix → absolute level (SPL, dBV, dBm, dBFS)

• Without suffix → relative ratio (input/output)

Decibel formulas

Power quantities:

Field quantities (magnitude):

• → Power, Intensity
• → Amplitude, Voltage, Current, Pressure

Power and magnitude relationship

Power is proportional to the square of field quantities:

When converting field quantities to decibels, this square relationship means:

→ The factor of 2 comes from the square relationship between power and field quantities.

Converting between linear values and decibels

Power ratio:

Amplitude ratio:

Example:

Two signals whose levels differ by 6 dB have an amplitude ratio of

Example: doubling voltage and power in dB

• Voltage doubles (1 V → 2 V):

• Power doubles (1 W → 2 W):

Decibel reference values

To express an absolute value, the suffix specifies the reference :

Example: sound pressure (SPL)

is the root mean square of the measured sound pressure
is the standard reference sound pressure of 20 micropascals in air

Example: calculating dB SPL

Reference: (threshold of hearing)

Example with :

Example: dBFS (Full Scale)

Digital audio systems:

• Reference = maximum quantizing level (0 dBFS)
• Values typically negative (counted down from maximum)

16-bit system: Range = –32768 to +32767, so

Example calculation:

→ Half maximum amplitude = –6 dBFS

Example: adding gain stages

→ Since dB is logarithmic, multiplying ratios in linear terms becomes addition in dB.

Adding independent sound sources

Convert decibel values to linear form, perform the summation, then reconvert to decibels.

Independent sources add powers, not pressures.

Adding two sound sources 80 and 85 dB

Decibel quick reference

Common misconceptions

Adding dB SPL values directly

• Wrong: 70 dB + 70 dB = 140 dB

• Correct: Must convert to linear, add, convert back (≈ 73 dB)

Confusing 10 log vs 20 log

• 20 log for: Voltage, Current, Pressure, Amplitude
• 10 log for: Power, Intensity

Example 1

Question: A sound measures 0.2 Pa. What is the SPL in dB?

Solution

Example 2

Question: Two independent sound sources each produce 70 dB SPL. What is the total SPL when both operate?

→ Independent sources add powers, not dB values.

Copyright and Licensing

Original content: © 2025 Lorenz Schwarz
Licensed under CC BY 4.0. Attribution required for all reuse.

Includes: text, diagrams, illustrations, photos, videos, and audio.

Third-party materials: Copyright respective owners, educational use.

Contact: lschwarz@hfg-karlsruhe.de

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