More About dB

🎯 Learning Objectives:

📘 This topic is further extension of the previous topic Log vs decibel scale. In this topic we will look at different example of dB scale and try to perceptually understand it.

We start this topic with some terms such as, $dB$, $dB(A)$, $dB(C)$, $dB(V)$, $dBm$ and $dBi$
Naturally the question arise here are:

• What are these all?
• How are they related to loudness, to phons and to sones?
• And how loud is loud?

This topic describes and compares them all and gives audio examples to understand them. Let us start with examples to explore more.
For instance, suppose we have two loudspeakers, the first playing a sound with power $P_1$, and another playing a louder version of the same sound with power $P2$, but everything else (i.e. how far away these sources are and the frequency) kept the same. Using the decibel unit, the difference in sound level, between the two is defined to be as follows

If the second produces twice as much power than the first, the difference in $dB$ is (to a good approximation):

This is illustrated in the following Figure.

What if the second loudspeaker has a million times the power of the first loudspeaker? What difference in dB would be? Here, it is.

In this example we see a feature of decibel scales that is useful in discussing sound: they can describe very big ratios using numbers of modest size as we have discussed in the topic "Log vs decibel scale".

Let us compare the sound pressure, sound power and sound intensity and see how they varies with respect to each others. For example what happens when we halve the sound power? The $\log(2)$ is 0.3010, so the $\log(12)$ is −0.3, to a good approximation. So, if we halve the power, we reduce the power and the sound level by 3dB. Halve it again (down to 14 of the original power) and we reduce the level by another 3dB. If you keep on halving the power, we have these ratios.

⭐ So, what happens if we add two identical sounds? Do we double the intensity (increase of 3dB)? Or do we double the pressure (increase of 6dB)?

This is the question that is discussed in exercise.

Let us now listen to the dBs and understand how perceptually we perceive sound with change in dBs.
As we have seen above that halving the power reduces the sound pressure by $\sqrt{2}$ and the sound level by 3dB, which is illustrated in the graph below. In the graph, the first chunk of sound is white noise (a mix of a broad range of audible frequencies, analogous to white light, which is a mix of all visible frequencies). The second sample is the same noise, with the voltage reduced by a factor of $\sqrt{2}$. Now $\sqrt{\frac{1}{2}}$ is approximately 0.7, in this way −3dB corresponds to reducing the voltage or the pressure to 70% of its original value. The red line shows a continuous exponential decay with time. The voltage falls by 50% for every second sample. Doubling of the power does not make a huge difference to the loudness. We will discuss this further below, but it is a useful thing to remember when choosing sound reproduction equipment.

Let us listen to the graph.

Now let us examine how big is the dB
In the example sounds, the successive samples are reduced by just one decibel ($1dB$), which means that one decibel is of the same order as the Just Noticeable Difference (JND) for sound level. As we listen to these sounds, we will notice that the last is quieter than the first, but it is rather less clear to the ear that the second of any pair is quieter than its predecessor i.e. $10\log(1.26)=1$, so to increase the sound level by 1dB, the power must be increased by 26%, or the voltage by 12%.
Let us listen.

Now let us see what if the difference is less than a decibel? Sound levels are rarely given with decimal places. The reason is that sound levels that differ by less than 1dB are hard to distinguish, as the next example shows. (This makes the dB a convenient size unit.) Here in this graph there is a 0.3dB step. We may notice that the last is quieter than the first, but it is difficult to notice the difference between successive pairs, i.e. $10\log(1.07)=0.3$, so to increase the sound level by 0.3dB, the power must be increased by 7%, or the voltage by 3.5%.
Let us listen it.

dB in Filters

The most widely used sound level filter is the A scale, which roughly corresponds roughly to the inverse of the 40dB (at 1kHz) equal-loudness curve. Using this filter, the sound level meter is thus less sensitive to very high and very low frequencies. Measurements made on this scale are expressed as dBA. The C scale varies little over several octaves and is thus suitable for subjective measurements only for moderate to high sound levels. Measurements made on this scale are expressed asdB(C). There is also a (rarely used) B weighting scale, intermediate between A and C. The figure below shows the response of the A filter (left) and C filter, with gains in dB given with respect to 1kHz

📝 Key Takeaways

• Decibels (dB) are a logarithmic ratio — they express relative changes, not absolute quantities.

• For power ratios, use:

  $$\Delta L = 10\log_{10}\!\left(\frac{P_2}{P_1}\right)$$

  For pressure or voltage ratios, use:

  $$\Delta L = 20\log_{10}\!\left(\frac{V_2}{V_1}\right)$$

• +3 dB ≈ doubling of power
  +6 dB ≈ doubling of pressure or voltage

• Decibel values can be negative when the measured level is below the reference.
  Remember that dB is dimensionless — it must be qualified (e.g., dB SPL, dBm, dBV).

• Small changes in dB can have noticeable perceptual effects:

  • 1 dB ≈ just noticeable difference (JND) in loudness.
  • 0.3 dB ≈ barely perceptible or often indistinguishable.

• Perceived loudness grows roughly logarithmically:

  • A 10 dB increase is perceived as twice as loud (approximation based on sones).

• Weighting curves like dB(A) and dB(C) adjust for human hearing sensitivity:

  • A-weighting → approximates human ear response at moderate levels.
  • C-weighting → used for higher levels with flatter response.

• Converting between dB and linear gain:

  $$\text{gain} = 10^{\frac{dB}{20}}$$

  Useful for setting levels in audio programming or DSP.

• RMS vs Peak levels:

  • RMS relates to perceived loudness.
  • Peaks can be much higher and cause clipping even if RMS seems fine.

• Visualization helps understanding:
  Combine:

  • A waveform view with a 0 dB reference line
  • A real-time dB meter
  • Interactive gain sliders (in dB and linear scale)

• Practical rule of thumb:
  When building audio demos, always show both:

  • dB scale (logarithmic)
  • Linear amplitude
    → so learners can see and hear how decibel changes map to perceived loudness.

🎧 Try it yourself:
Use the interactive tone demo — adjust the dB slider, observe how the waveform amplitude changes, and listen to how the perceived loudness follows the logarithmic scale.

🧪 Interactive Demo

🧠 Quick Quiz

Test your understanding - select and submit an answer.