Decibel & Level calculation

This is a short tutorial on the topic of levels and the decibel unit, including application examples and useful calculators. For a good understanding, it is recommended to read the subpages of this article in the order listed. But it is also possible to jump directly to topics.

Level calculation is useful where you want to describe a large range of numbers or do calculations with them. E.g. in acoustics or audio technology. Levels do not describe the peak values of signals, but always their effective values. The effective value of an alternating current or direct voltage corresponds, for example, to the direct current or direct voltage that would convert the same power over an ohmic resistor in the same time. In order to determine effective values, a signal is viewed over a certain period of time and determined using the following formula [1].

p_{RMS}=\sqrt{\frac{1}{T}\int_{t_{0}}^{t_{0+T}}p(t)^2 dt}

        [1]

Squaring and root extraction roughly corresponds to rectifying a signal with only an alternating component.

Therefore the effective value not only depends on the peak value, but also takes into account the individual behaviour of a signal over time respectively in a period of time of length T. The effective value is referred to as RMS (abbreviation for “Root Mean Square”).

For subsequent SPL (sound pressure level) calculations usually (see IEC 61672-1 standard) these values for T are chosen:

For the level calculation, let us look at human hearing as an illustrative example.

Our hearing can detect a sound pressure range from 20 µPa (effective value), which is the empirically determined hearing threshold at silence, up to the pain threshold of 100 to 200 Pa. There is at least a factor of 10 million in between. So it is quite a large range of numbers to take into account! In addition, our sound perception does not work linearly, but rather follows a logarithmic progression.

Calculating with so-called levels can help make this large range of numbers more manageable. The pseudo unit dB (decibel) is introduced, named after Alexander Graham Bell.

If it is about sound pressure formula [2]

L_{p}=20*log(\frac{p_{2}}{p_{2}})

             [2]

provides the ratio of two sound pressures p₁ and p₂. In the case of p₁ = 20 µPa and p₂ = 200 Pa, formula [2] gives Lp = 140 dB.

L_{p}=20*log(\frac{200 Pa}{20µPa})=140dB

So p₁ and p₂ have a relative level difference of 140 dB. Given the size of the number, that sounds much more manageable than a “difference of a factor of 10,000,000”.