Understanding decibels
Table of Contents
1. Introduction
The decibel is a unit that is mostly used for audio and signals, but it is often misunderstood. This article will attempt to explain precisely what it measures, its different types and how it can be calculated, along with a simple example.
First, it is important to understand that it is a relative unit of measurement, meaning that it always expresses the relationship between two quantities. Specifically, it measures the ratio of two values on a logarithmic scale.
Although it is often used to express absolute “volume” or “signal level”, it’s actually expresses the ratio between some variable over a constant. Some of these constants or “reference values” are listed here, along with their unit suffixes.
- dBW: Measures the relationship between some power in watts and 1 watt.
- dBm: Measures the relationship between some power in watts and 1 milliwatt.
- dBV: Measures the relationship between a RMS voltage and 1 volt.
- dBu: Measures the relationship between a RMS voltage and \(\sqrt{0.6}\) volts1.
- dBFS: Measures the relationship between the amplitude of a signal and the
maximum the device can handle before clipping occurs. The
FSstands for Full Scale. - dBSPL: Measures the relationship between a sound pressure and 20 micropascals (μPa). SPL stands for Sound Pressure Level.
Some of these units will be used below, specially those related to voltage and signal amplitude.
2. Calculating decibels
Given two values \(a\) and \(b\), the decibels can be calculated from their ratio using the following formula.
\begin{equation} \label{eq:ratio-to-decibels} \text{dB} = 20 \log_{10} \left( \frac{a}{b} \right) \\ \end{equation}Therefore, given the decibels, the ratio between the two values can also be obtained using a similar formula.
\begin{equation} \label{eq:decibels-to-ratio} \frac{a}{b} = 10^{\frac{\text{dB}}{20}} \end{equation}3. Example
To illustrate this, the following amplifier circuit will be used. An input signal of \(V_{in}\) volts is amplified using a specific gain of \(G\) decibels into an output signal of \(V_{out}\) volts2.
For this example, it will be assumed that the input voltage is 0.3 volts RMS, the gain of the amplifier is 24 decibels, and the maximum input voltage supported by the device is 5 volts RMS.
Note how the gain is expressed in decibels, not in “dBV” or “dBFS”. This is because there is no need for a constant or “reference value”, the value of the amplifier simply indicates the ratio that it is supposed to create between the input and the output voltage after the signal is amplified.
3.1. Calculating the input decibels
One common misunderstanding about decibels stems from the fact that the input signal is usually not measured in volts, but in “absolute” decibels using a constant reference value. In other words, the input is often expressed in dBV or dBu, but the amplification is expressed in dB, since it measures two variables.
Here are the formulas for calculating different decibels suffixes from a specific voltage.
| Suffix | Formula | Example |
|---|---|---|
| dBV | \[20 \log_{10} \left( \frac{V_{in}}{1} \right)\] | \[20 \log_{10} \left( \frac{0.3}{1} \right) = 20 \log_{10}(0.3) \approx -10.45\] |
| dBu | \[20 \log_{10} \left( \frac{V_{in}}{\sqrt{0.6}} \right)\] | \[20 \log_{10} \left( \frac{0.3}{0.774} \right) \approx 20 \log_{10}(0.387) \approx -8.23\] |
| dBFS | \[20 \log_{10} \left( \frac{V_{in}}{V_{max}} \right)\] | \[20 \log_{10} \left( \frac{0.3}{5} \right) = 20 \log_{10}(0.06) \approx -24.43\] |
All of these values express the same input voltage, 0.3 volts RMS, but they are expressed relative to different constants. The key detail is that the input value can’t be expressed in decibels without another reference value, which is usually constant.
Also note how the decibels are usually (but not necessarily always) negative, since the input voltage is smaller than the reference value. For example, the maximum dBFS value of a device should be zero, since:
\[ \log_{10} \left( \frac{V_{max}}{V_{max}} \right) = 0 \]
3.2. Calculating the output voltage
The following figure shows the previous circuit diagram, along with the known values: the input voltage and the amplifier gain.
By filling this know data in the formula from the previous section, the ratio between the input and output voltage can be obtained, and therefore the output voltage itself.
\begin{align*} \frac{V_{out}}{V_{in}} &= 10^{\frac{\text{dB}}{20}} \\ \frac{V_{out}}{0.3} &= 10^{\frac{24}{20}} \\ \frac{V_{out}}{0.3} &= 10^{1.2} \\ \frac{V_{out}}{0.3} &\approx 15.84 \\ V_{out} &\approx 0.3 \times 15.84 \\ V_{out} &\approx 4.75 \\ \end{align*}In the previous formula, 15.84 is the final ratio between the input and output voltage, which can be applied it to the input to obtain an output value of 4.75 volts. Notice how the output voltage grows logarithmically; it isn’t directly multiplied by 24, which would result in 7.2 output volts RMS.
3.3. Calculating the output dBV
Similarly to how the input voltage was converted to dBV, the same can be done to the output voltage.
\begin{split} \text{dBV} &= 20 \log_{10} \left( \frac{V_{out}}{1} \right) \\ &= 20 \log_{10} \left( \frac{4.75}{1} \right) \\ &= 20 \log_{10}(4.75) \\ &\approx 20 \times 0.67 \\ &\approx 13.53 \end{split}Just like with any other value expressed in decibel, this 13.53 expresses a ratio between two values, in this case the output voltage and a constant reference value of 1 volt, both RMS. This new value, although expressed in decibels, is completely independent from the input or the amplification, it simply indicates a ratio between a signal level and a constant.
Also note that, by subtracting the input and output dBV, the gain can be obtained. This can be done with a simple subtraction because the input and output dBV values were calculated using a logarithmic scale, which is also used to express the gain of the amplifier. In other words, a value expressed in “suffixed” decibels can be added to another value in “plain” decibels to obtain a result back in “suffixed” decibels because all decibels use the same logarithmic formula \eqref{eq:ratio-to-decibels}.