1.3. Decibels

Often when expressing the ratio of two measurements with respect to the same scale the decibel (dB) is used. In electronics the basic use of the decibel is to represent the ratio of power (energy). The ratio \(P_1/P_2\) is expressed as

\[10 \log_{10} \frac{P_1}{P_2}\quad (dB)\]

So when we increase the \(P_1\) power level with a factor 2 the ratio becomes

\[\begin{split}10 \log_{10} \frac{2P_1}{P_2} &= 10\left( \log_{10} 2 + \log_{10}\frac{P_1}{P_2}\right)\\ &= 10 \log_{10} 2 + 10 \log_{10}\frac{P_1}{P_2}\\ &\approx 3 + 10 \log_{10}\frac{P_1}{P_2}\end{split}\]

That is doubling the power \(P_1\) is equivalent with adding 3 to the power ratio expressed in decibel.

Well known is that sound level is often expressed in decibel. But wait… soundlevel is just one measurement and decibel is a measure for the ratio of two measurements. Indeed expressing the soundlevel in dB implicitly assumes a reference soundlevel which is chosen to be the sound level that is just hearable for the human ear (the sound threshold):

\[10 \log_{10} \frac{P}{P_\text{ref}}\]

For all types of measurements a lot of different reference levels are defined leading to different versions of dB scales. These are often indicated with an extra letter.

In signal processing we often compare signal levels, e.g. the level of the output level in relation to the level of the input signal or the level of a signal with respect to the noise level (the wellknown signal to noise ratio SNR). Another frequent use of the decibel is to express the dynamic range of a measurement or signal. The reference level than is the lowest level signal possible. For instance if you got a 16 bit quantization of your (audio) signal, e.g. on a compact disk or in a stream, the dynamic range is the ratio of the lowest (non nil) possible soundlevel and the highest possible sound level.

In this case the amplitude of an electric signal is measured. As the power is proportional to the square of the signal amplitude when we express the ratio of two signal amplitudes \(u_1\) and \(u_2\) in decibels we have for the power ratio:

\[10 \log_{10} \frac{u_1^2}{u_2^2} = 20 \log_{10} \frac{u_1}{u_2}\]

So for the ratio expressed in decibel of two signal amplitudes when we increase the amplitude of \(u_1\) with a factor of two the ratio expressed in decibel is increased with the additive term equal to 6.