Signal Amplitude Units

• Recorded signals and/or their mathematical representations correspond to amplitude values over time. If we record an audio signal with a calibrated microphone, we can associate specific pressure values (for example in units of Pascals) with each signal instance in time. In most instances when dealing with digital audio signals, however, the explicit reference to physical units is dropped.

• While digital audio signals can be represented using either fixed- or floating-point numerical formats, it is nowadays most common that audio signal levels are represented with floating-point numbers and constrained within a range of $\pm 1.0,$ at least when coming in from or being fed out to audio hardware systems.

• Sound signal amplitudes can vary over wide ranges. For example, the pressure “threshold of pain” of our auditory system is $10^6$ times greater than the “threshold of audibility.” Thus, it is sometimes convenient to express amplitudes in logarithmic units called decibels.

• The logarithmic expression $y = \log_b(x)$ is the inverse of the exponential expression $b^y = x$. The value $b$ is referred to as the “base”, which is chosen to be a positive real number. The most common bases for logarithms are 10, $e$, and 2.

• A decibel (dB) is defined as one tenth of a bel (so there are 10 decibels in a bel). A bel is an amplitude unit defined as the $\log_{10}$ of sound intensity relative to some reference intensity level. Since signal intensity, power, and energy are proportional to the square of signal amplitude, amplitude relationships in dB are given by $20 \log_{10}$ of sound amplitude to a reference amplitude.

• Signal amplitude changes by linear scale factors of 2, 1/2, 1/4 and 10 will correspond to decibel level changes of 6, -6, -12 and 20 dB.

• If the “theshold of audibility” is used as the reference amplitude, the “threshold of pain” would correspond to a level of 120 dB.

• In signal processing, the maximum signal amplitude is typically chosen as the reference amplitude. In other words, signals are normalized so that the maximum linear amplitude is 1, which corresponds to 0 dB and levels around -60 dB are considered inaudible.

• Another common reference amplitude is $10^5 = 0.00001$, in which case a maximum linear amplitude of 1 corresponds to 100 dB and 0 dB is considered inaudible (giving a convenient dB range from 0 to 100).

  Figure 2: Linear to decibel scales for different reference amplitudes.