The audible range is going to be divided into 2^16 discrete levels in order to record the audio signal. A loudness (corresponding to voltage) level must be assigned to each value. Humans hear perceive logarithmically changing levels of noise intensity as a linear change in noise volume. Therefore, a logarithmic scale is used to graph the perceived loudness of a sound as its intensity increases. The decibel (dB) is a measure of logarithmic ratio of intensity level. B is a decibel measure of the intensity level of the sound, I, with respect to a reference intensity level of sound, I0. The formula 4 shows the relationship.
For the reference level, we will use the low threshold of human hearing. This has a value of 10-12 Watts per square meter, and corresponds to a B of 0. Of course, there must be an upper limit to the sound level recorded on the CD-Audio device. There is one major limiting factor after the limitation of only having 16 bits to store the sound level. The first is to keep the maximum volume level from the signal at a recordable level while not losing the small amplitudes of the low frequencies. This imposes a restriction on the quantified level of volume/voltage that will be recorded. Recall the signal is being sampled and recorded digitally. The 16 bit value representing each sampled amplitude is a two's complement binary number. The maximum volume can be normalized to 1, and the minimum volume can be normalized to -1. Then, since a two's complement, 16 bit number can represent integers ranging from -32768 to +32767, each voltage level analyzed will be normalized with the maximum voltage represented by 1 volt, and the normalized value multiplied by 32767 if positive, and 32768 if negative. This value is rounded and then divided by the same number to get a value that is a multiple of 1/65536. This number is a quantized sample of the actual measured signal sample. The error will be less than detectable (we believe).
With sound, the intensity level is linearly proportional to power. Power is proportional to the voltage squared. If the voltage produced by the sound on a recording instrument is to be used to measure the sound, then we need to find the range of voltage levels to record. The logarithm of x^2 is equal to 2*log(x). 90dB (B) corresponds to an intensity level of 1.585 *10-3 (I) for an I0 of 10^-12 shown if Formula 5.
(I) = 10^(B/10) * I0
Now, the range of intensity is related to the range for voltage in the signal, since the square of the voltage is proportional to the power of the sound. Formula 6.
Where P is the power in the sound, and r is the distance from the source to location where the source is being measured with an intensity, I. Since the intensity of the signal is producing a voltage on a measuring device, we actually want the range for the voltage, not intensity, but the range is still logarithmic, and can be scaled as desired to match the equipment. If the intensity doubles, the power of the signal doubled. If the power doubled, the voltage on the measuring device will increase by a factor of sqr(2).
Max intensity - Min intensity = 1.585*10^-3 - 1*10^-12 ~ 1.585*10^-3.
Max Voltage - Min voltage = (depends on the minimum voltage level recorded)
Assuming a minimum voltage level corresponding to Formula 3-1 with a radius to the recording device of 1 cm and a recording resistance
of 1 ohm, the minimum magnitude of voltage is 3.545 *10^-8 volts. The
maximum voltage at 92 dB is given by Formula 7.
Where V0 is the minimum voltage for reference. The maximum voltage magnitude, then, is 1.411*10^-3 volts. The voltage range is effectively 2.822*10^-3 volts. Divided into 65,536 discrete values of equal distance, we get approximately 4.307*10^-8 volt. Every measured value of the signal's amplitude must be an integral multiple of this, from zero to the maximum voltage recordable. For simplification, we will use a discrete value of 4.3*10^-8 volt per division.