Copyright © 2013 by Wayne Stegall
Updated May 24, 2013. See Document History at end for
details.
Analog is Digital
Quantum mechanics defines quantization of
analog
signals comparable to that of digital audio.
Introduction
Even though analog signals are thought of as continuous, we know that
some point near the atomic level that they become granular. In
spite of this suspicion, we still disregard this idea because we think
these affects are infinitely small. It comes as a surprise then
that physics has already defined this analog granularity in specific
concrete terms at least for electronic and magnetic systems.
After establishing that electromagnetic radiation
is emitted in discrete packets know as photons, physicists determined
that all electromagnetic energy was composed of photonsized energy
quantums. Thus Plank's equation which defines the energy of an
individual photon also defines discrete levels at which analog energy
is quantized. This is likely related to the fact that electrons
move through conductors as waves as well as particles just as photons
do in transparent mediums.
Plank's Equation
where E is a discrete packet of energy,
h is Plank's constant of 6.62606896e34,
The Greek letter nu (ν) represents frequency in this branch of
physics. (I will use the conventional f for frequency from here
on.)
This equation represents a spectrum of quantization noise that rises in
direct proportion to frequency. The noise corner for quantum
noise is at kT/h or 6.25099THz at a temperature of ≈ 80°F. Below
this frequency quantization
is masked by white noise. Above this frequency white noise rolls
off with an exponential lowpass function and quantum noise takes over
and dominates.
The Real Resolution of Analog
The first deduction to be made is that analog resolution is not
determined by the thermal noise floor as is commonly thought by those
who compare it unfavorably with digital. The quantization level
is well below the thermal noise floor for all frequencies relevant to
audio. It might be informative to calculate white noise power and
quantization noise in the audio band to see what the actual difference
is. First calculate white noise.
(2)

P_{whitenoise} =

4kTΔf = 4 × 1.3806504e23
× 300K × (20kHz – 20Hz) = 3.31025e016 W

or –154.801dBW
Calculating the noise integral for white noise equated to just
multiplication because its spectrum is flat. Quantization noise
provides no such shortcuts. Therefore the full integral must be
solved.
^{1}
(3) 
P_{qn} = 4

∫ 
20000
20

hf 
df 
(4) 
P_{qn} = 4


20000
20

hf^{2}
2 
(5) 
P_{qn} = 2h


20000
20

f^{2} 
(6) 
P_{qn} =
2 × 6.62606896e34 × (20000^{2} – 20^{2}) =
5.30085e025 W 
or –242.757dBW
Therefore the quantization floor representing the actual resolution of
an analog signal is 242.757dB – 154.801dB or 87.956dB below the white
noise floor.
Comparison of Analog and Digital Processes
Next, interest seems to lead to comparing the quantization of analog
signals to some appropriate modern digital reproduction
chain. It will become quickly obvious why this comparison
should be made to DSD.
 The shape of the noise floor of both analog noise and DSD follow
the same pattern: To remain flat through a region and then rise
in a higher frequency region where the quantization process tends to
place the noise. (See noise
profiles in figures 1 and 2)
 The process by which the signal moves from input to output is
similar: Analog energy moves from the input to the output as
discrete quantum units of energy are available to move.
Similarly, input energy to a deltasigma modulator does not move to the
output until it can move in clockduration output pulses. The net
effect of both is that the energy density of the output represents the
input energy and quantization is filtered out by some lowpass
functionality in the receiving system.^{2}
 The claim by a physicist that the resolution of an analog system
can be increased by filtering out outofband quantum noise leaving
lower level inband quantum noise seems parallel to the idea of
converting DSD to PCM by means of a digital lowpass filter.^{3}
Figure 1: Analog noise
profile in a 1kΩ resistor. 

Figure 2: Noise floor of a hypothetical 2ndorder analog
deltasigma modulator.


Conclusion
The fact that analog is quantized at some low level boosts the idea
that increasing the resolution of digital audio systems holds real hope
to sound more analog.
^{1}Multiplication of the basic
noise equations by a factor of 4 for both thermal and quantization
noise seems to combine two factors of 2: one from adding negative
and positive frequencies, and the other to compensate for the fact that
only half of the noise power a resistor can be communicated into a
matched load.
^{2}See article DeltaSigma Modulators.
^{3}Jack Bybee is the physicist who makes
this claim for a product that he will sell you.
Document History
May 24, 2013 Created.
May 24, 2013 Corrected format of equation 3 and renumbered
equations.