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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 photon-sized 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
(1)
E = hν
where E is a discrete packet of energy,
h is Plank's constant of 6.62606896e-34,
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 ≈ 80F.  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)
Pwhitenoise =
4kTΔf = 4 1.3806504e-23 300K (20kHz 20Hz) = 3.31025e-016 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) Pqn =  4
20000


20

hf df

(4) Pqn =  4
20000


20

hf2
2

(5) Pqn =  2h
20000


20

f2

(6) Pqn =  2 6.62606896e-34 (200002 202) = 5.30085e-025 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.


Figure 1:  Analog noise profile in a 1kΩ resistor.
noise-analog

Figure 2:  Noise floor of a hypothetical 2nd-order analog delta-sigma modulator.
noise-dsd


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.





1Multiplication 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.
2See article Delta-Sigma Modulators.
3Jack 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.