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Great article! So that students of the subject can better knit their knowledge together, it might be worth mentioning here that your analysis of the R-2R DAC is an application of the Thevenin equivalent. Anyway, thank you for a clear explanation of this topic!

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ADC/DACs are still relevant, but I'd say digital is eating the world! Your two analog examples are actually great use-cases for all digital signal chains.

For image capture, you get much better dynamic range and lower noise if you capture pixels as single bits integrated over time. So, for example, you charge up a pixel array and then check all the pixels periodically to see which have flipped. Brightly illuminated pixels flip quickly. You get to pick how long you want to wait. At the limit, you can have pixels that are single photon detectors/counters (these actually exist today, just too low res for cameras... for now). Light itself is inherently digital, better to process it in its native domain! :)

The most efficient and lowest distortion way to drive a speaker is similarly digital. You basically send a series of bits at a much, much higher frequency than the highest audio frequency thought the entire amp chain and then let the inductance of the speaker coil integrate all the bits into the movement of the cone. Because the amps are driven rail to rail with very well defined edge transitions, they spend as little time as possible in their noisy and power hungry non-linear region. Here is also no noise to get amplified- you just get out much stronger versions of the exact same bits you put in.

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Feb 20·edited Feb 21Author

Fair! I mean, in some sense, we crossed that bridge long time ago with 1-bit delta-sigma ADCs and DACs. Whether the integration happens on the PCB or in the speaker coil is just an implementation detail - fundamentally, the conversion process is digital either way.

As to your examples - I know about time-to-first-count sensing (it's a popular paradigm for ionizing radiation detection with Geiger-Mueller tubes!), but I'm curious about your assertion that driving a speaker with (non-smoothed) 1-bit pulses is the lowest-distortion method of generating audio. I'd expect it to cause serious problems due to the speaker being a fairly nonlinear mechanical device (variable inductance / back-EMF due to moving coils, mechanical resonance, etc) - not to mention RF interference issues of sending high-frequency, high-current pulse modulation down a long run of speaker wire. Are there any hi-fi implementations of this?

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>Whether the integration happens on the PCB or in the speaker coil is just

>an implementation detail

But an important implementation detail! The longer you can keep the audio signal in the digital domain, the higher the fidelity of the resulting output.

If we take the limit, the human auditory system is itself a natively digital processor. There are a finite number of neurons in the cochlea, and each generates a digital pulse-density-coded signal for its frequency range. In the not too distant future we will be skipping the speaker along with all the non-linearities in the air + outer/middle ear system, and end up with the highest possible fidelity by going all digital, end to end!

>I'm curious about your assertion that driving a speaker with (non-smoothed) 1-bit

>pulses is the lowest-distortion method of generating audio

From first principles, we know that all analog amplifiers will amplify any incoming noise along with the incoming signal, and real-world components and transmission lines will add meaningful amounts of noise and distortion to the analog audio signal. There are hard, physical limits to how low you can get this noise in an analog system. Think about amplifying a high impedance 100KHz, 1V peak-to-peak analog audio signal into an 8 ohm, 50 watt speaker load. Even a tiny 0.1% noise/distortion on that incoming signal or in the amplifier is going to be *huge* and very audible in the output signal.

The beauty of moving the signal into the digital domain is that you get to pick how you use the available bandwidth to reduce noise and distortion, and there is a *lot* of available bandwidth. Now imagine that same 100KHz audio signal except now it is quantized into a 1 gigabit/s digital signal. With trivial current technology, we can move that signal along kilometers of transmission line with effectively zero bit error rate, and with a class I digital amplifier we can amplify it with exceedingly low and fixed distortion (mostly limited by transistor switching speeds and power supply quality, and both of those are mitigatable with technological process- and we are already far,far along this curve).

Also, as a practical matter all real-world hi-fi amplifiers today are built from transistors (silly tube amps notwithstanding!) and transistors have the physical property that they are very efficient and accurate when driven in cutoff or saturation mode (either on or off) and are very inefficient and relatively noisy and nonlinear in active mode (where output is input times some coefficient). As long as you use transistors that can switch high enough that they drive all of their artifacts above the audible range (and current transistors can easily switch much, much faster than this) then the best digital amp will be higher fidelity than the best analog one since, again, digital basically uses the extra available bandwidth to increase SNR whereas analog has no good way to do this.

>I'd expect it to cause serious problems due to the speaker being a fairly nonlinear

>mechanical device (variable inductance / back-EMF due to moving coils, mechanical

>resonance, etc)

Speakers are limited quality transducers no matter what is generating the driving signal. So the question is, can you do a better job compensating for their limitations in the analog domain or the digital domain?

In the analog domain you have very limited compensation tools at your disposal. You can keep adding gross filters and crossovers to the driver, but because each time you add components you are also increasing noise and distortion, in practice you are very limited how much correction you can do before the cure ends up being worse than the disease.

In the digital domain, you are limited only by processing power - and current hardware is more than capable of applying any arbitrary compensating transform to the output signal. We are lucky to live in a world where a $20 DSP can completely characterize and compensate out any real world speaker in real time.

> not to mention RF interference issues of sending high-frequency, high-current pulse

> modulation down a long run of speaker wire.

You are stuck in analog thinking!

In the analog world, we must put the amplifier as close to the signal source as possible and then send the amplified high current signal out over the long transmission lines (speaker wire). If we tried to insead send the low power signal out on the transmission lines, the noise that we picked up on those lines would be large relative to the signal (as anyone who has been to a middle school assembly with a long, analog mic wire knows first hand).

But in the digital world, we can send the low power signal without worrying about it picking up noise on the way. We can easily send 1 gigabit/s over a balanced twisted pair out to our speakers with astonishingly low EMI and effectively no loss in signal fidelity. We can then do the 1-bit digital amplification inside the speaker, directly next to the voice coil, inside a shielding cage, with very low EMI.

>Are there any hi-fi implementations of this?

About 20 years ago the technology crossed the threshold where it is was possible to make a better digital amp than the best analog amp for voice coil-based hi-fi speakers. Many products were released to *gushing* reviews. Google "TACT Millenium amplifier" to read some of them.

But then the tech fizzled out. My personal theory is that people who buy audiophile gear actually are not actually looking for the highest fidelity they can get. They are looking for a luxury good, and "digital" for some reason has low quality connotations in this context. How else do you justify people shelling out $10,000's for tube-based amps that add effectively random and even unpredictable distortions to their music? Because the warm glow of the tubes adds warmth to the sound! :)

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Hal Chamberlin mentions that a 24(?) bit DAC built from discrete components would lose its linearity just from the effect of a lit cigarette held a few inches from the resistors. (That’s my recollection, anyway, I read his book long ago)

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Yeah, I can believe that - although I imagine it would be drowned out by RF interference, Johnson–Nyquist noise, and so on before you can even measure it.

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