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I wasn't talking measuring 0-60 time in cars. I was talking about measuring steady 60mph. No one does that, because it's useless (although it would be easy to measure). 0-60 time would correlate somewhat to impulse response in audio, which is a useful measurement, and much more rarely seen.

I'm reminded of an experimental example a professor showed us in my college audio recording class. He hooked a square wave generator up to both analog tape and digital recorders, and recorded the square wave. Then he played back both sources on an oscilloscope. The tape had a sharp lead spike followed by a shrinkage on the tail of the square, from compression. The digital recording had a rippled top, from frequency-related distortion. Both were inaccurate, but they were inaccurate in very different ways.

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You can't accurately reproduce a square wave by using a Fourier series even mathematically in the infinitely wide frequency response limit, ignoring all physics:

https://en.wikipedia.org/wiki/Gibbs_phenomenon

Why are we talking about Fourier series? Uncompressed digital audio doesn't need them, it just records the raw voltage level samples. You still can't produce a perfect square wave, but there's no reason you shouldn't be able to reproduce whatever wave was provided as input (which by definition was produced by something), other than hardware limitations of the playback system.

> Why are we talking about Fourier series?

Because all real systems are bandlimited.

You are quite wrong about that statement. As the other comment mentioned, both analog and digital systems are bandwidth limited and can not reproduce infinite slopes like in a square wave. Fourier expansions are one of the easier ways to study bandwidth, even if there are other formalisms.

But that's the analog system which is band limited, not the digital! The problem is poor analog components after the digital decode phase in this case. A time domain digital representation absolutely can represent a perfect square wave. (There are other waves it can't represent.) That's completely different from a digital encoding that causes ringing in the square wave.

A DAQ that reconstructs a perfect square wave (or a perfect stair-step function) is employing a "Zero Order Hold", and it would make a god-awful audio DAC.

https://en.wikipedia.org/wiki/Zero-order_hold

With a 0OH you'd gain the ability to potentially reconstruct a infinite bandwidth signal (the squarewave), which is irrelevant in the audio case: With a 10kHz squarewave, the next harmonic would be at 30kHz which is already waaaay out of the human range of hearing.

But limiting the bandwidth of a DAC (or ADC) is the fundamental property which causes it to be able to perfectly reproduce the frequencies within its bandwidth (~up to 20kHz in HiFi Audio), so that's desirable. And the "ripples" on top of a squarewave are just the manifestation of this property: If you cut away the frequencies above Nyquist, you get a rippled squarewave. And conversely if you compute the difference between the perfect squarewave and the "rippled" squarewave you get out of an Audio DAC this only contains (non-audible) energy outside of the bandwith of the DAC!

Sure. But you're still talking about the analog portion of the circuit. If the analog output of a digital recording - supposedly the input was a near-perfect square wave - is different from the analog output of an analog recording, and neither looks like the input signal, then it is a) certainly possible to make an analog output stage that produces a more precise output that better matches the input, and b) possible to make an output that better matches the analog output. Remember, the input was supposedly a perfect square wave, and contained inaudible components. The recording/playback component had nothing to do with the fact that you can't hear the entire spectrum.

All the limitations are in the analog phase. As you point out, it depends on the design tradeoffs in the DAC, amplifiers, etc, and that's an important lesson to learn in the class that was being taught. Nevertheless, the point I was replying to is the claim that the digital representation could not represent a square wave. That's certainly not true, and no Fourier transforms are necessary to demonstrate it. A PCM recording is just a series of impulses, not a series of sine waves.

Absolutely wrong.

You should take a look at https://www.youtube.com/watch?v=cIQ9IXSUzuM to get a better idea of how digital audio works.

Inaccuracies visible on an oscilloscope might not be audible to humans. Try applying a 10Hz high-pass filter to some music and comparing the waveforms. There's obvious visible changes but it sounds identical.

True. And it certainly doesn't explain "better" or "worse". But it explains why music recorded on tape sounds different from music recorded digitally. And it puts a torch to the claims that digital is somehow "accurate", and all we like about analog is just evil bad naughty distortion.

Square waves are impossible to reproduce because they have infinite numbers of frequencies, but you don't listen to square waves. You listen to music with all its content below 20KHz and anything above that doesn't matter.

The only non-ideal thing about music is that it's not band-limited because time matters - that's why compressing music with a lot of cymbals etc smears it across time.

More simply, square waves are impossible to reproduce since they would require infinite velocities

What, you don't like it when your speakers cones blow a hole through the wall and possibly the neighbor's roof on their way to Saturn?

This is why you low-pass audio before quantizing. If you put the square-wave through a low-pass filter before recording it digital and on high-quality tape, the digital recording will be more accurate than the tape.

This was a digital recording onto a professional digital system. It was put through a Nyquist filter.

Then again, is "accurate" in some mathematical sense the right term? Which one sounds more like music? Among audio professionals, there's a broad consensus that tape sounds better - enough so that there is a strong market for digital plug-ins that emulate the (mis)behavior of tape.

If a pure-digital master can emulate tape, then isn't it superior to tape?

(I know that many analog systems have soft-rolloff allowing you more leeway in setting your headroom, but modern ADCs have so much dynamic range that it's a poor operator who doesn't allocate themselves sufficient headroom).

I realize you weren't talking about 0-60 time, it was an analogy I introduced. Not meant to be technically similar, only a similar fraction of the whole. I think "driv[ing] in a straight line at 60mph" is a smaller fraction of car performance than frequency response of headphone performance, and 0-60 time more similar.

I was talking about measuring steady 60mph.

Actually, this is kind of a thing: speed measurement and reliability for cruise control. We can get close enough to be useful and quite comfortable. However, most cars don't directly measure their own speed anymore. Instead, it's inferred from other measurements.

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