Why Vinyl Won — Yet Another Possible Explanation

Nov 1 20:37:01 View QTML

Abstract. A possible explanation for why audiophiles prefer vinyl: beats and other distortions due to not completely applying Nyquist's theorem.

Update 11/20/2023: Upon showing this to the big brain types who read Astral Codex Ten the concern described below may well be already taken care of by oversampling. It depends on how many delay lines there are in the Finite Impulse Response filter used for the upsampling. I have further research to do... (My prior research on oversampling didn't yield a good description of the process. The Internet has gotten better since then.)

Snap! Crackle! Pop! The venerable LP is now outselling the compact disk. How did that happen? Vinyl records are subject to scratches. They take up space. They warp. They need to be flipped. The needle skips if you dance too close to your record player.

Some possible explanations:

• Those square foot cardboard envelopes are an excuse for groovy artwork.

• Maybe there is a sense of ceremony that comes from putting a record on the turntable.
• Or maybe vinyl records sound better.

Over the years I have seen quite a few explanations for why vinyl records might sound better:

• The brick wall filtering applied to music before digitization (needed to prevent aliasing) produces a coldness.
• "Jitter." Something about the timing of the digital pulses in the D/A converter not being even.
• The low pass filtering needed for the output in order to avoid harmonics of the sampling frequency.
• The "warmth" of vinyl comes from the fact that it is less accurate than CDs. People like the imperfect frequency response.
• When CDs came out, producers compressed the volume range of music; i.e., the "Loudness Wars."

Here, I am going to suggest another possible explanation: beats and other distortions from not fully applying Nyquist's Theorem.

Beats Below the Nyquist Limit

Nyquist's sampling theorem states that if a signal is filtered so that there is nothing in the signal above the critical frequency, then it is possible to perfectly recreate the signal if you evenly sample the signal at two samples per period of the critical frequency. That is,

where $\Delta$ is the time between samples and$f_c$ is the Nyquist critical frequency.

For example, if we set $\Delta$ to be 1, the cosine of the critical frequency is sampled as follows:

The green curve is the actual function. The red dots are the sampled values. Eyeballing the picture, one can envision an interpolation function which takes those points and converts them back into the original cosine function.

But what if we do a phase shift? Behold the sine of the same frequency!

How do you interpolate a bunch of zeroes into a sine wave? And what should be the magnitude of that wave? This question bugged me greatly when I first encountered Nyquist's sampling theorem many years ago.

The answer is: Nyquist's theorem applies below the critical frequency. So let's dial down the frequency to 97% of the critical frequency:

Hmmm, any kind of localized spline is going to produce some serious beats! And we see the same thing for the cosine:

The reason why we can theoretically recreate the original signal is that Nyquist's theorem uses a very nonlocal spline! From Numerical Recipes:

The weight function for the sample at time equals zero is:

This function is zero at all other the sample points, but in between it is nonzero and only drops off as $\frac{1}{t}.$ (And, by the way, in order to have a sine wave that is close to the Nyquist frequency, it needs to run a long time and die off gently in order for the envelope to not broaden the signal beyond the Nyquist frequency.)

I know of no analog filter that would get rid of the beat artifacts seen earlier — other than filtering out the signal as well. To truly recover the full signal requires applying sinc functions from quite a few points to interpolate between the samples.

How many? I haven't done the experiment. I would note that for audio, the amplitude of the signal can vary by orders of magnitude. Remember that we have bass note fundamentals mixed in with the high harmonics we are trying to recover. We might have to go out thousands or even tens of thousands of points before the series above converges sufficiently.

Update 11/20/2023: OK, my understanding reading magazine articles back in the day was that A/D converters generated a jaggy curve which was then filtered by an analog filter to take out the jaggies. Upon finally finding a resource on how oversampling works, maybe the Finite Impulse Response filter used for oversampling approximates the Nyquist formula sufficiently. I need to do more research.

To strictly apply Nyquist's theorem, we should simply apply the convolution above to an entire track, to have at least one interpolated point between each sample point, and then have the analog filter working well outside the audible range. This is feasible for some types of players, this is not an acceptable solution for streaming. Exactly how much buffering would be required to up the quality of streaming audio is unknown at this point. I doubt that it is more than a second since the denominator hits the maximum range of a 16 bit integer in around 10,000 steps.

(And, by the way, computing over an entire track is an order$N\left(log\right(N\left)\right)$ job. I leave it as an exercise for the reader to figure out how.)

A Significant Problem?

If these beats are only significant when we are close to the Nyquist limit, then the simple solution is to just have an analog filter that trims off the top of the theoretically recoverable signal and let those with high frequency hearing buy vinyl LPs.

Let's look at the plots for some lower frequencies and use our eyes to estimate where the need for digital interpolation ends.

It appears to me that we need to get down around half the Nyquist frequency before we have enough points that a local spline would recover something close to the original function. Exactly how well an analog filter would behave like such an interpolator is unknown to me.

But if I were to numerically interpolate, I would like to compute more than one point between each sample and then have the analog filter operate well outside the audible range.

I do not know what A/D chips are available for taking advantage of such an interpolated string of numbers.

Conclusion

Unless modern CD players are playing some tricks I don't know about, digital music introduces significant spurious beats for the entire top octave of human hearing.

Whether people can hear these beats is unknown to me. We are talking about a range that is just for upper harmonics, snare drums, and crickets. But I do seem to recall audiophiles using the term "harsh". And a rapidly varying envelope can be harsh.

Tags: music nyquist frequency audio compact disk