Tim Wescott

If I may recommend "PID without a PHD" https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pdf

PID Without a PhD [1] is an excellent paper that gives a good intuition for PID controllers. In addition, it has concrete advice on how to go about implementing a practical control scheme, and how to tune the PID constants. It's really my go-to reference for any practical control theory application. [1]: https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pdf

Love this! I had read another paper from Wescott about sampling[1] years ago and loved it, but I didn't realize there's a handful of other great articles[2] that they've published. 1: https://www.wescottdesign.com/articles/Sampling/sampling.pdf 2: https://www.wescottdesign.com/articles.html

http://parlos.tamu.edu/MEEN651/E4.pdf Read page 3. It completely validates what I'm saying. Monty might be a really smart guy, and he might be worthy of lots of praise for a great many things. But in this particular very limited instance, he is not as correct as he could be. If all you can do is link me to the Wikipedia page about this, you're way out of your depth. I took multiple classes on this in college and I raised exactly this issue with my professors. They said that technically I was right, but in practice nobody samples at exactly the Nyquist rate (or even that close to it!) and it's really more of a guideline for the slowest sample rate you could possibly get away with, not the sample rate you should pick once you know your frequency content. Both Shannon and Nyquist were very smart and they've made a gigantic contribution to DSP. But knowing the phrases "Nyquist rate" and "2x the highest frequency" doesn't a signal processing expert make. You can read plenty more here too if you'd like to get more educated. http://www.wescottdesign.com/articles/Sampling/sampling.pdf Here's a snippet: "The theme of this paper can be summed up to this: the Nyquist rate isn’t a line in the sand that you can toe up to with complete safety. It is more like an electric fence or a hot poker; something that won’t hurt you if you keep your distance, but never something you want to saunter up to and lean against." I totally understand if you just want to argue on the internet…

I don't see how either of these papers solidly refute "because Nyquist" in the context of the article and sample rates applied to audio signals. They're both worth considering, though. The first[1], the more theoretical of the two, is focused on digital communications where a Nyquist rate derived from the signal rate isn't enough to recover the signal function. The numerical examples given highlight recovering a sine wave from two, three, and six samples. The millions of samples in a single song might as well be infinite compared to these. The second paper is much more practical and gives a number of real-world examples where naively applying Nyquist causes problems. The first, cutoff filters at half Nyquist to avoid aliasing, plauged many early CD players and other digital audio equipment. The second example is great as well, showing how understanding the signal being measured is important. The third example doesn't really apply to the subject at hand but is a good example of using aliasing for good. The fourth and fifth address the effect filters have on the signal. The sixth example refers to time response in control systems and minimizing phase delay to avoid oscillations. [1]http://www.academia.edu/8412078/Is_The_Nyquist_Rate_Enough [2]http://www.wescottdesign.com/articles/Sampling/sampling.pdf

Obligatory link to PID without a PhD: https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pdf

I think it is you who have misunderstood the Nyquist-Shannon theorem. Aliasing and noise are real concerns. Tim Wescott explains it very well [0] (Figures 3, 10 and 11). If your signal is below one half the sample rate but the noise isn't, you'll lose information about the signal. If your signal phase is shifted wrt. the sampling, you'll lose information. If your sampling period isn't representative, you'll lose information. These are not implementation details. [0] https://www.wescottdesign.com/articles/Sampling/sampling.pdf

I liked https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pdf though I’ve not looked at the OP so I don’t know how it compares.

The linked PDF appears to be an older version (2000), there's a newer one (2018): https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pdf

Did you try this? https://www.wescottdesign.com/articles/pid/pidWithoutAPhd.pdf

Incorrect. Nyquist-Shannon theorem does not guarantee phase! It only guarantees frequency reproduction. This paper illustrates the issue succinctly: http://www.wescottdesign.com/articles/Sampling/sampling.pdf

What's so funny is to see yet another occurrence of basically "because Nyquist" yet fails to address that Nyquist only holds true over infinite time. Over a window of any finite length perfect reproduction is NOT guaranteed. This paper, http://www.academia.edu/8412078/Is_The_Nyquist_Rate_Enough, for one, refutes this. More reading material: http://www.wescottdesign.com/articles/Sampling/sampling.pdf