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The Nyquist/Shannon theorum is one of the most mis-quoted and misunderstood concepts in audio!

24.214.89.70

Posted by TimNaff on April 13, 2001 at 23:11:56

In Reply to: interpolation posted by PaulN on April 13, 2001 at 13:53:43:

The theorum does not state that if your sample frequency is double your signal frequency, you will get all the information. In fact, it is trivial to prove by simple example (e.g., sample phase dependence when sampling at double the waveform frequency) that such is not the case. The theorum says that you will need to at least double the signal frequency to get the sample frequency.

As many have rushed to point out in response to my post, getting more samples makes it easier to generate an analog signal. I said that in the post, but lots of folks needed to say it again, with references to "brick wall" filters, etc.

I was hoping that someone might actually be able to offer a definition of upsampling and oversampling that would help clarify the picture. Given that no one did, I have to conclude that we, collectively, don't have an answer and that the audio gurus in the ivory towers at Stereophile and TAS probably don't either. Someone did mention “zero stuffing” as a definition of oversampling. I’ve heard that one too; I find it hard, however, to justify calling that “sampling.” It is bothersome, though that our little focus group seems to have a better grip than the Stereophile/TAS "experts."

Back to the sampling theorum. It is reasonable to ask just how many samples really are needed to cover a given analog upper frequency limit. And I'm not talking about D-to-A conversion facilitation; I'm talking about digitally capturing information about an analog waveform. Let me offer an experience that I had about 25 years ago, when digital computing was much younger. I was looking into the jamming susceptibility of a well-known US missile system. My assumption was that the threat would not know the exact frequency of my receiver, and would sweep his jammer frequency using a saw-tooth modulation of the frequency. (In other words, employing a rapidly repeating cycle of starting the jamming frequency at a lower frequency, f1 and sweeping to a higher frequency, f2.) What I wanted to do was to determine whether such a ploy would really fill up the f1-to-f2 bandwidth. What I did was to sample this analytically generated function and then do a digital fourier transform (FFT algorithms were around then) to see what it looked like in the frequency domain. If you aren't familiar with this jargon, think of it like this: I was generating a plot of the average signal amplitude versus frequency when the threat used his swept (or “chirped”) jammer. The plot that I generated versus frequency started up around f1, went through some wiggles, and came back down around f2. I was concerned with the wiggles: did it drop to zero anywhere, which would make the jammer less reliable?

The only part of this that is interesting is what I learned about the sample rate that I needed. When one is doing this type of analysis, and wants to find the minimum sample rate that will work, he can generally increase the sample rate until the result no longer changes. I used to do the same thing routinely with digital simulations of dynamic systems.

Here is what I found. When I used the so-called Nyquist criterion and used a sample rate of double the highest frequency (2 x f2), I got a totally erroneous result. I had to increase the rate to almost 50 percent higher than the Nyquist criterion number to get the result to stabilize. When it stabilized, it was a recognizable symmetric pattern in the frequency domain that could be verified theoretically. It is important to note that I was very careful to use a number of samples consistent with the requirements of the algorithm to avoid introducing artifacts. This was a very clean generation of samples and a very clean interpretation of samples. Doubling the highest frequency not only did not capture all of the information as perceived by the transform algorithm, it wasn’t even close. The transform fell totally apart in the upper half of the frequency spectrum. At the time, I made a mental note that the Nyquist criterion was not a practical guide for real-world applications. Since the advent of digital audio, I’ve discussed it with a lot of smart folks who, based on either similar experience or theory, agree. The ones that cite the theorem as gospel fall into the same trap that engineers so often do: taking some standard formula or rule of thumb out of context.

Now, I have to say that I think the situation in audio signal reconstruction is much more complex than in my example. Most adult hearing rolls off rapidly somewhere between 10 and 17 Khz. So, if my example applied, and that’s all there were to it, we might be able to get by with Redbook CDs – for some of us older farts at least. It may be that enough information is there, but we need more samples to make up for errors in extracting the info. It may be that our brains respond to higher frequencies than we consciously hear. It may be that the problem has more to do with the D-to-A conversion, including filtering baggage. The important thing is that scientific, double-blind tests indicate that digital artifacts are audible in Redbook CDs and that SACD and first-class analog sound less synthetic. Nothing is less scientific than spouting the results of some calculation as gospel and ignoring scientifically collected empirical results!

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