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| Letters -- March 2005 "Is the Brain an Analog or a Digital Device?" March 15, 2005 To Ross Mantle, I am enjoying your "Opinions." Keep them coming. Jon Digital brain March 7, 2005 To Ross Mantle, In your March "Opinion" you stated, "Theoretically, you can sample the real wave as often as you like, and make your digital representation as close to the real thing as the capacity of your storage medium allows." This is not quite right. In theory you could actually recover the signal untouched if you sampled it at any sampling rate greater than twice the higher frequency present in the signal. This is called the Nyquist rate. Humans hear from 20Hz-20kHz. This means if you had an ideal sampling system, you would need to sample at a frequency higher than 40kHz to be able to accurately represent the data. So in theory, 40,001Hz would be a perfectly acceptable sampling frequency under ideal circumstances. We know that theory is one thing, and then in practice there are other considerations (like the fact that there are no perfect samplers, there are timing errors, etc.), so this is basically a simplification of the real problem (before I get other engineers jumping on me). You know, even though I probably will never buy any of the stuff you guys review on Ultra Audio (I can't afford it), and even though I disagree with you in particular about a lot of things, especially with the whole "musical voluptuousness" opinion, every couple of weeks, I find myself wanting to read all the articles published here. You guys are doing a great job and make for very interesting reading. Have a good day, and keep up the good work. Felo You refer to the Nyquist frequency, which is half of the sampling rate. The relevance of this is that the signal being sampled must not exceed the Nyquist frequency or aliasing will occur. The classical example of aliasing is car wheels appearing to turn backwards when they rotate faster than half the frame rate of the camera. In the case of audio, you are making the traditional argument, which is that a sampling rate of greater than 40kHz will satisfy Nyquist, since the human ear is insensitive to frequencies above 20kHz. As you mention, there are other considerations. One of these is that inaudible frequencies above this cutoff will be aliased downward into the range of human hearing and interfere with what is heard. This makes filtering a necessity. Further, the traditional argument depends on the assumption that the frequencies we’re talking about are sine waves and that the ear is a Fourier transform-type analyzer. Other theories posit that the ear may analyze sound on the basis of "wavelets" or "wavefronts" and not Fourier transforms. These theories imply that the ear can detect the shape of a wave directly, without Fourier analysis. At a sampling rate of 40kHz, there will be only two data points to describe a 20kHz signal, so a sine wave would be rendered as an up-and-down step on a graph (i.e., a square wave). This kind of distortion, if audible, could be a source of the hardness people experience in the sound of CD. Mathematically speaking, you can indeed sample a wave as often as you like to get a digital approximation, which is arbitrarily close to the real thing, as I said in my opinion article. Practically speaking, current high-resolution formats (DVD-A and SACD) are based on sample rates much higher than required by the Nyquist theorem. To me, it’s pretty much undeniable that they sound better than CD. One might even say they sound more "voluptuous."...Ross Mantle
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