The October 2005 issue of “Hi Fi News” contained two letters (emails) from readers commenting on the “Good Resolutions” article which appeared in the April 2005 issue. Unfortunately, one of the letters (from Jim Carlyle) contained a number of serious errors and misunderstandings. These seem to be typical of the kinds of misunderstandings that arise surrounding the whole topic of digital sampled signals, so I though it would be of general interest to deal with them here in some detail. I hope this will help others not to fall prey to the same misunderstandings.
The letter from Mr Carlyle contains three specific assertions which should be corrected and clarified:
- Jim Lesurf's exposition (Hi-Fi News, April 2005) on the LP/CD resolution controversy was absorbing but he doesn't seem to acknowledge that the CD noise floor is granite, whereas the LP background noise can have the music still in it.
- He also blithely assumes that details quieter than noise are inaudible.
- Luckily stereo LP surface noise is mainly incoherent, and the pre-amp noise is totally incoherent (except for any common hum), but your CD player's DAC's granite noise floor is 100% coherent at the sampling frequency!
I will consider each of the above assertions in order.
a). I’m afraid that Mr Carlyle does not define his use here of the term “granite”. So far as I know it is not a standard engineering term relevant to this area. Given the lack of a definition I’d agree that I did not acknowledge that “... the CD noise floor is granite... ” because – so far as I can tell – this statement has no defined meaning at present to anyone other than Mr Carlyle! I am afraid that I feel no guilt for failing to acknowledge something which is either undefined or meaningless. If My Carlyle had defined his terms I might be able to do otherwise.
I would, however, say that – provided they are correctly recorded and reconstructed – digital systems including CD audio also have the property he ascribes here to LP. This means that that CD background noise can also, “have the music still in it”. On that basis CD and LP have similar behaviours if correctly recorded and replayed. Mr Carlyle’s implication that CD differs from LP in this respect is simply false. I will say more about this, lower down this webpage.
b). I’m afraid this assertion is simply an error on Mr Carlyle’s part. I did not say what he claims. I did not assume it. Nor did I imply it. Indeed, I do not think it. I am afraid that the “assumption” here is on Mr Carlyle’s part, and is based on a misunderstanding of what I actually wrote. This misunderstanding on his part is unfortunate, particularly as I think that – even if he misunderstood the wording of the article – if he’d checked and read the references it should have been obvious to him that he was making an error.
c). With this assertion we face a similar problem as with (a). This is that the terms Mr Carlyle employs are either undefined, inappropriate, or misused. Once again the undefined term in this context, “granite noise floor” is employed. The assertion that this is, “is 100% coherent at the sampling frequency” may therefore simply be meaningless. However if we ignore the word “granite” there are still problems with Mr Carlyle’s claim.
The sampling frequency for CD audio is 44,100 samples/sec. If the input when recorded was correctly dithered and noise shaped, and then correctly reconstructed, the resulting noise floor will show no coherent relationship with 44·1kHz. Indeed, if the system obeys the Sampling Theorem, neither the input for sampling, nor the output after reconstruction should contain any energy at this frequency, and the noise in the audio band will display the statistical properties we expect for random noise.
It is easy enough to check the above statements either by referring to standard texts on information theory, or by simply creating a correctly dithered input for sampling, and then performing a spectral analysis of the results.
It is hard to be sure given the undefined and ambiguous wording Mr Carlyle used, so I can only speculate as to what he might have in mind. For example: it is possible that he is thinking that a reconstruction that does not completely suppress artefacts outside the Nyquist bandwidth will permit out-of-band components that have a coherent or deterministic relationship with those in-band. However if this is what he has in mind, then his wording does not actually say what he means. Nor is is clear what audible significance (if any) such out-of-band images may have. If this is a concern, then it may be that we should also worry about out-of-band intermodulation products with LP. Either way, though, the question of any audible effects of nominally ultrasonic artefacts was not the topic of the article in question (although it was relevant to an earlier article on hearing, etc.)
In contrast to Mr Carlyle’s letter, I essentially agree with everything that Mr Geiss writes in his letter.
Mr Geiss makes two entirely valid points.
- That the use of sampling rate conversions during the recording and CD production process may lead to audible differences.
- That the ability to detect a signal below noise may be enhanced when the signal pattern extends over many samples.
Both of these points are quite consistent with the article. Indeed, some time before “Good Resolutions” appeared in Hi Fi News I wrote some webpages covering (b). These are at in a dither and show that CD systems are quite capable of providing signals whose power level is below the wideband noise floor. These pages were referenced in the published article, and are the reference I have mentioned elsewhere. However from the content of their letters I have the impression that neither Mr Carlyle or Mr Geiss read this reference. If they had done so, I suspect their letters to the magazine might have been different!
Although agreeing in general with him, there is one comment that Mr Geiss makes which I think may be usefully clarified. He says that:
- The Shannon theory quoted in the feature is correct concerning the ability of any system to detect a difference on a sample by sample basis. However the audible sounds that we listen to have a structure that straddle many samples.
Although I agree with the second sentence of the above, the first deserves some attention.
I think that Mr Geiss is referring to the Shannon Equation. This is the expression generally used to quantify the information carrying ‘capacity’ of a channel of communication. This tells us, in units like ‘bits per second’, the maximum rate at which information might be able to flow via a given channel. The actual amount of information communicated also depends upon the duration of the flow, and on how well a given system approaches the limiting rate specified by the Shannon Equation. Now, Mr Geiss’s second sentence about “... a structure that straddles many samples... ” is quite correct. However, what he says is actually entirely consistent with Shannon’s Equation, and hence with “Shannon Theory” as he terms it. A signal pattern which extends over many samples also has a duration consistent with that number of samples. The larger the number of samples (for a given channel/system) the longer the duration, and hence the larger the total amount of information that may have passed – at a rate limited by the Shannon Equation.
Thus the implication of the Shannon Equation is that the rate of information flow is limited. But this does not mean we cannot get more information by extending the duration provided that the pattern endures. Hence what I wrote in the “Good Resolutions” article. This point is covered in the reference mentioned above, and which was mentioned at the end of the article.
I will not consider Mr Geiss’s point (a) further as it deals with issued not covered in the article. Instead I will consider his point (b) as I think this may also shed some light on Mr Carlyle’s assertions.
It is a standard point in Information Theory that we can record and recover signal patterns whose power level is well below that of the noise. The ability to do this does, indeed, depend upon the signal pattern duration (i.e. number of samples) as well as some other factors. However despite Mr Carlyle’s “granite” comments about CD noise, this ability is available for CD and other digital systems just as it is for LP and other analog ones. Thus the ability to do this does not represent a distinction between LP and CD, despite Mr Carlyle seeming to believe that it does. That said, I regret to say that I fear that Mr Carlyle’s misunderstanding on this issue is a common one, even amongst some graduates and professionals who should perhaps know better. Given this, his mistake is understandable as he may have accepted it in good faith when told so by someone whose ability he felt he had reason to trust in such matters.
In the “Good Resolutions” article I was saying that at some suitably low signal power level the signals cease to be distinguishable (or audible) as a result of our ability to detect and recognise small details being limited by the noise. However I did not say that this occurs as soon as the signal power is less than the noise power. The level at which this occurs will depend to some extent on the duration of the signal as well as some other factors.
These issues are dealt with at much greater depth and detail in standard textbooks on Information Theory and Measurement - including my own textbook, “Information and Measurement” which was published in 1995 and remains in print. However I omitted a specific discussion of the points Mr Geiss correctly makes in order to keep the article down to a length that the Editor would accept and readers who are not students of the relevant areas might find digestible. Instead, I gave a set of references which included dealing with the background as I have outlined above.
8th Sep 2005