Modern Sampling Theory: https://link.springer.com/book/10.1007/978-1-4612-0143-4
Moreover:
https://link.springer.com/article/10.1007/s00034-018-0909-2
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Shanon and Nyquist proved mathematically that any bandwidth limited signal can be perfectly reconstructed from a set of discrete-time samples as long as the sampling rate is at least twice the bandwidth of the signal.
Let's let that sink in for a moment.
Proved mathematically. That's a pretty darn high bar. Probably the highest. I certainly give that a lot more credibility than experimental data. And by "experimental data" I mean properly executed scientific experiments. Which I give a lot more credibility to than someone's individual experience or the sighted, uncontrolled trials commonly performed by audiophiles (and many others).
The newer sampling theorems apply to a subset of signals. For example, compressed sensing requires that the signal satisfies the sparcity and incoherence criteria required for compressed sensing to work. According to Wikipedia, compressed sensing works well for MRI images. A "sparse" signal is a signal with lots of zero samples. I somehow doubt audio is an example of such a signal, but hey... Maybe you like to listen to silence. I'm not here to judge.
Tom
Let's let that sink in for a moment.
Proved mathematically. That's a pretty darn high bar. Probably the highest. I certainly give that a lot more credibility than experimental data. And by "experimental data" I mean properly executed scientific experiments. Which I give a lot more credibility to than someone's individual experience or the sighted, uncontrolled trials commonly performed by audiophiles (and many others).
The newer sampling theorems apply to a subset of signals. For example, compressed sensing requires that the signal satisfies the sparcity and incoherence criteria required for compressed sensing to work. According to Wikipedia, compressed sensing works well for MRI images. A "sparse" signal is a signal with lots of zero samples. I somehow doubt audio is an example of such a signal, but hey... Maybe you like to listen to silence. I'm not here to judge.
Tom
The signal itself doesn't have to be sparse, rather it has to be representable sparsely in some domain. The trick is finding the right representational domain. For example, x-ray images show tissue density but they may be represented sparsely in the gradient domain. That's just a way of saying that x-ray images don't have sharp edges in them. X-rays are kind of blurry due to scatter in tissue. That means some solutions to the reconstruction problem are not likely or not possible. Thus it is possible to discard planar ray images of a CT scanner that are badly affected by scatter from dental fillings in the head of a patient. With the problem rays having been discarded, the remaining CT planar ray images are now undersampled. However, if they are transformed into the gradient domain representation they can be reconstructed into a 3D image using sparse sensing techniques. The resulting reconstructed CT images look almost perfect without any streaking artifacts from metal fillings. This was from some work done by Stanford University Radiation Oncology Dept.
Also, sparse signals don't necessarily have to have a lot of zero samples. As a practical matter it is often sufficient to have equation term coefficients that are near zero. They may then be set to zero so that the system of equations can be solved.
Regarding the incoherence criteria, it is conceptually similar to the requirement for solving a system of simultaneous equations. The equations can't be equivalent to each other; there have to be enough non-equivalent equations as there are unknown variables to be solved for.
Also, sparse signals don't necessarily have to have a lot of zero samples. As a practical matter it is often sufficient to have equation term coefficients that are near zero. They may then be set to zero so that the system of equations can be solved.
Regarding the incoherence criteria, it is conceptually similar to the requirement for solving a system of simultaneous equations. The equations can't be equivalent to each other; there have to be enough non-equivalent equations as there are unknown variables to be solved for.
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Drug approvals very large sample sizes and the usual placebo tests. Golden ears give reasons why they can't provide large samples
Have they been tested to prove this or do you give out medals to each other when you meet?
Lots of people hear voices in their heads. They tend to get sectioned if they act on it. Are the voices real if enough people hear them?
none of which have been proven beyond some blokes having lols and claiming stuff.
vinyl is for enjoyment of the experience, not for fidelity. Anyway I have 2 very good vinyl setups provided you are not being snobbish and saying that the cost of the setup is what matters not the performance...
2 channel audio can never sound real as it cannot recreate a full soundfield. It can however provide good entertainment, invoke emotional responses, get the wife in the right mood etc. Just not real.
If you say so; I don't know any golden ears.
Also, I don't have a drug company budget nor do I know anyone who does. Would like to though.
I can't tell you how offensive I find that statement. Schizophrenia is a dead serious life changing illness. Nothing to make light of.
Yean, but if it is from master tapes, and if its optical cartridge playback with its exceptional transparency, then you might be surprised. You can get a pretty good idea of how the master tape sounds. You can also plainly hear if the master was recorded to digital, such as with old Telarc recordings.
That is true. But then again, nothing in real life is ideal. Nothing in real life has infinite bandwidth. Nothing in life has infinite dynamic range. Noise is a thing.
So it seems that it all boils down to which tradeoffs or non-idealities you're willing to live with.
Tom