What are the advantages, if any, of derivative sampling?

Question

In Five short stories about the cardinal series $[1]$, the author makes the following comment:

Interestingly enough, Shannon goes on to mention that other sets of data can also be used to determine the band-limited signal--for example, the values of ƒ and its first derivative at every other sample point, the values of ƒ and its first and second derivatives at every third sample point, and so on.

The paper mentions some historical developments, but I'm curious what the "killer apps" are for derivative sampling. Does it go by any other names? Are there further generalizations of this approach?

A simple overview, or pointers to some references would be great.

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1. J. R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. (N. S.) 12 (1985), no. 1, 45-89. http://bit.ly/plioNg

Answer 1

Papoulis introduced a generalization of the sampling theorem [1], of which derivative sampling approach is one case. The gist of the theorem, quoting from [2] is:

In 1977, Papoulis introduced a powerful extension of Shannon’s sampling theory, showing that a band-limited signal could be reconstructed exactly from the samples of the response of $m$ linear shift-invariant systems sampled at $1/m$ the reconstruction rate.

Perhaps one reason why it's hard to search for the term is because Papoulis' generalized sampling theorem is mentioned more often than "derivative sampling". [2] is also a very good article which presents a broad overview of the sampling approaches at the time of publication. [3], also by the same author is an extension of [1] to the class of non-bandlimited functions.

As for applications, in a recent paper [4], the derivative sampling approach is used to design wideband fractional delay filters and the authors show that sampling the derivative results in smaller errors. From the abstract:

In this paper, the design of wideband fractional delay filter is investigated. First, the reconstruction formula of derivative sampling method is applied to design wideband fractional delay filter by using index substitution and window method. ... Finally, numerical examples are demonstrated to show that the proposed method has smaller design error than the conventional fractional delay filter without sampling the derivative of signal.

While there certainly are more, I'll refrain from posting more references and application to keep it short (and avoid it turning into a list). A good point to start looking would be to check which papers have cited [1]-[3] and narrow down the list based on the abstract.

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[1]: A. Papoulis, “Generalized sampling expansion,” IEEE Trans. Circuits and Systems, vol. 24, no. 11, pp. 652-654, 1977.

[2]: M. Unser, "Sampling - 50 years after Shannon," Proceedings of the IEEE, vol. 88, num. 4, p. 569-587, 2000

[3]: M. Unser and J. Zerubia, "A generalized sampling theory without band-limiting constraints," IEEE Trans. Circuits and Systems II, vol. 45, num. 8, p. 959–969, 1998

[4]: C-C Tseng and S-L Lee, "Design of Wideband Fractional Delay Filters Using Derivative Sampling Method", IEEE Trans. Circuits and Systems I, vol. 57, num. 8, p. 2087-2098, 2010

Answer 2

I'm not aware of any applications of such a sampling scheme. It's typically more difficult to accurately sample a signal's derivative than its instantaneous value (differentiators are vulnerable to high-frequency noise due to their ramp-shaped frequency response). As endolith pointed out in the comment above, if you have enough information in your discrete samples to reconstruct the original signal, then you can calculate all of the derivatives that you would want.

Answer 3

That's a very nice article that you linked to (I hadn't read it before), and in fact, the answer that you seek is in that very article in §2.3! I've reproduced below a portion of §2.3 that is relevant.

2.3 Derivative sampling

In order to illustrate a practical sampling situation, J. Fogel (1955) has mentioned the example of an airplane pilot's instrument panel, which traditionally consists of dials with pointers giving information about the plane's altitude, attitude, speed, etc. Pilots scan their instruments, obtaining information from any one of them on a roughly periodic basis. It is possible that derivative information could be available to the pilot as well; for example, the altimeter would be noticed to be "unwinding" at an alarming rate if the plane were in a nose dive! It is just conceivable that the acceleration of the pointer could be observed as well; at any rate this little example does point out the general need for a sampling theorem which takes account of samples not only from the function itself but also from its first $r$ derivatives. When just the samples of $f$ (band-limited to $[-\pi W,\pi W]$) and $f'$ are available, the formula is

$$f(t)=\sum\left\{f\left(\frac{2\pi}{W}\right)+\left(t-\frac{2\pi}{W}\right)f'\left(\frac{2\pi}{W}\right)\right\}\left\{\frac{\sin \pi(Wt-2n)/2}{\pi(Wt-2n)/2}\right\}^2$$ and in this form it was first given by Jagerman and Fogel (1956).

I believe that this is still a very valid application of derivative sampling, as planes haven't gone out of fashion. There might have been several other technological advances (that I'm unaware of) that might make the use of derivative sampling unnecessary these days, but the point still remains.

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L. J. Fogel (1955), A note on the sampling theorem, IRE Trans. Inform. Theory 1, 47–48

D. L. Jagerman and L. J. Fogel (1956), Some general aspects of the sampling theorem, IEEE Trans. Inform. Theory 2, 139–156