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

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
• Isn't that just another way of representing the signal? [1,2,3,4] could also be written [1,+1,3,+1], where every other sample is the difference between the actual value and the previous value. Not sure what the point is. Aug 25 '11 at 19:27
• @endolith, that's the question--does it offer any surprising advantages, or is it really just a trivial transform? Aug 25 '11 at 20:53
• Is there any more context that explains it? Aug 26 '11 at 14:29
• @endolith, check yoda's answer below for an overview of what's mentioned in the paper. Aug 27 '11 at 2:03

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.

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

• Does this also go by the name "equivalent time sampling"? Jul 16 '12 at 15:07

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.

• If this method also goes by the name "Equivalent Time Sampling", then I think I may have seen it being used on radar applications. Essentially, instead of sampling at nyquist rate for such high frequency applications, multiple samplers all delayed in time can sample at a fraction of the nyquist rate and still reconstruct the radar receive signal. Apr 23 '12 at 4:41

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.

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

• Exactly, that's the "historical development" I was alluding to which makes me think more research might have been done in this direction (which I'm also unaware of). Thanks for referencing it here. I've only turned up a couple minor references so far other than that (in the context of non-uniform sampling, and fractional-delay filter design). Hoping more is out there. Aug 27 '11 at 1:02
• Oh, I thought you meant short story #1: "Historical notes" by that comment. I haven't been able to find many references for it either. I guess it was more of an issue back then, as they had to be picky about sampling just enough and nothing more. So they were trying to cut corners everywhere. Now-a-days, with the advent of increased computing power, that's not so much of a concern, although we have a different basket of problems now. Aug 27 '11 at 1:06
• Still great to have that section documented here though. I'm going to let this one percolate a bit to see if anything interesting turns up... Aug 27 '11 at 1:28
• The pilot does have 'derivative sampling': the vertical speed indicator gives the derivative of the altitude. Sep 5 '11 at 8:15
• You seem to be missing an $n$ somewhere under the sum (the $f$ and $f'$ terms are constant w.r.t. the summation as it stands). I would correct it myself, but the form given in the reference (Jagerman and Fogel) is completely different than what is here, so I am not quite sure what you were going for. Jan 11 '15 at 14:46