Nothing Is Better

A few months ago I commented on zero-filling on Glenn Facey's blog. I have reordered my own comments and here they are. In the following I mention Linear Prediction (LP). It's a technique I am going to explain tomorrow.

"Maybe a few readers of this blog, less fond of math, can better understand the following simple recipe. In high resolution 1D NMR, usually the signal has already decayed to (practically) zero at the end of the FID. In this case zero-filling and forward Linear Prediction should yield the same results, but the former is better (faster and more robust). When, for whichever reason, the signal is truncated, the preference goes instead to LP.

The statement "it is only forward linear prediction which adds new information to the spectrum" is not correct.
1. LP can't add any authentically NEW information because it simply extrapolates the information already contained into the FID.
2. Zero-filling has the property to recover the information contained into the imaginary part of the FID (uncorrelated to the real part) and to reverse this additional information onto the real part of the spectrum.
(see demonstration below)
3. The spectroscopist, to run an LP algorithm, feeds it with the hypothetical number of lines. While this is certainly NEW information not already present into the FID, it is arbitrary.
Bottom Line: LP is a good thing, yet things are more complicated.

DEMONSTRATION (?!?)
Let's say you collect a FID of n complex points. It's a sequence of n real and n imaginary numbers or measures. They are independent measures, because no single value has been calculated from the rest. The total information content of the experiment amounts to 2n measures. After FT and phase correction you have a real spectrum of n points while the imaginary spectrum is discarded. You may ask: "Why are we throwing away half of the information?". We could rescue it with an Hilbert transform (HT). It's a tool to calculate the imaginary spectrum when you only have the real part, or the real spectrum when you only have the imaginary part. If the noise of the imaginary part is uncorrelated to the noise of the real part we can add:
experimental real spectrum +
real spectrum regenerated from the imaginary part =
more signal/noise.
The HT works in this way: you set the real part to zero, apply an inverse FT. You reach an artificial FID which is (anti?-)symmetric (both the real and imaginary parts first decay, then grow again). Set the right half of the artificial FID to zero, then apply a direct FT. (I have omitted the adjustment of intensities, because I only want to show the concept).
As you can see, HT is akin to zero-filling. Although this is not a mathematical demonstration, it may convince you that zero-filling can move some information from the imaginary part onto the real one, so can enrich the latter with real information.
It only works, however, when the signal has already decayed to zero at the end of the FID, otherwise it's so unrealistic that generates unwelcome wiggles."
Here I wrote: "I am sorry I can't find a literature reference with the real demonstration".
Mike kindly added:
"I believe that the reference old swan refers to is a paper from Ernst's lab: JMR, 1973, 11, 9-19"

NOTE
When I commented on Glenn's blog, I wrote that the artificial FID, intermediate stage of the Hilbert Transform, is symmetric. This is the graphical impression. But, who can recognize if the sign of the FID changes? It looks just the same; (unless the oscillation is terribly slow, the envelope remains the same). This is a detail that doesn't affect at all the rest of the discussion. We are going to set to zero the second half of the imaginary FID, so it doesn't matter which sign it has before. The book Numerical Recipes contains a correspondence table (page 601 on the 3rd edition) between symmetries in the two domains. From the table I understand that, in our case (real spectrum = 0), the real FID is odd and the imaginary FID is even.