Looking Forward

The polygons are the idealized shapes of the FIDs of dioxane (green) and water (red). We suppose that the magnetization of water decays much faster, just to see what could happen in extreme cases. We also assume that, when the signal decays under a certain level, it has no more effect on the detector (in other words, it's zero). In both FIDs the evolution of the signal can be summarized with a single parameter, given by the ratio of any two contiguous points. They can be taken from the beginning or the end of the FID, it's just the same. Once we have the prediction parameter, we can prolong the FID indefinitely. (In the red case, there's no difference between forward linear prediction and zero-filling).
Now let's mix dioxane and water:


As explained yesterday, now we need 4 contiguous points at least to extract the two prediction coefficients. We can't choose them from the beginning of the FID (dominated by the water signal) because there is no more water in the portion of the FID we are going to predict. We must choose the final points. The lesson is: calculate the prediction coefficient from the same end of the FID you are going to prolong. Now let's consider another signal (tangerine) in the presence of noise (gray). Noise doesn't decay.


In this case it would be more convenient to exploit the initial points of the FID, because there is less noise (in percentage) and the estimated parameter will be more accurate. That would go against the only rule we imposed to ourselves a few lines above. To reduce the effect of the noise we can, however, make use of the whole FID. This can't be a rule. Sometimes you can see by inspection that a portion of the FID is irregular. Ever seen a FID coming from an Avance instrument? The very first points aren't useful for linear prediction.
In a fourth case we consider a spectrum with very little noise and a signal that decays to undetectable levels:


Most of the proton spectra acquired today correspond to this case. If we use the whole FID, we can calculate the coefficients with accuracy. When we prolong the FID, all we can prolong is, however, noise, because the true signal is already negligible at the end of the FID. The good thing is that the predicted noise will soon decays to zero (at the same rate of the blue signal); but zero-filling is even better, because it decays immediately!

LP is slower and less stable than FT. We had an hint yesterday. The formula to calculate the complex prediction coefficient would be:
p = S₂ / S₁.
Of the 4 arithmetic operations, when performed on a computer, division is the slowest and the less stable one. The formula for the FT contains a multiplication and an integral (that is, a sum), but no division.
Another inherent weakness of LP is the propagation of inaccuracy. Each point is predicted according to what comes before. The first point you predict is based exclusively on experimental data. The second point is also based on the first point, which is artificial. The third point is also based on the second point, a second generation artificial point. The 1001th point is also based on a 1000th generation artificial point. If there was a defect of precision, it has been amplified 1000 times.
If the last part of the experimental FID is dominated by noise, you risk to amplify it thousands of times.
In practice things aren't so bad. There is a single case I know when forward LP is useful. It's when you want to prolong the indirect dimension of a 2D/3D spectrum. Let's say your 2D spectrum requires 2 days of spectrometer time. You can decide to truncate it after 12 or 24 hours and predict the rest. From our discussions you can deduce the requirements: the number of points (in the indirect dimension) must be at least twice the number of signals (for the most populated column), the signals haven't decayed too much, noise should be low. The conditions are easily met in ethero-nuclear 2D (only a few cross-peaks per column; in the case of HSQC of small molecules they are 1 or 2). In the homo- case signals, when are much more, forward LP is often beneficial too. Evidently, it can work.
Next time we'll unleash the algorithms.