Processing: linear prediction

Linear prediction is the process of extending an FID by predicting additional points from experimental data points. It is often used in the indirect dimensions of multidimensional experiments where time restrictions prevent full sampling of the decay.


When used to add points to the end of an FID, linear prediction is similar to zero filling, but instead of just adding zeros the algorithm attempts to predict how the FID will evolve with time. The figure below shows an interferogram from the indirect ¹³C dimension of a HMBC and the results of applying linear prediction to it. The interferograms were extracted after fourier transformation of the directly detected ¹H dimension and correspond to a peak at 3.00 ppm in the ¹H dimension. Panel a shows the full 128 experimental data points that were collected, panel b shows the same experimental data truncated to 32 points, panel c shows the results of using those initial 32 points to linear predict to 128 points, and panel d shows the results of using the full 128 points to linear predict to 512 points.

In panels c and d the linear predicted points decay rapidly but still show the high and low frequencies present in the experimental points.

The second figure shows expansions from the fully transformed HMBC. Panels a and b below were processed without linear prediction and correspond to the interferograms of panels a and b in the first figure. Panel a used the full 128 experimental points, while panel b used just 32. The resolution loss in the ¹³C dimension is readily apparent. Panels c and d correspond to panels c and d of the first figure, where linear prediction has been applied. In panel c the first 32 experimental points have been used to predict an additional 96 points to give a total of 128 points for fourier transformation, the same number of points as in panel a, while panel d used all 128 experimental data points to extend the data to 512 points.

 

The spectrum in panel c is clearly better than that in panel b, despite both spectra using the same number of experimental data points. Panel c, however, is not as good a spectrum as panel a; in particular the peaks on the right of the expansion are not as well resolved. Linear prediction can improve resolution and the appearance of a spectrum, but is no substitute for collecting the experimental data if time permits. Panel d shows the resolution enhancement obtained by applying linear prediction to the full 128 data points. The peaks around 4.08,138.0 ppm are resolved with the help of linear prediction. Panel d also shows artifacts in the ¹³C dimension, but these could be reduced with a different window function. In general, the benefits of using linear prediction outweigh the drawbacks, and in the standard parameters used at the Skaggs NMR Facility four-fold linear prediction is used in the indirect dimensions of all 2D parameter sets.

In addition to extending interferograms in the indirect dimensions of multidimensional spectra, linear prediction can also be used to add data points to the start of an FID or to replace points in the middle. This can be useful when trying to remove instrumental artifacts or to repair corrupted data.