Reconstructing the Interferogram

The NMR signals are collected as waves in the audio-frequency range and can be manipulated with the armaments of Digital Signal Processing. If you have ever observed the waves of the sea, you know that their short time behavior can be predicted with good approximation. The same property applies to the NMR signals: you can reconstruct missing or damaged portions of the interferogram, using the information contained into the points nearby. The technique is called linear prediction (LP in the following) because involves the solution of a system of linear equations.

One common situation that requires LP is when the signal is not completely decayed by the end of the acquisition time. In this case you can predict the (missing) decay and, usually, multiply it with a decaying function (apodization). The alternative to LP is a stronger apodization which entails a loss of resolution.

Another possible application is when the baseline is severely distorted. In this case the first few points of the interferogram can be reconstructed with LP and the distortions disappear.

To Implement Linear Prediction:

Choose Process > Linear Prediction.

The dialog shows three column of values. You normally only use the first column. The other fields are reserved for the rare case in which you want to reconstruct two or three non contiguous portions of the interferogram.

Enter the index of the first point to be predicted/reconstructed.

Normally points are counted starting from zero.
Bruker digital data represent an important exception, because the first index is a negative value. Press “n” on the keyboard to see, from the scale labels, how points are numbered.

Enter the index of the last point to be predicted/reconstructed.

The third parameter (Points to Extrapolate from) is the number of equations that will solve the problem. It must be less than the number of points of the spectrum. In practice, you'll normally enter a value higher than allowed and let the program correct it automatically. Solving many equations takes time and memory, but can increase the accuracy. When there are really a lot of points in the spectrum, you may consider a lower value, for example 10-20 times the number of points to predict or 2-3 times the estimated number of signals (see below).

Enter the number of LP coefficients (Number of Signals). It corresponds, in first approximation, to the expected number of peaks into the interferogram. For example, an heteronuclear spectrum can contain hundreds of peaks, but only a few of them (or even zero) along each column. The number of coefficients should be equal, in theory, to the number of peaks into a single column. In practice, you need more coefficients to include, into the model, also the presence of noise and the inhomogeneity of the magnetic field.

Choose an algorithm. The best choice (fast and stable) is the Singular Value Decomposition. The Zhu and Bax method is stable, but only for forward prediction, and twice as slow. The Fast algorithm is the fastest one, but also the less stable.

Compared to the Fourier Transform, LP is much less stable. Even a minimum amount of noise, in unfavorable cases, can be amplified to give annoying artifacts. This is why the choice of parameters can be critical. If you prefer a default treatment, you can bypass the LP dialog and implement the option LP filling that you find inside the FT dialog.

The stand-alone LP and LP-filling, even when using the same algorithms, are not equivalent. When used in 2D (phase-sensitive) spectroscopy, stand-alone LP comes before shuffling (data-reduction) while LP-filling comes after it. It's unlikely, yet possible, that the outcomes are different.

It is also important which phase-sensitive protocol is adopted. If the next FT is a real FT (when the TPPI scheme is in use), the Fast LP creates a counter-diagonal. In the other cases (Ruben-States and States-TPPI), signals near the diagonal are marred by noise produced by the LP. In most cases this is preferable to the counter-diagonal.