Improved Linear Prediction of Damped NMR Signals Using Modified “Forward-Backward” Linear Prediction

Linear prediction (LP) has become a standard tool for enhancing the appearance of multidimensional NMR spectra (1-7). In principle, the method can be used to calculate the frequencies, amplitudes, damping factors (linewidths), and phases of all components contained in the time-domain signal. In practice, however, the method is not very robust in the presence of noise. For this reason, a more conservative approach, where the time-domain signal is extended using (imperfect) linear prediction coefficients, has been much more popular. In the latter case, the appearance of the final spectrum frequently is dominated by the acquired time-domain data, but truncation at the end of the acquired time domain is minimized by elongating the acquired data by predicted data. The weight of these predicted data is decreased by appropriate digital filtering prior to Fourier transformation. Here we present a simple modification of this commonly used procedure, based on the “forward-backward” LP (FB-LP) method (8, 9)) which improves significantly the quality of the prediction coefficients. The principles of linear prediction have been described many times before (Z-IO), and only the points salient to the present discussion will be briefly reiterated. In the case of forward linear prediction, a data point x,, is expressed as a linear combination of its K preceding data points: