Improved Linear Prediction for Truncated Signals of Known Phase

It has been suggested that increasing the dimensionality of protein NMR spectra may be a more efficient way of simplifying protein NMR spectra than increasing the resolution of conventional 2D NMR spectra (1). Because of measuring time constraints, severe truncation of the NMR signals frequently occurs in at least one and often in two dimensions of 3D and 4D data sets. Fourier transformation of such signals leads to truncation artifacts. and it has been suggested that linear prediction (LP) algorithms are more suitable for treatment of such signals. Linear prediction can be used either for measuring the frequencies, amplitudes, damping factors, and phases of all components of the NMR signal (Z-5) or, in a more conservative approach, for simply extending the time-domain data set, prior to Fourier transformation (5-9). In either case, no assumption is made about the shape of the signal apart from the assumption that it can be described by a sum of exponentially damped sinusoids. In multidimensional NMR experiments the phase of the NMR signal is well known in all the indirectly detected dimensions ( 10-12). Also, if signals are severely truncated, the small amount of signal decay during the short time domain can be described adequately by a single damping factor. However, apart from simplifying the calculation, this prior knowledge is of no immediate use for conventional linear prediction analysis of the time domain. Here we demonstrate a new method that effectively uses this prior information to obtain a better estimation of the NMR spectrum. In the case of forward linear prediction, a data point x, is expressed as a linear combination of its K preceding data points: