Processing linear prediction

Data shown as examples in this review were typically acquired as 2K X 128 or 2K x 160 point files. Data were processed with linear prediction or zero-filling prior to the first Fourier transform. Data were uniformly linear predicted to 512 points in the second dimension followed by zero-filling to afford final data matrices that were 2K x IK points. 

Later section.s of this chapter deal with more advanced and specialised processing options such as zero filling, linear prediction, deconvolution and the manipulation of 2D data sets. The chapter concludes with a set of tables containing recommendations for the type of processing function and the corresponding parameters to be used in a number of ID and 2D experiments. 

There are three main processing options based on the addition of a processing or correction function to the FID DC- or Baseline-Correction, Zero-Filling and Linear Prediction LP. 

Among the various processing options available to improve the quality of FIDs and the corresponding spectra, Linear Prediction (LP) and the Maximum Entropy Method (MEM) - not available with WIN-NMR - are probably the most exciting and powerful, even though they are not widely used. 

Knudsen, 1975] Knudsen, M. J. (1975). Real-Time Linear-Predictive Coding of Speech on the SPS-41 Triple-Microprocessor Machine. IEEE Trans. Acoustics Speech and Signal Processing, ASSP-23(1) 140-145. 

A model which has found application in many areas of time series processing, including audio restoration (see sections 4.3 and 4.7), is the autoregressive (AR) or allpole model (see Box and Jenkins [Box and Jenkins, 1970], Priestley [Priestley, 1981] and also Makhoul [Makhoul, 1975] for an introduction to linear predictive analysis) in which the current value of a signal is represented as a weighted sum of P previous signal values and a white noise term ... 

Another important consideration is the development of an efficient control strategy, because this minimizes the costs by maintaining the process in its optimal conditions. Costa et al. (5) determined the best control structures and studied the control of the process proposed by Silva et al. (3) using a linear predictive controller. Later, a nonlinear predictive controller using FLNs as the internal model was developed and implemented with the same process with promising results (6). 

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

One of the weaknesses of the linear prediction method is the need to find the M (complex) roots of a polynomial. In the presence of noise, it is difficult to separate the true signal-related roots from the spurious roots due to the noise. A rather different approach to the signal identification problem was proposed by Kung,122 and introduced into NMR by Barkhuijsen and colleagues.123 Despite its origins in state-space theory, many of the processing steps involved are markedly similar to those of the linear prediction method. Useful reviews are those by de Beer and colleagues.20 23... 

Nuclear Magnetic Resonance Concepts and Methods by Daniel Canet42 contains particularly clear presentations on techniques and data processing for Fourier transform NMR and related methods. Articles in the Encyclopedia of NMR on Fourier Transform Spectroscopy,43 Fourier Transform and Linear Prediction Methods, 39 and Maximum Entropy Reconstruction44 are also very informative. A Handbook of NMR includes a very clear description of the maximum entropy method and its limitations.19... 

The process models predicting bulk physical attributes have been augmented by a properties model describing the relations between the physical attribues and the sensory attributes. For each of the sensory attribute the lowest complexity model has been determined. Instead of an trial-and-error approach, a neural network with one hidden layer has been used as a generic non-linear function. The complexity of the neural network can thus be seen as the number of nodes in the hidden layer. The performance of the property function model obtained by the above described procedure has verified with the validation set and the property function model provides a good estimation of the sensory attributes. 

Like acquisition, data processing is performed differently in 2D, compared with ID, NMR experiments. The principal reason is that signal truncation is a much more serious problem in 2D than ID experiments. Zero filling also is used in 2D experiments, as is the relatively new technique of linear prediction. 

Process dynamics is another important factor that must be considered. In a distillation column, for instance, the time elapsed between changing the reflux rate and observing a change in a product composition could be measured in hours. With this response time, and in the absence of dynamic prediction capability, the controller will start taking action hours after a disturbance occurs, and it would take even longer for the correction to take effect. Linear predictions are commonly used to forecast trends of process variables but many processes, particularly multistage separations, are often highly nonlinear. Substantial improvement can be achieved with a nonlinear model. 

The method of linear prediction (LP) can play many roles in processing of NMR data [4,5], from the rectification of corrupted or distorted data, through to the complete generation of frequency-domain data from an FID an alternative to the FT. Here we consider its most popular usage, known as forward linear prediction, which extends a truncated FID. Rather than simply appending zeros, this method, as the name suggests, predicts the values of the missing data... 

The same raw data was used in each spectrum, with the F (carbon) dimension processed with (a) no data extension, (b) one zero-fill and (c) linear prediction in place of zero-filling. 

Figure 4.36. O spectra of ethyl acetate recorded (a) with and (b) without the RIDE sequence. The severe baseline distortion in (b) arises from acoustic ringing in the probehead. Spectrum (c) was from the same FED of (b) but this had the first 10 data points replaced with backward linear predicted points, computed from 256 uncorrupted points. The spectra are referenced to D2O and processed with 100 Hz line-broadening.

An alternative approach now available with modem processing software is to collect the distorted FID with the simple one-pulse-acquire sequence and to replace the early distorted data points with uncormpted points generated through backward linear prediction. Fig. 4.36c was produced from the same raw data as 4.36b, but with the first 10 data points of the FID replaced with predicted points. The baseline distortion is completely removed and there is no signal-to-noise loss through experimental imperfections, as occurs for the RIDE data set. 

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