Linear prediction

Linear prediction methods assume that the future time domain data can be represented as a linear combination of past time domain data, and vice versa—an assumption that is valid when the time domain data consist of exponentially damped sine functions and the signal/noise ratio is sufficiently large. FIDs from 

From N data points, N — k such equations can be generated, and as long as N — k is larger than the number of spectral lines, the problem is overdetermined and a set of coefficients af can be obtained by suitable mathematical procedures. With these coefficients the FID can be forward or back projected. For example, Fig. 3.9 illustrates the improvement in spectral quality as a result of replacement of a severely distorted data point. 

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. 

Backward LP (Fig. 5.21) is usually applied to repair the first few points of an FID, distorted by some spectrometer perturbation or a mis-set acquisition parameter, e.g. incorrect receiver gain. Backward LP is also used to reconstruct an FID back to t=0 in those cases where the start of data acquisition has been delayed, e.g. to exclude unwanted spectrometer noise such as the signals from acoustic ringing, and the first few data points are missing. In this case backward LP cancels or at least suppresses the corresponding spectral artefacts such as baseline roll etc. 

If the number of dummy scans in a 2D experiments is set too low, data acquisition may be started before a steady-state e.g. thermal equilibrium, has been established. Consequently the first few increments (data points in tl) may be distorted. With backward LP these wrong data points in tl can be recalculated and replaced using the information gained from the residual increments. 

With 2D data sets non-decayed FIDs in both time domains (Fig. 5.13) are very common and a simple Fourier transformation would give rise to truncation effects in the final spectrum. To circumvent such unwanted effects 2D FIDs are usually rigorously damped, using stringent weighting functions to smoothly bring the last part of the FID close to zero. However this simple procedure severely impairs spectral resolution and should be replaced by LP, followed by suitable weighting, which both improves spectral resolution and excludes any truncation effects. 

Using reasonable amounts of sample and combining 2D experiments with LP in tl can shorten drastically the total measuring time, since the many increments usually needed for adequate resolution in Fl may be omitted and the missing data points in tl may be predicted from a few measured increments. The dream to improve simultaneously the signal-to-noise ratio and the resolution in your 2D spectra seems to become a reality  

Note that this is the standard FIR form as discussed in Chapter 3, but the output is an estimate of the next sample in time. The task of linear prediction is to select the vector of predictor coefficients  

Usually as close as possible is defined by minimizing the Mem Square Error (MSB) citerion, which minimizes  

Many methods exist for arriving at the set of predictor coefficients a. which yield a minimum MSB One will be discussed in the next section. 

Once a set of predictor coefficients has been computed, the error signal can be computed as  

You should recognize this signal as the one that is being squared and summed over the frame to form the MSB. This signal is also sometimes called the innovations signal, because it represents all that is new about the signal that the linear prediction cannot capture. 

We have mentioned repeatedly that NMR spectroscopists are engaged in a seemingly never-ending conflict between sensitivity, on the one hand, and both time and resolution, on the other. Nowhere are these problems more acute than in the t dimension of 2D experiments. There must, of course, be a sufficient number of scans per time increment to observe a spectrum, but there must also be enough increments to resolve closely situated signals. Both requirements take precious spectrometer time, and spectroscopists are forced to compromise between sensitivity and resolution in a number of 2D experiments. 

Forward linear prediction (LP) represents an elegant solution to the -domain sensitivity-resolution-time dilemma. While the mathematical implementation of LP is not trivial, its underlying principle is surprisingly simple. LP can be likened to an ideal automobile race in which the vehicles travel at a constant rate of speed. If their relative positions after 256 laps are noted, a very good estimate can be made as to what their positions will be after 1,024 laps. 

In a time sequence of data points, the value of a particular data point, can be estimated from a linear combination (hence the name linear prediction) of the data points that immediately precede it (Hoch and Stem, 1996), as shown in the following equation, in which a, 02 the LP coejficients (also called the LP prediction filter)  

The number of coefficients used in the LP process is known as the order of the LP. The number of coefficients corresponds to the number of data points that are used to predict the value of the next data point in the series. 

The critical requirement for the employment of LP is that the interferograms must have sufficient S/N so that an accurate estimation can be made of the coefficients. This requirement presents a problem for both heteronucleus-detected 2D experiments and very dilute solutions in proton-detected experiments. LP, however, is less important for the former, in which protons, with their much smaller chemical-shift range, and hence intrinsically better data point resolution, constitute the uj dimension. Another important requirement is that the number of coefficients used in LP should be larger than the number of signals that make up each FID being extended. How much larger depends on the manufacturer of the spectrometer. 

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Figure B2.1.8 Dynamic absorption trace obtained with the dye IR144 in methanol, showing oscillations arising from coherent wavepacket motion (a) transient observed at 775 mn (b) frequency analysis of the oscillations obtained using a linear prediction, smgular-value-decomposition method.

Second, the resolution achieved in a 2-D experiment, particularly in the carbon domain is nowhere near as good as that in a 1-D spectrum. You might remember that we recommended a typical data matrix size of 2 k (proton) x 256 (carbon). There are two persuasive reasons for limiting the size of the data matrix you acquire - the time taken to acquire it and the shear size of the thing when you have acquired it This data is generally artificially enhanced by linear prediction and zero-filling, but even so, this is at best equivalent to 2 k in the carbon domain. This is in stark contrast to the 32 or even 64 k of data points that a 1-D 13C would typically be acquired into. For this reason, it is quite possible to encounter molecules with carbons that have very close chemical shifts which do not resolve in the 2-D spectra but will resolve in the 1-D spectrum. So the 1-D experiment still has its place. 

Linear prediction Method of enhancing resolution by artificially extending the FID using predicted valued based on existing data from the FID. 

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. 

Figure 9 1,/i-ADEQUATE spectrum of strychnine (1) optimized for 5 Hz. The data were acquired using a sample of 1.8 mg in 40 j.Lof deuterochloroform in a 1.7-mm NMR tube at 600 MHz using a 1.7-mm Micro CryoProbe. The data were acquired as IK x 160 points with 320 transients/q increment and a 3-s interpulse delay giving an acquisition time of 48 h 17 min. The data were linear predicted to IK points in the first dimension and from 160 to 512 point in the second frequency domain followed by zero-filling to give a final IK x IK data matrix. 

Direct chemical attack at the surface can be measured as the rate of loss of dimensions or mass and simple linear predictions made. The rate of degradation can be increased by raising the temperature and using an Arrhenius procedure as described in Section 8.6. 

The most commonly used fitting procedures in time-domain are the variable projection method (VARPRO) and the linear prediction singular value... 

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. 

Fig. 5.25 Parameters for a forward Linear Prediction with ID WIN-NMR (above the...

GHJCOSE 1D H GH 013001.FID. Note the baseline artifacts introduced by the truncated FID. In the Linear Prediction (LP) dialog box make sure that the Execute Backward LP option is enabled and the Execute Forward LP option disabled. Set LP backward to Point to 124. Following the rules given above vary the residual parameters First Point used for LP (recommended 196), Last Point used for for LP (recommended 2047) and Number of Coefficients (recommended 128 or larger). Carefully inspect the resulting spectra with respect to spectral resolution and signal shapes and compare it with the spectrum obtained without LP. 

Figure 8. Top Transient dichroism signal of I2 in n-hexane. Bottom Linear prediction singular-value decomposition (LP-SVD) for the pump and probe wavelengths of 580 and 475 nm, respectively. Note the overall delay of the negative signal, which is modulated by the B state frequency. (From Ref. 28.)...

CCITTrecG728, 1992] CCITTrecG728 (1992). Coding of speech at 16 kbit/s using low-delay code excited linear prediction. ITU-T. Recommendation G.728. 

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