Linear prediction spectra

Rgure 12.13 Examples of linear-prediction spectra, (a) LP spectrum superimposed on the DFT. (b) LP spectra for analysis of order 4, 6, 8,10, 12,14,16 and 18. In reality the plots lie on top of one another, but each has been separated vertically for purposes of clarity. The bottom plot is from LP analysis of order 4 the top plot is for order 18. 

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. 

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. 

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. 

Unfortunately, there is great scope for confusion, as two distinct techniques include the phrase maximum entropy in their names. The first technique, due to Burg,135 uses the autocorrelation coefficients of the time series signal, and is effectively an alternative means of calculating linear prediction coefficients. It has become known as the maximum-entropy method (MEM). The second technique, which is more directly rooted in information theory, estimates a spectrum with the maximum entropy (i.e. assumes the least about its form) consistent with the measured FID. This second technique has become known as maximum-entropy reconstruction (MaxEnt). The two methods will be discussed only briefly here. Further details can be found in references 24, 99, 136 and 137. Note that Laue et a/.136 describe the MaxEnt technique although they refer to it as MEM. 

Fig. 4. Spectra acquired for a 0.55 pmol sample of the indoloquinoline alkaloid cryptolepine (1) dissolved in 25 pL J6-DMSO. (a) Proton-carbon (HMQC) direct correlation spectrum recorded in 12m (16 x 2048 point files, 16 transients/ increment) shown with no linear prediction, (b) Long-range 1H-13C HMBC spectrum of the same sample optimized for 8 Hz. The data were acquired as 64 x 4096 files with 16 transients accumulated/ increment. Data are again shown without any linear prediction. (Reprinted with permission from Ref. 11. Copyright 1998, John Wiley Sons, Ltd.)...

The experiments based on proton detection of rare spin nuclei are usually the most sensitive methods of determining NMR parameters of magnetically diluted spin systems. Unfortunately, recording of 2D correlation maps is usually also a time consuming experiment, especially if wide spectral bands have to be covered in the indirectly detected dimension. The most frequently encountered situation is that only one or a few peaks are expected within a narrow spectral band. However, the position of this band is not known. Several attempts have been made to reduce the experimental time needed to perform such experiments. One approach would be to record a highly truncated data set and use the linear prediction [86,87] to reduce the effect of the data truncation on the appearance of the spectrum. This technique is now available with most... 

For the 2-D experiments described here, the spectral width of the proton dimension should be equal to that determined for the 1-D proton spectrum, and the spectral width of the carbon dimension should be equal to that determined for the 1-D carbon spectrum. If the 1-D carbon spectral width has not yet been determined, then 0 - 200 ppm is a reasonable default range. A good compromise of data set size versus adequate resolution is 2048 points in the t2 dimension and 512 points in the ti dimension. This data set size is appropriate for all the 2-D experiments described in this section. Linear prediction can be used to enhance the apparent spectral resolution of truncated data. 

A similar zone folding also occurs at Cs/Pt(lll) and the phonon mode appears as a small dip in the Fourier spectrum in Figure 19.4. A detailed analysis of the time domain data by linear prediction singular value decomposition has been performed and a decomposition of the time-domain data to phonon modes and alkali-substrate stretching modes has been carried out. Coherent nuclear motions have been observed on substrates other than Pt. Figure 19.5a shows time-resolved SHG traces... 

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.

While methods of spectrum analysis capable of super-resolution exist, that is, methods that can achieve resolution greater than l/Wx, the most common of these, linear prediction (LP) extrapolation, has substantial drawbacks. LP extrapolation is used to extrapolate signals beyond the measured interval. While this can dramatically suppress truncation artifacts associated with zero-filling as well as improve resolution, because LP extrapolation implicitly assumes exponential decay it can lead to subtle frequency bias when the signal decay is not perfectly exponential [8]. This bias can have detrimental consequences for applications that require the determination of small fi-equency differences, such as measurement of residual dipolar couplings (RDCs). 

Stem AS, Li K-B, Hoch JC (2002) Modem spectrum analysis in multidimensional NMR spectroscopy comparison of linear-prediction extrapolation and maximum-entropy reconstmction. J Am Chem Soc 124 1982-1993... 

If we find that the ringdown delay required to produce an artifact-free spectrum yields an unacceptably low signal-to-noise ratio (this is a subjective decision), then we can use backwards linear prediction to correct the first few corrupted points in the FID. Linear prediction is generally accepted in the NMR community as a reasonable method for avoiding choosing between a spectrum with a low signal-to-noise ratio and a spectrum marred by ringdown artifacts. 

Kumaresan etal. who are usually active in the field of linear prediction, proposed two approaches which were not too demanding in computational terms and applied to the frequency domain. The first consists in iteratively minimizing the difference between the experimental spectrum and a simulated one that may also require significant computer resource. The second consists in picking one line after the other, a method that should be quite fast. 

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