Apodization

Having recorded the FID, it is possible to treat it mathematically in many ways to make the information more useful by a process known as apodization (Ernst, 1966 Lindon and Ferrige, 1980). By choosing the right window function and multiplying the digitized FID by it, we can improve either the signal-to-noise ratio or the resolution. Some commonly used apodization functions are presented in Fig. 1.36. 

The signal-to-noise ratio can be increased by treating the data so as to bias the spectrum in favor of the signals and against the noise. This can be done by multiplying the FIDs by the proper apodization functions. What would happen if the spectrum is recorded without apodization  

Apodization is likely to change the relative intensities of signals with different line widths. Can it also affect the chemical shifts of the signals  

To increase the signal-to-noise ratio, we need to multiply the FIDs by a window function that will reduce the noise and lead to a relative increase in signal strength. Since most of the signals lie in the head of the FID while its tail contains relatively more noise, we multiply the FID by a mathematical function that will emphasize the head of the FID and suppress its tail.  

Most simply this can be done by multiplying each data point in the FID by an exponential decay term that starts at unity but decays to a negligible value at its end. Such an exponential multiplication (EM) is a simple and effective way to increase the signal-to-noise ratio at the expense of added line broadening (LB). The LB term in this function can be altered 

The sine x function is not a particularly usefiil lineshape for infrared spectrometry in view of its fairly large amplitude at wavenumbers well away from vi. The first minimum reaches below zero by an amount that is 22% of the height at vi. If a second weak line happened to be present in the spectmm at the wavenumber of this minimum, it would not be seen in the spectrum computed. One method to circumvent the problem of these secondary minima is through the process known as apodization. 

From Section 2.3 we know that when a cosine wave interferogram is unweighted, the shape of the spectral line is the convolution of the true spectrum and a sine function [i.e., the transform of the boxcar truncation function, 0(8)]. If instead of using the boxcar function, we used a simple triangular weighting function of the form 

Because of the small differences in resolution that depend on whether or not the interferogram is apodized, the particular apodization function and the resolution criterion that is applied, we prefer to call 1 /A the nominal resolution. 

Several general studies on apodization functions have been carried out. For example. Filler [8] investigated a variety of trigonometric functions, and Norton and Beer [9] tested over 1000 functions of the general form 

One other popular apodization function is the Happ-Genzel function, given by 

This has the effect of smoothing the FID away to zero, thus yielding lovely peaks. We call this exponential multiplication for obvious reasons  

It should be noted that Gaussian multiplication can severely distort peaks and also reduce signal-to-noise of the spectrum so it is not a good idea to do this if you have a very weak spectrum to start with. Spectrum 4.1 shows a real case where Gaussian multiplication has been able to resolve a triplet. Note that it is possible to just make out the triplet nature of the peak in the unmultiplied spectrum - Gaussian multiplication helps verify this and also allows us to measure the splitting pattern. 

There are many other apodization functions which are used for specific types of NMR data, in fact you can make up your own if you want to but for most data sets, the canned ones that are shipped with the instrument are more than adequate. 

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Once the basic work has been done, the observed spectrum can be calculated in several different ways. If the problem is solved in tlie time domain, then the solution provides a list of transitions. Each transition is defined by four quantities the mtegrated intensity, the frequency at which it appears, the linewidth (or decay rate in the time domain) and the phase. From this list of parameters, either a spectrum or a time-domain FID can be calculated easily. The spectrum has the advantage that it can be directly compared to the experimental result. An FID can be subjected to some sort of apodization before Fourier transfomiation to the spectrum this allows additional line broadening to be added to the spectrum independent of the sumilation. 

It is equivalent, when an ftk spectrometer is used, to re-apodization of the data. Curve fitting is a method of modeling a real absorption band on the assumption that it consists of a series of overlapped peaks having a specific lineshape. Typically the user specifies the number of peaks to attempt to resolve and the type of lineshape. The program then varies the positions, sizes, and widths of the peaks to minimize the difference between the model and the spectmm. The largest difficulty is in knowing the correct number of peaks to resolve. Derivative spectra are often useful in determining the correct number (18,53,54). 

Figure 1.36 Various selected apodization window functions (a) an unweighted FID (b) linear apodization (c) increasing exponential multiplication (d) trapezoidal multiplication (e) decreasing exponential multiplication (f) convolution differ-...

Apodization (exponential multiplication) is used to improve the signal-to-noise ratio, and it does not affect the chemical shifts of the NMR signals. 

At the end of the 2D experiment, we will have acquired a set of N FIDs composed of quadrature data points, with N /2 points from channel A and points from channel B, acquired with sequential (alternate) sampling. How the data are processed is critical for a successful outcome. The data processing involves (a) dc (direct current) correction (performed automatically by the instrument software), (b) apodization (window multiplication) of the <2 time-domain data, (c) Fourier transformation and phase correction, (d) window multiplication of the t domain data and phase correction (unless it is a magnitude or a power-mode spectrum, in which case phase correction is not required), (e) complex Fourier transformation in Fu (f) coaddition of real and imaginary data (if phase-sensitive representation is required) to give a magnitude (M) or a power-mode (P) spectrum. Additional steps may be tilting, symmetrization, and calculation of projections. A schematic representation of the steps involved is presented in Fig. 3.5. 

The apodization functions mentioned earlier have been applied extensively in ID NMR spectra, and many of them have also proved useful in 2D NMR spectra. Before discussing the apodization functions as employed in 2D NMR spectra we shall consider the kind of peak shapes we are dealing with. 

What are the common peak shapes, and why it is necessary to know the peak shapes before applying apodization ... 

Some of the apodization functions described in Section 1.3.11 can also be adopted for use in 2D spectra. The types of functions used will vary according to the experiment and the information wanted. 

Figure 3.10 Effect of different window functions (apodization functions) on the appearance of COSY plot (magnitude mode), (a) Sine-bell squared and (b) sine-bell. The spectrum is a portion of an unsymmetrized matrix of a H-COSY I.R experiment (400 MHz in CDCl, at 303 K) of vasicinone. (c) Shifted sine-bell squared with r/4. (d) Shifted sine-bell squared with w/8. (a) and (b) are virtually identical in the case of delayed COSY, whereas sine-bell squared multiplication gives noticeably better suppression of the stronger dispersion-mode components observed when no delay is used. A difference in the effective resolution in the two axes is apparent, with Fi having better resolution than F. The spectrum in (c) has a significant amount of dispersion-mode line shape.

The next step after apodization of the t time-domain data is Fourier transformation and phase correction. As a result of the Fourier transformations of the t2 time domain, a number of different spectra are generated. Each spectrum corresponds to the behavior of the nuclear spins during the corresponding evolution period, with one spectrum resulting from each t value. A set of spectra is thus obtained, with the rows of the matrix now containing Areal and A imaginary data points. These real and imagi-... 

There are generally three types of peaks pure 2D absorption peaks, pure negative 2D dispersion peaks, and phase-twisted absorption-dispersion peaks. Since the prime purpose of apodization is to enhance resolution and optimize sensitivity, it is necessary to know the peak shape on which apodization is planned. For example, absorption-mode lines, which display protruding ridges from top to bottom, can be dealt with by applying Lorentz-Gauss window functions, while phase-twisted absorption-dispersion peaks will need some special apodization operations, such as muliplication by sine-bell or phase-shifted sine-bell functions. 

Sine-beU An apodization function employed for enhancing resolution in 2D spectra displayed in the absolute-value mode. It has the shape of the first halfcycle of a sine function. 

It is instructive to consider a specific example of the method outline above. The triangle fimction (l/l) a (x/l) was discussed in Section 11.1.2. It was pointed out there that it arises in dispersive spectroscopy as the slit function for a monochromator, while in Fourier-transform spectroscopy it is often used as an apodizing function. Its Fourier transform is the function sine2, as shown in Fig. (11-2). The eight points employed to construct the normalized triangle fimction define the matrix... 

An apodizing function is employed to reduce oscillations in an observed spectrum due to discontinuities at the ends of an interferogram. 

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FIGURE 32. H NMR spectrum of filipin III, 3 mM in DMSO-dg, recorded at 400 MHz and 25 °C. The expanded region contains nine hydroxylic proton resonances that fully exchange with deuterium oxide and correspond to the nine hydroxyl groups of filipin III. No apodization functions were applied prior to the Fourier transformation. Reproduced by permission of John Wiley Sons from Reference 50... 

The raw band pass of an AOTF has a sine squared function lineshape with sidebands, which if ignored, may amount tol0% of the pass optical energy in off-centre wavelengths. This apodization issue is normally addressed by careful control of the transducer coupling to the crystal. 

ASOB2 is apolipoprotein D (apoD) of the lipocalin or Q 2n-microglobulin superfamily of carrier proteins (Preti etal, 1992 Zeng etal, 1996a,b). 

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