Processing 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. 

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. 

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It is found that multiplication of the Fourier transform of the data by a carefully chosen window function is very effective in removing the artifacts around peaked functions. This process is called apodization. Apodization with the triangular window function is often applied to Fourier transform spectroscopy interferograms to remove the ringing around the infrared... 

For FTS data, artifact removal is a consideration that is as important as resolution improvement for most researchers in this field. Interferogram continuation methods are not as yet widely known in this area. Methods currently in widespread use that are effective in artifact removal involve the multiplication of the interferogram by various window functions, an operation called apodization. A carefully chosen window function can be very effective in suppressing the artifacts. However, the peaks are almost always broadened in the process. This can be understood from the uncertainty principle. A window that reduces the function most strongly closest to the end points will yield a transform for the modified function that must be broader than it was originally. Alternatively we may employ the convolution... 

Of course, FT-IR has some inherent disadvantages. One of the most important is that the raw data, an interferogram, is for all practical purposes unintelligible to the analyst s eye. So a computer must be utilized to translate the raw data into interpretable data. In the process, a number of factors such as apodization and phasing are introduced which modify the raw data. The analyst must be knowledgeable about these modifications of his data and be prepared to control these modifications in a reproducible and optimum manner. 

Multiplication with a Processing Function s(t) fCt) Weighting , Filtering , Apodization ... 

In order to compensate for ringing, a mathematical process known as apodization is used. This multiplies the collected data by a function that gradually approaches zero and has a value of 1 at the ZPD. Using apodization effectively reduces the ringing in the spectrum, but unfortunately, also the resolution decreases. Several algorithms have been established to eliminate the effects of ringing while maintaining as much resolution in the spectrum as possible. 

Figure 6 Simulated 2H (73.58 MHz) QCPMG spectra corresponding to a 3-by-2-site jump process using parameter set P2b in Table 1 and the single-frame set of Euler angles were69 (0.0,124.0, 0.0), (57.594, 55.006, 91.716), (302.406, 55.006, 268.206), (0.0, 124.0,180.0), (57.594, 55.006, 271.716), (302.406, 55.006, 88.206). The logarithm of the rate constants and k2 are indicated at each row and column of the spectra. All spectra were apodized by Gaussian line broadening of 50 Hz prior to Fourier transformation.

Figure 7 Simulated 2H (73.58 MHz) MAS spectra corresponding to a 3-by-2-site jump process the same parameter set as in Figure 6. The logarithm of the rate constants and k2 are indicated at each row and column of the spectra. All spectra were apodized by Gaussian line broadening of 50 Hz prior to Fourier transformation.

A(8) is called the boxcar function. This limit on the retardation leads to a limit on the resolution of 1/2L, so if L - 100 cm, the highest resolution attainable is 0.005 cm-1. By the convolution theorem, the product of two functions in one space is the same as the convolution of the Fourier transforms of the two functions in the reciprocal space. The effect of multiplying by this boxcar function is to convolve each point in the reciprocal wavenumber space with a sine function [sinc(x) = sin(x)/x Figure 4], An undesirable feature of the sine function as a lineshape is the large amplitude oscillation (the first minimum is -22% of the maximum). This ringing can make it difficult to get information about nearby peaks and leads to anomalous values for intensities. This ringing can be removed by the process known as apodization. 

Figure 14 Stopped-flow TOCSY CEC-NMR spectra of paracetamol glucuronide. Acquisition parameters number of 144 scans for each of 256 increments. Spectral width of 4716 Hz, number of points 4096. Processing parameters zero filled and multiplied by apodization function of 3 Hz in both dimensions. (From Ref. 52 reproduced with permission from The Royal Society of Chemistry.)...

All the processing steps in this section are intended to enhance the time domain data leading to the suppression of distortions or artefacts and an improvement in the overall spectral quality. The suitability of the methods can be estimated from the appearance of the FID in the main spectrum window of ID WIN-NMR as illustrated in the schematic FID shown in Fig. 3.6. The envelope of the exponentially decaying FID has a dc offset as it is not symmetrical about the zero line whilst the spectrum lineshape may be improved by either zero filling or apodization or possibly both. 

Fig. 3.6 Time domain processing steps - dc offset correction, zero filling and apodization.

Apodization is the process of multiplying the FID prior to Fourier transformation by a mathematical function. The type of mathematical or window function applied depends upon the enhancement required the signal-to-noise ratio in a spectrum can be improved by applying an exponential window function to a noisy FID whilst the resolution can be improved by reducing the signal linewidth using a Lorentz-Gauss function. ID WIN-NMR has a variety of window functions, abbreviated to wdw function, such as exponential (EM), shifted sine-bell (SINE) and sine-bell squared (QSINE). Each window function has its own particular parameters associated with it LB for EM function, SSB for sine functions etc. 

The FIDs simulated using NMR-SIM should always be processed using the zero filling and apodization parameters shown in Table 3.1. This table also lists the abbreviations used for these parameters in the Check its. Occasionally parameters different from the recommended values may be used in a Check it this is particularly true in simulations where the relaxation option is disabled, the values of the parameters are chosen to improve the spectrum appearance by minimizing line distortions. The Check it may contain the phrase "use zero filling and an apodization (EM, LB 2.0 Hz)" implying zero filling of SI(r+i) 2 TD and an exponential apodization with a LB factor of 2.0. 

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