Data Manipulation After the Fourier Transform

12 DATA MANIPULATION AFTER THE FOURIER TRANSFORM 3.12.1 Phase Correction 

After you Fourier transform your FID, you get a frequency-domain spectrum with peaks, but the shape of the peaks may not be what you expected. Some peaks may be upside down, whereas others may have a dispersive (half up-half down) lineshape (Fig. 3.36). The shape of the peak in the spectrum (+ or — absorptive, + or — dispersive) depends on the starting point of the sine function in the time-domain FID (0° or 180°, 90° or —90°). The starting point of a sinusoidal function is called its phase. Phase errors come in all possible angles, including those intermediate between absorptive and dispersive (Fig. 3.37). The spectrum has to be phase corrected ( phased ) after the Fourier transform to obtain the 

Recall that the raw NMR data (FID) consists of two numbers for each data point one real value and one imaginary value. After the Fourier transform, there are also two numbers for each frequency point one real and one imaginary. In a perfect world, the real spectrum would be in pure absorptive mode (normal peak shape) and the imaginary spectrum would be in pure dispersive (up/down) mode. In reality, each spectrum is a mixture of absorptive and dispersive modes, and the proportions of each can vary with chemical shift (usually in a linear 

Absorptive spectrum = (real spectrum) x cos(0) + (imaginary spectrum) x sin(0) 

The angle 0 can be thought of as a rotation of the two mutually perpendicular vectors representing the real and imaginary spectra. The problem of phase correction boils down to finding the correct phase rotation angle 0. Well, actually it is a little more complicated because the phase correction 0 is usually a linear function of the chemical shift (8). Defining the line 

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After Fourier transformation, the effects on the spectrum of data manipulations, such as phase adjustments, can be controlled on a display before giving final calculating commands. Communication with the computer is generally via keyboard and graphic display. Light pen control via oscilloscope is also possible. 

Due to the fact that the first phase of manipulation of such data is usually a fast scanning of the entire collection, a highly compressed representation of uniformly coded data is essential in order to accelerate the handling. After the search reduces the collection to a smaller group in which the target object is supposed to be, the full (extended) representation of objects can be invoked if necessary for further manipulation. In the next sections we shall discuss the use of two methods, Fast Fourier Transformation (FFT) and Fast Hadamard Transformation (FHT), for the reduction of object representations and show by some examples in 1- and 2-dimensional patterns (spectra, images) how the explained procedures can be used... 

Before 1989, Fourier transform (FT) and fast Fourier transform (FFT) were employed mainly by chemists to manipulate data from analytical studies [5-7]. After the publication of an important paper by Daubechies [8] in 1988, a new transformation algorithm called wavelet transform (WT) became a popular method in various fields of science and engineering for signal processing. This new technique has been demonstrated to be fast in computation, with localization and quick decay properties, in contrast to existing methods such as FFT. A few chemists have applied this new method for... 

In the next section, a brief summary will be made of the various 2D NMR techniques that have been developed in recent years. Before proceeding to deal with the bewildering variety of pulse sequences and data manipulation methods available, it is instructive to consider a simple example of a double Fourier transform NMR experiment. One of the earliest techniques proposed was that of MUller, Kumar and Ernst,4 which allows multiplet structure to be separated from chemical shifts in proton-coupled carbon-13 NMR. The pulse sequence for this experiment, illustrated in Figure 1, consists simply of a 90" carbon-13 pulse followed by a delay tj, after which wideband proton decoupling is turned on and a proton-decoupled free induction decay recorded. Since no decoupling is used during tj, the resonance positions in f] in the resultant 2D spectrum are those expected for the proton-coupled conventional spectrum, and the positions in f2 are the chemical shift frequencies found in the normal decoupled spectrum. 

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