Fourier series and transforms in one dimension

We begin our discussion of Fourier analysis by considering the representation of a periodic function f t) with a period of 2P, f t + 2P) = f t). If f(f) has a finite number of local extrema and a finite number of times tj e [0,2P] at which it is discontinuous, Dirichlet s theorem states that it may be represented as the Fourier series 

A similar procedure, but multiplying by sminnt/P) instead, yields 

The summations above are over an infinite number of terms, but if we tnmcate the series to some finite order TV, we obtain an approximate Fonrier representation of the function  

To compute the 2N + 1 coefficients of this expansion, we might consider using numerical quadrature for the necessary integrals however, as TV increases, so does the required number of quadrature points, as the sine and cosine basis functions vary more rapidly witii increasing m. Thus, the amount of work necessary to obtain an approximate Fourier representation in this manner scales as TV. Below, we consider an alternative method that reqnires only NXogiN TV operations. 

Convergence of the Fonrier representation to the true function /(f) with increasing TV can be quite slow, particularly when the function is discontinuous or varies rapidly over a small interval. As an example, consider the square pulse function 

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By incrementing the interval of the second time dimension, a series of information-rich spectra is produced that contains the chemical shift frequency in one dimension and either the interaction frequency or interaction correlation in the second dimension. The projection of the frequency of these interactions, in the form of time-dependent intensity modulation or magnetization transfers along a second axis creates the two-dimensional spectrum. Fourier Transformation along this second dimension results in a tremendous resolution of spectral information. Several reviews of applications of two-dimensional NMR have appeared (37, 48, 105, 259, 277, 377). Only those experiments in routine use or of high potential value for natural products chemists are presented. 

In a 2D experiment one or more scans are acquired with a delay tl that is incremented in subsequent acquisitions to generate a time domain tl. The time domain tl in conjunction with the acquisition time domain t2 generates a 2D data set that upon double Fourier transform gives a 2D spectrum. In a very simplified view all 2D experiments can be described as series of ID experiments but in practise the situation is rather more complicated because to achieve quadrature detection in both dimensions phase cycling or pulse field gradients must be used. Consequently the processing of 2D data sets depends upon the detection mode and the experimental setup. 

Magnetization transfer occurs between coupled spins during the interval t, so that the amplitude of the signal for each nucleus is a sine function of which depends on its own frequency and the frequencies of nuclei coupled to it. In the 2D experiment, the above sequence is repeated for a series of time intervals, t, and the FID transformed for each as usual with respect to the data acquisition time. Then a second Fourier transformation with respect to t, is performed, giving a spectrum which has chemical shift data in two dimensions,/, and/2, corresponding to the two transformations. The peaks that appear in the spectrum can be described by co-ordinates in (/"i /2). Those on the diagonal reflect the one-dimensional spectrum and occur at the same chemical shift position (v, v) in both dimensions. Off-diagonal or cross-peaks... 

Figure 5.37. A schematic presentation of the mechanics of obtaining a 2D-NMR spectrum (a) FIDs are recorded at differing values of ti (b) the FIDs are schematically shown as signals in 1/ 2 dimensions (c) Fourier transformation of the rows of FIDs afford a series of spectra in ti and Wj dimensions (d) transposition and Fourier transformation of columns of FIDs afford the 2D-NMR spectrum with signals in the (1)1 and (O2 dimensions. The chemical shifts lie along one axis while the coupling constants lie along the other axis.

We now perform our second series of Fourier transformations on each of the 1024 interferograms to produce literally a transform of a transform. This result is the end product a 2-D spectrum we are now faced, however, with the challenge of visualizing our results. One way to plot the data is as a stacked plot similar to the plot that we have already seen in Figure 5.5a. This type of plot, shown in the left part of Figure 5.6, gives a sense of three-dimensions. Note that the two axes are now labeled F2 and FI, which is consistent with the rest of the text and is commonly used. For this spectrum, this... 

Similar to a one-dimensional experiment where is varied, in a two-dimensional experiment, a series of FIDs from experiments carried out at varying tj are processed. However, in the two-dimension experiment, the amplitudes of the signals at particular frequencies are read and Fourier-transformed a second time. The resulting signals are then a function of the two times tx and t2 and after Fourier transformation, frequencies, v and v2. The plot is usually given as a contour. 

The remarkable stability and controllability of NMR spectrometers permits not only the precise accumulation of FIDs over several hours, but also the acquisition of long series of spectra differing only in some stepped variable such as an interpulse delay. A peak at any one chemical shift will typically vary in intensity as this series is traversed. All the sinusoidal components of this variation with time can then be extracted, by Fourier transformation of the variations. For example, suppose that the normal ID NMR acquisition sequence (relaxation delay, 90° pulse, collect FID) is replaced by the 2D sequence (relaxation delay, 90° pulse, delay t-90° pulse, collect FID) and that x is increased linearly from a low value to create the second dimension. The polarization transfer process outlined in the previous section will then cause the peaks of one multiplet to be modulated in intensity, at the frequencies of any other multiplet with which it shares a coupling. 

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