FID Signal and NMR Spectrum as Fourier Transforms

In most pulsed NMR experiments, the rf field is applied off-resonance. Modulated pulse interferograms (Fig. 2.4(e), 2.5(a), 2.6(a), and 2.7(a)) arise because the vectors of transverse magnetization do not precess with a constant phase shift of itj2 relative to the vector Bj. This is demonstrated in Fig. 2.8. The transverse magnetization is then a resultant of two components, t (f) with a phase shift of n/2 relative to B, and u(r) in phase with Bt  

In mathematical treatments of FID and NMR signals, F (t) and f(cu), it is convenient to use complex quantities. The time domain signal is then defined by eq. (2.7). 

Since NMR spectra are not sequences of lines representing discrete Larrnor frequencies but sequences of Lorentzian frequency distributions f(to) (Fig. 1.9), eq. (2.10) must be replaced by eq. (2.11) M0 sin c is multiplied by the frequency function f(to), where a represents the difference between the frequency ojx and the Larrnor frequency distribution con + Aw, w = co1 — (w0 + Aro). Further, Mosin0f(tu)e must be integrated over the Larrnor frequency distribution. Given a Lorentzian line shape as in Fig. 1.9, the limits of integration are oo  

11) can be solved by developing it as a complex series of sines and cosines according to relation (2.9). This is a Fourier series [14]. Thus, an exponential in the time domain, F ((), and a Lorentzian in the frequency domain, f (op, are Fourier transforms of each other [15-17], 

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