Truncated Fourier domain signal

Highly truncated time domain signals are a feature of three- and higherdimensional NMR experiments. Much effort has therefore been put into finding alternatives to the Fourier transform which will generate spectra without these truncation artefacts. The popular methods are maximum entropy, linear prediction and FDM. Each has its merits and drawbacks they all need to be applied with great care. 

Figure 10.5. (a) Lorentzian band and its P ourier transform, (b) Same curves as in (a) but with a small amount of noise added the noise is particularly recognizable in the region where the Fourier domain signal is low, that is, at the end of the decay function, (c) Result of truncation of the Fourier domain signal to remove the low-SNR region and its transformation back to the spectral domain the noise on the Lorentzian band has been reduced at the expense of resolution. 

This will result in a cancellation of the decay, and the Fourier domain signal becomes a pure tmncated cosine wave, as shown in Figure 10.10. The Fourier transform of a pure truncated cosine wave is a sine function, as shown in Section 2.3. The sine function has a narrower FWHH than almost any other spectral waveform however, it does have large sidelobes. Of course, these could be removed with apo-dization (Section 2.4), but it is usually easier to change the rate of decay in Eq. 10.7. If the Fourier domain array is multiplied by an exponential function with a different FWHH, y such that y < y, that is. 

Figure 10.10. Full Fourier self-deconvolution of a Lorentizan band. The Lorentzian band (a) undergoes Fourier transformation to yield the Fourier domain signal b which has a decay, exp(—ya ). The signal in (b) is multiplied by the inverse exponential decay (c), expC-t-yx), to produce a truncated cosine wave (d). Upon inverse Fourier transformation of (d) a sine waveform is produced (e). The sine waveform has a narrower FWHH than that if the original Lorentzian band.

It should be noted that the use of an appropriate analytic function instead of discrete replica may be beneficial as this (1) is less influenced by noise and (2) allows one to reproduce the wings of the resonances, which are neglected in the original CLEAN algorithm due to intensity threshold. On the other hand, if the line widths in the Fourier domain are mainly due to signal truncation, the fitted parameters poorly reflect the tme signal properties, and this may affect the performance of described procedure. 

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