## Data Processing Methods: Fourier and Beyond

Vladislav Orekhov, Paweł Kasprzak, **Krzysztof Kazimierczuk**

This chapter presents a compact overview of both practical and rigorously mathematical aspects of modern NMR signal processing. It discusses the properties of the Fourier transform (FT), which will be later useful to explain the effects of the experimental imperfections and signal processing procedures. The fast FT algorithm, used to calculate the discrete FT requires the same number of points in the input and output. However, one can increase the number of spectral points to any desired value by *zero filling* , that is, extending the free induction decay by adding artificial data points equal to zero at the end. The quadrature detection in one-dimensional spectra is realized through the acquisition of the two modulations, interpreted as real and imaginary parts of a complex NMR signal. The Projection Theorem is a powerful tool, useful in accelerating NMR experiments of dimensionality three and more.