The Fourier transform

The Fourier transform is probably the best known and most commonly used data analysis tool in the domain of fluid mechanics and aeroacoustics (and indeed in engineering in general) – the Fourier power spectrum of the sound field radiated by an aeroacoustic system is the quantity that mod­
elling tools are required to reproduce; it is the quantity by which we most often endeavour to assess and understand the behaviour of the system. We recall it briefly in this section, simply so as to have it appear in juxtaposi­tion with a number of alternative, but less commonly used, data-processing tools. We do so because three of the latter (the wavelet transform, Proper Orthogonal Decomposition, and Dynamic Mode Decomposition), as evoked above, bear certain similarities to the Fourier transform in terms of the way their result can be useful as an aid to understanding and modelling; indeed these alternative processing techniques might be best thought of as surrogate tools for assessing complex data in situations where the Fourier transform may not necessarily be the best choice.

The Fourier transform involves the expansion of a given data set in terms of analytical basis functions that are specified a priori; there is no flexibility in this choice. The Fourier transform and its inverse are defined as

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