Review of some fundamentals of data processing

This chapter is devoted to reviewing some fundamental transforms and analysis procedures commonly used for both signal and data processing in fluid mechanics measurements. The chapter begins with a brief review of the Fourier transform and its digital counterpart the discrete Fourier transform. In particular its use for estimating power spectral density is discussed in detail. This is followed by an introduction of the correlation function and its relation to the Fourier transform. The Hilbert transform completes the introductory topics. The chapter then turns to a rigorous presentation of the proper orthogonal decomposition (POD) in the context of the approximation theory and as an application of singular value decomposition (SVD). The relationship between POD and SVD is discussed and POD is described in a statistical setting using an averaging operation for use with turbulent flows. The different POD approaches are briefly introduced, whereby the main differences between the classical POD and the snapshot POD are highlighted. This section closes with a presentation of the POD as a generalization of the classical Fourier analysis to inhomogeneous directions. The chapter continues with a discussion of conditional averages and stochastic estimation as a means of studying coherent structures in turbulent flows before moving in a final section to a comprehensive discussion of wavelets as a combination of data processing in time and frequency domain. After first introducing the continuous wavelet transform and orthogonal wavelet transform their application in experimental fluid mechanics is illustrated through numerous examples.