Hilbert transform revisited – Proper orthogonal decomposition applied to analytical signals of flow fields

Abstract

The modes delivered by proper orthogonal decomposition (POD) are uncorrelated as per definition; but interestingly, they are not necessarily independent in terms of spatio-temporal flow-pattern dynamics. For instance, periodic structures that travel as waves through a series of snapshots often consist of pairs of modes with harmonic functions shifted 90 degree in phase and/or a spatial offset by a quarter of the spatial wave length of the convective flow pattern. Identification of such pairs, however, largely builds upon experience, visual inspection and/or the analysis of the reconstructed coefficients in cyclograms (Lissajous figures). This effort becomes even more challenging if measurement noise or other spurious information contaminates the raw data under consideration. One possibility to automatically pair corresponding patterns with common POD algorithms is the immediate application of the POD method to complex data (see Pfeffer et al., 1990). As outlined by Horel (1984), the Hilbert transform is a well-known and straight forward means to obtain the required extension of the original signal with an appropriate 90 degrees phase shift, which is independent of the fundamental frequencies. The complex extension of the original (real) signal X_i and its (discrete) Hilbert transform HT{X_i} as the imaginary part X_i +iHT{X_i} with the imaginary unit i is commonly known as the so-called analytical signal.