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

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