Reconstruction of a Complex-Valued Field Using the Hilbert-Hankel Transform1

A well-known property in Fourier transform theory is that causality in one domain implies real-part sufficiency in the other domain. This property is the basis for the fact that the real and imaginary parts of a signal are related via the Hilbert transform, if the spectrum of the signal is causal. In wave propagation problems involving circular symmetry, it is the circularly symmetric two-dimensional Fourier transform, or equivalently the Hankel transform, which is of central importance. Because of the circular symmetry in such problems, the condition of causality is not applicable. However, in our work we have shown that under some circumstances, it is possible to relate the real and imaginary parts of a propagating field described by a Hankel transform. In this paper, an approximate real-part sufficiency condition for the Hankel transform is developed and an algorithm for reconstructing the real (or imaginary) component from the imaginary (or real) component is applied to synthetic and experimental underwater acoustic fields.