Gabor's “complex signal” revisited: Complexifying frames and bases

Abstract

AbstractIn 1946, Dennis Gabor introduced the analytic signal for real‐valued signals f. Here, H is the Hilbert transform. This complexification of functions allows for an analysis of their amplitude and phase information and has ever since given well‐interpretable insight into the properties of the signals over time. The idea of complexification has been reconsidered with regard to many aspects: examples are the dual tree complex wavelet transform, or via the Riesz transform and the monogenic signal, that is, a multi‐dimensional version of the Hilbert transform, which in combination with multi‐resolution approaches leads to Riesz wavelets, and others. In this context, we ask two questions: Which pairs of real orthonormal bases (ONBs), Riesz bases, frames and Parseval frames and can be “rebricked” to complex‐valued ones ? And which real operators A allow for rebricking via the ansatz ? In this short note, we give answers to these questions with regard to a characterization which linear operators A are suitable for rebricking while maintaining the structure of the original real valued family. Surprisingly, the Hilbert transform is not among them.