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

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