The GDST Frame and Inverse Transforms

Generalization of the Discrete Stockwell Transform (GDST) provides a constructive framework to unify and expand various existing discrete Stockwell transforms. Under the developed framework, we are able to flexibly adjust the time/frequency sampling resolution, tailor the amount of information redundancy and computational complexity, reserve the absolutely reference phases as well. In this paper, we propose an admissible condition and prove that any GDST satisfy this condition will constitute a frame in the Hilbert Space, which guides us to derive the explicit formula for the inverse of GDST through finding its canonical dual frame.