Superposition Frames for Adaptive Time-Frequency Analysis and Fast Reconstruction

In this paper, we introduce a broad family of adaptive, linear time-frequency representations termed superposition frames, and show that they admit desirable fast overlap-add reconstruction properties akin to standard short-time Fourier techniques. This approach stands in contrast to many adaptive time-frequency representations in the existing literature, which, while more flexible than standard fixed-resolution approaches, typically fail to provide for efficient reconstruction and often lack the regular structure necessary for precise frame-theoretic analysis. Our main technical contributions come through the development of properties which ensure that our superposition construction provides for a numerically stable, invertible signal representation. Our primary algorithmic contributions come via the introduction and discussion of two signal adaptation schemes based on greedy selection and dynamic programming, respectively. We conclude with two short enhancement examples that serve to highlight potential applications of our approach.