Frame Domain Signal Processing: Framework and Applications

Besides basis expansions, frames representations play a key role in signal processing. We thus consider the problem of frame domain signal processing, which is more complex and challenging than transform domain processing. Examples of such processing abound, from overlap-add/save convolution, to frequency domain LMS, and frame magnitude reconstruction. We develop a unified view of all these situations by using a common Hilbert space view of the problem, and consider algorithms in this common framework. In addition to a synthetic view of multiple signal processing methods in frames, we derive several original results. This include a direct solution to spectral modification (which usually uses an iterative algorithm) and a unicity condition for reconstruction from frame coefficient magnitudes.