Minimal Measurements And Almost Phase Retrieval

In this paper, we study the minimal measurements and phase retrieval in signal processing.We introduce the background and researches about measurements and phase re-trieval in Chapter1. And we show some basic definitions and propositions about frame and compressed sensing in Chapter2. LetΦ be an overcomplete frame. For all frame coefficients, there exists a matrix A such that Ac=0.(*) From equation (*), we study the following problems.In Chapter3, we study the minimal measurements in compressed sensing. For almost all K-sparse vectors, K+1measurements are enough for perfect reconstruction. As an application, we consider the impulse noise removal for the orthogonal frequency-division multiplexing (OFDM) system.In Chapter4, first we show the definitions and properties of frames who give phase retrieval. We show that almost all vectors of length N can be reconstruct with N N＋1intensity measurements by some frames. Then we give a characterization of such frames. We provide a method to test if a frame has this property or not. With our method, we can construct (tight) frames or modify any frames such that they give almost phase retrieval.In Chapter5, we study the phase retrieval by fusion frames. We give the definition of fusion frames who give almost phase retrieval. For almost all vectors of length N, we show that N+1measurements are enough for reconstruction. At the same time, we give a method to construct such fusion frames.In Chapter6, by equation (*), we study the problems of recovering vectors from the rearranged frame coefficients or frame coefficients with erasures at either known or unknown locations. For almost all vectors, we prove that DFT matrix can be used to recover vectors from the rearranged frame coefficients in some sense.