Estimates For Phase Retrieval From Gabor Transformation And Related Studies Of The Balian-Low Theorem

This paper mainly discuss the Stability estimates for phase retrieval from discrete linear canonical gabor transformation and generalizes Balian-Low theorem associated with the special affine fourier transform.Phase retrieval usually refers to the nonlinear inverse problem of recovering some signals from phase free measurements.Phase retrieval is an important issue in the field of engineering physics,which mainly recovers signals with phaseless measurements.Gabor systems and Gabor transforms are one of the important time-frequency analysis tools for non-stationary signals.This time spectrum obtained from Gabor transform coefficients reveals the characteristics of the frequency components and their evolution contained in the signal.The content of the Chapter 2,3,and 4 mainly starts from the measurement of discrete linear regular Gabor transform,and studies the stability estimation of phase retrieval(DLCGT).We first prove that we can recover the signal x on the connected component of G in the global phase sense.Then,we give stability estimates based on DLCGT measurements and autocorrelation relationships.Balian-Low theorem is a form of uncertainty principle,which is related to the time-frequency analysis field of signal and image processing.Special affine Fourier transform(SAFT)summarizes many well-known unitary transformation and mathematical operations of signal processing and optical correlation.Special Fourier Transform(SAFT),which relies on six independent parameter phases,extends the traditional Fourier transform to the time-frequency domain represented by a parameter matrix A(28)(a,b;c,d;p,q).In Chapter 5 of this paper,we mainly study the BalianLow theorem related to special affine Fourier transform domains.