We address the problem of signal reconstruction from intensity measurements with respect to a measurement frame. This nonconvex inverse problem is known as phase retrieval. The case considered in this paper concerns phaseless measurements taken with respect to a Gabor frame. It arises naturally in many practical applications, such as diffraction imaging and speech recognition. We present a reconstruction algorithm that uses a nearly optimal number of phaseless time-frequency structured measurements and discuss its robustness in the case when the measurements are corrupted by noise. We show how geometric properties of the measurement frame are related to the robustness of the phaseless reconstruction.

We investigate the impact of having additional information in a form of linear constraints in the channel identification problem. With those constraints taken into account, the problem turns into solving a linear system that is associated with a block matrix where each submatrix is either a Gabor system matrix or a matrix prescribed by the linear constraints. So, the identifiability hinges on whether one can find some generating windows of the Gabor systems for which the full linear system is solvable. We show that in single-input single-output (SISO) settings as well as in multiple-input multiple-output (MIMO) settings, linear constraints consisting of a single equation are beneficial for channel identification, as there always exist windows for which the corresponding full linear system is solvable. Concerning multiple linear constraints, however, there exists a set of linear constraints with two equations for which the full linear system is singular for all choices of windows. In the SISO case, we also provide some sufficient conditions on the linear side constraints under which the full linear system is solvable.

We study the recovery of operators with a bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as sampling of operators. Kailath, Bello, and later Kozek and the authors have shown that the sampling of operators is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper, we develop the theory of the sampling of operators in analogy with the classical theory of sampling of bandlimited functions. We define the notions of sampling set and sampling rate for operators and give necessary and sufficient conditions on the sampling rate that depend on the size and geometry of the bandlimiting set. We develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the bandlimiting set under which sampling of operators is possible by their action on a given periodically weighted delta train. We show that under mild geometric conditions on the bandlimiting set, classes of operators are bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. We generalize two results of the Heckel and Bölcskei concerning sampling of operators with area not greater than one-half and less than one, respectively, by finding a larger class of operators to which they apply. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.

This study addresses the problem of stochastic radar target measurement. The authors develop an algorithm that allows for the reconstruction of the scattering function of a wide‐sense stationarity with uncorrelated scattering radar target from the autocorrelation of the response of the target to a deterministic sounding signal. Although conventional methods are applicable only when the scattering function is supported on a ‘rectangle’ with area <1, the authors’ method can handle area 1 support sets of ‘arbitrary’ geometry. Based on the suggested theoretical recovery procedure, the authors propose an unbiased scattering function estimator.

We develop sampling methodology aimed at identifying stochastic operators that satisfy a support size restriction on the autocorrelation of the operators stochastic spreading function. The data that we use to reconstruct the operator (or, in some cases only the autocorrelation of the spreading function) is based on the response of the unknown operator to a known, deterministic test signal.

The goal of channel measurement or operator identification is to obtain complete knowledge of a channel operator by observing the image of a finite number of input signals. In this paper, it is shown that the spreading support of the operator (that is, the support of the symplectic Fourier transform of the Kohn-Nirenberg symbol of the operator) has area less than one then the operator is identifiable. If the spreading support is larger than one, then the operator is not identifiable. The shape of the support region is essentially arbitrary thereby proving a conjecture of Bello. The input signal considered is a weighted delta train where the weights are the window function of a finite Gabor system whose elements satisfy a certain robust completeness property

Let g be a continuous, compactly supported function on such that the integer translates of g constitute a partition of unity. We show that the Gabor system (g,a,b), with window g and time-shift and frequency-shift parameters a,b>0 has no lower frame bound larger than 0 if b=2,3,... and a>0. In particular, (g,a,b) is not a Gabor frame if g is a continuous, compactly supported wavelet scaling function and if b=2,3,... and a>0. We give an example for our result for the case that g=B/sub 1/, the triangle function supported by [-1,1], by showing pictures of the canonical dual corresponding to (g,a,b) where ab=1/4 and b crosses the lines N=2,3,.