A compendium of recent theoretical results associated with using higher-order statistics in signal processing and system theory is provided, and the utility of applying higher-order statistics to practical problems is demonstrated. Most of the results are given for one-dimensional processes, but some extensions to vector processes and multichannel systems are discussed. The topics covered include cumulant-polyspectra formulas; impulse response formulas; autoregressive (AR) coefficients; relationships between second-order and higher-order statistics for linear systems; double C(q,k) formulas for extracting autoregressive moving average (ARMA) coefficients; bicepstral formulas; multichannel formulas; harmonic processes; estimates of cumulants; and applications to identification of various systems, including the identification of systems from just output measurements, identification of AR systems, identification of moving-average systems, and identification of ARMA systems. >

Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications

It is the purpose of this tutorial paper to place bispectrum estimation in a digital signal processing framework in order to aid engineers in grasping the utility of the available bispectrum estimation techniques, to discuss application problems that can directly benefit from the use of the bispectrum, and to motivate research in this area Three general reasons are behind the use of bispectrum in signal processing and are addressed in the paper: to extract information due to deviations from normality, to estimate the phase of parametric signals, and to detect and characterize the properties of nonlinear mechanisms that generate time series

Bispectrum estimation: A digital signal processing framework

The strengths and limitations of correlation-based signal processing methods are discussed. The definitions, properties, and computation of higher-order statistics and spectra, with emphasis on the bispectrum and trispectrum are presented. Parametric and nonparametric expressions for polyspectra of linear and nonlinear processes are described. The applications of higher-order spectra in signal processing are discussed. >

Signal processing with higher-order spectra

i»?The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its Fourier transform, arises in various fields of science and engineering, including electron microscopy, crystallography, astronomy, and optical imaging. Exploring phase retrieval in optical settings, specifically when the light originates from a laser, is natural since optical detection devices [e.g., charge-coupled device (CCD) cameras, photosensitive films, and the human eye] cannot measure the phase of a light wave. This is because, generally, optical measurement devices that rely on converting photons to electrons (current) do not allow for direct recording of the phase: the electromagnetic field oscillates at rates of ~1015âHz, which no electronic measurement device can follow. Indeed, optical measurement/detection systems measure the photon flux, which is proportional to the magnitude squared of the field, not the phase. Consequently, measuring the phase of optical waves (electromagnetic fields oscillating at 1015âHz and higher) involves additional complexity, typically by requiring interference with another known field, in the process of holography.

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Phase Retrieval with Application to Optical Imaging: A contemporary overview

A simple iterative technique has been developed for blind deconvolution of two convolved functions. The method is described, and a number of results obtained from a computational implementation are presented. Some further possible applications are indicated.

Iterative blind deconvolution method and its applications

Due to the turbulent atmosphere the resolution of conventional astrophotography is limited to ∼1 sec of arc. However, the speckle-masking method can yield diffraction-limited resolution, i.e., 0.03 sec of arc with a 3.6-m telescope. Speckle masking yields true images of general astronomical objects. No point source is required in the isoplanatic field of the object. We present the theory of speckle masking; it makes use of triple correlations and their Fourier counterparts, the bispectra. We show algorithms for the recovery of the object from genuine astronomical bispectra data.

Bernhard Wirnitzer

Papers

Speckle masking in astronomy: triple correlation theory and applications