Reduced Ambiguity Calibration for LOFAR

Interferometric calibration always yields non unique solutions. It is therefore essential to remove these ambiguities before the solutions could be used in any further modeling of the sky, the instrument or propagation effects such as the ionosphere. We present a method for LOFAR calibration which does not yield a unitary ambiguity, especially under ionospheric distortions. We also present exact ambiguities we get in our solutions, in closed form. Casting this as an optimization problem, we also present conditions for this approach to work. The proposed method enables us to use the solutions obtained via calibration for further modeling of instrumental and propagation effects. We provide extensive simulation results on the performance of our method. Moreover, we also give cases where due to degeneracy, this method fails to perform as expected and in such cases, we suggest exploiting diversity in time, space and frequency.Comment: Draft version. Final version published on 10 April 201

Nonorthogonal approximate joint diagonalization with well-conditioned diagonalizers

To make the results reasonable, existing joint diagonalization algorithms have imposed a variety of constraints on diagonalizers. Actually, those constraints can be imposed uniformly by minimizing the condition number of diagonalizers. Motivated by this, the approximate joint diagonalization problem is reviewed as a multiobjective optimization problem for the first time. Based on this, a new algorithm for nonorthogonal joint diagonalization is developed. The new algorithm yields diagonalizers which not only minimize the diagonalization error but also have as small condition numbers as possible. Meanwhile, degenerate solutions are avoided strictly. Besides, the new algorithm imposes few restrictions on the target set of matrices to be diagonalized, which makes it widely applicable. Primary results on convergence are presented and we also show that, for exactly jointly diagonalizable sets, no local minima exist and the solutions are unique under mild conditions. Extensive numerical simulations illustrate the performance of the algorithm and provide comparison with other leading diagonalization methods. The practical use of our algorithm is shown for blind source separation (BSS) problems, especially when ill-conditioned mixing matrices are involved

Well-posedness of the permutation problem in sparse filter estimation with lp minimization

Convolutive source separation is often done in two stages: 1) estimation of the mixing filters and 2) estimation of the sources. Traditional approaches suffer from the ambiguities of arbitrary permutations and scaling in each frequency bin of the estimated filters and/or the sources, and they are usually corrected by taking into account some special properties of the filters/sources. This paper focusses on the filter permutation problem in the absence of scaling, investigating the possible use of the temporal sparsity of the filters as a property enabling permutation correction. Theoretical and experimental results highlight the potential as well as the limits of sparsity as an hypothesis to obtain a well-posed permutation problem

Bispectrum Inversion with Application to Multireference Alignment

We consider the problem of estimating a signal from noisy circularly-translated versions of itself, called multireference alignment (MRA). One natural approach to MRA could be to estimate the shifts of the observations first, and infer the signal by aligning and averaging the data. In contrast, we consider a method based on estimating the signal directly, using features of the signal that are invariant under translations. Specifically, we estimate the power spectrum and the bispectrum of the signal from the observations. Under mild assumptions, these invariant features contain enough information to infer the signal. In particular, the bispectrum can be used to estimate the Fourier phases. To this end, we propose and analyze a few algorithms. Our main methods consist of non-convex optimization over the smooth manifold of phases. Empirically, in the absence of noise, these non-convex algorithms appear to converge to the target signal with random initialization. The algorithms are also robust to noise. We then suggest three additional methods. These methods are based on frequency marching, semidefinite relaxation and integer programming. The first two methods provably recover the phases exactly in the absence of noise. In the high noise level regime, the invariant features approach for MRA results in stable estimation if the number of measurements scales like the cube of the noise variance, which is the information-theoretic rate. Additionally, it requires only one pass over the data which is important at low signal-to-noise ratio when the number of observations must be large

Penalty function-based joint diagonalization approach for convolutive blind separation of nonstationary sources

A new approach for convolutive blind source separation (BSS) by explicitly exploiting the second-order nonstationarity of signals and operating in the frequency domain is proposed. The algorithm accommodates a penalty function within the cross-power spectrum-based cost function and thereby converts the separation problem into a joint diagonalization problem with unconstrained optimization. This leads to a new member of the family of joint diagonalization criteria and a modification of the search direction of the gradient-based descent algorithm. Using this approach, not only can the degenerate solution induced by a unmixing matrix and the effect of large errors within the elements of covariance matrices at low-frequency bins be automatically removed, but in addition, a unifying view to joint diagonalization with unitary or nonunitary constraint is provided. Numerical experiments are presented to verify the performance of the new method, which show that a suitable penalty function may lead the algorithm to a faster convergence and a better performance for the separation of convolved speech signals, in particular, in terms of shape preservation and amplitude ambiguity reduction, as compared with the conventional second-order based algorithms for convolutive mixtures that exploit signal nonstationarity

Blind, MIMO system estimation based on PARAFAC decomposition of higher order output tensors

IEEE Transactions on Signal Processing, 54(11): pp. 4156-4168.We present a novel framework for the identification of a multiple-input multiple-output (MIMO) system driven by white, mutually independent unobservable inputs. Samples of the system frequency response are obtained based on parallel factorization (PARAFAC) of three- or four-way tensors constructed based on, respectively, third- or fourth-order cross spectra of the system outputs. The main difficulties in frequency-domain methods are frequency- dependent permutation and filtering ambiguities.We show that the information available in the higher order spectra allows for the ambiguities to be resolved up to a constant scaling and permutation ambiguities and a linear phase ambiguity. Important features of the proposed approach are that it does not require channel length information, needs no phase unwrapping, and unlike the majority of existing methods, needs no prewhitening of the system outputs

Convolutive Blind Source Separation Methods

In this chapter, we provide an overview of existing algorithms for blind source separation of convolutive audio mixtures. We provide a taxonomy, wherein many of the existing algorithms can be organized, and we present published results from those algorithms that have been applied to real-world audio separation tasks