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[Formulation of broadband array problems using polynomial matrix notation; polynomial matrix factorisations; broadband AoA estimation via polynomial matrix techniques; broadband MVDR adaptive beamforming.]

In Weiss (2018), we stated that any positive semi-definite parahermitian matrix R (z): C→CMxM that is analytic on an annulus containing at least the unit circle will admit a decomposition with analytic eigenvalues and analytic eigenvectors. In this note, we further qualify this statement, and define the class of matrices that fulfills the above properties yet does not admit an analytic EVD. We follow the notation in Weiss (2018).

This paper addresses the extension of the factorisation of a Hermitian matrix by an eigenvalue decomposition (EVD) to the case of a parahermitian matrix that is analytic at least on an annulus containing the unit circle. Such parahermitian matrices contain polynomials or rational functions in the complex variable z, and arise e.g. as cross spectral density matrices in broadband array problems. Specifically, conditions for the existence and uniqueness of eigenvalues and eigenvectors of a parahermitian matrix EVD are given, such that these can be represented by a power or Laurent series that is absolutely convergent, at least on the unit circle, permitting a direct realisation in the time domain. Based on an analysis on the unit circle, we prove that eigenvalues exist as unique and convergent but likely infinite-length Laurent series. The eigenvectors can have an arbitrary phase response, and are shown to exist as convergent Laurent series if eigenvalues are selected as analytic functions on the unit circle, and if the phase response is selected such that the eigenvectors are Hölder continuous with α>½ on the unit circle. In the case of a discontinuous phase response or if spectral majorisation is enforced for intersecting eigenvalues, an absolutely convergent Laurent series solution for the eigenvectors of a parahermitian EVD does not exist. We provide some examples, comment on the approximation of a parahermitian matrix EVD by Laurent polynomial factors, and compare our findings to the solutions provided by polynomial matrix EVD algorithms.

The increasing demand of spectrum resources and the need to keep the size, weight and power consumption of modern radar as low as possible, has led to the development of solutions like joint radar-communication systems. In this paper a novel Fractional Fourier Transform (FrFT) based multiplexing scheme is presented as joint radar-communication technique. The FrFT is used to embed data into chirp sub-carriers with different time-frequency rates. Some optimisation procedures are also proposed, with the objective of improving the bandwidth occupancy and the bit rate and/or Bit Error Ratio (BER). The generated waveform is demonstrated to have a good rejection to distortions introduced by the channel, leading to low BER, while keeping good radar characteristics compared to a widely used Linear Frequency Modulated (LFM) pulse with same duration and bandwidth.

Publication in the conference proceedings of EUSIPCO, Lisbon, Portugal, 2014

This paper demonstrates the ease with which broadband array problems can be generalised from their well-known, straightforward narrowband equivalents when using polynomial matrix formulations. This is here exemplified for the Capon beamformer, which presents a solution to the minimum variance distortionless response problem. Based on the space-time covariance matrix of the array and the definition of a broadband steering vector, we formulate a polynomial MVDR problem. Results from its solution in the polynomial matrix domain are presented.