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We investigate the singular value decomposition (SVD) of a rectangular matrix $\boldsymbol{\mathit{A}}(z)$ of functions that are analytic on an annulus that includes at least the unit circle. Such matrices occur, e.g., as matrices of transfer functions representing broadband multiple-input multiple-output systems. Our analysis is based on findings for the analytic SVD applicable to continuous time systems, and on the analytic eigenvalue decomposition. Using these, we establish two potentially overlapping cases where analyticity of the SVD factors is denied. Firstly, from a structural point of view, multiplexed systems require oversampling by the multiplexing factor in order to admit an analytic solution. Secondly, from an algebraic perspective, we state under which condition spectral zeros of any singular value require additional oversampling by a factor of two if an analytic solution is to be found. In all other cases, an analytic matrix admits an analytic SVD, whereby the singular values are unique up to a permutation, and the left- and right-singular vectors are coupled through a joint ambiguity w.r.t. an arbitrary allpass function. We demonstrate how some state-of-the-art polynomial matrix decomposition algorithms approximate this solution, motivating the need for dedicated algorithms.

This article is devoted to the polynomial eigenvalue decomposition (PEVD) and its applications in broadband multichannel signal processing, motivated by the optimum solutions provided by the EVD for the narrowband case [1], [2]. In general, we would like to extend the utility of the EVD to also address broadband problems. Multichannel broadband signals arise at the core of many essential commercial applications, such as telecommunications, speech processing, health-care monitoring, astronomy and seismic surveillance, and military technologies, including radar, sonar, and communications [3]. The success of these applications often depends on the performance of signal processing tasks, including data compression [4], source localization [5], channel coding [6], signal enhancement [7], beamforming [8], and source separation [9]. In most cases and for narrowband signals, performing an EVD is the key to the signal processing algorithm. Therefore, this article aims to introduce the PEVD as a novel mathematical technique suitable for many broadband signal processing applications.

An analytic parahermitian matrix admits in almost all cases an eigenvalue decomposition (EVD) with analytic eigenvalues and eigenvectors. We have previously defined a discrete Fourier transform (DFT) domain algorithm which has been proven to extract the analytic eigenvalues. The selection of the eigenvalues as analytic functions guarantees in turn the existence of unique one-dimensional eigenspaces in which analytic eigenvectors can exist. Determining such eigenvectors is not straightforward, and requires three challenges to be addressed. Firstly, one-dimensional subspaces for eigenvectors have to be woven smoothly across DFT bins where a non-trivial algebraic multiplicity causes ambiguity. Secondly, with the one-dimensional eigenspaces defined, a phase smoothing across DFT bins aims to extract analytic eigenvectors with minimum time domain support. Thirdly, we need to check whether the DFT length, and thus the approximation order, is sufficient. We propose an iterative algorithm for the extraction of analytic eigenvectors and prove that this algorithm converges to the best of a set of stationary points. We provide a number of numerical examples and simulation results, in which the algorithm is demonstrated to extract the ground truth analytic eigenvectors arbitrarily closely.

A number of algorithms are capable of iteratively calculating a polynomial matrix eigenvalue decomposition (PEVD), which is a generalisation of the EVD and will diagonalise a parahermitian polynomial matrix via paraunitary operations. While offering promising results in various broadband array processing applications, the PEVD has seen limited deployment in hardware due to the high computational complexity of these algorithms. Akin to low complexity divide-and-conquer (DaC) solutions to eigenproblems, this paper addresses a partially parallelisable DaC approach to the PEVD. A novel algorithm titled parallel-sequential matrix diagonalisation exhibits significantly reduced algorithmic complexity and run-time when compared with existing iterative PEVD methods. The DaC approach, which is shown to be suitable for multi-core implementation, can improve eigenvalue resolution at the expense of decomposition mean squared error, and offers a trade-off between the approximation order and accuracy of the resulting paraunitary matrices. [ABSTRACT FROM AUTHOR]

This paper expands the concept of the power method to polynomial para-Hermitian matrices in order to extract the principal analytic eigenpair. The proposed technique involves repeatedly multiplying the para-Hermitian matrix by a polynomial vector, followed by an appropriate normalization of the resulting product in each iteration, under the assumption that the principal analytic eigenvalue spectrally majorises the remaining eigenvalues. To restrain the growth in polynomial order of the product vector, truncation is performed after normalization in each iteration. The effectiveness of this proposed method has been verified through simulation results on an ensemble of randomly generated para-Hermitian matrices, demonstrating superior performance compared to existing algorithms.

There is an increasingly important requirement for day and night, wide field of view imaging and tracking for both imaging and sensing applications. Applications include military, security and remote sensing. We describe the development of a proof of concept demonstrator of an adaptive coded-aperture imager operating in the mid-wave infrared to address these requirements. This consists of a coded-aperture mask, a set of optics and a 4k x 4k focal plane array (FPA). This system can produce images with a resolution better than that achieved by the detector pixel itself (i.e. superresolution) by combining multiple frames of data recorded with different coded-aperture mask patterns. This superresolution capability has been demonstrated both in the laboratory and in imaging of real-world scenes, the highest resolution achieved being ½ the FPA pixel pitch. The resolution for this configuration is currently limited by vibration and theoretically ¼ pixel pitch should be possible. Comparisons have been made between conventional and ACAI solutions to these requirements and show significant advantages in size, weight and cost for the ACAI approach.

A novel waveform design technique for enabling a communication channel within a pulse radar is presented. The proposed waveform is composed of quasi-orthogonal chirp sub-carriers generated by means of the Fractional Fourier Transform (FrFT), with the aim of preserving the radar performance of a typical Linear Frequency Modulated (LFM) pulse while embedding data to be sent to a cooperative system. Waveform generation and demodulation are described, together with techniques aimed at optimising the design parameters and mitigating the Inter-Carrier Interference (ICI) caused by the quasi-orthogonality of the chirp sub-carriers. The proposed FrFT based communicating-radar (CoRadar) waveform design is compared with Orthogonal Frequency Division Multiplexing (OFDM) based CoRadar with respect to both radar and communication operations. Radar performance is evaluated through examination of the Ambiguity Function (AF) and by assessing the performance of a standard square law detector. Communication performance is shown in terms of Bit Error Ratio (BER) for different channel conditions. Results demonstrate that the proposed FrFT waveform presents performance close to a LFM pulse in terms of probability of detection and probability of false alarm, in exchange for slightly worse range and Doppler resolution. Furthermore, it is shown to maintain comparable communication performance with respect to the OFDM waveform. Finally, a hardware implementation is described that demonstrates the simultaneous radar and communication capabilities of the proposed system. [ABSTRACT FROM AUTHOR]