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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.]

The second order sequential best rotation (SBR2) algorithm is a popular algorithm to decompose a parahermitian matrix into approximated polynomial eigenvalues and eigen-vectors. The work horse behind SBR2 is a Givens rotation interspersed by delay operations. In this paper, we investigate and analyse the application of a fast Givens rotation in order to reduce the computation complexity of SBR2. The proposed algorithm inherits the SBR2's proven convergence to a diagonalised and spectrally majorised solution for the polynomial eigenvalues. We provide some analysis and examples for the execution speed of this fast Givens-based SBR2 compared to a standard SBR2 implementation.

We have proposed a broadband subspace approach to detect the presence of weak transient signals; this is based on second order statistics of sensor array data — the space-time covariance matrix — and a polynomial matrix EVD; this covariance matrix and its decomposition can be computed off-line; a subspace decomposition for the noise-only subspace determines a syndrome vector; in the absence of a transient signal, this syndrome only contains noise; a transient signal is likely to protrude into the noise-only subspace, and a change in energy can be detected even if the signal is weak; discrimination can be traded off against decision time; further work: (i) impact of time-varying channels, and (ii) forensic investigation of the transient source once detected.

As part of the last day of the UDRC 2021 summer school, this presentation provides an overview over polynomial matrix methods. The use of polynomial matrices is motivated through a number of broadband multichannel problems, involving space-time covariance matrices, filter banks, or wideband MIMO systems. We extend the utility of EVD from narrowband to broadband solutions via a number of factorisation algorithms belonging to the second order sequential rotation or sequential matrix diagonalisation families of algorithms. In a second part of this presentation, a number of application areas are explored, ranging from precoder and equaliser design for broadband MIMO communications systems, to broadband angle of arrival estimation, broadband beamforming, and the problem of identifying source-sensor transfer paths from the second order statistics of the sensor signals.

This paper introduces a cost function for the smoothness of a continuous periodic function, of which only some samples are given. This cost function is important e.g. when associating samples in frequency bins for problems such as analytic singular or eigenvalue decompositions. We demonstrate the utility of the cost function, and study some of its complexity and conditioning issues.

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.

To emit in the TV white space (TVWS) spectrum, the regulator has requested very strict spectral masks, which can be fulfilled using a FFT-modulated filter-bank multi-carrier system (FBMC) to extract one or several TVWS channels in the 470--790MHz range. Such a system reduces the channel dispersion, but even with near-perfectly reconstructing filter bank, the need for equalisation and synchronisation remains. In this work, we propose a per-band equalisation and synchronisation approach, performed by a constant modulus algorithms running concurrently with a direction-directed adaptation process for faster convergence and reduced phase ambiguity. We compare symbol- and fractionally-spaced versions, and investigate their fixed-point implementation on an FPGA. We compare the performance of the different systems in terms of mean squared error, computational cost, and robustness towards noise.

Filter bank-based multicarrier using offset quadrature amplitude modulation (FMBC/OQAM) systems suffer from inter-symbol-interference (ISI) and co-channel interference (CCI) if the transmission channel is dispersive, or simply if the overall system is not synchronised. This paper considers the solution to an equaliser that can mitigate both ISI and CCI by describing the equivalent transmultiplexed channel including the filter banks as a polyphase matrix of transfer functions, and by using polynomial matrix algebra to equalise this system. We demonstrate that the reduced-rank nature of this channel matrix requires a polynomial matrix pseudo-inverse, and demonstrate how the overall FBMC/OQAM system is orthogonalised, thus removing ISI and ICI.

State of the art narrowband noise cancellation techniques utilise the generalised eigenvalue decomposition (GEVD) for multichannel Wiener filtering which can be applied to independent frequency bins in order to achieve broadband processing. Here we investigate the extension of the GEVD to broadband, polynomial matrices, akin to strategies that have already been developed by McWhirter et. al on the polynomial matrix eigenvalue decomposition (PEVD).

Simplified broadband beamformers can be constructed by sharing a single tapped-delay-line within a narrowband subarray. This paper discusses the use of fractional delay filters to a steering in the digital domain. For the narrowband subarrays, an optimisation approach is proposed to maintain an off-broadside look direction constraint as best as possible across a given frequency range. We demonstrate the advantage that this approach has for generating beamformers with accurate off-broadside look direction compared to a benchmark.

This paper explores how the accuracy of short-term prediction of wind speed and direction can be enhanced by considering the diurnal variation of the wind. The wind speed and direction are modelled as the magnitude and phase of a complex-valued time series. The prediction is performed by a multichannel filter using the spatio-temporal correlation between measurements at different geographical locations and the past values of the target site. A multichannel complex-valued non-stationary prediction Wiener filter is proposed that takes into account both the seasonal and diurnal variation of the wind. Using hourly wind speed and direction measurements from over 22 Met Office weather stations distributed across the UK, we demonstrate that there can be a benefit for predicting one hour ahead when taking into account the diurnal and seasonal cyclo-stationary nature of the wind.

In this paper we show how 5 GHz and “TV White Space” wireless networks can be combined to provide fixed access for a rural community. Using multiple technologies allows the advantages of each to be combined to overcome individual limitations when assigning stations between networks. Specifically, we want to maximise throughput under the constraint of satisfying both the desired individual station data rate and the transmit power within regulatory limits. For this optimisation, we employ Pearl's algorithm, a Bayesian belief propagation implementation, which is informed by statistics drawn from network trials on Isle of Tiree with 100 households. The method confirms results obtained with an earlier deterministic approach.

The use of multiple frequency bands within a wireless network allows the advantages of each band to be exploited. In this paper we discuss how HopScotch, a rural wireless broadband access test bed running in the Scottish Highlands and Islands, uses both 5 GHz and ultra high frequency "white space" bands to offer large data rates and expansive coverage whilst reducing the number of base stations or required transmission power. This reduction in energy consumption allows HopScotch to provide a low-cost and green solution for rural broadband delivery.

We present three joint adaptive mean squared error receiver structures for a time division duplex down-link scenario in which a single mobile station is allocated several code-multiplexed channels. The first structure is a bank of single channel MMSE receivers. The second is composed of an equaliser common to all the received channels and a bank of matched filters for despreading. The equaliser can be updated using a multiple-error filtered-X LMS algorithm. The final proposed structure features both an equaliser and a bank of MMSE receivers. The performance of the three architectures is tested and compared under multipath fading channel propagation conditions and BER values are provided.

This paper reports on a student project performed at the University of Southampton jointly by 4th year MEng students within the course “Advanced Radio Communications”. The aim was to design a software modem capable of transmitting 16kb/s of data, whereby random number generation, advanced modulation, pulse shaping, synchronisation, and error counting techniques had to be applied. The ultimate aim was the implementation on a Texas Instruments TMS320-C6201 EVM board, which dictated some of the specifications of the design.