This letter addresses the problem of an adaptive linear combiner when using sample statistics. In this case, the Wiener-Hopf solution can be based on sample covariance matrices and cross-correlation vectors. Despite these estimates being inaccurate, the calculated, approximate solution can be very precise. We explore why this is so by interpreting the Wiener-Hopf solution as coming from a least squares problem. We compare this solution to the case where we exploit knowledge about some of the statistics. Surprisingly this has very limited benefits and often is detrimental. We also show why it is disadvantageous to separately estimate the two statistics.

We present a broadband unit-circle minimum variance distortionless response (UC-MVDR) beamformer and its extension with a spatial unit-circle generalized sidelobe canceller (UC-GSC). The proposed approach enforces per-bin unit-circle projection of polynomial zeros obtained from SMI-MVDR and GSC weights, thereby restoring the ensemble zero property while preserving the distortionless main-lobe constraint. Simulation results with multiple wideband interferers demonstrate that UC-MVDR achieves deeper and more stable notches than conventional spatial GSC, while UC-GSC further enhances interference suppression without compromising white noise gain. Both methods maintain positive WNG across the entire band, confirming unit-circle rectification as an efficient and robust strategy for broadband beamforming.

This paper addresses the analytic Procrustes problem, which aims to find the best least-squares paraunitary approximation of a square matrix of analytic transfer functions, or the best paraunitary transformation between two rectangular analytic matrices. This is accomplished by generalising the Procrustes solution from ordinary matrices to the case of matrices of analytic functions via their analytic singular value decomposition (SVD). Different from the ordinary matrix case, the analytic SVD is not restricted to singular values being nonnegative. In the case that singular values do not possess any zero crossings, we can find an analytic paraunitary matrix analogously to the standard Procrustes approach. In the case that singular values exhibit any zero crossings, the solution does not only depend on the left- and right-singular vectors, but also on a discontinuous and hence non-analytic switching function that forces those analytic singular values to become nonnegative real. We show that a close approximation of this switching function can be achieved via a complex-valued allpass filter, for which we suggest a new suitable design to minimise the overall least squares error of the fit. In addition, we propose a DFT domain algorithm to approximate this polynomial Procrustes solution, which avoids ambiguities in the analytic SVD, and possesses proven convergence. Generally, this solution requires a delay for causality, and this delay grows with the approximation order. Examples and simulations demonstrate our proposed method.

We extend the concept of Gram-Schmidt orthogonalisation from a set of ordinary vectors to vectors of analytic functions. Firstly, we address the normalisation of such a vector. Even if its Euclidean norm possesses spectral zeros, we show that it is always possible to find an analytic normalised vector that has unit norm everywhere on the unit circle. Secondly, a sequence of such normalisation steps can be utilised for a Gram-Schmidt procedure applied to analytic matrices of full spatial rank. Analogous to the case of ordinary matrices, where the Gram-Schmidt procedure leads to a QR factorisation, we prove that an analytic QR decomposition exists with an analytic paraunitary matrix and an upper-right triangular matrix of analytic functions as factors. Thirdly, we present algorithms with proven convergence for both the analytic vector normalisation and the analytic QR decomposition. This type of decomposition can find applications in broadband multiple-input multiple-output systems, and we compare our algorithmic realisation to existing solutions in the literature, which do not necessarily converge towards an analytic decomposition.

Uniform rectangular arrays (URA), structured non-uniform rectangular arrays (NURA), and parallelogram shaped (UPgA and NUPgA) arrays admit steering vectors that can be expressed as the Kronecker product of azimuth and elevation steering vectors. Accordingly, the full steering matrix can be represented as the Khatri Rao product of the corresponding azimuth and elevation steering matrices. This paper exploits this structure to develop an economical subspace decoupling framework for two dimensional angle of arrival (AoA) estimation. The proposed method first extracts the joint signal subspace from the spatial covariance matrix. Then it applies a low complexity decoupling scheme to recover the column spaces of the azimuth and elevation steering matrices. With the estimated decoupled subspaces, conventional one dimensional algorithms such as MUSIC, root MUSIC, and ESPRIT can be applied independently along each dimension, followed by pairing through a two dimensional spectral function. Monte Carlo simulations show that the proposed approach achieves higher accuracy than state of the art methods, i.e., two dimensional MUSIC, reduced-dimension MUSIC, and two-dimensional ESPRIT, for medium- and large scale arrays while requiring fewer snapshots, consequently with improved spectral efficiency.

We compare the existence and uniqueness of the analytic singular value decomposition (SVD) of a matrix $\boldsymbol{A}(z)$ to that of the analytic spectral or eigenvalue decomposition (EVD) of $\boldsymbol{R}(z)=\boldsymbol{A}(z) \boldsymbol{A}^{\mathrm{P}}(z)$. Both cases require oversampling if the matrices are connected to multiplexing operations. Additionally, the analytic SVD may require additional oversampling due to zero crossings of its singular values, which, different from ordinary matrices, cannot necessarily be constrained to be non-negative. It has recently been shown that oversampling can be compensated in parts by permitting singular values to be complex-valued holomorphic on the unit circle, and additionally for the SVD to perform a block-diagonalisation to pseudo-circulant subblocks. We demonstrate here that complex-valued singular values can also be motivated through fractional delay factors, link the SVDvariants to the analytic spectral decomposition, and discuss some aspects of the uniqueness of both the SVD and the spectral decomposition.

The Wiener-Hopf solution is based on estimates of the covariance matrix and the cross-correlation vector, which will likely be poor for small sample sizes. Yet the solution can be very accurate. We explore why this is, and why it is disadvantageous to separately estimate the two statistics, or try to include explicit knowledge of the input process to the adaptive system. The results are based on a least squares interpretation of Wiener-Hopf and an exploration of the variances of estimates, which we underpin by simulations.

The Khatri-Rao product is extensively used in array processing, tensor decomposition, and multi-way data analysis. Many applications require a least-squares (LS) Khatri-Rao factorization. In broadband sensor array problems, polynomial matrices effectively model frequency-dependent behaviors, necessitating extensions of conventional linear algebra techniques. This paper generalizes LS Khatri-Rao factorization from ordinary to polynomial matrices by applying it to the discrete Fourier transform (DFT) samples of polynomial matrices. Phase coherence across bin-wise Khatri-Rao factors is ensured via a phasesmoothing algorithm. The proposed method is validated through broadband angle-of-arrival (AoA) estimation for uniform planar arrays (UPAs), where the steering matrix is a polynomial matrix, which can be represented as a Khatri-Rao product between steering matrix in azimuth and elevation directions.

In this paper, we propose a scalable broadband angle-of-arrival (AoA) estimation approach based on an estimated uniform planar array (UPA) polynomial matrix. The proposed method employs least-squares (LS) Khatri-Rao factorization in discrete Fourier transform (DFT) frequency bins to compute samples of the space-time covariance matrix in both azimuth and elevation directions. This approach eliminates the need for computing a polynomial singular value decomposition (PSVD), thereby enhancing scalability with respect to the number of sources and array elements. The proposed method’s performance is showcased in a simulation where a polynomial steering matrix for three sources is deliberately perturbed with varying degrees of estimation error to simulate different signal to noise ratio (SNR) scenarios. With azimuth and elevation angles determined independently, these are paired using an economical angle pairing strategy specifically designed for broadband scenarios.

The Multiple Signal Classification (MUSIC) algorithm has been extended to broadband angle-of-arrival (AoA) estimation through the development of polynomial MUSIC, which relies on a polynomial eigenvalue decomposition (PEVD). However, a PEVD is computationally intensive. In this paper, we propose a novel approach that bypasses the need for a PEVD by directly computing the polynomial subspace projection matrix corresponding to the noise subspace by computing EVD within the discrete Fourier transform (DFT) bins of a space-time covariance. In simulations, we demonstrate that our approach can offer superior accuracy and computational efficiency compared to the existing polynomial MUSIC algorithm.

In this paper, we address the challenge of estimating the power spectral density of noise tied to a single source measured at an array of sensors. The purpose is to identify windows of opportunity when the noise possesses sufficiently low power to potentially observe weaker signals in the environment. The scenario is formulated via basis expansion models, which then define the time-varying ground truth power spectral densities. We compare this to a number of estimation methods based on data. This includes an averaged periodogram — the Welch method — as a baseline. We also apply a space-time covariance matrix estimation approach, where the matrix is perturbed since the estimate is based on finite data; a first advantage of this method is achieved by optimally limiting the lag support based on a recently proposed method; a second advantage is gained by performing a rank one approximation via an analytic eigenvalue decomposition. We discuss some of the relevant theoretical background, and demonstrate in examples and simulations how the number of sensors and some of the parameters of the basis expansion model influence the results.

This paper presents a flexible FPGA digital beamforming architecture to steer an array for wideband radio frequency signals. The architecture combines True Time-Delay (TTD) units and phase-shifting to beamform at digital baseband. The TTDs utilize low order, coefficient-symmetric Farrow structures to rapidly and flexibly adjust fractional sample delays applied to wideband signals with minimal FPGA resources. Bandpass sampling is employed to further reduce resource and power consumption. The designed Farrow structure’s group delay and magnitude characteristics are evaluated. Measured beam patterns are demonstrated through simulation of the proposed FPGA receive array using fixed-point arithmetic. The multiplier utilization of the proposed system is estimated and compared with the literature. Promising results open discussions to hardening Farrow structure cores on FPGAs to serve multiple signal processing techniques required in future communications systems without consuming programmable logic fabric.

This paper aims to design a jamming signal for a broadband array in the presence of known stationary users. If the transmit paths to the sources are known, we can adopt MIMO downlink schemes which can be extended to the broadband case using an analytic singular value decomposition of the different transmission matrices. If the transmit paths to the sources to the be jammed are not known but the source have previously been detected from their emissions, under the assumption of channel reciprocity we utilise a time reversal technique. We compare its performance and robustness to the case of channel knowledge in simulations.

We want to recover paraunitary matrices under small random perturbations. The polynomial Procrustes method, based on the analytic singular value decomposition of the perturbed system, in principle solves this. For small random perturbations, where the analytic singular values are close to unity, we propose a simplified polynomial Procrustes method that exploits this property, but show that the support of the solution is generally increased compared to the perturbed matrix. We therefore embed the simplified Procrustes method into an iterative truncation scheme, which can reduce the support while ensuring that a paraunitary approximation remains within a perimeter that is equivalent to the level of perturbation.

We investigate the detection of a weak transient broadband signal, and compare a polynomial subspace detection approach to a likelihood ratio test. The former is based on an analytic eigenvalue decomposition of the array data in order to derive a subspace projection away from stronger stationary sources that obscure the transient signal. An energy detection in the noise-only subspace has been demonstrated to work well in a number of broadband array applications. In this contribution, we aim to explore its comparison to a statistically optimum test, the likelihood ratio test (LRT). The LRT requires more information about the scenario than the subspace test — namely the data covariance due to the transient signal — but can still serve as a suitable benchmark. Somewhat surprisingly, simulation results show that the more dispersive the propagation environment and the weaker the transient signal is compared to any stationary sources, the better it is to base a test — either the LRT or even a simple energy criterion — on the data in the noise-only subspace. This is due to the reduced matrix dimensions and enhanced condition numbers of the involved space-time covariance matrices.

Spatial encoding, such as Hadamard multiplexing, reduces signal-independent noise by averaging multiple signals in parallel, thus improving the SNR. This study presents the implementation of a multi-element Laser Induced Phased Array (LIPA), synthesised by projecting orthogonally encoded binary laser patterns onto an aluminium block containing nine, 1 mm diameter side-drilled holes (SDHs) arranged radially. The elementary signals corresponding to each individual element of the LIPA were computationally reconstructed from the acquired multiplexed signals. The resulting Full Matrix Capture (FMC) was then processed using the Total Focusing Method (TFM) algorithm to generate TFM images.

In the context of broadband multichannel signal processing, problems can often be formulated using a space- time covariance matrix, and solved using a diagonalisation of this quantity via a polynomial or analytic eigenvalue decomposition (EVD). In this paper, we address the impact that an estimation of the space-time covariance has on the factors of such a decomposition. In order to address this, we consider a linear unbiased estimator based on Gaussian distributed data, and characterise the variance of this estimate, as well as the variance of the error between the estimate and the ground truth. These quantities in turn enable to find expressions for the bin-wise perturbation of the eigenvalues, which depends on the error variance of the estimate, and for the bin-wise perturbation of the eigenspaces, which depends on both the error variance but also on the eigenvalue distance. We adapt a number of known bounds for ordinary matrices and demonstrate the fit of these bounds in simulations. In order to minimise the error variance of the estimate, and hence the perturbation of the EVD factors, we discuss away to optimise the lag support of the space-time covariance estimate without access to the ground truth on which the estimate is based.

A matrix of transfer functions is, in most cases, known to admit an analytic singular value decomposition (SVD), with singular values that are real-valued but potentially negative on the unit circle. In this contribution, we propose an algorithm to retrieve such analytic singular values. We propose approach operates in the frequency domain, and first computes a standard SVD of the given polynomial matrix in each discrete Fourier transform (DFT) bin. Thereafter, in order to re-establish their association across bins, the bin-wise singular values are permuted by assessing the orthogonality of singular vectors in adjacent DFT bins. In addition, the proposed algorithm determines whether bin-wise singular value should become negative, which can be required for analyticity. The proposed algorithm is validated through an ensemble simulation involving polynomial matrices with known analytic SVD factors.

In order to determine the analytic eigenvalues of a parahermitian matrix, the state-of-the-art algorithm offers proven convergence but its complexity grows factorially with the matrix dimension. Operating in the discrete Fourier transform (DFT) domain, its computational bottleneck is a maximum likelihood (ML) sequence estimation, that investigates a set of paths of likely associations across DFT bins. Therefore, this paper investigates an algorithm that remains covered by its predecessor's proof of convergence but offers a significant reduction in complexity by trading the number of retained paths versus the DFT length. We motivate this, and also introduce an enhanced initialisation point for the ML sequence estimation. The benefits of this proposed scalable analytic extraction algorithm are illustrated in simulations.

This paper presents a novel method for calculating a compact order singular value decomposition (SVD) of polynomial matrices, building upon the recently proven existence of an analytic SVD for analytic, non-multiplexed polynomial matrices. The proposed method calculates a conventional SVD in sample points on the unit circle, and then applies phase smoothing Algorithms to establish phase-coherence between adjacent frequency bins. This results in the extraction of compact order singular values and their corresponding singular vectors. The method is evaluated through experiments conducted on an ensemble of randomised polynomial matrices, demonstrating its superior performance in terms of higher decomposition accuracy and lower polynomial order compared to state-of-the-art techniques.