Broadband angle of arrival estimation using polynomial matrix decompositions

This thesis is concerned with the problem of broadband angle of arrival (AoA) estimation for sensor arrays. There is a rich theory of narrowband solutions to the AoA problem, which typically involves the covariance matrix of the received data and matrix factorisations such as the eigenvalue decomposition (EVD) to reach optimality in various senses. For broadband arrays, such as found in sonar, acoustics or other applications where signals do not fulfil the narrowband assumption, working with phase shifts between different signals - as sufficient in the narrowband case - does not suffice and explicit lags need to be taken into account. The required space-time covariance matrix of the data now has a lag dimension, and classical solutions such as those based on the EVD are no longer directly applicable. There are a number of existing broadband AoA techniques, which are reviewed in this thesis. These include independent frequency bin processors, where the broadband problem is split into several narrowband ones, thus loosing coherence across bins. Coherent signal subspace methods effectively apply a pre-steering by focussing matrices in the assumed directions of existing sources, and sum the narrowband covariance matrices coherently. Subsequently, classical narrowband methods can be applied. A recent auto-focussing approach dispenses with the requirement of knowing the approximate direction of sources, and calculates the focussing matrices on a data-dependent fashion. A recent parametric covariance matrix approach for broadband AoA estimation is also reviewed, and it is shown that this can only detect a single - the strongest - source. Based on a polynomial EVD (PEVD) factorisation of polynomial matrices such as created by a space-time covariance matrix emerging from a broadband problem, this thesis proposes an extension of the powerful high-resolution but narrowband multiple signal classification (MUSIC) algorithm. While narrowband MUSIC is based on an EVD to identify signal and noise subspaces, the PEVD can extract polynomial subspaces. This also requires the definition of broadband steering vectors, which are used in the proposed polynomial MUSIC (P-MUSIC) method to scan the noise-only subspace. Two different P-MUSIC versions are proposed here: a spatio-spectral P-MUSIC (SSP-MUSIC) is capable to resolve sources with respect to the AoA and frequency range, and a spatial P-MUSIC (SP-MUSIC) extracts the AoA alone. Broadband steering vectors are proposed as polynomial vectors containing fractional delay filters. For the implementation, a number of methods are reviewed and compared, including windowed sinc functions and Farrow structures. All these techniques show degraded performance as the frequency approaches half of the sampling rate. Therefore, this dissertation also proposes a highly accurate fractional delay filter implementation based on undecimated filter banks, whereby the subband signals are modulated to lower frequency ranges, where individual fractional delay filters can operate with high accuracy. For the implementation of P-MUSIC, we demonstrate that the broadband steering vector accuracy is important. We also apply different iterative PEVD algorithms belonging to the families of second order sequential best rotation (SBR2) and sequential matrix diagonalisation (SMD) algorithms. We demonstrate the SMD familly, which offers a better diagonalisation of the space-time covariance matrix, is also capable of providing a more accurate subspace decomposition than SBR2. This is evidenced by a higher resolution that can be achieved if SP-MUSIC and SSP-MUSIC are based on SMD rather than SBR2. The thesis concludes with an extensive set of simulations for both toy problems and realistic scenarios. This is to explain and highlight the operation of the P-MUSIC algorithms, but also compares their performance to other state-of-the-art broadband AoA methods. For the closest competitor, the auto-focussing approach, an analysis in a polynomial matrix framework is provided, which highlights similarities and differences to P-MUSIC. The simulations indicate that PMUSIC is a powerful and robust extension of MUSIC to the broadband case.This thesis is concerned with the problem of broadband angle of arrival (AoA) estimation for sensor arrays. There is a rich theory of narrowband solutions to the AoA problem, which typically involves the covariance matrix of the received data and matrix factorisations such as the eigenvalue decomposition (EVD) to reach optimality in various senses. For broadband arrays, such as found in sonar, acoustics or other applications where signals do not fulfil the narrowband assumption, working with phase shifts between different signals - as sufficient in the narrowband case - does not suffice and explicit lags need to be taken into account. The required space-time covariance matrix of the data now has a lag dimension, and classical solutions such as those based on the EVD are no longer directly applicable. There are a number of existing broadband AoA techniques, which are reviewed in this thesis. These include independent frequency bin processors, where the broadband problem is split into several narrowband ones, thus loosing coherence across bins. Coherent signal subspace methods effectively apply a pre-steering by focussing matrices in the assumed directions of existing sources, and sum the narrowband covariance matrices coherently. Subsequently, classical narrowband methods can be applied. A recent auto-focussing approach dispenses with the requirement of knowing the approximate direction of sources, and calculates the focussing matrices on a data-dependent fashion. A recent parametric covariance matrix approach for broadband AoA estimation is also reviewed, and it is shown that this can only detect a single - the strongest - source. Based on a polynomial EVD (PEVD) factorisation of polynomial matrices such as created by a space-time covariance matrix emerging from a broadband problem, this thesis proposes an extension of the powerful high-resolution but narrowband multiple signal classification (MUSIC) algorithm. While narrowband MUSIC is based on an EVD to identify signal and noise subspaces, the PEVD can extract polynomial subspaces. This also requires the definition of broadband steering vectors, which are used in the proposed polynomial MUSIC (P-MUSIC) method to scan the noise-only subspace. Two different P-MUSIC versions are proposed here: a spatio-spectral P-MUSIC (SSP-MUSIC) is capable to resolve sources with respect to the AoA and frequency range, and a spatial P-MUSIC (SP-MUSIC) extracts the AoA alone. Broadband steering vectors are proposed as polynomial vectors containing fractional delay filters. For the implementation, a number of methods are reviewed and compared, including windowed sinc functions and Farrow structures. All these techniques show degraded performance as the frequency approaches half of the sampling rate. Therefore, this dissertation also proposes a highly accurate fractional delay filter implementation based on undecimated filter banks, whereby the subband signals are modulated to lower frequency ranges, where individual fractional delay filters can operate with high accuracy. For the implementation of P-MUSIC, we demonstrate that the broadband steering vector accuracy is important. We also apply different iterative PEVD algorithms belonging to the families of second order sequential best rotation (SBR2) and sequential matrix diagonalisation (SMD) algorithms. We demonstrate the SMD familly, which offers a better diagonalisation of the space-time covariance matrix, is also capable of providing a more accurate subspace decomposition than SBR2. This is evidenced by a higher resolution that can be achieved if SP-MUSIC and SSP-MUSIC are based on SMD rather than SBR2. The thesis concludes with an extensive set of simulations for both toy problems and realistic scenarios. This is to explain and highlight the operation of the P-MUSIC algorithms, but also compares their performance to other state-of-the-art broadband AoA methods. For the closest competitor, the auto-focussing approach, an analysis in a polynomial matrix framework is provided, which highlights similarities and differences to P-MUSIC. The simulations indicate that PMUSIC is a powerful and robust extension of MUSIC to the broadband case