Graph signal processing is an emerging field which aims to model processes that exist on the nodes of a network and are explained through diffusion over this structure. Graph signal processing works have heretofore assumed knowledge of the graph shift operator. Our approach is to investigate the question of graph filtering on a graph about which we only know a model. To do this we leverage the theory of graphons proposed by L. Lovasz and B. Szegedy. We make three key contributions to the emerging field of graph signal processing. We show first that filters defined over the scaled adjacency matrix of a random graph drawn from a graphon converge to filters defined over the Fredholm integral operator with the graphon as its kernel. Second, leveraging classical findings from the theory of the numerical solution of Fredholm integral equations, we define the Fourier-Galerkin shift operator. Lastly, using the Fourier-Galerkin shift operator, we derive a graph filter design algorithm which only depends on the graphon, and thus depends only on the probabilistic structure of the graph instead of the particular graph itself. The derived graphon filtering algorithm is verified through simulations on a variety of random graph models.@en

Source localization is among the most fundamental problems in statistical signal processing. Methods which rely on the orthogonality of the signal and noise subspaces, such as Pisarenko’s method, MUSIC, and root-MUSIC are some of the most widely used algorithms to solve this problem. As a common feature, these methods require both a-priori knowledge of the number of sources, and an estimate of the noise subspace. Both requirements are complicating factors to the practical implementation of the algorithms, and sources of potentially severe error. In this paper, we propose a new localization criterion based on the algebraic structure of the noise subspace. An algorithm is proposed which adaptively learns the number of sources and estimates their locations. Simulation results show significant improvement over root-MUSIC, even when the correct number of sources is provided to the root-MUSIC algorithm.@en

Source localization and spectral estimation are among the most fundamental problems in statistical and array signal processing. Methods that rely on the orthogonality of the signal and noise subspaces, such as Pisarenko's method, MUSIC, and root-MUSIC, are some of the most widely used algorithms to solve these problems. As a common feature, these methods require both a priori knowledge of the number of sources and an estimate of the noise subspace. Both requirements are complicating factors to the practical implementation of the algorithms and, when not satisfied exactly, can potentially lead to severe errors. In this paper, we propose a new localization criterion based on the algebraic structure of the noise subspace that is described for the first time to the best of our knowledge. Using this criterion and the relationship between the source localization problem and the problem of computing the greatest common divisor (GCD), or more practically approximate GCD, for polynomials, we propose two algorithms, which adaptively learn the number of sources and estimate their locations. Simulation results show a significant improvement over root-MUSIC in challenging scenarios such as closely located sources, both in terms of detection of the number of sources and their localization over a broad and practical range of signal-to-noise ratios. Furthermore, no performance sacrifice in simple scenarios is observed.@en

The detection and localization of multiple targets is a fundamental research area for multiple input multiple output (MIMO) radar. In many civilian applications of MIMO technology, for example, automotive radar, high resolution direction of arrival (DOA) estimation is required. In this paper, a novel DOA estimation algorithm based on tensor decomposition is proposed for collocated transmit beamspace MIMO radar. First, we introduce the flipped-conjugate version of the transmit beamspace matrix, which focuses the transmit energy into fixed region. This can increase the signal to noise ratio (SNR) of targets. Then we reshape the received data into a tensor form, the structure of which provides the estimations of the transmit and receive steering matrices. The alternating least squares (ALS) algorithm is applied to find the tensor components. The DOA estimation is conducted in transmitters via the rotational invariance property achieved by beamspace matrix. It is proved that at most M-2 grating lobes exist during the process of DOA estimation, where M is the number of the transmitters. These grating lobes can be eliminated by finite trials of spectrum search. The performance of our proposed DOA estimation method surpasses several conventional algorithms in terms of accuracy and resolution.@en

The detection and localization of multiple targets is a fundamental research area for multiple input multiple output (MIMO) radar. In many civilian applications of MIMO technology, for example, automotive radar, high resolution direction of arrival (DOA) estimation is required. In this paper, a novel DOA estimation algorithm based on tensor decomposition is proposed for collocated transmit beamspace MIMO radar. First, we introduce the flipped-conjugate version of the transmit beamspace matrix, which focuses the transmit energy into fixed region. This can increase the signal to noise ratio (SNR) of targets. Then we reshape the received data into a tensor form, the structure of which provides the estimations of the transmit and receive steering matrices. The alternating least squares (ALS) algorithm is applied to find the tensor components. The DOA estimation is conducted in transmitters via the rotational invariance property achieved by beamspace matrix. It is proved that at most M-2 grating lobes exist during the process of DOA estimation, where M is the number of the transmitters. These grating lobes can be eliminated by finite trials of spectrum search. The performance of our proposed DOA estimation method surpasses several conventional algorithms in terms of accuracy and resolution.@en

We present the theory of sequences of random graphs and their convergence to limit objects. Sequences of random dense graphs are shown to converge to their limit objects in both their structural properties and their spectra. The limit objects are bounded symmetric functions on [0,1]2. The kernel functions define an equivalence class and thus identify collections of large random graphs who are spectrally and structurally equivalent. As the spectrum of the graph shift operator defines the graph Fourier transform (GFT), the behavior of the spectrum of the underlying graph has a great impact on the design and implementation of graph signal processing operators such as filters. The spectra of several graph limits are derived analytically and verified with numerical examples.@en

Generalized sidelobe canceler (GSC) uses a two step procedure in order to produce a beampattern with a fixed mainlobe and suppressed sidelobes. In the first step, a beampattern with a fixed response in the look direction is produced by convolving a vector of constraints with a normalized beamforming vector with the desired mainlobe response. In the second step, the signals in the look direction are blocked out using so-called blocking matrix, while the output power is minimized. Observing that for Griffiths-Jim GSC the beamforming vector contains the coefficients of a polynomial with at least one root at 1, we find here that all rows of a blocking matrix should be the coefficients of polynomials from the polynomial ideal with a root at 1. This allows us to reveal and exploit the underlying algebraic structure for GSC blocking matrix design using methods from computational algebraic geometry. It also allows to arrive to and prove several generalized statements. For example, the necessary and sufficient condition for a signal to be blocked can be easily found. The condition to a row-space of blocking matrix for blocking multiple signals impinging upon the array from multiple directions can also be easily formulated. The linear independence of rows of blocking matrix implies that all the corresponding polynomial share a single root. In general, understanding the algebraic structure that GSC's blocking matrix has to satisfy makes the GSC's design simpler and more intuitive.@en

Generalized sidelobe canceler (GSC) uses a two step procedure in order to produce a beampattern with a fixed mainlobe and suppressed sidelobes. In the first step, a beampattern with a fixed response in the look direction is produced by convolving a vector of constraints with a normalized beamforming vector with the desired mainlobe response. In the second step, the signals in the look direction are blocked out using so-called blocking matrix, while the output power is minimized. Observing that for Griffiths-Jim GSC the beamforming vector contains the coefficients of a polynomial with at least one root at 1, we find here that all rows of a blocking matrix should be the coefficients of polynomials from the polynomial ideal with a root at 1. This allows us to reveal and exploit the underlying algebraic structure for GSC blocking matrix design using methods from computational algebraic geometry. It also allows to arrive to and prove several generalized statements. For example, the necessary and sufficient condition for a signal to be blocked can be easily found. The condition to a row-space of blocking matrix for blocking multiple signals impinging upon the array from multiple directions can also be easily formulated. The linear independence of rows of blocking matrix implies that all the corresponding polynomial share a single root. In general, understanding the algebraic structure that GSC's blocking matrix has to satisfy makes the GSC's design simpler and more intuitive.@en