Florian Römer

Enhancements of unitary ESPRIT for non-circular sources

Estimating the directions of arrival of several wavefronts impinging on an array of sensors is a requirement in a variety of applications including radar, mobile communications, sonar, and seismology. Subspace based high-resolution parameter estimation schemes like ESPRIT and MUSIC have become very popular. Such parameter estimation algorithms often use forward-backward averaging to enhance their resolution, especially in the case of correlated sources. Further enhancements can be achieved if the source signals are non-circular. We derive an efficient subspace estimation scheme that exploits the non-circularity of the sources and already includes forward-backward averaging. Moreover, appropriate spatial smoothing techniques are introduced. Completely real-valued implementations of 1D and 2D unitary ESPRIT for non-circular sources are presented as examples. In these cases, NC unitary ESPRIT improves the resolution capability and the noise robustness of standard ESPRIT as well as unitary ESPRIT and can handle more sources than sensors.

Multi-dimensional space-time-frequency component analysis of event related EEG data using closed-form PARAFAC

The efficient analysis of electroencephalographic (EEG) data is a long standing problem in neuroscience, which has regained new interest due to the possibilities of multidimensional signal processing. We analyze event related multi-channel EEG recordings on the basis of the time-varying spectrum for each channel. It is a common approach to use wavelet transformations for the time-frequency analysis (TFA) of the data. To identify the signal components we decompose the data into time-frequency-space atoms using Parallel Factor (PARAFAC) analysis. In this paper we show that a TFA based on the Wigner-Ville distribution together with the recently developed closed-form PARAFAC algorithm enhance the separability of the signal components. This renders it an attractive approach for processing EEG data. Additionally, we introduce the new concept of component amplitudes, which resolve the scaling ambiguity in the PARAFAC model and can be used to judge the relevance of the individual components.

Robust methods based on the hosvd for estimating the model order in PARAFAC models

Parallel factor (PARAFAC) analysis represents a decomposition of a tensor into a minimum sum of rank one tensors. For this task, one crucial problem is the estimation of the number of rank one components that are required to represent the tensor. This problem is also known as model order estimation. Recently we have developed new R-dimensional techniques based on the HOSVD to estimate the number of components in multi-dimensional harmonic retrieval problems (i.e., R-D EFT, R-D AIC, and R-D MDL). In this paper, we apply these R-D methods to the PARAFAC model, which is a more general multi-way data model, and show that they outperform T-CORCONDIA, a nonsubjective form of CORCONDIA, in terms of the probability of detection as well as the required computational complexity.

Enhanced Model Order Estimation using Higher-Order Arrays

Frequently, R-dimensional subspace-based methods are used to estimate the parameters in multi-dimensional harmonic retrieval problems in a variety of signal processing applications. Since the measured data is multi-dimensional, traditional approaches require stacking the dimensions into one highly structured matrix. Recently, we have shown how an HOSVD based low-rank approximation of the measurement tensor leads to an improved signal subspace estimate, which can be exploited in any multi-dimensional subspace-based parameter estimation scheme. To achieve this goal, it is required to estimate the model order of the multi-dimensional data. In this paper, we show how the HOSVD of the measurement tensor also enables us to improve the model order estimation step. This is due to the fact that only one set of eigenvalues is available in the matrix case. Applying the HOSVD, we obtain R + 1 sets of n-mode singular values of the measurement tensor that are used jointly to improve the accuracy of the model order selection significantly.

On the estimation of grid offsets in CS-based direction-of-arrival estimation

Compressed Sensing (CS) has been recently applied to direction of arrival (DOA) estimation, leveraging the fact that a superposition of planar wavefronts corresponds to a sparse angular power spectrum. However, to apply the CS framework we need to construct a finite dictionary by sampling the angular domain with a predefined sampling grid. Therefore, the target locations are almost surely not located exactly on a subset of these grid points. This leads to a model mismatch which deteriorates the performance of the estimators. In this paper we take an analytical approach to investigate the effect of such grid offsets on the recovered spectra. We show that each off-grid source can be well approximated by the closest two neighboring points on the grid. We propose a simple and efficient scheme to estimate the grid offset for a single source or multiple well-separated sources. We also discuss a numerical procedure for the joint estimation of the grid offsets of closer sources. Simulation results demonstrate the effectiveness of the proposed methods.

3-D Unitary ESPRIT: Accurate attitude estimation for unmanned aerial vehicles with a hexagon-shaped ESPAR array

Accurate estimation of the attitude of unmanned aerial vehicles (UAVs) is crucial for their control and displacement. Errors in the attitude estimate may misuse the limited battery energy of UAVs or even cause an accident. For attitude estimation, proprioceptive sensors such as inertial measurement units (IMUs) are widely applied, but they are susceptible to inertial guidance error. With antenna arrays currently being installed in UAVs for communication with ground base stations, we can take advantage of the array structure in order to improve the estimates of IMUs via data fusion. In this paper, we therefore propose an attitude estimation system based on a hexagon-shaped 7-element electronically steerable parasitic antenna radiator (ESPAR) array. The ESPAR array is well-suited for installment in the UAVs with broad wings and short bodies. Our proposed solution returns an estimation for the pitch and roll based on the inter-element phase delay estimates of the line-of-sight path of the impinging signal over the antenna array. By exploiting the parallel and centrosymmetric structure in the hexagon-shaped ESPAR array, the 3-dimensional Unitary ESPRIT algorithm is applied for phase delay estimation to achieve high accuracy as well as computational efficiency. We devise an attitude estimation algorithm by exploiting the geometrical relationship between the UAV attitude and the estimated phase delays. An analytical closed-form expression of the attitude estimates is obtained by solving the established simultaneous nonlinear equations. Simulations results show the feasibility of our proposed solution for different signal-to-noise ratio levels as well as multipath scenarios.

Sparsity order estimation for single snapshot compressed sensing

In this paper we discuss the estimation of the spar-sity order for a Compressed Sensing scenario where only a single snapshot is available. We demonstrate that a specific design of the sensing matrix based on Khatri-Rao products enables us to transform this problem into the estimation of a matrix rank in the presence of additive noise. Thereby, we can apply existing model order selection algorithms to determine the sparsity order. The matrix is a rearranged version of the observation vector which can be constructed by concatenating a series of non-overlapping or overlapping blocks of the original observation vector. In both cases, a Khatri-Rao structured measurement matrix is required with the main difference that in the latter case, one of the factors must be a Vandermonde matrix. We discuss the choice of the parameters and show that an increasing amount of block overlap improves the sparsity order estimation but it increases the coherence of the sensing matrix. We also explain briefly that the proposed measurement matrix design introduces certain multilinear structures into the observations which enables us to apply tensor-based signal processing, e.g., for enhanced denoising or improved sparsity order estimation.

Efficient source enumeration for accurate direction-of-arrival estimation in threshold region

Estimation of the number of signals impinging on an array of sensors, also known as source enumeration, is usually required prior to direction-of-arrival (DOA) estimation. In challenging scenarios such as the presence of closely-spaced sources and/or high level of noise, using the true source number for nonlinear parameter estimation leads to the threshold effect which is characterized by an abnormally large mean square error (MSE). In cases that sources have distinct powers and/or are closely spaced, the error distribution among parameter estimates of different sources is unbalanced. In other words, some estimates have small errors while others may be quite inaccurate with large errors. In practice, we will be only interested in the former and have no concern on the latter. To formulate this idea, the concept of effective source number (ESN) is proposed in the context of joint source enumeration and DOA estimation. The ESN refers to the actual number of sources that are visible at a given noise level by a parameter estimator. Given the numbers of sensors and snapshots, number of sources, source parameters and noise level, a Monte Carlo method is designed to determine the ESN, which is the maximum number of available accurate estimates. The ESN has a theoretical value in that it is useful for judging what makes a good source enumerator in the threshold region and can be employed as a performance benchmark of various source enumerators. Since the number of sources is often unknown, its estimate by a source enumerator is used for DOA estimation. In an effort to automatically remove inaccurate estimates while keeping as many accurate estimates as possible, we define the matched source number (MSN) as the one which in conjunction with a parameter estimator results in the smallest MSE of the parameter estimates. We also heuristically devise a detection scheme that attains the MSN for ESPRIT based on the combination of state-of-the-art source enumerators.

Spatially resolved sub-Nyquist sensing of multiband signals with arbitrary antenna arrays

In recent years it has been shown that wideband analog signals can be sampled significantly below the Nyquist rate without loss of information, provided that the unknown frequency support occupies only a small fraction of the overall bandwidth. The modulated wideband converter (MWC) is a particular architecture that implements this idea. In this paper we discuss how the use of antenna arrays allows to extend this concept towards spatially resolved wideband spectrum sensing by leveraging the sparsity in the angular-frequency domain. In our system each antenna element of the array is sampled at a sub-Nyquist rate by an individual MWC block. This results in a trade-off between the number of antennas and MWC channels per antenna. We derive bounds on the minimal total number of channels and minimal sampling rate required for perfect recovery of the 2D angular-frequency spectrum of the incoming signal and present a concrete reconstruction approach. The proposed system is applicable to arbitrary antenna arrays, provided that the array manifold is ambiguity-free.

Speeding up 3D SAFT for ultrasonic NDT by sparse deconvolution

In this paper we propose to pre-process ultrasonic measurements (A-scans) in Non-Destructive Testing (NDT) by sparse deconvolution before post-processing the data with the Synthetic Aperture Focusing Technique (SAFT). Compared to state-of-the-art SAFT post-processing of raw A-scan measurements, pre-processing by sparse deconvolution can improve NDT in the following ways: First, the temporal resolution of signal reflections is increased. Second, because the A-scans appear as a sparse signal of spikes, it is possible to formulate the time-domain SAFT algorithm in a new fashion that is both faster compared to conventional SAFT and the deconvolved input data can be focussed better leading to a higher resolution. Since sparse deconvolution could be implemented directly into the ultrasonic probe hardware/software measurement setup, this approach can significantly speed up measurements in time-critical environments. We test the proposed scheme on CIVA simulation data as well as measurements and show B- and C-images of raw SAFT vs. Orthogonal Matching Pursuit (OMP) + SAFT and Basis Pursuit Denoising (BPDN) + SAFT.