Sparse multidimensional exponential analysis with an application to radar imaging

We present a d-dimensional exponential analysis algorithm that offers a range of advantages compared to other methods. The technique does not suffer the curse of dimensionality and only needs O((d + 1)n) samples for the analysis of an n-sparse expression. It does not require a prior estimate of the sparsity n of the d-variate exponential sum. The method can work with sub-Nyquist sampled data and offers a validation step, which is very useful in low SNR conditions. A favourable computation cost results from the fact that d independent smaller systems are solved instead of one large system incorporating all measurements simultaneously. So the method also lends itself easily to a parallel execution. Our motivation to develop the technique comes from 2D and 3D radar imaging and is therefore illustrated on such examples

Regular sparse array direction of arrival estimation in one dimension

Traditionally regularly spaced antenna arrays follow the spatial Nyquist criterion to guarantee an unambiguous analysis. We present a novel technique that makes use of two sparse non-Nyquist regularly spaced antenna arrays, where one of the arrays is just a shifted version of the other. The method offers several advantages over the use of traditional dense Nyquist spaced arrays, while maintaining a comparable algorithmic complexity for the analysis. Among the advantages we mention: an improved resolution for the same number of receivers and reduced mutual coupling effects between the receivers, both due to the increased separation between the antennas. Because of a shared structured linear system of equations between the two arrays, as a consequence of the shift between the two, the analysis of both is automatically paired, thereby avoiding a computationally expensive matching step as is required in the use of so-called co-prime arrays. In addition, an easy validation step allows to automatically detect the precise number of incoming signals, which is usually considered a difficult issue. At the same time, the validation step improves the accuracy of the retrieved results and eliminates unreliable results in the case of noisy data. The performance of the proposed method is illustrated with respect to the influence of noise as well to the effect of mutual coupling

VEXPA: Validated EXPonential Analysis through regular sub-sampling

We present a procedure that adds a number of desirable features to standard exponential analysis algorithms , among which output reliability, a divide-and-conquer approach, the automatic detection of the exponential model order, robustness against some outliers, and the possibility to parallelize the analysis. The key enabler for these features is the introduction of uniform sub-Nyquist sampling through decima-tion of the dense signal data. We actually make use of possible aliasing effects to recondition the problem statement rather than that we avoid aliasing. In Section 2 the standard exponential analysis is described, including a sensitivity analysis. In Section 3 the ingredients for the new approach are collected, of which good use is made in Section 4 where we essentially bring everything together in what we call VEXPA. Some numerical examples of the new procedure illustrate in Section 5 that the additional features are indeed realized and that VEXPA is a valuable add-on to any stand-alone exponential analysis. While returning a lot of additional output, it maintains a favourable comparison to the CRLB of the underlying method, for which we here choose a matrix pencil method. Moreover, the output reliability of VEXPA is similar to that of atomic norm minimization, whereas its computational complexity is far less

Antenna position estimation through sub-sampled exponential analysis of harmonically related input signals

Accurate placement of elements in large antenna arrays is a difficult and costly process. We explore the use of the validated exponential analysis (VEXPA) technique that was previously formulated to solve a direction-of-arrival (DOA) estimation problem, to find the antenna element positions in an array after the installation phase, so that cost-savings can be realised during placement of the antenna elements. Measurements are taken from harmonically related input signals transmitted from an Unmanned Aerial Vehicle (UAV) for which the position in the sky is known. It is shown how the UAV's zenith angle can be manipulated to generate parameters required for VEXPA's de-aliasing step. A simple simulation illustrates the functioning of the proposed method

CAD-Based Design Optimization of Four-Bar Mechanisms: An Emergency Ventilator Case Study

The design optimization of mechanisms is promising as it results in more energy-efficient machines without compromising performance. However, machine builders do not apply state-of-the-art methods, as these algorithms require case-specific theoretical analysis. Moreover, the design synthesis approaches in the literature predominantly utilize heuristic optimizers, leading to suboptimal local minima. This paper introduces a widely applicable workflow, guaranteeing the global optimum. The constraints describing the feasible region of the possible designs are essential to find the global optimum. Therefore, kinematic analysis of the point-to-point planar four-bar mechanism is discussed. Within the feasible design space, objective value samples were generated through the CAD multi-body software. These motion simulations determine the required torque to fulfill the movement for a combination of design parameters. This replaces the cumbersome analytic derivation of the torque. This paper introduces sparse interpolation techniques to avoid brute force sampling of the design space. The advantage of this approach is that it is easily scalable to more design parameters, as the interpolation method minimizes the number of necessary samples. This paper explains the mathematical background of our developed interpolation approach. However, a step-by-step procedure is introduced to allow the employment of the interpolation technique by machine designers without the necessity to understand the underlying mathematics. Finally, the mathematical expression, obtained from the interpolation, enables applying global optimizers. In a case study of an emergency ventilator mechanism with three design parameters, 1870 CAD motion simulations allowed reducing the RMS torque of the mechanism by 67

Antenna Position Estimation Through Subsampled Exponential Analysis of Signals in the Near Field

In a previous article we explored the use of a subsampled exponential analysis algorithm to find the antenna-element positions in a large irregular planar array after the installation phase. The application requires an unmanned aerial vehicle to be flown over the antenna array while transmitting several odd harmonic signals. The received signal samples at a chosen reference antenna element are then compared to those at every other element in the array in order to find its position. Previously, the far-field approximation was used to calculate the time delay between received signals. In this article the method is reconsidered for the more realistic case of when the source is in the near field of the array. A number of problems that arise are addressed, and results from a controlled simulation are presented to illustrate that the computational method works

CAD-Based Design Optimization of Four-Bar Mechanisms: An Emergency Ventilator Case Study

The design optimization of mechanisms is promising as it results in more energy-efficient machines without compromising performance. However, machine builders do not apply state-of-the-art methods, as these algorithms require case-specific theoretical analysis. Moreover, the design synthesis approaches in the literature predominantly utilize heuristic optimizers, leading to suboptimal local minima. This paper introduces a widely applicable workflow, guaranteeing the global optimum. The constraints describing the feasible region of the possible designs are essential to find the global optimum. Therefore, kinematic analysis of the point-to-point planar four-bar mechanism is discussed. Within the feasible design space, objective value samples were generated through the CAD multi-body software. These motion simulations determine the required torque to fulfill the movement for a combination of design parameters. This replaces the cumbersome analytic derivation of the torque. This paper introduces sparse interpolation techniques to avoid brute force sampling of the design space. The advantage of this approach is that it is easily scalable to more design parameters, as the interpolation method minimizes the number of necessary samples. This paper explains the mathematical background of our developed interpolation approach. However, a step-by-step procedure is introduced to allow the employment of the interpolation technique by machine designers without the necessity to understand the underlying mathematics. Finally, the mathematical expression, obtained from the interpolation, enables applying global optimizers. In a case study of an emergency ventilator mechanism with three design parameters, 1870 CAD motion simulations allowed reducing the RMS torque of the mechanism by 67

On the conditioning of some structured generalized eigenvalue problems

This work continues the analysis of the conditioning of a Hankel structured generalized eigenvalue problem (GEP) started in [1]. The considered generalized eigenvalue problem appears in exponential analysis and sparse interpolation. We generalize the proof in [1] and add expressions for the relative condition numbers of two reformulations of the GEP, a reformulation as a Loewner GEP valid for general complex data, and a compression to a Hankel+Toeplitz GEP in the case of real data. Both reformulations are compared to the original Hankel GEP. The analysis is concluded with ample numerical illustrations