TU CE GSC CE People at the Graduate School CE Alumni Christian Weiß Research Project of Christian Weiss

Sparse signal processing methods with applications to radar and photonics

Sparse signal processing is an emerging technology which serves a wide range of applications dealing with compressive signals. For most practical signals, it is possible to identify a certain basis in which the signal can be represented by only a few non-zero coefficients. One of the simplest examples is a sine signal with only one frequency component. Using the Fourier transform, it is completely described by only one coefficient - it is compressive. Thus, we obtain a sparse vector with only one non-zero entry. This fact can be exploited in many ways, e.g. for signal compression or noise suppression.

Key element of sparse signal processing is an appropriate basis, suited to the specific application at hand. It can be obtained by accurate system modeling which requires profound knowledge about the underlying physics. Once such a basis has been found, the sparse coefficients can be estimated by solving a sparse regularization problem. The constraints of this optimization problem have to be defined by the operator according to the desired properties and features of the signal to be estimated / reconstructed. Hence, they are highly dependent on the specific application. The regularization parameter renders strong impact on the performance as it has to account for any kind of model perturbation or noise. Choosing a good value for the regularization parameter is a highly non-trivial task and one of the central problems of this research.

Main goals of sparse signal processing are to

- improve the signal quality
- reduce noise or suppress unwanted effects
- robustify a system (model uncertainties, inaccurate physical parameters, etc.)

Proper modeling plays a fundamental role and the overall performance relies significantly on the quality of the obtained signal model and the selected basis with respect to the real physical system.

One important branch of sparse signal processing and focus of recent research is called 'compressive sensing'. It allows for signal compression on the fly, i.e. without prior storing of the signal, in order to reduce the sampling rate. Thus, there is no more need for costly A/D converters working at high frequencies. Also the required amount of storage can be tremendously reduced.

Fiber sensors are smart devices used e.g. for structural health monitoring. An optical fiber is introduced into the structure of interest such as the concrete of buildings/dikes, in order to measure perturbations, i.e. strain, temperature variations or vibrations. A number of Fiber-Bragg-Gratings are imprinted into the fiber core. They act as mirrors at certain wavelengths. Strain or temperature fluctuations at the locations of these gratings cause a shift in the reflected wavelength. A wavelength-swept laser is used to sample those mirrors at a certain sweep rate. The time delays of the reflected signals provides information about the amount of pertubation. However, high sweep rates require high sampling rates. The system can be modeled as a coupled multi-physical system, including laser, optical path (fiber), interaction with and reflection at the gratings and the opto-electrical conversion. From this model, an appropriate basis can be derived and used for sparse signal processing. Compressive sensing offers the possibility to replace costly high-speed A/D converters by more economical devices since the average sampling rate can be significantly reduced.

A fundamental problem in radar/sonar is the estimation of the bearing of a source. Sparse array signal processing can be applied to achieve super-resolution DOA estimation. The set of array steering vectors (at a certain angular accuracy) serves as a sparse basis when the number of sources is small as compared to the signal dimension (number of sensors). Then, each source corresponds to a certain steering or basis vector. Ideally, all coefficients in the sparse domain are zero except those belonging to the source steering vectors. However, model errors such as non-uniform sensor gain along the array or uncertainties in the exact sensor locations introduce model errors which render negative impact on the DOA estimation performance. Using a statistical framework, these errors can be modeled to adjust the regularization parameter of the sparse optimization problem. As a figure of merit, the mean-squared error between the perturbed measurements and the assumed underlying model can be applied. This technique was shown to robustify the system against the joint effects of model errors and noise while preserving its super-resolution capabilities for DOA estimation.

Multi-Physics; Photonics; Radar/Sonar; Statistical Signal Processing

Christian Weiß

M.Sc.

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