In this paper we introduce a highly adaptive continuous wavelet transform using Gaussian wavelets multiplied by an appropriate rational term. The zeros and poles of this rational modifier act as free parameters and their choice highly influences the shape of the mother wavelet. This allows the proposed construction to approximate signals with complex morphology using only a few wavelet coefficients. We show that the proposed rational Gaussian wavelets are admissible and provide numerical approximations of the wavelet coefficients using variable projection operators. In addition, we show how the proposed variable projection based rational Gaussian wavelet transform can be used in neural networks to obtain a highly interpretable feature learning layer. We demonstrate the effectiveness of the proposed scheme through a number of numerical experiments including biomedical applications, and the detection of abnormal road surface based on tire sensor signals.

In this work we propose proof-of-concept methods to detect malfunctions of the braking system in passenger vehicles. In particular, we investigate the problem of detecting deformations of the brake disc based on data recorded by acceleration sensors mounted on the suspension of the vehicle. Our core hypothesis is that these signals contain vibrations caused by brake disc deformation. Since faults of this kind are typically monitored by the driver of the vehicle, the development of automatic fault-detection systems becomes more important with the rise of autonomous driving. In addition, the new brake boosters separate the brake pedal from the hydraulic system which results in less significant effects on the brake pedal force. Our paper offers two important contributions. Firstly, we provide a detailed description of our novel measurement scheme, the type and placement of the used sensors, signal acquisition and data characteristics. Then, in the second part of our paper we detail mathematically justified signal representations and different algorithms to distinguish between deformed and normal brake discs. For the proper understanding of the phenomenon, different brake discs were used with measured runout values. Since, in addition to brake disc deformation, the vibrations recorded by our accelerometers are nonlinearly dependent on a number of factors (such as the velocity, suspension, tire pressure, etc.), data-driven models are considered. Through experiments, we show that the proposed methods can be used to recognize faults in the braking system caused by brake disc deformation.

We propose a novel method to identify the transfer functions of single-input-single-output linear time invariant (SISO-LTI) dynamic systems. Our approach makes use of the operator based generalization of Prony's method. In particular, the operator based Prony algorithm is used to reconstruct the transfer function of the system as a linear combination of rational basis functions. A considerable benefit of the proposed method is its robustness against the estimated system order. That is, if system order is over estimated, the correct system order can be found naturally. Another important benefit is that the proposed method is shown to be asymptotically robust towards zero expectation noise with the correct choice of certain evaluation functionals. The effectiveness of the proposed method is demonstrated through numerical experiments.