Expected Supremum of a Random Linear Combination of Shifted Kernels

We address the expected supremum of a linear combination of shifts of the sinc kernel with random coefficients. When the coefficients are Gaussian, the expected supremum is of order \sqrt{\log n}, where n is the number of shifts. When the coefficients are uniformly bounded, the expected supremum is of order \log\log n. This is a noteworthy difference to orthonormal functions on the unit interval, where the expected supremum is of order \sqrt{n\log n} for all reasonable coefficient statistics.Comment: To appear in the Journal of Fourier Analysis and Application

Algorithmic Computability of the Capacity of Gaussian Channels with Colored Noise

Designing capacity-achieving coding schemes for the band-limited additive colored Gaussian noise (ACGN) channel has been and is still a challenge. In this paper, the capacity of the band-limited ACGN channel is studied from a fundamental algorithmic point of view by addressing the question of whether or not the capacity can be algorithmically computed. To this aim, the concept of Turing machines is used, which provides fundamental performance limits of digital computers. t is shown that there are band-limited ACGN channels having computable continuous spectral densities whose capacity are non-computable numbers. Moreover, it is demonstrated that for those channels, it is impossible to find computable sequences of asymptotically sharp upper bounds for their capacities

Characterization of the Complexity of Computing the Capacity of Colored Noise Gaussian Channels

This paper explores the computational complexity involved in determining the capacity of the band-limited additive colored Gaussian noise (ACGN) channel and its capacity-achieving power spectral density (p.s.d.). The study reveals that when the noise p.s.d. is a strictly positive computable continuous function, computing the capacity of the band-limited ACGN channel becomes a $\#\mathrm{P}_1$-complete problem within the set of polynomial time computable noise p.s.d.s. Meaning that it is even more complex than problems that are $\mathrm{NP}_1$-complete. Additionally, it is shown that the capacity-achieving distribution is also $\#\mathrm{P}_1$-complete. Furthermore, under the widely accepted assumption that $\mathrm{FP}_1 \neq \#\mathrm{P}_1$, it has two significant implications for the ACGN channel. The first implication is the existence of a polynomial time computable noise p.s.d. for which the computation of its capacity cannot be performed in polynomial time, i.e., the number of computational steps on a Turing Machine grows faster than all polynomials. The second one is the existence of a polynomial time computable noise p.s.d. for which determining its capacity-achieving p.s.d. cannot be done within polynomial time

Signal and System Approximation from General Measurements

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley-Wiener space PW_\pi^1. We consider approximation processes, where the input signal is not directly used to generate the system output, but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.Comment: This paper will be published as part of the book "New Perspectives on Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's 85th birthday" in the Applied and Numerical Harmonic Analysis Series, Birkhauser (Springer-Verlag). Parts of this work have been presented at the IEEE International Conference on Acoustics, Speech, and Signal Processing 2014 (ICASSP 2014

Neuromorphic Twins for Networked Control and Decision-Making

We consider the problem of remotely tracking the state of and unstable linear time-invariant plant by means of data transmitted through a noisy communication channel from an algorithmic point of view. Assuming the dynamics of the plant are known, does there exist an algorithm that accepts a description of the channel's characteristics as input, and returns 'Yes' if the transmission capabilities permit the remote tracking of the plant's state, 'No' otherwise? Does there exist an algorithm that, in case of a positive answer, computes a suitable encoder/decoder-pair for the channel? Questions of this kind are becoming increasingly important with regards to future communication technologies that aim to solve control engineering tasks in a distributed manner. In particular, they play an essential role in digital twinning, an emerging information processing approach originally considered in the context of Industry 4.0. Yet, the abovementioned questions have been answered in the negative with respect to algorithms that can be implemented on idealized digital hardware, i.e., Turing machines. In this article, we investigate the remote state estimation problem in view of the Blum-Shub-Smale computability framework. In the broadest sense, the latter can be interpreted as a model for idealized analog computation. Especially in the context of neuromorphic computing, analog hardware has experienced a revival in the past view years. Hence, the contribution of this work may serve as a motivation for a theory of neuromorphic twins as a counterpart to digital twins for analog hardware