On the Need of Analog Signals and Systems for Digital-Twin Representations

We consider the task of converting different digital descriptions of analog bandlimited signals and systems into each other, with a rigorous application of mathematical computability theory. Albeit very fundamental, the problem appears in the scope of digital twinning, an emerging concept in the field of digital processing of analog information that is regularly mentioned as one of the key enablers for next-generation cyber-physical systems and their areas of application. In this context, we prove that essential quantities such as the peak-to-average power ratio and the bounded-input/bounded-output norm, which determine the behavior of the real-world analog system, cannot generally be determined from the system's digital twin, depending on which of the above-mentioned descriptions is chosen. As a main result, we characterize the algorithmic strength of Shannon's sampling type representation as digital twin implementation and also introduce a new digital twin implementation of analog signals and systems. We show there exist two digital descriptions, both of which uniquely characterize a certain analog system, such that one description can be algorithmically converted into the other, but not vice versa

Generalized Poisson Summation Formulas for Continuous Functions of Polynomial Growth

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of $L _{ 1 }$ signals. This $L _{ 1 }$ assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that d∕2 + ε derivatives of the signal are bounded by some polynomial in the $L _{ 2 }$ sense

The Inversion of Sampling Solved Algebraically

We show that Shannon's reconstruction formula can be written as a ∗ (b · c) = c = (a ∗ b) · c with tempered distributions a, b, c where ∗ is convolution, · is multiplication, c is the function being sampled and restored after sampling, b· is sampling and a∗ its inverse. The requirement a ∗ b = 1 which describes a smooth partition of unity where b = III is the Dirac comb implies that a is satisfied by unitary functions introduced by Lighthill (1958). They form convolution inverses of the Dirac comb. Choosing a = sinc yields Shannon's reconstruction formula where the requirement a ∗ b = 1 is met approximately and cannot be exact because sinc is not integrable. In contrast, unitary functions satisfy this requirement exactly and stand for the set of functions which solve the problem of inverse sampling algebraically

A Sampling Theory for Non-Decaying Signals

The classical assumption in sampling and spline theories is that the input signal is square-integrable, which prevents us from applying such techniques to signals that do not decay or even grow at infinity. In this paper, we develop a sampling theory for multidimensional non-decaying signals living in weighted $L _{ p }$ spaces. The sampling and reconstruction of an analog signal can be done by a projection onto a shift-invariant subspace generated by an interpolating kernel. We show that, if this kernel and its biorthogonal counterpart are elements of appropriate hybrid-norm spaces, then both the sampling and the reconstruction are stable. This is an extension of earlier results by Aldroubi and Gröchenig. The extension is required because it allows us to develop the theory for the ideal sampling of non-decaying signals in weighted Sobolev spaces. When the d-dimensional signal and its d∕p + ε derivatives, for arbitrarily small ε > 0, grow no faster than a polynomial in the $L _{ p }$ sense, the sampling operator is shown to be bounded even without a sampling kernel. As a consequence, the signal can also be interpolated from its samples with a nicely behaved interpolating kernel

Nonuniform Sampling Of Band-limited Functions

In this thesis, we will study certain generalizations of the classical Shannon Sampling Theorem, which allows for the reconstruction of a pi-band-limited, square-integrable function from its samples on the integers. J. R. Higgins provided a generalization where the integers can be perturbed by less than 1/4, which includes nonuniform and nonperiodic sampling sets. We generalize Higgins’ theorem by allowing for sampling sets that are perturbations of the set of zeros of a π-sine-type function. A second type of generalization allows for functions f that, while still band-limited, need not be square-integrable but may have polynomial growth when restricted to the real line. We investigate two ways to achieve this goal, again using nonuniform sampling sets. The first is an approximate method that uses the multiplication of f by a smooth and rapidly decaying auxiliary function. The second method is exact and uses oversampling by finitely many additional points. It is also shown that oversampling by finitely many points is not only economical and may lead to faster convergence of the series, but also enables the perturbed sampling points to go beyond a quarter from the integers. Furthermore, oversampling by finitely many points is applied to control the error stemming from a quantization of the sampled function values. The final topic considered is the so-called peak value problem, where one seeks to find an upper bound for the infinity norm of a function from knowledge of the supremum of its sampled values. We generalize an existing approach by first proving and then applying a nonuniform version of the Valiron-Tschakaloff sampling theorem