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This paper is an attempt to formulate a unique mathematical theory of the sampling operation of analog signals by connecting all of its scattered in the literature fragments into a one whole. We think that there exists a need for performing this, since many of the fragments mentioned appear to be inconsistent with each other, especially in the cases when they are misunderstood and misinterpreted. Our hope is that this paper will meet expectations of many people working in the area of digital signal processing to have an ordered mathematical theory that describes the signal sampling process.

It is common knowledge that no signals having the form of a sequence of weighted Dirac deltas, as they are currently modelled so, are present, in reality, at the output of analog-to-digital converters (ADC). No such signals are available in the otherwise large variety of physical signals. This has been illustrated in many works and shown by means of measurements. However, it is highly intriguing that the same papers (with the exception of this article) consider it necessary to model the signals at the output of ADC via sequences of Dirac deltas because only such an approach allows to describe the aliasing effect occurring in the spectra of these signals. But due to this incorrect description such an approach is obviously incomprehensible to everyone. In accepting this, however, it has been overlooked that this does not have to be the case at all. The effect of aliasing in the spectrum of a sampled signal may be explained by modeling this signal as a weighted step function. Moreover, such an approach offers also a quantitatively more accurate result for this spectrum than the one obtained using the current method. All that is illustrated in this paper that focuses on this specific theme.

This paper crowns efforts, made by its author, aiming in showing and proving that the current formula for calculation of the spectra of output signals at A/D converters requires a correcting factor in it. A number of partial results obtained and published in the last years are referred to here. They paved the way to a fully satisfactory and correct result; it is presented in this work. The corrected formula for spectrum calculation is derived using a description of the output signal of an A/D converter by means of the so-called Dirac comb, however not in a direct form, but with taking into account physical reality. In addition, the paper contains a number of interpretative remarks, comments, and explanations - clarifying those matters that have so far been omitted in analyses of the sampling process, despite the fact that they raised various types of doubts.

In this paper, we show why the descriptions of the sampled signal used in calculation of its spectrum, that are used in the literature, are not correct. And this finding applies to both kinds of descriptions: the ones which follow from an idealized way of modelling of the signal sampling operation as well as those which take into account its non-idealities. The correct signal description, that results directly from the way A/D converters work (regardless of their architecture), is presented and dis-cussed here in detail. Many figures included in the text help in its understanding.

In this paper, we compare behaviors of two possible descriptions of the sampled signal in the case when the sampling period tends to zero, but remains all the time greater than zero. Note that this is the case we are dealing with in analog-to-digital conversion in microwave photonic systems. From this comparison, it follows that the description with the weighted step function is superior to the description with the weighted Dirac comb. A couple of useful comments and remarks associated with the results presented are also provided in the context of microwave photonic systems.

In this paper, we show that the signal sampling operation considered as a non-ideal one, which incorporates finite time switching and operation of signal blurring, does not lead, as the researchers would expect, to Dirac impulses for the case of their ideal behavior.

The problem of an inconsistent description of an “interface” between the A/D converter and the digital signal processor that implements, for example, a digital filtering (described by a difference equation) – when a sequence of some hypothetical weighted Dirac deltas occurs at its input, instead of a sequence of numbers – is addressed in this paper. Digital signal processors work on numbers, and there is no “interface” element that converts Dirac deltas into numbers. The output of the A/D converter is directly connected to the input of the signal processor. Hence, a clear conclusion must follow that sampling devices do not generate Dirac deltas. Not the other way around. Furthermore, this fact has far-reaching implications in the spectral analysis of discrete signals, as discussed in other works referred to in this paper.

A new model of ideal signal sampling operation is developed in this paper. This model does not use the Dirac comb in an analytical description of sampled signals in the continuous time domain. Instead, it utilizes functions of a continuous time variable, which are introduced in this paper: a basic Kronecker time function and a Kronecker comb (that exploits the first of them). But, a basic principle behind this model remains the same; that is it is also a multiplier which multiplies a signal of a continuous time by a comb. Using a concept of a signal object (or utilizing equivalent arguments) presented elsewhere, it has been possible to find a correct expression describing the spectrum of a sampled signal so modelled. Moreover, the analysis of this expression showed that aliases and folding effects cannot occur in the sampled signal spectrum, provided that the signal sampling is performed ideally.

In this paper, a new proof of ambiguity of the formula describing the aliasing and folding effects in spectra of sampled signals is presented. It uses the model of non-ideal sampling operation published by Vetterli et al. Here, their model is modified and its black-box equivalent form is achieved. It is shown that this modified model delivers the same output sequences but of different spectral properties. Finally, a remark on two possible understandings of the operation of non-ideal sampling is enclosed as well as fundamental errors that are made in perception and description of sampled signals are considered.

In this paper, the problem of aliasing and folding effects in spectrum of sampled signals in view of Information Theory is discussed. To this end, the information content of deterministic continuous time signals, which are continuous functions, is formulated first. Then, this notion is extended to the sampled versions of these signals. In connection with it, new signal objects that are partly functions but partly not are introduced. It is shown that they allow to interpret correctly what the Whittaker– Shannon reconstruction formula in fact does. With help of this tool, the spectrum of the sampled signal is correctly calculated. The result achieved demonstrates that no aliasing and folding effects occur in the latter. Finally, it is shown that a Banach–Tarski-like paradox can be observed on the occasion of signal sampling.