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Abstract

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.
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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Department of Marine Telecommunications, Faculty of Electrical Engineering, Gdynia Maritime University, Poland
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Abstract

This paper tries to get a response to the following question: When can a narrowband power amplifier (PA) be considered to be memoryless and when can it not be considered memoryless? To this end, a thorough and consistent analysis of the notions and definitions related with the above topic is carried out. In the considerations presented, two models of the narrowband PA are exploited interchangeably: the black box model widely used in the literature and a model developed here, which is based on the Volterra series. These two models complement each other. In this paper, the conditions for a linear or nonlinear narrowband PA to be memoryless or approximately memoryless or possessing memory are derived and illustrated. They are formulated in terms of the signal delay as well as in terms of the amplitude-to-phase (AM/PM) conversion of the amplifier. Furthermore, the two possible interpretations of the amplitude-to-amplitude (AM/AM) and AM/PM conversions are given a mathematical framework. That is these conversions are presented through some operations. One set of these operations allows to treat the AM/AM and AM/PM conversions as distortions of the modulating signals. Or equivalently as distortions of a given signal constellation when it passes through the PA. Finally, it is proved that the Saleh’s and Ghorbani’s models of the AM/AM and AM/PM conversions occurring in the PAs, which were published in the literature, are not memoryless ones.
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Authors and Affiliations

Andrzej Borys
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Abstract

The convolution operation used in deterministic network calculus differs from its counterpart known from the classic systems theory. A reason for this lies in the fact that the former is defined in terms of the so-called min-plus algebra. Therefore, it is oft difficult to realize how it really works. In these cases, its graphical interpretation can be very helpful. This paper is devoted to a topic of construction of the min-plus convolution curve. This is done here in a systematic way to avoid arriving at non-transparent figures that are presented in publications. Contrary to this, our procedure is very transparent and removes shortcomings of constructions known in the literature. Some examples illustrate its usefulness.
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Authors and Affiliations

Andrzej Borys
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Abstract

This paper describes a “distributed method” of introducing the humanitarian engineering principles and concepts to the curriculum of telecommunications at a maritime university. That is by modifying appropriately the syllabi of the telecommunications subjects taught. The propositions made in this area are illustrated by the concrete examples taken from the current Polish Qualifications Framework for the higher education system in Poland. And, for clarity and consistency of presentation, fundamentals and principles as well as a basic terminology and features of this Framework are also highlighted here shortly. Moreover, it has been shown that the approach presented in this paper is more useful compared to a method based on organization of some special courses for students on the humanitarian engineering, in particular when this regards a maritime university.

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Authors and Affiliations

Andrzej Borys
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Abstract

In this paper, we present some useful results related with the sampling theorem and the reconstruction formula. The first of them regards a relation existing between bandwidths of interpolating functions different from a perfectreconstruction one and the bandwidth of the latter. Furthermore, we prove here that two non-identical interpolating functions can have the same bandwidths if and only if their (same) bandwidth is a multiple of the bandwidth of an original unsampled signal. The next result shows that sets of sampling points of two nonidentical (but not necessarily interpolating) functions possessing different bandwidths are unique for all sampling periods smaller or equal to a given period (calculated in a theorem provided). These results are completed by the following one: in case of two different signals possessing the same bandwidth but different spectra shapes, their sets of sampling points must differ from each other.

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Authors and Affiliations

Andrzej Borys
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Abstract

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.
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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Department of Marine Telecommunications, Electrical Engineering Faculty, Gdynia Maritime University, ul. Morska 81-87, 81-225 Gdynia, Poland
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Abstract

In this paper, it has been shown that any measuring process can be modeled as a process of sampling of signals. Also, a notion of a special kind of functions, called here functions with attributes, has been introduced. The starting point here, in the first of the above themes, is an observation that in fact we are not able to measure and record truly continuously in time any physical quantity. The measuring process can be viewed as going stepwise that is in steps from one instant to another, similarly as a sampling of signals proceeds. Therefore, it can be modeled as the latter one. We discuss this in more detail here. And, the notion of functions with attributes, we introduced here, follows in a natural way from the interpretation of both the measuring process as well as the sampling of signals that we present in this paper. It turns out to be useful.

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Authors and Affiliations

Andrzej Borys
ORCID: ORCID
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Abstract

We present here a few thoughts regarding topological aspects of transferring a signal of a continuous time into its discrete counterpart and recovering an analog signal from its discrete-time equivalent. In our view, the observations presented here highlight the essence of the above transformations. Moreover, they enable deeper understanding of the reconstruction formula and of the sampling theorem. We also interpret here these two borderline cases that are associated with a time quantization step going to zero, on the one hand, and approaching its greatest value provided by the sampling theorem, on the other

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Authors and Affiliations

Andrzej Borys
ORCID: ORCID
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Abstract

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.
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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Gdynia Maritime University, Poland
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Abstract

In this paper, it has been shown that the spectrum aliasing and folding effects occur only in the case of non-ideal signal sampling. When the duration of the signal sampling is equal to zero, these effects do not occur at all. In other words, the absolutely necessary condition for their occurrence is just a nonzero value of this time. Periodicity of the sampling process plays a secondary role.
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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Department of Marine Telecommunications, Faculty of Electrical Engineering, Gdynia Maritime University, Gdynia, Poland
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Abstract

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.
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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Department of Marine Telecommunications, Faculty of Electrical Engineering, Gdynia Maritime University, Gdynia, Poland
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Abstract

It is shown that a number of equivalent choices for the calculation of the spectrum of a sampled signal are possible. Two such choices are presented in this paper. It is illustrated that the proposed calculations are more physically relevant than the definition currently in use.
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Bibliography

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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Department of Marine Telecommunications, Faculty of Electrical Engineering, Gdynia Maritime University, Gdynia, Poland
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Abstract

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.
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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Department of Marine Telecommunications, Faculty of Electrical Engineering, Gdynia Maritime University, Gdynia, Poland
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Abstract

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.
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Authors and Affiliations

Andrzej Borys
1
ORCID: ORCID

  1. Department of Marine Telecommunications, Faculty of Electrical Engineering, Gdynia Maritime University, Gdynia, Poland
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Abstract

This paper is devoted to some problems that appear in derivations of the discrete time Fourier transform from a formula for its continuous time counterpart for transformation from the time into the frequency domain as well as to those regarding transformation in the inverse direction. In particular, the latter ones remained so far an unresolved problem. It is solved for the first time here. Many detailed explanations accompanying the solution found are presented. Finally, it is also worth noting that our derivations do not exploit any of such sophisticated mathematical tools as the so-called Dirac delta and Dirac comb.

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Authors and Affiliations

Andrzej Borys
ORCID: ORCID

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