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1
Content available Revision of the formula describing the spectrum of output signals at A/D converters
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2024
|
Vol. 70, No. 1
183--190
EN
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.
2
Content available Sampled Signal Description That Is Used in Calculation of Spectrum of This Signal Needs Revision
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2023
|
Vol. 69, No. 2
319--324
EN
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.
3
Content available An Unexpected Result on Modelling the Behavior of A/D Converters and the Signals They Produce
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2023
|
Vol. 69, No. 1
185--191
EN
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.
4
Content available On modelling and description of the output signal of a sampling device
Borys Andrzej
Archives of Electrical Engineering
|
2023
|
Vol. 72, nr 1
147--155
EN
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.
5
Content available The Problem of Aliasing and Folding Effects in Spectrum of Sampled Signals in View of Information Theory
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2022
|
Vol. 68, No. 2
315--322
EN
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.
6
Content available Another Proof of Ambiguity of the Formula Describing Aliasing and Folding Effects in Spectrum of Sampled Signal
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2022
|
Vol. 68, No. 1
115--122
EN
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.
7
Content available Extended Definitions of Spectrum of a Sampled Signal
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2021
|
Vol. 67, No. 3
395--401
EN
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.
8
Content available Some Topological Aspects of Sampling Theorem and Reconstruction Formula
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2020
|
Vol. 66, No. 2
301--307
EN
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.
9
Content available On Derivation of Discrete Time Fourier Transform from Its Continuous Counterpart
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2020
|
Vol. 66, No. 2
355-360
EN
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.
10
Content available Measuring Process via Sampling of Signals, and Functions with Attributes
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2020
|
Vol. 66, No. 2
309--314
EN
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.
11
Content available Further Discussion on Modeling of Measuring Process via Sampling of Signals
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2020
|
Vol. 66, No. 3
507--513
EN
In this paper, we continue a topic of modeling measuring processes by perceiving them as a kind of signal sampling. And, in this respect, note that an ideal model was developed in a previous work. Whereas here, we present its nonideal version. This extended model takes into account an effect, which is called averaging of a measured signal. And, we show here that it is similar to smearing of signal samples arising in nonideal signal sampling. Furthermore, we demonstrate in this paper that signal averaging and signal smearing mean principally the same, under the conditions given. So, they can be modeled in the same way. A thorough analysis of errors related to the signal averaging in a measuring process is given and illustrated with equivalent schemes of the relationships derived. Furthermore, the results obtained are compared with the corresponding ones that were achieved analyzing amplitude quantization effects of sampled signals used in digital techniques. Also, we show here that modeling of errors related to signal averaging through the so-called quantization noise, assumed to be a uniform distributed random signal, is rather a bad choice. In this paper, an upper bound for the above error is derived. Moreover, conditions for occurrence of hidden aliasing effects in a measured signal are given.
12
Content available Filtering Property of Signal Sampling in General and Under-Sampling as a Specific Operation of Filtering Connected with Signal Shaping at the Same Time
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2020
|
Vol. 66, No. 3
589--594
EN
In this paper, we show that signal sampling operation can be considered as a kind of all-pass filtering in the time domain, when the Nyquist frequency is larger or equal to the maximal frequency in the spectrum of a signal sampled. We demonstrate that this seemingly obvious observation has wide-ranging implications. They are discussed here in detail. Furthermore, we discuss also signal shaping effects that occur in the case of signal under-sampling. That is, when the Nyquist frequency is smaller than the maximal frequency in the spectrum of a signal sampled. Further, we explain the mechanism of a specific signal distortion that arises under these circumstances. We call it the signal shaping, not the signal aliasing, because of many reasons discussed throughout this paper. Mainly however because of the fact that the operation behind it, called also the signal shaping here, is not a filtering in a usual sense. And, it is shown that this kind of shaping depends upon the sampling phase. Furthermore, formulated in other words, this operation can be viewed as a one which shapes the signal and performs the low-pass filtering of it at the same time. Also, an interesting relation connecting the Fourier transform of a signal filtered with the use of an ideal low-pass filter having the cut frequency lying in the region of under-sampling with the Fourier transforms of its two under-sampled versions is derived. This relation is presented in the time domain, too.
13
Content available Modelling of a Wireless Channel with Time Dependent Parameters
Mieczyńska Marta, Borys Andrzej
Scientific Journal of Gdynia Maritime University
|
2019
|
nr 111
24--38
EN
It is convenient to analyse wireless channels and links by exploiting their input-output description. This approach relies on treating the system as a black box, whose behaviour can be fully described by the relationship between the input and output signals. In this paper, we study a relationship of the above type for linear wireless channels having time-dependent parameters, which also takes a multipath propagation environment into account. A starting point for the derivations presented here is a relationship derived in the literature for this type of model with the application of a single sinusoidal input signal. The subject of this paper is the generalization of that relationship for periodic input signals and then of non-periodic signals. To the best of our knowledge, the literature lacks a suitably convincing generalization. The derivations of this paper exploit a principle of superposition valid for linear systems as well as the relations existing between Fourier series and the Fourier integrals. The discussion is illustrated by the results of simulations performed with the help of the MATLAB program.
14
Content available A Note on Treatment of Signals in Digital Signal Processing and in Network Calculus
Borys Andrzej
Scientific Journal of Gdynia Maritime University
|
2019
|
nr 109
21--29
EN
This short paper presents from the perspective of the operator theory some basic operations performed on signals in the digital signal processing as well as in the network calculus. These are the following operations: signal sampling, amplitude quantization, and signal recovery from its samples – in the digital signal processing. And regarding the network calculus, building up an auxiliary (continuous) traffic flow and recovery of a real traffic that possesses a non-continuous structure (with some granularity) after manipulations that were carried out with the use of a flow model are discussed in this paper. Some interesting results achieved and interpretations regarding the aforementioned stuff are presented.
15
Content available Some Useful Results Related with Sampling Theorem and Reconstruction Formula
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2019
|
Vol. 65, No. 3
471--475
EN
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.
16
Content available Introducing Humanitarian Engineering Concepts to the Curriculum of Telecommunications
Borys Andrzej
International Journal of Electronics and Telecommunications
|
2019
|
Vol. 65, No. 1
125--131
EN
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.
17
Content available Bandwidth Estimation Using Network Calculus in Practice
Wasielewska Katarzyna, Borys Andrzej
International Journal of Electronics and Telecommunications
|
2019
|
Vol. 65, No. 1
133--138
EN
An available bandwidth at a link is an unused capacity. Its measuring and/or estimation is not simple in practice. On the other hand, we know that its continuous knowledge is crucial for the operation of almost all networks. Therefore, there is a continuous effort in improving the existing and developing new methods of available bandwidth measurement and/or estimation. This paper deals with these problems. Network calculus terminology allows to express an available bandwidth in terms of a service curve. The service curve is a function representing a service available for a traffic flow which can be measured/estimated in a node as well as at an end-to-end connection of a network. An Internet traffic is highly unpredictable what hinders to a large extent an execution of the tasks mentioned above. This paper draws attention to pitfalls and difficulties with application of the existing network calculus methods of an available bandwidth estimation in a real Internet Service Provider (ISP) network. The results achieved in measurements have been also confirmed in simulations performed as well as by mathematical considerations presented here. They give a new perspective on the outcomes obtained by other authors and on their interpretations.
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