VIRTUAL STAND FOR SIGNALS DISCRETISATION RESEARCH
E. Berezkin
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russian Federation
Table of contents
2. The complex of technical means pertaining to the virtual stand
3. Visualisation parameters and spectral characteristics of signals
Abstract
The program model of the research stand for studying the main principles of the discretisation of continuous signals on time is offered. The result is a technical scheme of the virtual stand, with the possibility of supervision of signals at different points on a virtual oscillograph. The substantiation of the accepted decisions is given. The process of restoration of the initial determined signal on discrete readouts is shown. Spectral characteristics of signals by means of algorithm fast Fourier transform (FFT) are calculated.
Keywords: discretisation, frequency criterion, discretisation interval.
1. Introduction
Much less information is usually required for practical tasks of data processing than there is received from the sensors in the form of a continuous analog signal. Rational performance of discretisation and quantisation of initial data provides an opportunity to reduce the cost of storing and processing information [1].
In communications technology, when sending different signals
we usually have to deal with functions of time, the spectrum of which is limited,
i.e. does not contain frequencies above a certain boundary. Such functions have
a unique feature established for the first time in 1933 by V.A. Kotelnikov and expressed
in his theorem – a theorem which plays a fundamental role in the theory of
communication and, in particular, in the pulse of communication. The theorem in the formulation of the
author says: «In any function 
, consisting of frequencies from 0 to 
, it is possible to transfer with any
accuracy by means of the numbers following one after another through 
seconds».
In studying theoretical elements of the digitisation of continuous functions, we can utilise the extremely useful visualisation of the process of digitisation [2]. The curricula of many technical directions of preparation, of both bachelors and specialists, include disciplines that contain chapters on the study of the principles of discretisation of continuous signals. As a rule, practical support training on digital signal processing is based on the use of professional packages MatLAB [3], MathCAD [4] and LabVIEW [5]. However, this approach requires substantial investment of resources and classroom time. The alternative to such training is the establishment of specialised research laboratory stands [6,7]. An essential obstacle to the realisation of this variant is the necessary presence of a corresponding laboratory base.
In this work the task is formulated as follows: to support the learning process and practical sessions by the most economical way in conditions of limited resources and classroom hours with the least risk of reducing the quality of education. Such a decision is important considering the preparation of students who do not specialize in the digital processing of signals but who should possess corresponding knowledge and skills.
This can potentially be solved by the creation of a program model for the research stand. This program model is developed by the author in the environment of programming DELPHI, with the use of hypertext help systems HTML Help Workshop [8].
The main decisions which have been accepted by development of the virtual research stand are justified by known theoretical positions and mathematical models [9,10].
2. The complex of technical means pertaining to the virtual stand
The technical means of the research stand are realised as a program complex, the functional model of which is shown in Fig. 1. In the functional model of the software complex are 10 objects, each of which has its own properties and parameters. Each object has its own algorithm depending on its function. They will transform a signal by the set rules, or remove it in the display interface.
Fig. 1. Functional work model of a program complex
It should be noted that these software tools of scientific visualisation complex essentially implements the research stand (Fig. 2) with wide functional capabilities, which allow the generation of various signals, performance of digitisation, repair of discrete readouts’ initial form and calculation of their spectral characteristics with the help of algorithm FFT.
Fig. 2. Generalised structure of technical means designed for the study of signals
On the generalised structure of technical means one can see:
generator, reproducing the signal under investigation;
· analog-to-digital converter (ADC), discretizing signal in time and quantizing level;
· ideal filter of low frequencies (FLF), which is necessary for signal restoration on discrete readouts;
· COMPUTERS realising algorithms of an estimation and visualisation for signal spectrum, correlation function and spectral density of an average power;
· oscillograph which is used to display an initial continuous signal, digitisation of a signal and a signal restored by ideal FLF.
For the generator we set: signal type (a choice from the
list – a harmonious signal, rectangular impulses, Gaussian noise and the mixed
signal); length of a signal 
(multiplication of
quantity of points of digitisation on 
); signal
parameters. For a harmonious signal the amplitude, frequency and a signal phase
are entered The amplitude, the period and duration of a rectangular impulse are
entered for impulses. The mathematical expectation and mean squared deviation
of a random variable are entered for Gaussian noise.
For ADC the interval of digitisation and a size of one
quantum are introduced. For FLF frequency of a cut 
is introduced.
Besides, the parameter «Show components» is set for FLF. At its installation
the oscillograph will display the coordinate determined functions of time
,
which addition with weight factors 
also
restores an initial signal. It is necessary for clearer understanding of a
statement of the theorem as function with the limited spectrum can be presented
by a series
,
called series of Kotelnikov. For example, the expansion in a series of Kotelnikov of exponential impulse will look like that presented on Fig. 3.
Fig. 3. Expansion of exponential impulse abreast Kotelnikova
3. Visualisation parameters and spectral characteristics of signals
The button ‘Model’ is activated after input of the necessary data and parameters. As a result the simulated signal and its spectral characteristics will be constructed on tabs ‘Oscillograph’, «Spectrum of amplitudes», «Correlative function» and «Spectral density power ».
It is possible to observe a signal on a virtual oscillograph at each stage of handling: the continuous, discrete and restored signal.
The tab «Spectrum of amplitudes» displays 
.
For discrete representation of signals argument 
is
substituted by numbers of readouts 
, and Fourier transforms
are performed by the argument 
(frequency step) on the
main periods. Transformations
are named as discrete Fourier transforms (DFT) [1]. For great volumes of data files of an evaluation can lead to essential temporary expenditures. Acceleration of evaluations is reached at by fast Fourier transformation. The FFT is based on the fact that there are many periodically repeating values (owing to periodicity of functions) for evaluations among multipliers (sine and cosines).
The algorithm of FFT groups is composed with identical
multipliers in a pyramidal algorithm, considerably reducing the number of
multiplications due to eliminating of repeated evaluations. As a result
depending on 
FFT speed can exceed hundreds of
times the speed of standard algorithm. Thus, it is necessary to underline that
FFT algorithm is even more exact than standard one because the reducing of
operations number leads to smaller round-off errors.
FFT algorithms are based on properties of a complex
exponential function 
: its symmetries 
and
periodicity 
with the period equal to the
length of the treated realisation of a signal 
. The essence of
an FFT’s algorithm consists in partition DFT of initial sequence on DFT
subsequences of smaller length. The partition means decimation on time or in
frequency area. Unlike DFT, the FFT can be calculated only on certain number of
points` , corresponding to the whole degree
of its foundation 
, where 
is a
number of stages decimation, 
. FFT on the foundation 
can
be considered as the most used ones. The FFT algorithm with decimation on time
on the foundation 2 is realised in the given model.
Singularity of algorithm of an FFT with decimation on time
is the unnatural order of references of the entering signal, caused by its
multiple partitions on even and odd subsequences (
= 0, 4,
2, 6, 1, 5, 3, 7 for 
). It leads to necessity of
preliminary permutation of
discrete readouts of the initial sequence prior to the beginning of evaluations.
The tab «Correlative function» displays 
. The
correlation analysis gives the chance to establish in the series of digital
data of signals the presence of certain connection of a modification of values
of signals on an explanatory variable. For a function space of signals this
degree of connection can be expressed in normalised units of correlation coefficient
which accepts values from 1 (full coincidence of signals) to -1 (complete
antithesis) and does not depend on the value of units of measurements.
The autocorrelation function of a signal 
,
final on energy, is quantitative integral characteristic of the form of a
signal and is defined by an integral from multiplication of two copies of the
set signal 
, shifted relative to each other at
the time 
,
.
The tab «Spectral density power» displays 
.
Spectral density power or the energy spectrum is a function defined in the strict mathematical sense as
,
where 
is the spectral density of the -realization
on the segment 
.
From theorem Khinchin-Wiener energy spectrum and correlation function are a pair of Fourier transform:
This means that the spectral density power of the stationary random process is the amplitude spectrum of the autocorrelation function.
Fig. 4 and Fig. 5 show electronic forms of display and research of test signals (piece of harmonic oscillations and packet of rectangular pulses) at different stages of their processing; their spectral characteristics are also presented.
| a) | b) |
 |
 |
| c) | d) | e) |
 |
 |
 |
| f) | g) | h) |
 |
 |
 |
Fig. 4. Screen forms of a piece of harmonic signal investigation:
a – restored signal at the output of an ideal FLF; b – coordinate deterministic functions with weights
factors 
; c – timing
values of the initial signal taken through 
; d – correlation
function of a deterministic signal 
; e – spectral
characteristics of the digitisation signal 
; f – signal to
noise; g – spectral characteristics a signal
to noise ; h – spectral
density of average power 
| a) | b) |
 |
 |
| c) | d) | e) |
 |
 |
 |
Fig. 5. Screen forms of a pack of rectangular impulses investigation:
a – initial pack of
rectangular impulses; b – restored signal at the
output of an ideal FLF ; c – coordinate deterministic functions with weights
factors ;
d – spectral characteristics of the
digitisation signal ;
e – correlation function of a deterministic
signal
A dimensional grid with the signed values of vertical and horizontal readouts is provided on the form of display of spectral characteristics. There are tools for easy viewing in the program: motion charts in either direction, zoom in a certain area of the graph, graph scaling on visibility area. The implemented program model was debugged with purpose to reach the most possible correct response to modelled situations and the full coincidence of theoretical calculations to received results. Digital processing of test signals is shown in Fig. 4 and Fig. 5, highlighting the validity of this assertion.
4. Conclusion
In general, virtual research bench intensifies the learning process and provides for the formation of knowledge, abilities and skills at the application level and at the level of creativity. The whole software complex of educational materials implements a system-active approach, which focuses on education. As result we do not consider the sum of learned information (mathematical models), we consider ability to act in certain situations.
In addition, a virtual stand can reduce classroom time on the study of the relevant section of the technical subjects without sacrificing the quality of education. It is indispensable, if there is a deficiency of pedagogical and laboratory resources for quality teaching in high schools of the distributed university.
References
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