RU178399U1 - OWN VECTOR SPECTRUM ANALYZER AND SIGNAL COMPONENTS - Google Patents

Abstract

The utility model relates to measuring technique, in particular to signal spectrum analyzers. The structure of the spectrum analyzer of eigenvectors and signal components includes: ensemble formation unit 1, covariance matrix calculation unit 2, eigenvector and eigenvalue calculation unit 3, eigenvector selection unit 4, memory and program control unit 5, spectral characteristics estimation and interpretation unit and errors 6. The spectrum analyzer of eigenvectors and signal components can significantly increase the sensitivity and selectivity of the analysis by eliminating the negative effect of mixing I am an uncorrelated component arising from ordinary spectral analysis. 12 ill.

Description

Technical field

The spectrum analyzer of eigenvectors and signal components (ASCViKS) is a universal measuring and research device (device), separate or built-in, the purpose of which is visual and / or automated (including automatic) analysis (including retrospective) of the studied signal, regardless of its physical nature.

ASCViKS opens up new possibilities for solving the problems of identifying energy-non-dominant components of the analyzed signal that are independent of other components and spectrally localized at predetermined frequencies, the detection of which is impossible using well-known methods of spectral analysis. ASCViKS opens up new possibilities in solving the problems of radio and geophysics, acoustics, sound, radio and sonar, medical and other diagnostics, when new physical phenomena are detected.

Table 1 shows the abbreviations used in the description of the utility model.

The level of technology. Analogs and their disadvantages

Periodic, almost periodic, quasiperiodic [2, 3] processes occupy a special place in the study of natural and technical objects. To describe such processes, spectral analysis has been developed and is widely used, in which the signal is represented as the sum of simple harmonic components.

The transition to digital processing methods has expanded the scope of algebraic representations of signals. In this regard, technical solutions have appeared in which algebraic and spectral representations are used together. Among such solutions is an eigenvector analyzer and signal components - an aigenoscope [1] - FIG. 1 a).

Among the classical, most frequently used, spectral descriptions, one should note the amplitude and phase spectra (used in the analysis of deterministic signals) and energy spectra (used in the analysis of random signals and deterministic chaos).

In research practice, the task of identifying a component in an analyzed time series that is localized at a predetermined frequency is often encountered. This task is trivial if the detected component is a harmonic oscillation with a known frequency and an unknown initial phase, and the signal-to-noise ratio is quite large. In this case, the so-called incoherent harmonic signal receiver against the background of noise is used [5]. the structure of such a receiver coincides with the structure of a classical amplitude spectrum analyzer [4] tuned to a known frequency. This is the so-called quadrature scheme, the construction of which is shown in FIG. 1 b). The identification of the periodic component using this scheme is the more effective the greater the signal-to-noise ratio and the interval of spectral analysis.

In the case when the task arises of identifying spectrally localized (colored) noise against the background of unpainted (white) noise, the design changes and coincides with the energy spectrum analyzer [4]. This is the same quadrature scheme in which filters are included behind the multipliers (the impulse response of which is determined by the correlation function of colored noise), quadrators and integrators - see Fig. 1 c).

The situation is complicated when the solution to the identification problem is associated with high a priori uncertainty, both about the properties of the detected spectrally localized component and about the processes against which the identification problem is solved.

For example, the use of energy and amplitude spectrum analyzers in identifying tidal phenomena associated with periodic movements of the Moon and the Earth does not always allow us to identify components in the vertical component of the Earth’s electric field at known frequencies of lunar tides, although they are well detected using gravimetry. Further, we show that this is due to two circumstances: the low energy of the detected spectrally localized component and the complex structure of the signal from which the component is extracted. As will be shown below, the use of ASCViKS allows the identification of components that correspond to lunar tides.

Referring ACC & A by the signs of calculating the covariance matrix and its eigenvectors and eigenvalues to aigenoscopes in it, it should be noted that the presence in ACC & C of a very specific unit for evaluating and interpreting spectral characteristics and errors 6 gives the device new qualities and allows solving new problems.

ASCViKS, unlike a prototype (aigenoscope), allows you to eliminate and at the same time evaluate the value of the negative effect of mixing in the analyzer of the energy spectrum of uncorrelated components, thereby increasing the sensitivity and selectivity of the analysis several times.

It should be noted that the design of ACC & A (combination of blocks and their connections) provides (in contrast to the prototype) the formation of:

- new previously unused spectral characteristics, allowing to evaluate the energy contribution of both the group and individual detected components;

- assess the applicability of the analyzers of the amplitude and energy spectra.

As a prototype and analogues for the claimed device, the structures shown in FIG. 1 a), 1 b) and 1 c), namely: an aigenoscope, a quadrature scheme for estimating the amplitude spectrum and an energy spectrum analyzer.

Utility Model Disclosure

ASCViKS is a universal device for identifying, analyzing, and interpreting energetically non-dominant spectrally localized components of time series (signals) using eigenvectors of the covariance matrix of a time series at a given finite analysis interval.

All the work of ACCS & AC units is aimed at ensuring the actions of the unit for evaluating and interpreting spectral characteristics and errors 6 and processing in it the results obtained from other units, which will be described later.

In the course of processing at ACC & A:

1. The covariance matrix of the time series is estimated, its eigenvectors and eigenvalues are calculated.

2. A spectral analysis of the eigenvectors and interpretation of the amplitude spectra of the eigenvectors is carried out in order to identify those eigenvectors that determine the uncorrelated components of the time series localized at given frequencies.

3. Based on the amplitude spectra of the selected eigenvectors, spectral characteristics are formed.

The technical result from the use of ASCViKS consists in the possibility of identifying and separate spectral analysis of uncorrelated (at a given analysis interval) components of the time series, which allows (as will be shown later) to increase the selectivity and sensitivity of the analysis by several times, as well as form an assessment of applicability in a specific research situation standard methods of spectral analysis.

The proposed device - ASCViKS is designed to identify and analyze on a finite time interval analysis of time series containing spectrally localized energetically non-dominant uncorrelated components that cannot be detected using classical methods of spectral analysis.

Among these investigated time series is the set of time series produced including geophysical, astrophysical and technical objects.

In FIG. 2 shows the design of the spectrum analyzer of eigenvectors and signal components, where the following are connected to the memory and program control unit 5:

- block formation of the ensemble 1,

- block calculation of the covariance matrix 2,

- unit for calculating eigenvectors and eigenvalues 3,

- block selection of eigenvectors 4,

- unit for the assessment and interpretation of spectral characteristics and errors 6.

In this case, the double-sided output of block 5 serves to:

- receiving the initial processed information,

- communication with external devices to produce results,

- receiving new tasks.

The results of the operation of blocks 1, 2, 3, 4, 6 are received in the memory of block 5, from where they are taken for further processing.

Let us dwell on the work of the design blocks of the spectrum analyzer of eigenvectors and signal components and their interaction, which is described in detail in Table 6.

, а вектор, записанный в виде матрицы-столбца («кет») как

. В этой нотации скалярное произведение двух действительных векторов [6, 7] записывается в виде

или в виде

.Recall that a vector is an ordered sequence of numbers [6]. When writing vectors, we will use the “bracket” notation used in quantum mechanics [7], denoting a vector written in the form of a row matrix (“brace”) as

, and the vector written in the form of a column matrix (“ket”) as

. In this notation, the scalar product of two real vectors [6, 7] is written as

or in the form

.

In the unit of formation of ensemble 1, the time series, which is a sequence of samples S ₁ , S ₂ , ..., S _L , is converted into the matrix of the ensemble A _{M × N} , where M is the size of the final analysis interval, and N≤ (L-M + 1) - the number of elements of the ensemble. Each of the elements of the ensemble is a segment of a time series of length M and is recorded in the matrix of the ensemble in the form of a column.

If N = (L-M + 1), then the ensemble matrix includes all segments of a time series of length M that do not coincide with each other. If each subsequent element of the ensemble matrix is obtained by a unit shift with respect to the previous one, then the ensemble matrix A _{M × N} coincides with the so-called trajectory matrix [8]. In the general case, when forming an ensemble matrix, a shift not equal to unity can be used.

The ensemble matrix formed in block 1 is transferred to storage in block 5, from where it enters block 2.

The covariance matrix in block 2 is calculated based on the matrix of the ensemble A _{M × N} using the relation

Where

- транспонированная матрица A_M×N,

- transposed matrix A _{M × N} ,

- парциальная ковариационная матрица,

- partial covariance matrix,

и

- векторы отсчетов на i-ом конечном интервале анализа в форме «бра» и «кет», соответственно.

and

are the sample vectors on the i-th final analysis interval in the form of “sconces” and “ket”, respectively.

From the output of block 2, the covariance matrix goes to block 5, from where it is transferred to the block of the calculator of eigenvectors and eigenvalues 3.

At the output of the block of the calculator of eigenvectors and eigenvalues 3 are formed (using any suitable linear algebra algorithm [9]) eigenvectors and eigenvalues corresponding to equivalent relations

Where

K _{M × M} - covariance matrix of size M × M,

- собственные значения,λ _i

- eigenvalues

- собственные векторы в форме «бра»,〈Ψ _i ⎜,

- eigenvectors in the form of a sconce,

- собственные векторы в форме «кет»,⎜ψ _i 〉

- eigenvectors in the form of "ket",

M is the size of the final analysis interval.

спектром собственных значений [1].As a rule, eigenvectors are numbered in descending order of eigenvalues. In this case, we will call the sequence λ _i ,

spectrum of eigenvalues [1].

, получаемых из элементов ансамбля

.In the general case, as can be seen from relation (1), the covariance matrix is obtained by averaging the partial covariance matrices

derived from ensemble elements

.

The eigenvectors of the covariance matrix form an orthonormal basis [6], that is

Where

δ _{i, j} is the Kronecker symbol [6].

From relations (1-3), it follows that the covariance matrix can be represented through its own vectors and eigenvalues in the form

Relation (5) is called the polarization relation [10]; it will be further needed to justify causal relationships that provide the claimed technical result.

The eigenvectors and eigenvalues obtained in block 3 are stored in the memory and program control unit 5 for further use.

Let us dwell on the options for analyzing the amplitude spectra of eigenvectors in the eigenvector selection unit 4, the unit for evaluating and interpreting spectral characteristics and errors 6, and also on the interaction of these units with the memory and program control unit 5, with which they are connected by two-way communications designed to control the units transferring stored information in them and transmitting the processing results to storage.

The eigenvector selection block 4 receives from block 5 a set of all eigenvectors, as well as control information about the type of selection criterion used and the selection parameters and a list of desired frequencies. Block 4 returns as results of processing to block 5 an estimate of the amplitude spectra of the eigenvectors, the numbers of those eigenvectors that are recognized spectrally localized at the desired frequencies, as well as additional information about the criteria for the spectral localization of eigenvectors.

In the unit for estimating and interpreting spectral characteristics and errors 6 using a selected list of eigenvectors, spectral characteristics are formed on the basis of estimates of the amplitude spectra of the eigenvectors and eigenvalues, the properties of which will be discussed below.

The memory and program control unit 5 is connected by a two-way output to the peripheral ASCViKS and therefore receives the necessary source and control information, and also transfers the results of the ASCViKS operation to the external environment. Block 5 transfers to the remaining blocks the control and initial information necessary for their operation.

Let us dwell on the work of block 4.

,

. Поскольку каждое из БПФ представляет собой комплексную функцию целочисленного аргумента

такую, что ее действительная часть представляет собой четную функцию, а мнимая - нечетную функцию относительно начала координат, перенесенного в точку M/2, то далее обрабатывается только последовательность отсчетов БПФ с номерами от 1 до целой части величины М/2. Таким образом, областью определения

для i-го собственного вектора

будет последовательность дискретных частот

, где ceil (…) - функция округления числа сверху. Дискрет частоты БПФ собственного вектора определяется соотношениемIn block 4, a fast Fourier transform (FFT) of each of the M eigenvectors is performed. We will further designate these FFTs as

,

. Since each of the FFTs is a complex function of an integer argument

such that its real part is an even function, and the imaginary part is an odd function relative to the origin transferred to M / 2, then only the sequence of FFT samples with numbers from 1 to the integer part of M / 2 is processed. Thus, the scope

for the i-th eigenvector

there will be a sequence of discrete frequencies

, where ceil (...) is the function of rounding a number from above. The discrete frequency of the FFT of the eigenvector is determined by the relation

Where

ΔТ is the time series discretization time.

FFT module values corresponding to frequencies

together with these frequencies are considered as an estimate of the amplitude spectrum of the eigenvector, which is transferred to storage in block 5.

Next, the numbers of those eigenvectors are determined for which the amplitude spectra are localized at frequencies close to those sought. For this, criteria can be used:

- the criterion of "minimum frequency detuning" between the desired frequency and the frequency of the maximum amplitude spectrum of the eigenvector;

- the criterion "coherence index";

- the criterion "maximum correlation coefficient".

The eigenvector coherence index is a numerical criterion whose value determines the degree of spectral localization of the eigenvector. The value of the coherence index is defined as the ratio of the maximum value of the amplitude spectrum of the eigenvector to the average value of the amplitude spectrum of the same eigenvector, i.e.

Where

max (...) - maximum

mean (...) - average.

The minimum value of the coherence index is unity and is achieved when the maximum value is equal to the average value. The maximum value of the coherence index is achieved when the FFT has one nonzero sample (all the others are equal to zero). Let the value of this reference (numerator (8)) be equal to A, then the denominator will be equal to A / ceil (M / 2). Thus, the maximum value of the coherence index is limited by ceil (M / 2).

After determining the coherence indices for all eigenvectors, they are transferred to storage in block 5 and are used further in block 6 for interpretation and decision making.

Let us consider the use of the criterion “minimum frequency detuning” in the eigenvector block 4.

When using it, for each eigenvector the position of the maximum of its amplitude spectrum is determined, and the value of the maximum frequency is compared with the list of sought frequencies. If the difference between the frequency of the maximum amplitude spectrum of the eigenvector and the desired frequency is less than the permissible value ΔF _DOP , then the number of the eigenvector is included in the list of selected eigenvectors. After comparing all the frequencies of the maxima of the amplitude spectra of the eigenvectors with all the desired frequencies, duplicate numbers are removed from the list of numbers of the selected eigenvectors, and this list is transferred to the memory of block 5 for further use in block 6.

,

, определяемого по формулеThe second option for the selection of eigenvectors in block 4 is the use of the criterion for the maximum correlation coefficient between eigenvectors and segments of harmonic oscillations with the desired frequencies

,

defined by the formula

Where

max _φ (...) is the maximum of the expression in brackets, achieved by varying the parameter φ,

и начальной фазой φ,- normalized harmonic vector with frequency

and initial phase φ,

K ^(H) is the normalization coefficient.

The value ρ _{m, n} ≤1.

.The larger this value, the closer the mth eigenvector to the segment of the harmonic signal with a given frequency

.

The value of the criterion “maximum correlation coefficient”, determined for all frequencies from the list of sought frequencies, form a sample of M × N _CSI , which determines the reasonable value of the threshold of the criterion. As such a threshold, it is appropriate to use the quantile value of a given level (for example, the median of a criterion) [6].

The values of the criterion “maximum correlation coefficient” for all eigenvectors and all frequencies from the list of sought frequencies are compared with the threshold value. If the criterion exceeds the threshold, then the eigenvector number falls into the list of selected eigenvectors. The repeating values are eliminated from the list and it is transferred to the memory of block 5 for further calculation of the spectral characteristics in block 6.

Let us describe the operation of the unit for evaluating and interpreting spectral characteristics and errors 6, which determines the operation of the device and its novelty. This block executes:

- calculation and visualization of spectral characteristics presented in table 2;

- calculation and visualization of assessments of applicability of analyzers of energy and amplitude spectra leading to the achievement of the claimed technical result (the meaning of these estimates will be explained later in the description of cause-effect relationships);

- on the instructions of the user, an assessment of the criteria determining the degree of spectral localization of uncorrelated components, as well as the likelihood of a false detection of a group of spectrally localized uncorrelated components;

- preparation of data for visualization and visualization itself.

Note that the aigenoscope, taken as a prototype, does not use the spectral characteristics that are calculated in block 6 of the ASSViKS.

In (13-15), the normalized eigenvalues determined by the relation

совпадает со средним квадратом отсчетов временного ряда, поэтому правая часть в соотношениях (15) и (16) представляет средний квадрат отобранных компонент временного ряда, определяемый соотношениемIn [1] it is shown that the quantity

coincides with the average square of the samples of the time series; therefore, the right-hand side in relations (15) and (16) represents the average square of the selected components of the time series, determined by the relation

In FIG. 11 shows the spectral characteristics corresponding to (14) obtained using ACC & A for sets of double frequencies of rotation of binary star systems from table 4 (Fig. 11 a)), as well as for some individual double frequencies (Fig. 11 b), FIG. 11 c)). Spectral characteristics indicate the successful detection of uncorrelated components spectrally localized at given frequencies.

Consider a causal relationship leading to a technical result.

To demonstrate the technical result obtained using ACC & A, we compare this result with the analogs shown in FIG. 1 b) and 1 c).

To compare ACC & A with the energy spectrum analyzer, we introduce the block diagram shown in FIG. 1 g). It can be shown that the circuit of FIG. 1 d) has the same properties as scheme 1 c), and is the equivalent algebraic diagram of the energy spectrum analyzer.

Thus, it is possible, on the one hand, to show the advantages of ASCViKS in relation to the energy spectrum analyzer, and, on the other hand, to introduce spectral characteristics that allow ASCViKS to estimate errors associated with the use of the energy spectrum analyzer in a specific situation.

. Обозначим эти векторы как

и

, соответственно.The introduced equivalent algebraic scheme of the energy spectrum analyzer (see Fig. 1 g)), like ACCSiX, contains blocks 1 and 2. From the block of reference signals, normalized harmonic vectors corresponding to the desired frequency and having initial phases shifted by

. We denote these vectors as

and

, respectively.

The signals at the outputs of blocks 16 of the circuit of FIG. 1 g) are equivalent to the signals at the inputs of block 15 of the circuit of FIG. 1 c).

Based on the polarization relation (6), we obtain the expressions for the output quantities P _S (ƒ _И ) and P _C (ƒ _И ) of the blocks 16 (Fig. 1 g)) supplied to the inputs of the adder 15, and for the quantity P _Σ (ƒ _И ) at the output of the adder 15, respectively:

Where

ƒ _And - the desired frequency.

For further it is important that the basis of eigenvectors on a finite analysis interval defines a representation of the form [6]

Where

и

- конкретный сегмент временного ряда на конечном интервале анализа,

and

- a specific segment of the time series at the final analysis interval,

- коэффициенты разложения сегмента временного ряда на конечном интервале анализа по собственным векторам ковариационной матрицы.

- the coefficients of the expansion of the segment of the time series in the finite analysis interval according to the eigenvectors of the covariance matrix.

для всех сегментов временного ряда, входящих в матрицу ансамбля A_M×N, имеет видMatrix of expansion coefficients for a specific eigenvector

for all segments of the time series included in the matrix of the ensemble A _{M × N} , has the form

The covariance coefficient of the ith and jth coefficients is determined by the ratio

It follows from relation (30) that the covariance coefficients for i ≠ j are always equal to zero, and for i = j the coefficients are equal to λ _i .

Thus, the necessary prerequisites are identified to justify the technical result.

If the time series under study is formed from additive components independent of each other in the finite analysis interval, then we should expect that the covariance coefficients of these components will (statistically large N) converge to zero, that is, correspond to relation (30). It follows that such independent components must be sought among the eigenvectors of the covariance matrix, and the average square of these components will coincide with the eigenvalues of the covariance matrix.

Conclusion 1: If the signal is formed by components independent on a finite analysis interval, then they should be searched for among the eigenvectors of the covariance matrix.

Corollary 1: All other components orthonormalized on a finite analysis interval (for example, harmonic) are correlated mixtures of eigenvectors of the covariance matrix.

Corollary 2: If among the eigenvectors of the covariance matrix there are no harmonic eigenvectors, then the harmonic orthonormal components on the finite analysis interval are correlated mixtures of the eigenvectors of the covariance matrix.

Corollary 3: If there are no harmonic vectors among the eigenvectors of the covariance matrix at the output of block 3 of the ACVCS, then they are not among the independent components in the finite analysis interval in the analyzed time series. If among the eigenvectors of the covariance matrix there are vectors similar to harmonic by some criteria, then they are uncorrelated components.

Relations (20-25) describe the process of mixing uncorrelated components in the energy spectrum analyzer.

уместно назвать функцией распределения оценки энергетического спектра по некоррелированным компонентам (собственным векторам). Сокращенно это распределение будем обозначать ФРОЭС/СВ. Как будет показано далее, ФРОЭС/СВ является характеристикой применимости анализатора энергетического спектра при анализе некоррелированных компонент временного ряда.The terms included in the sum (22)

it is appropriate to call the distribution function of the estimate of the energy spectrum over uncorrelated components (eigenvectors). This distribution will be abbreviated as FROEC / NE. As will be shown below, the FRES / SW is a characteristic of the applicability of the energy spectrum analyzer in the analysis of uncorrelated components of the time series.

и

уместно назвать распределениями квадратурных компонент оценки энергетического спектра по некоррелированным компонентам (собственным векторам). Сокращенно эти распределения будем обозначать ФРОЭС^(S)/CB и ФРОЭС^(С)/СВ, соответственно.Similar to

and

it is appropriate to call the distributions of the quadrature components of the energy spectrum estimate for uncorrelated components (eigenvectors). We will abbreviate these distributions as FROES ^(S) / CB and FROES ^(C) / CB, respectively.

If discrete frequencies are used as the desired frequencies, such that sinusoidal and cosine vectors of dimension M form an orthonormal basis, then the sum

for all frequencies ƒ _k can be used to form an estimate of the normalized energy spectrum

In the case when the normalized energy spectrum is considered as the energy spectrum (32), we will use the abbreviated names FRONES / SV, FRONES ^(S) / SV and FRONES ^(C) / SV for the introduced distributions.

, где i - номер по порядку в списке, а величина g_i - номер собственного вектора.Let R eigenvectors whose numbers form the list g _i

, where i is the number in the order in the list, and g _i is the number of the eigenvector.

In this case, in the spectral estimate (22), the selected eigenvectors localized at a frequency ƒ _AND will correspond to

Component (33) represents the “useful” part of the estimate, which is formed by the energy spectrum analyzer. Thus, the ratio of the “useless” part (that is, the error) to the “useful” part at the output of the energy spectrum analyzer will be equal to

The larger the value of (34), the greater the error estimated the uncorrelated component using the energy spectrum analyzer. The value (34) will be called the pseudo-estimation coefficient in the energy spectrum analyzer. This coefficient depends on the processed time series and signals how appropriate it is to use ACCWiKS when processing the time series. Small values of the pseudo-estimation coefficient in the energy spectrum analyzer indicate that when processing the time series, you can use the energy spectrum analyzer, and large ones indicate that the energy spectrum analyzer will give incorrect estimates, and it is necessary to use ACC & A.

It follows from the foregoing that the pseudo-estimation coefficient in the energy spectrum analyzer and FRONES is expediently included in the number of spectral characteristics, which are evaluated in the unit for estimating and interpreting spectral characteristics and errors 6.

In FIG. Figure 8-10 shows the FRONES / CB for the specific frequencies sought. The upper graphs show FRONES / CB ordered by eigenvector numbers, on average FRONES / CB for the case when the eigenvectors are ordered by frequency, at which the maximum of the amplitude spectrum of the eigenvector is located. In the lower FRONES / CB graphs, ordered by frequency, are shown in more detail in the region of the maximum of the function. The specific values of FRONES / CB are shown by circles. Values located on the same vertical line correspond to eigenvectors having a maximum of amplitude spectra at the same discrete frequency.

Tables 3 and 4 show the values of (34) calculated for one of the long-term time series of the vertical component of the electric field at the lunar tidal frequencies and doubled revolution frequencies of binary stellar systems.

. Это означает, что действующее значение погрешности оценки по крайней мере вдвое меньше действующего значения выявляемой компоненты. Тогда из таблицы 3 следует, что анализатор энергетического спектра может быть использован в 1/3 случаев при выявлении спектрально локализованных компонент, связанных с лунными приливами и частотой вращения Земли (приливы Q1, Р1, М2, K2, O1, M1, OO1, K1 и СВЗ). Аналогично, при выявлении составляющих, спектрально локализованных на удвоенных частотах двойных звездных систем, анализатор энергетического спектра применим только в 1/5 случаев. В отличие от анализатора энергетического спектра, АССВиКС обеспечивает выявление во всех рассмотренных случаях.We assume that for the justified use of the energy spectrum analyzer in identifying uncorrelated components localized at previously known frequencies, it suffices to have the value

. This means that the effective value of the estimation error is at least half the effective value of the detected component. Then it follows from Table 3 that the energy spectrum analyzer can be used in 1/3 of the cases in detecting spectrally localized components associated with the lunar tides and the Earth's rotation frequency (tides Q1, P1, M2, K2, O1, M1, OO1, K1 and SVZ). Similarly, when identifying components spectrally localized at double frequencies of binary stellar systems, the energy spectrum analyzer is applicable only in 1/5 of cases. Unlike the energy spectrum analyzer, ASCViKS provides detection in all cases considered.

FIG. Figures 3 and 4 show the appearance of the uncorrelated components detected by AHRCC for those cases when the gains from the use of AHRCC are small. As can be seen from the graphs, in this case the eigenvectors are similar to harmonic. These graphs correspond to lines 12 and 4 from table 3, respectively.

In FIG. Figures 5 and 6 show a variant of uncorrelated components identified by ACC & A for the case presented in rows 13 and 37 of table 4, respectively. As can be seen from the graphs, in these cases the eigenvectors differ significantly from the harmonic ones.

From the aforesaid, it can be concluded that the use of ASCViKS can significantly increase the likelihood of detecting a spectrally localized component at the desired frequency and the more difficult the situation, the higher (in comparison with the energy spectrum analyzer) the ASCViKS gain. As follows from the foregoing, FRONES / SV and the pseudo-estimation coefficient allow forming in block 6 a general assessment of the applicability of the energy spectrum analyzer in a specific situation, which is a new technical result.

Let us compare the ACCiX with the quadrature circuit shown in FIG. 1 b). FIG. 7 illustrates the behavior of the quadrature scheme when processing the same time series at three frequencies corresponding to the lunar tide frequency K1 and the Earth’s daily rotation frequency (upper graph), the lunar tide frequency 2N2 (middle graph) and the frequency of the double star system UV Leo [11] (bottom graph). The ordinates of all three graphs show the magnitudes of the estimates of the amplitude spectrum (average and effective values of the estimate, as well as its standard deviation) obtained using the quadrature scheme. The abscissa is the analysis time in hours.

As a comparison of the graphs shows, for the case of lunar tide frequencies K1 and the Earth’s daily rotation frequency, the estimate converges to a non-zero value with an increase in the analysis interval, and for the other two cases it decreases monotonically. This means that the quadrature scheme reacts to the components at the frequency of the lunar tide 2N2 and the frequency of the binary star system UV Leo as noise and does not reveal those components that can be detected using ASCViKS [14].

Table 5 summarizes the generalized characteristics of the three analyzers when analyzing the three types of components of the vertical component of the Earth’s electric field at the Voeikovo observation station. As can be seen from the table, ASCViKS always allows the identification of spectrally localized uncorrelated components.

It can be stated that the use of ASCViKS gives a clear advantage compared to analyzers of amplitude and energy spectra.

Suppose there is reason to believe that all uncorrelated components localized at the desired frequencies do not arise randomly, but as a result of the action of a certain physical phenomenon, so it is necessary to decide on the presence or absence of such a phenomenon. We will show how to use the criterion “coherence index” calculated in block 4 to make such a decision in block 6 based on the results of identifying several eigenvectors spectrally localized at known frequencies.

Table 3

Comparison results of the ASCViKS and the energy spectrum analyzer when detecting spectrally localized components at the lunar tidal frequencies and the Earth's rotation frequency when processing a long-term time series of the vertical component of the electric field at the Voeikovo observation station. Values M = 1000, Δt = 1 hour. The median value of the coherence index is 64.

Table 4

суток. Вероятность ложной тревоги при обнаружении удвоенных частот обращения двойных звездных систем, представленных в данной таблице, определенная при использовании схемы Бернулли, составляет

.Comparison results of ASCViKS and an energy spectrum analyzer in detecting spectrally localized components at double revolution frequencies of binary stellar systems when processing a long-term time series of the vertical component of the electric field at the Voeikovo observation station. The value of M = 1000, the sampling time is 1 hour. The median value of the coherence index is 64. The double frequencies are calculated according to the data of [13] for binary stellar systems with revolution periods of more than

days. The probability of false alarm when detecting double frequencies of the binary star systems presented in this table, determined using the Bernoulli scheme, is

.

Table 5

Comparison of three types of analyzers in identifying and analyzing three types of components of the vertical component of the electric field in the surface layer of the atmosphere at the Voyeykovo observation station. + means that the component is detected, - means that the component is not detected.

As an example, we consider a group of frequencies equal to the double revolution frequency of binary star systems presented in the third column of Table 4. In the classical work [11], binary star systems are proposed to be considered as sources of gravitational waves. It is shown that the radiation of gravitational waves should occur mainly at the doubled revolution frequency of binary stellar systems.

We pose the question: are there uncorrelated components in the electric and magnetic fields of the Earth that are spectrally localized at double the revolution frequencies of binary stellar systems?

To answer this question, it is necessary to use some generally accepted scheme of statistical inference. The easiest way to use a widespread test scheme is Bernoulli [6].

The probability of k success in a series of n Bernoulli trials is determined by the formula [6]

where S _n is the number of successes in a series of n Bernoulli trials,

p is the probability of success,

q = 1-p is the probability of failure.

Let us evaluate the results presented in the sixth column of Table 4 from the point of view of the Bernoulli scheme.

We will consider it success if the selected eigenvector has a coherence index greater than the median of the coherence index determined from all eigenvectors. We consider failure to be the event opposite to success.

, формула (35) упрощается и принимает видIn this case

, formula (35) is simplified and takes the form

Relation (36) shows what is the probability of a random series of successes.

Таким образом, случайное превышение индексом когерентности медианного значения в 78 случаях из 89 весьма маловероятно.What is the probability of the random occurrence of those values of the coherence indices of the eigenvectors that are listed in the last column of table 4? This value

Thus, accidental excess of the median value by the coherence index in 78 cases out of 89 is highly unlikely.

Of course, the obtained very small probability value is not yet proof of the presence of gravitational waves, but the result indicates the presence of a physical effect, the same as that predicted for gravitational waves.

(Душети) и

(Верхнее Дуброво).The effect is also present on other long-term time series. Checks made at the double revolution frequencies of binary star systems shown in Table 4 for long-term time series of the vertical component of the electric field at the observation stations of Dusheti and Verkhnyaya Dubrovo led to probabilities

(Dusheti) and

(Upper Dubrovo).

Thus, with the help of ASCViKS, it is possible, with a vanishingly small probability of error, to make a decision that a spectral localization effect of uncorrelated components of the vertical component of the Earth’s electric field at doubled revolution frequencies of binary stellar systems was found, which does not contradict the hypothesis about the observation of gravitational waves.

Consequently, ASViKS - as a research device - is able to identify those effects that cannot be detected using analyzers of amplitude and energy spectra.

During the interpretation in block 6 visualize:

1. Selected eigenvectors (as shown in FIG. 3-6).

2. Spectral characteristics determined by the ratios of table 2 (according to the example of Figs. 11 and 12).

3. FRONES / NE (as an example, Fig. 8-10).

4. The behavior of the spectral estimation when using the analyzer of the amplitude spectrum (according to the example of Fig. 7).

5. Other graphs: spectra of eigenvalues and normalized spectra of eigenvalues.

6. Amplitude spectra of individual eigenvectors.

7. Distribution functions of the criteria “coherence index” and “maximum correlation coefficient”.

In FIG. 11 and 12 show some typical visualization options for eigenvectors, normalized spectra of eigenvalues, the criterion “coherence index”, as well as spectral characteristics that can be implemented in block 6.

To summarize:

1. The technical result from the use of ASCViKS is that ASCViKS allows the detection of energetically non-dominant uncorrelated spectrally localized components at given frequencies that are not detected by known spectral analysis tools - an amplitude spectrum analyzer and an energy spectrum analyzer.

2. The loss of the energy spectrum analyzer in relation to ACCS & A is due to the fact that when generating spectral estimates, the energy spectrum analyzer mixes various uncorrelated components in the generated estimate. Such mixing reduces the selectivity of the analyzer and leads to an unacceptable increase in the error of the spectral estimate; this error increases as the correlation properties of the analyzed signal become more complex.

3. ASCViKS allows making a reasonable statistical decision on the presence of a group of uncorrelated components spectrally localized at given frequencies. In the event that there is a physically based hypothesis about the occurrence of such a group of components, such a statistical solution should be considered as a valid confirmation of the physical hypothesis.

4. The technical result in ACCS & A is achieved due to the fact that the time series at the final analysis interval is represented as eigenvectors (uncorrelated components), which are processed and interpreted in block 6 in order to form spectral estimates and make decisions, as well as a reasonable estimate of the ACC & S gain in relation to analyzers of amplitude and energy spectra.

Thus, it is advisable to use the ASCViKS in cases where the known methods of spectral analysis do not give a result. At the same time, during the use of ASCViKS, an assessment of the applicability of the amplitude and energy spectrum analyzers in a specific situation is automatically generated.

ASCViKS, as a universal device expanding the capabilities of an aigenoscope [1], will certainly find its application in solving very urgent problems of exploration geophysics (some possibilities are presented in [15]), in solving problems of searching for precursors of strong earthquakes (preliminary data are given in [16]) and in many other applications.

Information sources

1. Isakevich V.V., Isakevich D.V., Grunskaya L.V. Eigenvector analyzer and signal components. Utility Model No. 116242 RU.

2. Bor G. Almost periodic functions: Trans. with him. / Under. ed. A.I. Plesner. Ed. 3rd - M .: LIBROCOM, 2009. - 128 p.

3. Levitan B.M., Zhikov V.V. Almost periodic functions and differential equations. M .: Publishing house of Moscow University, 1978. - 205 p.

4. Kharkevich A.A. Spectra and analysis. Ed. 5th. - M .: LIBROCOM, 2009. - 240 p.

5. Kharkevich A.A. Fighting interference. Ed. 3rd - M .: LIBROCOM, 2009. - 280 p.

6. Korn G., Korn T. Handbook of mathematics (for scientists and engineers). M .: Nauka, 831 p.

7. Sasskind A., Friedman A. Quantum mechanics. Theoretical minimum / per. from English A. Sergeev. - SPb. : Peter. 2015 .-- 400 p.

8. The main components of the time series: the "Caterpillar" method. / Under. ed. D.L. Danilova and A.A. Zhiglyavsky. Publishing house of St. Petersburg State University. 1997.

9. Faddeev D.K., Faddeeva V.N. Computational methods of linear algebra. M .: Publishing house "Lan". 2009 .-- 736 p.

10. Holevo A.S. Quantum systems, channels, information. ICMMO, 2010.

11. Zeldovich Ya.B., Novikov I.D. The theory of gravity and the evolution of stars. M .: Science. 1971. -486 p.

12. Cupronickel P. Earth tides. M .: World. 1968 .-- 482 p.

13. http://www.johnstonsarchive.net/relativity/binpulstable

14. Grunskaya L.V., Isakevich V.V., Isakevich D.V., Rubay D.V. Analysis of eigenvectors and the main components of the vertical component of the electric field in the surface layer of the atmosphere: Frequencies of lunar tides. M .: Publishing House "Pero", 2015. - 356 p.

15. Isakevich V.V., Isakevich D.V., Grunskaya L.V. About some possibilities of aigenoscopy / Instruments and systems for exploration geophysics. 2016, No. 2, p. 69-75.

16. Isakevich D.V. Fundamentals of the analysis of eigenvectors and components of regular oscillations. M .: Publishing house "Pero", 2014, ISBN 9-785000-862803? 138 p.

17. Firstov P.P., Isakevich V.V., Makarov E.O., Isakevich D.V., Grunskaya L.V. Application of aigenoscopy technique to search for precursors of strong earthquakes in the field of soil radon ( ²²² Rn) in Kamchatka (August 2012 - August 2013) / Seismic Instruments, 2014, v. 50, No. 3, p. 63-75.

Description of figures and drawings

FIG. 1. Prototypes and analogues:

but). The eigenvector analyzer and signal components according to patent No. 116242: 8 — the scaling unit, 9 — the mixed-matrix matrix calculator block, 3 — the eigenvector and eigenvalue calculator block, 10 — the scalar product calculator block and the feature analyzer.

b) The quadrature diagram of the amplitude spectrum analyzer: 11 - multiplier, 12 - block of reference signals, 13 - integrator, 14 - quadrator, 15 - adder.

at). Energy spectrum analyzer.

d). Equivalent algebraic diagram of the energy spectrum analyzer: 16 - a quadratic form calculator.

FIG. 2. The spectrum analyzer of eigenvectors and signal components according to the formula of the utility model: 1 - ensemble formation unit; 2 - block calculation of the covariance matrix;

3 - unit for calculating eigenvectors and eigenvalues; 4 - block selection of eigenvectors; 5 - a block of memory and program control; 6 - unit assessment and interpretation of spectral characteristics and errors.

FIG. 3. Eigenvectors detected using an eigenvector spectrum analyzer and signal components from a long-term time series of the vertical component of the Earth’s electric field at the Voyeykovo observation station at the lunar tide frequency K1 and the daily Earth rotation frequency. Eigenvectors are close to harmonic. Components corresponding to these eigenvectors are detected both using the energy spectrum analyzer (see row 12 of table 3) and using the quadrature scheme (see Fig. 7 a)).

FIG. 4. Eigenvectors detected using an eigenvector spectrum analyzer and signal components from a long-term time series of the vertical component of the Earth’s electric field at the Voeikovo observation station at the lunar tide frequency M2. Eigenvectors are close to harmonic. Components corresponding to these eigenvectors can also be detected using the energy spectrum analyzer (see row 4 of table 3).

FIG. 5. Eigenvectors detected using an eigenvector spectrum analyzer and signal components from a long-term time series of the vertical component of the Earth’s electric field at the Voyeykovo observation station at twice the rotation frequency of the binary star system J1012 + 5307. Eigenvectors are not similar to harmonic. The uncorrelated components corresponding to these eigenvectors are detected only using the spectrum analyzer of eigenvectors and signal components.

FIG. 6. The eigenvector detected using the spectrum analyzer of eigenvectors and signal components from a long-term time series of the vertical component of the Earth’s electric field at the Voeikovo observation station at twice the rotation frequency of the double star system J1909-3744. The eigenvector does not look like a harmonic one. The uncorrelated component corresponding to this eigenvector is detected only using the spectrum analyzer of eigenvectors and signal components.

FIG. 7. The dependence of the assessment of the amplitude spectrum of the vertical component of the electric field in the surface layer of the atmosphere on the duration of the analysis interval at the observation station Dusheti, carried out using the prototype:

but). At the frequency of the lunar tide K1 and the frequency of the daily rotation of the Earth.

b) At the frequency of the lunar tide 2N2.

at). At the frequency of the double star system UV Leo.

FIG. 8. FRONES at the lunar tide frequency K1 and the Earth's daily rotation frequency for a long-term time series of the vertical component of the electric field in the surface layer of the atmosphere at the Voyeykovo observation station (the final analysis interval is 1000 hours):

but). FRONES according to the numbers of eigenvectors.

b) The distribution function of the responses by the frequencies of the maxima of the amplitude spectra of the eigenvectors.

at). The distribution function of the responses over the frequencies of the maxima of the amplitude spectra of the eigenvectors in the vicinity of the maximum.

FIG. 9. FRONES at the lunar tide frequency 2N2 for a long-term time series of the vertical component of the electric field in the surface layer of the atmosphere at the Voyeykovo observation station (the final analysis interval is 1000 hours):

but). FRONES according to the numbers of eigenvectors.

b) The distribution function of the responses by the frequencies of the maxima of the amplitude spectra of the eigenvectors.

at). The distribution function of the responses over the frequencies of the maxima of the amplitude spectra of the eigenvectors in the vicinity of the maximum.

FIG. 10. FRONES at the frequency of the binary star system UV Leo for a long-term time series of the vertical component of the electric field in the surface layer of the atmosphere at the Voyeykovo observation station (the final analysis interval is 1000 hours):

but). FRONES according to the numbers of eigenvectors.

b) The distribution function of the responses by the frequencies of the maxima of the amplitude spectra of the eigenvectors.

at). The distribution function of the responses over the frequencies of the maxima of the amplitude spectra of the eigenvectors in the vicinity of the maximum.

.FIG. 11. Spectral characteristics (14) for a long-term time series of the vertical component of the electric field in the surface layer of the atmosphere at the Voyeykovo observation station. Criterion "minimum frequency detuning"

.

but). The spectral characteristic for the doubled frequencies of all binary star systems from Table 4. Vertical thin lines show doubled frequency of revolution (third column of table 4).

b) Spectral response for the doubled frequency of the J1911-5958A binary star system. The vertical dashed line shows the doubled reference frequency (third column, row 38 of table 4).

at). The spectral response for the doubled frequency of the binary star system J2140-2310 V. The vertical dotted line shows the doubled frequency of revolution (third column, row 41 of table 4).

FIG. 12. Illustration of the operation of the unit 6. In the upper left window there is an eigenvector No. 592, localized at the second harmonic of the revolution frequency of the binary star system J1756-2251 and having a normalized eigenvalue of -37.9 dB. In the upper right window is the amplitude spectrum of eigenvector No. 592 normalized to its maximum. The abscissa axis represents the frequency in Hz, the ordinate axis represents the normalized amplitude spectrum. The coherence index is 97.5 (shown above the window). The vertical dashed lines show the allowable boundaries of the position of the maximum amplitude spectrum. The vertical solid line shows the desired frequency (in this case, 7.242 10 ^-5 Hz). In the lower left window is the normalized spectrum of eigenvalues. A normalized eigenvalue corresponding to the selected eigenvector is noted (has a value of -37.9 dB). The abscissa axis is the eigenvalue number, the ordinate axis is the normalized eigenvalue in dB. In the lower right window are the coherence indices of all eigenvectors of the covariance matrix of the investigated time series. The value of the coherence index, the value of which is 97.5, is noted for the selected eigenvector No. 592. The abscissa axis is the number of the eigenvector, and the ordinate axis is the coherence index. The horizontal line is the median of the coherence index (equal to 63.9; the numerical value is given above the window). As can be seen from the graphs, the magnitude of the coherence index of the eigenvector No. 592 exceeds the median level.

The implementation of the invention

The spectrum analyzer of eigenvectors and signal components is implemented in the form of a hardware or software-hardware universal complex in which the blocks interact as described in table 6.

Claims (1)

An eigenvector spectrum analyzer and signal components comprising an eigenvector and eigenvalue calculation unit 3, characterized in that the device consists of an ensemble formation unit 1, a covariance matrix calculation unit 2, an eigenvector and eigenvalue calculation unit 3, an eigenvector selection unit 4 , a memory block and program control 5, a block for estimating and interpreting spectral characteristics and errors 6, wherein: the ensemble formation block 1, the covariance calculation unit 2, the unit for calculating the eigenvectors and eigenvalues 3, the unit for selecting the eigenvectors 4, the unit for estimating and interpreting spectral characteristics and errors 6 are connected to the memory and program control unit 5, the two-way output of which serves to obtain the initial processed information, for communication with external devices for the issuance of results and for new tasks.

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