# The method of spectral analysis of quasiperiodic signals

(57) Abstract:

The invention relates to the field of spectral analysis and can be used in the classification of quasi-periodic signals. The technical result is to increase the accuracy. The method is based on obtaining a two-dimensional matrix of the source signal, and performing spectral transformations on rows, columns, matrix spectra lines of the original signal, and then produces the visualization of the obtained spectral representation, then compare this image with a model image. 1 Il. The invention relates to the field of spectral analysis and can be used in the identification and classification of quasi-periodic signals in medicine, physics, astronomy, non-destructive testing of materials and products, processing and analysis of speech.The known method of spectrum analysis, namely, that of the analyzed signal using parallel analog filtering, based on band-pass filters or lo produce spectral components, corresponding to the analyzed frequency bands. By amplitude detection of these spectral components get set for the same. Then, switch this voltage is sequentially fed to the input of videoeditor (cathode ray tube), horizontal scanning which synchronizes the switching of the switch (see Max J., Methods and techniques of signal processing at the physical changes. In 2 volumes. TRANS. with Franz. - M.: Mir, 1983. So 2, S. 79-80).The disadvantage of this method of analysis is that this method determines the instantaneous spectrum, i.e. the spectrum on a finite interval, which necessitates the use of "window" and leads to distortion of the spectral pattern.Closest to the proposed method - the method of spectral analysis of quasiperiodic processes, namely, that of the analyzed signal using a digital filter emit spectral components, corresponding to the analyzed frequency bands, erect it in a square and integrate into discrete intervals [nt,(n+1)t], and the result is presented in the form of a matrix, the k-th column of which corresponds to the k-th frequency band, and n-th row, n-th time (see Max J., Methods and techniques of signal processing in the physical dimensions. In 2 volumes. TRANS. with Franz. - M.: Mir, 1983, So 2, S. 220-222). Note that this method penalize spectrum associated with the what a grayscale image pinbalance to the length of the "window" t, but also by the fact that the image is not reflected objectively the relationship between spectral components in "boxes".An object of the invention is to improve the performance and reliability analysis.The task is solved by the fact that the analyzed signal emit k related samples (quasiperiodic) of the analyzed signal, and each of the adjacent sampled signal is mapped to one quasiperiod the process under investigation.The essence of the proposed method of analysis is that the two-dimensional discrete spectral transform of a sequence of samples x(n_{1}n

_{2}), calculated as

< / BR>

for 0<k

<N

-1 and 0<k<N-1,

where W

presented as two transformations

< / BR>

< / BR>

where

< / BR>

here Fi (G(n1, k2 )) - approximating polynomials built on discrete samples of a function G (n1, k2,

< / BR>

where N

< / BR>

icon ! marked the join operation between two vectors into one by adding the first vector of the second to the right, [O

< / BR>

used approximately polynomial (for example, a cubic spline)

< / BR>

tab this function (approximating polynomial).The use of these approaches, the original signal is converted into a set of vectors of strings of equal length. This set of vectors of strings called the matrix of the original signal. The matrix of the source signal is the output of block 2 of Fig.1 (it is called "Segmentation on quasiperiodic, align). The rows of this matrix form of the vector-line

< / BR>

Defining the spectra of rows of the original matrix, obtain the spectral characteristics of quasiperiodic the process under investigation. It is also a matrix, which we will call the matrix of one-dimensional spectral characteristics. Practice has shown that a number of processes, especially in biology and medicine, it is very difficult to identify using spectral or correlation methods, as they are tied to their spatial temporal cycles. In this regard, the interest is not instantaneous spectra, and their dynamics of change over time. To identify these dynamics, the proposed method of spectral analysis is the second spectral transformation matrix columns of one-dimensional spectral characteristics. The columns of this matrix determines the energy components of the same frequency spectra of quasiperiodic, provided of course that the outcome of the m two-dimensional spectral plane of the investigated process.For analysis (visualization) results of two-dimensional spectral conversion, split raster videoeditor on the matrix elements of the raster corresponding to a two-dimensional matrix of spectral coefficients, and put the brightness and or color of each element of the raster in accordance with the magnitude of the corresponding spectral coefficient, and then compare the resulting image with the image obtained in the same manner for the reference signal.In the analysis of the functional state of the investigated system, the obtained image is compared with a model for a specific set of criteria S. On the comparison of S(S= {s

1) that each row of the matrix of samples of the analyzed signal corresponds to one of its quasiperiod,

2) is the alignment of the number of samples in quasiperiodic,

3) the analysis is performed on a specified number of quasiperiodic,

4) the spectral image plane reflects the dynamics of changes of spectral components of quasiperiodic in the individual time scale. The method of spectral analysis of quasiperiodic signals, which consists in the computation related To the instantaneous spectra of the original signal x(t) with the width of the "window" t, wherein when receiving adjacent samples each adjacent sample of the original signal set according to one of his quasiperiod, then perform alignment of counts in adjacent samples, resulting in a gain of a two-dimensional matrix of the original signal, then perform spectral transformation, in this case, first determine the spectra of each of the i-th (i= 0,1, ..., K-1) rows of a two-dimensional matrix of the source signal, and then the spectra of the dynamics of changes in the frequency components of the instantaneous spectra - spectra of columns of the matrix spectra lines of the original signal, and then produces the visualization of the obtained spectral PR is

where W

_{N1}W_{N2}- some orthogonal basis,presented as two transformations

< / BR>

< / BR>

where

< / BR>

here Fi (G(n1, k2 )) - approximating polynomials built on discrete samples of a function G (n1, k2,

_{i}).As samples x(n1, n2) select, as a rule, the samples obtained on the same quasiperiodic. And the value specifies the maximum number of samples in quasiperiodic moom analysis of quasiperiodic signals, to implement the transformation (1, 2, 3). The first stage of the transformation (2) - select "open" instantaneous observations (unit 2 - segmentation on quasiperiodic Fig.1). The window is selected without overlapping, and its length is adapted to the studied process. The essence of the adaptation length of the window under the analyzed signal is to allocate one or more quasiperiodic and use their length to determine the coordinates of the boundaries of the "window" instant of observation. In the segmentation vector-line reports of the original signal is split into a vector of strings where i is the number of the i-th vector of strings_{i}and_{i+1}coordinates of the i-th window, and where N_{1}- the total number of quasiperiodic ("Windows").Since the process is quasi-periodic, then the number of elements in the vectors are the rows is different, that is_{i+1}-_{i}const. In each of the vectors of strings, you can apply the transformation (2) and get the other set of vectors of strings. This operation is used in the known methods, but at the same time_{i+1}-_{i}= const. So here it is not possible to perform the transformation (3), as for its implementation, that is, to obtain a two-dimensional spectrum, it is necessary to build the vectors [x_{j}^{i}] so that j-I JV is the number of spectral components in the vectors [x_{j}^{i}] not identical, you need to bring them to a common scale, that is, to equalize the number of samples in quasiperiodic.Investigated several approaches to alignment based on obtaining a priori information about the process under study. Can be used a group in the frequency and time domains.Depending on what component of the quasi - rhythmic or energy is the most informative or closely associated with the classification signs developed two methods of alignment vectors of the lines.If the classification process requires information on frequency modulation (rhythmic quasi-periodicity), the alignment process can be represented as< / BR>

where N

_{2}the maximum length of the vector in the set< / BR>

icon ! marked the join operation between two vectors into one by adding the first vector of the second to the right, [O

_{j}] - zero vector is a string.If the classification process is based on the analysis of the energy characteristics of a quasi-periodic process, to obtain vectors< / BR>

used approximately polynomial (for example, a cubic spline)

< / BR>

tab this function (approximating polynomial).The use of these approaches, the original signal is converted into a set of vectors of strings of equal length. This set of vectors of strings called the matrix of the original signal. The matrix of the source signal is the output of block 2 of Fig.1 (it is called "Segmentation on quasiperiodic, align). The rows of this matrix form of the vector-line

< / BR>

Defining the spectra of rows of the original matrix, obtain the spectral characteristics of quasiperiodic the process under investigation. It is also a matrix, which we will call the matrix of one-dimensional spectral characteristics. Practice has shown that a number of processes, especially in biology and medicine, it is very difficult to identify using spectral or correlation methods, as they are tied to their spatial temporal cycles. In this regard, the interest is not instantaneous spectra, and their dynamics of change over time. To identify these dynamics, the proposed method of spectral analysis is the second spectral transformation matrix columns of one-dimensional spectral characteristics. The columns of this matrix determines the energy components of the same frequency spectra of quasiperiodic, provided of course that the outcome of the m two-dimensional spectral plane of the investigated process.For analysis (visualization) results of two-dimensional spectral conversion, split raster videoeditor on the matrix elements of the raster corresponding to a two-dimensional matrix of spectral coefficients, and put the brightness and or color of each element of the raster in accordance with the magnitude of the corresponding spectral coefficient, and then compare the resulting image with the image obtained in the same manner for the reference signal.In the analysis of the functional state of the investigated system, the obtained image is compared with a model for a specific set of criteria S. On the comparison of S(S= {s

_{1},s_{2},...,s_{n}}) it is possible to judge the presence of a deviation from the norm. To research the source and the reference spectral planes need to bring them to a common scale as the abscissa axis and the ordinate axis. The scale on the x-axis is determined by the sampling rate of the signal. The scale on the ordinate axis is determined by the number of quasiperiodic in the reference and test samples. As a set of criteria 8 can be used energy spectrum in the local area spectral plane defined in the learning process systemsim is:1) that each row of the matrix of samples of the analyzed signal corresponds to one of its quasiperiod,

2) is the alignment of the number of samples in quasiperiodic,

3) the analysis is performed on a specified number of quasiperiodic,

4) the spectral image plane reflects the dynamics of changes of spectral components of quasiperiodic in the individual time scale. The method of spectral analysis of quasiperiodic signals, which consists in the computation related To the instantaneous spectra of the original signal x(t) with the width of the "window" t, wherein when receiving adjacent samples each adjacent sample of the original signal set according to one of his quasiperiod, then perform alignment of counts in adjacent samples, resulting in a gain of a two-dimensional matrix of the original signal, then perform spectral transformation, in this case, first determine the spectra of each of the i-th (i= 0,1, ..., K-1) rows of a two-dimensional matrix of the source signal, and then the spectra of the dynamics of changes in the frequency components of the instantaneous spectra - spectra of columns of the matrix spectra lines of the original signal, and then produces the visualization of the obtained spectral PR is

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