CN107632963A - Length is the power system sampled signal Hilbert transform method based on FFT of composite number - Google Patents
Length is the power system sampled signal Hilbert transform method based on FFT of composite number Download PDFInfo
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- CN107632963A CN107632963A CN201710669170.2A CN201710669170A CN107632963A CN 107632963 A CN107632963 A CN 107632963A CN 201710669170 A CN201710669170 A CN 201710669170A CN 107632963 A CN107632963 A CN 107632963A
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Abstract
A kind of length is Hilbert transform implementation method of the power system sampled signal based on FFT of composite number, the sequence x (n) that length to collection is composite number N, using mixed base and prime factor (PFA) algorithm, the harmonic signal X (k) of each time of signal is tried to achieve；To X (k), further ranking operation asks IFFT computings, obtains the signal of 90 degree of phase shiftPrecision is reduced for the fence effect brought in fft algorithm, spectral leakage is suppressed using Hanning windows.The inventive method simple, intuitive is easy to Digital Implementation, has not only expanded the application of traditional FFT, also make it that Hilbert transform is more convenient, is laid the first stone for it in the processing of non-stationary signal.
Description
Technical Field
The invention belongs to the technical field of analysis and processing of sampled signals of an electric power system, and relates to a power system sampled signal Hilbert transform method with the length of a composite number and based on FFT (fast Fourier transform).
Background
Signals of the power system can be divided into normal periodic signals and non-stationary signals in a fault process, the non-stationary signals contain various non-periodic information, and rich contents of the non-stationary signals have important values for theoretical research and engineering application of the system. At present, the power grid is complex in operation environment and serious in electromagnetic interference, various noises can be mixed in a sampling signal, an error is generated, and effective signal extraction becomes more and more complex.
In the research and practice of the smart grid, the popularization of various sensors and measuring equipment and the online monitoring system of the equipment are all based on data acquisition. Therefore, the analysis and processing of signals in power systems are becoming more and more important, and methods related to time domain analysis, frequency domain analysis and time-frequency analysis of signals are being researched and developed.
The Hilbert transform, as a new time-frequency domain analysis method, has played a great role in the fields of low-frequency oscillation analysis, harmonic analysis, high-voltage transmission line protection, fault diagnosis and the like of a power system, but the method depends on empirical analysis and complex signal decomposition process. At present, the digital implementation of the method is directly realized based on definition, and the solution is most accurate, but with the increase of the complexity of the signal, the analytic expression of h (t) is difficult to directly give, and the subsequent derivation cannot be continued. The EMD decomposition-based implementation method has the properties of adaptivity, orthogonality, completeness and the like, but the method needs iterative computation, the selection of an iterative threshold is complex, an excessively small threshold increases the computation amount, and an excessively large threshold causes the final result to be inaccurate.
Disclosure of Invention
In order to overcome the defects of large operation amount, low operation speed and poor universality of the conventional power system sampling signal conversion method, the invention provides a power system sampling signal Hilbert conversion implementation method with the length of a composite number and based on FFT (fast Fourier transform).
The technical scheme adopted by the invention for solving the technical problems is as follows:
a length is electric power system Hilbert transform implementation method based on FFT of the complex number, including the following steps:
1) Multifactorial representation of a complex number N, N = r 1 r 2 …r L Then any n&Positive integers of N are all expressed as
2) Actually acquired voltage and current are finite-length signals, and windowing is carried out to obtain an original signal x (n);
3) Number of combinations N = r 1 r 2 Two numbers can have a common factor, corresponding mapping is carried out, and according to DFT definition, X (k) = X (r) 1 k 2 +k 1 ) Introducing twiddle factors to carry out mixed base FFT operation;
4) If r 1 And r 2 Mutiple, elimination of twiddle factors by selection of coefficientsUpper labelRepresenting powers, for further simplification;
5) Continue to r 1 And r 2 Decomposing until each layer is a DFT algorithm with 7, 5, 4, 3 and 2 decimal points proposed by a WFTA algorithm;
6) Weighting the transformed sequence X (k), solving inverse Fourier transform, wherein the inverse transform is consistent with the forward transform, and changing the sign of the twiddle factor;
7) After inverse transformationNamely a signal which is subjected to 90-degree phase shift, namely a signal after Hilbert transformation.
Further, in step 1), the expansion equation representing the numerical value is as follows:
(n) 10 ＝n L-1 (r 2 r 3 …r L )+n L-2 (r 3 r 4 …r L )+…+n 1 r L +n 0 (1)
in the formula n L-1 ＝0,1,…r 1 -1，n L-2 ＝0,1,…r 2 -1，…，n 0 ＝0,1,…r L -1。
Still further, in step 2), the expression of the window function used is as follows:
in step 3), the mapping relationship adopted is as follows:
in step 3), the operation process of DFT is as follows:
wherein X (k) is a converted signal,for twiddle factors, the superscript nk represents the power;
substituting (3) and (4) into (5) yields:
wherein, the first and the second end of the pipe are connected with each other,
consider X (k) as a two-dimensional array X (k) 2 ,k 1 ) Let x (n) be regarded as a two-dimensional array x (n) 1 ,n 2 ) DFT is performed for each row and column.
In step 4), if the two numbers are relatively prime, the simplification is as follows:
PFA can provide a corresponding reduction in computation and memory compared to mixed-base algorithms.
In step 7), the obtained hilbert signal is defined as follows:
as can be seen in the equation, the hilbert transform of the signal can be seen as the output of the original signal through a filter.
The invention has the following beneficial effects: 1) The method related by the invention has relatively small calculation amount and high calculation speed;
2) On the basis of certain performance of the processor, the algorithm is improved, the operation of the long sequence is changed into the realization of the cascade small point number by adopting a plurality of FFT realization methods, and the principle is simple. The method for realizing Hilbert by positive and negative FFT operation has more universality than the mode of application definition.
Drawings
FIG. 1 is a flow chart of the present invention.
Fig. 2 is a 200-point algorithm scheme decomposition.
FIG. 3 calculates an 8-point DFT for 2-point and 4-point DFTs.
Fig. 4 shows a 200-point FFT operation.
Fig. 5 is a 4-point and 2-point FFT plot.
Fig. 6 is a diagram of FFT analysis.
Fig. 7 is a Hilbert operation analysis chart.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 7, a long-sequence FFT-based power system sampling signal hilbert transform implementation method includes the following steps:
1) And (4) performing decomposition operation on the sequence length N, and finally splitting the sequence length N into decimal points such as 7, 5, 4, 3 and 2. Multifactor representation of complex number N, N = r 1 r 2 …r L Then any n&Positive integers of N are all expressed asThe values represented are expanded as follows:
(n) 10 ＝n L-1 (r 2 r 3 …r L )+n L-2 (r 3 r 4 …r L )+…+n 1 r L +n 0 (1)
in the formula n L-1 ＝0,1,…r 1 -1，n L-2 ＝0,1,…r 2 -1，…，n 0 ＝0,1,…r L -1。
2) The actually collected voltage and current are signals with finite length, and the original signal x (n) is obtained by windowing to eliminate the problem of precision reduction caused by the fence effect, and the window function is as follows:
obtaining new sequence x after original signal x (n) is windowed 1 (n)：
x 1 (n)＝x(n)*ω(n)
3) For finite long sequence x with length N 1 (N), defining an N-point discrete fourier transform thereof:
3.1 Composite number N = r) 1 r 2 The two numbers may have a common factor, mapped as follows:
by DFT definition, X (k) = X (r) 1 k 2 +k 1 ) And introducing a twiddle factor to perform mixed base FFT operation:
wherein X (k) is a converted signal,for twiddle factors, the superscript nk represents the power;
substituting (3) and (4) into (5) yields:
wherein the content of the first and second substances,
consider X (k) as a two-dimensional array X (k) 2 ,k 1 ) Let x (n) be regarded as a two-dimensional array x (n) 1 ,n 2 ) DFT is performed for each row and column.
3.2 If r) 1 And r 2 Mutualin, elimination of twiddle factors by selection of coefficientsFurther simplification is achieved:
compared with the mixed base algorithm, the PFA can reduce the computation amount and the storage space correspondingly. If r is 1 And r 2 If not, continuing the decomposition operation, and selecting the corresponding optimal FFT algorithm according to whether the decomposition result of each layer is relatively prime or not.
4) And (3) carrying out weighted operation on the transformed sequence X (k), solving the inverse Fourier transform, and changing the sign of a twiddle factor:
comparing the DFT and IDFT definitions it can be seen that the twiddle factor is defined byBecome intoTherefore, the sign can be changed when the number is implemented, and the algorithm is consistent with the algorithm.
5) After inverse transformationNamely a signal which is subjected to 90-degree phase shift, namely a signal after Hilbert transformation:
it can be seen that the hilbert transform of the signal can be seen as the output of the original signal through a filter.
The equation shows that the sum of squares of errors between the obtained data and the actual data is minimum.
6) When the fan runs, collecting stator side phase current and rotating speed information, and solving the characteristic amplitude of the impeller at the frequency doubling position of the rotating speed 1 according to the methods in the steps 2) and 3).
The method for implementing the Hilbert transform through the FFT is described above.
The method related to the implementation has relatively small computation amount and high computation speed.
On the basis of certain processor performance, the algorithm is improved, the operation of the long sequence is changed into the realization of the cascaded small point number by adopting a plurality of FFT realization methods, and the principle is simple. The method for realizing Hilbert by positive and negative FFT operation has more universality than the method using definition.
The sample rate of 200 points per cycle is taken as a research object, and corresponding FFT and Hilbert operations are carried out on the sample rate. 200 can be decomposed intoCombining with 7, 5, 4, 3 and 2 decimal point DFT algorithms proposed by the WFTA algorithm, according to the advantages of calculation amount and storage space of the PFA algorithm, if two numbers are relatively prime between the same layer, the PFA algorithm is preferentially selected, fig. 2 is an algorithm exploded view, and fig. 4 is a 200-point FFT algorithm flow. Firstly, 200 points are decomposed into the components by a prime factor algorithmDecomposing the points into 25 points and 8 points by a mixed base algorithm respectivelyDot sumAnd points, wherein 2, 4 and 5 point FFT are calculated by using WFTA algorithm and traditional base-2 algorithm. Wherein, the 8-point FFT unit is shown in fig. 3, the 4-point and 2-point WFTA units perform DFT operation with 4-way and 2-way inputs, respectively, as shown in fig. 3, and in consideration of coordination problems of different bases, 3 2-point DFT operation units may be used in specific implementation. The method can reduce the storage time of the twiddle factors and the calculation times of multiplication and addition.
It should be noted that, the sign of the twiddle factor may be changed directly in the inverse fourier transform mentioned herein, and in the process of digital implementation, a flag bit may be set, and if the inverse transform is performed to set 1, the corresponding processing function and the like are all kept unchanged.
Now, according to an example to verify the effectiveness of the algorithm, the periodic signal y (t) = sin (100 π t) +0.5sin (80 π t) +0.5sin (120 π t), 200 points per cycle are sampled, with a sampling frequency of 4kHz. The FFT analysis is performed as described above and the time domain and frequency spectrum of the signal are shown in fig. 6. As can be seen from the spectrogram, the frequency quantities (40 Hz, 50Hz and 60 Hz) correspond to the time domain.
For signal y 1 (t) = sin (100 π t) by Hilbert transform, which means that after Hilbert transform, the signal has a constant amplitude, 90 degree phase shift for negative frequency, and-90 degree phase shift for positive frequency, and fig. 7 shows that the original signal and the transformed signal have 90 degree phase difference, corresponding to the theoretical result.
Compared with a wavelet analysis method, the Hilbert-Huang transformation method has more ideal and more effective noise reduction effect and is widely applied to the analysis of abnormal sound signals of the bearing bush of the engine; the method is also applied to analysis and extraction of harmonic waves in a power system, and the method is convenient for engineering realization.
The method for realizing the Hilbert calculation through twice positive and negative Fourier transformation avoids the difficulty in realizing the Hilbert definition, effectively reduces the calculation amount by flexibly using various Fourier algorithms, is convenient for realizing digital signal processing, is simple and effective in the whole method, has low software and hardware cost, and is an effective and reliable Hilbert calculation method.
Finally, it should be noted that the above-mentioned embodiments illustrate rather than limit the invention, and that those skilled in the art will be able to make variations in form and detail without departing from the scope of the invention as defined by the appended claims.
It should also be noted that the examples disclosed herein describe various parameters for the purpose of better describing the invention, and the skilled person will appreciate that the same diagnostic results can be achieved by modifying the parameter values of the invention, but such implementation should not be considered as beyond the scope of the present invention.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. The general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
Claims (6)
1. A length is the electric power system Hilbert transform implementation method based on FFT of the complex number, characterized by, including the following step:
1) Multifactor representation of complex number N, N = r 1 r 2 …r L Then any n&N is expressed as
2) Actually acquired voltage and current are finite-length signals, and windowing is carried out to obtain an original signal x (n);
3) Number of combinations N = r 1 r 2 Two numbers can have a common factor, corresponding mapping is carried out, and according to DFT definition, X (k) = X (r) 1 k 2 +k 1 ) Introducing twiddle factors to carry out mixed base FFT operation;
4) If r 1 And r 2 Mutiple, elimination of twiddle factors by selection of coefficientsUpper label n2k1 Expressing power, for further simplification;
5) Continue to r 1 And r 2 Decomposing until each layer is a DFT algorithm with 7, 5, 4, 3 and 2 decimal points proposed by a WFTA algorithm;
6) Weighting the transformed sequence X (k), solving inverse Fourier transform, wherein the inverse transform is consistent with the forward transform, and changing the sign of the twiddle factor;
7) After inverse transformationNamely a signal subjected to 90-degree phase shift, namely a Hilbert transformed signal.
2. The method for implementing FFT-based hilbert transform of sampled signals of an electric power system with complex length as claimed in claim 1, wherein in the step 1), the expansion equation representing the numerical value is as follows:
(n) 10 ＝n L-1 (r 2 r 3 …r L )+n L-2 (r 3 r 4 …r L )+…+n 1 r L +n 0 (1)
in the formula n L-1 ＝0,1,…r 1 -1，n L-2 ＝0,1,…r 2 -1，…，n 0 ＝0,1,…r L -1。
3. The FFT-based hilbert transform realization method for a sampling signal of an electric power system with a complex length as set forth in claim 2, wherein in the step 2), the window function expression is as follows:
4. the FFT-based hilbert transform implementation method for sampled signals of an electric power system with complex lengths according to claim 3, wherein in step 3), the mapping relationship is as follows:
in step 3), the operation process of DFT is as follows:
wherein X (k) is a converted signal,for twiddle factors, the superscript nk represents the power;
substituting (3) and (4) into (5) yields:
wherein the content of the first and second substances,
consider X (k) as a two-dimensional array X (k) 2 ,k 1 ) Let x (n) be regarded as a two-dimensional array x (n) 1 ,n 2 ) The DFT is performed for each row and column.
5. The method as claimed in claim 4, wherein in step 4), if the two numbers are relatively prime, the method is simplified as follows:
6. the method as claimed in claim 5, wherein in step 7), the obtained hilbert signal is defined as follows:
the hilbert transform of a signal can be viewed as the output of the original signal through a filter.
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