CN104897961B - Three spectral line interpolation FFT harmonic analysis methods and system based on cross multiplication window function - Google Patents
Three spectral line interpolation FFT harmonic analysis methods and system based on cross multiplication window function Download PDFInfo
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Abstract
The invention discloses a kind of three spectral line interpolation FFT harmonic analysis methods and system based on cross multiplication window function, it is related to frequency analysis field.This method comprises the following steps：Construct cross multiplication window function；Signal Pretreatment；Determine three spectral lines；Calculate the correction formula of three spectral line interpolation algorithms；Calculate fundamental wave parameter；Determine harmonic parameters；Carry out error analysis.The present invention proposes a kind of make of new window function, and different multiplicative combinations is carried out using conventional window function, realizes construction cross multiplication window function first, cross multiplication window function is applied in three spectral line interpolation FFT algorithms, carry out frequency analysis, calculate harmonic parameters.A variety of conventional cosine window function calculated examples show that the cross multiplication window function that the present invention constructs has the higher degree of accuracy, can realize the high-acruracy survey of harmonic wave relative to custom window function interpolation algorithm.
Description
Technical field
The present invention relates to frequency analysis field, is specifically related to a kind of three spectral line interpolation FFTs based on cross multiplication window function
Harmonic analysis method and system.
Background technology
With the use of a large amount of non-linear power electronic devices so that harmonic pollution problems getting worse in power network.Harmonic wave
Problem not only deteriorates the quality of power supply, and considerable influence is also resulted in the safety and stability and economical operation of power network.Therefore, to humorous in system
Wave parameter carries out high-acruracy survey, and for reducing Harmfulness Caused by Harmonics, it is very necessary to safeguard electricity net safety stable, Effec-tive Function.
The key issue of harmonic detecting is：Under non-synchronous sampling, how to solve spectral leakage and fence effect.At present, often
The detection harmonic wave and the method for m-Acetyl chlorophosphonazo seen mainly have：Fourier transformation, machine learning method, wavelet transformation, power Spectral Estimation, wish
That Bert-Huang.
In Fourier transformation analyzes harmonic wave, method therefor is：With various special window functions, signal is blocked,
Frequency analysis is carried out then in conjunction with spectral line interpolation FFT (Fast Fourier Transform, Fast Fourier Transform (FFT)).
Conventional window function has：The Chinese peaceful (Hanning) window function, Blacknam (Blackman) window function, Blacknam
The Chinese this (Blackman-Harris) window function, Nuttall (Nuttall) window function, Lay husband Vincent (Rife-Vincent) window
Function and various combination window functions.
On the basis of adding window, D.Agrez and Pang Hao et al. each propose the correction algorithm of bispectrum line, Wu Jing, Niu Sheng
Lock and Huang Dongmei et al. propose three spectral line correction algorithms.These improvement reduce the influence of spectrum leakage and fence effect, carry
The high accuracy of frequency analysis.
However, in engineering actual use, conventional window function interpolation algorithm still can not meet high-precision harmonic wave point
Analysis requires.
The content of the invention
The invention aims to overcome the shortcomings of above-mentioned background technology, there is provided it is a kind of based on cross multiplication window function three
Spectral line interpolation FFT harmonic analysis method and system, with custom window function ratio, there is higher precision.
The present invention provides a kind of three spectral line interpolation FFT harmonic analysis methods based on cross multiplication window function, including following step
Suddenly：
S1, construction cross multiplication window function：
The general formula of multiplication window function is multiplication window function w (n) the general public affairs as caused by multiple window function products
Formula is：
Wherein, n is the ordinal number of sampled point, and n is natural number；M is for the basic window function for participating in construction multiplication window function
Number, m is positive integer；wi(n) it is i-th of basic window function expression formula, i=1,2...m；ciFor i-th of basic window function of participation
Number, referred to as wi(n) sub- order；C is total order of cross multiplication window function；According to c1、c2...cmDifference
Value, form different combination (c1c2...cm), that is, construct various forms of multiplication window functions；
Work as wi(n) when expression formula is identical, i.e., used by basic window function it is identical when, w (n) is multiplication window function, works as wi
(n) during expression formula difference, i.e., used by basic window function difference when, w (n) is cross multiplication window function；
The expression formula of three conventional window functions is respectively w1(n), w2(n), w3(n), total order c=c1+c2+c3, to c mono-
Individual maximum, order：C=3；
According to c1、c2、c3Different values, form different combination (c1c2c3), construct following 9 kinds of multiplication window functions：
Work as c1=3, c2=0, c3When=0, construct and mutually multiply 300 window functions；
Work as c1=2, c2=1, c3When=0, construct and mutually multiply 210 window functions；
Work as c1=2, c2=0, c3When=1, construct and mutually multiply 201 window functions；
Work as c1=1, c2=2, c3When=0, construct and mutually multiply 120 window functions；
Work as c1=1, c2=0, c3When=2, construct and mutually multiply 102 window functions；
Work as c1=0, c2=3, c3When=0, construct and mutually multiply 030 window function；
Work as c1=1, c2=1, c3When=1, construct and mutually multiply 111 window functions；
Work as c1=0, c2=2, c3When=1, construct and mutually multiply 021 window function；
Work as c1=0, c2=0, c3When=3, construct and mutually multiply 003 window function；
Wherein, mutually multiply 300 window functions, mutually multiply 030 window function, mutually multiply 003 window function, these three window functions belong to multiplication
Window function, remaining the 6 kinds cross multiplication window functions to construct；
S2, Signal Pretreatment：
Transformer gathers power network signal, the power network signal x (n) that transformer is collected, is transferred to host computer；Power network is believed
Number x (n) is carried out plus cross multiplication window function w (n) is blocked, and obtains windowing signal xw(n)：
xw(n)=x (n) w (n) (2)
After the windowing signal progress FFT of formula (2), windowing FFT frequency spectrum is obtained：
Wherein, W () is the frequency spectrum of window function, and k is positive integer, and X (k) represents the frequency spectrum of kth subharmonic, AkFor kth time
The amplitude of harmonic wave, j represent imaginary unit, and e is the truth of a matter of natural logrithm,It is humorous for the first time for the initial phase of kth subharmonic
Ripple is fundamental wave, fsFor sample frequency, f0For fundamental frequency, Δ f is discrete frequency intervals, and Δ f=fs/N；
Order：k0=f0/ Δ f, k0For the position of spectral line of real frequency spectrum,
Ignore the influence of secondary lobe at negative frequency point, formula (3) is changed into：
S3, determine three spectral lines：
In the windowing FFT spectrum peak near zone that S2 is obtained, k0Locating three larger spectral lines of Frequency point is respectively：Kth1、
k2、k3Root, k1、k2、k3It is positive integer, k1~k3Relation be：k2=k1+ 1, k3=k2+ 1, amplitude corresponding to this three spectral lines
Respectively y1、y2、y3；
Remember variable α=k-k2, then -0.5≤α≤0.5；
Another note variable
S4, the correction formula for calculating three spectral line interpolation algorithms：
Obtained according to formula (4) and (5)：
Odd function α=g is calculated using Polynomial Approximation Method-1(β), expression formula are：
α≈p11×β+p13×β3+...+p1pβp (7)
p11, p13；…p1pFor the odd term coefficient of approximation by polynomi-als, p is odd number；
According to formula (4), the amplitude A of power network signal ith harmonic wave is tried to achievei：
Wherein, i is positive integer, yiFor the amplitude of ith harmonic wave after windowing FFT；
In view of y2It is from the nearest spectral line of true spectral line point, obtains：
During N ＞ 1000, window function coefficient is real coefficient, and formula (9) is expressed as：
A1=N-1(y3+2y2+y1)u(α)
Wherein, u (α) is correction formula, and is even function, and approximating polynomial is free of odd item；
Three spectral line amendment approximating polynomials are as follows：
U (α)=(p20+p22α2+…+p2dαd) (10)
In formula (10), p20,p22…p2dFor the even term coefficient of approximation by polynomi-als, d is the highest order of fitting, and d is
Even number；
S5, calculate fundamental wave parameter：
Calculate fundamental frequency f0, fundamental voltage amplitude A1：
f0=k Δ f=(k2+α)Δf (11)
A1=N-1(y3+2y2+y1)(p20+p22α2+…+p2dαd) (12)
According to formula (4), the phase of fundamental wave is calculated：
Asking for for fundamental wave parameter is copied, each harmonic ginseng is carried out according to formula (6), (7), (9), (11), (12), (13)
Several analyses；
S6, determine harmonic parameters：Determine fundamental frequency f0Afterwards, in scope (kf0-5,kf0+ 5) in, repeat step S3~
S5, finished until all harmonic parameters calculate；
S7, carry out error analysis：Analyze the three spectral line interpolation FFT algorithms based on cross multiplication window function error, and with it is normal
Rule window function interpolation algorithm precision is compared.
On the basis of above-mentioned technical proposal, the power network signal includes current signal, voltage signal.
On the basis of above-mentioned technical proposal, the position of spectral line k of the real frequency spectrum0For decimal.
The present invention also provides a kind of three spectral line interpolation FFT frequency analysis systems based on cross multiplication window function, including mutually multiplies
Method window function structural unit, Signal Pretreatment unit, spectral line determining unit, correction formula computing unit, fundamental wave parameter calculate single
Member, harmonic parameters determining unit, error analysis unit, wherein：
The cross multiplication window function structural unit, for constructing cross multiplication window function：
The general formula of multiplication window function is multiplication window function w (n) the general public affairs as caused by multiple window function products
Formula is：
Wherein, n is the ordinal number of sampled point, and n is natural number；M is for the basic window function for participating in construction multiplication window function
Number, m is positive integer；wi(n) it is i-th of basic window function expression formula, i=1,2...m；ciFor i-th of basic window function of participation
Number, referred to as wi(n) sub- order；C is total order of cross multiplication window function；According to c1、c2...cmDifference
Value, form different combination (c1c2...cm), that is, construct various forms of multiplication window functions；
Work as wi(n) when expression formula is identical, i.e., used by basic window function it is identical when, w (n) is multiplication window function, works as wi
(n) during expression formula difference, i.e., used by basic window function difference when, w (n) is cross multiplication window function；
The expression formula of three conventional window functions is respectively w1(n), w2(n), w3(n), total order c=c1+c2+c3, to c mono-
Individual maximum, order：C=3；
According to c1、c2、c3Different values, form different combination (c1c2c3), construct following 9 kinds of multiplication window functions：
Work as c1=3, c2=0, c3When=0, construct and mutually multiply 300 window functions；
Work as c1=2, c2=1, c3When=0, construct and mutually multiply 210 window functions；
Work as c1=2, c2=0, c3When=1, construct and mutually multiply 201 window functions；
Work as c1=1, c2=2, c3When=0, construct and mutually multiply 120 window functions；
Work as c1=1, c2=0, c3When=2, construct and mutually multiply 102 window functions；
Work as c1=0, c2=3, c3When=0, construct and mutually multiply 030 window function；
Work as c1=1, c2=1, c3When=1, construct and mutually multiply 111 window functions；
Work as c1=0, c2=2, c3When=1, construct and mutually multiply 021 window function；
Work as c1=0, c2=0, c3When=3, construct and mutually multiply 003 window function；
Wherein, mutually multiply 300 window functions, mutually multiply 030 window function, mutually multiply 003 window function, these three window functions belong to multiplication
Window function, remaining the 6 kinds cross multiplication window functions to construct；
The Signal Pretreatment unit, for being pre-processed to signal：
Transformer gathers power network signal, the power network signal x (n) that transformer is collected, is transferred to host computer；Power network is believed
Number x (n) is carried out plus cross multiplication window function w (n) is blocked, and obtains windowing signal xw(n)：
xw(n)=x (n) w (n) (2)
After the windowing signal progress FFT of formula (2), windowing FFT frequency spectrum is obtained：
Wherein, W () is the frequency spectrum of window function, and k is positive integer, and X (k) represents the frequency spectrum of kth subharmonic, AkFor kth time
The amplitude of harmonic wave, j represent imaginary unit, and e is the truth of a matter of natural logrithm,It is humorous for the first time for the initial phase of kth subharmonic
Ripple is fundamental wave, fsFor sample frequency, f0For fundamental frequency, Δ f is discrete frequency intervals, and Δ f=fs/N；
Order：k0=f0/ Δ f, k0For the position of spectral line of real frequency spectrum,
Ignore the influence of secondary lobe at negative frequency point, formula (3) is changed into：
The spectral line determining unit, for determining three spectral lines：
In the windowing FFT spectrum peak near zone that the Signal Pretreatment unit obtains, k0Locate Frequency point it is larger three
Root spectral line is respectively：Kth1、k2、k3Root, k1、k2、k3It is positive integer, k1~k3Relation be：k2=k1+ 1, k3=k2+ 1, this
Amplitude corresponding to three spectral lines is respectively y1、y2、y3；
Remember variable α=k-k2, then -0.5≤α≤0.5；
Another note variable
The correction formula computing unit, for calculating the correction formula of three spectral line interpolation algorithms：
Obtained according to formula (4) and (5)：
Odd function α=g is calculated using Polynomial Approximation Method-1(β), expression formula are：
α≈p11×β+p13×β3+…+p1pβp (7)
p11, p13；…p1pFor the odd term coefficient of approximation by polynomi-als, p is odd number；
According to formula (4), the amplitude A of power network signal ith harmonic wave is tried to achievei：
Wherein, i is positive integer, yiFor the amplitude of ith harmonic wave after windowing FFT；
In view of y2It is from the nearest spectral line of true spectral line point, obtains：
During N ＞ 1000, window function coefficient is real coefficient, and formula (9) is expressed as：
A1=N-1(y3+2y2+y1)u(α)
Wherein, u (α) is correction formula, and is even function, and approximating polynomial is free of odd item；
Three spectral line amendment approximating polynomials are as follows：
U (α)=(p20+p22α2+…+p2dαd) (10)
In formula (10), p20,p22…p2dFor the even term coefficient of approximation by polynomi-als, d is the highest order of fitting, and d is
Even number；
The fundamental wave parameter calculation unit, for calculating fundamental wave parameter：
Calculate fundamental frequency f0, fundamental voltage amplitude A1：
f0=k Δ f=(k2+α)Δf (11)
A1=N-1(y3+2y2+y1)(p20+p22α2+…+p2dαd) (12)
According to formula (4), the phase of fundamental wave is calculated：
Asking for for fundamental wave parameter is copied, each harmonic ginseng is carried out according to formula (6), (7), (9), (11), (12), (13)
Several analyses；
The harmonic parameters determining unit, for determining harmonic parameters：Determine fundamental frequency f0Afterwards, in scope (kf0-5,
kf0+ 5) in, the spectral line determining unit, correction formula computing unit, fundamental wave parameter calculation unit repeat to calculate, until
All harmonic parameters are calculated and finished；
The error analysis unit, for carrying out error analysis：Analyze three spectral line interpolation FFTs based on cross multiplication window function
The error of algorithm, and compared with custom window function interpolation arithmetic accuracy.
On the basis of above-mentioned technical proposal, the power network signal includes current signal, voltage signal.
On the basis of above-mentioned technical proposal, the position of spectral line k of the real frequency spectrum0For decimal.
Compared with prior art, advantages of the present invention is as follows：
The present invention proposes a kind of make of new window function, and different multiplicative combinations is carried out using conventional window function,
Construction cross multiplication window function is realized first, cross multiplication window function is applied in three spectral line interpolation FFT algorithms, carries out harmonic wave point
Analysis, calculate harmonic parameters.A variety of conventional cosine window function calculated examples show, the cross multiplication window function phase that the present invention constructs
For custom window function interpolation algorithm, there is the higher degree of accuracy, realize the high-acruracy survey of harmonic wave, there is higher practical valency
Value.
Brief description of the drawings
Fig. 1 is the flow of the three spectral line interpolation FFT harmonic analysis methods based on cross multiplication window function in the embodiment of the present invention
Figure.
Fig. 2 is the oscillogram of harmonic signal.
Fig. 3 is the amplitude versus frequency characte figure for mutually multiplying 210 window functions.
Fig. 4 is the amplitude versus frequency characte figure for mutually multiplying 201 window functions.
Fig. 5 is the amplitude versus frequency characte figure for mutually multiplying 120 window functions.
Fig. 6 is the amplitude versus frequency characte figure for mutually multiplying 102 window functions.
Fig. 7 is the amplitude versus frequency characte figure for mutually multiplying 111 window functions.
Fig. 8 is the amplitude versus frequency characte figure for mutually multiplying 021 window function.
Embodiment
Below in conjunction with the accompanying drawings and specific embodiment the present invention is described in further detail.
Shown in Figure 1, the embodiment of the present invention provides a kind of three spectral line interpolation FFT harmonic waves point based on cross multiplication window function
Analysis method, comprises the following steps：
S1, construction cross multiplication window function：
The general formula of multiplication window function is multiplication window function w (n) the general public affairs as caused by multiple window function products
Formula is：
Wherein, n is the ordinal number of sampled point, and n is natural number；M is for the basic window function for participating in construction multiplication window function
Number, m is positive integer；wi(n) it is i-th of basic window function expression formula, i=1,2...m；ciFor i-th of basic window function of participation
Number, referred to as wi(n) sub- order；C is total order of cross multiplication window function.According to c1、c2...cmNo
Same value, form different combination (c1c2...cm), that is, construct various forms of multiplication window functions.
Work as wi(n) when expression formula is identical, i.e., used by basic window function it is identical when, w (n) is multiplication window function, works as wi
(n) during expression formula difference, i.e., used by basic window function difference when, w (n) is cross multiplication window function.
Below by taking three conventional window functions as an example, expression formula is respectively w1(n), w2(n), w3(n).Total order c=c1+c2+
c3.Because c is bigger, it is meant that the basic window function of participation multiplication window is more, and the multiplication window function item number of composition is more, can increase
The complexity of calculating.Mono- maximum of c typically is given, for example, order：C=3.
According to c1、c2、c3Different values, form different combination (c1c2c3), construct following 9 kinds of multiplication window functions：
Work as c1=3, c2=0, c3When=0, construct and mutually multiply 300 window functions；
Work as c1=2, c2=1, c3When=0, construct and mutually multiply 210 window functions；
Work as c1=2, c2=0, c3When=1, construct and mutually multiply 201 window functions；
Work as c1=1, c2=2, c3When=0, construct and mutually multiply 120 window functions；
Work as c1=1, c2=0, c3When=2, construct and mutually multiply 102 window functions；
Work as c1=0, c2=3, c3When=0, construct and mutually multiply 030 window function；
Work as c1=1, c2=1, c3When=1, construct and mutually multiply 111 window functions；
Work as c1=0, c2=2, c3When=1, construct and mutually multiply 021 window function；
Work as c1=0, c2=0, c3When=3, construct and mutually multiply 003 window function；
It can be seen that from above-mentioned 9 kinds of multiplication window functions：Mutually multiply 300 window functions, mutually multiply 030 window function, mutually multiply 003 window letter
Number, these three window functions actually belong to multiplication window function (present invention not discusses), remaining the 6 kinds cross multiplications to construct
Window function.
S2, Signal Pretreatment：
Transformer gathers power network signal, and power network signal includes current signal, voltage signal, the power network that transformer is collected
Signal x (n), it is transferred to host computer；Power network signal x (n) is carried out plus cross multiplication window function w (n) is blocked, obtains adding window letter
Number xw(n)：
xw(n)=x (n) w (n) (2)
After the windowing signal progress FFT of formula (2), windowing FFT frequency spectrum is obtained：
Wherein, W () is the frequency spectrum of window function, and k is positive integer, and X (k) represents the frequency spectrum of kth subharmonic, AkFor kth time
The amplitude of harmonic wave, j represent imaginary unit, and e is the truth of a matter of natural logrithm,For the initial phase of kth subharmonic, the first subharmonic
For fundamental wave, fsFor sample frequency, f0For fundamental frequency, Δ f is discrete frequency intervals, and Δ f=fs/N。
Order：k0=f0/ Δ f, k0For the position of spectral line of real frequency spectrum, because non-synchronous sampling or non-integer-period block, k0One
As be decimal, be not integer.
If ignoring the influence of secondary lobe at negative frequency point, formula (3) is changed into：
S3, determine three spectral lines：
In the windowing FFT spectrum peak near zone that S2 is obtained, k0Locating three larger spectral lines of Frequency point is respectively：Kth1、
k2、k3Root, k1、k2、k3It is positive integer, k1~k3Relation be：k2=k1+ 1, k3=k2+ 1, amplitude corresponding to this three spectral lines
Respectively y1、y2、y3。
For convenience of calculating, variable α=k-k is remembered2, then -0.5≤α≤0.5；
Another note variable
S4, the correction formula for calculating three spectral line interpolation algorithms：
It can be obtained according to formula (4) and (5)：
When N values are bigger, such as：During N ＞ 1000, formula (6) can be using abbreviation as a function β=g (α), its inverse function
For α=g-1(β).Because used Cosine Window coefficient is real coefficient, its frequency response is even symmetry, thus g () and
g-1() is odd function.
Odd function α=g is calculated using Polynomial Approximation Method-1(β), expression formula are：
α≈p11×β+p13×β3+…+p1pβp (7)
p11, p13；…p1pFor the odd term coefficient of approximation by polynomi-als, p is odd number.
According to formula (4), the amplitude A of power network signal ith harmonic wave is tried to achievei：
Wherein, i is positive integer, yiFor the amplitude of ith harmonic wave after windowing FFT；
By taking fundamental wave as an example, it is contemplated that y2It is from the nearest spectral line of true spectral line point, gives greater weight, can obtain：
The approach method of similar formula (7), when N is bigger, such as during N ＞ 1000, window function coefficient is real coefficient, formula
(9) it is represented by：
A1=N-1(y3+2y2+y1)u(α)
Wherein, u (α) is correction formula, and is even function, and approximating polynomial is free of odd item.
Three spectral line amendment approximating polynomials are as follows：
U (α)=(p20+p22α2+…+p2dαd) (10)
In formula (10), p20,p22…p2dFor the even term coefficient of approximation by polynomi-als, d is the highest order of fitting, and d is
Even number.
S5, calculate fundamental wave parameter：
In view of y2It is the two piece spectral lines nearest from true spectral line point, gives greater weight, fundamental frequency f can be calculated0、
Fundamental voltage amplitude A1：
f0=k Δ f=(k2+α)Δf (11)
A1=N-1(y3+2y2+y1)(p20+p22α2+…+p2dαd) (12)
According to formula (4), the phase of fundamental wave can also be calculated：
Copy asking for for fundamental wave parameter, according to formula (6), (7), (9), (11), (12), (13) can carry out each time it is humorous
The analysis of wave parameter.
S6, determine harmonic parameters：Determine fundamental frequency f0Afterwards, in scope (kf0-5,kf0+ 5) in, repeat step S3~
S5, finished until all harmonic parameters calculate；
S7, carry out error analysis：Analyze the three spectral line interpolation FFT algorithms based on cross multiplication window function error, and with it is normal
Rule window function interpolation algorithm precision is compared.
It is described in detail below by a specific embodiment.Select the harmonic wave of a power network signal, the ripple of harmonic wave
Parameter is as shown in Figure 2.
The embodiment of the present invention provides a kind of three spectral line interpolation FFT harmonic analysis methods based on cross multiplication window function, specifically
Comprise the following steps：
The construction of step 1, cross multiplication window function：
By taking Hanning windows, Hamming windows, Blackman windows as an example, the structural model of cross multiplication window function is provided, is constructed
6 kinds of multiplication window function performances it is as shown in table 1.The problem of in view of amount of calculation, the order of multiplication window function be not suitable for it is too high, because
This limits order c=3.The sub- order of every kind of window function may take 0,1,2 three value.For the aspect of writing, simplify table below
Up to form：Hanning is abbreviated as Hn, and Hamming is abbreviated as Hm, and Blackman is abbreviated as Bm.
Table 1, the performance based on normal function construction cross multiplication window function
| Hn(c1) | Hm(c2) | Bm(c3) | Main lobe width (× π) | Sidelobe reduction (dB) |
| 2 | 1 | 0 | 0.0054 | -62.8 |
| 2 | 0 | 1 | 0.0059 | -84.5 |
| 1 | 2 | 0 | 0.0051 | -64.3 |
| 1 | 0 | 2 | 0.0061 | -100.6 |
| 1 | 1 | 1 | 0.0056 | -84.9 |
| 0 | 2 | 1 | 0.0056 | -86.1 |
The pretreatment of step 2, signal：
The discrete power network signal x (n) that transformer is collected, signal model ginseng are shown in Table 2.In order to verify carried algorithm
Precision, carry out 10 subharmonic simulation analysis.Signal model is：
Wherein：Fundamental frequency f0For 50.5Hz, sample frequency fsFor 5120Hz, sampling number N is 1024.
The harmonic parameters of table 2, power network signal
Signal spectrum is obtained after windowing FFT conversion：
Step 3, determine three spectral lines：
Seek maximum three spectral lines of mould, respectively y in 45~55Hz1, y2, y3。
So
Step 4, the correction formula for calculating three spectral line interpolation algorithms：
According to the derivation of step S4 in embodiment, correction formula α=g of 6 kinds of window functions in table 1-1(β), u
(α) is as follows respectively：
(1) mutually 210 window functions are multiplied：
β -0.0908720 β of α=1.114587153+0.01413650β5
U (α)=1.80783068+0.41014621 α2+0.05139583α4
It is shown in Figure 3 mutually to multiply 210 window function amplitude versus frequency charactes.
(2) mutually 201 window functions are multiplied：
β -0.09843530 β of α=1.280765373+0.01499840β5
U (α)=1.95369960+0.38423865 α2+0.04102745α4
It is shown in Figure 4 mutually to multiply 201 window function amplitude versus frequency charactes.
(3) mutually 120 window functions are multiplied：
β -0.08842400 β of α=1.085163003+0.0140181β5
U (α)=1.78667431+0.41627357 α2+0.05339022α4
The amplitude versus frequency characte for mutually multiplying 120 window functions is shown in Figure 5.
(4) mutually 102 window functions are multiplied：
β -0.10399918 β of α=1.423549573+0.01605665β5
U (α)=2.07275558+0.36590106 α2+0.03458435α4
The amplitude versus frequency characte for mutually multiplying 102 window functions is shown in Figure 6.
(5) mutually 111 window functions are multiplied：
β -0.09572246 β of α=1.25306403+0.01451709β5
U (α)=1.93442536+0.38883257 α2+0.04234880α4
The amplitude versus frequency characte for mutually multiplying 111 window functions is shown in Figure 7.
(6) mutually 021 window function is multiplied：
β -0.09322158 β of α=1.224471783+0.01445149β5
U (α)=1.91516084+0.39387119 α2+0.04376915α4
The amplitude versus frequency characte for mutually multiplying 021 window function is shown in Figure 8.
Step 5, calculate fundamental wave parameter：
f0=k Δ f=(k2+α)Δf
A1=N-1(y3+2y2+y1)(p20+p22α2+…+p2dαd)
Step 6, determine harmonic parameters：Determine fundamental frequency f0Afterwards, in scope (kf0-5,kf0+ 5) in, repeat step 3~
5, finished until all harmonic parameters calculate.
Step 7, carry out error analysis：The error of the three spectral line interpolation FFT algorithms based on cross multiplication window function is analyzed, and
Compared with custom window function interpolation arithmetic accuracy.Error analysis comparative result referring to shown in 3~table of table 5, wherein, DAiRepresent
The relative error of fundamental wave and each harmonic amplitude measure；Df0Represent the relative error of fundamental wave frequency measurement value；Represent base
The relative error of ripple and each harmonic initial phase measured value, is expressed as a percentage.
Table 3, cross multiplication window amplitude, frequency measurement relative error
Table 4, cross multiplication window phase measurement relative error
And individually use Hanning, Hamming and Blackman window function amplitude, frequency and phase measurement error such as table 5
It is shown.
Table 5, the amplitude of Hn, Hm, Bm window, frequency and phase relative error
As can be seen here, the cross multiplication window function in the embodiment of the present invention is in fundamental frequency, higher than basic window function by 1~
3 orders of magnitude, high 2~3 orders of magnitude are counted in crest meter, advantage is more prominent in phase calculation, high 3~6 orders of magnitude.
Result of calculation is generally better than common window function spectral line interpolation algorithm it can be seen from simulation result more than.
In summary, being constructed mutually based on cross multiplication window function harmonic analysis method, first realization in the embodiment of the present invention
Multiplication window function, cross multiplication window function is applied in three spectral line interpolation FFT algorithms, calculates harmonic parameters, can obtain than normal
Advise the higher precision of window function.
The embodiment of the present invention also provides a kind of three spectral line interpolation FFT frequency analysis systems based on cross multiplication window function, bag
Include cross multiplication window function structural unit, Signal Pretreatment unit, spectral line determining unit, correction formula computing unit, fundamental wave parameter
Computing unit, harmonic parameters determining unit, error analysis unit, wherein：
Cross multiplication window function structural unit, for constructing cross multiplication window function：
The general formula of multiplication window function is multiplication window function w (n) the general public affairs as caused by multiple window function products
Formula is：
Wherein, n is the ordinal number of sampled point, and n is natural number；M is for the basic window function for participating in construction multiplication window function
Number, m is positive integer；wi(n) it is i-th of basic window function expression formula, i=1,2...m；ciFor i-th of basic window function of participation
Number, referred to as wi(n) sub- order；C is total order of cross multiplication window function；According to c1、c2...cmDifference
Value, form different combination (c1c2…cm), that is, construct various forms of multiplication window functions；
Work as wi(n) when expression formula is identical, i.e., used by basic window function it is identical when, w (n) is multiplication window function, works as wi
(n) during expression formula difference, i.e., used by basic window function difference when, w (n) is cross multiplication window function；
Below by taking three conventional window functions as an example, expression formula is respectively w1(n), w2(n), w3(n).Total order c=c1+c2+
c3.Because c is bigger, it is meant that the basic window function of participation multiplication window is more, and the multiplication window function item number of composition is more, can increase
The complexity of calculating.Mono- maximum of c typically is given, for example, order：C=3.
According to c1、c2、c3Different values, form different combination (c1c2c3), construct following 9 kinds of multiplication window functions：
Work as c1=3, c2=0, c3When=0, construct and mutually multiply 300 window functions；
Work as c1=2, c2=1, c3When=0, construct and mutually multiply 210 window functions；
Work as c1=2, c2=0, c3When=1, construct and mutually multiply 201 window functions；
Work as c1=1, c2=2, c3When=0, construct and mutually multiply 120 window functions；
Work as c1=1, c2=0, c3When=2, construct and mutually multiply 102 window functions；
Work as c1=0, c2=3, c3When=0, construct and mutually multiply 030 window function；
Work as c1=1, c2=1, c3When=1, construct and mutually multiply 111 window functions；
Work as c1=0, c2=2, c3When=1, construct and mutually multiply 021 window function；
Work as c1=0, c2=0, c3When=3, construct and mutually multiply 003 window function；
It can be seen that from above-mentioned 9 kinds of multiplication window functions：Mutually multiply 300 window functions, mutually multiply 030 window function, mutually multiply 003 window letter
Number, these three window functions actually belong to multiplication window function (present invention not discusses), remaining the 6 kinds cross multiplications to construct
Window function.
Signal Pretreatment unit, for being pre-processed to signal：
Transformer gathers power network signal, and power network signal includes current signal, voltage signal；The power network that transformer is collected
Signal x (n), it is transferred to host computer；Power network signal x (n) is carried out plus cross multiplication window function w (n) is blocked, obtains adding window letter
Number xw(n)：
xw(n)=x (n) w (n) (2)
After the windowing signal progress FFT of formula (2), windowing FFT frequency spectrum is obtained：
Wherein, W () is the frequency spectrum of window function, and k is positive integer, and X (k) represents the frequency spectrum of kth subharmonic, AkFor kth time
The amplitude of harmonic wave, j represent imaginary unit, and e is the truth of a matter of natural logrithm,For the initial phase of kth subharmonic, the first subharmonic
For fundamental wave, fsFor sample frequency, f0For fundamental frequency, Δ f is discrete frequency intervals, and Δ f=fs/N；
Order：k0=f0/ Δ f, k0For the position of spectral line of real frequency spectrum, because non-synchronous sampling or non-integer-period block, k0One
As be decimal, be not integer.
Ignore the influence of secondary lobe at negative frequency point, formula (3) is changed into：
Spectral line determining unit, for determining three spectral lines：
In the windowing FFT spectrum peak near zone that Signal Pretreatment unit obtains, k0Locate three larger spectrums of Frequency point
Line is respectively：Kth1、k2、k3Root, k1、k2、k3It is positive integer, k1~k3Relation be：k2=k1+ 1, k3=k2+ 1, this three
Amplitude corresponding to spectral line is respectively y1、y2、y3；
Remember variable α=k-k2, then -0.5≤α≤0.5；
Another note variable
Correction formula computing unit, for calculating the correction formula of three spectral line interpolation algorithms：
Obtained according to formula (4) and (5)：
When N values are bigger, such as：During N ＞ 1000, formula (6) can be using abbreviation as a function β=g (α), its inverse function
For α=g-1(β).Because used Cosine Window coefficient is real coefficient, its frequency response is even symmetry, thus g () and
g-1() is odd function.
Odd function α=g is calculated using Polynomial Approximation Method-1(β), expression formula are：
α≈p11×β+p13×β3+…+p1pβp (7)
p11, p13；…p1pFor the odd term coefficient of approximation by polynomi-als, p is odd number；
According to formula (4), the amplitude A of power network signal ith harmonic wave is tried to achievei：
Wherein, i is positive integer, yiFor the amplitude of ith harmonic wave after windowing FFT；
By taking fundamental wave as an example, it is contemplated that y2It is from the nearest spectral line of true spectral line point, gives greater weight, can obtain：
During N ＞ 1000, window function coefficient is real coefficient, and formula (9) is expressed as：
A1=N-1(y3+2y2+y1)u(α)
Wherein, u (α) is correction formula, and is even function, and approximating polynomial is free of odd item；
Three spectral line amendment approximating polynomials are as follows：
U (α)=(p20+p22α2+…+p2dαd) (10)
In formula (10), p20,p22…p2dFor the even term coefficient of approximation by polynomi-als, d is the highest order of fitting, and d is
Even number；
Fundamental wave parameter calculation unit, for calculating fundamental wave parameter：
In view of y2It is the two piece spectral lines nearest from true spectral line point, gives greater weight, fundamental frequency f can be calculated0、
Fundamental voltage amplitude A1：
f0=k Δ f=(k2+α)Δf (11)
A1=N-1(y3+2y2+y1)(p20+p22α2+…+p2dαd) (12)
According to formula (4), the phase of fundamental wave is calculated：
Asking for for fundamental wave parameter is copied, each harmonic ginseng is carried out according to formula (6), (7), (9), (11), (12), (13)
Several analyses；
Harmonic parameters determining unit, for determining harmonic parameters：Determine fundamental frequency f0Afterwards, in scope (kf0-5,kf0+5)
Interior, the spectral line determining unit, correction formula computing unit, fundamental wave parameter calculation unit repeat to calculate, until all humorous
Wave parameter is calculated and finished；
Error analysis unit, for carrying out error analysis：Analyze the three spectral line interpolation FFT algorithms based on cross multiplication window function
Error, and compared with custom window function interpolation arithmetic accuracy.
Those skilled in the art can carry out various modifications and variations to the embodiment of the present invention, if these modifications and change
Type is within the scope of the claims in the present invention and its equivalent technologies, then these modifications and variations are also in protection scope of the present invention
Within.
The prior art that the content not being described in detail in specification is known to the skilled person.
Claims (6)
1. a kind of three spectral line interpolation FFT harmonic analysis methods based on cross multiplication window function, it is characterised in that including following step
Suddenly：
S1, construction cross multiplication window function：
The general formula of multiplication window function is as caused by multiple window function products, and multiplication window function w (n) general formula is：
Wherein, n is the ordinal number of sampled point, and n is natural number；M constructs the number of the basic window function of multiplication window function, m for participation
For positive integer；wi(n) it is i-th of basic window function expression formula, i=1,2...m；ciTo participate in of i-th of basic window function
Number, referred to as wi(n) sub- order；C is total order of cross multiplication window function；According to c1、c2…cmDifference take
Value, forms different combination (c1 c2…cm), that is, construct various forms of multiplication window functions；
Work as wi(n) when expression formula is identical, i.e., used by basic window function it is identical when, w (n) is multiplication window function, works as wi(n)
During expression formula difference, i.e., used by basic window function difference when, w (n) is cross multiplication window function；
The expression formula of three conventional window functions is respectively w1(n), w2(n), w3(n), total order c=c1+c2+c3, give c mono- maximum
Value, order：C=3；
According to c1、c2、c3Different values, form different combination (c1 c2 c3), construct following 9 kinds of multiplication window functions：
Work as c1=3, c2=0, c3When=0, construct and mutually multiply 300 window functions；
Work as c1=2, c2=1, c3When=0, construct and mutually multiply 210 window functions；
Work as c1=2, c2=0, c3When=1, construct and mutually multiply 201 window functions；
Work as c1=1, c2=2, c3When=0, construct and mutually multiply 120 window functions；
Work as c1=1, c2=0, c3When=2, construct and mutually multiply 102 window functions；
Work as c1=0, c2=3, c3When=0, construct and mutually multiply 030 window function；
Work as c1=1, c2=1, c3When=1, construct and mutually multiply 111 window functions；
Work as c1=0, c2=2, c3When=1, construct and mutually multiply 021 window function；
Work as c1=0, c2=0, c3When=3, construct and mutually multiply 003 window function；
Wherein, mutually multiply 300 window functions, mutually multiply 030 window function, mutually multiply 003 window function, these three window functions belong to multiplication window letter
Number, remaining the 6 kinds cross multiplication window functions to construct；
S2, Signal Pretreatment：
Transformer gathers power network signal, the power network signal x (n) that transformer is collected, is transferred to host computer；To power network signal x
(n) carry out plus cross multiplication window function w (n) is blocked, obtain windowing signal xw(n)：
xw(n)=x (n) w (n) (2)
After the windowing signal progress FFT of formula (2), windowing FFT frequency spectrum is obtained：
Wherein, W () is the frequency spectrum of window function, and k is positive integer, and X (k) represents the frequency spectrum of kth subharmonic, AkFor kth subharmonic
Amplitude, j represent imaginary unit, and e is the truth of a matter of natural logrithm,For the initial phase of kth subharmonic, the first subharmonic is base
Ripple, fsFor sample frequency, f0For fundamental frequency, Δ f is discrete frequency intervals, and Δ f=fs/N；
Order：k0=f0/ Δ f, k0For the position of spectral line of real frequency spectrum,
Ignore the influence of secondary lobe at negative frequency point, formula (3) is changed into：
S3, determine three spectral lines：
In the windowing FFT spectrum peak near zone that S2 is obtained, k0Locating three larger spectral lines of Frequency point is respectively：Kth1、k2、k3
Root, k1、k2、k3It is positive integer, k1~k3Relation be：k2=k1+ 1, k3=k2+ 1, amplitude corresponding to this three spectral lines is distinguished
For y1、y2、y3；
Remember variable α=k-k2, then -0.5≤α≤0.5；
Another note variable
S4, the correction formula for calculating three spectral line interpolation algorithms：
Obtained according to formula (4) and (5)：
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α≈p11×β+p13×β3+…+p1pβp (7)
p11, p13；…p1pFor the odd term coefficient of approximation by polynomi-als, p is odd number；
According to formula (4), the amplitude A of power network signal ith harmonic wave is tried to achievei：
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Wherein, i is positive integer, yiFor the amplitude of ith harmonic wave after windowing FFT；
In view of y2It is from the nearest spectral line of true spectral line point, obtains：
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During N ＞ 1000, window function coefficient is real coefficient, and formula (9) is expressed as：
A1=N-1(y3+2y2+y1)u(α)
Wherein, u (α) is correction formula, and is even function, and approximating polynomial is free of odd item；
Three spectral line amendment approximating polynomials are as follows：
U (α)=(p20+p22α2+…+p2dαd) (10)
In formula (10), p20,p22…p2dFor the even term coefficient of approximation by polynomi-als, d is the highest order of fitting, and d is even
Number；
S5, calculate fundamental wave parameter：
Calculate fundamental frequency f0, fundamental voltage amplitude A1：
f0=k Δ f=(k2+α)Δf (11)
A1=N-1(y3+2y2+y1)(p20+p22α2+…+p2dαd) (12)
According to formula (4), the phase of fundamental wave is calculated：
Asking for for fundamental wave parameter is copied, point of each harmonic parameter is carried out according to formula (6), (7), (9), (11), (12), (13)
Analysis；
S6, determine harmonic parameters：Determine fundamental frequency f0Afterwards, in scope (kf0-5,kf0+ 5) in, repeat step S3~S5, until
All harmonic parameters are calculated and finished；
S7, carry out error analysis：Analyze the error of the three spectral line interpolation FFT algorithms based on cross multiplication window function, and and custom window
Function interpolation arithmetic accuracy is compared.
2. the three spectral line interpolation FFT harmonic analysis methods based on cross multiplication window function, its feature exist as claimed in claim 1
In：The power network signal includes current signal, voltage signal.
3. the three spectral line interpolation FFT harmonic analysis methods based on cross multiplication window function, its feature exist as claimed in claim 1
In：The position of spectral line k of the real frequency spectrum0For decimal.
A kind of 4. three spectral line interpolation FFT frequency analysis systems based on cross multiplication window function, it is characterised in that：Including cross multiplication window
It is construction of function unit, Signal Pretreatment unit, spectral line determining unit, correction formula computing unit, fundamental wave parameter calculation unit, humorous
Wave parameter determining unit, error analysis unit, wherein：
The cross multiplication window function structural unit, for constructing cross multiplication window function：
The general formula of multiplication window function is as caused by multiple window function products, and multiplication window function w (n) general formula is：
Wherein, n is the ordinal number of sampled point, and n is natural number；M constructs the number of the basic window function of multiplication window function, m for participation
For positive integer；wi(n) it is i-th of basic window function expression formula, i=1,2...m；ciTo participate in of i-th of basic window function
Number, referred to as wi(n) sub- order；C is total order of cross multiplication window function；According to c1、c2...cmDifference take
Value, forms different combination (c1 c2...cm), that is, construct various forms of multiplication window functions；
Work as wi(n) when expression formula is identical, i.e., used by basic window function it is identical when, w (n) is multiplication window function, works as wi(n)
During expression formula difference, i.e., used by basic window function difference when, w (n) is cross multiplication window function；
The expression formula of three conventional window functions is respectively w1(n), w2(n), w3(n), total order c=c1+c2+c3, give c mono- maximum
Value, order：C=3；
According to c1、c2、c3Different values, form different combination (c1 c2 c3), construct following 9 kinds of multiplication window functions：
Work as c1=3, c2=0, c3When=0, construct and mutually multiply 300 window functions；
Work as c1=2, c2=1, c3When=0, construct and mutually multiply 210 window functions；
Work as c1=2, c2=0, c3When=1, construct and mutually multiply 201 window functions；
Work as c1=1, c2=2, c3When=0, construct and mutually multiply 120 window functions；
Work as c1=1, c2=0, c3When=2, construct and mutually multiply 102 window functions；
Work as c1=0, c2=3, c3When=0, construct and mutually multiply 030 window function；
Work as c1=1, c2=1, c3When=1, construct and mutually multiply 111 window functions；
Work as c1=0, c2=2, c3When=1, construct and mutually multiply 021 window function；
Work as c1=0, c2=0, c3When=3, construct and mutually multiply 003 window function；
Wherein, mutually multiply 300 window functions, mutually multiply 030 window function, mutually multiply 003 window function, these three window functions belong to multiplication window letter
Number, remaining the 6 kinds cross multiplication window functions to construct；
The Signal Pretreatment unit, for being pre-processed to signal：
Transformer gathers power network signal, the power network signal x (n) that transformer is collected, is transferred to host computer；To power network signal x
(n) carry out plus cross multiplication window function w (n) is blocked, obtain windowing signal xw(n)：
xw(n)=x (n) w (n) (2)
After the windowing signal progress FFT of formula (2), windowing FFT frequency spectrum is obtained：
Wherein, W () is the frequency spectrum of window function, and k is positive integer, and X (k) represents the frequency spectrum of kth subharmonic, AkFor kth subharmonic
Amplitude, j represent imaginary unit, and e is the truth of a matter of natural logrithm,For the initial phase of kth subharmonic, the first subharmonic is base
Ripple, fsFor sample frequency, f0For fundamental frequency, Δ f is discrete frequency intervals, and Δ f=fs/N；
Order：k0=f0/ Δ f, k0For the position of spectral line of real frequency spectrum,
Ignore the influence of secondary lobe at negative frequency point, formula (3) is changed into：
The spectral line determining unit, for determining three spectral lines：
In the windowing FFT spectrum peak near zone that the Signal Pretreatment unit obtains, k0Locate three larger spectral lines of Frequency point
Respectively：Kth1、k2、k3Root, k1、k2、k3It is positive integer, k1~k3Relation be：k2=k1+ 1, k3=k2+ 1, this three spectrums
Amplitude corresponding to line is respectively y1、y2、y3；
Remember variable α=k-k2, then -0.5≤α≤0.5；
Another note variable
The correction formula computing unit, for calculating the correction formula of three spectral line interpolation algorithms：
Obtained according to formula (4) and (5)：
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According to formula (4), the amplitude A of power network signal ith harmonic wave is tried to achievei：
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</mrow>
</mfrac>
<mo>-</mo>
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<mn>9</mn>
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During N ＞ 1000, window function coefficient is real coefficient, and formula (9) is expressed as：
A1=N-1(y3+2y2+y1)u(α)
Wherein, u (α) is correction formula, and is even function, and approximating polynomial is free of odd item；
Three spectral line amendment approximating polynomials are as follows：
U (α)=(p20+p22α2+…+p2dαd) (10)
In formula (10), p20,p22…p2dFor the even term coefficient of approximation by polynomi-als, d is the highest order of fitting, and d is even
Number；
The fundamental wave parameter calculation unit, for calculating fundamental wave parameter：
Calculate fundamental frequency f0, fundamental voltage amplitude A1：
f0=k Δ f=(k2+α)Δf (11)
A1=N-1(y3+2y2+y1)(p20+p22α2+…+p2dαd) (12)
According to formula (4), the phase of fundamental wave is calculated：
Asking for for fundamental wave parameter is copied, point of each harmonic parameter is carried out according to formula (6), (7), (9), (11), (12), (13)
Analysis；
The harmonic parameters determining unit, for determining harmonic parameters：Determine fundamental frequency f0Afterwards, in scope (kf0-5,kf0+5)
Interior, the spectral line determining unit, correction formula computing unit, fundamental wave parameter calculation unit repeat to calculate, until all humorous
Wave parameter is calculated and finished；
The error analysis unit, for carrying out error analysis：Analyze the three spectral line interpolation FFT algorithms based on cross multiplication window function
Error, and compared with custom window function interpolation arithmetic accuracy.
5. the three spectral line interpolation FFT frequency analysis systems based on cross multiplication window function, its feature exist as claimed in claim 4
In：The power network signal includes current signal, voltage signal.
6. the three spectral line interpolation FFT frequency analysis systems based on cross multiplication window function, its feature exist as claimed in claim 4
In：The position of spectral line k of the real frequency spectrum0For decimal.
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| CN109633262A (en) * | 2019-01-29 | 2019-04-16 | 国网湖南省电力有限公司 | Three phase harmonic electric energy gauging method, device based on composite window multiline FFT |
| CN110031552B (en) * | 2019-05-27 | 2021-10-22 | 嘉兴博传科技有限公司 | Structural health monitoring damage characteristic value calculation method |
| CN115825557A (en) * | 2022-11-25 | 2023-03-21 | 国网四川省电力公司映秀湾水力发电总厂 | Generalized harmonic analysis method, device and medium based on harmonic component zero setting |
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