CN104062500A - Signal harmonic analysis method and system based on Hamming product window - Google Patents

Abstract

The invention discloses a signal harmonic analysis method and system based on a Hamming product window. The method includes the steps that discrete signals are windowed through the Hamming product window, and FFT is conducted on the windowed signals and the Hamming product window; actual fundamental frequency and the amplitude and the phases of sub-harmonics are acquired according to a signal spectrogram obtained after FFT and the Hamming product window obtained after FFT. The Hamming product window is combined with FFT, and harmonic analysis with extremely high accuracy can be achieved in an asynchronous sampling state; meanwhile, the signal harmonic analysis method and system further have the advantages of being small in calculated amount, short in consumed time, high in practicability and the like.

Description

Signal harmonic analytical approach and system based on Hamming product window

Technical field

The present invention relates to a kind of Harmonious Waves in Power Systems frequency spectrum analysis method, be specially a kind of signal harmonic analytical approach and system based on Hamming product window.

Background technology

Nonlinear-load in electric system can cause the deterioration of the quality of power supply.In actual signal processing procedure, signal is processed degree of accuracy can be subject to the impact that the quality of power supply changes, the non-synchronous sampling that the fluctuation of fundamental frequency causes will cause spectral leakage and hurdle effect, make electric system detect harmonic amplitude, phase angle and frequency and all have larger error, can not meet accuracy requirement.Hurdle effect can adopt method of interpolation to solve, but the measuring error causing because of spectrum leakage phenomenon still exists.To carrying out the error that direct computation of DFT leaf analysis can reduce spectral leakage after signal windowing, but process is complicated, calculated amount is large again.

For addressing the above problem, Chinese scholars has been done corresponding research, has proposed high precision harmonic detecting method based on FFT, based on Nuttall (Nuttall) window and 5 Lai Fu-Vincents (Rife-Vincent) (I) the Electric Power Harmonic Analysis method, rectangle convolution window etc. of window double spectral line interpolation FFT.Yet because the frequency spectrum sidelobe performance of above window function is not ideal enough, limited to spectral leakage inhibiting effect, make frequency analysis error larger, be difficult to realize high-precision frequency measurement and frequency analysis, and computation process is complicated, length consuming time, is difficult to meet the requirement of harmonic analysis in power system real-time and accuracy.

Summary of the invention

The problem existing for prior art, the invention provides a kind of signal harmonic frequency spectrum analysis method and system based on Hamming product window that can effectively suppress spectral leakage, the method calculated amount is little, consuming time short, degree of accuracy is high, can realize signal in real time and Accurate Analysis.

The present invention is based on Hamming product window (Hamming product window) and realize, take Hamming window function as basic window function, basic window function is multiplied each other and obtains Hamming product window w _m(n):

In formula (1), w _i(n) represent i basic window function, i=1,2 ..., p.

The signal harmonic frequency spectrum analysis method based on Hamming product window that the present invention proposes, comprises step:

Step 1, carries out analog to digital conversion to power system signal x (t) and obtains discretize signal x (n), and described power system signal x (t) is voltage signal or current signal;

Step 2, adopts Hamming product window to discretize signal x (n) windowing, and signal after windowing and Hamming product window are carried out to FFT conversion, obtains signal spectrum figure and Hamming product window after FFT conversion; Described Hamming product window is the acquisition of multiplying each other of p Hamming window, and p value is set according to actual conditions;

Step 3, obtains actual fundamental frequency according to signal spectrum figure and Hamming product window after FFT conversion;

Step 4, according to amplitude, the phase place of the signal spectrum figure after FFT conversion and actual fundamental frequency acquisition first-harmonic and each harmonic.

Hamming window described in step 2 is cosine Hamming window.

Step 3 is specially:

By put the discrete spectral line sequence number in the nearest left and right sides from theoretical peak, be designated as respectively k ₁and k ₂, discrete spectral line k ₁and k ₂corresponding amplitude is respectively y ₁and y ₂, according to obtain the discrete spectral line sequence number k of actual first-harmonic ₀thereby, obtain fundamental frequency; Wherein, W () is the Hamming product window after FFT conversion.

Step 4 is specially:

[(the i*f of signal spectrum figure after FFT conversion ₀-a), (i*f ₀+ a)] look for the discrete spectral line k that i subharmonic is corresponding in frequency range _i1and k _i2, i is overtone order; The harmonic wave maximum deviation that a allows according to electric system arranges; Based on discrete spectral line k _i1and k _i2corresponding amplitude y _i1and y _i2, adopt amplitude correction formula and phase place correction formula to obtain amplitude and the phase angle of i subharmonic.

The signal harmonic analytic system based on Hamming product window that the present invention proposes, comprising:

Analog-to-digital conversion module, is used for that power system signal x (t) is carried out to analog to digital conversion and obtains discretize signal x (n), and described power system signal x (t) is voltage signal or current signal;

Windowing and FFT change module, are used for adopting Hamming product window to discretize signal x (n) windowing, and signal after windowing and Hamming product window are carried out to FFT conversion, obtain signal spectrum figure and Hamming product window after FFT conversion; Described Hamming product window is the acquisition of multiplying each other of p Hamming window, and p value is set according to actual conditions;

Actual fundamental frequency obtains module, is used for obtaining actual fundamental frequency according to signal spectrum figure and Hamming product window after FFT conversion;

Harmonic amplitude and harmonic phase obtain module, are used for according to amplitude, the phase place of the signal spectrum figure after FFT conversion and actual fundamental frequency acquisition first-harmonic and each harmonic.

Above-mentioned harmonic amplitude and harmonic phase obtain module and further comprise:

The discrete spectral line that harmonic wave is corresponding is looked for module, is used for the [(i*f of the signal spectrum figure after FFT conversion ₀-a), (i*f ₀+ a)] look for the discrete spectral line k that i subharmonic is corresponding in frequency range _i1and k _i2, i is overtone order; The harmonic wave maximum deviation that a allows according to electric system arranges;

Amplitude and phase solution module, be used for based on discrete spectral line k _i1and k _i2corresponding amplitude y _i1and y _i2, adopt amplitude correction formula and phase place correction formula to obtain amplitude and the phase angle of i subharmonic.

Compared with prior art, the present invention has the following advantages and beneficial effect:

(1) based on Hamming product window, realize, under non-synchronous sampling, the inhibition of spectral leakage is much better than to classical window function.

(2) by Hammng product window and FFT conversion combination, under non-synchronous sampling state, can realize the frequency analysis of very high degree of precision.

(3) calculated amount is little, and consuming time short, practicality is high, can guarantee real-time and the degree of accuracy of frequency analysis simultaneously.

Accompanying drawing explanation

Fig. 1 is the time domain specification of different Hamming product windows;

Fig. 2 is the frequency domain characteristic of different Hamming product windows;

Fig. 3 is the concrete implementing procedure figure of the inventive method;

Fig. 4 is the spectrogram of windowing signal after FFT conversion;

Fig. 5 is the amplitude relative error comparison diagram of the inventive method and classic method;

Fig. 6 is the phase place relative error comparison diagram of the inventive method and classic method.

Embodiment

The present invention is based on Hamming product window and realize the harmonic wave detection of power system signal, Hamming product window w _hamming-m(n) by some Hamming window w _hamming(n) acquisition of multiplying each other:

In formula (1), p is the Hamming window number multiplying each other.In concrete enforcement, p value is selected according to actual conditions and experience, and generally speaking, p value is larger, and the side lobe peak height of Hamming product window is lower, but main lobe is wider.And suitable window function requires, main lobe is narrow, secondary lobe is low.

In this concrete enforcement, adopt cosine Hamming window w _hamming(n), it has very little side lobe peak:

$w_{\mathrm{ham}\min g}\left(n\right)=0.54-0.46\cos \left(\frac{2\mathrm{\πn}}{N-1}\right)---\left(2\right)$

In formula (2), N is window function length, 0≤n≤N-1.

When basic window function type and number change in Hamming product window, can obtain different Hamming product windows.Fig. 1 and Fig. 2 have shown respectively time domain specification and the frequency domain characteristic of different Hamming product windows, and the characteristic parameter of each Hamming product window is in Table 1.

The spectral characteristic parameter of table 1Hamming product window

In table 1, Hamming represents cosine Hamming window, Hamming ², Hamming ³, Hamming ⁴, Hamming ⁵represent respectively the Hamming product window that 2,3,4,5 cosine Hamming windows multiply each other and obtain.

The main lobe width (MB) of Hamming product window, side lobe attenuation speed (SRR) have following approximation relation with parameter p:

$\left\{\begin{array}{c}{w}_{\mathrm{ham}\min g-m}{\left(n\right)}_{\mathrm{MB}}=(1+\frac{1}{2}p)\×w_{\mathrm{ham}\min g}{\left(n\right)}_{\mathrm{MB}}\\ w_{\mathrm{ham}\min g-m}{\left(n\right)}_{\mathrm{SSR}}=6\end{array}\right.---\left(3\right)$

In formula (3): w _hamming-m(n) _mBthe main lobe width that represents Hamming product window, w _hamming(n) _mBthe main lobe width that represents Hamming window, w _hamming-m(n) _sSRthe side lobe attenuation speed that represents Hamming product window.

In Table 1, when p increases, the side lobe peak height (PSL) of Hamming product window diminishes gradually, and along with basic window function amount increases, Hamming product window suppresses spectral leakage ability can be strengthened.

Fig. 3 is the particular flow sheet of the inventive method, comprises step:

Step 1, sampling power system signal x (t) also obtains discretize signal x (n).

With fixed sampling frequency 4kHz sampling power system signal x (t), signal x (t) can be voltage signal or current signal; Signal x (t) is obtained to discretize signal x (n) through analog to digital conversion, send discretize signal x (n) to merge cells after merging combination, transfer to digital information network.

Discretize signal x (n) is as follows:

x(n)＝Asin(2πnf ₀/f _s+θ) (4)

In formula (4), n is sampled point numbering; A and θ are respectively amplitude and the phase place of harmonic wave; f ₀for signal x (t) crest frequency, fundamental frequency to be asked; f _sfor sample frequency, be 4kHz herein.

Step 2, adopts Hamming product window to carry out windowing to discretize signal x (n), and signal after windowing is carried out to the signal spectrum figure after FFT conversion acquisition conversion.

In electric energy quality monitor, discretize signal x (n) is carried out to windowing: adopt Hamming product window w _hamming-m(n) block discretize signal x (n), signal x after acquisition windowing _m(n)=x (n) * w _hamming-m(n), n=0,1 ... N-1, N is window length, carries out according to actual needs value, N=1000 in this concrete enforcement.

Ignore the peak secondary lobe impact frequently of negative frequency peak point place, signal after windowing is carried out to FFT conversion, after conversion, signal X (k Δ f) is:

$X(k\·\mathrm{\Δf})=\frac{A}{2j}\×\left[e^{\mathrm{j\θ}}W(\frac{2\mathrm{\π}(k\·\mathrm{\Δf}-f_0)}{f_s}\right]---\left(5\right)$

In formula (5), j represents imaginary part; A and θ are respectively amplitude and the phase place of nth harmonic; W () is the Hamming product window after FFT conversion; Δ f is discrete frequency interval, Δ f=f _s/ N, N is window length; K is independent variable, represents discrete spectral line sequence number; Crest frequency f ₀=k ₀Δ f generally there will not be on the Frequency point of discrete spectral line, k ₀discrete spectral line sequence number for first-harmonic.

Step 3, obtains actual fundamental frequency according to the signal spectrum figure after Fast Fourier Transform (FFT) (FFT) and Hamming product window value.

By put the discrete spectral line sequence number in the nearest left and right sides from theoretical peak, be designated as respectively k ₁and k ₂, spectral line k ₁and k ₂corresponding amplitude is respectively y ₁and y ₂, from formula (5):

$\frac{y_1}{y_2}=\left|\frac{W(2\mathrm{\π}(k_1-k_0)/N)}{W(2\mathrm{\π}(k_2-k_0)/N)}\right|---\left(6\right)$

Because window function W () is known, according to formula (6), can obtain unknown quantity k ₀thereby, obtain fundamental frequency.

Step 4, obtains amplitude and the phase angle of each harmonic based on fundamental frequency.

Because of 0≤k ₀-k ₁≤ 1, for simplifying, calculate defined parameters α=k ₀-k ₁-0.5, wherein, α ∈ [0.5,0.5], when window length N is larger, formula (6) can be reduced to:

α＝g ^-1(β) (7)

Adopt polynomial approach algorithm to solve formula (7), by control, approach number of times, can control flexibly approximation accuracy.The weight of supposing spectral line is directly proportional to its amplitude, approaches in the approximate treatment formula of obtaining with polynomial expression,, containing odd term, does not obtain amplitude correction formula (8) and phase place correction formula (9):

A＝N ^-1(y ₁+y ₂)(b ₀+b ₂β ²+…+b _2iβ ²ⁱ) (8)

θ＝arg[X(k _i·Δf)]+π/2-arg[W(2π(k _i-k ₀)/N)] (9)

In formula (8)～(9), i represents overtone order.

Therefore, adopt amplitude correction formula (8) and phase place correction formula (9) can directly obtain amplitude and the phase place of each harmonic.

The amplitude of each harmonic and the acquisition methods of phase angle are specific as follows:

[(the i*f of signal spectrum figure after FFT conversion ₀-a), (i*f ₀+ a)] look for the discrete spectral line k that i subharmonic is corresponding in frequency range _i1and k _i2, i is overtone order, i=1,2 ..., n, during i=1, be first-harmonic; The harmonic wave maximum deviation that a allows according to electric system arranges, and this embodiment makes a=5; By discrete spectral line k ₁and k ₂corresponding amplitude y ₁and y ₂substitution amplitude correction formula (8) and phase place correction formula (9) obtain amplitude and the phase angle of i subharmonic.

To further illustrate technical solution of the present invention and technique effect by embodiment below.

Power system signal x (t) with fixed sampling frequency 4kHz sampling containing 11 subharmonic, fundamental frequency f ₁=50.1Hz, each harmonic amplitude and phase place are in Table 2.

The amplitude of table 2 each harmonic and phase place

Above-mentioned power system signal x (t) is carried out to harmonic wave detection, and step is as follows:

Step 1, carries out analog to digital conversion to original signal x (t) and obtains discretize signal x (n), adopts Hamming product window to carry out windowing to discretize signal x (n) and obtains truncated signal x _m(n)=x (n) * w _hamming-m(n), wherein, n=0,1 ..., N-1, Hamming product window length N=1000.

Truncated signal is carried out to FFT conversion, and after conversion, the spectral characteristic of signal is shown in Fig. 4.

Step 2 is looked for the discrete spectral line k nearest from theoretic frequency point both sides in [45Hz, 55Hz] frequency range ₁and k ₂, spectral line k ₁and k ₂corresponding amplitude is designated as y ₁and y ₂, wherein:

$\frac{y_1}{y_2}=\left|\frac{W(2\mathrm{\π}(k_1-k_0)/N)}{W(2\mathrm{\π}(k_2-k_0)/N)}\right|---\left(10\right)$

Obtain factor alpha and β:

$\mathrm{\β}=\frac{y_2-y_2}{y_2+y_1}---\left(11\right)$

α＝0.1314β+0.2865β ³+0.4671β ⁵(12)

Step 3, obtains amplitude correction formula (13) and the phase angle correction formula (14) of each harmonic according to factor alpha and β value:

A＝N ^-1·(y ₁+y ₂)·(0.3065+0.6624*β ²+0.9504*β ⁴) (13)

θ＝arg[X(k _i·Δf)]-π·(α-(-1) ⁱ·0.5)...(i＝1,2) (14)

Step 4, can obtain fundamental frequency f according to formula (10) ₁=f ₀; Amplitude and the phase place of each harmonic obtain based on fundamental frequency, are specially: at [(i*f ₀-5), (i*f ₀+ 5)] in frequency range, find the corresponding discrete spectral line k of i subharmonic _i1and k _i2, by corresponding amplitude y _i1and y _i2substitution formula (13)～(14), amplitude and the phase place of acquisition i subharmonic.Repeatedly repeat this step until obtain amplitude and the phase place of all harmonic waves.

Frequency analysis result to the present embodiment is carried out error analysis, and fundamental frequency relative error is in Table 3, and the amplitude of harmonic wave and the relative error of phase place are in Table 4～5.

The fundamental frequency relative error of the different product windows of table 3

The each harmonic amplitude relative error of the different product windows of table 4

The each harmonic phase place relative error of the different product windows of table 5

The present embodiment also adopts respectively traditional Hamming (Hamming) window, peaceful (Hanning) window of the Chinese and this (Blackman-Harris) window of haab to carry out frequency analysis to above-mentioned power system signal x (t), and obtain amplitude relative error and the phase angle relative error that various window functions are corresponding, see respectively Fig. 5 and Fig. 6.Result shows, compares with classical window function, and Hamming multiplication window is with the obvious advantage: side lobe peak is lower.When N basic window multiplies each other, the sidelobe performance of Hamming multiplication window is obviously better than basic window, can better suppress spectral leakage, obviously improves harmonic parameters computational accuracy.

Claims (6)

1. the signal harmonic analytical approach based on Hamming product window, is characterized in that, comprises step:

Step 1, carries out analog to digital conversion to power system signal x (t) and obtains discretize signal x (n), and described power system signal x (t) is voltage signal or current signal;

Step 2, adopts Hamming product window to discretize signal x (n) windowing, and signal after windowing and Hamming product window are carried out to FFT conversion, obtains signal spectrum figure and Hamming product window after FFT conversion; Described Hamming product window is the acquisition of multiplying each other of p Hamming window, and p value is set according to actual conditions;

Step 3, obtains actual fundamental frequency according to signal spectrum figure and Hamming product window after FFT conversion;

Step 4, according to amplitude, the phase place of the signal spectrum figure after FFT conversion and actual fundamental frequency acquisition first-harmonic and each harmonic.

2. the signal harmonic analytical approach based on Hamming product window as claimed in claim 1, is characterized in that:

Described Hamming window is cosine Hamming window.

3. the signal harmonic analytical approach based on Hamming product window as claimed in claim 1, is characterized in that:

Described signal spectrum figure and Hamming product window according to after FFT conversion obtain actual fundamental frequency, are specially:

By put the discrete spectral line sequence number in the nearest left and right sides from theoretical peak, be designated as respectively k ₁and k ₂, discrete spectral line k ₁and k ₂corresponding amplitude is respectively y ₁and y ₂, according to obtain the discrete spectral line sequence number k of actual first-harmonic ₀thereby, obtain fundamental frequency; Wherein, W () is the Hamming product window after FFT conversion.

4. the signal harmonic analytical approach based on Hamming product window as claimed in claim 1, is characterized in that:

Described amplitude, the phase place that obtains first-harmonic and each harmonic according to the signal spectrum figure after FFT conversion and actual fundamental frequency, is specially:

[(the i*f of signal spectrum figure after FFT conversion ₀-a), (i*f ₀+ a)] look for the discrete spectral line k that i subharmonic is corresponding in frequency range _i1and k _i2, i is overtone order; The harmonic wave maximum deviation that a allows according to electric system arranges; Based on discrete spectral line k _i1and k _i2corresponding amplitude y _i1and y _i2, adopt amplitude correction formula and phase place correction formula to obtain amplitude and the phase angle of i subharmonic.

5. the signal harmonic analytic system based on Hamming product window, is characterized in that, comprising:

Analog-to-digital conversion module, is used for that power system signal x (t) is carried out to analog to digital conversion and obtains discretize signal x (n), and described power system signal x (t) is voltage signal or current signal;

Windowing and FFT change module, are used for adopting Hamming product window to discretize signal x (n) windowing, and signal after windowing and Hamming product window are carried out to FFT conversion, obtain signal spectrum figure and Hamming product window after FFT conversion; Described Hamming product window is the acquisition of multiplying each other of p Hamming window, and p value is set according to actual conditions;

Actual fundamental frequency obtains module, is used for obtaining actual fundamental frequency according to signal spectrum figure and Hamming product window after FFT conversion;

Harmonic amplitude and harmonic phase obtain module, are used for according to amplitude, the phase place of the signal spectrum figure after FFT conversion and actual fundamental frequency acquisition first-harmonic and each harmonic.

6. the signal harmonic analytic system based on Hamming product window as claimed in claim 5, is characterized in that:

Described harmonic amplitude and harmonic phase obtain module and further comprise:

The discrete spectral line that harmonic wave is corresponding is looked for module, is used for the [(i*f of the signal spectrum figure after FFT conversion ₀-a), (i*f ₀+ a)] look for the discrete spectral line k that i subharmonic is corresponding in frequency range _i1and k _i2, i is overtone order; The harmonic wave maximum deviation that a allows according to electric system arranges;

Amplitude and phase solution module, be used for based on discrete spectral line k _i1and k _i2corresponding amplitude y _i1and y _i2, adopt amplitude correction formula and phase place correction formula to obtain amplitude and the phase angle of i subharmonic.