CN101603985B - Method for measuring sine signal with high accuracy - Google Patents

Abstract

The invention provides a method for measuring a sine signal with high accuracy, which is realized by steps of data acquisition system sampling, discrete Fourier transform, peak searching, frequency measurement, and amplitude and phase measurement. The invention provides an interpolating method capable of correcting leakage effect in a long-range and a short-range simultaneously to realize high accuracy measurement on parameters of the sine signal by the Fourier transform algorithm. The method can improve the accuracy for measuring the sine signal to over 10<-8>.

Description

Method for measuring sine signal with high accuracy

Technical field

The invention belongs to the signal measurement technique field, relate to a kind of method for measuring sine signal with high accuracy.

Background technology

The measurement of sinusoidal ac signal has important use in electrical engineering, electrician's metering and other fields.For example, the metering of electric energy, the fluctuation of mains frequency, the reflection high voltage electric equipment makes moist, deterioration is rotten or insulation in the assessment of dielectric loss angle of defective such as gas discharge, the calibrating of AC signal generator and calibration, or the like, all need offset of sinusoidal signal frequency, amplitude, initial phase etc. accurately to measure.

Sinusoidal signal has frequency, amplitude, three parameters of initial phase, and existing method for measuring sine signal is varied.In order to record frequency, can adopt zero passage detection method based on hardware circuit; Under the known condition of frequency, can adopt the correlation analysis software algorithm to measure amplitude and initial phase; In order to record frequency, amplitude and initial phase simultaneously, can adopt Fourier Transform Algorithm.Comparatively speaking, Fourier Transform Algorithm is fast with its measuring speed, antijamming capability is strong, can record advantage such as a plurality of parameters simultaneously, is used widely on engineering.

Utilize Fourier Transform Algorithm to measure sinusoidal signal, at first need data acquisition system (DAS) that continuous sinusoidal signal is sampled, and then utilize microprocessor that the discrete digital signal that collects is carried out discrete Fourier transformation (Discrete FourierTransform is called for short DFT).If it is integer-period sampled that data acquisition system (DAS) can be accomplished in sampling process, promptly time of covering of sample is the integral multiple in sinusoidal signal cycle, utilizes DFT to measure sinusoidal signal so and will not have error on the algorithm principle.But because of the restriction of hardware device performance and the influence of other random disturbance factors, desirable integer-period sampled being difficult to accomplished in actual measurement.At this moment, utilize DFT to measure sinusoidal signal and error on the algorithm principle will occur, comprising: short scope leakage effect, i.e. the observed deviation that causes by the fence effect of discrete spectrum signal frequency; Long scope leakage effect is promptly because the phase mutual interference between the signal spectrum secondary lobe that the signal brachymemma causes.

In order to improve accuracy of measurement, must revise or compensate these errors.Proposed multiple modification method at present, be mainly used in and revise the measuring error that short scope leakage effect causes, revised accuracy can reach 10 ^-5～10 ^-4About, but this accuracy grade can not satisfy the requirement of some precision measurement occasion.For long scope leakage effect, the measuring error that particularly negative frequency leakage effect causes has not yet to see effective modification method.But this part error can reach 10 sometimes ^-4About, it has reduced the accuracy based on the surveying instrument of DFT to a great extent.

As fully visible, under non-integer-period sampled condition, carry out pin-point accuracy for the parameter of utilizing DFT offset of sinusoidal signal and measure, need all revise long and short scope leakage effect.

Summary of the invention

Measure for the pin-point accuracy that realizes the sinusoidal signal parameter, the invention provides a kind of method for measuring sine signal with high accuracy, can all revise long and short scope leakage effect.

Technical scheme of the present invention is as follows:

Method for measuring sine signal with high accuracy comprises the step of obtaining N sample from sinusoidal signal, and N is a natural number, f _sSample frequency for data acquisition system (DAS);

And the step of the sample that obtains being carried out the DFT processing.

Also comprise the steps:

A, search spectrum peak: from the discrete spectrum that utilizes sample to obtain, choose the p root spectral line and the amplitude time maximum q root spectral line of amplitude maximum, write down the real part R of p root spectral line _PWith imaginary part I _P, write down the real part R of q root spectral line _qWith imaginary part X _q

B, survey frequency.

Set up formula

$F(N,p,q)=\frac{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;q}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;p}}{N}\right)}{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)}$

Wherein $\mathrm{\&lambda;}=\frac{I_p}{I_q};$

Calculate τ:

$\mathrm{\&tau;}=\frac{N}{2\mathrm{\&pi;}}\arccos \left[F(N,p,q)\right]$

The frequency f that obtains sinusoidal signal is: $f=\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;N}};$

C, measurement amplitude and phase place:

Calculate $\mathrm{\&alpha;}=\frac{2\mathrm{\&pi;}}{N}(\mathrm{\&tau;}-p),$ $\mathrm{\&beta;}=-\frac{2\mathrm{\&pi;}}{N}(\mathrm{\&tau;}+p),$ $L_1=\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}},$ $L_2=\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}},$ $b=\frac{2I_p}{L_1+L_2},$ $a=\frac{2(R_p-b)}{L_1-L_2},$

Then the amplitude A of sinusoidal signal is:

$A=\frac{N\sqrt{a^2+b^2}}{1-\cos 2\mathrm{\&pi;\&tau;}};$

The initial phase of sinusoidal signal

For:

Wherein j is an imaginary unit.

Described sampling can be integer-period sampled, also can be non-integer-period sampled, and promptly sampling period and sinusoidal signal can be the integral multiple relation between the cycle, also can be non-integral multiple relations.

The above-mentioned method for measuring sine signal based on DFT can further specify by following theoretical derivation:

Tested sinusoidal signal can be expressed as form usually

Wherein, f is a frequency, and A is an amplitude,

Be initial phase, t is the time.Ignore the quantization error in the analog-digital conversion process, and the various stochastic errors in the measuring process, utilize sample frequency to be f _sData acquisition system (DAS) adopt N sample

x _nDFT be

$X_m=\mathrm{DFT}\left(x_n\right)=\frac{1}{N}\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{x}_n{e}^{-j\left(\frac{2\mathrm{\&pi;}}{N}\right)n\&CenterDot;m}$

Wherein, m=0,1 ..., N-1, τ=Nf/f _s, j is an imaginary unit.For integer-period sampled, when promptly τ=p and p are integer,

X _-=0.So the frequency of this sinusoidal signal, amplitude, initial phase can be utilized x _nDiscrete spectrum in p root spectral line try to achieve

$f=\frac{p{f}_s}{N}---\left(4\right)$

A＝2NX _P (5)

Under non-integer-period sampled condition, promptly τ=p+ ε and | ε |＜1 o'clock,

Be x _nDiscrete spectrum in have short scope spectrum leakage, Be x _nDiscrete spectrum in have long scope spectrum leakage (being also referred to as the negative frequency spectrum leakage).At this moment, the spectral line corresponding to this sinusoidal signal will be positioned at x _nDiscrete spectrum between two spectral lines of the maximum and inferior maximum of amplitude, suppose that these two spectral lines are respectively p root spectral line and q root spectral line.

In order to improve accuracy of measurement, existing interpolation correcting method is to ignore X _-, think X _m=X ₊So,, p root spectral line is Q root spectral line is

Utilize this principle formula structure interpolation algorithm.Owing to ignored X _-, thereby the accuracy of this interpolation correcting method can not surpass X _-Size.

In order to obtain the higher interpolation correcting method of accuracy, do not ignore X _-, promptly think X _m=X ₊+ X _-, according to formula (3), have so

Order

$\frac{2\mathrm{\&pi;}}{N}(\mathrm{\&tau;}-p)=\mathrm{\&alpha;},$ $-\frac{2\mathrm{\&pi;}}{N}(\mathrm{\&tau;}+p)=\mathrm{\&beta;},$ So

$X_p=\{\frac{b}{2}(\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}+\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}})j+[b+\frac{a}{2}(\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}-\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}})\left]\right\}---\left(9\right)$

X _pImaginary part be

$I_p=\frac{b}{2}(\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}+\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}})=\frac{b}{2}\frac{2\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)}{\cos \left(\frac{2\mathrm{\&pi;p}}{N}\right)-\cos \left(\frac{2\mathrm{\&pi;\&tau;}}{N}\right)}---\left(10\right)$

X _pReal part be

$R_p=b+\frac{a}{2}(\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}-\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}})=b+\frac{a}{2}\frac{2\sin \left(\frac{2\mathrm{\&pi;\&tau;}}{N}\right)}{\cos \left(\frac{2\mathrm{\&pi;p}}{N}\right)-\cos \left(\frac{2\mathrm{\&pi;\&tau;}}{N}\right)}---\left(11\right)$

In like manner, X _qImaginary part be

$I_q=\frac{b}{2}\frac{2\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)}{\cos \left(\frac{2\mathrm{\&pi;q}}{N}\right)-\cos \left(\frac{2\mathrm{\&pi;\&tau;}}{N}\right)}---\left(12\right)$

Order

$\mathrm{\&lambda;}=\frac{I_p}{I_q}=\frac{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;q}}{N}\right)-\cos \left(\frac{2\mathrm{\&pi;\&tau;}}{N}\right)}{\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;p}}{N}\right)-\cos \left(\frac{2\mathrm{\&pi;\&tau;}}{N}\right)}---\left(13\right)$

Then

$\cos \left(\frac{2\mathrm{\&pi;\&tau;}}{N}\right)=\frac{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;q}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;p}}{N}\right)}{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)}=F(N,p,q)---\left(14\right)$

In following formula, N is a sample number, and is known in advance; Discrete signal is carried out after DFT analyzes, and the size of p, q can be obtained from amplitude-versus-frequency curve, i.e. the position of the spectral line of amplitude maximum; I _pBe p root spectral line X _pImaginary part; I _qBe q root spectral line X _qImaginary part; λ is I _pWith I _qThe ratio.Therefore, F (N, p, q) big or small known.So can get according to formula (14)

$\mathrm{\&tau;}=\frac{N}{2\mathrm{\&pi;}}\arccos \left[F(N,p,q)\right]---\left(15\right)$

Again according to τ=Nf/f _s, the frequency of sinusoidal signal is as can be known

$f=\frac{\mathrm{\&tau;}{f}_s}{N}---\left(16\right)$

After τ obtained, α, β all can obtain.According to

$\left\{\begin{array}{c}{R}_p=b+\frac{a}{2}(\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}-\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}})=b+\frac{a}{2}(L_1-L_2)\\ I_p=\frac{b}{2}(\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}+\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}})=\frac{b}{2}(L_1+L_2)\end{array}\right.---\left(17\right)$

Can get

$\left\{\begin{array}{c}b=\frac{2I_p}{L_1+L_2}\\ a=\frac{2(R_p-b)}{L_1-L_2}\end{array}\right.---\left(18\right)$

According to

The amplitude of tested sinusoidal signal and initial phase are respectively as can be known

$A=\frac{N\sqrt{a^2+b^2}}{1-\cos 2\mathrm{\&pi;\&tau;}}---\left(19\right)$

Beneficial effect of the present invention:

No matter be under sampling of complete cycle device or the non-integer-period sampled condition, can both utilize DFT and long and short scope leakage effect is revised.Under the condition of not considering the data acquisition system (DAS) hardware error, the accuracy of this method can reach 10 ^-8More than.

Description of drawings

Fig. 1 is the process flow diagram of the inventive method.

Embodiment

Below in conjunction with Fig. 1 technical scheme of the present invention is elaborated.

Process step as shown in Figure 1, method for measuring sine signal with high accuracy provided by the invention comprises the steps:

Step 1, data acquisition system sampling: the sinusoidal signal that the needs of input are measured is sampled, obtain N sample, N is a natural number, and the sample frequency of data acquisition system (DAS) is f _s

Step 2, discrete Fourier transformation: N the sample that collects carried out discrete Fourier transformation;

Step 3, search spectrum peak: from the discrete spectrum that sample constitutes, choose the p root spectral line and the amplitude time maximum q root spectral line of amplitude maximum, write down X _pReal part R _PWith imaginary part I _P, and X _qReal part R _qWith imaginary part I _q

Step 4, survey frequency: calculate $\frac{I_p}{I_q}=\mathrm{\&lambda;},$ Set up following formula:

$F(N,p,q)=\frac{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;q}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;p}}{N}\right)}{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)}$

Calculate $\mathrm{\&tau;}=\frac{N}{2\mathrm{\&pi;}}\arccos \left[F(N,p,q)\right].$

The frequency that then obtains sinusoidal signal is $f=\frac{\mathrm{\&tau;}{f}_s}{2\mathrm{\&pi;N}}.$

Step 5, measurement amplitude and phase place: calculate $\mathrm{\&alpha;}=\frac{2\mathrm{\&pi;}}{N}(\mathrm{\&tau;}-p),$ $\mathrm{\&beta;}=-\frac{2\mathrm{\&pi;}}{N}(\mathrm{\&tau;}+p);$ And

$L_1=\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}$

$L_2=\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}}$

And

$b=\frac{2I_p}{L_1+L_2}$

$a=\frac{2(R_p-b)}{L_1-L_2}$

The amplitude that obtains sinusoidal signal is:

$A=\frac{N\sqrt{a^2+b^2}}{1-\cos 2\mathrm{\&pi;\&tau;}};$

The initial phase of sinusoidal signal is:

Embodiment: below provide an emulation testing example that utilizes the inventive method to carry out.

Utilize MATLAB software that the measuring method that the present invention proposes is carried out emulation testing.For frequency is that f=50.5Hz, amplitude A are 5, initial phase Be the sinusoidal signal of 0.1 radian, utilize sample frequency to be f _sThe data acquisition system (DAS) of=500Hz is adopted to such an extent that N=16 sample analyzed.Ignore the influence of quantization error and other stochastic errors of analog-digital conversion process, 16 samples are respectively:

Sequence number	Sample size
		0	0.499167083234141
1	3.351458215784571
		2	4.898744376371143
3	4.538536736964815
		4	2.411094471363638
5	-0.655189954997563
		6	-3.466353746160143
7	-4.927774057896540
		8	-4.470396869493416
9	-2.272317644626180
		10	0.810566233377442
11	3.577828403886208
		12	4.951940621364167
13	4.397845260029381
		14	2.131298314713716
15	-0.965142580745820

After handling through DFT, 16 samples obtain 16 discrete spectrum values and amplitude is respectively:

Sequence number	The discrete spectrum real part	The discrete spectrum imaginary part	Amplitude
				0	0.925706553948097	0	0.925706553948097
1	1.436032857696581	-0.422083125355980	1.496778050712423
				2	-1.445232022183261	0.941689678557234	1.724956535291584
3	-0.235422973732956	0.285890618776699	0.370347704820473
				4	-0.061403116989602	0.162396710873847	0.173617494736799
5	0.000503940501871	0.101704755446294	0.101706003934968
				6	0.028448912309876	0.061143923407858	0.067438267929426
7	0.041277163716393	0.028942548813747	0.050413047679207
				8	0.045051006648239	0	0.045051006648239
9	0.041277163716393	-0.028942548813747	0.050413047679207
				10	0.028448912309876	-0.061143923407858	0.067438267929426
11	0.000503940501871	-0.101704755446294	0.101706003934968
				12	-0.061403116989602	-0.162396710873847	0.173617494736799
13	-0.235422973732956	-0.285890618776699	0.370347704820473
				14	-1.445232022183261	-0.941689678557234	1.724956535291584
15	1.436032857696581	0.422083125355980	1.496778050712423

The maximum spectral line of amplitude is the 2nd spectral line in 16 discrete spectrum values, and amplitude time maximum spectral line is the 1st spectral line, thereby p=2, q=1,

X ₂＝R ₂+jI ₂＝-1.445232022183261+j0.941689678557234

X ₁＝R ₁+jI ₁＝1.436032857696581-j0.422083125355980

So

$\mathrm{\&lambda;}=\frac{I_2}{I_1}=-2.231052657608675$

$F\left(\mathrm{16,2,1}\right)=\frac{\sin \left(\frac{4\mathrm{\&pi;}}{16}\right)\cos \left(\frac{2\mathrm{\&pi;}}{16}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;}}{16}\right)\cos \left(\frac{4\mathrm{\&pi;}}{16}\right)}{\sin \left(\frac{4\mathrm{\&pi;}}{16}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;}}{16}\right)}=0.805307885711122$

The frequency of sinusoidal signal is

$f=\frac{500}{2\mathrm{\&pi;}}\arccos \left[F\left(\mathrm{16,2,1}\right)\right]=50.499999999999972$

τ, α, β are respectively

τ＝16f/500＝1.615999999999999

$\mathrm{\&alpha;}=\frac{2\mathrm{\&pi;}}{16}(\mathrm{\&tau;}-2)=-0.150796447372310$

$\mathrm{\&beta;}=-\frac{2\mathrm{\&pi;}}{N}(\mathrm{\&tau;}+2)=-1.419999879422586$

Thereby

$L_1=\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}=-13.237769652807650$

$L_2=\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}}=-1.163428472404786$

A, b are respectively

$b=\frac{2I_p}{L_1+L_2}=0.130779351880258$

$a=\frac{2(R_p-b)}{L_1-L_2}=0.261051323714696$

The amplitude and the phase place of sinusoidal signal are respectively

$A=\frac{16\sqrt{a^2+b^2}}{1-\cos 2\mathrm{\&pi;\&tau;}}=5.000000000000006$

Therefore the accuracy of this algorithm can reach 10 ^-8More than.In addition, in this example, when p=2, long scope is leaked

Size be 0.447932936245234.Obviously, if ignore X _-, then no matter adopt which kind of interpolation algorithm, all can not obtain higher accuracy.

Claims (2)

1. method for measuring sine signal with high accuracy comprises the steps:

Step 1, data acquisition system sampling: the sinusoidal signal that the needs of input are measured is sampled, obtain N sample, N is a natural number, and the sample frequency of data acquisition system (DAS) is f _s

Step 2, discrete Fourier transformation: N the sample that collects carried out discrete Fourier transformation;

It is characterized in that, also comprise the steps:

Step 3, search spectrum peak: the p root spectral line X that from the discrete spectrum that sample constitutes, chooses the amplitude maximum _pWith amplitude time maximum q root spectral line X _q, write down X _pReal part R _PWith imaginary part I _P, and X _qReal part R _qWith imaginary part I _q

Step 4, survey frequency: calculate

Set up following formula:

$F(N,p,q)=\frac{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;q}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)\cos \left(\frac{2\mathrm{\&pi;p}}{N}\right)}{\sin \left(\frac{2\mathrm{\&pi;p}}{N}\right)-\mathrm{\&lambda;}\sin \left(\frac{2\mathrm{\&pi;q}}{N}\right)}$

Calculate $\mathrm{\&tau;}=\frac{N}{2\mathrm{\&pi;}}\arccos \left[F(N,p,q)\right],$

The frequency that then obtains sinusoidal signal is

Step 5, measurement amplitude and phase place: calculate

And

$L_1=\frac{\sin \mathrm{\&alpha;}}{1-\cos \mathrm{\&alpha;}}$

$L_2=\frac{\sin \mathrm{\&beta;}}{1-\cos \mathrm{\&beta;}}$

And

$b=\frac{{2I}_p}{L_1+L_2}$

$a=\frac{2(R_p-b)}{L_1-L_2}$

The amplitude that obtains sinusoidal signal is:

The initial phase of sinusoidal signal is:

Wherein j is an imaginary unit.

2. method for measuring sine signal with high accuracy according to claim 1 is characterized in that describedly being sampled as non-integer-period sampledly, and promptly sampling period and sinusoidal signal are not integral multiple relation between the cycle.