CN102004186B - High-accuracy sinusoidal signal measuring method for removing frequency spectrum leakage - Google Patents

Abstract

The invention discloses a high-accuracy sinusoidal signal measuring device for removing frequency spectrum leakage, comprising an analog signal pre-processing unit, a data acquisition unit, a digital signal processing unit, a measuring result display unit, and a communication interface unit, wherein the analog signal pre-processing unit is connected with the data acquisition unit through multipath signal wires, and the digital signal processing unit is connected with the data acquisition unit, the measuring result display unit and the communication interface unit respectively through an SPI (Serial Peripheral Interface) bus. The measuring method comprises the steps of: 1, continuously sampling a signal to be measured to obtain sequences S1 and S2; 2, carrying out fast Fourier transform on the S1, extracting a peak frequency spectrum and obtaining the peak frequency spectrum of the S2 through recursion; 3, calculating the frequency of the signal to be measured by applying the peak frequency spectrums of the S1 and the S2; 4, calculating the phase of the signal to be measured according to the phase angle information of the peak frequency spectrums of the S1 and S2; 5, solving an equation set relevant to positive and negative frequency spectrum amplitudes to obtain the measured value of the amplitude of the signal to be measured; and 6, removing the influence of the frequency spectrum leakage to obtain the accurate measured valve of a direct-current component.

Description

A kind of method for measuring sine signal with high accuracy of eliminating spectrum leakage

One, technical field

The present invention relates to a kind of method for measuring sine signal with high accuracy and device of eliminating spectrum leakage, belong to the signal measurement technique field.

Two, background technology

In industrial circles such as electrotechnical measurement, power network monitoring, vibration noise measurement, audio frequency processing and Radar Signal Processing, often need each parameter of offset of sinusoidal signal to carry out high-acruracy survey.The amplitude that needs accurately to measure sinusoidal signal as the electric energy metrical of precision, the frequency jitter of monitoring electrical network needs accurately to measure in real time sinusoidal wave frequency, flow and radar precision ranging with the Coriolis flowmeter tracer liquid need the accurately phase differential of measurement sinusoidal signal, and the loss of calculating transformer direct current needs the DC component in the measuring-signal.

Method for measuring sine signal can be divided into hardware mensuration and software measurement method.Traditional measuring method and device mainly realize that by hardware circuit its measuring accuracy depends on the performance of metering circuit.The software measurement method is with computing machine digital signal to be handled after analog signal conversion is become digital signal, and it is the new development trend of current input, and its measuring accuracy and performance depend primarily on digital signal processing algorithm.

1, the subject matter of traditional sinusoidal signal measurement mechanism

Mainly there is following problem in traditional measurement device for detection of the sinusoidal signal parameter:

(1) the traditional measurement device mainly realizes that by hardware circuit be vulnerable to noise, measuring accuracy is lower, can not satisfy more and more higher commercial measurement requirement.

(2) the traditional measurement device is big to the hardware-dependent degree, and clock stability, device have seriously restricted the measuring accuracy of traditional measurement device to the ardware features such as delay of signal.

(3) traditional measurement plant automation degree is low, and is poor to the input signal amplitude adaptive faculty, can not be according to the automatic conditioning signal Amplifier Gain coefficient of the size of measured signal.When signal amplitude is big, may exceed measurement range, shown in Fig. 4 a and Fig. 4 b; When the signal amplitude signal quantization error that hour sampling obtains bigger, shown in Fig. 5 a and Fig. 5 b.

2, present main sinusoidal signal Measurement Algorithm and the problem of existence thereof

Signal processing algorithm among the present invention belongs to the category of Spectral Analysis Method.The most basic principle of Spectral Analysis Method is that signal is carried out discrete Fourier transform (DFT) (DFT), obtains the parameter information of signal from frequency field.Discrete Fourier transform (DFT) has ripe fast algorithm FFT, i.e. fast fourier transform already.Under non-integer-period sampled condition, directly use the FFT method can produce spectrum leakage, the frequency spectrum of gained is the frequency content of reflected signal truly, short scope is leaked and long scope is leaked two parts error but comprised, shown in Fig. 6 b and Fig. 6 d.

It is because the discrete spectrum peak value that is obtained by the continuous frequency spectrum sampling under the non-integer-period sampled condition not at the crest frequency place of continuous spectrum, has caused the spectrum amplitude distortion that short scope is leaked.The main method that overcomes short scope leakage at present has two kinds: frequency spectrum interpolation and windowing intercepting.Long scope leakage is that the negative frequency composition in the frequency spectrum causes, and the error that it causes under following two kinds of situations be can not ignore.The first, when measured signal frequency during much smaller than sample frequency or near Nyquist frequency (half of sample frequency); The second, when sampling number hour.The main method that overcomes long scope leakage at present has the method for approximation of simplification and analytical method.These two kinds of errors have had a strong impact on the measuring accuracy of FFT method.

Use interpolation method and windowing intercept method can overcome the influence that short scope is leaked.In the Korean Patent database, the patent No. is KR20030089995A, name is called patent and document " based on the high precision FFT frequency analysis of the bimodal spectral line of the windowing " [Liu Min of " adopting frequency and phase shift interpolation to improve the method for discrete Fourier transform (DFT) or fast fourier transform precision " (METHOD FOR INTERPOLATING FREQUENCY AND PHASE OFFSET TO IMPROVE RESOLUTION OF DFT OR FFT), Wang Keying .[J]. electrical measurement and instrument, 2006,43 (4): 20-23] all adopt the method for bimodal spectral line interpolation to overcome the influence that short scope is leaked, but all ignored the negative frequency composition, thereby can't overcome the influence that long scope is leaked.Although the latter adopts window adding technology to suppress the influence of negative frequency to frequency spectrum, measuring accuracy is higher than the former, but still can't fundamentally eliminate the influence of negative frequency, especially when the measured signal frequency during much smaller than sample frequency or near the Nyquist frequency measuring error more obvious.And the precision of windowed interpolation method depends primarily on window function, and when the measured signal parameter changed, the parameter of window function also needed to change thereupon, need determine the parameter of window function according to signal parameter, and implementation method is complicated.

It is the bottleneck of restriction sinusoidal signal measuring accuracy that long scope is leaked.In the Inpadoc, the patent No. is WO03058261A1, name is called the influence that " method of eliminating non-integer-period sampled signal discrete Fourier transform spectrum leakage " (DFT LEAKAGE REMOVAL FOR NON-COHERENTLY SAMPLED SIGNALS) considered negative frequency, has provided the accurate measurement method of sinusoidal signal amplitude when accurately predicting the measured signal frequency.But it is inapplicable fully under measured signal frequency condition of unknown.

In the United States Patent (USP) database, the patent No. is US6965068B2, the patent that name is called " system and method that input signal spectrum is estimated " (SYSTEM AND METHOD FOR ESTIMATING TONES IN AN INPUT SIGNAL) has adopted iterative algorithm, at first ignoring the amplitude that three spectral line approximate evaluations that utilize frequency spectrum under the situation of negative frequency influence go out sinusoidal signal, frequency, phase parameter, then according to the parameter reconstruct negative frequency composition frequency spectrum that estimates, the negative frequency frequency spectrum that deducts reconstruct from former frequency spectrum obtains new frequency spectrum, estimate the parameter of sinusoidal signal again with three frequency spectrums, iterate, till the precision that reaches setting.There is following shortcoming in said method: the first, and iterative process needs the frequency spectrum of reconstruct negative frequency repeatedly, and calculated amount is bigger.The second, need use three spectral lines and just can measure, thereby can't tell the signal of frequency between first and the 3rd spectral line, reduced the frequency resolution of algorithm.

Document " improving the frequency domain interpolation method of phase measurement " [Dusan Agrez.[J]. measure, 2008,41 (2): 151-159] (" Interpolation in the frequency domain to improve phase measurement " [Dusan Agrez.[J] .Measurement, 2008,41 (2): 151-159]) proposed under the situation of considering negative frequency, to utilize three spectral lines to be similar to the method for interpolation, can overcome the influence that short scope is leaked and long scope is leaked, improve measuring accuracy.But the method has following shortcoming: the first, and need use three spectral lines and just can measure, thereby can't tell the signal of frequency between first and the 3rd spectral line respective frequencies point, reduced the frequency resolution of algorithm.The second, the method is approximate data, only is applicable to the situation that sampling number is bigger, when sampling number more after a little while, Algorithm Error is bigger.

The present invention adopts a kind of analytic method that is easy to realize, can eliminate the influence that short scope is leaked and long scope is leaked fully, has fundamentally solved non-integer-period and has blocked the spectrum leakage problem that causes.The digital sequence of N point of continuous sampling is carried out FFT obtain spectrum peak, obtain postponing the spectrum peak of signal after the sampling interval by one step of simple algebraic operation recursion, utilize two spectrum peaks and flip-flop frequency spectrum to parse amplitude, frequency, phase place and the DC component of sinusoidal signal accurately.

Name is called " sinusoidal signal four parameters testing method and virtual instrument signal supervisory instrument ", the patent No. is that the Chinese patent of ZL200810101338.0 discloses a kind of simplification approximate measure method of considering short scope leakage and long scope leakage contribution simultaneously, and the method that proposes with the present invention has following difference.The first, above-mentioned patent is that the digital sequence of 2N-1 point is split into two sections, carries out FFT respectively, in fact will carry out the FFT that twice N ordered, and the present invention only need carry out N point FFT one time.The second, above-mentioned patent has adopted simplifies approximate method, is exact method and the present invention adopts, and without simplification, thereby the measurement result precision is higher than foregoing invention, sees Table 1.The 3rd, it is that sampling number N is bigger that foregoing invention is simplified approximate precondition, and the present invention is not subjected to this condition restriction.N is more big, and the operand of FFT is also just more big.In addition, foregoing invention will be carried out twice FFT computing, and calculated amount is about twice of the present invention.The 4th, one of foregoing invention feature is sampling number " N gets 2 positive integer time power ", and the sampling number in this method does not have this restriction.

Name is called " method for measuring sine signal with high accuracy ", and application number is that 200910089315.7 Chinese patent also discloses and a kind ofly considers that simultaneously short scope is leaked and the measuring method of long scope leakage contribution, and measuring process also is similar to simplification.But with the present invention tangible difference is arranged.First, one of feature of above-mentioned patent is " described be sampled as non-integer-period sampled, namely sampling period and sinusoidal signal are not integral multiple relation between the cycle ", and that the present invention both had been applicable to was integer-period sampled, be applicable to non-integer-period sampledly again, be not subjected to the restriction of above-mentioned measuring condition.The second, above-mentioned patented method utilization be maximum and time two big spectral lines of amplitude in the frequency spectrum, thereby can't tell the signal of frequency between these two spectral line respective frequencies points, reduced the frequency resolution of algorithm.And utilization of the present invention is two peaks spectrum of same discrete point in frequency, can not reduce the Measurement Resolution to signal frequency.The 3rd, measure low frequency signal with above-mentioned patented method, when sampling number is low, can occur unusual, cause data to be overflowed, parameter that can't measuring-signal, and this type of unusual situation can not appear in this method, still can keep high accuracy, shown in table 2a, table 2b and table 2c.The 4th, above-mentioned patented method does not provide the measuring method of DC component, uses the inventive method can obtain the pin-point accuracy measured value of whole four parameters of sinusoidal signal.

Name is called " high-efficiency measurement method of the sinusoidal signal frequency under owing to sample and device ", and the Chinese patent of application number 201010162687.0 discloses a kind of method of owing to measure under the sampling condition sinusoidal signal frequency.Said method and the inventive method have following difference.The first, one of said method feature is to be applicable to owe sampling condition, does not namely satisfy the situation of sampling thheorem, and the inventive method is applicable to the situation that satisfies sampling thheorem.The second, said method only provides the measuring method of sinusoidal signal frequency, and the inventive method can be measured the whole parameter of sinusoidal signal; The 3rd, said method another feature is " being weighted with the long individual data of the center of convolution window Wc sampling point x (0) forward and backward (2N-1) for (2N-1); the data that will be spaced apart N are then carried out overlap-add in twos; again the data behind the overlap-add are carried out DFT ", and the inventive method does not need data are carried out windowing process, directly the N point data is done the FFT computing.The 4th, reach same frequency resolution, the data length of the required sampling of the inventive method is less than said method.For example, when sample frequency all is fs, reach the frequency resolution of fs/128, said method 255 point data of need sampling, 129 point data and the inventive method only need be sampled.

Three, summary of the invention

1, purpose: at the deficiency that has method for measuring sine signal and device both at home and abroad now, the purpose of this invention is to provide a kind of method for measuring sine signal with high accuracy and device of eliminating spectrum leakage, it can realize the measurement of offset of sinusoidal signal amplitude, frequency, phase place and DC component pin-point accuracy.

2, technical scheme:

(1) a kind of sine signal with high accuracy measurement mechanism of eliminating spectrum leakage of the present invention, it is made up of simulating signal pretreatment unit, data acquisition unit, digital signal processing unit, measurement result display unit, communications interface unit, as shown in Figure 1.Position annexation between them is: connect by the multiple signals line between simulating signal pretreatment unit and the data acquisition unit, digital signal processing unit is connected with data acquisition unit, measurement result display unit, communications interface unit respectively by spi bus.

Described simulating signal pretreatment unit is made up of buffer circuit and filtering circuit, is connected by the multiple signals line between its buffer circuit and the filtering circuit; This buffer circuit adopts accurate isolated amplifier ISO124, and it realizes the isolation of input stage and output stage signal by the isolation capacitance that is encapsulated in device inside; This filtering circuit adopts continuous time filter MAX274, the high frequency noise that its function produces when being filtering process buffer circuit; This unit mainly isolates and low-pass filtering simulating signal.

Described data acquisition unit is made up of multi-channel analog selector switch, programmable gain amplifier, Programmable Logic Controller CPLD and AD converter.The multiple signals line that multi-channel analog selector switch input termination filtering circuit comes out, output terminal is one road signal wire, is connected to the programmable gain amplifier input end, the programmable gain amplifier output terminal is connected to AD converter by one road signal wire.This multi-channel analog selector switch adopts DG201, is used for the signal that will sample from the multichannel analog signals selection; This programmable gain amplifier adopts AD256 to realize, it can change the Amplifier Gain coefficient according to the control code that Programmable Logic Controller is sent here; This Programmable Logic Controller CPLD is used for finishing the function of controlling of sampling and adjusting, controls the gain coefficient of programmable gain amplifier according to the range of each input; This AD converter adopts AD677, is used for pretreated simulating signal is converted into digital signal.

Described digital signal processing unit adopts dsp chip TMS320C5402, and signalization process of measurement in the built-in RAM of DSP mainly comprises FFT computing and a spot of algebraic operation, is used for realizing signal measurement algorithm proposed by the invention.

Described measurement result display unit, adopting model is LCD demonstration signal waveform and the measurement result of LCM320240.

Described communications interface unit adopts asynchronous communication interface chip 16C550, and measurement mechanism is communicated by RS232 bus interface and PC, realizes the debugging to measurement mechanism, also signal measurement data and result can be delivered to PC from measurement mechanism.

A kind of sine signal with high accuracy measurement mechanism of eliminating spectrum leakage, its course of work is as follows:

Multiple signals to be measured are delivered to isolator by the multiple signals line and are carried out the external noise isolation, through the high frequency noise jamming in the wave filter filtered signal, deliver to the multi-channel analog selector switch then.Programmable Logic Controller CPLD and AD677 carry out exchanges data by spi bus and dsp chip.The CPLD sending controling instruction is to multi-channel analog selector switch and AD converter, make AD converter with the multiple signals of fixed sampling frequency sampling input, AD converter is delivered to the data of sampling among the DSP by solidifying the multichannel buffered serial port McBSP that arranges in spi bus and the DSP then.DSP calculates suitable amplifier gain coefficient according to the magnitude range that receives data, send to CPLD by spi bus, CPLD produces steering order and delivers to programmable gain amplifier, changes its gain coefficient, and CPLD delivers to DSP with amended gain coefficient by spi bus simultaneously.The resulting measurement result of dsp operation signal processing algorithm is delivered to LCD through spi bus and is shown, simultaneously can pass through RS232 asynchronous communication interface chip 16C550, signal measurement data and result are delivered to PC via standard serial interface, carry out visualization processing and demonstration.

(2) a kind of method for measuring sine signal with high accuracy of eliminating spectrum leakage of the present invention, adopt a kind of analytic method that is easy to realize, can eliminate the influence that short scope is leaked and long scope is leaked fully, fundamentally solve non-integer-period and blocked the spectrum leakage problem that causes.It at first carries out FFT to the digital sequence of N point of continuous sampling and obtains spectrum peak, the spectrum peak of signal utilized the geometric vector method to parse amplitude, frequency, phase place and the DC component of sinusoidal signal accurately according to two spectrum peaks again after one step, recursion obtained postponing a sampling period by simple algebraic operation then.As shown in Figure 2, the concrete steps of this method are as follows:

Step 1: continuous sampling to measured signal, obtain the digital sequence S1 of N point, and the Serial No. S2 after it is postponed a sampling period;

N is sampling number, and f is sinusoidal signal frequency, and T is the sampling period,

Be sample frequency.

Be the normalization numerical frequency.

The Nyquist sampling thheorem is satisfied in sampling.Wherein [] represents integral part.

Step 2: sequence S1 is carried out fast Fourier transform (FFT), extract peaks spectrum, and obtain the peaks spectrum of sequence S2 through a step recursion;

The frequency spectrum of S1 $X_1\left(k\right)=\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{x}_1\left(n\right)\&CenterDot;\exp (-\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{2\mathrm{\&pi;}}{N}\mathrm{nk})k=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,N-1$

Following formula can be write as

Wherein: first is positive frequency composition frequency spectrum; Second is negative frequency composition frequency spectrum; Third part

Be the DC component frequency spectrum.Preceding two frequency spectrums that constitute alternating component jointly.

The alternating component peaks spectrum is

The peaks spectrum X of S2 ₂(q) do not need sequence S2 is done the FFT computing, only need utilize the peaks spectrum X of S1 ₁(q) obtain through a step recursion

$X_2\left(q\right)=\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{2\mathrm{\&pi;}}{N}q\right)[X_1\left(q\right)-x_1\left(0\right)+x_2(N-1)]---\left(3\right)$

S1 peak values of ac frequency spectrum with positive and negative frequency content frequency spectrum designation is

X ₁(q)＝A ₁·exp(jα ₁)+A ₂·exp(-jβ ₁) (4)

Wherein positive and negative frequency content spectral magnitude is respectively A ₁, A ₂, corresponding phase place is α respectively ₁, (β ₁).

S2 peak values of ac frequency spectrum with positive and negative frequency content frequency spectrum designation is

X ₂(q)＝A ₁·exp(jα ₂)+A ₂·exp(-jβ ₂) (5)

Wherein positive and negative frequency content spectral magnitude is respectively A ₁, A ₂, corresponding phase angle is α respectively ₂, (β ₂).

$A_1=\frac{A}{2}\&CenterDot;\frac{\sin \mathrm{\&pi;\&theta;}}{\sin \frac{\mathrm{\&pi;\&theta;}}{N}}$

${\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">A</figure-callout>}}_2=\frac{A}{2}\&CenterDot;\frac{\sin \mathrm{\&pi;\&theta;}}{\sin \frac{\mathrm{\&pi;}(2q+\mathrm{\&theta;})}{N}}$

α ₂＝α ₁+ω

β ₂＝β ₁+ω

By on can get $\mathrm{\&lambda;}={\mathrm{\&beta;}}_1-{\mathrm{\&alpha;}}_1={\mathrm{\&beta;}}_2-{\mathrm{\&alpha;}}_2=2\mathrm{\&pi;}\frac{N-1}{N}q$

When the spectrum peak subscript was determined, λ was a known quantity.

Step 3: the amplitude of integrated use S1 and S2 peaks spectrum and the coupled relation between the phase angle accurately calculate the frequency of measured signal;

Not independent of each other between the amplitude of S1 and the positive and negative frequency content of S2 and the phase angle, but have coupled relation.

Order $\mathrm{\&lambda;}={\mathrm{\&beta;}}_1-{\mathrm{\&alpha;}}_1=2\mathrm{\&pi;}\frac{N-1}{N}q,$ $k_A=\frac{A_1}{{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}$

Then S1 is had

${\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="HSA00000340633400031.png,HSA00000340633400041.png"\; state="\{\{state\}\}">U</figure-callout>}}_1=\frac{\mathrm{Re}[X_1\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}{\mathrm{Im}[X_1\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}=\frac{A_1-A_2}{{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="HSA00000340633400031.png,HSA00000340633400041.png"\; state="\{\{state\}\}">A</figure-callout>}}_1+A_2}\&CenterDot;\mathrm{tg}\frac{{\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1}{2}---\left(6\right)$

$\mathrm{tg}\frac{{\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1}{2}=U_1\frac{{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="HSA00000340633400031.png,HSA00000340633400041.png"\; state="\{\{state\}\}">A</figure-callout>}}_1+A_2}{A_1-{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}=U_1\frac{k_A+1}{k_A-1}---\left(7\right)$

In like manner, S2 is had

${\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">U</figure-callout>}}_2=\frac{\mathrm{Re}[X_2\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}{\mathrm{Im}[X_2\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}=\frac{A_1-A_2}{{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="HSA00000340633400031.png,HSA00000340633400041.png"\; state="\{\{state\}\}">A</figure-callout>}}_1+A_2}\&CenterDot;\mathrm{tg}\frac{{\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2}{2}---\left(8\right)$

$\mathrm{tg}\frac{{\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2}{2}=U_2\frac{{\mathrm{<figure-callout\; id="1"\; label="A"\; filenames="HSA00000340633400031.png,HSA00000340633400041.png"\; state="\{\{state\}\}">A</figure-callout>}}_1+A_2}{A_1-{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}=U_2\frac{k_A+1}{k_A-1}---\left(9\right)$

Obtained by formula (7) and formula (9)

$\mathrm{tg\&omega;}=\mathrm{tg}(\frac{{\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2}{2}-\frac{{\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1}{2})=\frac{(k_A^2-1)(U_2-U_1)}{{(k_A-1)}^2+U_1{U}_2{(k_A+1)}^2}---\left(10\right)$

$(U_2-U_1)\mathrm{ctg\&omega;}=\frac{k_A-1}{k_A+1}+\frac{k_A+1}{k_A-1}{U}_1{U}_2---\left(11\right)$

By

$k_A=\frac{A_1}{{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">A</figure-callout>}}_2}=\frac{\sin \frac{\mathrm{\&pi;}(2q+\mathrm{\&theta;})}{N}}{\sin \frac{\mathrm{\&pi;\&theta;}}{N}}---\left(12\right)$

Can be with formula (11) arrangement

$(U_2-U_1)\frac{1-{\mathrm{tg}}^2\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}{2\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}=\frac{\mathrm{tg}\frac{\mathrm{q\&pi;}}{N}}{\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}+\frac{\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}{\mathrm{tg}\frac{\mathrm{q\&pi;}}{N}}{U}_1{U}_2---\left(13\right)$

Order $p=\mathrm{tg}\frac{\mathrm{q\&pi;}}{N},y=\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}$

Formula (13) is

$(U_2-U_1)\frac{1-{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">y</figure-callout>}}^2}{2y}=\frac{p}{y}+\frac{y}{p}{U}_1{U}_2---\left(14\right)$

Can get

$y=\sqrt{\frac{p(U_2-U_1-2p)}{2U_1{\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">U</figure-callout>}}_2+p(U_2-U_1)}}---\left(15\right)$

The normalization numerical frequency of spectrum peak place Frequency point correspondence

Sinusoidal signal normalization numerical frequency measured value

The accurate measured value of sinusoidal signal frequency

The accurate measured value of θ then

Step 4: accurately calculate the phase place of measured signal according to the peaks spectrum phase angle information of S1 and S2 with the geometric vector method, as shown in Figure 3;

Note δ ₁=Arg[X ₁(q)], δ ₂=Arg[X ₂(q)], then

$\mathrm{tg}({\mathrm{\&delta;}}_1+{\mathrm{\&beta;}}_1)=\frac{A_1\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}{A_2+A_1\cos ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}=\frac{k_A\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}{1+k_A\cos ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}---\left(19\right)$

$\mathrm{tg}({\mathrm{\&delta;}}_2+{\mathrm{\&beta;}}_2)=\frac{A_1\sin ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}{A_2+A_1\cos ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}=\frac{k_A\sin ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}{1+k_A\cos ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}---\left(20\right)$

By formula (19) and formula (20), cancellation k _A

sin(2α ₁+λ+ω)sin(δ ₂-δ ₁)＝sin(δ ₁+δ ₂+λ)sinω (21)

Namely

Following formula has disclosed the phase angle of S1 and S2 AC compounent spectrum peak and the frequency of signal, the relation of phase place.

The accurate measured value of sinusoidal signal phase place

Step 5: use the geometric vector method, obtain the accurate measured value of measured signal amplitude by solution about the linear equation in two unknowns group of positive and negative frequency spectrum amplitude;

By (4) Shi Kede

$\left\{\begin{array}{c}\mathrm{Re}\left[X_1\left(q\right)\right]=\left|\right|X_1\left(q\right)\left|\right|\cos {\mathrm{\&delta;}}_1=A_1\cos \mathrm{\&alpha;}+A_2\cos \mathrm{\&beta;}\\ \mathrm{Im}\left[X_1\left(q\right)\right]=\left|\right|X_1\left(q\right)\left|\right|\sin {\mathrm{\&delta;}}_1=A_1\sin \mathrm{\&alpha;}-{\mathrm{<figure-callout\; id="2"\; label="A"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">A</figure-callout>}}_2\sin \mathrm{\&beta;}\end{array}\right.---\left(24\right)$

Solve

$\left\{\begin{array}{c}{A}_1=\left|\right|X_1\left(q\right)\left|\right|\frac{\sin ({\mathrm{\&beta;}}_1+{\mathrm{\&delta;}}_1)}{\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}\\ A_2=\left|\right|X_1\left(q\right)\left|\right|\frac{\sin ({\mathrm{\&alpha;}}_1-{\mathrm{\&delta;}}_1)}{\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}\end{array}\right.---\left(25\right)$

The phase angle accurate Calculation value of positive and negative frequency content is in the peaks spectrum

By

Can get

The accurate measured value of AC compounent amplitude

$\hat{A}=2\frac{\sin ({\widehat{\mathrm{\&beta;}}}_1+{\mathrm{\&delta;}}_1)\sin \frac{\widehat{\mathrm{\&omega;}}-{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">q</figure-callout>}}}{2}}{\sin ({\widehat{\mathrm{\&alpha;}}}_1+{\mathrm{\&beta;}}_1)\sin \frac{N(\widehat{\mathrm{\&omega;}}-{\mathrm{\&omega;}}_q)}{2}}\left|\right|X_1\left(q\right)\left|\right|---\left(26\right)$

Step 6: remove spectrum leakage to the influence of DC component frequency spectrum, obtain the accurate measured value of DC component;

Considered by (1) formula

The accurate measured value of DC component

(3) advantage and effect:

A kind of method for measuring sine signal with high accuracy and device of eliminating spectrum leakage of the present invention has following advantage:

1) measuring method can be measured whole four parameters of amplitude, frequency, phase place and DC component of sinusoidal signal under the condition of the complete the unknown of measured signal.

2) measuring method precision height, antijamming capability is strong.

3) the measuring method calculated amount is little.Measuring method does not have iterative process, only need carry out FFT and a spot of algebraic operation.Wherein topmost calculated amount is the FFT computing, and ripe fast algorithm has been arranged, and can realize dynamic real-time measurement through this Measurement Algorithm of practice test.

4) measuring condition is loose, and measuring method is good to signal parameter adaptability.

5) will reach identical frequency resolution, the data volume of the required sampling of this method is few.

All be f in sample frequency _S, the frequency resolution of algorithm all is f _SUnder/128 the condition, the data volume of existing method and the required sampling of this method is compared.In the United States Patent (USP) database, the patent No. is US6965068B2, and name is called the patent of " input signal spectrum estimate system and method " (SYSTEM AND METHOD FOR ESTIMATING TONES IN AN INPUT SIGNAL) need be to signal sampling 384 points.Application number is that 200910089315.7 the disclosed method of Chinese patent " method for measuring sine signal with high accuracy " need be to signal sampling 256 points.The patent No. is that the disclosed method of Chinese patent " sinusoidal signal four parameters testing method and virtual instrument signal supervisory instrument " of ZL200810101338.0 need be to signal sampling 255 points.The disclosed method of the Chinese patent of application number 201010162687.0 " high-efficiency measurement method of the sinusoidal signal frequency under owing to sample and device " need be to data 255 points of sampling.And this method only need be to signal sampling 129 points.

6) the measurement mechanism cost is low, the cost performance height.

7) the anti-external noise interference performance of measurement mechanism is strong.

8) measurement mechanism automaticity height can be according to the automatic conditioning signal Amplifier Gain coefficient of the size of measured signal, and good to input signal amplitude adaptability, quantization error is little.

The amplifier gain of traditional measurement device can not be regulated automatically, may exceed measurement range when signal amplitude is big, shown in Fig. 4 a and Fig. 4 b; When the signal amplitude signal quantization error that hour sampling obtains is bigger, distorted signals is serious, shown in Fig. 5 a and Fig. 5 b.And adopt this measurement device can be according to the automatic conditioning signal Amplifier Gain coefficient of the size of measured signal, good to input signal amplitude adaptability, quantization error is little.

Four, description of drawings

Fig. 1 is that the hardware of measurement mechanism is formed structural drawing

Fig. 2 is the Measurement Algorithm structural representation of measurement mechanism

Fig. 3 is each composition phase plane synoptic diagram of peaks spectrum

Fig. 4 a is the traditional measurement device to signals sampling oscillogram significantly

Fig. 4 b is apparatus of the present invention to signals sampling oscillogram significantly

Fig. 5 a is that the traditional measurement device is to the sample waveform figure of small amplitude signal

Fig. 5 b is that apparatus of the present invention are to the sample waveform figure of small amplitude signal

Fig. 6 a is continuous frequency spectrum under the integer-period sampled condition

Fig. 6 b is continuous frequency spectrum under the non-integer-period sampled condition

Fig. 6 c is discrete spectrum under the integer-period sampled condition

Fig. 6 d is discrete spectrum under the non-integer-period sampled condition

Symbol description is as follows among the figure:

The FFT fast fourier transform;

The Re real;

The imaginary part of Im plural number;

The spectrum peak vector

The phase angle of δ spectrum peak vector

Positive frequency composition vector in the spectrum peak

The phase angle of positive frequency composition vector in the α spectrum peak

Negative frequency composition vector in the spectrum peak

The opposite number of negative frequency composition vector phase angle in the β spectrum peak

Five, embodiment

The present invention is further illustrated below in conjunction with accompanying drawing and listed examples.

(1) sees Fig. 1, a kind of sine signal with high accuracy measurement mechanism of eliminating spectrum leakage of the present invention, it is made up of simulating signal pretreatment unit, data acquisition unit, digital signal processing unit, measurement result display unit, communications interface unit, as shown in Figure 1.Position annexation between them is: connect by the multiple signals line between simulating signal pretreatment unit and the data acquisition unit, digital signal processing unit is connected with data acquisition unit, measurement result display unit, communications interface unit respectively by spi bus.

Described simulating signal pretreatment unit is made up of buffer circuit and filtering circuit, is connected by the multiple signals line between its buffer circuit and the filtering circuit; This buffer circuit adopts accurate isolated amplifier ISO124, and it realizes the isolation of input stage and output stage signal by the isolation capacitance that is encapsulated in device inside; This filtering circuit adopts continuous time filter MAX274, the high frequency noise that its function produces when being filtering process buffer circuit; This unit mainly isolates and low-pass filtering simulating signal.

Described data acquisition unit is made up of multi-channel analog selector switch, programmable gain amplifier, Programmable Logic Controller CPLD and AD converter.The multiple signals line that multi-channel analog selector switch input termination filtering circuit comes out, output terminal is one road signal wire, is connected to the programmable gain amplifier input end, the programmable gain amplifier output terminal is connected to AD converter by one road signal wire.This multi-channel analog selector switch adopts DG201, is used for the signal that will sample from the multichannel analog signals selection; This programmable gain amplifier adopts AD256 to realize, it can change the Amplifier Gain coefficient according to the control code that Programmable Logic Controller is sent here; This Programmable Logic Controller CPLD is used for finishing the function of controlling of sampling and adjusting, controls the gain coefficient of programmable gain amplifier according to the range of each input; This AD converter adopts AD677, is used for pretreated simulating signal is converted into digital signal.

Described digital signal processing unit adopts dsp chip TMS320C5402, and signalization process of measurement in the built-in RAM of DSP mainly comprises FFT computing and a spot of algebraic operation, is used for realizing signal measurement algorithm proposed by the invention.

Described measurement result display unit, adopting model is LCD demonstration signal waveform and the measurement result of LCM320240.

Described communications interface unit adopts asynchronous communication interface chip 16C550, and measurement mechanism is communicated by RS232 bus interface and PC, realizes the debugging to measurement mechanism, also signal measurement data and result can be delivered to PC from measurement mechanism.

(2) see Fig. 2, a kind of method for measuring sine signal with high accuracy of eliminating spectrum leakage of the present invention, these method concrete steps are as follows:

Step 1: continuous sampling to measured signal, obtain the digital sequence S1 of N point, and the Serial No. S2 after it is postponed a sampling period;

N is sampling number, and f is sinusoidal signal frequency, and T is the sampling period, Be sample frequency.

Be the normalization numerical frequency.

The Nyquist sampling thheorem is satisfied in sampling.Wherein [] represents integral part.

Step 2: sequence S1 is carried out fast Fourier transform (FFT), extract peaks spectrum, and obtain the peaks spectrum of sequence S2 through a step recursion;

The frequency spectrum of S1 is X ₁(k), k=0,1 ... N-1.Its peaks spectrum is X ₁(q), q is the subscript of peaks spectrum correspondence.

The peaks spectrum of S2 $X_2\left(q\right)=\exp \left(\mathrm{<figure-callout\; id="2"\; label="j"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">j</figure-callout>}\frac{2\mathrm{\&pi;}}{N}q\right)[X_1\left(q\right)-x_1\left(0\right)+x_2(N-1)]$

Step 3: the frequency that the amplitude of integrated use S1 and S2 peaks spectrum and phase angle information accurately calculate measured signal;

Order $\mathrm{\&lambda;}=2\mathrm{\&pi;}\frac{N-1}{N}q$ $p=\mathrm{tg}\frac{\mathrm{q\&pi;}}{N}$

${\mathrm{<figure-callout\; id="1"\; label="U"\; filenames="HSA00000340633400031.png,HSA00000340633400041.png"\; state="\{\{state\}\}">U</figure-callout>}}_1=\frac{\mathrm{Re}[X_1\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}{\mathrm{Im}[X_1\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}$ ${\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">U</figure-callout>}}_2=\frac{\mathrm{Re}[X_2\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}{\mathrm{Im}[X_2\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}$

$y=\sqrt{\frac{p(U_2-U_1-2p)}{2U_1{\mathrm{<figure-callout\; id="2"\; label="U"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">U</figure-callout>}}_2+p(U_2-U_1)}}$

The normalization numerical frequency of spectrum peak place Frequency point correspondence

Sinusoidal signal normalization numerical frequency measured value

The accurate measured value of sinusoidal signal frequency

Step 4: accurately calculate the phase place of measured signal according to the peaks spectrum phase angle information of S1 and S2 with the geometric vector method, as Fig. 3;

X ₁(k) phase angle δ ₁=Arg[X ₁(q)]

X ₂(k) phase angle δ ₂=Arg[X ₂(q)]

The accurate measured value of non-integer-period sampled coefficient

The accurate measured value of sinusoidal signal phase place

Step 5: use the geometric vector method, obtain the accurate measured value of measured signal amplitude by solution about the linear equation in two unknowns group of positive and negative frequency spectrum amplitude;

The phase angle of positive and negative frequency content is respectively α in the peaks spectrum ₁, (β ₁)

Its calculated value is

The accurate measured value of AC compounent amplitude

$\hat{A}=2\frac{\sin ({\widehat{\mathrm{\&beta;}}}_1+{\mathrm{\&delta;}}_1)\sin \frac{\widehat{\mathrm{\&omega;}}-{\mathrm{\&omega;}}_{\mathrm{<figure-callout\; id="2"\; label="q"\; filenames="HSA00000340633400042.png,HSA00000340633400051.png"\; state="\{\{state\}\}">q</figure-callout>}}}{2}}{\sin ({\widehat{\mathrm{\&alpha;}}}_1+{\mathrm{\&beta;}}_1)\sin \frac{N(\widehat{\mathrm{\&omega;}}-{\mathrm{\&omega;}}_q)}{2}}\left|\right|X_1\left(q\right)\left|\right|$

Step 6: remove spectrum leakage to the influence of DC component frequency spectrum, obtain the accurate measured value of DC component;

(3) measuring method of the present invention and existing method measurement effect are relatively

Under the condition of not considering noise and AD quantization error, sample frequency is 500Hz, sampling number is 512 points, signal x (n)=5cos (50.3t+160 °)+1 is detected, be that the measurement result of the disclosed method of Chinese patent " sinusoidal signal four parameters testing method and virtual instrument signal supervisory instrument " of ZL200810101338.0 compares with the inventive method with the patent No., the said method measuring accuracy is 10 ^-4Magnitude, this method measuring accuracy are 10 ^-14Magnitude is higher than said method far away, illustrates that this method is a kind of accurate measurement method.Constant when other condition, when sampling number was reduced at 32, the said method measuring accuracy was reduced to 10 ^-2Magnitude, and this method measuring accuracy is 10 ^-13Magnitude illustrates that this method measuring accuracy can not reduce with the minimizing of sampling number.If consider that noise and AD quantization error and other condition are constant, sampling number is 512 points, and signal to noise ratio (S/N ratio) is 20dB, under 12 AD conversion accuracy conditions above-mentioned signal is detected, and said method amplitude measurement precision is 0.03, and this method is 0.005; The said method frequency measurement accuracy is 0.0004Hz, and this method is 0.0002Hz; 0.1 ° of said method phase measurement accuracy, this method are 0.08 °; Said method DC component measuring accuracy is 0.003, and this method is 0.0005.Explanation is in the actual measurement process, and the antijamming capability of this method is better than said method

Table 1 is the measurement result that has method and this method under the different measuring condition

Table 1

Under the condition of not considering noise and AD quantization error, sample frequency is 1000Hz, sampling number from 8 to 128, signal x (n)=10cos (44.75t+30 °)+2 is detected, and is that the measurement result of 200910089315.7 the disclosed method of Chinese patent " method for measuring sine signal with high accuracy " compares with this method and application number.When sampling number more than or equal to 32 the time, said method is suitable with this method precision; But when sampling number less than 32 the time, said method occurs unusual, can not measure signal parameter, and this method still can be carried out high-acruracy survey to signal.

Table 2a is the amplitude measurement result who has method and this method under the different sampling number conditions

Table 2b is the frequency measurement that has method and this method under the different sampling number conditions

Table 2c is the phase measurement that has method and this method under the different sampling number conditions

Table 2a

Table 2b

Table 2c

Claims (1)

1. method for measuring sine signal with high accuracy of eliminating spectrum leakage, it is characterized in that: these method concrete steps are as follows:

Step 1: the continuous sampling of offset of sinusoidal signal, obtain the digital sequence S1 of N point, and the Serial No. S2 after it is postponed a sampling period; S1:

N=0,1 ..., N-1S2:

N=0,1 ..., N-1N is sampling number, and A is the amplitude of sinusoidal signal, and D is DC component, and f is sinusoidal signal frequency, T is the sampling period,

Be sample frequency; $\mathrm{\&omega;}=2\mathrm{\&pi;}\frac{f}{f_S}=2\mathrm{\&pi;}\frac{q+\mathrm{\&theta;}}{N}$ Be the normalization numerical frequency; $q=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;,\left[\frac{N}{2}\right],$ $-\frac{1}{2}<\mathrm{\&theta;}\&le;\frac{1}{2},$ The Nyquist sampling thheorem is satisfied in sampling; Wherein [] represents integral part;

Step 2: it is FFT that sequence S1 is carried out fast fourier transform, extracts peaks spectrum, and obtains the peaks spectrum of sequence S2 through a step recursion;

The frequency spectrum of S1 $X_1\left(k\right)=\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{x}_1\left(n\right)\&CenterDot;\exp (-j\frac{2\mathrm{\&pi;}}{N}\mathrm{nk})$ K=0,1 ..., the N-1 following formula is write as

(1)

Wherein: first is positive frequency composition frequency spectrum; Second is negative frequency composition frequency spectrum; The 3rd $X_D\left(k\right)=\left\{\begin{array}{cc}\mathrm{DN},& k=0\\ 0,& k=1,\&CenterDot;\&CenterDot;\&CenterDot;,N-1\end{array}\right.,$ Be the DC component frequency spectrum, preceding two frequency spectrums that constitute alternating component jointly; The alternating component peaks spectrum is

The peaks spectrum X of S2 ₂(q) do not need sequence S2 is done the FFT computing, only need utilize the peaks spectrum X of S1 ₁(q) obtain through a step recursion

$X_2\left(q\right)=\exp \left(j\frac{2\mathrm{\&pi;}}{N}q\right)[X_1\left(q\right)-x_1\left(0\right)+x_2(N-1)]---\left(3\right)$

S1 peak values of ac frequency spectrum with positive and negative frequency content frequency spectrum designation is

X ₁(q)=A ₁·exp(jα ₁)+A ₂·exp(-jβ ₁) (4)

Wherein positive and negative frequency content spectral magnitude is respectively A ₁, A ₂, corresponding phase place is α respectively ₁, (β ₁);

S2 peak values of ac frequency spectrum with positive and negative frequency content frequency spectrum designation is

X ₂(q)=A ₁·exp(jα ₂)+A ₂·exp(-jβ ₂) (5)

Wherein positive and negative frequency content spectral magnitude is respectively A ₁, A ₂, corresponding phase angle is α respectively ₂, (β ₂);

$A_1=\frac{A}{2}\&CenterDot;\frac{\sin \mathrm{\&pi;\&theta;}}{\sin \frac{\mathrm{\&pi;\&theta;}}{N}}$

$A_2=\frac{A}{2}\&CenterDot;\frac{\sin \mathrm{\&pi;\&theta;}}{\sin \frac{\mathrm{\&pi;}(2q+\mathrm{\&theta;})}{N}}$

α ₂=α ₁+ω

β ₂=β ₁+ω

By on draw $\mathrm{\&lambda;}={\mathrm{\&beta;}}_1-{\mathrm{\&alpha;}}_1={\mathrm{\&beta;}}_2-{\mathrm{\&alpha;}}_2=2\mathrm{\&pi;}\frac{N-1}{N}q$

When the spectrum peak subscript was determined, λ was a known quantity;

Step 3: the amplitude of integrated use S1 and S2 peaks spectrum and the coupled relation between the phase angle accurately calculate the frequency of measured signal;

Not independent of each other between the amplitude of S1 and the positive and negative frequency content of S2 and the phase angle, but have coupled relation;

Order $\mathrm{\&lambda;}={\mathrm{\&beta;}}_1-{\mathrm{\&alpha;}}_1=2\mathrm{\&pi;}\frac{N-1}{N}q,$ $k_A=\frac{A_1}{A_2}$

Then S1 is had

$U_1=\frac{\mathrm{Re}[X_1\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}{\mathrm{Im}[X_1\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}=\frac{A_1-A_2}{A_1+A_2}\&CenterDot;\mathrm{tg}\frac{{\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1}{2}---\left(6\right)$

$\mathrm{tg}\frac{{\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1}{2}=U_1\frac{A_1+A_2}{A_1-A_2}=U_1\frac{k_A+1}{k_A-1}---\left(7\right)$

In like manner, S2 is had

$U_2=\frac{\mathrm{Re}[X_2\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}{\mathrm{Im}[X_2\left(q\right)\&CenterDot;\exp \left(j\frac{\mathrm{\&lambda;}}{2}\right)]}=\frac{A_1-A_2}{A_1+A_2}\&CenterDot;\mathrm{tg}\frac{{\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2}{2}---\left(8\right)$

$\mathrm{tg}\frac{{\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2}{2}=U_2\frac{A_1+A_2}{A_1-A_2}=U_2\frac{k_A+1}{k_A-1}---\left(9\right)$

Obtained by formula (7) and formula (9)

$\mathrm{tg\&omega;}=\mathrm{tg}(\frac{{\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2}{2}-\frac{{\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1}{2})=\frac{(k_A^2-1)(U_2-U_1)}{{(k_A-1)}^2+U_1{U}_2{\left(k_{A+1}\right)}^2}---\left(10\right)$

$(U_2-U_1)\mathrm{ctg\&omega;}=\frac{k_A-1}{k_A+1}+\frac{k_A+1}{k_A-1}{U}_1{U}_2---\left(11\right)$

By

$k_A=\frac{A_1}{A_2}=\frac{\sin \frac{\mathrm{\&pi;}(2q+\mathrm{\&theta;})}{N}}{\sin \frac{\mathrm{\&pi;}\mathrm{\&theta;}}{N}}---\left(12\right)$

With formula (11) arrangement be

$(U_2-U_1)\frac{1-t{g}^2\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}{2\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}=\frac{\mathrm{tg}\frac{\mathrm{q\&pi;}}{N}}{\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}+\frac{\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}}{\mathrm{tg}\frac{\mathrm{q\&pi;}}{N}}{U}_1{U}_2---\left(13\right)$

Order $p=\mathrm{tg}\frac{\mathrm{q\&pi;}}{N},$ $y=\mathrm{tg}\frac{q+\mathrm{\&theta;}}{N}\mathrm{\&pi;}$

Formula (13) is

$(U_2-U_1)\frac{1-y^2}{2y}=\frac{p}{y}+\frac{y}{p}{U}_1{U}_2---\left(14\right)$

Obtain

$y=\sqrt{\frac{p(U_2-U_1-2p)}{2U_1{U}_2+p(U_2-U_1)}}---\left(15\right)$

The normalization numerical frequency of spectrum peak place Frequency point correspondence ${\mathrm{\&omega;}}_q=\frac{2\mathrm{\&pi;}}{N}q---\left(16\right)$

Sinusoidal signal normalization numerical frequency measured value $\widehat{\mathrm{\&omega;}}=2\mathrm{arctgy}---\left(17\right)$

The accurate measured value of sinusoidal signal frequency $\hat{f}=\frac{\widehat{\mathrm{\&omega;}}}{2\mathrm{\&pi;}}{f}_S$

The accurate measured value of θ then $\widehat{\mathrm{\&theta;}}=\frac{N}{2\mathrm{\&pi;}}(\widehat{\mathrm{\&omega;}}-{\mathrm{\&omega;}}_q)---\left(18\right)$

Step 4: the phase place that accurately calculates measured signal according to the peaks spectrum phase angle information of S1 and S2 with the geometric vector method;

Note δ ₁=Arg[X ₁(q)], δ ₂=Arg[X ₂(q)], then

$\mathrm{tg}({\mathrm{\&delta;}}_1+{\mathrm{\&beta;}}_1)=\frac{A_1\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}{A_2+A_1\cos ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}=\frac{k_A\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}{1+k_A\cos ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}---\left(19\right)$

$\mathrm{tg}({\mathrm{\&delta;}}_2+{\mathrm{\&beta;}}_2)=\frac{A_1\sin ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}{A_2+A_1\cos ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}=\frac{k_A\sin ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}{1+k_A\cos ({\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_2)}---\left(20\right)$

By formula (19) and formula (20), cancellation k _A

sin(2α ₁+λ+ω)sin(δ ₂-δ ₁)=sin(δ ₁+δ ₂+λ)sinω (21)

Namely

Following formula has disclosed the phase angle of S1 and S2 AC compounent spectrum peak and the frequency of signal, the relation of phase place;

The accurate measured value of sinusoidal signal phase place

Step 5: use the geometric vector method, obtain the accurate measured value of measured signal amplitude by solution about the linear equation in two unknowns group of positive and negative frequency spectrum amplitude;

Drawn by (4) formula

$\left\{\begin{array}{c}\mathrm{Re}\left[X_1\left(q\right)\right]=\left|\right|X_1\left(q\right)\left|\right|\cos {\mathrm{\&delta;}}_1=A_1\cos \mathrm{\&alpha;}+A_2\cos \mathrm{\&beta;}\\ \mathrm{Im}\left[X_1\left(q\right)\right]=\left|\right|X_1\left(q\right)\left|\right|\sin {\mathrm{\&delta;}}_1=A_1\sin \mathrm{\&alpha;}-A_2\sin \mathrm{\&beta;}\end{array}\right.---\left(24\right)$

Solve

$\left\{\begin{array}{c}{A}_1=\left|\right|X_1\left(q\right)\left|\right|\frac{\sin ({\mathrm{\&beta;}}_1+{\mathrm{\&delta;}}_1)}{\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}\\ A_2=\left|\right|X_1\left(q\right)\left|\right|\frac{\sin ({\mathrm{\&alpha;}}_1-{\mathrm{\&delta;}}_1)}{\sin ({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)}\end{array}\right.---\left(25\right)$

The phase angle accurate Calculation value of positive and negative frequency content is in the peaks spectrum

By $A_1=\frac{A}{2}\&CenterDot;\frac{\sin \mathrm{\&pi;\&theta;}}{\sin \frac{\mathrm{\&pi;\&theta;}}{N}}$ Obtain

The accurate measured value of AC compounent amplitude

$\hat{A}=2\frac{\sin ({\widehat{\mathrm{\&beta;}}}_1+{\mathrm{\&delta;}}_1)\sin \frac{\widehat{\mathrm{\&omega;}}-{\mathrm{\&omega;}}_q}{2}}{\sin ({\widehat{\mathrm{\&alpha;}}}_1+{\mathrm{\&beta;}}_1)\sin \frac{N(\widehat{\mathrm{\&omega;}}-{\mathrm{\&omega;}}_q)}{2}}\left|\right|X_1\left(q\right)\left|\right|---\left(26\right)$

Step 6: remove spectrum leakage to the influence of DC component frequency spectrum, obtain the accurate measured value of DC component;

Considered by (1) formula

The accurate measured value of DC component

Images