CN1206510C - Method for utilizing frequency domain method to make defference measurement of accurate reconfiguration - Google Patents

Abstract

The present invention relates to a method for utilizing a frequency domain method to make difference measurement of accurate reconfiguration, which is characterized in that two groups of difference values are in continuation periodically by the present invention according to two different separation distances or difference values obtained by shearing measurement of a measured object; two initial evaluating curves are obtained by the fourier positive and negative transition; then, the ratio difference value of the zeroing errors and the separation distance or the shearing quantity of a corresponding sensor is calculated; the two initial evaluating curves are integrated to eliminate the influence of the zeroing errors; therefore, the evaluating curve of the measured parameter is obtained; then, the linear trend of the evaluating curves is removed; finally, a precise evaluating curve without theoretical errors is obtained; therefore, the precise reconfiguration of the measured parameter is realized. Measured parameters under general conditions can be precisely reconstructed by the present invention without any 'prior ' knowledge. The present invention is suitable for the conditions that the sampling interval is not equal to the separation distance or the shearing quantity of a sensor; the size of a measured piece is not equal to the integer multiple of the separation distance or the shearing quantity of the sensor; the measured parameters are periodic, non-periodic, smooth and rough, etc.

Description

Utilize frequency domain method to carry out the precise reconstruction method of difference measurement

Technical field:

The present invention relates to detect applied technical field, be specifically related to a kind of differential measurement values and come the detection method of the former measured parameter of reconstruct, comprise linearity, flatness, cylindricity, the detection of free form surface face shape and the wavefront reconstruction of lateral shearing interference of precision, ultraprecise surface of the work according to measured parameter.

Background technology:

In ultraprecise processing, for linearity, cylindricity, flatness and the surface figure accuracy that detects the work piece surface, usually adopt many gauge heads scanning method, this method is installed in displacement gauge head (commonly used non-contact sensor such as current vortex or electric capacity etc.) or angle detector (as autocollimator etc.) on the slide carriage, tested profile moving linearly along workpiece, perhaps sensor is fixed, workpiece moves with slide carriage.When the workpiece profile error of measuring when needs and the rectilinear motion error of slide carriage are in same class, expect accurate workpiece profile parameter such as linearity, cylindricity, flatness and surface figure accuracy etc., just need to adopt the way of error separating.On-position measure such as workpiece is exactly directly to utilize machine slide to make the measuring method of scanning motion, and must separate the surface profile error of workpiece and the kinematic error of slide carriage this moment.The lateral shearing interference method is basic identical with many gauge head scanning ratio juris, all is based on the measurement of difference value, and the accurate reconstruct problem that has therefore solved based on difference has just solved many gauge heads scanning method and the problems of measurement of walking crosswise shearing interference method.

Surface figure accuracy reconstruct problem when detecting for linearity, cylindricity, flatness and employing displacement transducer, its the most basic problem is exactly the reconstruct problem that solves linearity, and therefore the problems of measurement based on difference value just can be summarised as linearity simply and walk crosswise two aspects of shear interference measurement.

In order to detect the surface of the work profile accurately, people have begun the research of this respect very early.Such as measurement, just carry out the research of this respect since the beginning of the eighties, and until also constantly proposing the whole bag of tricks now in the hope of reconstruct surface of the work linearity profile accurately to the workpiece linearity.For the verticality measuring method of scanning feeler method, the difference according to adopting the sensor number is divided into one-point method and multipoint method (comprising two-point method, three point method even four-point method etc.).When measuring with one-point method, want error is separated, usually need workpiece is overturn or translation is done just can carry out behind twice scanning survey.The degree of consistency of the relative workpiece of sensor installation site is all influential to final evaluation result in the repeatable accuracy of slide carriage motion and twice measurement like this, and this influence can't be eliminated fully.So in ultra precise measurement, this method operability is bad.Different with inversion method or translation method, multipoint method uses more than one gauge head, carries out error separating, also can not carry out duplicate measurements to workpiece even therefore slide carriage motion repeatability is high.

Two-point method uses two gauge heads, is the simplest multipoint method.Respectively at the profile height of two spaced points detection measured surfaces, the difference output of adopting two gauge heads is with the translational error of eliminating the scanning slide carriage and the difference value that provides the profile height on measured workpiece surface with two gauge heads for this method.Three point method is used three displacement transducers, and a sensor that has more generally is used for eliminating the influence of slide carriage Run-out error.But, owing to can there be the zero error that is difficult to eliminate (being sensor moved and detected the surface of the work time institute perception amount of a desirable no straightness error on the guide rail of desirable no kinematic error difference) in the sensor actual installation, thereby can make difference value produce skew, special when workpiece size is longer, this zero error just becomes three point method and measures maximum error source.Adopt the differential laser autocollimator method of two angle gauge heads and the mixing method that adopts two displacement transducers and an angular transducer, all have same problem.

According to the difference of error separating method, the scanning feeler method can be divided into time domain method and frequency domain method again.Frequency domain method promptly is based on the measuring method of discrete Fourier transform (DFT), because straightness error does not have periodically, so frequency domain method need take certain measure to remove to construct periodic straightness error, the problem that existing frequency domain two-point method and three point method all exist harmonic wave to suppress, existing people has proposed frequency domain optimization error component method (OEST), the workpiece linearity is made rotation peace to be removed and approaches periodically, try to achieve these two rotations and translation parameters by optimal algorithm again, thereby try to achieve suitable linearity to approach the value that obtains by two-point method.The OEST three point method has been made the OEST two-point method some again and has been improved, and has proposed the thought of the fusion of verticality measuring method in conjunction with the advantage of time-domain and frequency-domain method, has enriched the measuring method of linearity.Adopt three sensor fixed installations, choose the right sensors spacing and take certain method aperiodic surface profile can be constructed periodic function and measure, also belong to the frequency domain three point method, but this method has equally been ignored the influence of slide carriage Run-out error with two-point method.

People study in the measuring method to linearity always, in the hope of on enough little sampling interval point, reconstructing the linearity profile of workpiece accurately, but existing these algorithms all can't reach this purpose, promptly can't accurately reconstruct to gear shaper without theoretical errors the linearity of any workpiece on less than the enough little sampling interval each point of transducer spacing.

And for the lateral shearing interference method, because it has simple and stable structure, do not need characteristics such as high precision reference surface, detect so be widely used in stress, strain and vibration analysis in liquids and gases Study on Flow, the Experimental Mechanics and high-precision optical surface face shape, particularly the detection of free form surfaces such as aspheric surface etc.But simultaneously, to the reconstruct based on difference but is a very complicated problems, people have developed multiple computing method over more than 20 year, previous method is based on the reconstruct of consecutive point difference value, up to the reconstructing method that just propose to adapt to big shearing displacement in recent years, make nearest research cause interest widely to lateral shearing interference, but all there is certain limitation in all these methods, suppose that the wavefront that is studied has certain " priori " knowledge such as slickness etc., adopt the little shearing displacement that equals the measurement point interval to measure, even can only obtain separating of approaching.Because all these methods can't accurately reconstruct various surface profiles, make and adopt lateral shear wavefront interference technique also not have remarkable advantages than the measuring method that adopts reference surface.Up to developing a kind of new " natural continuation " (Natural Extension) method in recent years, realized the accurate continuation of difference value, just might be in frequency domain accurate reconstruct measured surface.Thereby injected new vitality again for lateral shear wavefront interference technique, this method does not need any " priori " knowledge just (to comprise the cycle in the various surfaces of restructural, non-periodic, smooth and non-smooth surface etc.), also developed a kind of ESAD (Extended Shear Angle Difference) surface topography pick-up unit on this basis, the part but this method also comes with some shortcomings, have only when the size of workpiece equals this special circumstances of multiple of shearing displacement as it, accurately reconstruct, otherwise need carry out smooth continuation, thereby bring some influences to reconstruction accuracy, and the relation between the sensor zero error when not calculating twice shearing displacement and measuring, thereby this method is being used for displacement sensor such as linearity, cylindricity, also can have bigger error during flatness etc., thereby need further to expand and perfect.

Summary of the invention:

Technical matters to be solved by this invention is the defective that overcomes prior art, on basis based on the measured parameter difference value, problem at existing difference reconfiguration technique exists does not need any " priori " knowledge, just can accurately reconstruct to gear shaper without theoretical errors measured parameter generally speaking.

The present invention solves the problems of the technologies described above by following technical scheme.It is the difference value that records according to two different spacing on the measurand or shearing displacement, two groups of difference values are carried out periodic extension, again by the positive inverse transformation of Fourier, obtain calculating behind two initial evaluation curves the difference of zero error and the ratio of corresponding sensor spacing or shearing displacement, comprehensive these two initial evaluation curves are also eliminated the influence of zero error, obtain the appraisal curve of measured parameter, remove the linear trend in the appraisal curve again, obtain the exact evaluation curve of a gear shaper without theoretical errors at last, thereby realize the accurate reconstruct of measured parameter.The present invention is characterised in that it comprises the steps: that (1) select certain sampling interval Δ and transducer spacing r Δ, v Δ; (2) measure difference value according to two different spacing of measurand or shearing displacement; (3) calculate the difference value of measuring; (4) two groups of difference values are carried out periodic extension; (5) fourier series of difference value after the calculating continuation; (6) ask the fourier series of measured parameter; (7) ask the fourier series of sawtooth wave; (8) selecting frequency component in the frequency of the Fourier analysis that twice measurement tried to achieve calculates the difference of zero error and the ratio of corresponding sensor spacing or shearing displacement then; (9) select weighting coefficient; (10) in conjunction with the measurement result under two kinds of spacer conditions, draw the evaluation result after the weighting: (11) ask the appraisal curve of measured parameter: (12) remove the linear deflection amount of appraisal curve, try to achieve final appraisal curve.

The present invention has following technique effect:

1, applicable to two-point method or three point method the workpiece linearity is estimated, and the various shear interferences measurements that are applicable to variable shearing displacement.Its principle can be used for estimating cylindricity, flatness and surface figure accuracy etc.Do not need to adopt complicated apparatus.

2, can adopt big transducer spacing (or shearing displacement).For the evaluation of linearity etc., because the influence of sensor physical dimension, cause the transducer spacing can not be too little, for the lateral shearing interference method, too small shearing displacement can cause resolution to reduce, and can influence evaluation effect equally.And the method for the present invention of sampling then can adopt big transducer spacing (or shearing displacement).

3,, need not suppose that promptly measured parameter has slickness or periodicity etc., so can be suitable for general measurand without any need for " priori " knowledge.

4, can try to achieve the exact evaluation of measurand but not the resulting approximate or evaluation that approaches of conventional method.

5, owing to accurately calculated of the influence of sensor zero error, therefore when sensor installation, needn't return to zero accurately or zero error accurately be measured evaluation result.And in fact when accurate, ultra precise measurement, in general the accurate zeroing of sensor or the accurate measurement of zero error is time-consuming and rely on more high-precision device all is difficult to accomplish.Therefore the present invention has improved detection efficiency greatly and has reduced cost.

6, carrying out on the basis of accurate reconstruct based on " natural continuation ", calculated the influence of sensor zero error first, thereby this frequency domain precise reconstruction method is generalized to more generally detection occasion as linearity, cylindricity, flatness and surface figure accuracy etc., the control device that does not need stability sensor in the surface topography pick-up unit has been simplified measuring mechanism greatly.

7, being not equal to integral multiple, measured parameter that transducer spacing (or shearing displacement), measured piece size need not equal transducer spacing (or shearing displacement) in sampling interval is under cycle, non-periodic, the smooth and non-situation such as smooth, also can carry out accurate reconstruct, smooth continuation method is improved and improved.

Description of drawings:

Fig. 1 is a measuring process synoptic diagram of the present invention;

Fig. 2 is the process flow diagram of exact evaluation of the present invention.

Embodiment:

The present invention is the difference value that records according to two different spacing (or shearing displacement) on the measurand, carries out in frequency domain that accurate reconstruct realizes.It at first adopts " natural continuation " method that two groups of difference values are carried out accurate periodic extension, again by the positive inverse transformation of Fourier, obtain two initial evaluation curves, but it is the harmonic component in cycle that these two initial evaluation curves lack with transducer spacing (or shearing displacement) separately respectively, and because the influence of zero error, has different side-play amounts respectively, gone out the difference of the ratio of zero error and corresponding sensor spacing (or shearing displacement) then by theoretical analysis and calculation, thereby on the basis of eliminating the zero error influence, one group of harmonic component that appraisal curve lacked is replenished by the corresponding component of another sets of curves, obtains the exact evaluation curve of a gear shaper without theoretical errors at last.And the generalized case when will this accurate restructing algorithm to be generalized to the measured piece size be not the long-pending integral multiple of two kinds of transducer spacings (or shearing displacement), expand and perfect precise reconstruction method of the prior art.

As shown in Figure 1, establish sensor and be respectively s at interval ₁And s ₂, sensor mount 4 moves on guide rail 5, and workpiece 6 surperficial linearitys are Ms (x), and workpiece calibration length is p, and total measure dot number is N on the workpiece, and sampling interval is Δ=p/N, and the zero error of sensor is respectively e ₁, e ₂Because there is zero error in sensor, the relative primary curve of surface of the work linearity curve that the historical facts or anecdotes border is measured has produced skew.If two zero errors equate with sensor ratio at interval, i.e. e ₁/ s ₁=e ₂/ s ₂, then Pian Yi result can not bring influence to the evaluation of linearity, but above-mentioned equation is difficult to set up in the actual measurement, i.e. e ₁/ s ₁≠ e ₂/ s ₂, also be difficult to accurately record e ₁And e ₂, but want reconstruct workpiece linearity exactly, then need to know between the two relation or its difference at least.Do twice measurement or adopt three sensors to make the synoptic diagram of one-shot measurement for two sensors of employing (this moment is without sensor 3) among Fig. 1.In the practice, desirable s ₁=A, and s ₂Can get s as required ₂=B, or s ₂=C, the zero error e of sensor ₁=e _A, e ₂=e _BOr e ₂=e _CNote when adopting three sensors, as get s ₂During=B, the initial detection position of slide carriage is the starting point of sensor 2 at workpiece, and 1 pair of sky of sensor this moment moves s up to slide carriage ₁The time sensor 1 just begin image data.Work as s ₂During=C, then the initial detection position of slide carriage is the starting point 7 of sensor 1 at workpiece, and the final position of slide carriage motion is that sensor 2 moves to workpiece terminating point 8, image data not then when sensor 3 exceeds the tested zone of workpiece.

Because the existence of zero error, the appraisal curve that makes two shearing displacements measure has different linear deflection, and as not eliminating the influence of zero error, then final evaluation result can produce mistake equally.The present invention anatomize two shearing displacements record the data mutual relationship get basic on, the influence of zero error is accurately calculated, concrete computing method are provided below.

If the linearity of workpiece is represented with function f (x).After the discrete sampling, make f (m)=f (x _m), m=0, K, N-1.

Then spacing is s ₁The curve that records should be

$f_1\left(m\right)=f\left(m\right)+\frac{e_1}{s_1}\mathrm{m\; m}=0,K,N-1---\left(1\right)$

Spacing is s ₂The time curve that records should be

$f_2\left(m\right)=f\left(m\right)+\frac{e_2}{s_2}\mathrm{m\; m}=0,k,N-1---\left(2\right)$

Respectively above two formulas are made discrete Fourier transform (DFT):

$\mathrm{DFT}\left[f_1\left(m\right)\right]=\mathrm{DFT}[f\left(m\right)+\frac{e_1}{s_1}m]=\mathrm{DFT}\left[f\left(m\right)\right]+\mathrm{DFT}\left[\frac{e_1}{s_1}m\right]=\mathrm{DFT}\left[f\left(m\right)\right]+\frac{e_1}{s_1}\mathrm{DFT}\left[m\right]---\left(3\right)$

$\mathrm{DFT}\left[f_2\left(m\right)\right]=\mathrm{DFT}[f\left(m\right)+\frac{e_2}{s_2}m]=\mathrm{DFT}\left[f\left(m\right)\right]+\mathrm{DFT}\left[\frac{e_2}{s_2}m\right]=\mathrm{DFT}\left[f\left(m\right)\right]+\frac{e_2}{s_2}\mathrm{DFT}\left[m\right]---\left(4\right)$

DFT[in the formula] the expression discrete Fourier transform (DFT).DFT[m] for being that 45 ° oblique line is that sawtooth wave is made Fourier transform for the N inclination angle to the cycle.

Subtract each other and can get by (3) and (4):

$\mathrm{DFT}\left[f_1\left(m\right)\right]-\mathrm{DFT}\left[f_2\left(m\right)\right]=(\frac{e_1}{s_1}-\frac{e_2}{s_2})\mathrm{DFT}\left[m\right]---\left(5\right)$

Choose identical frequency component, by the detailed process of following frequency domain algorithm as can be known, for non-s ₁And s ₂The component in cycle can be tried to achieve respectively in (3) and (4) two formulas, so the value of left end is known in the formula, thereby the difference that can try to achieve sensor zero error and the ratio of transducer spacing is:

$\frac{e_1}{s_1}-\frac{e_2}{s_2}=\frac{\mathrm{DFT}\left[f_1\left(m\right)\right]-\mathrm{DFT}\left[f_2\left(m\right)\right]}{\mathrm{DFT}\left[m\right]}---\left(6\right)$

For all non-s ₁And s ₂The frequency component in cycle all can calculate the difference of following formula, during emulation, choosing any such frequency component gets final product, but in actual measurement because The noise, the value that at every turn calculates may be inconsistent, averages to suppress The noise so all ratio difference can be calculated.

The concrete application process of the present invention is as shown in Figure 2:

For the measurement of double-shear amount, still discuss in two kinds of situation: a kind of is as workpiece length overall p=s ₁S ₂The time special circumstances, another kind is as workpiece length overall p ≠ s ₁S ₂The time generalized case.Hereinafter provide concrete simulation calculation process respectively.

One, as workpiece length overall p=s ₁S ₂The time special circumstances

Step 1: select integer r, the v of suitable relatively prime (no common divisor), and definition:

N=rv is total measure dot number; Δ=p/N is the spacing distance between the neighbouring sample point, and wherein p is a workpiece calibration length; s ₁=v Δ, s ₂=r Δ is respectively two kinds of transducer spacings; $n_{s_1}=(r-1)v,$ $n_{s_2}=(v-1)r$ Be respectively under the different shearing displacement situations counting of on Workpiece length actual samples;

Convenience for expressing makes Δ=1, then p=N, s ₁=v, s ₂=r;

The surface of the work measured parameter makes with function f (x) expression

f(m)＝f(x _m)，m＝0，...，N-1 (7)

Step 2: according to shown in Fig. 1 measurand is measured, comprised workpiece calibration length p, total measure dot number N on the workpiece, sampling interval Δ, the zero error e of sensor ₁, e ₂

Step 3: calculate the difference value of measuring: Δ f (x)=f (x+s)-f (x), promptly

$\mathrm{\Δ}{f}_{j,\mathrm{\σ}}\left(\mathrm{\α}\right)=f_{j,\mathrm{\σ}}(\mathrm{\α}+s_j)-f_{j,\mathrm{\σ}}\left(\mathrm{\α}\right),\mathrm{\α}=0,...,n_{s_j}-1,j=\mathrm{1,2}---\left(8\right)$

Step 4: difference value is carried out accurate periodic extension:

Step 5: the fourier series of difference value after the calculating continuation:

${\tilde{y}}_{j,\mathrm{\σ}}\left(k\right)=\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}{y}_{j,\mathrm{\σ}}\left(m\right)e^{-2\mathrm{\πikm}/N},k=0,...,N-1,j=\mathrm{1,2}---\left(10\right)$

Step 6: the fourier series of asking measured parameter:

k＝0，...，N-1，j＝1，2

For all All can try to achieve unique solution.Because s ₁, s ₂There is not common divisor, for all k=1 ..., N-1 is at least at two number [e ^{2 π iks1/N}-1] and [e ^{2 π iks2/N}-1] there is a nonzero value, so can be by accurately reconstruct (except that constant arbitrarily);

Step 7: the fourier series of asking sawtooth wave t (x)=x (cycle is p)

Step 8:, in the frequency of the Fourier analysis that twice measurement tried to achieve, choose

The time the same frequency component, calculate the difference of zero error and the ratio of corresponding sensor spacing or shearing displacement then:

Δe＝e ₁/s ₁-e ₂/s ₂ (12)

S in the formula ₁, s ₂Be respectively two different spacings or shearing displacement, e ₁, e ₂Zero error when being respectively corresponding spacing or shearing displacement

Step 9: select suitable weighting coefficient w _S1, w _S2:

k＝1，...，N-1，j＝1，2

Step 10:, draw the evaluation result after the weighting in conjunction with the measurement result under two kinds of spacer conditions:

$\hat{\tilde{f}}\left(k\right)=\frac{w_{s_1}\left(k\right){\tilde{f}}_{1,\mathrm{\σ}}\left(k\right)+(w_{s_2}\left(k\right)+\mathrm{\Δe}\·\tilde{t}\left(k\right)){\tilde{f}}_{2,\mathrm{\σ}}\left(k\right)}{w_{s_1}\left(k\right)+w_{s_2}\left(k\right)},k=1,...,N-1---\left(13\right)$

Step 11: the appraisal curve of asking measured parameter:

$\hat{f}\left(m\right)=\frac{1}{N}\underset{k=0}{\overset{N-1}{\mathrm{\Σ}}}\hat{\tilde{f}}\left(k\right)e^{2\mathrm{\πikm}/N},m=0,...,N-1---\left(14\right)$

Step 12: remove linear trend, try to achieve final appraisal curve.

Two, as workpiece length overall p ≠ s ₁S ₂The time generalized case:

Generalized case is that p is not s ₁S ₂Multiple, and signal f (x) do not satisfy f (p)=f (0), still can handle by above way this moment.Enough little so long as not workpiece size, always can find p '＜p, make p '=s ₁S ₂Then p can be divided into two sections and handle, first segment length is p '=s ₁S ₂, 0≤x＜p ', second segment length also are p '=s ₁S ₂And p-p '＜x＜p obtains twice evaluation function respectively as stated above, because it is the same part of workpiece for what survey that p-p '≤x＜p ' section is arranged in this twice evaluation function, so with this transposition of partial with overlapping, thereby can obtain exact evaluation result on the whole work-piece length.

Be noted that in addition, when above employing displacement transducer detects the linearity, cylindricity, flatness of workpiece and surface figure accuracy, the deflection angle of relative workpiece was to the influence of evaluation result when we had all ignored the slide carriage motion, this influence is very little because in experimenting, we find out that, can ignore, when if the influence of this deflection angle can not be ignored, we then can be provided with an angular transducer (as autocollimator etc.) the deflection angle γ of detecting sensor erecting frame in real time on slide carriage _i(wherein i=0, N-s, s are shearing displacement) deducting stg (γ then in corresponding difference value _i), promptly Δ f ' (i)=Δ f (i)-stg (γ _i), (i) replace corresponding Δ f (i) with Δ f ', get final product by method calculating of the present invention again.

Claims (1)

1, a kind of precise reconstruction method that utilizes frequency domain method to carry out difference measurement is characterized in that it comprises the steps:

(1) select certain sampling interval Δ and transducer spacing r Δ, v Δ, the surface of the work measured parameter is represented with function f (x):

f(m)＝(x _m)，m＝0，...，N-1

(2) measure difference value according to two different spacing of measurand or shearing displacement;

(3) calculate the difference value of measuring: Δ f (x)=f (x+s)-f (x), promptly

$\mathrm{\Δ}{f}_{j,\mathrm{\σ}}\left(\mathrm{\α}\right)=f_{j,\mathrm{\α}}(\mathrm{\α}+s_j)-f_{j,\mathrm{\σ}}\left(\mathrm{\α}\right),\mathrm{\α}=0,...,n_{s_j}-1,j=\mathrm{1,2}$

(4) two groups of difference values are carried out periodic extension:

(5) fourier series of difference value after the calculating continuation:

${\tilde{y}}_{j,\mathrm{\σ}}\left(k\right)=\underset{m=0}{\overset{N-1}{\mathrm{\Σ}}}{y}_{j,\mathrm{\σ}}\left(m\right)e^{-2\mathrm{\πikm}/N},k=0,...,N-1,j=\mathrm{1,2}$

(6) ask the fourier series of measured parameter:

(7) ask the fourier series of sawtooth wave t (x)=x Wherein the cycle of t (x)=x is p;

(8) in the frequency of the Fourier analysis that twice measurement tried to achieve, choose

The time the same frequency component, calculate the difference of zero error and the ratio of corresponding sensor spacing or shearing displacement then:

Δe＝e ₁/s ₁-e ₂/s ₂

S in the formula ₁, s ₂Be respectively two different spacings or shearing displacement, e ₁, e ₂Zero error when being respectively corresponding spacing or shearing displacement (9) is selected weighting coefficient w _S1, w _S2:

k＝1，...，N-1，j＝1，2

(10), draw the evaluation result after the weighting in conjunction with the measurement result under two kinds of spacer conditions:

$\stackrel{\widehat{~}}{f}\left(k\right)=\frac{w_{s_1}\left(k\right){\tilde{f}}_{1,\mathrm{\σ}}\left(k\right)+(w_{s_2}\left(k\right)+\mathrm{\Δe}\·\tilde{t}\left(k\right)){\tilde{f}}_{2,\mathrm{\σ}}\left(k\right)}{w_{s_1}\left(k\right)+w_{s_2}\left(k\right)},k=1,...,N-1$

(11) ask the appraisal curve of measured parameter:

$\hat{f}\left(m\right)=\frac{1}{N}\underset{k=0}{\overset{N-1}{\mathrm{\Σ}}}\stackrel{\widehat{~}}{f}\left(k\right)e^{2\mathrm{\πikm}/N},m=0,...,N-1$

(12) remove the linear deflection amount of appraisal curve, try to achieve final appraisal curve.

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