CN103345569A - Noise coefficient computing method used for Allan variance analysis technology - Google Patents

Abstract

The invention discloses a noise coefficient computing method used for the Allan variance analysis technology. The noise coefficient computing method used for the Allan variance analysis technology solves the problems that matrix inversion is singular, noise coefficients have negative values, and the noise coefficients are not optimal, wherein the problems exist in five types of typical random noise coefficient processes in the prior art. The noise coefficient computing method used for the Allan variance analysis technology mainly comprises the following steps that 1) an Allan variance sequence and a corresponding relevant time sequence are preset; 2) a relevant time matrix is computed; 3) a halt condition is set, and linearization is conducted on a target function; 4) initial values of loop iteration related parameters are set; 5) matrix transformation is conducted; 6) sphere transformation is conducted, and an unconstrained optimal increment is computed; 7) a hyperplane nonnegative restriction optimal increment is computed; 8) the noise coefficients are updated; 9) a halt index is computed; 10) whether a halt criterion is met is judged. Random noise coefficients estimated by the noise coefficient computing method can be used for inertial device performance evaluation or accurate modeling of random measurement errors, and has significance for improving navigation accuracy of an inertia matrix integrated navigation system.

Description

A kind of noise figure calculating method for the Allan technique of variance analysis

Technical field

The present invention relates to the evaluation method of noise intensity coefficient, particularly a kind of noise figure calculating method for the Allan technique of variance analysis.

Background technology

The Allan technique of variance analysis refers to a kind of time domain noise analysis technology that the Allan variance of output data is analyzed the random noise composition that contains in the instrument output data and strength factor thereof when utilizing instrument static, and it is widely used at aspects such as clock degree of stability analysis frequently and inertia sensitive element (as gyro and accelerometer) performance evaluations.May contain 5 kinds of typical random noises in the data of output when generally thinking instrument static state, i.e. quantizing noise, white noise, zero inclined to one side instability, speed random walk and rate ramp.The estimation of these 5 kinds of random noise strength factors is piths of Allan technique of variance analysis, the at present existing typical noise figure evaluation method that is used for the Allan technique of variance analysis mainly contains least square method [1] (referring to Lawrence C.Ng, Darryll J.Pines.Characterization of Ring Laser Gyro Performance Using the Allan Variance Method[J] .Journal of Guidance Control and Dynamics, Vol.20, No.1,1997:211-214), piecewise fitting method [2] is (referring to Naser El-Sheimy, Haiying Hou, Xiaoji Niu.Analysis and Modeling of Inertial Sensors Using Allan Variance[J] .IEEE Transaction on Instrument and Measurement, Vol.57, No.1,2008:140-149) and normalization error least squares method [3] (referring to Brian E.Grantham P.E., Mark A.Bailey.A Least-Squares Normalized Error Regression Algorithm with Application to the Allan Variance Noise Analysis Method[C] .IEEE/ION Position, Location and Navigation Symposium, 2006:750-756), these three kinds of typical methods all can not effectively solve following problem: it is unusual matrix inversion to occur in the coefficient estimate process; The noise figure that estimates has negative value; The noise figure that estimates is not optimum.

Summary of the invention

The present invention is directed to the problem of the estimation existence of 5 kinds of typical random noise figures in the Allan technique of variance analysis, proposed the optimization coefficient estimate method based on the nonnegative number constraint.This method synthesis matrix analysis theory, Optimum Theory and lineoid theory, measures such as matrixing, spheroid conversion and lineoid constraint have been taked, the coefficient that has guaranteed 5 kinds of random noises estimating is non-negative and optimum, and can not occur the unusual phenomenon of matrix inversion in the noise figure estimation process.

Technical scheme of the present invention is as follows:

A kind of noise figure calculating method for the Allan technique of variance analysis, carry out according to the following steps:

The Allan variance sequence of output data and corresponding time related sequence when 1) given instrument to be analyzed is static;

2) calculate matrix correlation time: utilize time related sequence to calculate matrix A correlation time according to the relation of Allan variance theoretical value and 5 kinds of typical random noise figure C;

3) set halt condition and to the objective function linearization: set the halt condition of loop iteration according to optimization objective function, and objective function is carried out the expression formula that the intermediate vector b of optimum increment under the matrix of coefficients E of random noise coefficient increment dC and the unconfined condition is obtained in linearization;

4) initial value of iterative loop correlation parameter is set: mainly comprise 5 kinds of typical random noise figure C initial value, calculating Allan variance theoretical value sequence initial value, objective function initial value and shut down the initial value of index;

5) matrixing: the Allan variance theoretical value sequence σ that utilizes correlation matrix A and calculating ²Compute matrix E, it is matrix that matrix E is carried out matrixing

Unusual to prevent that follow-up matrix inversion from occurring, and increment dC, coefficient C and vectorial b carried out corresponding being transformed to

With

To keep the balance of objective function;

6) spheroid conversion is calculated no constrained optimum increment: according to matrix

Calculate spheroid transformation matrix H, with increment

Carry out spheroid and be transformed to x, the optimum increment r after the conversion of calculating spheroid under the unconfined condition;

7) calculate the optimum increment of lineoid nonnegativity restrictions: with the inverse matrix H of spheroid transformation matrix H ^-1Each row be considered as normal vector of lineoid, determine by H ^-1With

The non-negative region that the lineoid that determines retrains is calculated the nearest x of the no constrained optimum increment r of distance in non-negative region;

8) noise figure upgrades: the constrained optimum increment x that has that will obtain carries out obtaining coefficient increment dC after spheroid inverse transformation and the matrix inversion conversion successively, calculates new random noise coefficient C;

9) calculate the shutdown index: utilize given Allan variance sequence and Allan variance theoretical value sequence calculating target function value f (C), and then calculate and shut down exponent e;

10) judge whether satisfy to shut down criterion: shut down criterion if satisfy, new coefficient C is non-negative optimum, shuts down criterion if do not satisfy, and returns step 5) and continues to calculate.

Wherein:

In step 1), with given Allan variance sequence

Be designated as , time related sequence { τ ₁, τ ₂..., τ _nBe designated as τ.

In step 2) described in Allan variance theoretical value

With the pass of 5 kinds of typical random noise figure C be:

${\mathrm{\&sigma;}}_i^2=\frac{3}{{\mathrm{\&tau;}}_i^2}{\mathrm{<figure-callout\; id="2"\; label="Q"\; filenames="773678DEST\_PATH\_IMAGE001.png,DEST\_PATH\_IMAGE001.png"\; state="\{\{state\}\}">Q</figure-callout>}}^2+\frac{1}{{\mathrm{\&tau;}}_i}{\mathrm{<figure-callout\; id="2"\; label="N"\; filenames="773678DEST\_PATH\_IMAGE001.png,DEST\_PATH\_IMAGE001.png"\; state="\{\{state\}\}">N</figure-callout>}}^2+\frac{2\mathrm{<figure-callout\; id="2"\; label="ln"\; filenames="773678DEST\_PATH\_IMAGE001.png,DEST\_PATH\_IMAGE001.png"\; state="\{\{state\}\}">ln</figure-callout>}2}{\mathrm{\&pi;}}{\mathrm{<figure-callout\; id="2"\; label="B"\; filenames="773678DEST\_PATH\_IMAGE001.png,DEST\_PATH\_IMAGE001.png"\; state="\{\{state\}\}">B</figure-callout>}}^2+\frac{{\mathrm{\&tau;}}_i}{3}{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="773678DEST\_PATH\_IMAGE001.png,DEST\_PATH\_IMAGE001.png"\; state="\{\{state\}\}">K</figure-callout>}}^2+\frac{{\mathrm{\&tau;}}_i^2}{2}{R}^2---\left(1\right)$

In the formula (1), Q ²Be quantizing noise coefficient, N ²Be white noise coefficient, B ²Be zero inclined to one side instability coefficient, K ²Speed random walk coefficient, R ²Be the rate ramp coefficient, For being correlation time τ _iThe time Allan variance theoretical value.Then random noise coefficient C is:

C＝[Q ²,N ²,B ²,K ²,R ²] ^T

Subscript T represents vector or transpose of a matrix.Note Then for time related sequence { τ ₁, τ ₂... τ _n, correlation time, matrix A was:

$A=\left(\begin{array}{c}a\left({\mathrm{\&tau;}}_1\right)\\ a\left({\mathrm{\&tau;}}_2\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ a\left({\mathrm{\&tau;}}_n\right)\end{array}\right)---\left(2\right)$

Note Allan variance theoretical value sequence Be σ ², according to formula (1), Allan variance theoretical value sequence σ ²Can be expressed as::

σ ²＝A·C (3)

At the optimization objective function described in the step 3) be:

$f\left(C\right)={[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]}^T\&CenterDot;[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]---\left(4\right)$

In the formula (4), σ ²Be the Allan variance theoretical value sequence of calculating according to formula (3),

Be the Allan variance sequence of importing in the step 1).

Exponent e is shut down in definition:

$e=\frac{|f\left(C\right)-f(C\_)|}{f(C\_)}---\left(5\right)$

In the formula (5), the target function value that the noise figure that f (C) obtains for this circulation utilizes formula (4) to calculate, f (C_) utilize the target function value of formula (4) calculating for the noise figure that last time, circulation obtained.

The shutdown criterion of getting loop iteration is:

e≤e ₀ (6)

In the formula (6), e ₀Be halt condition, can determine according to the precision of required noise figure, generally can be taken as one and be not more than 0.001 positive number.

Formula (3) substitution formula (4) line linearity of going forward side by side is turned to:

f(C+dC)＝dC ^T·E ^T·E·dC+2·b ^T·dC+f(C) (7)

In the formula (7), the element computing method of the matrix of coefficients E of increment dC are:

$E(i,j)=\frac{A(i,j)}{{\mathrm{\&sigma;}}_i^2}---\left(8\right)$

E (i, the j) element of the capable j row of the i of expression matrix of coefficients E, A (i, the j) element of the capable j row of the i of expression matrix A correlation time, Expression sequence σ ²In i element.

In the formula (7), the computing method of the intermediate vector b of optimum increment are under the unconfined condition:

$b=E^T\&CenterDot;[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]---\left(9\right)$

Being initialized as of the loop iteration correlation parameter described in the step 4): the initial value C=[0 of random noise coefficient C, 0,0,0,0] ^TThe initial value of the Allan variance theoretical value sequence of calculating The initial value f (C_) of objective function should make that circulation can be carried out first, generally can be taken as one and is not less than 10 ⁵Positive number; Shut down criterion e initial value and be taken as one greater than e ₀Positive number get final product.

Carry out as follows in the operation of the matrixing described in the step 5):

The mould of the j column vector of note matrix E is l _j, that is:

l _j＝|E(:,j)|

Wherein E (:, j) the j column vector of representing matrix E, || expression is to the modulo operation of vector, then the matrix that obtains through the matrixing operation

Element be:

$\hat{E}(i,j)=\frac{E(i,j)}{l_j}---\left(10\right)$

After the element of increment dC, coefficient C and the conversion

Element between the pass be:

$d\hat{C}\left(j\right)=l_j\&CenterDot;\mathrm{dC}\left(j\right),\hat{C}\left(j\right)=l_j\&CenterDot;C\left(j\right)$

According to formula (9) and formula (10) as can be known

Expression formula be:

$\hat{b}={\hat{E}}^T\&CenterDot;[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]---\left(11\right)$

Then formula (7) can be rewritten as:

$f(C+\mathrm{dC})=d{\hat{C}}^T\&CenterDot;({\hat{E}}^T\&CenterDot;\hat{E})\&CenterDot;d\hat{C}+2\&CenterDot;{\hat{b}}^T\&CenterDot;d\hat{C}+f\left(C\right)---\left(12\right)$

According to formula (12) as can be known by the objective function equivalence The set that constitutes be one with Spheroid with dimension.

Carry out as follows in the spheroid map function described in the step 6): because

Be positive definite matrix, certainly exist invertible matrix H and make:

$H^T\&CenterDot;H={\hat{E}}^T\&CenterDot;\hat{E}---\left(13\right)$

Therefore can be right

Carry out matrix decomposition to obtain spheroid transformation matrix H.

Will

Carry out spheroid and be transformed to x:

$x=H\&CenterDot;d\hat{C}$

Order:

$r={\left(H^{-1}\right)}^T\&CenterDot;\hat{b}---\left(14\right)$

Then formula (12) can be rewritten as:

f(C+dC)＝x ^T·x+2·r ^T·x+f(C) (15)

The set that is constituted by the x of objective function equivalence as can be known according to formula (15) be one with the spheroid of x with dimension, make that then the x of target function value minimum should be nearest apart from this spheroid centre of sphere r, under abandoned condition, the minor increment of x and r is 0, so r is no constrained optimum increment.

Calculate as follows at the optimum increment of the lineoid nonnegativity restrictions described in the step 7): because each element of random noise coefficient C all is non-negative, that is:

C≥0

So should guarantee for the x that asks:

$H^{-1}\&CenterDot;x+\hat{C}\&GreaterEqual;0---\left(16\right)$

The represented constraint of formula (16) can be understood like this: the note matrix H ^-1The capable vector of i be n _i, I element and n _iThe ratio of mould length be d _i, that is:

$n_i=H^{-1}(i,:),d_i=\frac{\hat{C}\left(i\right)}{\left|n_i\right|}$

N then _iCan be regarded as the normal vector of lineoid i, d _iCan be regarded as initial point to the distance of lineoid i.Yi Zhi is by H ^-1With

Determined lineoid has 5, if x is in lineoid i, then

If x is on lineoid i, then

If x is under lineoid i, then

So formula (16) expression x is by H ^-1With

Among determined 5 lineoid or on, that is to say that the zone that is made of the x that satisfies formula (16) is the non-negative region of lineoid constraint.

It is nearest apart from r that the target function value minimum is equivalent to x, that is:

(x-r) ^T·(x-r)＝min (17)

If r is not in the represented constraint of formula (16), the x that then satisfies formula (16) and formula (17) must be r borderline projection in the constraint, i.e. the optimum increment x of lineoid nonnegativity restrictions or r itself, or r is in the borderline projection of non-negative region.

Then the optimum increment x of lineoid nonnegativity restrictions can be undertaken by following step:

A. judge that r is whether in the constraint

Order

If all elements of y all is not less than 0, i.e. y 〉=0, then r is the x that satisfies formula (16) and formula (17), otherwise carries out view field's calculating;

B. calculate the view field of r

Suppose among the y less than the { n that is numbered that equals 0 the corresponding lineoid of element ₁..., n _j| j＜5}, the x that then satisfies formula (16) and formula (17) is inevitable in these lineoid, namely

$n_{n_k}\&CenterDot;x+\hat{C}\left(n_k\right)=0,n_k\&Element;\{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="DEST\_PATH\_IMAGE001.png"\; state="\{\{state\}\}">n</figure-callout>}}_1,\&CenterDot;\&CenterDot;\&CenterDot;,n_j|j<5\}---\left(18\right)$

Satisfy zone that the x of formula (18) constitutes and be the view field of r;

As if j=5, namely all elements of y is all less than 0 in addition, and the x that satisfies formula (16) and formula (17) this moment is the intersection point of these 5 lineoid, namely

But in fact this situation is impossible take place, because this moment, all noise figures were 0, this means that the static state of instrument is exported 5 kinds of noise components that do not contain supposition in the data, and this starting condition with the Allan variance analysis contradicts.

C. calculate the subpoint of r

The intersection point of remembering these 5 lineoid is x ₀, then

If x be r at the subpoint of view field, then x is except satisfying formula (18), for the arbitrfary point x in the view field _iFollowing relation must be arranged:

(x _i-x ₀) ^T·(x-r)＝0 (19)

Need (5-j) individual incoherent some x as can be known by formula (18) _i, some x _iCan calculate like this: establish

Be x _iN _iIndividual element is for n _i∈ { n ₁..., n _j| j＜5}, look

Be element to be asked, for remaining (5-j) individual element, one of them put 1 all the other elements successively put 0, then with x _iSubstitution formula (18) is found the solution and is got final product, and then formula (18) and formula (19) simultaneous can be found the solution subpoint x;

D. judge that subpoint is whether in the constraint

Order

If all elements of y all is not less than 0, i.e. y 〉=0, then subpoint x is the r that asks in the borderline projection of non-negative region; Otherwise return step b recomputates r according to y view field.

Described in the step 8) x is being carried out the spheroid inverse transformation, its computing method are:

$d\hat{C}=H^{-1}\&CenterDot;x---\left(20\right)$

Right

Carry out the matrix inversion conversion, its computing method are:

$\mathrm{dC}\left(j\right)=\frac{d\hat{C}\left(j\right)}{l_j}---\left(21\right)$

Then new random noise coefficient C is:

C＝C+dC (22)

At the Allan variance theoretical value sequence σ described in the step 9) ²Utilize new random noise coefficient C and correlation time matrix A calculate according to formula (3), target function value f (C) calculates according to formula (4), shuts down exponent e and calculates according to formula (5).

Refer to formula (6) establishment in the satisfied shutdown criterion described in the step 10), do not satisfy the shutdown criterion and refer to that formula (6) is false.

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

1) the present invention directly with the non-negative characteristic of noise figure as constraint condition, therefore the noise figure that estimates negative value can not occur, makes to carry out rational physical interpretation to the estimated value of noise figure;

2) the present invention adopts the Optimum Theory of belt restraining condition that noise figure is estimated, the estimated value of acquisition is optimum in theory;

3) the present invention shifts by the matrixing singularity that matrix is possible, has guaranteed that the matrix inversion in the coefficient estimate process unusual appearance can not occur;

4) the present invention finds the solution the calculating that is transformed to apart from centre of sphere closest approach by the spheroid conversion with the optimum of complexity, has reduced the difficulty that optimal value is found the solution, and has reduced the calculated amount that noise figure is found the solution;

5) the present invention adopts logarithmic function as objective function, has effectively reduced the subsidiary weights influence that the orders of magnitude different in the Allan variance yields causes objective function.

Description of drawings

Fig. 1 is noise figure estimation process flow diagram of the present invention;

Fig. 2 is optimum solution calculation flow chart under the lineoid nonnegativity restrictions of the present invention;

Fig. 3 schemes Allan variance and the correlation time of the static output of certain type optical fibre gyro data;

Fig. 4 is noise figure estimation convergence process figure of the present invention;

Fig. 5 estimates Allan variance and actual measurement Allan variance comparison diagram for the present invention.

Need to prove that the color of the lines in the accompanying drawing only is used for the more clear contrast of expressing between corresponding curve and the curve, is convenient to be more readily understood the present invention, is not for restriction protection scope of the present invention.

Embodiment

Below in conjunction with the drawings and specific embodiments, the evaluation method of noise figure among the present invention is further described.

The present invention's " a kind of noise figure calculating method for the Allan technique of variance analysis ", its concrete steps are as follows:

1) given Allan variance sequence and time related sequence

As shown in Figure 3, transverse axis is represented correlation time, and unit is second, and the longitudinal axis is represented the Allan variance, and unit is (degree/time) ², solid line represents that 2.5 hours the data computation of frequency sampling with 400Hz when given Allan variance sequence is static according to certain type optical fibre gyro obtains according to the Allan variance of the static state output data computation of optical fibre gyro.

2) calculate matrix correlation time

Utilize the time related sequence among Fig. 3 to calculate matrix A correlation time according to formula (2).

3) set halt condition and to the objective function linearization

Getting halt condition is: e ₀=0.001, stop iteration when namely the increment of target function value is millesimal less than target function value.According to formula (8) compute matrix E.

4) initial value of iterative loop correlation parameter is set

Get C=[0,0,0,0,0] ^T, initial Allan variance estimated value σ ²Be taken as the Allan variance sequence among Fig. 3, the initial target function is taken as f (C_)=10 ⁵, initially shut down index and be taken as e=1.

5) matrixing

Carry out the matrixing compute matrix according to formula (10)

, and by

Calculate according to formula (11)

6) no constrained optimum increment is calculated in spheroid conversion

Calculate spheroid transformation matrix H and its inverse matrix H according to formula (13) ^-1, by

And H ^-1According to the optimum increment r under formula (14) the calculating unconfined condition.

7) calculate the optimum increment of lineoid nonnegativity restrictions

Here be example with the cyclic process first time, introduce the computation process of the optimum increment of lineoid nonnegativity restrictions in detail.In the cyclic process first time, the noise figure after the inverse matrix of spheroid transformation matrix, no constrained optimum increment and the matrixing is respectively:

$H^{-1}=\left(\begin{array}{ccccc}1.0000& -0.2551& 0.0803& -0.0669& 0.0737\\ 0& 1.0320& -0.3260& 0.2721& -0.3004\\ 0& 0& 1.0487& -1.6355& 2.7100\\ 0& 0& 0& 1.8667& -6.8692\\ 0& 0& 0& 0& 4.9593\end{array}\right),$ $r=\left(\begin{array}{c}112.1880\\ -0.6094\\ 0.1956\\ -0.1570\\ 0.1738\end{array}\right),$ $\hat{C}=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}\right)$

A. judge that r is whether in the constraint

Yi Zhi $y=H^{-1}\&CenterDot;r+\hat{C}={\left[\begin{array}{ccccc}112.3825& -0.7876& 0.9329& -1.4870& 0.8620\end{array}\right]}^T,$ Owing to have the element less than 0 among the y, so r does not satisfy formula (16) and formula (17), need carry out view field and calculate.

B. calculate the view field of r

Among the y less than equal being numbered of 0 the corresponding lineoid of element 2,4}, the x that then satisfies formula (16) and formula (17) is inevitable in these two lineoid, can get according to formula (18):

$\left(\begin{array}{ccccc}0& 1.0320& -0.3260& 0.2721& -0.3004\\ 0& 0& 0& 1.8667& -6.8692\end{array}\right)\&CenterDot;x+\left(\begin{array}{c}0\\ 0\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)---\left(23\right)$

Satisfy zone that the x of formula (23) constitutes and be the view field of r.

C. calculate the subpoint of r

The intersection point of these 5 lineoid is $x_0={\left[\begin{array}{ccccc}0& 0& 0& 0& 0\end{array}\right]}^T,$ 3 incoherent some x that need in addition _iCan get like this:

$x_1={\left[\begin{array}{ccccc}1& x_{12}& 0& x_{14}& 0\end{array}\right]}^T,$ $x_2={\left[\begin{array}{ccccc}0& x_{22}& 1& x_{24}& 0\end{array}\right]}^T,$ $x_3={\left[\begin{array}{ccccc}0& x_{32}& 0& x_{34}& 1\end{array}\right]}^T$

With x _iSubstitution formula successively (23) can get:

$x_1={\left[\begin{array}{ccccc}1& 0& 0& 0& 0\end{array}\right]}^T,$ $x_2={\left[\begin{array}{ccccc}0& 0.3158& 1& 0& 0\end{array}\right]}^T,$ $x_3={\left[\begin{array}{ccccc}0& -0.6791& 0& 3.6799& 1\end{array}\right]}^T$

So can get according to formula (19):

$\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0.3158& 1& 0& 0\\ 0& -0.6791& 0& 3.6799& 1\end{array}\right)(x-r)=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)---\left(24\right)$

With r substitution formula (24), merge and can get with formula (23) behind the abbreviation:

$\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1.0320& -0.3260& 0.2721& -0.3004\\ 0& 0.3158& 1& 0& 0\\ 0& 0& 0& 1.8667& -6.8692\\ 0& -0.6791& 0& 3.6799& 1\end{array}\right)\&CenterDot;x+\left(\begin{array}{c}-112.1880\\ 0\\ -0.0032\\ 0\\ -0.0100\end{array}\right)=\left(\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}\right)---\left(25\right)$

Solution formula (25) gets final product to such an extent that r at the subpoint of view field is:

$x={\left[\begin{array}{ccccc}112.1880& 0.0005& 0.0030& 0.0026& 0.0007\end{array}\right]}^T$

D. judge that subpoint x is whether in the constraint

Yi Zhi $y=H^{-1}\&CenterDot;x+\hat{C}={\left[\begin{array}{ccccc}112.1880& 0& 0.0008& 0& 0.0035\end{array}\right]}^T,$ Owing to there is not the element less than 0 among the y, so subpoint x satisfies formula (16) and formula (17), then x is the optimum solution that satisfies the nonnegative number constraint.

8) noise figure upgrades

, according to formula (22) noise figure C is upgraded by dC according to formula (20) and formula (21) calculating noise coefficient increment dC by subpoint x.Be example with circulation for the first time, the noise figure after the renewal is:

C＝[0.0162,0,1.7940×10 ^-7,0,2.9127×10 ^-13] ^T

9) calculate the shutdown index

By the noise figure C after upgrading and correlation time matrix A calculate Allan variance theoretical value sequence σ according to formula (3) ², by given Allan variance

With calculating Allan variance theoretical value sequence σ ²According to formula (4) calculating target function value f (C), calculate the shutdown exponent e by f (C) and f (C_) according to formula (5).Be example with circulation for the first time, can get target function value according to formula (4) is f (C)=5.8641 * 10 ³, then shut down index and be:

$e=\frac{|f\left(C\right)-f(C\_)|}{f(C\_)}=\frac{|5.8641\&times;{10}^3-1\&times;{10}^5|}{1\&times;{10}^5}=0.9414$

10) judge whether to satisfy the shutdown criterion

Then know: e＞e ₀, namely do not satisfy and shut down criterion, so make f (C_)=5.8641 * 10 ³, and return step 5) and continue to calculate.

" shutdown index " among Fig. 4 refers to the e in the formula (5).

Fig. 4 has shown the cyclic process of present embodiment noise figure estimation, and wherein transverse axis is represented the loop iteration number of times, and the longitudinal axis represents to shut down the index variation process, and solid line is represented halt condition, and dotted line represents to shut down index.Satisfy through the noise figure that calculates after 9 circulations as can be seen and shut down criterion.

Among Fig. 5, transverse axis is represented correlation time, and unit is second, the longitudinal axis is represented the Allan variance, unit is (degree/time) 2, and solid line represents that dotted line represents to utilize according to the noise figure of estimation the Allan variance of formula (3) calculating according to the Allan variance of the static state output data computation of optical fibre gyro.Both are close to validity and feasibility that table of coincidences is understood noise figure evaluation method of the present invention.

The comparison of the different noise figure of table 1 calculating method

Listed least square method, piecewise fitting method, normalization error least squares method and the inventive method estimation result to noise figure among the embodiment in the table 1 respectively, the situation of negative value can appear in the noise figure that estimates of least square method as can be seen, this does not meet the actual physics situation, also has the unusual possibility of matrix inversion in addition when the estimated noise coefficient; Segmentation has bigger subjectivity to the piecewise fitting method to the Allan variance in estimation process, therefore the possibility of result instability that obtains; Normalization error least squares method just forces to put 0 in the noise figure estimation process when negative value occurring, and this point is difficult to that proper explanations is arranged, and also has the unusual possibility of matrix inversion in estimation process.Comparison by data in the his-and-hers watches 1 can draw that noise figure evaluation method of the present invention estimates 5 in typical random noise coefficient be non-negative and optimum.

Claims (4)

1. a noise figure calculating method that is used for the Allan technique of variance analysis is characterized in that, may further comprise the steps:

The Allan variance sequence of output data and corresponding time related sequence when 1) given instrument to be analyzed is static: with given Allan variance sequence

Be designated as

Time related sequence { τ ₁, τ ₂..., τ _nBe designated as τ;

2) calculate matrix correlation time: utilize time related sequence τ, according to Allan variance theoretical value

(subscript i represents i element of sequence, if no special instructions, down together) calculate matrix A correlation time with the relation of 5 kinds of typical random noise figure C;

Described

With the pass of C be:

${\mathrm{\&sigma;}}_i^2=\frac{3}{{\mathrm{\&tau;}}_i^2}{Q}^2+\frac{1}{{\mathrm{\&tau;}}_i}{N}^2+\frac{2\ln 2}{\mathrm{\&pi;}}{B}^2+\frac{{\mathrm{\&tau;}}_i}{3}{K}^2+\frac{{\mathrm{\&tau;}}_i^2}{2}{R}^2---\left(1\right)$

In the formula (1), Q ²Be quantizing noise coefficient, N ²Be white noise coefficient, B ²Be zero inclined to one side instability coefficient, K ²Be speed random walk coefficient, R ²Be the rate ramp coefficient,

For being correlation time τ _iThe time Allan variance theoretical value, then random noise coefficient C is:

C＝[Q ²,N ²,B ²,K ²,R ²] ^T

Subscript T represents vector or transpose of a matrix, note Then for time related sequence { τ ₁, τ ₂..., τ _n, correlation time, matrix A was:

$A=\left(\begin{array}{c}a\left({\mathrm{\&tau;}}_1\right)\\ a\left({\mathrm{\&tau;}}_2\right)\\ \&CenterDot;\\ \&CenterDot;\\ \&CenterDot;\\ a\left({\mathrm{\&tau;}}_n\right)\end{array}\right)---\left(2\right)$

Note Allan variance theoretical value sequence

Be σ ², according to formula (1), Allan variance theoretical value sequence σ ²Can be expressed as:

σ ²＝A·C (3)

3) set halt condition and to the objective function linearization: set the halt condition of loop iteration according to optimization objective function, and objective function is carried out the expression formula that the intermediate vector b of optimum increment under the matrix of coefficients E of random noise coefficient increment dC and the unconfined condition is obtained in linearization;

Described optimization objective function f (C) is:

$f\left(C\right)={[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]}^T\&CenterDot;[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]---\left(4\right)$

In the formula (4), σ ²Be the Allan variance theoretical value sequence of calculating according to formula (3),

Be the Allan variance sequence of importing in the step 1);

Exponent e is shut down in definition:

$e=\frac{|f\left(C\right)-f(C\_)|}{f(C\_)}---\left(5\right)$

In the formula (5), the target function value that the noise figure that f (C) obtains for this circulation utilizes formula (4) to calculate, f (C_) utilize the target function value of formula (4) calculating for the noise figure that last time, circulation obtained;

The shutdown criterion of getting loop iteration is:

e≤e ₀ (6)

In the formula (6), e is for shutting down index, e ₀Be halt condition, can determine according to the precision of required noise figure;

Formula (3) substitution formula (4) line linearity of going forward side by side is turned to:

f(C+dC)＝dC ^T·E ^T·E·dC+2·b ^T·dC+f(C) (7)

In the formula (7), the element E of the matrix of coefficients E of increment dC (i, j) computing method are:

$E(i,j)=\frac{A(i,j)}{{\mathrm{\&sigma;}}_i^2}---\left(8\right)$

E (i, the j) element of the capable j row of the i of expression matrix of coefficients E, A (i, the j) element of the capable j row of the i of expression matrix A correlation time,

Expression sequence σ ²In i element;

In the formula (7), the computing method of the intermediate vector b of optimum increment are under the unconfined condition:

$b=E^T\&CenterDot;[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]---\left(9\right)$

4) initial value of loop iteration correlation parameter is set: comprise the initial value C=[0 of 5 kinds of typical random noise figure C, 0,0,0,0] ^T, the Allan variance theoretical value sequence σ of calculating ²Initial value

The initial value of objective function initial value and shutdown index, the initial value of objective function f (C_) should make that circulation can be carried out first, shut down the exponent e initial value and are taken as one greater than e ₀Positive number get final product;

5) matrixing: the Allan variance theoretical value sequence σ that utilizes correlation matrix A and calculating ²Compute matrix E, it is matrix that matrix E is carried out matrixing

Unusual to prevent that follow-up matrix inversion from occurring, and increment dC, coefficient C and vectorial b carried out corresponding being transformed to

With

To keep the balance of objective function;

Described matrixing operation is carried out as follows:

The mould of the j column vector of note matrix E is l _j, that is:

l _j＝|E(:,j)|

Wherein E (:, j) the j column vector of representing matrix E, || expression is to the modulo operation of vector, then the matrix that obtains through the matrixing operation

Element For:

$\hat{E}(i,j)=\frac{E(i,j)}{l_j}---\left(10\right)$

After the element of increment dC, coefficient C and the conversion

Element between the pass be:

$d\hat{C}\left(j\right)=l_j\&CenterDot;\mathrm{dC}\left(j\right),\hat{C}\left(j\right)=l_j\&CenterDot;C\left(j\right)$

Wherein, dC (j) and

Represent respectively dC and

J element, C (j) and

Represent respectively C and J element;

According to formula (9) and formula (10) as can be known

Expression formula be:

$\hat{b}={\hat{E}}^T\&CenterDot;[\ln \left({\mathrm{\&sigma;}}^2\right)-\ln \left({\widetilde{\mathrm{\&sigma;}}}^2\right)]---\left(11\right)$

Variable being changed to of formula (7) then:

$f(C+\mathrm{dC})=d{\hat{C}}^T\&CenterDot;({\hat{E}}^T\&CenterDot;\hat{E})\&CenterDot;d\hat{C}+2\&CenterDot;{\hat{b}}^T\&CenterDot;d\hat{C}+f\left(C\right)---\left(12\right)$

According to formula (12) as can be known by the objective function equivalence

The set that constitutes be one with

Spheroid with dimension;

6) spheroid conversion is calculated no constrained optimum increment: according to matrix Calculate spheroid transformation matrix H, with increment Carry out spheroid and be transformed to x, the optimum increment r after the conversion of calculating spheroid under the unconfined condition;

Described spheroid map function is carried out as follows: because

Be positive definite matrix, certainly exist invertible matrix H and make:

$H^T\&CenterDot;H={\hat{E}}^T\&CenterDot;\hat{E}---\left(13\right)$

Therefore can be right

Carry out matrix decomposition to obtain spheroid transformation matrix H, will Carry out spheroid and be transformed to x:

$x=H\&CenterDot;d\hat{C}$

Order:

$r={\left(H^{-1}\right)}^T\&CenterDot;\hat{b}---\left(14\right)$

Then formula (12) can be rewritten as:

f(C+dC)＝x ^T·x+2·r ^T·x+f(C) (15)

The set that is constituted by the x of objective function equivalence as can be known according to formula (15) be one with the spheroid of x with dimension, make that then the x of target function value minimum should be nearest apart from this spheroid centre of sphere r, under abandoned condition, the minor increment of x and r is 0, so r is no constrained optimum increment;

7) calculate the optimum increment of lineoid nonnegativity restrictions: with the inverse matrix H of spheroid transformation matrix H ^-1Each row be considered as normal vector of lineoid, determine by H ^-1With

The non-negative region that the lineoid that determines retrains is calculated the nearest x of the no constrained optimum increment r of distance in non-negative region;

The optimum increment of described lineoid nonnegativity restrictions carries out as follows: because each element of random noise coefficient C all is non-negative, that is:

C≥0

So should guarantee for the x that asks:

$H^{-1}\&CenterDot;x+\hat{C}\&GreaterEqual;0---\left(16\right)$

The represented constraint of formula (16) can be understood like this: the note matrix H ^-1The capable vector of i be n _i,

I element and n _iThe ratio of mould length be d _i, that is:

n _i＝H ^-1(i,:)

$d_i=\frac{\hat{C}\left(i\right)}{\left|n_i\right|}$

N then _iCan be regarded as the normal vector of lineoid i, d _iCan be regarded as initial point to the distance of lineoid i, easily know by H ^-1With

Determined lineoid has 5;

If x is in lineoid i, then $n_i\&CenterDot;x+\hat{C}\left(i\right)=0;$

If x is on lineoid i, then $n_i\&CenterDot;x+\hat{C}\left(i\right)>0;$

If x is under lineoid i, then $n_i\&CenterDot;x+\hat{C}\left(i\right)<0;$

So formula (16) expression x is by H ^-1With

Among determined 5 lineoid or on, that is to say that the zone that is made of the x that satisfies formula (16) is the non-negative region of lineoid constraint;

It is nearest apart from r that the target function value minimum is equivalent to x, namely requires (x-r) ^T(x-r) get minimum value, be expressed as (min refers to minimum value in the formula):

(x-r) ^T·(x-r)＝min (17)

If r is not in the represented constraint of formula (16), the x that then satisfies formula (16) and formula (17) must be r borderline projection in the constraint, i.e. the optimum increment x of lineoid nonnegativity restrictions is exactly r itself, and perhaps x is that r is in the borderline projection of non-negative region;

8) noise figure upgrades: the constrained optimum increment x that has that will obtain carries out obtaining coefficient increment dC after spheroid inverse transformation and the matrix inversion conversion successively, calculates new random noise coefficient C;

Described x is carried out the spheroid inverse transformation, its computing method are:

$d\hat{C}=H^{-1}\&CenterDot;x---\left(18\right)$

Right

Carry out the matrix inversion conversion, its computing method are:

$\mathrm{dC}\left(j\right)=\frac{d\hat{C}\left(j\right)}{l_j}---\left(19\right)$

Then the value of C+dC is as the new random noise coefficient C after upgrading, that is:

C＝C+dC (20)

9) calculate to shut down index: utilize new random noise coefficient C and correlation time matrix A calculate Allan variance theoretical value sequence σ according to formula (3) ², according to formula (4) calculating target function value f (C), calculate according to formula (5) again and shut down exponent e;

10) judge whether satisfy to shut down criterion: shut down criterion if formula (6) is set up then to satisfy, new random noise coefficient C is non-negative optimum; If formula (6) is false, then do not satisfy and shut down criterion, return step 5) and continue to calculate.

2. the noise figure calculating method for the Allan technique of variance analysis according to claim 1 is characterized in that: the described r of step 7) calculates as follows at the borderline projection x of non-negative region:

1) judges that r is whether in the constraint: order

If all elements of y all is not less than 0, i.e. y 〉=0, then r is the x that satisfies formula (16) and formula (17), otherwise enters next step, carries out view field's calculating;

2) calculate the view field of r: among the supposition y less than the { n that is numbered that equals 0 the corresponding lineoid of element ₁..., n _j| j＜5}, the x that then satisfies formula (16) and formula (17) is inevitable in these lineoid, namely

$n_{n_k}\&CenterDot;x+\hat{C}\left(n_k\right)=0,n_k\&Element;\{n_1,\&CenterDot;\&CenterDot;\&CenterDot;,n_j|j<5\}---\left(21\right)$

Satisfy zone that the x of formula (21) constitutes and be the view field of r;

3) subpoint of calculating r: the intersection point of remembering these 5 lineoid is x ₀, then

If x be r at the subpoint of view field, then x is except satisfying formula (21), for the arbitrfary point x in the view field _iFollowing relation must be arranged:

(x _i-x ₀) ^T·(x-r)＝0 (22)

Need (5-j) individual incoherent some x as can be known by formula (21) _i, some x _iCan calculate like this: establish Be x _iN _iIndividual element is for n _i∈ { n ₁..., n _j| j＜5}, look

Be element to be asked, for remaining (5-j) individual element, one of them put 1 all the other elements successively put 0, then with x _iSubstitution formula (21) is found the solution and is got final product, and then formula (21) and formula (22) simultaneous can be found the solution subpoint x;

4) judge that subpoint is whether in the constraint: order

If all elements of y all is not less than 0, i.e. y 〉=0, then subpoint x is the r that asks in the borderline projection of non-negative region; Otherwise return step 2) recomputate the view field of r according to y.

3. the noise figure calculating method for the Allan technique of variance analysis according to claim 1 and 2 is characterized in that: halt condition e in the described step 3) ₀Be taken as one and be not more than 0.001 positive number.

4. the noise figure calculating method for the Allan technique of variance analysis according to claim 1 and 2, it is characterized in that: the initial value of objective function f (C_) is taken as one and is not less than 10 in the described step 4) ⁵Positive number.

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