CN103560984B

CN103560984B - Channel self-adapting method of estimation based on multi-model weighting soft handover

Info

Publication number: CN103560984B
Application number: CN201310529529.8A
Authority: CN
Inventors: 杨睿哲; 宗亮; 张琳; 张延华; 孙恩昌
Original assignee: Beijing University of Technology
Current assignee: Beijing University of Technology
Priority date: 2013-10-31
Filing date: 2013-10-31
Publication date: 2017-12-15
Anticipated expiration: 2033-10-31
Also published as: CN103560984A

Abstract

The invention belongs to radio communication field, discloses a kind of channel self-adapting method of estimation based on multi-model weighting soft handover.Channel system model is initially set up, time-frequency doubly selective channel is selected, determines channel parameter.It is then based on channel model and method of estimation establishes channel estimation submodel and multi-model channel estimation storehouse, analyzes and calculate the model error and evaluated error of channel estimation model.The switching index finally combined according to model error and evaluated error, the switching of model is completed by LUMV weighting multi-model adaptive estimation algorithm.The present invention is proposed under transmission channel model condition of uncertainty, with reference to channel model and method of estimation, multi-model thought in multi-model Adaptive Control theory is incorporated into channel estimation, and complete to switch using a kind of weighting multi-model adaptive estimation method of linearity error minimum variance, make channel estimation methods that there is high robust and accuracy in the range of Complex Channel.

Description

Channel self-adaptive estimation method based on multi-model weighted soft handover

Technical Field

The invention belongs to the field of radio communication, and relates to a channel self-adaptive estimation method based on multi-model weighted soft handover.

Background

The channel estimation algorithm designs the estimation algorithm according to different criteria for the established channel model to obtain model parameter values. Common channel estimation algorithms include maximum likelihood ML estimation, EM estimation (the EM algorithm is an iterative algorithm that implements progressive ML estimation when observed data is incomplete), LS estimation, RLS estimation, LMMSE estimation, Kalman filtering, and the like. The ML/EM estimation is based on the maximum likelihood criterion, under the condition that the estimated parameter does not have any prior knowledge, the parameter is estimated by utilizing a plurality of known observation values (complete or incomplete), the effectiveness is high, and the complexity is high; LS takes the minimization of the error of the two-times of the estimated value and the target value as the target, the algorithm is simple, but the influence of the signal-to-noise ratio and the CIR length is large, so the LS is often used as the initial estimation step and is improved by combining other algorithms; based on the observed value and the channel statistical characteristic, the LMMSE estimation takes the mean square error minimization (MMSE) of the estimated value and the target value as the target for estimation, the performance is relatively good, the Kalman filtering is combined with an AR model to be used as an MMSE estimator, the characteristic of self-adaptive tracking is achieved, and the estimation can still be effectively carried out under the condition that the channel estimation characteristic is uncertain.

Based on the above analysis, it can be seen that establishing a channel model, designing an estimation algorithm, and effectively tracking sudden changes of a channel state become the key to improve the channel estimation performance. At present, the research analyzes the errors under different models and estimation methods and continuously improves the methods to reduce the errors, but the performance of the channel estimation errors under different channel environments in different models and estimation processes is very different, and the estimation method with relatively good performance is closely related to the acquisition of the channel statistical characteristics, so that the research of a robust channel estimation scheme applied to an actual unknown complex environment is urgently needed.

Disclosure of Invention

In order to overcome the defects that different channel models and estimation processes have great difference in the performance of channel estimation errors in different channel environments, and an estimation method with relatively good performance is closely related to the acquisition of channel statistical characteristics, the invention provides a channel estimation multi-model weighted soft switching method.

In order to achieve the purpose, the invention adopts the technical scheme that: firstly, the performance and condition factors of various estimation algorithms of a wireless channel are analyzed, and the estimation algorithm is determined to be used. Then, a channel system model is established, time-frequency double-channel selection is selected, and channel parameters are determined. And establishing a channel estimation sub-model and a multi-model channel estimation library with high coverage and flexibility based on the channel model and the estimation method. And analyzing and calculating the model error and the estimation error of the channel estimation model. And finally, according to a switching index combining the model error and the estimation error, completing the switching of the model by a LUMV weighted multi-model adaptive estimation algorithm so as to achieve the aim of optimization.

The technical scheme adopted by the invention comprises the following steps:

step 1, establishing a channel system model.

The invention adopts time-frequency double channel selection. The influence coefficient of the transmitting symbol at the time (T-tau) on the receiving symbol at the time (T-tau) is represented by h (T; tau), tau is time delay, and the sampling intervals of the transmitting sequence and the receiving sequence are assumed to be T_sIn combination with each otherInstead of tau. The transmission symbols of the data block transmission system are u (i),taking N (1-N200) continuous symbols in u (i) to form a k (k 1-N) data block u_k。

Assuming that the channel is a Rayleigh channel, the maximum time delay and the maximum Doppler shift of the path of the channel are tau_maxAnd f_maxAnd satisfy 2 τ_maxf_max< 1 and its channel state is slowly changing over a range of transmission data blocks. The data block u_kChannel h on_k(n; l) can be expressed by the Base Extension Model (BEM) as follows:

in the formula,n is more than or equal to 1 and less than or equal to N, omega_q＝π(q-(Q+1)/2)/N，1≤q≤Q,The number of the basis coefficients; c_k,q(l) L is more than or equal to 1 and less than or equal to L,

taking a data block u (i) as an estimation unit, and adopting (Q +1) × (L +1) C of a channel base model_k,q(l) To represent N × (L +1) unknown quantities h_k(n; l) so that the channel model and the estimated parameters are simplified. According to the Rayleigh channel characteristics, C_k,_q(l) Is a mean of zero and a variance ofComplex gaussian random variables.

And 2, estimating the channel response by adopting a pilot frequency symbol assisted channel estimation method.

And 2.1, establishing a signal sending and receiving model.

Sending end sends sequence u_kComprising two parts, i.e. information symbols s_gAnd a pilot symbol p_g. Uniformly inserting G pilot symbols p in one data block_gData block u_kExpressed as:

wherein the ZP (zero padding) pilot sequence is in the form ofTogether comprising N_sAn information symbol and N_pA pilot sequence symbol, N ═ N_s+N_p。

The symbol y (i) received after the transmission symbol u (i) passes through the multipath channel h (i; l) is expressed as:

wherein w (i) represents a mean value of zero and a variance ofAdditive White Gaussian Noise (AWGN).

And 2.2, acquiring pilot frequency information.

Based on a base extension model, corresponding to a transmitted data block u_kReceived data block y of_kCan be expressed as:

in the formula,is zero mean value,AWGN vector of variance, H of N × N_kIs a lower triangular matrix and satisfies [ H_k]_n,m＝h_k(n；n-m)，n≥m，C_k,qIs the first column of [ C_k,q(0),C_k,q(1),...,C_k,q(L),0,…0]^TToeplitz matrix of (D), F_q＝diag[f_q(1),f_q(2),…,f_q(N)]Is a pair of angular arrays.

Receiving a data block y_kOf which one is affected only by the pilot sequenceThe expression is as follows:

in the formula,andare respectively H_kAnd w_kIncludes (Q +1) × (L +1) unknown parameters C_k,q(l) Corresponding to each p_gAt the receiving end, only (L +1) symbols affected only by pilot are obtained, so at least (Q +1) pilot sequences are inserted into one data block to obtain C_k,q(l) Namely G is more than or equal to Q + 1. Combining the formula (5) and the principle of mutual conversion between Toeplitz matrix and vector to obtain the vector:

in the formula phi^pIs a pilot frequency information conversion matrix, and the expression is as follows:

in the formula,is F_qCorresponds to a pilot sequence p_gThe expression of the submatrix is:

P_gis formed by a pilot sequence p_gPilot symbol p in (1)_g,l(L +1) × (L +1) type Toeplitz matrix of construction, L1, …, L +1, due to p_gThe structure is the same hereIgnoring subscript g, i.e. P₁＝P,…,P_GWhen P is equal to P, then [ P]_n,m＝p_L-m+n。

Step 2.3, the channel response is calculated.

Channel response matrix C_kComprises the following steps:

and 3, establishing a channel estimation multi-model.

Based on the analysis of the channel model and the estimation method, the invention takes the channel model and the estimation method as a channel estimation model, and covers complex and variable channel environment by establishing a multi-model library of a plurality of channel estimation models, which is different from the structures of the model and the controller in the multi-model control theory.

And 3.1, establishing a channel model set combination.

The channel estimation model library is M ═ B, E, where B is the channel model set and its expression is:

B＝{b_j,j＝1,2|P-BEM,CE-BEM-AR} (10)

redefining the basis functions in equation (1) toω_qPi (Q- (Q +1)/2)/N andsubscripts 1,2 are added to distinguish between different models, and subsequently indicate the same.

In addition, E ═ E₁＝LS,e₂Kalman, which is a set of estimation methods.

With two models M₁And M₂For example, the model library is:

M＝{M₁(b₁,e₁),M₂(b₂,e₂)} (11)

step 3.2, calculate model M₁The channel response of (2).

Model M₁The LS estimation value of the coefficient corresponding to the expansion base obtained based on the LS estimation algorithm of the expansion and the receiving and sending signal of the pilot frequency point and the least square method is as follows:

the estimated value of the channel response is then:

andthe relationship of (2) is the same as the formula (9), and the superscript is correspondingly added.

Step 3.3, calculate model M₂The channel response of (2).

Based on the BEM model and the sending and receiving signals, a Kalman filtering state equation and an observation equation can be constructed:

wherein A is_k，v_kAs a known Doppler estimate f_dThe obtained state transition matrix and the state transition noise SNR.

The kalman estimation procedure thus results as:

in the formula,in order to be a Kalman channel estimation value,is a zero matrix, e_kFor measurement of errors, G_kIs Kalman gain, P_kTo estimate the covariance matrix of the error, then:

The established channel estimation model library covers different channel parameter conditions under Doppler and signal-to-noise ratio conditions, and the models based on P-BEM and CE-BEM-AR have good performance on low Doppler frequency shift and high Doppler frequency shift; meanwhile, the LS estimation has very good performance at high SNR, while the Kalman estimation works well at low SNR.

And 4, performing channel estimation multi-model weighted soft handover.

In the multi-model control of the present invention, a soft handover strategy is adopted because the hard handover strategy generally requires that each model has the same structure and does not conform to the model set composition. The invention adopts a weighted multi-model adaptive estimation method of linear error minimum variance (Linear inactive minimum variance-LUMV) to complete the switching, and can effectively achieve the performance improvement of the online switching.

And 4.1, calculating the LUMV.

Assume the model is M_jJ is 1,2, and the variance of the estimation output, estimation error, and estimation error are:v_k,l,jandthe relationship between the estimated output and the error is then:

wherein, correspond toIs given by the true channel value of h_k,lIt is shown that,for the covarianceI is an identity matrix, J represents the number of models currently under consideration, and J is 2.

And 4.2, calculating the error of the model.

Combining with the multi-model weighted adaptive control theory, the channel model switching in the invention adopts a weight switching algorithm and uses an estimation errorVariance of (2)Calculating the weight:

estimation error v_k,l,jThe method mainly comprises a model error and an estimation error. The statistical variance used for the model error is:

wherein,is v is_k,l,jSome of these, calculated values of model error; s ═ F^T(FF^T)^-1F, and [ F]_n,q＝f_q，j(n)。

And 4.3, calculating the LS estimation error and the estimation error variance of the weight.

The covariance of the LS estimation error comes from its noise variance basis, due to:

obtaining:

wherein phi⁺＝((Φ^p)^HΦ^p)^-1(Φ^p)^H. As can be seen from equation (12), the following is provided here

Model M₁The estimation error variance used to calculate the weights is:

in the formula,as model M₁Total estimation error of E_LS,lAs model M₁The estimated error variance of (2);

and 4.4, calculating the Kalman estimation error and the estimation error variance of the weight.

The Kalman estimated error covariance is calculated by an online a posteriori error covariance matrix:

E_kal＝BP_clB (23)

wherein, B ═ F₂，[P_cl]_m',n'＝E{[e_cl,k]_q(l)[e_cl,k]_q'(l') } is the posterior error covariance of the base coefficient, [ P }_k]_m,n＝E{[e_c,k]_q(l)[e_c,k]_q'(L ') }, corresponding transformations are performed according to the relationship of m ' ═ L (Q +1) + Q +1, n ' ═ L ' (Q +1) + Q ' +1, and m ═ Q (L +1) + L +1, n ═ Q ' (L +1) + L ' + 1;

model M₂The estimation error variance used to calculate the weights is:

in the formula,as model M₂Total estimation error of E_kal,lAs model M₂The estimated error variance of (2);

during handover, for ease of calculation, it is assumed that each h is_k(n; l) have the same covariance; this can be improvedAnd obtaining:

according to the above and the LUMV estimation theory, assuming that a plurality of sub-models obtain independent channel estimation results and perform weighted summation, a lower Cramer-Rao bound can be obtained and the channel estimation performance can be improved.

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

the invention provides a multi-model channel estimation self-adaptive control theory which is used for establishing and selecting an estimation model to automatically switch to a performance optimal state under the uncertain condition of a transmission channel model by combining a channel model and an estimation method. The multi-model thought in the multi-model adaptive control theory is introduced into channel estimation, and a weighted multi-model adaptive estimation method with the minimum variance of linear errors is adopted to complete switching, so that the channel estimation method has high robustness and accuracy in a complex channel range by utilizing the high coverage and flexibility of the multi-model.

Drawings

FIG. 1 is a flow chart of a channel adaptive estimation method based on multi-model weighted soft handover;

fig. 2 is a channel estimation error curve obtained by simulation in the embodiment of the present invention, in which: curve 1 is M₁Error curve of channel model, curve 2 being M₂And 3, an error curve of a channel model, namely an error curve obtained by a channel estimation multi-model weighting soft switching method.

Detailed Description

The invention is further described below with reference to the accompanying drawings.

In this embodiment, Matlab simulation software is adopted, and a method flowchart is shown in fig. 1, and includes the following steps:

step 1, establishing a channel system model.

A random time-frequency selective rayleigh fading channel is established.

Setting channel parameters: carrier frequency of f_c2GHz with a sampling interval T_s25 mus and L2, N140, maximum doppler shift of the channel is f_max＝370Hz。

And 2, estimating the channel response by using a pilot frequency symbol auxiliary method.

The pilot frequency symbol aided channel estimation method is adopted, the sending symbol of the data block transmission system is u (i),

in u (i), 200 consecutive symbols are taken as N to form a k (k 60) th data block u_k。

Each data block comprising two parts, i.e. information symbols s_gAnd a pilot symbol p_g. Uniformly inserting 8 pilot symbols p into one data block_g。

And 3, establishing a channel estimation multi-model.

Establishing two models M₁And M₂The channel estimation model of (1).

And 4, performing channel estimation multi-model weighted soft handover.

FIG. 2 is M obtained by simulation₁And M₂Error curves of the channel estimation model and the switched model. In the figure, the abscissa is the Doppler shift formed by the mobile station in simulation, the ordinate is the minimum mean square error generated by the various models, and curve 1 is M₁Error curve of channel modelCurve 2 is M₂Curve 3 is the error curve after this channel estimation multi-model weighted soft handover method. As can be seen from FIG. 2, M₁The minimum mean square error of the LS estimation method in the channel model under low Doppler frequency shift is between-16 dB and-17 dB, and the LS estimation method has better performance. But due to the limitation of the estimation method, the estimation method is seriously influenced by noise, so that the performance is seriously reduced under high Doppler frequency shift. M₂The Kalman estimation method in the channel model has obvious effect of resisting noise, the minimum mean square error value can be about-16 dB, but the performance does not have M under the medium and low Doppler frequency shift₁The effect is good. Curve 3 is significantly lower than curves 1 and 2, indicating the performance of the channel estimation multi-model weighted soft handoff method of the present invention, compared to M alone₁And M₂Compared with an estimation model, the method has obvious improvement, and particularly achieves good optimization effect (a few dB of remarkable improvement) at the intersection.

Claims

1. A channel self-adaptive estimation method based on multi-model weighted soft handover is characterized by comprising the following steps:

step 1, establishing a channel system model;

adopting a time-frequency double channel selection; the influence coefficient of the transmitting symbol at the time (T-tau) on the receiving symbol at the time (T-tau) is represented by h (T; tau), tau is time delay, and the sampling intervals of the transmitting sequence and the receiving sequence are assumed to be T_sBy usingReplacing tau; the transmission symbols of the data block transmission system are u (i),taking N consecutive symbols in u (i) to form the k-th data block u_k，1≤N≤200，k≥1；

Assuming that the channel is a Rayleigh channel, the maximum time delay and the maximum Doppler shift of the path of the channel are tau_maxAnd f_maxSatisfy 2 τ_maxf_max< 1 and its channel state is slowly varying over a range of transmission data blocks; the data block u_kChannel h on_k(n; l) is represented by the basis extension model as follows:

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taking a data block u (i) as an estimation unit, and adopting (Q +1) × (L +1) C of a channel base model_k,q(l) Represents N × (L +1) unknown quantities h_k(n; l) simplifying the channel model and the estimation parameters; according to the Rayleigh channel characteristics, C_k,q(l) Is a mean of zero and a variance ofComplex gaussian random variables of (a);

step 2, estimating channel response by adopting a pilot frequency symbol-assisted channel estimation method;

step 2.1, establishing a signal sending and receiving model;

sending end sends sequence u_kComprising two parts, i.e. information symbols s_gAnd a pilot symbol p_g(ii) a Uniformly inserting G pilot symbols p in one data block_gData block u_kExpressed as:

wherein the ZP (zero padding) pilot sequence is in the form ofTogether comprising N_sAn information symbol and N_pA pilot sequence symbol, N ═ N_s+N_p；

<mrow> <mi>y</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>L</mi> </munderover> <mi>h</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>;</mo> <mi>l</mi> <mo>)</mo> </mrow> <mi>u</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>-</mo> <mi>l</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>w</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

wherein w (i) represents a mean value of zero and a variance ofAdditive white gaussian noise of (1);

step 2.2, pilot frequency information is obtained;

based on a base extension model, corresponding to a transmitted data block u_kReceived data block y of_kExpressed as:

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>H</mi> <mi>k</mi> </msub> <msub> <mi>u</mi> <mi>k</mi> </msub> <mo>+</mo> <msub> <mi>w</mi> <mi>k</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>H</mi> <mi>k</mi> </msub> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>Q</mi> </munderover> <msub> <mi>F</mi> <mi>q</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

in the formula,is zero mean value,Additive white gaussian noise vector of variance, H_kIs a lower triangular matrix and satisfies [ H_k]_n,m＝h_k(n；n-m)，n≥m，C_k,qIs the first column of [ C_k,q(0),C_k,q(1),...,C_k,q(L),0,…0]^TToeplitz matrix of (D), F_q＝diag[f_q(1),f_q(2),…,f_q(N)]Is a pair of angular arrays;

receiving a data block y_kOne of which is affected only by the pilot sequenceThe expression is as follows:

in the formula,andare respectively H_kAnd w_kThe sub-matrix corresponding to the pilot p in (1), wherein the (Q +1) × (L +1) unknown parameters C are included_k,q(l) Corresponding to each p_gAt the receiving end, only (L +1) symbols affected only by pilot are obtained, so at least (Q +1) pilot sequences are inserted into one data block to obtain C_k,q(l) G is more than or equal to Q + 1; combining the formula (5) and the principle of mutual conversion between Toeplitz matrix and vector to obtain the vector:

P_gis formed by a pilot sequence p_gPilot symbol p in (1)_g,l(L +1) × (L +1) type Toeplitz matrix of construction, L1, …, L +1, due to p_gThe structures are identical, here omitting the subscript g, i.e., P₁＝P,…,P_GWhen P is equal to P, then [ P]_n,m＝p_L-m+n；

Step 2.3, calculating channel response;

channel response matrix C_kComprises the following steps:

step 3, establishing a channel estimation multi-model;

based on the analysis of a channel model and an estimation method, the channel model and the estimation method are used as a channel estimation model, and a multi-model base of a plurality of channel estimation models is established to cover a complex and variable channel environment, wherein the structure of the model and the controller is different from that of a model and a controller in a multi-model control theory;

step 4, channel estimation multi-model weighted soft switching is carried out;

in multi-model control, a soft handover strategy is adopted because a hard handover strategy generally requires that each model has the same structure and does not conform to the structure of a model set; the switching is completed by adopting a weighted multi-model self-adaptive estimation method of the minimum variance of the linear error, so that the performance improvement of the online switching can be effectively achieved;

the method for establishing the channel estimation multiple models in the step 3 comprises the following steps:

(1) establishing a channel model set combination;

B＝{b_j,j＝1,2|P-BEM,CE-BEM-AR} (10)

in addition, E ═ E₁＝LS,e₂Kalman is the set of estimation methods, two models M₁And M₂The model library of (a) is:

M＝{M₁(b₁,e₁),M₂(b₂,e₂)} (11)

(2) calculation model M₁The channel response of (a);

the estimated value of the channel response is then:

andthe relationship of (A) is the same as the formula (9);

(3) calculation model M₂The channel response of (a);

based on the basis of the basis expansion model and the sending and receiving signals, a state equation and an observation equation of Kalman filtering can be constructed:

wherein A is_k，v_kAs a known Doppler estimate f_dObtaining a state transition matrix and a state transition noise SNR; the kalman estimation procedure thus results as:

in the formula,in order to be a Kalman channel estimation value,is a zero matrix, e_kFor measuring errors, P_kTo estimate the covariance matrix of the error, then:

andthe relationship of (A) is the same as the formula (9);

the method for performing channel estimation multi-model weighted soft handover in step 4 is as follows:

(1) calculating a linear error minimum variance;

assume the model is M_jJ is 1,2, and the variance of the estimation output, the estimation error, and the estimation error are:v_k,l,jandthe relationship between the estimated output and the error is then:

wherein,for the covarianceRepresents; due to the sub-module tracking algorithm, the unit of switching is converted into a sub-block,actually expressed as the estimated channel for each of the aforementioned submodelsThe channel coefficient corresponding to the first sub-module in (1);

(2) calculating a model error;

combining the multi-model weighted adaptive control theory, the channel model switching adopts a weight switching algorithm, and estimation errors are usedVariance of (2)Calculating the weight:

<mrow> <mtable> <mtr> <mtd> <mrow> <msubsup> <mover> <mi>h</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> <mi>o</mi> </msubsup> <mo>=</mo> <msub> <mi>W</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>W</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>Z</mi> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <msubsup> <mover> <mi>&Delta;</mi> <mo>~</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>Z</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>Z</mi> <mi>T</mi> </msup> <msup> <mrow> <mo>(</mo> <msubsup> <mover> <mi>&Delta;</mi> <mo>~</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> <mn>2</mn> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>

estimation error v_k,l,jThe method comprises a model error and an estimation error; the statistical variance used for the model error is:

<mrow> <msub> <mi>E</mi> <msub> <mi>M</mi> <mi>j</mi> </msub> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>v</mi> </msub> <mo>-</mo> <mi>S</mi> <mo>)</mo> </mrow> <msubsup> <mi>R</mi> <mi>&alpha;</mi> <mn>0</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>I</mi> <mi>v</mi> </msub> <mo>-</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>

wherein E is_MjIs v is_k,l,jSome of these, calculated values of model error; s ═ F^T(FF^T)^-1F, and [ F]_n,q＝f_q，j(n)。

(3) Calculating an LS estimation error and an estimation error variance of the weight;

therefore, the first and second electrodes are formed on the substrate,obtaining:

wherein phi⁺＝(Φ^HΦ)^-1Φ^H；

Model M₁The estimation error variance used to calculate the weights is:

<mrow> <msubsup> <mi>&Delta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>E</mi> <msub> <mi>M</mi> <mn>1</mn> </msub> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow> <mi>L</mi> <mi>S</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>

in the formula,as model M₁Total estimation error of E_LSAs model M₁The estimated error variance of (2);

(4) calculating the kalman estimation error and the estimation error variance of the weight;

E_kal＝BP_clB (23)

wherein [ P ]_cl]_m',n'＝E{[e_cl,k]_q(l)[e_cl,k]_q'(l') } is the posterior error covariance of the base coefficient, [ P }_c]_m,n＝E{[e_c,k]_q(l)[e_c,k]_q'(l ') }, according to m' ═ l (Q +1) + Q +1, n '═ l' (Q +1)The corresponding conversion is carried out on the relations of + q ' +1 and m ═ q (L +1) + L +1, and n ═ q ' (L +1) + L ' + 1;

model M₂The estimation error variance used to calculate the weights is:

<mrow> <msubsup> <mi>&Delta;</mi> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <msub> <mi>E</mi> <msub> <mi>M</mi> <mn>2</mn> </msub> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow> <mi>k</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>

in the formula,as model M₂Total estimation error of E_kalAs model M₂The estimated error variance of (2);

<mrow> <msubsup> <mover> <mi>h</mi> <mo>^</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> <mi>o</mi> </msubsup> <mo>=</mo> <munder> <mo>&Sigma;</mo> <mi>j</mi> </munder> <msup> <mrow> <mo>(</mo> <munder> <mo>&Sigma;</mo> <mi>j</mi> </munder> <msubsup> <mover> <mi>&Delta;</mi> <mo>~</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msubsup> <mover> <mi>&Delta;</mi> <mo>~</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> <msub> <mover> <mi>h</mi> <mo>~</mo> </mover> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>4