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; ***; 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-adapting method of estimation based on multi-model weighting soft handover

Technical field

The invention belongs to radio communication field, is related to a kind of channel self-adapting estimation based on multi-model weighting soft handover Method.

Background technology

Channel estimation method designs algorithm for estimating to obtain model ginseng for the channel model established according to different criterions Numerical value.Conventional channel estimation method has maximum likelihood ML estimations, EM estimations, and (EM algorithms are when observation data are incomplete, in fact The iterative algorithm of existing progressive ML estimations), LS estimations, RLS estimations, LMMSE estimations, Kalman filter etc..ML/EM estimations are based on most Maximum-likelihood criterion, be in the case where there is no any priori to estimative parameter, it is (complete using known some observations It is standby or incomplete) estimate the parameter, its validity is higher, and complexity is also higher；LS multiplies error with the two of estimate and desired value Target is minimised as, algorithm is simple, but is had a great influence by signal to noise ratio and CIR length, therefore passes through frequently as initial estimation step again It is improved with reference to other algorithms；Based on observation and channel statistic, LMMSE estimations are with the square of estimate and desired value It is that target is estimated that performance is relatively preferable that error, which minimizes (MMSE), and Kalman filter combination AR models are as a kind of MMSE Estimator, there is the characteristic of adaptive tracing, still can effectively be estimated under uncertain channel estimation quality event.

Analyze based on more than visible, establish channel model, design algorithm for estimating and effectively track channel status and be mutated into To improve the key of channel estimating performance.Current has researched and analysed the error under different models and method of estimation and has updated Method is to reduce error, but the performance of channel estimation errors of the different models and estimation procedure under different channel circumstances In the presence of very big difference, and the relatively good method of estimation of performance and channel statistic are closely related, and therefore, compel to be essential Strong channel estimation scheme is studied, the unknown complex environment applied to reality.

The content of the invention

In order to overcome the property of the channel estimation errors of different channel models and estimation procedure under different channel circumstances Can there are very big difference, and the deficiency that the acquisition of the relatively good method of estimation of performance and channel statistic is closely related, this Invention provides a kind of channel estimation multi-model weighting soft handover method.

To achieve these goals, the technical solution adopted by the present invention is：The various algorithm for estimating of wireless channel are analyzed first Performance and condition element, it is determined that using algorithm for estimating.Then, channel system model is established, selects time-frequency doubly selective channel, it is determined that Channel parameter.Channel estimation submodel and high spreadability and the multi-model of flexibility letter are established based on channel model and method of estimation Estimate storehouse in road.Analyze and calculate the model error and evaluated error of channel estimation model.Finally missed according to model error and estimation The switching index that difference combines, completes the switching of model, to reach excellent by LUMV weighting multi-model adaptive estimation algorithm The purpose of change.

The technical solution adopted in the present invention comprises the following steps：

Step 1, channel system model is established.

The present invention uses time-frequency doubly selective channel.With h (t；τ) represent that the transmission symbol at (t- τ) moment receives symbol to t Caused influence coefficient, τ are time delay, it is assumed that the sampling interval for sending sequence and receiving sequence is T_s, it is used in combinationReplace For τ.The transmission symbol of transmission of data blocks system is u (i),N (1≤N≤200) individual continuous symbol is taken in u (i) Form kth (k >=1) individual data block u_k。

Assuming that channel is Rayleigh channel, the path maximum delay and maximum doppler frequency of channel are τ_maxAnd f_max, and it is full 2 τ of foot_maxf_max＜ 1, and its channel status is slowly varying in the scope of a transmission data block.Then data block u_kOn Channel h_k(n；L) (Basic ExpendModel, BEM) model can be extended with base and represents as follows：

In formula,For basic function, 1≤n≤N, ω_q=π (q- (Q+1)/2)/N, 1≤q≤Q,For the number of base system number；C_k,q(l) it is the coefficient of basis expansion model, 1≤l≤L,

Using a data block u (i) as estimation unit, using the individual C of (Q+1) × (L+1) of channel basic mode type_k,q(l) table is carried out Show the individual unknown quantity h of N × (L+1)_k(n；L) so that channel model and estimation parameter are simplified.Can according to Rayleigh channel characteristic Know, C_k,_q(l) it is that average is zero, variance isMultiple Gauss stochastic variable.

Step 2, channel response is estimated using the channel estimation methods of Pilot symbols aided.

Step 2.1, transmission, receipt signal model are established.

Transmitting terminal sends sequence u_kIncluding two parts, i.e. information symbol s_gWith frequency pilot sign p_g.In a data block uniformly Insert G frequency pilot sign p_g, data block u_kIt is expressed as：

In formula, ZP (zero padding) pilot frequency sequence form isInclude N altogether_sIndividual information symbol and N_pIndividual sequence of pilot symbol, N=N_s+N_p。

Send symbol u (i) and pass through multipath channel h (i；L) the symbol y (i) received after is expressed as：

In formula, w (i) expression averages are zero, variance isAdditive white Gaussian noise (AWGN).

Step 2.2, pilot frequency information is obtained.

Based on basis expansion model, corresponding to transmission data block u_kReception data block y_kIt is represented by：

In formula,For zero-mean,The AWGN vectors of variance, N × N H_kFor lower triangular matrix and satisfaction [H_k]_n,m=h_k(n；N-m), n >=m, C_k,qIt is first to be classified as [C_k,q(0),C_k,q(1),...,C_k,q(L),0,…0]^T's Toeplitz matrixes, F_q=diag [f_q(1),f_q(2),…,f_q(N)] it is pair of horns battle array.

Receive data block y_kIn have it is a kind of only to be influenceed by pilot frequency sequenceIts expression formula is：

In formula,WithRespectively H_kAnd w_kIn the submatrix corresponding to pilot tone p.It is individual that formula includes (Q+1) × (L+1) Unknown parameter C_k,q(l) each p, is corresponded to_gReceiving terminal can only obtain (L+1) it is individual only by the symbol of pilot contribution, therefore need At least (Q+1) individual pilot frequency sequence is inserted in a data block and obtains C_k,q(l), i.e. G >=Q+1.With reference to (5) formula and Toeplitz matrixes obtain with vectorial mutually transfer principle：

In formula, Φ^pFor pilot frequency information transformed matrix, its expression formula is：

In formula,For F_qIn correspond to pilot frequency sequence p_gSubmatrix, its expression formula is：

P_gIt is by pilot frequency sequence p_gIn frequency pilot sign p_g,l(L+1) × (L+1) the type Toeplitz matrixes of construction, l= 1 ..., L+1, due to p_gStructure is identical, ignores subscript g, i.e. P herein₁=P ..., P_G=P, then [P]_n,m=p_L-m+n。

Step 2.3, calculating channel response.

Channel response matrix C_kFor：

Step 3, channel estimation multi-model is established.

Analysis based on channel model and method of estimation, with the structure of model and controller in Multiple model control theory not Together, the present invention is using channel model and method of estimation as a channel estimation model, by establishing multiple channel estimation models Multi-model storehouse covers the channel circumstance of complicated change.

Step 3.1, the combination of channel model collection is established.

Channel estimation model storehouse is M=(B, E), wherein, B is channel model collection, and its expression formula is：

B={ b_j, j=1,2 | P-BEM, CE-BEM-AR } (10)

Respectively by the basic function in formula (1) redefine forω_q=π (q- (Q+1)/2)/N andWherein increase subscript 1,2 to distinguish different models, it is follow-up to represent similarly.

In addition, E={ e₁=LS, e₂=Kalman }, it is method of estimation collection.

With two model Ms₁And M₂Exemplified by, model library is：

M={ M₁(b₁,e₁),M₂(b₂,e₂)} (11)

Step 3.2, calculation model M₁Channel response.

Model M₁LS algorithm for estimating and pilot point based on extension receive to send signal and least square method is obtained and extended The LS estimates of the corresponding coefficient of base are：

Then the estimate of channel response is：

WithRelation with (9) formula, it is corresponding to increase subscript.

Step 3.3, calculation model M₂Channel response.

Based on BEM models and reception signal is sent, the state equation and observational equation of Kalman filter can be constructed：

Wherein, A_k, v_kFor known Doppler's estimate f_dState-transition matrix and state transfer the noise SNR of acquisition.

Thus obtaining kalman estimation procedures is：

In formula,For Kalman channel estimation values,Initial value be null matrix, e_kFor measurement error, G_kFor Kalman gains, P_kFor estimation error covariance matrix, then：

WithRelation with (9) formula, it is corresponding to increase subscript.

The channel estimation model storehouse that the above is established covers the different channels parameter situation of Doppler and signal to noise ratio condition, It can be seen that the model based on P-BEM and CE-BEM-AR has good performance on low Doppler frequency shift and high doppler shift； Meanwhile LS estimations have extraordinary performance under high SNR, and Kalman estimations effect in the case of low SNR is pretty good.

Step 4, channel estimation multi-model weighting soft handover is carried out.

In the Multiple model control of the present invention, because direct-cut operation strategy usually requires that each model is respectively provided with identical knot Structure, Models Sets composition is not met, so using Soft Switch Method.The present invention uses a kind of linearity error minimum variance (Linear Unbiased MinimumVariance-LUMV) weighting multi-model adaptive estimation method complete to switch, can effectively reach To the performance boost switched online.

Step 4.1, LUMV is calculated.

Hypothesized model is M_j, j=1,2, it estimates that output, evaluated error and variance of estimaion error are respectively： v_k,l,jWithThen the relation of estimation output and error is：

Wherein, correspond toActual channel values with h_k,lRepresent,Covariance use Represent, I is unit matrix, and J represents the model quantity currently considered, there is J=2.

Step 4.2, computation model error.

With reference to multi-model Weighted adaptive control theory, the channel model switching in the present invention uses weights handoff algorithms, Use evaluated errorVarianceCalculate weights：

Evaluated error v_k,l,jMainly it is made up of model error and evaluated error two parts.The statistical variance that model error uses For：

Wherein,For v_k,l,jThe calculated value of a part therein, i.e. model error；S=F^T(FF^T)^-1F, and [F]_n,q =f_{Q, j}(n)。

Step 4.3, LS evaluated errors and the estimation error variance of weights are calculated.

On the basis of LS estimation error covariances come from its noise variance, due to：

Obtain：

Wherein, Φ⁺=((Φ^p)^HΦ^p)^-1(Φ^p)^H.From formula (12), this is sentenced

Model M₁Estimation error variance for calculating weights is：

In formula,For model M₁Overall estimate error, E_LS,lFor model M₁Estimation error variance；

Step 4.4, kalman evaluated errors and the estimation error variance of weights are calculated.

Kalman evaluated errors covariance is calculated by online 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') it is } the posteriori error covariance of base system number, [P_k]_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)+q'+1 and m=q (L+1) + l+1, n=q'(L+1)+l'+1 relation carry out corresponding conversion；

Model M₂Estimation error variance for calculating weights is：

In formula,For model M₂Overall estimate error, E_kal,lFor model M₂Estimation error variance；

In handoff procedure, for the ease of calculating, it is assumed that each h_k(n；L) covariance is identical；It can thus change EnterAnd obtain：

In accordance with the above with LUMV estimation theories, it is assumed that multiple submodels obtain independent channel estimation results, and will Its weighted sum, then lower Cramer-Rao circle can be obtained, and improve channel estimating performance.

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

The present invention is proposed under transmission channel model condition of uncertainty, with reference to channel model and method of estimation, establish and Selection estimation model automatically switches to the multi-model channel estimation Adaptive Control Theory of best performance state to reach.By multimode Multi-model thought in type Adaptive Control Theory is incorporated into channel estimation, and using a kind of linearity error minimum variance Multi-model adaptive estimation method is weighted to complete to switch, using the high spreadability of multi-model and flexibility, makes channel estimation side Method has high robust and accuracy in the range of Complex Channel.

Brief description of the drawings

Fig. 1 is the channel self-adapting method of estimation flow chart that soft handover is weighted based on multi-model；

Fig. 2 is that the embodiment of the present invention emulates obtained channel estimation errors curve, in figure：Curve 1 is M₁Channel model Error curve, curve 2 are M₂The error curve of channel model, curve 3 are after channel estimation multi-model weights soft handover method Error curve.

Embodiment

The present invention is further described below in conjunction with the accompanying drawings.

The present embodiment uses Matlab simulation softwares, and method flow diagram is as shown in figure 1, comprise the following steps：

Step 1, channel system model is established.

Establish the Ruili fading channel of a random time frequency selection.

Set channel parameter：Carrier frequency is f_c=2GHz, sampling interval T_s=25 μ s, and L=2, N=140, letter The maximum doppler frequency in road is f_max=370Hz.

Step 2, channel response is estimated using Pilot symbols aided method.

Using the channel estimation methods of Pilot symbols aided, the transmission symbol of transmission of data blocks system is u (i),

N=200 continuous symbol is taken to form kth (k=60) individual data block u in u (i)_k。

Each data block includes two parts, i.e. information symbol s_gWith frequency pilot sign p_g.Uniformly inserted in a data block G=8 frequency pilot sign p_g。

Step 3, channel estimation multi-model is established.

Establish two model Ms₁And M₂Channel estimation model.

Step 4, channel estimation multi-model weighting soft handover is carried out.

Fig. 2 is by emulating obtained M₁And M₂The error curve of model after channel estimation model and switching.In figure, horizontal seat The Doppler frequency shift that the mobile station being designated as in emulation is formed, ordinate is least mean-square error caused by various models, and curve 1 is M₁The error curve of channel model, curve 2 are M₂Error curve, curve 3 soft is cut by the weighting of this channel estimation multi-model Error curve after the method changed.As seen from Figure 2, M₁LS methods of estimation in channel model are under low Doppler frequency shift Least mean-square error has preferable performance between -16 arrive -17dB.But due to the limitation of method of estimation, its is affected by noise Than more serious, cause hydraulic performance decline under high doppler shift serious.M₂Kalman methods of estimation in channel model for The resistance significant effect of noise, its least mean-square error value can be in -16dB or so, but the performance under low Doppler frequency shift in There is no M₁Effect is good.Curve 3 is significantly lower than curve 1 and curve 2, shows to weight by channel estimation multi-model of the present invention Performance after the method for soft handover, with single M₁And M₂Estimation model is compared, and is had more obvious raising, is especially crossed at it Place has reached relatively good effect of optimization (several dB's significantly improves).

Claims

1. a kind of channel self-adapting method of estimation based on multi-model weighting soft handover, it is characterised in that comprise the following steps：

Step 1, channel system model is established；

Using time-frequency doubly selective channel；With h (t；τ) represent the transmission symbol at (t- τ) moment on influence caused by t reception symbol Coefficient, τ are time delay, it is assumed that the sampling interval for sending sequence and receiving sequence is T_s, useSubstitute τ；Data block passes The transmission symbol of defeated system is u (i),N number of continuous symbol is taken to form k-th of data block u in u (i)_k, 1≤N≤ 200, k >=1；

Assuming that channel is Rayleigh channel, the path maximum delay and maximum doppler frequency of channel are τ_maxAnd f_max, meet 2 τ_maxf_max＜ 1, and its channel status is slowly varying in the scope of a transmission data block；Then data block u_kOn channel h_k (n；L) represent as follows with basis expansion model：

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In formula,For basic function, 1≤n≤N, ω_q=π (q- (Q+1)/2)/N, 1≤q≤Q, For the number of base system number；C_k,q(l) it is the coefficient of basis expansion model, 1≤l≤L,

Using a data block u (i) as estimation unit, using the individual C of (Q+1) × (L+1) of channel basic mode type_k,q(l) represent N × (L+1) individual unknown quantity h_k(n；L), it is simplified channel model and estimation parameter；Understood according to Rayleigh channel characteristic, C_k,q(l) It is that average is zero, variance isMultiple Gauss stochastic variable；

Step 2, channel response is estimated using the channel estimation methods of Pilot symbols aided；

Step 2.1, transmission, receipt signal model are established；

Transmitting terminal sends sequence u_kIncluding two parts, i.e. information symbol s_gWith frequency pilot sign p_g；Uniformly inserted in a data block G frequency pilot sign p_g, data block u_kIt is expressed as：

In formula, ZP (zero padding) pilot frequency sequence form isInclude N altogether_sIndividual information symbol and N_pIt is individual Sequence of pilot symbol, N=N_s+N_p；

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In formula, w (i) expression averages are zero, variance isAdditive white Gaussian noise；

Step 2.2, pilot frequency information is obtained；

Based on basis expansion model, corresponding to transmission data block u_kReception data block y_kIt is expressed 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 formula,For zero-mean,The additive white Gaussian noise vector of variance, H_kFor lower triangular matrix and satisfaction [H_k]_n,m=h_k(n；N-m), n >=m, C_k,qIt is first to be classified as [C_k,q(0),C_k,q(1),...,C_k,q(L),0,…0]^T's Toeplitz matrixes, F_q=diag [f_q(1),f_q(2),…,f_q(N)] it is pair of horns battle array；

Receive data block y_kIn have and a kind of only influenceed by pilot frequency sequenceIts expression formula is：

In formula,WithRespectively H_kAnd w_kIn the submatrix corresponding to pilot tone p；It is individual unknown that formula includes (Q+1) × (L+1) Parameter C_k,q(l) each p, is corresponded to_gReceiving terminal can only obtain (L+1) it is individual only by the symbol of pilot contribution, therefore need to be one At least (Q+1) individual pilot frequency sequence is inserted in individual data block and obtains C_k,q(l), i.e. G >=Q+1；With reference to (5) formula and Toeplitz squares Battle array obtains with vectorial mutually transfer principle：

P_gIt is by pilot frequency sequence p_gIn frequency pilot sign p_g,l(L+1) × (L+1) the type Toeplitz matrixes of construction, l=1 ..., L+ 1, due to p_gStructure is identical, ignores subscript g, i.e. P herein₁=P ..., P_G=P, then [P]_n,m=p_L-m+n；

Step 2.3, calculating channel response；

Channel response matrix C_kFor：

Step 3, channel estimation multi-model is established；

Analysis based on channel model and method of estimation, it is different from the structure of model and controller in Multiple model control theory, will Channel model and method of estimation are covered as a channel estimation model by establishing the multi-model storehouse of multiple channel estimation models Lid is complicated, the channel circumstance of change；

Step 4, channel estimation multi-model weighting soft handover is carried out；

In Multiple model control, because direct-cut operation strategy usually requires that each model is respectively provided with identical structure, Models Sets are not met Form, so using Soft Switch Method；Using a kind of weighting multi-model adaptive estimation method of linearity error minimum variance come Switching is completed, can effectively reach the performance boost switched online；

The method that the step 3 establishes channel estimation multi-model is as follows：

(1) combination of channel model collection is established；

B={ b_j, j=1,2 | P-BEM, CE-BEM-AR } (10)

In addition, E={ e₁=LS, e₂=Kalman }, it is method of estimation collection, two model Ms₁And M₂Model library be：

M={ M₁(b₁,e₁),M₂(b₂,e₂)} (11)

(2) calculation model M₁Channel response；

Model M₁LS algorithm for estimating and pilot point receiving transmission signal and least square method based on extension obtain relative with extension base The LS estimates for the coefficient answered are：

Then the estimate of channel response is：

WithRelation with (9) formula；

(3) calculation model M₂Channel response；

Based on basis expansion model and reception signal is sent, the state equation and observational equation of Kalman filter can be constructed：

Wherein, A_k, v_kFor known Doppler's estimate f_dState-transition matrix and state transfer the noise SNR of acquisition；Thus It is to kalman estimation procedures：

In formula,For Kalman channel estimation values,Initial value be null matrix, e_kFor measurement error, P_kFor evaluated error Covariance matrix, then：

WithRelation with (9) formula；

The method that the step 4 carries out channel estimation multi-model weighting soft handover is as follows：

(1) linearity error minimum variance is calculated；

Hypothesized model is M_j, j=1,2, it estimates that output, evaluated error and variance of estimaion error are respectively：v_k,l,jWithThen the relation of estimation output and error is：

Wherein,Covariance useRepresent；Due to using submodule tracing algorithm, switching Unit is changed into a sub-block,Actually it is expressed as each submodel estimation channel of foregoing correspondenceIn first submodule Corresponding channel coefficients；

(2) computation model error；

With reference to multi-model Weighted adaptive control theory, channel model switching uses weights handoff algorithms, uses evaluated error VarianceCalculate weights：

<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>

Evaluated error v_k,l,jIt is made up of model error and evaluated error two parts；The statistical variance that model error uses 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_MjFor v_k,l,jThe calculated value of a part therein, i.e. model error；S=F^T(FF^T)^-1F, and [F]_n,q=f_{Q, j} (n)。

(3) LS evaluated errors and the estimation error variance of weights are calculated；

SoObtain：

Wherein, Φ⁺=(Φ^HΦ)^-1Φ^H；

Model M₁Estimation error variance for calculating 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 formula,For model M₁Overall estimate error, E_LSFor model M₁Estimation error variance；

(4) kalman evaluated errors and the estimation error variance of weights are calculated；

E_kal=BP_clB (23)

Wherein, [P_cl]_m',n'=E { [e_cl,k]_q(l)[e_cl,k]_q'(l') it is } the posteriori error covariance of base system number, [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)+q'+1 and m=q (L+1)+l+1, n= Q'(L+1)+l'+1 relation carries out corresponding conversion；

Model M₂Estimation error variance for calculating 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 formula,For model M₂Overall estimate error, E_kalFor model M₂Estimation error variance；

In handoff procedure, for the ease of calculating, it is assumed that each h_k(n；L) covariance is identical；It can thus improveAnd obtain：

<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