CN104536017B - A kind of navigation neceiver STAP method of Beam synthesis after first subspace projection - Google Patents

Abstract

The invention discloses the navigation neceiver STAP algorithm of Beam synthesis after a kind of first subspace projection, first reception data covariance matrix is carried out SVD decomposition, big eigenvalue characteristic of correspondence vector composition interference space, little eigenvalue characteristic of correspondence vector composition noise subspace, reception data vector is realized AF panel to noise subspace projection, then the local pseudo-code using navigation signal carries out Beam synthesis as reference signal to array weight, makes array antenna main beam point to navigation signal arrival bearing.The advantage that the present invention has the tracking accuracy improving array output Signal to Interference plus Noise Ratio and signal.

Description

A kind of navigation neceiver STAP method of Beam synthesis after first subspace projection

Technical field

The present invention relates to terminal unit and develop field, be specifically related to a kind of raising navigation terminal equipment anti-interference Can be with the method for tracking accuracy, it can be used in and use satellite navigation system of array antenna to receive terminal class etc. In equipment.

Background technology

Array antenna has the superiority that traditional antenna is incomparable, increases as obtained higher space diversity Benefit, it is often more important that can effectively realize airspace filter, and the acquisition of information in space can be greatly increased Amount, detection and estimation for spacing wave provide strong support.Digital beam forming technology uses exactly Aerial array uses the method for Digital Signal Processing that spacing wave is made the technology of optimum reception, has and makes antenna Array beam maximum gain direction alignment desired signal, zero falls into alignment interference arrives signal with Wave beam forming simultaneously The advantage of directional information, plays pivotal role in Array Signal Processing direction, and at communication, radar, sound Receive, navigate and the field such as electronic countermeasure obtains and extensively uses and develop rapidly.

Space-time adaptive processes (Space-Time Adaptive Processing, STAP) technology is not increasing array element On the premise of, by increasing delay unit after each array element, adaptive array can be increased largely The degree of freedom of row, it is possible to simultaneously realize arrowband and the purpose of broadband interference suppression, STAP answering in navigation field With there being the most wide prospect.

Array signal STAP is through seeking optimum weight vector under certain optimization criterion and reaches to suppress dry Disturb signal and protect the purpose of useful signal.Conventional optimization criterion mainly has output to minimize (MV) Criterion, criterion that minimum variance is undistorted (MVDR), least mean-square error (MMSE) criterion and line Property constraint minimum variance (LCMV) criterion etc..Wherein MV criterion is much larger than according to interfering signal power and leads The hypothesis of boat signal power adjusts weights makes output minimum to realize the purpose of AF panel, and this algorithm is not Needing other prior informations, range is wide, can effectively carry out AF panel, but this criterion can not make The main beam of array antenna points to navigation signal arrival bearing, it is impossible to strengthen navigation signal, Er Qiexu Will invert data covariance, operand is bigger.MVDR criterion makes the fixing increasing of signal arrival bearing satisfied Making output minimum strengthen to realize AF panel and signal on the premise of benefit, this criterion needs known signal The array steering vector of arrival bearing, when direction of signal and aerial array geometry are unknown or there is bigger mistake During difference, algorithm performance cannot ensure.LCMV criterion makes output minimum under conditions of multiple constraints, with MVDR criterion is similar to, and needs known multiple prior information.MMSE criterion makes array defeated by adjusting weights Go out the mean square error residual quantity with reference signal minimum, the lowest, often owing to receiving the power of navigation signal In the case of rule, MMSE criterion will not obtain good interference suppressioning effect, and MMSE criterion needs known ginseng Examine signal.

Summary of the invention

Considering the strongly disturbing feature of satellite navigation low signal-to-noise ratio, the present invention proposes a kind of first subspace projection postwave The navigation neceiver STAP method of Shu Hecheng.First reception data covariance matrix is carried out SVD decomposition, Big eigenvalue characteristic of correspondence vector composition interference space, little eigenvalue characteristic of correspondence vector composition noise Subspace, realizes AF panel by reception data vector to noise subspace projection.Then navigation letter is used Number local pseudo-code as reference signal, array weight is carried out Beam synthesis, make array antenna main beam point to Navigation signal arrival bearing, thus improve array output Signal to Interference plus Noise Ratio and the tracking accuracy of signal, emulate and real Test the effectiveness of result verification algorithm.

Specifically, the navigation neceiver STAP side of Beam synthesis after the present invention provides a kind of first subspace projection Method, it is characterised in that first reception data covariance matrix being carried out SVD decomposition, big eigenvalue is corresponding Characteristic vector composition interference space, little eigenvalue characteristic of correspondence vector composition noise subspace, will receive Data vector realizes AF panel to noise subspace projection, then uses the local pseudo-code of navigation signal to make For reference signal, array weight is carried out Beam synthesis, make array antenna main beam point to navigation signal incoming wave side To.

Detailed process is as follows: set N number of snap data set that L array element of aerial array receives become column vector as

X (k)=[x₀(k)^T,x₁(k-T₀)^T,…,x_N-1(k-(N-1)T₀)^T]^T (1)

Wherein T₀For unit snap data delay, x_m(k-mT₀) be k moment m-th snap L tie up array data Column vector, then have

$x\left(k\right)=a\left(\mathrm{\θ}\right)s\left(k\right)+\underset{i=1}{\overset{M-1}{\mathrm{\Σ}}}b\left({\mathrm{\θ}}_i\right)j_i\left(k\right)+n\left(k\right)---\left(2\right)$

Wherein s (t), j (t) represent useful signal waveform and interference signal waveform, θ respectively_iAntenna array is incided for signal The elevation angle of row, a (θ) is navigation signal steering vector, b (θ when inciding array empty_i) it is that i-th disturbs signal Empty time steering vector, incoherent additivity white complex gaussian noise column vector when n (k) is empty.

Array weight vector is expressed as

W=[w₁₁,w₂₁,…,w_L1,…,w_1N,w_2N,…,w_LN]^T (3)

Then k moment antenna array receiver data output y after STAP can be expressed as

Y (k)=w^Hx(k) (4)

Array signal STAP is through seeking optimum weight vector w under certain optimization criterion and reaches to suppress dry Disturb signal and protect the purpose of useful signal.

Further, the process of subspace projection is as follows:

Array received data covariance matrix is carried out feature decomposition (SVD) and by eigenvalue matrix from greatly to Minispread, then have

Wherein R_XFor array received data covariance matrix, λ_j(j=1 ..., LN) it is the eigenvalue of array covariance matrix, And meet

λ₁≥λ₂≥…≥λ_P＞ λ_P+1=...=λ_LN (6)

u_jIt is characterized value λ_jCharacteristic of correspondence vector and mutually orthogonal.It is far smaller than dry owing to navigation signal receives power Disturbing signal and noise floor, therefore defining the subspace that P big eigenvalue characteristic of correspondence vector open is Interference space U_I, the subspace that LN-P corresponding little eigenvalue characteristic of correspondence vector is opened is that signal is made an uproar Phonon space U_N.Interference space and signal subspace orthocomplement, orthogonal complement subspace each other, and interference space is with dry Disturbing signal and inciding the subspace that the steering vector of array opens is same subspace, i.e. has

$U_I{U}_I^H+U_N{U}_N^H=I---\left(7\right)$

span{u₁,…,u_P}=span{b (θ₁),…,b(θ_M-1)} (8)

Therefore by antenna array receiver data vector is divided to signal noise subspace projection, high reject signal Amount can effectively be suppressed, and the non-interference data vector after subspace projection will only comprise noise and flood Navigation signal therein, makes P_⊥For projection operator, then have

$x_s\left(k\right)=P_{\⊥}x\left(k\right)=(I-U_I{U}_I^H)x\left(k\right)---\left(9\right)$

For improving the Signal to Interference plus Noise Ratio of array output further, propose first with the nothing after Orthogonal subspace projection Interference reference antenna array element data complete navigation signal pseudo-code phase and carrier doppler is estimated, then use same Local pseudo-code after step uses least mean-square error (MMSE) criterion to complete Beam synthesis as reference signal, And then the main beam of space-time adaptive array is pointed to navigation signal arrival bearing.

While receiver pseudo-code DLL loop is to the tracking of navigation signal, beamforming algorithm calculates and completes Renewal to space-time adaptive filter weights, thus the main ripple of aerial array in aeronautical satellite motor process Bundle points to the arrival bearing of navigation signal all the time, and lifting navigation is received by the array output signal of high Signal to Interference plus Noise Ratio The stability of machine track loop and tracking accuracy.

Further, the process of Beam synthesis is as follows:

Output data after subspace projection and Beam synthesis are believed for reference with the most synchronize local pseudo-code The margin of error between number can be expressed as

$\begin{array}{c}e\left(k\right)=r\left(k\right)-y\left(k\right)\\ =d(k-\mathrm{\τ})-w^H{P}_{\⊥}x\left(k\right)\end{array}---\left(10\right)$

Wherein r (t) is local reference signal, and d (t) is local pseudo-code, τ be navigation signal from satellite to receiver antenna The propagation delay of phase center, is obtained by reference antenna receives the capture of data essence.

Convenient in order to represent, time variable k in above formula to be omitted, such mean square error residual quantity can be expressed as

MSE=E{e²}=E{ | d-w^HP_⊥x|²} (11)

Utilize formula to seek the mean square error residual quantity MSE gradient to weight vector w, can obtain

${\▿}_w\left(MSE\right)=\frac{\mathrm{\δ}MSE}{{\mathrm{\δw}}^{*}}=-2P_{\⊥}{R}_{Xd}+2P_{\⊥}{R}_X{P}_{\⊥}w---\left(12\right)$

Wherein R_Xd=E{x^HD} is the cross-correlation vector of array received data and reference signal.Wave beam forming institute foundation MMSE criterion i.e. finds optimization weighted vector makes mean square error minimum, uses minimal gradient iterative algorithm, The recursion more new formula that can obtain array weight vector is

W (k+1)=w (k)+μ_w(MSE) (13)

In formula, μ is the step factor of minimal gradient algorithm, uses beamforming algorithm after therefore can obtaining subspace projection Right value update formula is

W (k+1)=(I+2 μ P_⊥R_XP_⊥)w(k)-2μP_⊥R_Xd (14)

Further, simulation analysis is finally carried out:

Simulation means is used to verify the effectiveness of the method.Investigating and using 5 array elements is 0.5 λ composition with spacing Even linear array (ULA) is receiving the navigation neceiver performance of GPS B3 frequency navigation signal, and wherein λ is The wavelength of B3 frequency mid frequency 1268.52MHz.After each array element, the exponent number of adaptive temporal filter device is 7.Setting simulating scenes and incide the elevation angle of array as 34 ° as signal, carrier-to-noise ratio is 50dBHz, 1 list Frequently continuous wave CO_2 laser (CWI) and two arrowbands interference (NBI) are respectively with 15 °, and 62 ° and 4 ° of elevations angle enter Penetrating, dry making an uproar than all is 50dB, and three interfering frequencies are respectively relative to the deviation of B3 frequency mid frequency -3MHz, 0 and 6MHz.The jamming bandwidth of arrowband interference is all 2MHz.By to using subspace projection to calculate Method RELATED APPLICATIONS array element receives the power spectral density contrast of data, it can be seen that through subspace projection interference letter Number effectively suppressed.Beam direction is processed from using the subspace projection space-time adaptive with Wave beam forming Figure, it can be seen that fall into defining the point-like degree of depth zero at the angle and frequency location of mono-tone interference, and two Individual arrowband disturbs out and defines banding zero and fall into, and zero to fall into the degree of depth be about 45dB, and the bandwidth of interference is the widest, is formed Banding zero fall into the widest.And through Beam synthesis, the maximum gain of array pattern is positioned at coming of signal Near 34 ° of ripple direction, maximum gain amplitude is about 5dB.

Accompanying drawing explanation

The principle process schematic diagram of Fig. 1 present invention.

Detailed description of the invention

The present invention provides the concrete of a kind of navigation neceiver STAP method of Beam synthesis after first subspace projection Embodiment, first carries out SVD decomposition, big eigenvalue characteristic of correspondence vector to reception data covariance matrix Composition interference space, little eigenvalue characteristic of correspondence vector composition noise subspace, data vector will be received Realize AF panel to noise subspace projection, then use the local pseudo-code of navigation signal as with reference to letter Number array weight is carried out Beam synthesis, make array antenna main beam point to navigation signal arrival bearing.Such as Fig. 1 Shown in, comprise the following steps:

Step S1, uses subspace projection algorithm to realize AF panel, and its calculation procedure is

Array received data vector is

X (k)=[x₀(k)^T,x₁(k-T₀)^T,…,x_N-1(k-(N-1)T₀)^T]^T (15)

Data covariance matrix calculates

R_X=E{xx^H} (16)

Data covariance matrix is carried out SVD decomposition

And eigenvalue is arranged from big to small

λ₁≥λ₂≥…≥λ_P＞ λ_P+1=...=λ_LN (18)

Wherein u_jIt is characterized value λ_jCharacteristic of correspondence vector and mutually orthogonal, will receive data vector empty to noise Between be projected as

$x_s\left(k\right)=P_{\⊥}x\left(k\right)=(I-U_I{U}_I^H)x\left(k\right)---\left(19\right)$

Step S2, least mean-square error Beam synthesis, its calculation procedure is

Output data after subspace projection and Beam synthesis and with local pseudo-code as reference signal between The margin of error is

E (k)=r (k)-y (k)=d (k-τ)-w^HP_⊥x(k) (20)

Seek the mean square error residual quantity MSE gradient to weight vector w

${\▿}_w\left(MSE\right)=\frac{\mathrm{\δ}MSE}{{\mathrm{\δw}}^{*}}=-2P_{\⊥}{R}_{Xd}+2P_{\⊥}{R}_X{P}_{\⊥}w---\left(21\right)$

The recursion more new formula that can obtain array weight vector is

W (k+1)=w (k)+μ_w(MSE)=(I+2 μ P_⊥R_XP_⊥)w(k)-2μP_⊥R_Xd (22)

Detailed process is as follows: set N number of snap data set that L array element of aerial array receives become column vector as

X (k)=[x₀(k)^T,x₁(k-T₀)^T,…,x_N-1(k-(N-1)T₀)^T]^T (1)

Wherein T₀For unit snap data delay, x_m(k-mT₀) be k moment m-th snap L tie up array data Column vector, then have

$x\left(k\right)=a\left(\mathrm{\θ}\right)s\left(k\right)+\underset{i=1}{\overset{M-1}{\Σ}}b\left({\mathrm{\θ}}_i\right)j_i\left(k\right)+n\left(k\right)---\left(2\right)$

Wherein s (t), j (t) represent useful signal waveform and interference signal waveform, θ respectively_iAntenna array is incided for signal The elevation angle of row, a (θ) is navigation signal steering vector, b (θ when inciding array empty_i) it is that i-th disturbs signal Empty time steering vector, incoherent additivity white complex gaussian noise column vector when n (k) is empty.

Array weight vector is expressed as

W=[w₁₁,w₂₁,…,w_L1,…,w_1N,w_2N,…,w_LN]^T (3)

Then k moment antenna array receiver data output y after STAP can be expressed as

Y (k)=w^Hx(k) (4)

Array signal STAP is through seeking optimum weight vector w under certain optimization criterion and reaches to suppress dry Disturb signal and protect the purpose of useful signal.

Further, the process of subspace projection is as follows:

Array received data covariance matrix is carried out feature decomposition (SVD) and by eigenvalue matrix from greatly to Minispread, then have

Wherein R_XFor array received data covariance matrix, λ_j(j=1 ..., LN) it is the eigenvalue of array covariance matrix, And meet

λ₁≥λ₂≥…≥λ_P＞ λ_P+1=...=λ_LN (6)

u_jIt is characterized value λ_jCharacteristic of correspondence vector and mutually orthogonal.It is far smaller than dry owing to navigation signal receives power Disturbing signal and noise floor, therefore defining the subspace that P big eigenvalue characteristic of correspondence vector open is Interference space U_I, the subspace that LN-P corresponding little eigenvalue characteristic of correspondence vector is opened is that signal is made an uproar Phonon space U_N.Interference space and signal subspace orthocomplement, orthogonal complement subspace each other, and interference space is with dry Disturbing signal and inciding the subspace that the steering vector of array opens is same subspace, i.e. has

$U_I{U}_I^H+U_N{U}_N^H=I---\left(7\right)$

span{u₁,…,u_P}=span{b (θ₁),…,b(θ_M-1)} (8)

Therefore by antenna array receiver data vector is divided to signal noise subspace projection, high reject signal Amount can effectively be suppressed, and the non-interference data vector after subspace projection will only comprise noise and flood Navigation signal therein, makes P_⊥For projection operator, then have

$x_s\left(k\right)=P_{\⊥}x\left(k\right)=(I-U_I{U}_I^H)x\left(k\right)---\left(9\right)$

For improving the Signal to Interference plus Noise Ratio of array output further, propose first with the nothing after Orthogonal subspace projection Interference reference antenna array element data complete navigation signal pseudo-code phase and carrier doppler is estimated, then use same Local pseudo-code after step uses least mean-square error (MMSE) criterion to complete Beam synthesis as reference signal, And then the main beam of space-time adaptive array is pointed to navigation signal arrival bearing.

While receiver pseudo-code DLL loop is to the tracking of navigation signal, beamforming algorithm calculates and completes Renewal to space-time adaptive filter weights, thus the main ripple of aerial array in aeronautical satellite motor process Bundle points to the arrival bearing of navigation signal all the time, and lifting navigation is received by the array output signal of high Signal to Interference plus Noise Ratio The stability of machine track loop and tracking accuracy.

Further, the process of Beam synthesis is as follows:

Output data after subspace projection and Beam synthesis are believed for reference with the most synchronize local pseudo-code The margin of error between number can be expressed as

$\begin{array}{c}e\left(k\right)=r\left(k\right)-y\left(k\right)\\ =d(k-\mathrm{\τ})-w^H{P}_{\⊥}x\left(k\right)\end{array}---\left(10\right)$

Wherein r (t) is local reference signal, and d (t) is local pseudo-code, τ be navigation signal from satellite to receiver antenna The propagation delay of phase center, is obtained by reference antenna receives the capture of data essence.

Convenient in order to represent, time variable k in above formula to be omitted, such mean square error residual quantity can be expressed as

MSE=E{e²}=E{ | d-w^HP_⊥x|²} (11)

Utilize formula to seek the mean square error residual quantity MSE gradient to weight vector w, can obtain

${\▿}_w\left(MSE\right)=\frac{\mathrm{\δ}MSE}{{\mathrm{\δw}}^{*}}=-2P_{\⊥}{R}_{Xd}+2P_{\⊥}{R}_X{P}_{\⊥}w---\left(12\right)$

Wherein R_Xd=E{x^HD} is the cross-correlation vector of array received data and reference signal.Wave beam forming institute foundation MMSE criterion i.e. finds optimization weighted vector makes mean square error minimum, uses minimal gradient iterative algorithm, The recursion more new formula that can obtain array weight vector is

W (k+1)=w (k)+μ_w(MSE) (13)

In formula, μ is the step factor of minimal gradient algorithm, uses beamforming algorithm after therefore can obtaining subspace projection Right value update formula is

W (k+1)=(I+2 μ P_⊥R_XP_⊥)w(k)-2μP_⊥R_Xd (14)

Further, simulation analysis is finally carried out:

Simulation means is used to verify the effectiveness of the method.Investigating and using 5 array elements is 0.5 λ composition with spacing Even linear array (ULA) is receiving the navigation neceiver performance of GPS B3 frequency navigation signal, and wherein λ is The wavelength of B3 frequency mid frequency 1268.52MHz.After each array element, the exponent number of adaptive temporal filter device is 7.Setting simulating scenes and incide the elevation angle of array as 34 ° as signal, carrier-to-noise ratio is 50dBHz, 1 list Frequently continuous wave CO_2 laser (CWI) and two arrowbands interference (NBI) are respectively with 15 °, and 62 ° and 4 ° of elevations angle enter Penetrating, dry making an uproar than all is 50dB, and three interfering frequencies are respectively relative to the deviation of B3 frequency mid frequency -3MHz, 0 and 6MHz.The jamming bandwidth of arrowband interference is all 2MHz.By to using subspace projection to calculate Method RELATED APPLICATIONS array element receives the power spectral density contrast of data, it can be seen that through subspace projection interference letter Number effectively suppressed.Beam direction is processed from using the subspace projection space-time adaptive with Wave beam forming Figure, it can be seen that fall into defining the point-like degree of depth zero at the angle and frequency location of mono-tone interference, and two Individual arrowband disturbs out and defines banding zero and fall into, and zero to fall into the degree of depth be about 45dB, and the bandwidth of interference is the widest, is formed Banding zero fall into the widest.And through Beam synthesis, the maximum gain of array pattern is positioned at coming of signal Near 34 ° of ripple direction, maximum gain amplitude is about 5dB.

In sum, although the present invention is disclosed above with preferred embodiment, and so it is not limited to the present invention, Any those of ordinary skill in the art, without departing from the spirit and scope of the present invention, when making various change With retouching, therefore protection scope of the present invention is when defining in the range of standard depending on claims.

Claims (1)

1. the navigation neceiver STAP method of Beam synthesis after a first subspace projection, it is characterised in that

S1. reception data covariance matrix is carried out SVD decomposition, big eigenvalue characteristic of correspondence vector composition interference space, little eigenvalue characteristic of correspondence vector composition noise subspace, realizes AF panel by reception data vector to noise subspace projection, and its step is as follows:

S1.1 set N number of snap data set that L array element of aerial array receives become column vector as

X (k)=[x₀(k)^T, x₁(k-T₀)^T..., x_N-1(k-(N-1)T₀)^T]^T (1)

Wherein T₀For unit snap data delay, x_m(k-mT₀) be k moment m-th snap L tie up array data column vector, then have

Wherein s (t), j (t) represent useful signal waveform and interference signal waveform, θ respectively_iIncide the elevation angle of aerial array for signal, a (θ) is navigation signal steering vector, b (θ when inciding array empty_i) be i-th interference signal empty time steering vector, n (k) be sky time incoherent additivity white complex gaussian noise column vector；

Array weight vector is expressed as

W=[w₁₁, w₂₁..., w_L1..., w_1N, w_2N..., w_LN]^T (3)

Then k moment antenna array receiver data output y after STAP can be expressed as

Y (k)=w^Hx(k) (4)

Array signal STAP is through seeking optimum weight vector w under optimization criterion and reaches suppression interference signal and protect useful signal；

S1.2 carries out feature decomposition and eigenvalue matrix is arranged from big to small array received data covariance matrix, then have

Wherein R_XFor array received data covariance matrix, λ_j, j=1 ..., LN, for the eigenvalue of array covariance matrix, and meet

λ₁≥λ₂≥…≥λ_P＞ λ_P+1=...=λ_LN (6)

u_jIt is characterized value λ_jCharacteristic of correspondence vector and mutually orthogonal, defining the subspace that P big eigenvalue characteristic of correspondence vector open is interference space U_I, the subspace that LN-P corresponding little eigenvalue characteristic of correspondence vector is opened is signal noise subspace U_N, interference space and signal subspace orthocomplement, orthogonal complement subspace each other, and interference space incide the subspace that the steering vector of array opens with interference signal be same subspace, i.e. have

span{u₁..., u_P}=span{b (θ₁) ..., b (θ_M-₁)} (8)

Therefore by can effectively be suppressed to signal noise subspace projection, high reject signal component by antenna array receiver data vector, the non-interference data vector after subspace projection will only comprise noise and flood navigation signal therein, make P_⊥For projection operator, then have

S2. utilize the noiseless reference antenna array element data after Orthogonal subspace projection to complete navigation signal pseudo-code phase and carrier doppler is estimated, then use the local pseudo-code after synchronizing to use minimum mean square error criterion to complete Beam synthesis as reference signal, make array antenna main beam point to navigation signal arrival bearing；

Output data after subspace projection and Beam synthesis with the most synchronize local pseudo-code as reference signal between the margin of error can be expressed as

Wherein r (t) is local reference signal, and d (t) is local pseudo-code, τ be navigation signal from satellite to the propagation delay of receiver antenna phase center, obtained by reference antenna being received the capture of data essence；

Time variable k in above formula being omitted, mean square error amount is expressed as

MSE=E{e²}=E{ | d-w^HP_⊥x|²} (11)

Utilize formula to seek the mean square error residual quantity MSE gradient to weight vector w, can obtain

Wherein R_Xd=E{x^HD} is the cross-correlation vector of array received data and reference signal；The MMSE criterion of Wave beam forming institute foundation i.e. finds optimization weighted vector makes mean square error minimum, uses minimal gradient iterative algorithm, and the recursion more new formula that can obtain array weight vector is

In formula, μ is the step factor of minimal gradient algorithm, and the right value update formula using beamforming algorithm after therefore can obtaining subspace projection is

W (k+1)=(I+2 μ P_⊥R_XP_⊥)w(k)-2μP_⊥R_Xd (14)。