CN106802403A - Acoustic vector sensors two-dimensional array MUSIC decorrelation LMS method for parameter estimation - Google Patents

Abstract

Acoustic vector sensors two-dimensional array MUSIC decorrelation LMS method for parameter estimation, K far field of array received arrowband coherent signal obtains the n times snapshot data that all array elements are exported using receiving array；Four submatrix data of acoustic pressure and the x, y, z axle velocity of sound are extracted, the data covariance matrix after decorrelation LMS is constructed by converting covariance matrix and the front and rear Cross-covariance of conversion after preceding data covariance matrix, data are converted；Singular value decomposition is carried out to the data covariance matrix after decorrelation LMS and obtains noise subspace, noise subspace is divided into x-axis and y-axis noise subspace according to the characteristics of L-type array, and corresponding MUSIC spatial spectrums are constructed, by two linear search direction cosines estimated matrix；Pairing computing is carried out to the direction cosines estimate in x-axis and y-axis direction using full array corresponding noise subspace, so as to obtain the estimate of angle of arrival；One two-dimensional search is divided into two linear searches by the method, greatly reduces amount of calculation, improves Parameter Estimation Precision.

Description

Acoustic vector sensors two-dimensional array MUSIC decorrelation LMS method for parameter estimation

Technical field

Coherent source two dimension the invention belongs to signal processing technology field, more particularly to a kind of acoustic vector-sensor array row is arrived Up to angular estimation method.

Background technology

In actual information transmitting procedure, due to multipath reflection and artificial disturbance, Correlated Signals have turned into generally existing As, in the case of there is strong correlation or coherent signal in space, the subspace class method failure with MUSIC as representative.Spatial domain angle Estimation can produce very big error, or even can not estimate the angle of arrival of signal, and the research of coherent Mutual coupling is The important subject of signal transacting.

Decorrelation LMS method with space smoothing as representative reduces array aperture, increases the beam angle of array, reduces The resolution capability of array.And space smoothing is typically only applicable to even linear array, the range of application of method is seriously limited.Sound is sweared Quantity sensor is a kind of new sound-source signal direction-finding equipment, and it is by three mutually orthogonal particle vibration velocity sensors and one Sound pressure sensor is constituted, it is thus possible to the sound pressure in somewhere and particle vibration velocity in synchro measure sound field.Acoustic vector-sensor array Compared with scalar sensors array, acoustic vector-sensor array row can not only obtain array aperture information to row, and contain arrow Quadrature information between each component of quantity sensor, thus with spatial resolution and direction finding precision higher, turned into recent years The hot issue of domestic and foreign scholars research.Existing method primarily directed to one-dimensional even linear array one-dimensional method for parameter estimation, For two-dimentional angle estimation technique study seldom, and often extremely complex, the paper that Wu little Qiang is delivered " is based on ESPRIT algorithms Arrival direction estimation technique study " have studied improved two dimension in (Harbin Engineering University's master thesis in 2008) ESPRIT algorithms and the two dimensional ESPRIT algorithm based on fourth order statistic treatment and space time processing, the method have certain solution phase Dry ability, but the method needs the extremely complex matrix of construction, and subsequent algorithm is also extremely complex；The present invention is directed to existing method Deficiency propose non-homogeneous L-type acoustic vector-sensor array row decorrelation LMS MUSIC methods, the method takes full advantage of acoustic vector biography The invariable rotary characteristic decorrelation LMS of sensor array itself, the method for referred to as rotating decorrelation LMS.And using cross covariance before and after conversion Matrix combines decorrelation LMS, and noise subspace is obtained using data covariance matrix after decorrelation LMS, by matrix-block computing by signal Subspace is divided into x-axis noise subspace block and y-axis noise subspace block, to each noise subspace construction MUSIC spectrums, by two Individual linear search completes the estimation of x-axis and y-axis direction cosines, then using the noise subspace of full array to both direction cosine Carry out matching the estimation that computing obtains angle of arrival, the two-dimensional search of two dimensional arrival angles is divided into two linear searches by the method, greatly Reduce amount of calculation greatly.

The content of the invention

It is an object of the invention to provide a kind of decorrelation LMS two dimension MUSIC method for parameter estimation.

To achieve these goals, the present invention takes following technical solution：

Acoustic vector sensors two-dimensional array MUSIC decorrelation LMS method for parameter estimation, K relevant arrowband, steady far field sound source Signal (θ from different directions_k, φ_k) incide on the receiving array, θ_k∈ [0, pi/2] is k-th angle of pitch of signal, φ_k ∈ [0,2 π] is k-th azimuth of signal, and the array is made up of non-homogeneous L-type array 2M-1 acoustic vector sensors, its Middle M acoustic vector sensors are located at x-axis, and M acoustic vector sensors are located at y-axis, and the axle of acoustic vector sensors two of the origin of coordinates is total to With the array element is the acoustic vector sensing with space concurrent synchro measure acoustic pressure and x-axis, y-axis and z-axis direction vibration velocity component Device, the respective channel of all the sensors is parallel to each other：All of sound pressure sensor is parallel to each other, and all of x-axis direction vibration velocity is passed Sensor is parallel to each other, and all of y-axis direction vibration velocity sensor is parallel to each other, and all of z-axis direction vibration velocity sensor is mutual It is parallel；Adjacent array element is smaller than being equal to λ_min/ 2, λ_minIt is the minimum wavelength of incident acoustic wave signal；

Two-dimensional array MUSIC decorrelation LMS method for parameter estimation steps are as follows：

Step one, non-homogeneous L-type acoustic vector-sensor array are arranged as receiving array, receive K far field arrowband coherent signal, Receiving array output n times synchronously sampled data Z；

Step 2, extraction acoustic pressure and the corresponding data of the submatrix of the velocity of sound of reference axis four of x, y, z three, by submatrix data Covariance matrix treatment recovers the order of data covariance matrix, the data covariance matrix R before being converted_Z；

Data are divided into acoustic pressure and x-axis, y-axis and z-axis direction vibration velocity four by the arrangement rule according to array data Z Submatrix data Z_p、Z_x、Z_yAnd Z_z, calculate 4 covariance matrixes of submatrix dataWithWherein,Assisted by 4 submatrix data The arithmetic average of variance matrixFull rank data covariance matrix R before being converted_Z；

Step 3, to submatrix receive data enter line translation, ask conversion after covariance data matrix R_YBefore and after conversion Cross covariance data matrix R_ZY, by matrix R_Z、R_YAnd R_ZYThe total data covariance matrix R after decorrelation LMS is obtained, so as to recover number According to the order of covariance matrix；

To submatrix data Z_p、Z_x、Z_y、Z_zEnter line translation Wherein, J_MIt is the opposition angular transformation matrix of M × M, j_{I, M-i+1}=1 (i=1 ..., M) is J_MThe i-th row M-i+ The element of 1 row, J_MOther elements all zero,Represent to submatrix data Z_p、Z_x、Z_y、Z_zTake conjugation Data afterwards, seek submatrix data Y after conversion_p、Y_x、Y_y、Y_zData covariance matrixWherein, Seek the cross covariance square of the front and rear data of conversion Battle array：Q_p、Q_x、Q_y、Q_z, wherein, Data covariance matrix R after the arithmetic average of covariance matrix is converted after conversion_Y, the cross covariance square of data before and after conversion Battle array ask arithmetic average to be converted after data Cross-covariance R_ZY,R_ZY=(Q_p+Q_x +Q_y+Q_z)/4, the total data covariance matrix R=[R after construction decorrelation LMS_ZY R_Z R_Y]；

Step 4, singular value decomposition is carried out by the data covariance matrix R after decorrelation LMS obtain signal subspace U_sWith make an uproar Phonon space U_n；According to L gusts of design feature by noise subspace U_nPiecemeal is carried out, is divided into x-axis and the corresponding noise of y-axis submatrix Subspace U_nxAnd U_ny, construct x-axis and the corresponding MUSIC spatial spectrums P (u of y-axis noise subspace_x) and P (u_y), it is one-dimensional by two Spectrum peak search obtains the estimated matrix of the direction cosines in x-axis and y-axis directionWith

Wherein,u_x=sin θ Cos φ and u_y=sin θ sin φ are respectively the direction cosines in x-axis and y-axis direction, and θ ∈ [0, pi/2] are to search for the angle of pitch, φ ∈ [0, 2 π] it is search azimuth, It is respectively the corresponding direction cosine matrix of the spectral peak of x-axis and y-axis；

Step 5, using the corresponding noise subspace U of full array_nConstruction MUSIC spectrums, more than the direction in x-axis and y-axis direction String estimated matrixWithPairing computing is carried out, using the direction cosines after pairingWithObtain estimating for angle of arrival EvaluationWith

Because search is independently carried out twice,WithCorresponding element not necessarily correspond to same signal, whenWithCorresponding element when not corresponding to same signal, it is impossible to carry out the estimation of angle of arrival, it is necessary to carry out pairing computing, Steering vector according to successful matching is perpendicular to noise subspace so as to the principle for having highest spectral peak is matched；It is MUSIC spectral functions, for's K-th elementWithIn each elementIt is combined, is existed using the maximum method of MUSIC spectral functionsFind withThe element of matchingIt is achieved thereby that k-th signal x-axis direction cosines With the direction cosines in y-axis directionPairing, then the estimate of angle of arrival be：

K=1 ..., K, l=1 ..., K in abovementioned steps.

The non-homogeneous L-type array that the present invention is used, the array element of array is by sound pressure sensor and x-axis, y-axis and z-axis direction The acoustic vector sensors that constitute of vibration velocity sensor, and all of sound pressure sensor is parallel to each other, all of x-axis direction vibration velocity Sensor is parallel to each other, and all of y-axis direction vibration velocity sensor is parallel to each other, and all of z-axis direction vibration velocity sensor is mutually put down OK.The inventive method makes full use of acoustic vector-sensor array to arrange the vector structure characteristic of itself, it is proposed that based on submatrix rotation not Become the decorrelation LMS method of characteristic, breach the limitation of existing spatial domain smoothing solution coherent approach, by existing based on one-dimensional equal The angle-of- arrival estimation method of even linear array has been generalized to the two-dimentional angle estimation of two-dimentional nonuniform noise, and using L-type array Two-dimentional MUSIC spectrum peak searches are decomposed into two one-dimensional MUSIC spectrum peak searches by design feature, and complete by simple pairing computing Into the estimation of angle of arrival, the method greatly reduces amount of calculation on the premise of Parameter Estimation Precision is not reduced.

Brief description of the drawings

In order to illustrate more clearly about the embodiment of the present invention or technical scheme of the prior art, below will be to embodiment or existing The accompanying drawing for having technology to be needed to use in describing does simple introduction, it should be apparent that, drawings in the following description are only the present invention Some embodiments, for those of ordinary skill in the art, on the premise of not paying creative work, can also basis These accompanying drawings obtain other accompanying drawings.

Fig. 1 is the schematic diagram of embodiment of the present invention acoustic vector-sensor array row；

Fig. 2 is the flow chart of the inventive method；

Fig. 3 is the MUSIC spatial spectrum curve maps that the inventive method is based on pitching angular direction；

Fig. 4 is the MUSIC spatial spectrum curve maps that space smoothing decorrelation LMS method is based on pitching angular direction；

Fig. 5 is the MUSIC spatial spectrum curve maps that the inventive method is based on azimuth direction；

Fig. 6 space smoothing decorrelation LMSs method is based on the MUSIC spatial spectrum curve maps of azimuth direction；

Specific embodiment

In order to above and other objects of the present invention, feature and advantage can be become apparent from, the embodiment of the present invention cited below particularly, And coordinate appended diagram, it is described below in detail.

Fig. 1 show the schematic diagram of the acoustic vector-sensor array row of the embodiment of the present invention.Acoustic vector sensors of the invention Array is made up of non-homogeneous L-type array 2M-1 acoustic vector sensors, and wherein M acoustic vector sensors are located at x-axis, M sound arrow Quantity sensor is located at y-axis, and the axle of acoustic vector sensors two of the origin of coordinates is shared, and the array element is with space concurrent synchro measure The acoustic vector sensors of acoustic pressure and x-axis, y-axis and z-axis direction vibration velocity component, the respective channel of all the sensors is parallel to each other： All of sound pressure sensor is parallel to each other, and all of x-axis direction vibration velocity sensor is parallel to each other, and all of y-axis direction vibration velocity is passed Sensor is parallel to each other, and all of z-axis direction vibration velocity sensor is parallel to each other；Adjacent array element is smaller than being equal to λ_min/ 2, λ_minIt is the minimum wavelength of incident acoustic wave signal；

The step of reference picture 2, acoustic vector sensors two-dimensional array MUSIC decorrelation LMS method for parameter estimation of the invention, is such as Under：Non-homogeneous L-type acoustic vector-sensor array row receive K relevant arrowband, steady far field sound-source signal, and K is incident sound-source signal Quantity,

Step one, non-homogeneous L-type acoustic vector-sensor array are arranged as receiving array, receive K far field arrowband coherent signal, Receiving array output n times synchronously sampled data Z；

Step 2, extraction acoustic pressure and the corresponding data of the submatrix of the velocity of sound of reference axis four of x, y, z three, by submatrix data Covariance matrix treatment recovers the order of data covariance matrix, the data covariance matrix R before being converted_Z；

Data are divided into four submatrixs of acoustic pressure and x-axis, y-axis and z-axis direction vibration velocity by the arrangement rule according to array data Z Data Z_p、Z_x、Z_yAnd Z_z, calculate 4 covariance matrixes of submatrix dataWithWherein,Assisted by 4 submatrix data The arithmetic average of variance matrixFull rank data covariance matrix R before being converted_Z；

Step 3, to submatrix receive data enter line translation, ask conversion after covariance data matrix R_YBefore and after conversion Cross covariance data matrix R_ZY, by matrix R_Z、R_YAnd R_ZYThe total data covariance matrix R after decorrelation LMS is obtained, so as to recover number According to the order of covariance matrix；

To submatrix data Z_p、Z_x、Z_y、Z_zEnter line translation Wherein, J_MIt is the opposition angular transformation matrix of M × M, j_{I, M-i+1}=1 (i=1 ..., M) is J_MThe i-th row M-i+ The element of 1 row, J_MOther elements all zero,Represent to submatrix data Z_p、Z_x、Z_y、Z_zTake conjugation Data afterwards, seek submatrix data Y after conversion_p、Y_x、Y_y、Y_zData covariance matrixWherein, Ask the mutual association side of the front and rear data of conversion Difference matrix：Q_p、Q_x、Q_y、Q_z, wherein, Data covariance matrix R after the arithmetic average of covariance matrix is converted after conversion_Y, the cross covariance square of data before and after conversion Battle array ask arithmetic average to be converted after data Cross-covariance R_ZY,R_ZY=(Q_p+Q_x +Q_y+Q_z)/4, the total data covariance matrix R=[R after construction decorrelation LMS_ZY R_Z R_Y]；

Step 4, singular value decomposition is carried out by the data covariance matrix R after decorrelation LMS obtain signal subspace U_sWith make an uproar Phonon space U_n；According to L gusts of design feature by noise subspace U_nPiecemeal is carried out, is divided into x-axis and the corresponding noise of y-axis submatrix Subspace U_nxAnd U_ny, construct x-axis and the corresponding MUSIC spatial spectrums P (u of y-axis noise subspace_x) and P (u_y), it is one-dimensional by two The estimated matrix of the direction cosines for obtaining x-axis and y-axis direction is estimated in searchWith

Wherein,u_x=sin θ Cos φ and u_y=sin θ sin φ are respectively the direction cosines in x-axis and y-axis direction, and θ ∈ [0, pi/2] are the search angle of pitch, φ ∈ [0,2 π] it is search azimuth, It is respectively the corresponding direction cosine matrix of the spectral peak of x-axis and y-axis；

Step 5, using the corresponding noise subspace U of full array_nConstruction MUSIC spectrums, more than the direction in x-axis and y-axis direction String estimated matrixWithPairing computing is carried out, using the direction cosines after pairingWithObtain estimating for angle of arrival EvaluationWith

Because search is independently carried out twice,WithCorresponding element not necessarily correspond to same signal, whenWithCorresponding element when not corresponding to same signal, it is impossible to carry out the estimation of angle of arrival, it is necessary to carry out pairing computing, Steering vector according to successful matching is perpendicular to noise subspace so as to the principle for having highest spectral peak is matched；It is MUSIC spectral functions, for's K-th elementWithIn each elementIt is combined, is existed using the maximum method of MUSIC spectral functionsFind withThe element of matchingIt is achieved thereby that k-th signal x-axis direction cosines With the direction cosines in y-axis directionPairing, then the estimate of angle of arrival be：

K=1 ..., K, l=1 ..., K in abovementioned steps.

The present invention gives based on non-homogeneous L-type acoustic vector sensors MUSIC decorrelation LMS method for parameter estimation, make full use of Acoustic vector-sensor array arranges the vector structure characteristic of itself, it is proposed that Space Rotating decorrelation LMS method, and utilizes cross covariance square Battle array joint decorrelation LMS, obtains noise subspace, by matrix-block computing by signal subspace using data covariance matrix after decorrelation LMS Space is divided into x-axis noise subspace block and y-axis noise subspace block, to each noise subspace construction MUSIC spectrums, by two Linear search completes the estimation of x-axis and y-axis direction cosines, and both direction cosine is entered using the noise subspace of full array then Row pairing computing obtains the estimation of angle of arrival, and the two-dimensional search of two dimensional arrival angles is divided into two linear searches by the method, significantly Amount of calculation is reduced, the inventive method is set up to uniform non-homogeneous L-type array.

Effect of the invention can be further illustrated by following simulation result：

Emulation experiment condition is as follows：

Two relevant arrowbands, steady far field sound-source signal are incided and are equidistantly spaced from the array element in x-axis and 8 by 8 The L-type acoustic vector-sensor array row constituted in the array element in y-axis are equidistantly spaced from, as shown in figure 1, the receiving array is by 15 battle arrays Unit's composition, array element interval is less than or equal to 0.5 λ_minRandom distribution, the parameter of incoming signal is：(θ₁, φ₁30 °, 20 ° of)=(), (θ₂, φ₂80 °, 60 ° of)=(), fast umber of beats is 512 times.

The simulation experiment result as shown in Figures 3 to 6, Fig. 3 for signal to noise ratio be 10dB when, the inventive method angle of arrival spatial spectrum The result of estimation, Fig. 4 for signal to noise ratio be 10dB when, the result of space smoothing decorrelation LMS method angle of arrival Estimation of Spatial Spectrum, from Fig. 3 With Fig. 4 it can be seen that the spatial spectrum of the inventive method angle-of- arrival estimation is very sharp, there are excellent Sidelobe Suppression effect and high point Resolution, illustrates that the angle-of- arrival estimation precision of the inventive method is very high.Fig. 5 and Fig. 6 for signal to noise ratio be 10dB when, the inventive method and The result of space smoothing decorrelation LMS method angle of arrival Estimation of Spatial Spectrum, from figs. 5 and 6, it can be seen that space smoothing decorrelation LMS side The spectral peak of method is not obvious, and the angle of arrival of signal is unable to estimate substantially, and the angle of arrival spectral peak of the inventive method is sharply with relatively low Secondary lobe and spatial resolution higher, it is very high up to angular estimation precision so as to demonstrate that the inventive method obtains.

The above, is only presently preferred embodiments of the present invention, and any formal limitation is not done to the present invention, though So the present invention is disclosed above with preferred embodiment, but is not limited to the present invention, any to be familiar with this professional technology people Member, without departing from the scope of the present invention, when making a little change or modification using the technology contents of the disclosure above It is the Equivalent embodiments of equivalent variations, as long as being the content without departing from technical solution of the present invention, according to technical spirit of the invention Any simple modification, equivalent variations and the modification made to above example, still fall within the range of technical solution of the present invention.

Claims (1)

1. acoustic vector sensors two-dimensional array MUSIC decorrelation LMS method for parameter estimation, it is characterised in that：

The array is made up of non-homogeneous L-type array 2M-1 acoustic vector sensors, and wherein M acoustic vector sensors are located at x-axis, M acoustic vector sensors are located at y-axis, and the axle of acoustic vector sensors two of the origin of coordinates is shared, and the array element is with space concurrent The acoustic vector sensors of synchro measure acoustic pressure and x-axis, y-axis and z-axis direction vibration velocity component, the respective channel phase of all the sensors It is mutually parallel：All of sound pressure sensor is parallel to each other, and all of x-axis direction vibration velocity sensor is parallel to each other, all of y-axis direction Vibration velocity sensor is parallel to each other, and all of z-axis direction vibration velocity sensor is parallel to each other；Adjacent array element is smaller than being equal to λ_min/ 2, λ_minIt is the minimum wavelength of incident acoustic wave signal；

The step of two-dimensional array MUSIC decorrelation LMS method for parameter estimation, is as follows：Non-homogeneous L-type acoustic vector-sensor array row receive K Individual relevant arrowband, steady far field sound-source signal,

Step one, non-homogeneous L-type acoustic vector-sensor array are arranged as receiving array, receive K far field arrowband coherent signal, are received Array output n times synchronously sampled data Z；

Step 2, extraction acoustic pressure and the corresponding data of the submatrix of the velocity of sound of reference axis four of x, y, z three, by submatrix data association side Difference matrix disposal recovers the order of data covariance matrix, the data covariance matrix R before being converted_Z；

Data are divided into four submatrix data of acoustic pressure and x-axis, y-axis and z-axis direction vibration velocity by the arrangement rule according to array data Z Z_p、Z_x、Z_yAnd Z_z, calculate 4 covariance matrixes of submatrix dataWithWherein, By 4 submatrix data The arithmetic average of covariance matrixFull rank data covariance matrix before being converted R_Z；

Step 3, to submatrix receive data enter line translation, ask conversion after covariance data matrix R_YWith the mutual association side before and after conversion Difference data matrix R_ZY, by matrix R_Z、R_YAnd R_ZYThe total data covariance matrix R after decorrelation LMS is obtained, so as to recover data association side Difference rank of matrix；

To submatrix data Z_p、Z_x、Z_y、Z_zEnter line translation Wherein, J_MIt is the opposition angular transformation matrix of M × M, j_{I, M-i+1}=1 (i=1 ..., M) is J_MThe i-th row M-i+1 row element, J_MOther elements all zero,Represent to submatrix data Z_p、Z_x、Z_y、Z_zThe data after conjugation are taken, Seek submatrix data Y after conversion_p、Y_x、Y_y、Y_zData covariance matrixWherein, Seek the cross covariance square of the front and rear data of conversion Battle array：Q_p、Q_x、Q_y、Q_z, wherein,After conversion The arithmetic average of covariance matrix converted after array covariance matrix R_Y, the Cross-covariance of data seeks arithmetic before and after conversion Data Cross-covariance R after averagely being converted_ZY, Total data covariance matrix R=[R after construction decorrelation LMS_ZY R_Z R_Y]；

Step 4, singular value decomposition is carried out by the data covariance matrix R after decorrelation LMS obtain signal subspace U_sIt is empty with noise Between U_n；According to L gusts of design feature by noise subspace U_nPiecemeal is carried out, is divided into x-axis and the corresponding noise subspace of y-axis submatrix U_nxAnd U_ny, construct x-axis and the corresponding MUSIC spatial spectrums P (u of y-axis noise subspace_x) and P (u_y), estimated by two linear searches Meter obtains the estimated matrix of the direction cosines in x-axis and y-axis directionWith

Wherein,u_x=sin θ cos φ and u_y=sin θ sin φ are respectively the direction cosines in x-axis and y-axis direction, and θ ∈ [0, pi/2] are the search angle of pitch, φ ∈ [0,2 π] it is search azimuth, It is respectively the corresponding direction cosine matrix of the spectral peak of x-axis and y-axis；

Step 5, using the corresponding noise subspace U of full array_nConstruction MUSIC spectrums, the direction cosines to x-axis and y-axis direction are estimated Meter matrixWithPairing computing is carried out, using the direction cosines after pairingWithObtain the estimate of angle of arrivalWith

Because search is independently carried out twice,WithCorresponding element not necessarily correspond to same signal, when WithCorresponding element when not corresponding to same signal, it is impossible to carry out the estimation of angle of arrival, it is necessary to carry out pairing computing, Steering vector according to successful matching is perpendicular to noise subspace so as to the principle for having highest spectral peak is matched；It is MUSIC spectral functions, for's K-th elementWithIn each elementIt is combined, is existed using the maximum method of MUSIC spectral functionsFind withThe element of matchingIt is achieved thereby that k-th signal x-axis direction cosines With the direction cosines in y-axis directionPairing, then the estimate of angle of arrival be：

${\widehat{\mathrm{\θ}}}_k=\arcsin \left(\sqrt{{\hat{u}}_{kx,opt}^2+{\hat{u}}_{ky,opt}^2}\right),$

${\widehat{\mathrm{\&phi;}}}_k=\left\{\begin{array}{cc}arctan\left(\frac{{\hat{u}}_{ky,opt}}{u_{kx,opt}}\right),& {\hat{u}}_{kx,opt}\&GreaterEqual;0\\ \mathrm{\&pi;}+\arctan \left(\frac{{\hat{u}}_{ky,opt}}{u_{kx,opt}}\right),& {\hat{u}}_{kx,opt}<0\end{array}\right.;$

K=1 ..., K, l=1 ..., K in abovementioned steps.