CN105242236B - Sensor position uncertainties bearing calibration in broadband signal super-resolution direction finding - Google Patents

Abstract

Sensor position uncertainties bearing calibration in broadband signal super-resolution direction finding, the present invention relates to the bearing calibration of array error present in broadband signal super-resolution direction finding.The problem of existing operand is big, convergence rate is slow and the existing alignment technique to more small-bore arrays is not suitable for broadband signal when the present invention is in order to solve the problems, such as the optimization of existing sensor position uncertainties bearing calibration solution high dimensional nonlinear.The present invention utilizes majorized function corresponding to the signal structure on each frequency, it is openness using the spatial domain of signal afterwards, optimization processing is iterated to the function on each frequency by management loading method respectively, finally carrying out fusion to the information on all frequencies estimates direction of arrival of signal.This method can effectively realize the array error correction in the presence of sensor position uncertainties, and the speed of service of algorithm is effectively improved using multi-disc digital signal processor.The present invention is applied to the correction field of array error present in broadband signal super-resolution direction finding.

Description

Array element position error correction method in broadband signal super-resolution direction finding

Technical Field

The invention relates to a method for correcting array errors existing in broadband signal super-resolution direction finding.

Background

Super-resolution direction finding is an important research content in array signal processing, and has a wide application in the fields of radio monitoring, internet of things, electronic countermeasure and the like. Most of the current direction-finding methods are based on the premise of accurately mastering the array flow pattern. In an actual direction-finding system, due to the disturbance of the position of the array element or inaccurate measurement, a position error of the array element is often caused during direction-finding estimation, which directly causes the performance of many super-resolution direction-finding methods to deteriorate or even fail, so that it is necessary to correct the errors.

The correction methods of the parameter class can be generally classified into active correction and self-correction. Active correction can be performed by off-line estimation of the array perturbation parameters by arranging auxiliary sources with known orientations in space, while self-correction methods generally estimate the orientation of a spatial source and the array perturbation parameters jointly according to some optimization function. Friedlander B and Weiss A J provide an array self-correction technology for alternately and iteratively estimating information source orientation, mutual coupling between array elements and array element gain and phase disturbance based on a subspace principle. However, the technology needs to solve the optimization problem of high-dimensional nonlinearity, the operation amount is large, the convergence rate is low, and the array disturbance parameters have the fuzzy problem for the uniform linear array. The Mavrychev and other scholars research partial correction technologies of the multi-small-aperture array, and effectively solve the problem that a single small-aperture array cannot meet the multi-target resolution and orientation precision requirements. Since the constructed estimator does not need to accurately know the position information among the sub-arrays, the influence of false peaks and position estimation errors on the azimuth estimation is avoided. However, they are only suitable for narrow-band signals, and published documents are not rare for array element position error correction technology in super-resolution direction finding of wide-band signals.

Disclosure of Invention

The invention aims to solve the problems of large computation amount and low convergence speed when the existing array element position error correction method is used for solving the high-dimensional nonlinear optimization and the problem that the existing correction technology for a multi-small-aperture array is not suitable for broadband signals.

The array element position error correction method in the broadband signal super-resolution direction finding comprises the following steps:

step 1: establishing an array signal model containing array element position errors:

when array element position error exists in the array, the frequency point f_iThe output of the array at (1) can be expressed as

X'(f_i)＝A'(f_i,α)S(f_i)+N(f_i)，i＝1,2,…,J (12)

A'(f_iα) is the frequency point f when array element position error exists_iAn array flow pattern matrix above; s (f)_i) Is a signal s_k(t) a signal vector matrix after fourier transformation; n (f)_i) Is noise n_m(t) noise vector matrix after Fourier transform, mean 0, variance μ²(f_i)；

Time frequency point f with array element position error_iA received signal covariance matrix of

R'(f_i)＝E{X'(f_i)(X'(f_i))^H}，i＝1,2,…,J (13)

A(f_i,α)＝[a(f_i,α₁),…,a(f_i,α_k),…,a(f_i,α_K)]Is the frequency point f under the ideal condition_iArray flow pattern matrix of_i,α_k) Is the frequency point f under the ideal condition_iAn array steering vector for the k-th signal;

when the position of the m-th array element has an error △ d_mWhen the method is used, the method can be equivalent to the method that azimuth-dependent phase disturbance is introduced into the array steering vector, namely

Wherein, W (f)_i,α_k) Is a frequency point f_iUp and down directionα_kArray element position error disturbance matrix of (a') (f)_i,α_k) Time frequency point f for indicating array element position error_iAn array steering vector for the k-th signal;

from direction α for the k signal_kWhen the m-th array element is reached, the signal source propagation delay error introduced by the disturbance of the position error of the array element is introduced; the time frequency point f with array element position error_iAn array flow pattern matrix of

A'(f_i,α)＝[a'(f_i,α₁),…,a'(f_i,α_k),…,a'(f_i,α_K)]

(17)

＝W(f_i,α)·A(f_i,α)

Wherein, W (f)_i,α)＝[W(f_i,α₁),…,W(f_i,α_k),…,W(f_i,α_K)]Representing frequency point f_iAn array element position error disturbance matrix is arranged;

step 2: estimating array signal parameters containing array element position errors:

the search space is first divided into a number of discrete angular gridsCorresponding to the L directions in which the signal may arrive; thereby obtaining the frequency point f_iSparse representation of an over-array streaming matrix

Wherein,is a frequency point f_iThe array steering vector of the last ith sparse signal can correspondingly obtain the time frequency point f with array element position error_iSparse representation of an over-array streaming matrix Is a frequency point f_iSparse representation of the position error perturbation matrix of the upper array element,is a frequency point f_iArray element position error disturbance matrixes of the upper and the l-th sparse signals,when the ith sparse signal reaches the mth array element, the signal source propagation delay error is introduced by the disturbance of the position error of the array element,for existence of array element position error time frequency point f_iThe array steering vector of the last ith sparse signal can obtain the time frequency point f with array element position error_iSparse representation of upper array output signal

Wherein, Λ (f)_i) Is a parameter only related to the original signal and is not related to errors;is Λ (f)_i) A sparse representation of (c); w (f)_i)＝[△d₂,…,△d_M]^TRepresenting the position error disturbance vector of the array element, taking the 1 st array element as a position reference point,△d₂,…,△d_Mare respectively frequency points f_iDeviation of true and measured positions of the upper 2 nd to Mth array elements from the signal frequency f_iIrrelevant;

has a covariance matrix of

In the formula (18)Is S (f)_i) Is represented by a sparse representation of (a) a,

wherein,is a sparse matrix, is S (f)_iKp) is used in the sparse representation,contains only K non-zero elements,is composed ofIf and only ifTime of flightWherein the elements are not all zero and haveThus, it isCan be regarded as S (f)_i) A matrix obtained by adding a plurality of 0 elements;

is provided (f)_i)＝[₁(f_i),…,_l(f_i),…,_L(f_i)]^TIs composed ofThe variance of the medium element reflects the energy of the signal, i.e. has

Wherein, Σ (f)_i)＝diag((f_i) That is to sayObedience mean 0, variance (f)_i) (ii) a gaussian distribution of;

due to the fact thatCan be regarded as S (f)_i) A vector obtained by adding a plurality of 0 elements, so (f)_i) Contains K non-zero elements and has K<<L is according to (f)_i) In combination with w (f)_i) Sum noise variance μ²(f_i) Estimate outThus reconstructing the original signal and correcting the error;

as can be seen from equation (18), the frequency f when there is an array element position error_iHas a probability density of

The combination of formula (18), (20) and (21) gives

Wherein, I_MIs a unit matrix of dimension M × M;

the Expectation Maximization (EM) method is adopted to pair w (f)_i)、μ²(f_i) And_l(f_i) Performing iterative estimation to obtain an estimated valueAndis correspondingly obtainedAnd

and step 3: by usingAndcorrecting array errors and solving the signal arrival direction;

let X be the vector formed by the sum of all frequency point signals received by the array in a period of observation time, and because the signals of each frequency point have statistical independence, the joint probability density of the signals received by each frequency point is

The logarithm at both ends of the pair formula (35) is

Thus, by maximizing equation (36), the direction of arrival of the signal, i.e., the estimated value of the direction of arrival of the signal, can be obtainedK is 1,2, …, K, i.e. can pass

Obtaining;

by derivation have

Wherein Re {. is the real part of the solution {. cndot.); omega_-k、Respectively expressed from Ω andremoving the k-th element from the first element; k is 1,2, …, K;

according to△ d is found from the expression₂,…,△d_MThen, W (f) is obtained from the equations (16) and (15)_i,α_k) And W (f)_iΩ), then array correction is performedDe a' (f)_i,α_k) And A' (f)_i,Ω_-k) (ii) a Based on the above parameters and equation (38), an estimate of the direction of arrival of the array-corrected signal can be obtained

The invention has the following beneficial effects:

the invention provides a broadband signal super-resolution direction finding error correction method when array element position errors exist, which comprises the steps of constructing a corresponding optimization function by using signals on all frequency points, then performing iterative optimization processing on the functions on all the frequency points by using the space domain sparsity of the signals and a sparse Bayesian learning method, and finally fusing information on all the frequency points to estimate the arrival direction of the signals. The array error correction method can effectively realize array error correction when array element position errors exist, and when the signal-to-noise ratio is 10dB and the sampling fast beat number of each frequency point is 40, the accuracy can reach 0.6 degree/sigma.

The method of the invention can use a plurality of digital signal processors to process, and can effectively improve the running speed of the algorithm.

Drawings

FIG. 1 is a schematic diagram of a broadband signal super-resolution direction-finding array signal model;

FIG. 2 is a diagram of a broadband signal detection system;

FIG. 3 is a diagram of a broadband signal super-resolution direction-finding device according to a fourth embodiment;

FIG. 4 is a diagram of a broadband signal super-resolution direction-finding device according to a fifth embodiment;

fig. 5 is a diagram of a broadband signal super-resolution direction finding device according to a sixth embodiment.

Detailed Description

The first embodiment is as follows:

the array element position error correction method in the broadband signal super-resolution direction finding comprises the following steps:

step 1: establishing an array signal model containing array element position errors:

when array element position error exists in the array, the frequency point f_iThe output of the array at (1) can be expressed as

X'(f_i)＝A'(f_i,α)S(f_i)+N(f_i)，i＝1,2,…,J (12)

A'(f_iα) is the frequency point f when array element position error exists_iAn array flow pattern matrix above; s (f)_i) Is a signal s_k(t) a signal vector matrix after fourier transformation; n (f)_i) Is noise n_m(t) noise vector matrix after Fourier transform, mean 0, variance μ²(f_i)；

Time frequency point f with array element position error_iA received signal covariance matrix of

R'(f_i)＝E{X'(f_i)(X'(f_i))^H}，i＝1,2,…,J (13)

A(f_i,α)＝[a(f_i,α₁),…,a(f_i,α_k),…,a(f_i,α_K)]Is the frequency point f under the ideal condition_iArray flow pattern matrix of_i,α_k) Is the frequency point f under the ideal condition_iAn array steering vector for the k-th signal;

when the position of the m-th array element has an error △ d_mWhen the method is used, the method can be equivalent to the method that azimuth-dependent phase disturbance is introduced into the array steering vector, namely

Wherein, W (f)_i,α_k) Is a frequency point f_iUp and direction α_kArray element position error disturbance matrix of (a') (f)_i,α_k) Time frequency point f for indicating array element position error_iAn array steering vector for the k-th signal;

from direction α for the k signal_kWhen the m-th array element is reached, the signal source propagation delay error introduced by the disturbance of the position error of the array element is introduced; the time frequency point f with array element position error_iAn array flow pattern matrix of

A'(f_i,α)＝[a'(f_i,α₁),…,a'(f_i,α_k),…,a'(f_i,α_K)]

(17)

＝W(f_i,α)·A(f_i,α)

Wherein, W (f)_i,α)＝[W(f_i,α₁),…,W(f_i,α_k),…,W(f_i,α_K)]Representing frequency point f_iAn array element position error disturbance matrix is arranged;

step 2: estimating array signal parameters containing array element position errors:

the search space is first divided into a number of discrete angular gridsCorresponding letterL directions in which the number may arrive; thereby obtaining the frequency point f_iSparse representation of an over-array streaming matrix

Wherein,is a frequency point f_iThe array steering vector of the last ith sparse signal can correspondingly obtain the time frequency point f with array element position error_iSparse representation of an over-array streaming matrix Is a frequency point f_iSparse representation of the position error perturbation matrix of the upper array element,is a frequency point f_iArray element position error disturbance matrixes of the upper and the l-th sparse signals,when the ith sparse signal reaches the mth array element, the signal source propagation delay error is introduced by the disturbance of the position error of the array element,for existence of array element position error time frequency point f_iThe array steering vector of the last ith sparse signal can obtain the time frequency point f with array element position error_iSparse representation of upper array output signal

Wherein, Λ (f)_i) Is a parameter only related to the original signal and is not related to errors;is Λ (f)_i) A sparse representation of (c); w (f)_i)＝[△d₂,…,△d_M]^TRepresenting the position error disturbance vector of the array element, taking the 1 st array element as the position reference point, △ d₂,…,△d_MAre respectively frequency points f_iDeviation of true and measured positions of the upper 2 nd to Mth array elements from the signal frequency f_iIrrelevant;

has a covariance matrix of

In the formula (18)Is S (f)_i) Is represented by a sparse representation of (a) a,

wherein,is a sparse matrix, is S (f)_iKp) is used in the sparse representation,contains only K non-zero elements,is composed ofThe first one ofElements, if and only ifTime of flightWherein the elements are not all zero and haveThus, it isCan be regarded as S (f)_i) A matrix obtained by adding a plurality of 0 elements;

is provided (f)_i)＝[₁(f_i),…,_l(f_i),…,_L(f_i)]^TIs composed ofThe variance of the medium element reflects the energy of the signal, i.e. has

Wherein, Σ (f)_i)＝diag((f_i) That is to sayObedience mean 0, variance (f)_i) (ii) a gaussian distribution of;

due to the fact thatCan be regarded as S (f)_i) A vector obtained by adding a plurality of 0 elements, so (f)_i) Contains K non-zero elements and has K<<L is according to (f)_i) In combination with w (f)_i) Sum noise variance μ²(f_i) Estimate outThus reconstructing the original signal and correcting the error;

as can be seen from equation (18), the frequency f when there is an array element position error_iHas a probability density of

The combination of formula (18), (20) and (21) gives

Wherein, I_MIs a unit matrix of dimension M × M;

the Expectation Maximization (EM) method is adopted to pair w (f)_i)、μ²(f_i) And_l(f_i) Performing iterative estimation to obtain an estimated valueAndis correspondingly obtainedAnd

and step 3: by usingAndcorrecting array errors and solving the signal arrival direction;

let X be the vector formed by the sum of all frequency point signals received by the array in a period of observation time, and because the signals of each frequency point have statistical independence, the joint probability density of the signals received by each frequency point is

The logarithm at both ends of the pair formula (35) is

Thus, by maximizing equation (36), the direction of arrival of the signal, i.e., the estimated value of the direction of arrival of the signal, can be obtainedK is 1,2, …, K, i.e. can pass

Obtaining;

by derivation have

Wherein Re {. is the real part of the solution {. cndot.); omega_-k、Respectively expressed from Ω andremoving the k-th element from the first element; k is 1,2, …, K;

according to△ d is found from the expression₂,…,△d_MThen, W (f) is obtained from the equations (16) and (15)_i,α_k) And W (f)_iΩ), then array correction is performed to find a' (f)_i,α_k) And A' (f)_i,Ω_-k) (ii) a Based on the above parameters and equation (38), an estimate of the direction of arrival of the array-corrected signal can be obtained

The second embodiment is as follows:

the specific steps of establishing an array signal model containing array element position errors in step 1 of the present embodiment are as follows:

step 1.1: establishing an ideal array signal model:

as shown in FIG. 1, K far-field broadband signals s are provided_k(t), K is 1,2, …, K, and is incident on a broadband uniform linear array of M omnidirectional array elements, arriving at α ═ α₁,…,α_k,…,α_K]The array element interval is d; far field broadband signal s_k(t), broadband signal s for short_k(t)；

Taking the 1 st array element as the phase reference point, ideally, the output of the m-th array element is expressed as

Wherein,representing the kth broadband signal s_k(t) the time delay for reaching the mth array element relative to its arrival at the phase reference point, c is the propagation speed of the electromagnetic wave in vacuum, n_m(t) Gaussian white noise received by the mth array element;

suppose the frequency range of the wideband signal is f_Low,f_High]The broadband signal is divided into J frequency points by using discrete Fourier transform, and the J frequency points are separated by a narrow-band filter bank, so that the output signal of the filter array of the ith group is expressed as

X(f_i)＝A(f_i,α)S(f_i)+N(f_i)，i＝1,2,…,J (2)

Wherein f is_Low≤f_i≤f_High，i＝1,2,…,J；

Assuming at each frequency point f_iHaving performed KP subsampling, X (f)_i) Is expressed in matrix form as

X(f_i)＝[X(f_i,1),…,X(f_i,kp),…,X(f_i,KP)]，i＝1,2,…,J (3)

Wherein, X (f)_iKp) is X (f)_i) The kp-th time data sampling matrix of (c),

X(f_i,kp)＝[X₁(f_i,kp),…,X_m(f_i,kp),…,X_M(f_i,kp)]^T，i＝1,2,…,J， (4)

X_m(f_ikp) is the m-th array element at frequency point f_iThe kp-th data sampling value obtained above;

A(f_iα) is the ideal case frequency point f_iThe array flow pattern matrix on the upper surface,

A(f_i,α)＝[a(f_i,α₁),…,a(f_i,α_k),…,a(f_i,α_K)]，i＝1,2,…,J， (5)

a(f_i,α_k) Is the frequency point f under the ideal condition_iThe array of the top k-th signal leads to the vector,

wherein,is the phase of the kth signal; j is a complex number flag;

S(f_i)＝[S(f_i,1),…,S(f_i,kp),…,S(f_i,KP)]，i＝1,2,…,J， (8)

is a signal s_k(t) a fourier transformed signal vector matrix, K being 1,2, …, K;

wherein, S (f)_iKp) is S (f)_i) The kp-th sub-sampling matrix of the signal,

S(f_i,kp)＝[S₁(f_i,kp),…S_k(f_i,kp),…,S_K(f_i,kp)]^Ti＝1,2,…,J (9)

S_k(f_ikp) is the k signal at frequency f_iThe kpth signal sampling value obtained above;

N(f_i)＝[N(f_i,1),…,N(f_i,kp),…,N(f_i,KP)]i＝1,2,…,J (10)

is noise n_m(t) noise vector matrix after Fourier transform, mean 0, variance μ²(f_i) (ii) a M is 1,2, …, M; wherein, N (f)_iKp) is N (f)_i) The kp-th time of the noise sampling matrix,

N(f_i,kp)＝[N₁(f_i,kp),…,N_m(f_i,kp),…,N_M(f_i,kp)]^Ti＝1,2,…,J (11)

N_m(f_ikp) is the m-th array element at frequency point f_iThe kp-th noise sampling value obtained above;

step 1.2: establishing an array signal model containing array element position errors on the basis of an ideal array signal model:

when array element position error exists in the array, the frequency point f_iThe output of the array at (1) can be expressed as

X'(f_i)＝A'(f_i,α)S(f_i)+N(f_i)，i＝1,2,…,J (12)

Time frequency point f with array element position error_iA received signal covariance matrix of

R'(f_i)＝E{X'(f_i)(X'(f_i))^H}，i＝1,2,…,J (13)

In formula (12), A' (f)_i,α)＝[a'(f_i,α₁),…,a'(f_i,α_k),…,a'(f_i,α_K)]For existence of array element position error time frequency point f_iArray flow pattern matrix of (a') (f)_i,α_k) Steering vectors for the corresponding arrays;

when the position of the m-th array element has an error △ d_mWhen the method is used, the method can be equivalent to the method that azimuth-dependent phase disturbance is introduced into the array steering vector, namely

Wherein,

is a frequency point f_iThe up and arrival direction is α_kThe position error of the array element disturbs the matrix;

wherein

When the kth signal reaches the mth array element, the signal source propagation delay error is introduced by the disturbance of the position error of the array element;

then the array flow pattern matrix in the presence of array element position error is

A'(f_i,α)＝[a'(f_i,α¹),…,a'(f_i,α_k),…,a'(f_i,α_K)]

(17)

＝W(f_i,α)·A(f_i,α)

Wherein W (f)_i,α)＝[W(f_i,α₁),…,W(f_i,α_k),…,W(f_i,α_K)]Time frequency point f for indicating array element position error_iAnd (3) disturbing the matrix by the position error of the array element.

Other steps and parameters are the same as in the first embodiment.

The third concrete implementation mode:

in step 2 of the present embodiment, w (f) is maximized by the expectation maximization method_i)、μ²(f_i) And_l(f_i) The specific steps for performing the iterative estimation are as follows:

in the E-step of expectation maximization method, first, the method is performedIs calculated by the distribution function of

Wherein the operator < · > represents the solution condition expectation;

in the M-step of the expectation-maximization method, the distribution functions are respectively obtainedDerivatives of unknown parameters, i.e. pairsSolving each unknown parameter by taking an extreme value;

respectively making the above derivatives be 0, so as to obtain the estimated value of every unknown parameter in p-th iteration

Wherein (p) represents the number of iterations, in equation (27)

Is a matrixRow r1, column r2, where tr [ ·]Representing trace-solving operation;

in formula (30);

O(f_i)＝Σ(f_i)(A'(f_i,Ω))^H(μ²(f_i)I_M+A'(f_i,Ω)Σ(f_i)(A'(f_i,Ω))^H)^-1X(f_i) (31)

is an intermediate variable;

Ξ(f_i)

(32)

＝Σ(f_i)-Σ(f_i)(A'(f_i,Ω))^H(μ²(f_i)I_M+A'(f_i,Ω)Σ(f_i)(A'(f_i,Ω))^H)^-1A'(f_i,Ω)Σ(f_i)

is an intermediate variable;

in formula (27);

in the formula (28)

In formula (33), Ψ_r(f_i) The intermediate variable is a matrix with M × M dimensions, only the elements at the r +1 th row and the r +1 th column are all 1, and the rest elements are all 0;

w (f) is calculated by directly using equations (27) to (28)_i) And mu²(f_i) Since it is relatively complicated, w (f) can be degenerated by substituting equations (30) to (34) for equivalents in equations (27) to (28)_i) And mu²(f_i) Solving;

after several iterations, w (f)_i)、μ²(f_i) And_l(f_i) The three estimated values vary towards 0, when they can be considered to have converged, and the final estimate can be derivedAndcorrespond to obtainAnd

other steps and parameters are the same as in the second embodiment.

The fourth concrete implementation mode: this embodiment is described in detail with reference to figures 2 and 3,

this embodiment is a wideband signal detection system and a method for implementing detection according to the first to third embodiments,

as shown in fig. 2, the broadband signal detection system includes: the system comprises a broadband uniform linear array 1, a multi-channel broadband digital receiver 2 and a broadband signal super-resolution direction-finding device 3;

as shown in fig. 3, the broadband signal super-resolution direction-finding device 3 includes 6 digital signal processors, i.e., DSPs, and adopts fast serial input/output ports, i.e., SRIO ports, to form a multiprocessor system to implement parallel processing. Wherein, the DSP3-1 is a master DSP, and the DSPs 3-2-3-6 are slave DSPs; the broadband signal super-resolution direction-finding device 3 further comprises a CPLD3-7, a PROM3-8, a FLASH3-9, an SRAM3-10, a JTAG3-11, a power supply, a crystal oscillator and a reset.

The digital signal processor adopts TMS320C6678 of Texas Instruments (TI) company, adopts 6 processors for parallel processing, 6 DSPs are connected through SRIO ports, PROM3-8 loads programs to CPLD3-7 after power-on, FLASH3-9 also loads programs to the 6 DSPs (3-1 to 3-6), then the main DSP3-1 starts to receive observed data of J frequency points transmitted by the multichannel broadband digital receiver 2, the observed data are divided into W groups, if J is 30 and W is 6, each DSP can process observed data of U30/6 to 5 frequency points, the main DSP3-1 transmits the observed data processed by other auxiliary DSPs (3-2 to 3-6) to the auxiliary DSPs through the SRIO ports, then each DSP (3-1 to 3-6) solves the problem according to the steps of theoretical derivation, and then 5 auxiliary DSPs (3-2 to 3-6) transmit respective error estimation values to SRIO ports through SRIO ports 3 The host DSP3-1 reuses these results and the combination (38) yields the signal angle of arrival. The SRAM3-10 is responsible for storing data, the JTAG3-11 is responsible for debugging the DSP (3-1 to 3-6), the power supply is responsible for overall power supply, the crystal oscillator is responsible for providing a clock, and the reset is responsible for providing a reset signal.

The fifth concrete implementation mode: this embodiment is described in detail with reference to figures 2 and 4,

this embodiment is a wideband signal detection system and a method for implementing detection according to the first to third embodiments,

as shown in fig. 2, the broadband signal detection system includes: the system comprises a broadband uniform linear array 1, a multi-channel broadband digital receiver 2 and a broadband signal super-resolution direction-finding device 3;

as shown in fig. 4, the broadband signal super-resolution direction-finding device 3 includes 6 digital signal processors, i.e., DSPs, and adopts a shared bus tight coupling mode to form a multiprocessor system to implement parallel processing. Wherein, the DSP3-1 is a master DSP, and the DSPs 3-2-3-6 are slave DSPs; the broadband signal super-resolution direction-finding device 3 further comprises a CPLD3-7, a PROM3-8, a FLASH3-9, an SRAM3-10, a JTAG3-11, a power supply, a crystal oscillator and a reset.

The digital signal processor adopts ADSP-TS201S of Analog Device Instruments (ADI), 6 DSPs are processed in parallel, the 6 DSPs are connected in a tight coupling mode through a shared bus, after the power is on, PROM3-8 loads a program to CPLD3-7 to configure the DSPs (3-1 to 3-6), then FLASH3-9 loads the program to the 6 DSPs (3-1 to 3-6), main DSP3-1 starts to receive observed data of J frequency points transmitted by the multichannel broadband digital receiver 2, the observed data are divided into W groups, if J is 30 and W is 6, each DSP can process observed data of U30/6 which is 5 frequency points, main DSP3-1 transmits other observed data which are processed by the auxiliary DSPs (3-2 to 3-6) to the DSP through the bus, and then each DSP (3-1 to 3-6) performs theoretical derivation solving according to the theoretical derivation steps, then 5 slave DSPs (3-2 to 3-6) transmit respective error estimation values to the master DSP3-1 through a bus, and the master DSP3-1 reuses the results to obtain the signal arrival angle by combining the formula (38). The SRAM3-10 is responsible for storing data, the JTAG3-11 is responsible for debugging the DSP (3-1 to 3-6), the power supply is responsible for overall power supply, the crystal oscillator is responsible for providing a clock, and the reset is responsible for providing a reset signal.

The sixth specific implementation mode: this embodiment is described in detail with reference to figures 2 and 5,

this embodiment is a wideband signal detection system and a method for implementing detection according to the first to third embodiments,

as shown in fig. 2, the broadband signal detection system includes: the system comprises a broadband uniform linear array 1, a multi-channel broadband digital receiver 2 and a broadband signal super-resolution direction-finding device 3;

as shown in fig. 5, the broadband signal super-resolution direction-finding device 3 includes 6 digital signal processors, i.e., DSPs, and adopts a link port cascade loose coupling manner to form a multiprocessor system to implement parallel processing. Wherein, the DSP3-1 is a master DSP, and the DSPs 3-2-3-6 are slave DSPs; the broadband signal super-resolution direction-finding device 3 further comprises a CPLD3-7, a PROM3-8, a FLASH3-9, an SRAM3-10, a JTAG3-11, a power supply, a crystal oscillator and a reset.

The digital signal processor adopts ADSP-TS201S of Analog Device Instruments (ADI), adopts 6 processors to process in parallel, 6 DSPs are connected in a cascade loose coupling mode through link ports, after power-on, PROM3-8 loads programs to CPLD3-7, FLASH3-9 loads programs of the 6 DSPs to main DSP3-1, main DSP3-1 sequentially transmits programs of other slave DSPs (3-2 to 3-6) to the DSPs through the link ports in a first-level and first-level manner, then main DSP3-1 starts to receive observation data of J frequency points transmitted from multi-channel broadband digital receiver 2, divides the observation data into W groups, assuming that J is 30 and W is 6, each DSP can process observation data of U30/6 is 5 frequency points, main DSP3-1 transmits the observation data of other DSPs (3-2 to 3-6) in a first-level manner to the observation data sequentially through a chain, then each DSP (3-1 to 3-6) is solved according to the steps deduced by the theory, then 5 slave DSPs (3-2 to 3-6) successively upload respective error estimation values to a master DSP3-1 through a link port in a first-level and first-level mode, and the master DSP3-1 reuses the results and combines the formula (38) to obtain the signal arrival angle. The SRAM3-10 is responsible for storing data, the JTAG3-11 is responsible for debugging the DSP (3-1 to 3-6), the power supply is responsible for overall power supply, the crystal oscillator is responsible for providing a clock, and the reset is responsible for providing a reset signal.

Claims (3)

1. The array element position error correction method in the broadband signal super-resolution direction finding is characterized by comprising the following steps of:

step 1: establishing an array signal model containing array element position errors:

when array element position error exists in the array, the frequency point f_iThe output of the array at is represented as

X'(f_i)＝A'(f_i,α)S(f_i)+N(f_i)，i＝1,2,…,J (12)

A'(f_iα) is the frequency point f when array element position error exists_iAn array flow pattern matrix above; s (f)_i) Is a signal s_k(t) a signal vector matrix after fourier transformation; n (f)_i) Is noise n_m(t) noise vector matrix after Fourier transform, mean 0, variance μ²(f_i) J is the number of frequency points, α ═ α₁,…,α_k,…,α_K]The arrival direction of K far-field broadband signals incident on a broadband uniform linear array consisting of M omnidirectional array elements is determined;

time frequency point f with array element position error_iA received signal covariance matrix of

R'(f_i)＝E{X'(f_i)(X'(f_i))^H}，i＝1,2,…,J (13)

A(f_i,α)＝[a(f_i,α₁),…,a(f_i,α_k),…,a(f_i,α_K)]Is the frequency point f under the ideal condition_iArray flow pattern matrix of_i,α_k) Is the frequency point f under the ideal condition_iAn array steering vector for the k-th signal; k is the number of far-field broadband signals;

when the position of the m-th array element has an error delta d_mWhen the method is equivalent to introducing azimuth-dependent phase disturbance into the array steering vector, the method has the following advantages

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>W</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mi>W</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>

Wherein, W (f)_i,α_k) Is a frequency point f_iUp and direction α_kArray element position error disturbance matrix of (a') (f)_i,α_k) Time frequency point f for indicating array element position error_iAn array steering vector for the k-th signal;

<mrow> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;d</mi> <mi>m</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>sin&amp;alpha;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>

from direction α for the k signal_kWhen the m-th array element is reached, the signal source propagation delay error introduced by the disturbance of the position error of the array element is introduced;

the time frequency point f with array element position error_iAn array flow pattern matrix of

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>&amp;lsqb;</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>K</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>W</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

Wherein, W (f)_i,α)＝[W(f_i,α₁),…,W(f_i,α_k),…,W(f_i,α_K)]Representing frequency point f_iAn array element position error disturbance matrix is arranged;

step 2: estimating array signal parameters containing array element position errors:

the search space is first divided into a number of discrete angular gridsCorresponding to the L directions in which the signal may arrive; thereby obtaining a frequency point f_iSparse representation of an over-array streaming matrix

<mrow> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>&amp;lsqb;</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mover> <mi>&amp;alpha;</mi> <mo>&amp;OverBar;</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mover> <mi>&amp;alpha;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mover> <mi>&amp;alpha;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>L</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow>

Wherein,is a frequency point f_iCorrespondingly obtaining the time frequency point f with array element position error by the array steering vector of the last ith sparse signal_iSparse representation of an over-array streaming matrixd is the array element spacing;is a frequency point f_iSparse representation of the position error perturbation matrix of the upper array element,is a frequency point f_iArray element position error disturbance matrixes of the upper and the l-th sparse signals,when the ith sparse signal reaches the mth array element, the source propagation delay error is introduced by the disturbance of the position error of the array element, and c is the propagation speed of the electromagnetic wave in vacuum;for existence of array element position error time frequency point f_iObtaining the frequency point f when the array element position error exists by the array steering vector of the last ith sparse signal_iSparse representation of upper array output signal

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>w</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>N</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>J</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>

Wherein, Λ (f)_i) Is a parameter related to the original signal only;is Λ (f)_i) A sparse representation of (c); w (f)_i)＝[Δd₂,…,Δd_M]^TRepresenting the position error disturbance vector of the array element, taking the 1 st array element as a position reference point, delta d₂,…,Δd_MAre respectively frequency points f_iDeviation of the actual position and the measured position of the upper 2 nd array element to the Mth array element;

has a covariance matrix of

<mrow> <msup> <mover> <mi>R</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>E</mi> <mo>{</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>}</mo> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>J</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>

In the formula (18)Is S (f)_i) KP is sampling frequency, and KP represents KP sampling;

wherein,is a sparse matrix, is S (f)_iKp) is used in the sparse representation,contains only K non-zero elements,is composed ofIf and only ifTime of flightWherein the elements are not all zero and havel＝1,2,…,L，k＝1,2, …, K; thus, it isIs regarded as S (f)_i) A matrix obtained by adding a plurality of 0 elements; s (f)_iKp) is S (f)_i) Of the kth signal sampling matrix, S_k(f_iKp) is the k signal at frequency f_iThe kpth signal sampling value obtained above;

is provided (f)_i)＝[₁(f_i),…,_l(f_i),…,_L(f_i)]^TIs composed ofThe variance of the medium element reflects the energy of the signal, i.e. has

<mrow> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>~</mo> <mi>N</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>&amp;Sigma;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>

Wherein, Σ (f)_i)＝diag((f_i) That is to sayObedience mean 0, variance (f)_i) (ii) a gaussian distribution of;

according to the formula (18), the frequency f when the position error of the array element exists_iHas a probability density of

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>|</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>;</mo> <mi>w</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>|</mo> <mi>&amp;pi;</mi> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <msup> <mo>|</mo> <mrow> <mo>-</mo> <mi>K</mi> <mi>P</mi> </mrow> </msup> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>|</mo> <mo>|</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>w</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>

In combination with (18), (20) and (21)

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>;</mo> <mi>&amp;delta;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mi>w</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>|</mo> <mi>&amp;pi;</mi> <mrow> <mo>(</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mi>&amp;Sigma;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mrow> <mo>-</mo> <mi>K</mi> <mi>P</mi> </mrow> </msup> <mo>&amp;times;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> <mi>&amp;Sigma;</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>R</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>

Wherein, I_MM is a unit array with M × dimension, wherein M is the number of omnidirectional array elements;

using expectation maximization method to pair w (f)_i)、μ²(f_i) And_l(f_i) Performing iterative estimation to obtain an estimated value Andcorresponding to getAnd

and step 3: by usingAndcorrecting array errors and solving the signal arrival direction;

let X be the vector formed by the sum of all frequency point signals received by the array in a period of observation time, and because the signals of each frequency point have statistical independence, the joint probability density of the signals received by each frequency point is

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&amp;Pi;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mi>P</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>;</mo> <mover> <mi>&amp;delta;</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>w</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>|</mo> <mi>&amp;pi;</mi> <msup> <mo>|</mo> <mrow> <mo>-</mo> <mi>J</mi> <mo>&amp;times;</mo> <mi>K</mi> <mi>P</mi> </mrow> </msup> <munderover> <mi>&amp;Pi;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mo>|</mo> <mrow> <mo>(</mo> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mrow> <mo>-</mo> <mi>K</mi> <mi>P</mi> </mrow> </msup> <mo>&amp;times;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>exp</mi> <mo>{</mo> <mo>-</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>R</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>

The logarithm at both ends of the pair formula (35) is

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>I</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>P</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <mi>J</mi> <mo>&amp;times;</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <mi>I</mi> <mi>n</mi> <mi>&amp;pi;</mi> <mo>-</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <mrow> <mo>(</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mi>I</mi> <mi>n</mi> <mo>|</mo> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>|</mo> <mo>)</mo> </mrow> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>R</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>

Maximizing equation (36) yields the direction of arrival of the signal, i.e., an estimate of the direction of arrival of the signalK is 1,2, …, K, i.e. by

<mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>I</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>P</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>&amp;part;</mo> <mi>&amp;alpha;</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow>

Obtaining;

by derivation have

<mrow> <msub> <mover> <mi>&amp;alpha;</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <mi>arg</mi> <munder> <mi>max</mi> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </munder> <mo>|</mo> <mi>Re</mi> <mfenced open = "{" close = "}"> <mtable> <mtr> <mtd> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <msub> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>&amp;times;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mrow> <mo>(</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>R</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mrow> <mo>(</mo> <msup> <mover> <mi>R</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> </mrow> <mo>)</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;times;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mover> <mi>&amp;mu;</mi> <mo>^</mo> </mover> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <mi>A</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mover> <mi>&amp;Sigma;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;Omega;</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&amp;times;</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <msup> <mo>|</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mrow>

Wherein Re {. is the real part of the solution {. cndot.); omega_-k、Respectively expressed from Ω andremoving the k-th element from the first element; k is 1,2, …, K;

according toIs calculated as Δ d₂,…,Δd_MThen, W (f) is obtained from the equations (16) and (15)_i,α_k) And W (f)_iΩ), then array correction is performed to find a' (f)_i,α_k) And A' (f)_i,Ω_-k) (ii) a Based on the above parameters and equation (38), an estimate of the direction of arrival of the array-corrected signal can be obtained

2. The method for correcting array element position errors in broadband signal super-resolution direction finding according to claim 1, wherein the specific steps of establishing the array signal model containing the array element position errors in step 1 are as follows:

step 1.1: establishing an ideal array signal model:

provided with K far-field broadband signals s_k(t), K is 1,2, …, K, and is incident on a broadband uniform linear array of M omnidirectional array elements, arriving at α ═ α₁,…,α_k,…,α_K]The array element interval is d; far field broadband signal s_k(t), broadband signal s for short_k(t)；

Taking the 1 st array element as the phase reference point, ideally, the output of the m-th array element is expressed as

<mrow> <msub> <mi>x</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <msub> <mi>s</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>&amp;tau;</mi> <mi>m</mi> </msub> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>n</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> <mo>,</mo> <mi>M</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>

Wherein,representing the kth broadband signal s_k(t) the time delay of the m-th array element relative to its arrival at the phase reference point, c is the propagation speed of the electromagnetic wave in vacuumDegree, n_m(t) Gaussian white noise received by the mth array element;

suppose the frequency range of the wideband signal is f_Low,f_High]The broadband signal is divided into J frequency points by using discrete Fourier transform, and the J frequency points are separated by a narrow-band filter bank, so that the output signal of the filter array of the ith group is expressed as

X(f_i)＝A(f_i,α)S(f_i)+N(f_i)，i＝1,2,…,J (2)

Wherein f is_Low≤f_i≤f_High，i＝1,2,…,J；

Assuming at each frequency point f_iHaving performed KP subsampling, X (f)_i) Is expressed in matrix form as

X(f_i)＝[X(f_i,1),…,X(f_i,kp),…,X(f_i,KP)]，i＝1,2,…,J (3)

Wherein, X (f)_iKp) is X (f)_i) The kp-th time data sampling matrix of (c),

X(f_i,kp)＝[X₁(f_i,kp),…,X_m(f_i,kp),…,X_M(f_i,kp)]^T，i＝1,2,…,J， (4)

X_m(f_ikp) is the m-th array element at frequency point f_iThe kp-th data sampling value obtained above;

A(f_iα) is the ideal case frequency point f_iThe array flow pattern matrix on the upper surface,

A(f_i,α)＝[a(f_i,α₁),…,a(f_i,α_k),…,a(f_i,α_K)]，i＝1,2,…,J， (5)

a(f_i,α_k) Is the frequency point f under the ideal condition_iThe array of the top k-th signal leads to the vector,

wherein,is the phase of the kth signal; j is a complex number flag;

S(f_i)＝[S(f_i,1),…,S(f_i,kp),…,S(f_i,KP)]，i＝1,2,…,J， (8)

is a signal s_k(t) a fourier transformed signal vector matrix, K being 1,2, …, K;

wherein, S (f)_iKp) is S (f)_i) The kp-th sub-sampling matrix of the signal,

S(f_i,kp)＝[S₁(f_i,kp),…S_k(f_i,kp),…,S_K(f_i,kp)]^Ti＝1,2,…,J (9)

S_k(f_ikp) is the k signal at frequency f_iThe kpth signal sampling value obtained above;

N(f_i)＝[N(f_i,1),…,N(f_i,kp),…,N(f_i,KP)]i＝1,2,…,J (10)

is noise n_m(t) noise vector matrix after Fourier transform, mean 0, variance μ²(f_i) (ii) a M is 1,2, …, M; wherein, N (f)_iKp) is N (f)_i) The kp-th time of the noise sampling matrix,

N(f_i,kp)＝[N₁(f_i,kp),…,N_m(f_i,kp),…,N_M(f_i,kp)]^Ti＝1,2,…,J (11)

N_m(f_ikp) is the m-th array element at frequency point f_iThe kp-th noise sampling value obtained above;

step 1.2: establishing an array signal model containing array element position errors on the basis of an ideal array signal model:

when array element position error exists in the array, the frequency point f_iThe output of the array at is represented as

X'(f_i)＝A'(f_i,α)S(f_i)+N(f_i)，i＝1,2,…,J (12)

Time frequency point f with array element position error_iA received signal covariance matrix of

R'(f_i)＝E{X'(f_i)(X'(f_i))^H}，i＝1,2,…,J (13)

In formula (12), A' (f)_i,α)＝[a'(f_i,α₁),…,a'(f_i,α_k),…,a'(f_i,α_K)]For existence of array element position error time frequency point f_iArray flow pattern matrix of (a') (f)_i,α_k) Steering vectors for the corresponding arrays;

when the position of the m-th array element has an error delta d_mWhen the method is equivalent to introducing azimuth-dependent phase disturbance into the array steering vector, the method has the following advantages

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>W</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mrow>

Wherein,

<mrow> <mi>W</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mo>...</mo> <mo>,</mo> <msup> <mi>e</mi> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>M</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> </mrow> </msup> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>

is a frequency point f_iThe up and arrival direction is α_kThe position error of the array element disturbs the matrix;

wherein

<mrow> <msub> <mi>&amp;Delta;&amp;tau;</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;d</mi> <mi>m</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msub> <mi>sin&amp;alpha;</mi> <mi>k</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>

When the kth signal reaches the mth array element, the signal source propagation delay error is introduced by the disturbance of the position error of the array element;

then the array flow pattern matrix in the presence of array element position error is

<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>&amp;lsqb;</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msup> <mi>a</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>&amp;alpha;</mi> <mi>K</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>W</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

Wherein W (f)_i,α)＝[W(f_i,α₁),…,W(f_i,α_k),…,W(f_i,α_K)]Time frequency point f for indicating array element position error_iAnd (3) disturbing the matrix by the position error of the array element.

3. The method for correcting array element position error in broadband signal super-resolution direction finding according to claim 2, wherein the method of maximizing the expectation is adopted in step 2 to pair w (f)_i)、μ²(f_i) And_l(f_i) The specific steps for performing the iterative estimation are as follows:

in the E-step of expectation maximization method, first, the method is performedIs calculated by the distribution function of

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>F</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>;</mo> <mi>&amp;delta;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mi>w</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mrow> <mo>&amp;lang;</mo> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mi>M</mi> <mo>&amp;times;</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <msup> <mi>In&amp;mu;</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>w</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>J</mi> </munderover> <mrow> <mo>(</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <msub> <mi>In&amp;delta;</mi> <mi>l</mi> </msub> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mo>(</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>k</mi> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>K</mi> <mi>P</mi> </mrow> </munderover> <mo>|</mo> <msub> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mi>l</mi> </msub> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>k</mi> <mi>p</mi> <mo>)</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mrow> <msub> <mi>&amp;delta;</mi> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>&amp;rang;</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>

Wherein the operator < · > represents the solution condition expectation;

in the M-step of the expectation-maximization method, the distribution functions are respectively obtainedDerivatives of unknown parameters, i.e. pairsSolving each unknown parameter by taking an extreme value;

<mrow> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>F</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>;</mo> <mi>&amp;delta;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mi>w</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>w</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mn>2</mn> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;lsqb;</mo> <mo>&lt;</mo> <msup> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&gt;</mo> <mi>w</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>&lt;</mo> <msup> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>-</mo> <mi>A</mi> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>&gt;</mo> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>F</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>;</mo> <mi>&amp;delta;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mi>w</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>M</mi> <mo>&amp;times;</mo> <mi>K</mi> <mi>P</mi> </mrow> <mrow> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <msup> <mrow> <mo>(</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>&lt;</mo> <mo>|</mo> <mo>|</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>&gt;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <mi>F</mi> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>;</mo> <mi>&amp;delta;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <mi>w</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>,</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;delta;</mi> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mi>K</mi> <mi>P</mi> </mrow> <mrow> <msub> <mi>&amp;delta;</mi> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <msubsup> <mi>&amp;delta;</mi> <mi>l</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&lt;</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>k</mi> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>K</mi> <mi>P</mi> </mrow> </munderover> <mo>|</mo> <msub> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>k</mi> <mi>p</mi> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>&gt;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>

respectively making the above derivatives be 0, i.e. obtaining the estimated value of each unknown parameter at the p-th iteration

<mrow> <msup> <mi>w</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mo>&lt;</mo> <msup> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mo>&gt;</mo> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>&lt;</mo> <msup> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>-</mo> <mi>A</mi> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>&gt;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msup> <mrow> <mo>(</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>M</mi> <mo>&amp;times;</mo> <mi>K</mi> <mi>P</mi> </mrow> </mfrac> <mo>&lt;</mo> <mo>|</mo> <mo>|</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>&gt;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msubsup> <mi>&amp;delta;</mi> <mi>l</mi> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>K</mi> <mi>P</mi> </mrow> </mfrac> <mo>&lt;</mo> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>k</mi> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>K</mi> <mi>P</mi> </mrow> </munderover> <mo>|</mo> <msub> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mi>l</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>k</mi> <mi>p</mi> <mo>)</mo> </mrow> <msup> <mo>|</mo> <mn>2</mn> </msup> <mo>&gt;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>

Wherein (p) represents the number of iterations, in equation (27)

<mrow> <mtable> <mtr> <mtd> <mrow> <mo>&lt;</mo> <msup> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mo>&gt;</mo> <mrow> <mi>r</mi> <mn>1</mn> <mo>,</mo> <mi>r</mi> <mn>2</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msubsup> <mi>&amp;Psi;</mi> <mrow> <mi>r</mi> <mn>1</mn> </mrow> <mi>H</mi> </msubsup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>&amp;Psi;</mi> <mrow> <mi>r</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>sin&amp;alpha;</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>sin&amp;alpha;</mi> <mi>K</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> <mo>)</mo> </mrow> <mo>&amp;times;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mi>O</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msup> <mi>O</mi> <mi>H</mi> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>+</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <mi>&amp;Xi;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> <msup> <mrow> <mo>(</mo> <mi>A</mi> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>sin&amp;alpha;</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>sin&amp;alpha;</mi> <mi>K</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>

Is a matrixRow r1, column r2, where tr [ ·]Representing the movement of a targetCalculating;

in formula (30);

O(f_i)＝Σ(f_i)(A'(f_i,Ω))^H(μ²(f_i)I_M+A'(f_i,Ω)Σ(f_i)(A'(f_i,Ω))^H)^-1X(f_i) (31)

is an intermediate variable;

<mrow> <mtable> <mtr> <mtd> <mrow> <mi>&amp;Xi;</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>&amp;Sigma;</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mi>&amp;Sigma;</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>I</mi> <mi>M</mi> </msub> <mo>+</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> <mi>&amp;Sigma;</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>H</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mi>&amp;Sigma;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>

is an intermediate variable;

in formula (27);

<mrow> <mtable> <mtr> <mtd> <mrow> <mo>&lt;</mo> <msup> <mover> <mi>&amp;Lambda;</mi> <mo>&amp;OverBar;</mo> </mover> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>-</mo> <mi>A</mi> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mo>&gt;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mi>t</mi> <mi>r</mi> <mo>&amp;lsqb;</mo> <msubsup> <mi>&amp;Psi;</mi> <mi>r</mi> <mi>H</mi> </msubsup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>X</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mi>O</mi> <mi>H</mi> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mi>A</mi> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>sin&amp;alpha;</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>sin&amp;alpha;</mi> <mi>K</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>&amp;rsqb;</mo> <mo>-</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>t</mi> <mi>r</mi> <mo>&amp;lsqb;</mo> <msubsup> <mi>&amp;Psi;</mi> <mi>r</mi> <mi>H</mi> </msubsup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mi>A</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>O</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msup> <mi>O</mi> <mi>H</mi> </msup> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>+</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <mi>&amp;Xi;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mi>A</mi> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>j</mi> <mn>2</mn> <msub> <mi>&amp;pi;f</mi> <mi>i</mi> </msub> </mrow> <mi>c</mi> </mfrac> <msup> <mrow> <mo>&amp;lsqb;</mo> <msub> <mi>sin&amp;alpha;</mi> <mn>1</mn> </msub> <mo>,</mo> <mn>...</mn> <mo>,</mo> <msub> <mi>sin&amp;alpha;</mi> <mi>K</mi> </msub> <mo>&amp;rsqb;</mo> </mrow> <mi>T</mi> </msup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>&amp;rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mrow>

in the formula (28)

<mrow> <mtable> <mtr> <mtd> <mrow> <mo>&lt;</mo> <mo>|</mo> <mo>|</mo> <msup> <mover> <mi>X</mi> <mo>&amp;OverBar;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mover> <mi>S</mi> <mo>&amp;OverBar;</mo> </mover> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mn>2</mn> <mn>2</mn> </msubsup> <mo>&gt;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mi>O</mi> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>|</mo> <msubsup> <mo>|</mo> <mi>F</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mi>K</mi> <mi>P</mi> <mo>&amp;times;</mo> <mi>t</mi> <mi>r</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mi>&amp;Xi;</mi> <mo>(</mo> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>)</mo> <msup> <mrow> <mo>(</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mo>&amp;prime;</mo> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mi>i</mi> </msub> <mo>,</mo> <mi>&amp;Omega;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>p</mi> <mo>)</mo> </mrow> </msup> <mo>)</mo> </mrow> <mi>H</mi> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>

In formula (33), Ψ_r(f_i) The intermediate variable is a matrix with M × M dimensions, only the elements at the r +1 th row and the r +1 th column are all 1, and the rest elements are all 0;

substitution of equations (30) to (34) into equivalent equations of equations (27) to (28) to degenerate w (f)_i) And mu²(f_i) Solving;

after several iterations, w (f)_i)、μ²(f_i) And_l(f_i) If the variation of the three estimated values approaches 0, the final estimated value is obtainedAndcorrespond to obtainAnd