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

Abstract

The invention relates to a method for correcting array element position error in broadband signal super-resolution direction finding, and the invention relates to a method for correcting array error existing in broadband signal super-resolution direction finding. The present invention aims to solve the problems of large amount of computation and slow convergence speed existing in the existing array element position error correction method for high-dimensional nonlinear optimization and the problem that the existing correction technology for multi-small aperture arrays is not suitable for broadband signals. The present invention utilizes the signals on each frequency point to construct corresponding optimization functions, and then utilizes the spatial sparsity of the signals to iteratively optimize the functions on each frequency point through a sparse Bayesian learning method, and finally optimizes the functions on all frequency points The information is fused to estimate the direction of arrival of the signal. This method can effectively realize the array error correction when the array element position error exists, and use multi-chip digital signal processors to effectively improve the running speed of the algorithm. The invention is applicable to the field of correcting array errors existing 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 technique

Super-resolution direction finding is an important research content in array signal processing, and has a wide range of applications in the fields of radio monitoring, Internet of Things and electronic countermeasures. Most of the current direction finding methods are based on the premise of accurately grasping the flow pattern of the array. In the actual direction finding system, due to the disturbance of the position of the array element or the inaccurate measurement, the direction finding estimation is often accompanied by the position error of the array element, which directly leads to the deterioration of the performance of many super-resolution direction finding methods, and even invalid, so it is necessary to correct it.

Calibration methods for parametric classes can generally be divided into active calibration and self-calibration. Active correction can estimate array disturbance parameters offline by setting auxiliary sources with known orientation in space, while self-calibration methods usually jointly estimate the orientation of space sources and array disturbance parameters according to some optimization function. Based on the principle of subspace, Friedlander B and Weiss A J proposed an array self-calibration technique for alternate iterative estimation of source azimuth, inter-array mutual coupling, array element gain and phase disturbance. However, this technology needs to solve high-dimensional nonlinear optimization problems, which requires a large amount of calculation and slow convergence speed, and for uniform linear arrays, there are fuzzy problems in the array disturbance parameters. Scholars such as Mavrychev have studied the partial correction technology of multi-small aperture arrays, which effectively solve the multi-target resolution and orientation accuracy requirements that are usually difficult to meet with a single small-aperture array. Since the constructed estimator does not need to accurately know the location information between sub-arrays, the influence of false peaks and location estimation errors on orientation estimation is avoided. However, they are only suitable for narrowband signals, and there are few published literatures on the correction technology of array element position error in super-resolution direction finding of broadband signals.

Contents of the invention

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

A method for correcting array element position errors in broadband signal super-resolution direction finding, comprising the following steps:

Step 1: Establish an array signal model including array element position error:

When there is an element position error in the array, the array output at frequency f _i can be expressed as

X'(f _i )=A'(f _i ,α)S(f _i )+N(f _i ), i=1,2,...,J (12)

A'(f _i ,α) is the array flow pattern matrix at the time-frequency point f _i when there is an element position error; S(f _i ) is the signal vector matrix of the signal s _k (t) after Fourier transform; N(f _i ) is the noise vector matrix of the noise n _m (t) after Fourier transform, the mean value is 0, and the variance is μ ² (f _i );

The covariance matrix of the received signal at the time-frequency point _fi when there is an element position error is

R'(f _i )=E{X'(f _i )(X'(f _i )) ^H }, i=1,2,...,J (13)

A(f _i ,α)=[a(f _i ,α ₁ ),…,a(f _i ,α _k ),…,a(f _i ,α _{K )} ] is ideally the Array flow pattern matrix, a(f _i ,α _k ) is the array steering vector of the kth signal on the frequency point f _i under ideal conditions;

When there is an error △d _m in the position of the mth array element, it can be equivalent to introducing an azimuth-dependent phase disturbance into the array steering vector, then

Among them, W(f _i ,α _k ) is the array element position error perturbation matrix in the direction α _k at the frequency point f _{i ,} a'(f _i ,α _k ) means that when there is an array element position error array-steering vector for k signals;

When the kth signal arrives at the mth array element from the direction α _k , the signal source propagation delay error introduced by the disturbance of the array element position error; then the array flow pattern matrix at the time-frequency point _fi when there is an array element position error is

A'(f _i ,α)=[a'(f _i ,α ₁ ),...,a'(f _i ,α _k ),...,a'(f _i ,α _K )]

(17)

＝W(f _i ,α)·A(f _i ,α)

Among them, W(f _i ,α)=[W(f _i ,α ₁ ),…,W(f _i ,α _k ),…,W(f _i ,α _K )] represents the matrix on the frequency point f _i element position error perturbation matrix;

Step 2: Estimate the array signal parameters including array element position error:

First, the search space is divided into several discrete angle grids Corresponds to the L directions that the signal may arrive; thus, the sparse representation of the array flow matrix on the frequency point f _i can be obtained

in, is the array-steering vector of the l-th sparse signal on frequency point f _i , correspondingly, the sparse representation of the array flow pattern matrix on frequency point f _i can be obtained when there is an array element position error is the sparse representation of the element position error perturbation matrix at the frequency point f _i , is the array element position error perturbation matrix of the lth sparse signal at frequency f _i , When the lth sparse signal arrives at the mth array element, the source propagation delay error introduced by the array element position error disturbance, is the array steering vector of the lth sparse signal on time-frequency point f _i with array element position error, then the sparse representation of the array output signal at time-frequency point f _i with array element position error can be obtained

Among them, Λ(f _i ) is a parameter only related to the original signal, and has nothing to do with the error; is the sparse representation of Λ(f _i ); w(f _i )=[△d ₂ ,…,△d _M ] ^T represents the element position error disturbance vector, taking the first array element as the position reference point, △d ₂ ,...,△d _M are the deviations between the real position and the measured position of the second array element to the Mth array element on the frequency point fi _{, which have nothing to do with the signal frequency f i ;}

The covariance matrix of is

In formula (18) is the sparse representation of S(f _i ),

in, is a sparse matrix, which is the sparse representation of S(f _i ,kp), contains only K non-zero elements, for The l-th element in , if and only if Time The elements in are not all zero and have Therefore It can be regarded as a matrix obtained by adding many 0 elements to S(f _i );

Suppose δ(f _i )=[δ ₁ (f _i ),…,δ _l (f _i ),…,δ _L (f _i )] ^T is The variance of the elements in reflects the energy of the signal, that is,

Among them, Σ(f _i )=diag(δ(f _i )), namely Obey the Gaussian distribution with mean value 0 and variance δ(f _i );

because It can be seen as a vector obtained by adding many 0 elements to S(f _i ), so δ(f _i ) contains K non-zero elements, and K<<L, according to δ(f _i ), combined with w (f _i ) and noise variance μ ² (f _i ) to estimate So as to reconstruct the original signal and correct the error at the same time;

According to formula (18), it can be seen that the probability density of the array output signal of the frequency point _fi when there is an array element position error is

Combining formulas (18), (20) and (21) can be obtained

Wherein, I _M is the unit matrix of M * M dimensions;

Use the Expectation Maximization (EM) method to iteratively estimate w(f _i ), μ ² (f _i ) and δ _l (f _i ), and obtain the estimated value with corresponding available as well as

Step 3: Take advantage of with Correct the array error and solve the signal arrival direction;

Let X be a vector composed of the sum of all frequency point signals received by the array within a period of observation time. Since the signals of each frequency point are statistically independent, the joint probability density of the received signals at each frequency point is

Taking the logarithm of both ends of the formula (35) has

Therefore, by maximizing the formula (36), the direction of arrival of the signal can be obtained, that is, the estimated value of the direction of arrival of the signal k=1,2,...,K, that is, it can pass

obtain;

After derivation there are

Among them, Re{·} is to find the real part of {·}; Ω _-k , Respectively from Ω and Remove the kth element in it; k=1,2,...,K;

according to Calculate △d 2 ,…,△d M from the expressions of △d ₂ ,…,△d _M , and then calculate W(f _i ,α _k ) and W(f _i ,Ω) according to formulas (16) and (15), and then perform array correction to obtain a'(f _i ,α _k ) and A'(f _i ,Ω _-k ); according to the above parameters and formula (38), the estimated value of the arrival direction of the signal after array correction can be obtained

The present invention has the following beneficial effects:

The present invention proposes a broadband signal super-resolution direction-finding error correction method when there is an array element position error, using the signals on each frequency point to construct a corresponding optimization function, and then using the spatial sparsity of the signal to learn through sparse Bayesian The method iteratively optimizes the functions on each frequency point respectively, and finally fuses the information on all frequency points to estimate the signal arrival direction. The present invention can effectively realize the array error correction when the array element position error exists, and when the signal-to-noise ratio is 10dB and the sampling snapshot number of each frequency point is 40, the accuracy can reach 0.6°/σ.

Moreover, the method of the invention can be processed by multi-chip digital signal processors, which can effectively improve the running speed of the algorithm.

Description of drawings

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

Fig. 2 is the device diagram of broadband signal detection system;

3 is a diagram of a broadband signal super-resolution direction finding device in Embodiment 4;

4 is a diagram of a broadband signal super-resolution direction finding device in Embodiment 5;

FIG. 5 is a diagram of a broadband signal super-resolution direction finding device according to Embodiment 6. FIG.

detailed description

Specific implementation mode one:

A method for correcting array element position errors in broadband signal super-resolution direction finding, comprising the following steps:

Step 1: Establish an array signal model including array element position error:

When there is an element position error in the array, the array output at frequency f _i can be expressed as

X'(f _i )=A'(f _i ,α)S(f _i )+N(f _i ), i=1,2,...,J (12)

A'(f _i ,α) is the array flow pattern matrix at the time-frequency point f _i when there is an element position error; S(f _i ) is the signal vector matrix of the signal s _k (t) after Fourier transform; N(f _i ) is the noise vector matrix of the noise n _m (t) after Fourier transform, the mean value is 0, and the variance is μ ² (f _i );

The covariance matrix of the received signal at the time-frequency point _fi when there is an element position error is

R'(f _i )=E{X'(f _i )(X'(f _i )) ^H }, i=1,2,...,J (13)

A(f _i ,α)=[a(f _i ,α ₁ ),…,a(f _i ,α _k ),…,a(f _i ,α _{K )} ] is ideally the Array flow pattern matrix, a(f _i ,α _k ) is the array steering vector of the kth signal on the frequency point f _i under ideal conditions;

When there is an error △d _m in the position of the mth array element, it can be equivalent to introducing an azimuth-dependent phase disturbance into the array steering vector, then

Among them, W(f _i ,α _k ) is the array element position error perturbation matrix in the direction α _k at the frequency point f _{i ,} a'(f _i ,α _k ) means that when there is an array element position error array-steering vector for k signals;

When the kth signal arrives at the mth array element from the direction α _k , the signal source propagation delay error introduced by the disturbance of the array element position error; then the array flow pattern matrix at the time-frequency point _fi when there is an array element position error is

A'(f _i ,α)=[a'(f _i ,α ₁ ),...,a'(f _i ,α _k ),...,a'(f _i ,α _K )]

(17)

＝W(f _i ,α)·A(f _i ,α)

Among them, W(f _i ,α)=[W(f _i ,α ₁ ),…,W(f _i ,α _k ),…,W(f _i ,α _K )] represents the matrix on the frequency point f _i element position error perturbation matrix;

Step 2: Estimate the array signal parameters including array element position error:

First, the search space is divided into several discrete angle grids Corresponds to the L directions that the signal may arrive; thus, the sparse representation of the array flow matrix on the frequency point f _i can be obtained

in, is the array-steering vector of the l-th sparse signal on frequency point f _i , correspondingly, the sparse representation of the array flow pattern matrix on frequency point f _i can be obtained when there is an array element position error is the sparse representation of the element position error perturbation matrix at the frequency point f _i , is the array element position error perturbation matrix of the lth sparse signal at frequency f _i , When the lth sparse signal arrives at the mth array element, the source propagation delay error introduced by the array element position error disturbance, is the array steering vector of the lth sparse signal on the time-frequency point f _i with the array element position error, then the sparse representation of the array output signal at the time-frequency point f _i with the array element position error can be obtained

Among them, Λ(f _i ) is a parameter only related to the original signal, and has nothing to do with the error; is the sparse representation of Λ(f _i ); w(f _i )=[△d ₂ ,…,△d _M ] ^T represents the element position error disturbance vector, taking the first array element as the position reference point, △d ₂ ,...,△d _M are the deviations between the real position and the measured position of the second array element to the Mth array element on the frequency point fi _{, which have nothing to do with the signal frequency f i ;}

The covariance matrix of is

In formula (18) is the sparse representation of S(f _i ),

in, is a sparse matrix, which is the sparse representation of S(f _i ,kp), contains only K non-zero elements, for The l-th element in , if and only if Time The elements in are not all zero and have Therefore It can be regarded as a matrix obtained by adding many 0 elements to S(f _i );

Suppose δ(f _i )=[δ ₁ (f _i ),…,δ _l (f _i ),…,δ _L (f _i )] ^T is The variance of the elements in reflects the energy of the signal, that is,

Among them, Σ(f _i )=diag(δ(f _i )), namely Obey the Gaussian distribution with mean value 0 and variance δ(f _i );

because It can be regarded as a vector obtained by adding many 0 elements to S(f _i ), so δ(f _i ) contains K non-zero elements, and K<<L, according to δ(f _i ), combined with w (f _i ) and noise variance μ ² (f _i ) to estimate So as to reconstruct the original signal and correct the error at the same time;

According to formula (18), it can be seen that the probability density of the array output signal of the frequency point _fi when there is an array element position error is

Combining formulas (18), (20) and (21) can be obtained

Wherein, I _M is the unit matrix of M * M dimensions;

Use the Expectation Maximization (EM) method to iteratively estimate w(f _i ), μ ² (f _i ) and δ _l (f _i ), and obtain the estimated value with corresponding available as well as

Step 3: Take advantage of with Correct the array error and solve the signal arrival direction;

Let X be a vector composed of the sum of all frequency point signals received by the array within a period of observation time. Since the signals of each frequency point are statistically independent, the joint probability density of the received signals at each frequency point is

Taking the logarithm of both ends of the formula (35) has

Therefore, by maximizing the formula (36), the direction of arrival of the signal can be obtained, that is, the estimated value of the direction of arrival of the signal k=1,2,...,K, that is, it can pass

obtain;

After derivation there are

Among them, Re{·} is to find the real part of {·}; Ω _-k , Respectively from Ω and Remove the kth element in it; k=1,2,...,K;

according to Calculate △d 2 ,…,△d M from the expressions of △d ₂ ,…,△d _M , and then calculate W(f _i ,α _k ) and W(f _i ,Ω) according to formulas (16) and (15), and then perform array correction to obtain a'(f _i ,α _k ) and A'(f _i ,Ω _-k ); according to the above parameters and formula (38), the estimated value of the arrival direction of the signal after array correction can be obtained

Specific implementation mode two:

The specific steps for establishing the array signal model containing the array element position error described in step 1 of this embodiment are as follows:

Step 1.1: Build an ideal array signal model:

As shown in Figure 1, there are K far-field broadband signals s _k (t), k=1, 2,..., K, incident on a broadband uniform linear array composed of M omnidirectional array elements, and the arrival direction is α ＝[α ₁ ,…,α _k ,…,α _K ], the array element spacing is d; the far-field broadband signal s _k (t), referred to as the broadband signal s _k (t);

Taking the first array element as the phase reference point, ideally, the output of the mth array element is expressed as

in, Indicates the delay of the k-th broadband signal s _k (t) arriving at the m-th array element relative to its arrival at the phase reference point, c is the propagation speed of electromagnetic waves in vacuum, and n _m (t) is the receiving time of the m-th array element to Gaussian white noise;

Assuming that the frequency range of the broadband signal is [f _Low , f _High ], the broadband signal is divided into J frequency points by discrete Fourier transform, and separated by a narrowband filter bank, then the i-th filter array output signal is expressed as

X(f _i )=A(f _i ,α)S(f _i )+N(f _i ), i=1,2,...,J (2)

Among them, f _Low ≤ f _i ≤ f _High , i=1,2,...,J;

Assuming that KP samples are taken on each frequency point f _i , the matrix form of X(f _i ) is expressed as

X(f _i )=[X(f _i ,1),...,X(f _i ,kp),...,X(f _i ,KP)], i=1,2,...,J (3)

Among them, X(f _i ,kp) is the kpth data sampling matrix of X(f _i ),

X(f _i ,kp)=[X ₁ (f _i ,kp),…,X _m (f _i ,kp),…,X _M (f _i ,kp)] ^T ,i=1,2,…, J, (4)

X _m (f _i , kp) is the kpth data sampling value obtained by the mth array element at the frequency point f _i ;

A(f _i ,α) is ideally the array flow pattern matrix on the frequency point f _i ,

A(f _i ,α)=[a(f _i ,α ₁ ),…,a(f _i ,α _k ),…,a(f _i ,α _K )], i=1,2,…,J , (5)

a(f _i ,α _k ) is the array steering vector of the kth signal on the frequency point f _i under ideal conditions,

in, is the phase of the k-th signal; j is a complex sign;

S(f _i )=[S(f _i ,1),...,S(f _i ,kp),...,S(f _i ,KP)], i=1,2,...,J, (8)

is the signal vector matrix of signal s _k (t) after Fourier transform, k=1,2,...,K;

Among them, S(f _i ,kp) is the kpth signal sampling matrix of S(f _i ),

S(f _i ,kp)＝[S ₁ (f _i ,kp),...S _k (f _i ,kp),...,S _K (f _i ,kp)] ^T i=1,2,...,J ( 9)

S _k (f _i , kp) is the kp-th signal sampling value obtained by the k-th signal at the frequency point f _i ;

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

is the noise vector matrix of the noise n _m (t) after Fourier transformation, the mean value is 0, and the variance is μ ² (f _i ); m=1,2,...,M; where, N(f _i ,kp) is N The kpth noise sampling matrix of (f _i ),

N(f _i ,kp)=[N ₁ (f _i ,kp),…,N _m (f _i ,kp),…,N _M (f _i ,kp)] ^T i=1,2,…,J (11)

N _m (f _i , kp) is the kpth noise sampling value obtained by the mth array element at the frequency point f _i ;

Step 1.2: Based on the ideal array signal model, an array signal model including array element position error is established:

When there is an element position error in the array, the array output at frequency f _i can be expressed as

X'(f _i )=A'(f _i ,α)S(f _i )+N(f _i ), i=1,2,...,J (12)

The covariance matrix of the received signal at the time-frequency point _fi when there is an element position error is

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 ,α ₁ ),...,a'(f _i ,α _k ),...,a'(f _i ,α _K )] is the array flow pattern matrix at the time-frequency point f _i where there is an array element position error, and a'(f _i ,α _k ) is the corresponding array steering vector;

When there is an error △d _m in the position of the mth array element, it can be equivalent to introducing an azimuth-dependent phase disturbance into the array steering vector, then

in,

is the array element position error perturbation matrix at frequency f _i and direction of arrival α _k ;

in

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

Then the array flow pattern matrix when there is an element position error is

A'(f _i ,α)=[a'(f _i ,α ¹ ),...,a'(f _i ,α _k ),...,a'(f _i ,α _K )]

(17)

＝W(f _i ,α)·A(f _i ,α)

Where W(f _i ,α)=[W(f _i ,α ₁ ),…,W(f _i ,α _k ),…,W(f _i ,α _K )] means that there is a time-frequency point where there is an element position error Array element position error perturbation matrix on f _i .

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

Specific implementation mode three:

The specific steps for iteratively estimating w(f _i ), μ ² (f _i ) and δ _l (f _i ) using the expectation maximization method described in step 2 of this embodiment are as follows:

In the E-step step in the expectation maximization method, the first Calculate the distribution function of

The operator <·> represents the expectation of the solution condition;

In the M-step step in the expectation maximization method, the distribution functions are obtained respectively Derivatives for each unknown parameter, that is, for Take extreme values to solve for each unknown parameter;

Let the above derivatives be 0 respectively, then the estimated value of each unknown parameter at the pth iteration can be obtained

where (p) represents the number of iterations, in formula (27)

for the matrix The elements in row r1 and column r2, where tr[·] represents the trace operation;

In formula (30);

O(f _i )＝Σ(f _i )(A'(f _i ,Ω)) ^H (μ ² (f _i )I _M +A'(f _i ,Ω)Σ(f _i )(A'(f _i ,Ω)) ^H ) ^-1 X(f _i ) (31)

is an intermediate variable;

Ξ(f _i )

(32)

＝Σ(f _i )-Σ(f _i )(A'(f _i ,Ω)) ^H (μ ² (f _i )I _M +A'(f _i ,Ω)Σ(f _i )(A'( f _i ,Ω)) ^H ) ^-1 A'(f _i ,Ω)Σ(f _i )

is an intermediate variable;

In formula (27);

In formula (28)

In formula (33), Ψ _r (f _i ) is an intermediate variable, which is an M×M dimensional matrix, only the elements at row r+1 and column r+1 are all 1, and the rest of the elements are all 0;

Since it is more complicated to calculate w(f _i ) and μ ² (f _i ) directly using formulas (27)～(28), formulas (30)～(34) can be substituted into formulas (27)～(28) and equivalent Simplify the formula and solve for w(f _i ) and μ ² (f _i );

After iterating several steps, the changes of the estimated values of w(f _i ), μ ² (f _i ) and δ _l (f _i ) tend to 0, and they can be considered to have converged at this time, then the final estimated value with correspond to get as well as

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

Specific Embodiment Four: This embodiment is described in detail with reference to FIG. 2 and FIG. 3 ,

This implementation mode is a broadband signal detection system and a detection method for implementing the methods described in the first to third specific implementation modes.

As shown in Figure 2, the broadband signal detection system includes: 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 Figure 3, the broadband signal super-resolution direction finding device 3 includes 6 pieces of digital signal processors, ie DSP, using fast serial input and output ports, ie SRIO port, to form a multi-processor system to realize parallel processing. Among them, DSP3-1 is the main DSP, and DSP3-2～DSP3-6 are slave DSPs; the broadband signal super-resolution direction finding device 3 also includes CPLD3-7, PROM3-8, FLASH3-9, SRAM3-10, JTAG3-11, Power, Crystal Oscillator and Reset.

The digital signal processor adopts TMS320C6678 of Texas Instruments (TI) company, adopts 6 processors for parallel processing, and 6 DSPs are connected through SRIO ports. After power on, PROM3-8 first loads the program to CPLD3-7, and FLASH3-9 The program is loaded to these 6 DSPs (3-1~3-6), and then the main DSP3-1 starts to receive the observation data of J frequency points from the multi-channel wideband digital receiver 2, and divides them into W groups, assuming J=30, W=6, then each piece of DSP can process the observation data of U=30/6=5 frequency points, and the main DSP3-1 is responsible for processing other slave DSPs (3-2～3-6) through the SRIO port After that, each DSP (3-1 ~ 3-6) will solve the problem according to the steps of theoretical derivation above, and then the 5 chips will get their error estimates from DSP (3-2 ~ 3-6) It is transmitted to the main DSP3-1 through the SRIO port, and the main DSP3-1 uses these results again to obtain the signal arrival angle by combining formula (38). Among them, SRAM3-10 is responsible for storing data, JTAG3-11 is responsible for debugging DSP (3-1～3-6), power supply is responsible for overall power supply, crystal oscillator is responsible for providing clock, and reset is responsible for providing reset signal.

Specific Embodiment Five: This embodiment is described in detail with reference to FIG. 2 and FIG. 4 ,

This implementation mode is a broadband signal detection system and a detection method for implementing the methods described in the first to third specific implementation modes.

As shown in Figure 2, the broadband signal detection system includes: 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, ie, DSPs, which form a multi-processor system to realize parallel processing by using shared bus tightly coupled. Among them, DSP3-1 is the main DSP, and DSP3-2～DSP3-6 are slave DSPs; the broadband signal super-resolution direction finding device 3 also includes CPLD3-7, PROM3-8, FLASH3-9, SRAM3-10, JTAG3-11, Power, Crystal Oscillator and Reset.

The digital signal processor adopts the ADSP-TS201S of Analog Devices Instruments (ADI), and adopts 6 DSPs for parallel processing, and the 6 DSPs are connected through a shared bus tightly coupled. After power-on, PROM3-8 first loads the program to CPLD3-7 pair The DSP (3-1～3-6) is configured, and then FLASH3-9 loads the program to these 6 DSPs (3-1～3-6), and the main DSP3-1 starts to receive the multi-channel broadband digital receiver 2. The observed data of J frequency points are divided into W groups, assuming J=30, W=6, then each DSP can process the observed data of U=30/6=5 frequency points, and the main DSP3-1 passes The bus transmits the observation data that other slave DSPs (3-2~3-6) are responsible for processing to them, and then each DSP (3-1~3-6) solves according to the steps of the above theoretical derivation, and then the 5 slaves DSPs (3-2～3-6) transmit their respective error estimation values to the main DSP3-1 through the bus, and the main DSP3-1 uses these results again to obtain the signal arrival angle by combining formula (38). Among them, SRAM3-10 is responsible for storing data, JTAG3-11 is responsible for debugging DSP (3-1～3-6), power supply is responsible for overall power supply, crystal oscillator is responsible for providing clock, and reset is responsible for providing reset signal.

Specific embodiment six: This embodiment is described in detail with reference to Fig. 2 and Fig. 5 ,

This implementation mode is a broadband signal detection system and a detection method for implementing the methods described in the first to third specific implementation modes.

As shown in Figure 2, the broadband signal detection system includes: 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, ie DSPs, which are composed of a multi-processor system by cascading and loosely coupling link ports to realize parallel processing. Among them, DSP3-1 is the main DSP, and DSP3-2～DSP3-6 are slave DSPs; the broadband signal super-resolution direction finding device 3 also includes CPLD3-7, PROM3-8, FLASH3-9, SRAM3-10, JTAG3-11, Power, Crystal Oscillator and Reset.

The digital signal processor adopts the ADSP-TS201S of Analog Devices Instruments (ADI), and uses 6 processors for parallel processing. The 6 DSPs are connected through the link port cascading and loose coupling. CPLD3-7, FLASH3-9 load the programs of these 6 DSPs to the master DSP3-1, and the master DSP3-1 sequentially loads the programs of other slave DSPs (3-2～3-6) through the link port level by level Pass them, main DSP3-1 starts to receive the observation data of J frequency points that multi-channel broadband digital receiver 2 transmits later, they are divided into W group, assuming J=30, W=6, then every chip DSP can Process the observation data of U=30/6=5 frequency points, and the main DSP3-1 passes the observation data that other DSPs (3-2～3-6) are responsible for processing to them one by one through the link port. After that, each DSP (3-1~3-6) will solve the problem according to the steps of the above theoretical derivation, and then the 5 chips will send their error estimates from the DSP (3-2~3-6) through the link port level by level. The stages are uploaded to the main DSP3-1 one by one, and the main DSP3-1 uses these results again to obtain the signal arrival angle by combining formula (38). Among them, SRAM3-10 is responsible for storing data, JTAG3-11 is responsible for debugging DSP (3-1～3-6), power supply is responsible for overall power supply, crystal oscillator is responsible for providing clock, and reset is responsible for providing 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