FFRA diagonal ULA phase difference estimation and fuzzy solution method based on degraded space ARMA model

Abstract

Based on a degraded space ARMA model, the invention provides an FFRA diagonal ULA phase difference estimation and fuzzy solution method based on the degraded space ARMA model, a corrected Yule-Walker (MYW) linear equation set and a rooting polynomial are deduced firstly, and due to the fact that the diagonal ULA array element interval is larger than half of the carrier wave length, the diagonal phase difference generates the fuzzy problem. The invention provides a simple and effective diagonal phase difference fuzzy elimination scheme by utilizing an explicit linear combination relation met by a diagonal phase difference and an axial phase difference, and an actual interval of a non-fuzzy diagonal phase difference is deduced from an axial estimation phase difference. According to the method, the diagonal ULA of the FFRA on the X-Y plane is utilized, RMSE (root-mean-square error) and CRB (Cramer-Rao bound) numerical simulation results show that the method is effective for independent and coherent signal sources, and the RMSE is slowly converged to the corresponding CRB after ambiguity elimination. Due to the fact that the axial ULA and the diagonal ULA of the FFRA can be fully utilized, the estimation method provides a basis for two-dimensional DoA estimation based on the ULA combination.

Description

Technical Field

[0001] The invention belongs to the technical field of two-dimensional source measurement based on multidimensional array in array signal processing, and in particular relates to an FFRA diagonal ULA phase difference estimation and fuzzy resolution method based on a degenerate space ARMA model. Background Art

[0002] As a representative parameter estimation method, 2D DoA estimation of source or target based on multidimensional (MD) sensor array has been a traditional and interesting research topic in array signal processing. In the past few decades, this subject has been widely used in different engineering fields, including wireless communications, radar, sonar and seismic exploration.

[0003] In MD sensor arrays, dense rectangular area arrays (FFRA) such as massive MIMO (multiple input multiple output) systems are increasingly used as underlying physical devices. Due to the equal spacing of the sensor elements, the FFRA array structure provides a rich variety of uniform linear arrays (ULA) for 2D DoA estimation. The well-known ULAs are mainly set on the axis and diagonal or parallel to the axis and diagonal. For simplicity, they are called axial ULA and diagonal ULA respectively.

[0004] At present, the popular 2D DoA estimation methods mainly include: subspace-based methods, matrix bundle method (MPM) and propagator method (PM). Since the source azimuth and elevation are directly extracted from ULA data or transformed data after eigenvalue decomposition (EVD), the above estimation methods are generally based on a few special ULA formations, among which the L-shaped formation is a commonly used formation. The L-shaped array usually consists of two orthogonal ULAs located on the axis and axis in the Cartesian coordinate system. Another commonly used formation is the double parallel line formation. Compared with the number of axial ULAs in FFRA, this utilization is quite insufficient. As another important ULA type of FFRA, diagonal ULA is rarely mentioned and applied in 2D DoA estimation.

[0005] By analyzing the phase structure associated with ULA in FFRA, we can find the reason behind the current status of 2D DoA estimation. Unlike the widely studied pure azimuth ULA model (The azimuth-only ULA, the source phase is simply proportional to the phase difference), in 2D FFRA, the phase and phase difference of most ULAs satisfy a linear relationship, and a constant phase induced by the position of the ULA is usually added. The phase varies with the ULA involved and becomes diversified in the functional form. At the same time, the phase difference and constant phase in the phase are not independent of each other. They are both functions of azimuth and elevation. The existence of constant phase causes the existing estimation methods characterized by directly extracting the azimuth and elevation of the source to rely on specific ULAs. Due to the absence of constant phase and the simple functional form of phase and phase difference, ULA located on the coordinate axis is widely recommended to reduce the complexity of estimation.

[0006] In addition, the single or multinomial function form of the phase difference also determines the utilization of ULA in FFRA. The multinomial function form generally appears on the diagonal ULA, and the diagonal phase difference is a complex function of the elevation angle and the azimuth angle. Since the diagonal ULA array element spacing is far more than half of the carrier wavelength, the multinomial function form will cause the estimation of elevation and azimuth angles to be blurred at the same time. The ambiguity elimination of elevation and azimuth angles becomes another problem and difficulty in the utilization of diagonal ULA, which further hinders the utilization of diagonal ULA by existing methods.

[0007] In addition to the axial ULA, FFRA provides a variety of diagonal ULAs in the array structure. For 2D DoA estimation methods, it is necessary to have the basic ability to flexibly handle various ULAs of FFRA. Obviously, so far, the existing methods are still far from this ability. Summary of the invention

[0008] In order to overcome the above problems, the present invention provides a FFRA diagonal ULA phase difference estimation and fuzzy resolution method based on the degenerate space ARMA model, which can be widely used in research related to target parameter estimation, positioning, identification and tracking in the fields of radar, sonar, communication, seismic exploration and autonomous driving.

[0009] The FFRA diagonal ULA phase difference estimation and fuzzy resolution method based on the degraded space ARMA model specifically includes the following contents:

[0010] (I) 2D array model:

[0011] (1) Without loss of generality, a 2D array is used. The array is composed of M×N isotropic sensor elements. The number of sensor elements parallel to the X and Y axes are N and M respectively. K far-field narrow-band stationary sources are incident on the 2D array. Their incident azimuth and elevation angles are φ respectively. k ,θk , where k = 1, ..., K, the sensor array element located at the origin of the array coordinate is assumed to be a zero-phase array element, and the phase of the kth signal source propagating to the (m, n)th array element of the 2D array is assumed to be but for:

[0012]

[0013] Among them, m and n represent the spatial position index of the array element along the Y axis and X axis respectively. and are the phase differences of the kth source along the X-axis and Y-axis respectively, and the calculation formula is as follows:

[0014]

[0015] The scaling factor f and c are carrier frequency and propagation speed, Δd x , Δd y are the sensor array element spacing along the X and Y axis directions, For the defined parameters, the calculation formula is as follows:

[0016]

[0017] where θ k and φ k are the elevation and azimuth angles of the kth source;

[0018] (2) For the axial ULAs in FFRA, the phase difference is equal to Δξ x and Δξ y , Δξ x and Δξ y denotes the phase difference of ULAs located on / parallel to the X-axis and Y-axis respectively, and for diagonal ULAs in FFRA, the corresponding phase difference Δξ D and Δξ D is Δξ x and Δξ y The linear combination of x , Δξ y and Δξ z A linear combination of , so it has a polynomial function form;

[0019] where Δξ D and Δξ D They represent the phase difference of ULAs on / parallel to the main diagonal and the phase difference of ULAs on / parallel to the secondary diagonal, Δξ z is the phase difference of the sensor element along the Z axis;

[0020] (II) Spatial spectrum model of FFRA diagonal ULAs and its diagonal phase difference estimation

[0021] Step 2.1, the sensor data for ULAs located / parallel to the main diagonal is given by:

[0022]

[0023] Among them, d l (t) is the data received by the sensors of the ULAs located / parallel to the main diagonal, and L are the fixed phase and number of sensors generated by the kth source, respectively, s k (t) and w l (t) are the signal of the kth source and the additive noise on the lth sensor, where k = 1,…,K; is the phase difference of the ULAs on / parallel to the main diagonal corresponding to the kth source, and is the phase difference of the ULAs located on / parallel to the main diagonal corresponding to the kth source along the X-axis and Y-axis respectively;

[0024] Step 2.2, construct and solve the noise-free correlation or variance MYW linear equation system of the selected diagonal ULA, and use the total least squares (TLS) method to give the solution of the MYW linear equation system; the specific contents are as follows:

[0025] Define the phase difference of the ULAs located on / parallel to the main diagonal corresponding to the kth source The source propagator ν k for:

[0026]

[0027] First, the correlation sequence of the ULA sensor data under study is constructed. Assuming that the external source and sensor noise are stable, a phase difference is extracted by performing spatial difference operation on the phase of the same source in two adjacent elements of the correlation sequence. After repeating the spatial difference operation on all K phase differences, a linear equation system is derived and expressed in a matrix as follows:

[0028] H D γ=r D (14)

[0029] Among them, H D , γ and r D are the coefficient matrix, the parameters to be determined, and the column vector of the constant term, respectively. D , γ and r D The dimensioning and construction of is the same as the dense rectangular array of azimuth-only ULA, where:

[0030]

[0031] γ=[ γ K γ K-1 … γ1] T , (16)

[0032] r D =-[h K+1 h K+2 … h L ] T , (17)

[0033] in,(·) T is the transposition operator. In formula (15), H D and r D Elements used in It is given by the following formula:

[0034] h l =R l -R wl =E{d l (t)d ′H (t)}-E{w l (t)d ′H (t)}, l=1,…,L (18)

[0035] Where: R l is the correlation function of the sensor data, R wl is the correlation function of the noise, d′(t) is the reference signal, which is used to form a correlation function under the assumption of stationary external source and sensor noise. and d ′H (t) is the orthogonal transpose of the reference signal, R Sk Yes k (t), and R Sk =E{s k (t)d ′H (t)}; In addition, when the source signal and the sensor noise are uncorrelated and the sensor noise is a white Gaussian process with complex space and time, by selecting an appropriate d′(t) so that becomes zero, thus constructing MYW linear equations with h being 0: l =R l ;Therefore, it is possible to make an estimate without considering the sensor noise;

[0036] Step 2.3, for ULAsD located on / parallel to the main diagonali , whether i ≥ 0 or i < 0, the unknown number of column vector γ All have the following explicit form:

[0037]

[0038] In order to determine the K source propagators Construct the following polynomial:

[0039] f(ν)=ν K +γ1ν K-1 +…+γ k v K-k +…+γ K , (20)

[0040] Where v represents the unknown independent variable of the source propagator;

[0041] In formula (19), Substituting into formula (20) we get

[0042] f(v)=(v-v1)(v-v2)…(v-ν K ), (twenty one)

[0043] It is easy to prove that the source propagator is the root of the polynomial f(v);

[0044] Taking the root of f(v), we get but Use the following formula to find:

[0045]

[0046] For ULAsD located / parallel to the minor diagonal i , the phase difference of the ULAs on the sub-diagonal line corresponding to the kth signal source is The source propagator v k Given by:

[0047]

[0048] Assuming that the external source signal and sensor noise are stable, a phase difference is extracted by performing spatial difference operation on the phase of the same source in two adjacent elements of the correlation sequence; after repeating the spatial difference operation on all K phase differences, a linear equation system is derived and expressed in a matrix as follows:

[0049] H D c =r D , (twenty four)

[0050] Among them, H D , γ and r D are the coefficient matrix, the parameters to be determined, and the column vector of the constant term, respectively. D , γ and r D The dimension setting and construction method of are the same as the corresponding items in formula (14);

[0051] H D and r D Elements in Given by the following formula

[0052] h l =E{d l (t)d ′H (t)}-E{w l (t)d ′H (t)},l=1,…,L. (25)

[0053] where d L (t) and w l (t) is ULAD i The lth sensor data and noise;

[0054] The elements in the column vector γ to be found With the following explicit expression:

[0055]

[0056] In order to determine the K source propagators Construct the following polynomial:

[0057] f(v)=v K +γ1ν K-1 +…+γ k ν K-k +…+γ K , (27)

[0058] Taking the root of f(v), we get for Given by:

[0059]

[0060] (III) Diagonal phase blur problem

[0061] 3.1, Δξ is the phase difference applicable to all ULAs, assuming is the estimated value of the phase difference Δξ, where and Δξ satisfy the following equation:

[0062]

[0063] Where v is the source propagator of Δv. As can be seen from equation (29), due to the introduction of a non-zero integer p, there are multiple estimated values ​​for the same Δξ, which is of blur;

[0064] 3.2, let Δξ be the phase difference Δξ of ULAs along the X-axis x , replace Δξ in equation (2) x Substituting the explicit expression into equation (29), we get:

[0065]

[0066] where u x =cosφsinθ, set for u x The estimated value u x The value range is [-1,1], the value range of the azimuth angle φ is [0,π], and the value range of the elevation angle θ is λ is the carrier wavelength. In order to avoid Blurred, and u x The one-to-one mapping should be maintained in the interval [-1,1]; only when Δd x In line with tradition Rules, u x The one-to-one mapping is established;

[0067] 3.3, let Δξ in equation (29) be the phase difference Δξ of ULAs along the Y axis y , then similarly:

[0068]

[0069] where u y = sinφsinθ, set for u y To avoid The blur, Δd y Also need to meet rule;

[0070] 3.4, first rewrite the diagonal difference into a monomial;

[0071] Let Δξ be ULAD i The external signal source on the D , we get Δξ D Estimated value of

[0072]

[0073] where the definition definition ΔdD is the diagonal array element spacing, So:

[0074]

[0075] This shows that the monomial u in equation (34) D The value of is also in the range of [-1,1]; Substituting formula (34) into formula (33) yields:

[0076]

[0077] Which set for u D Therefore, The blur depends on the diagonal element spacing Δd D ; Similarly, without considering Δd x and Δd y For Δd D In the case of constraints, when Δd D conform to Rules, In the interval

[0078] There is no blur problem in [-1,1]; due to the diagonal array element spacing Δd D same, The same rules apply to ULAs located on / parallel to the secondary diagonal;

[0079] 3.5, Phase difference blur:

[0080] Let Δd x and Δd y are equal to but and With maximum resolution, but

[0081] and Blurring may occur.

[0082] (IV) Solution to the diagonal phase blur problem:

[0083] In actual estimation, and has its actual interval, here we assume [a xL ,a xU ]for The actual interval, a xL and a xU are the corresponding lower and upper bounds; similarly, [a yL ,a yU ]for The actual interval of; In the case of no ambiguity, for The interval is reduced from [-1,1] to [a xL ,a xU ],for The interval is reduced to [a yL ,a yU ];

[0084] Likewise, along the main diagonal There is also a practical interval [a DL ,a DU ], according to Δξ D With Δξ x and Δξ y The linear combination relationship of [a yL ,a yU ] and [a xL ,a xU ], the actual interval [a DL ,a DU ] is determined by the following formula:

[0085]

[0086] From formula (36), it can be concluded that, considering and After the actual interval of The interval is reduced from the original [-1,1] to [a DL ,a DU ]; due to [a yL ,a yU ] and [a xL ,a xU ] within and There is no blur, In [a DL ,a DU ] also does not have any ambiguity, so [a DL ,a DU The estimated value within the interval is judged as The true estimated value of is, and the estimated value outside this interval is judged as The false estimates of The fuzzy problem;

[0087] Through similar analysis, for the The actual interval is reduced from the original interval [-1,1] to [a D L ,a D U ], and meet the following conditions:

[0088]

[0089] Using [a D L ,a D U ], so it will belong to [[a D L ,a D U The estimated value within the interval is judged as The true estimated value of is, and the estimated value outside this interval is judged as spurious estimates; after distinguishing true and false estimates, solve fuzzy problem.

[0090] The specific description of part (2) of part (a) is as follows:

[0091] According to the changes in the sensor corner mark, ULAs in FFRA are divided into the following two types:

[0092] Type 1 refers to the axial ULAs in FFRA. In equation (2), the phase difference of any source along the X-axis and Y-axis is Δξ x and Δξ y ;;

[0093] Type 2 refers to ULAs located on / parallel to the main / minor diagonal;

[0094] D i Used to describe ULAs located on / parallel to the main diagonal, where the integer i represents the sequence number of the ULA, i>0 and i<0 represent D i Located in D 0 above and below, where D 0 The ULA representing the main diagonal;

[0095] For D i , when M = N, its phase Given by:

[0096]

[0097] In the formula, Δξ D =Δξ x +Δξ y ,l is D i The spatial position of the sensor element, Indicated by D i The fixed phase caused by the spatial position of i The two-dimensional coordinates of the starting sensor are (i+1,1), and (1,|i|+1) when i<0; therefore, and l are given by:

[0098]

[0099] Δξ D Is minus What you get

[0100]

[0101] Δξ D For ULAD i From equations (4) and (5), we know that the ULAs located on / parallel to the main diagonal have the same phase difference Δξ D and different fixed phases; therefore, all ULAs located / parallel to the main diagonal, regardless of i ≥ 0 or i < 0, can be classified into the same ULA category; the phase difference is Δξ D as the corresponding category marker;

[0102] Similarly, the ULAD on / parallel to the secondary diagonal i , for D i , when M = N, its phase Given by:

[0103]

[0104] In the formula, Δξ D For ULAD i The external signal source difference on D =Δξ x -Δξ y ,l is D i The spatial position of the sensor element, Because D i The fixed phase caused by the spatial position, D i The two-dimensional coordinates of the starting sensor are (M, i+1) when i ≥ 0 and (M-|i|, 1) when i < 0, so and l are given by:

[0105]

[0106] Similarly, all ULAs located on / parallel to the secondary diagonal, regardless of i ≥ 0 or i < 0, can be classified into the same ULA category with a difference of Δξ D as the corresponding category marker;

[0107] From equations (4) and (8), it can be concluded that for FFRA axial ULAs, the corresponding phase difference is equal to Δξ xand Δξ y , and the corresponding phase differences with the diagonal ULAs of the FFRA are Δξ x and Δξ y Linear combinations of

[0108] Similar analysis shows that the phase difference along the surface and volume diagonal can be summarized as Δξ x , Δξ y and Δξ z A linear combination of z is the phase difference of the sensor array element along the Z axis, and its function form is as follows:

[0109] Δξ z =bΔd z cosθ, (11)

[0110] where Δd z is the array element spacing between sensors along the Z axis;

[0111] For convenience, Δξ x , Δξ y and Δξ z represents the axial phase difference, and Δξ D and Δξ D Represents the diagonal phase difference; since all diagonal phase differences of the MD sensor array are Δξ x , Δξ y and Δξ z So it has a polynomial function form.

[0112] The beneficial effects of the present invention are:

[0113] (1) The spatial spectrum model of diagonal ULA is still a degraded spatial ARMA model under the assumption that the external source and sensor noise are stationary. The modified Yule-Walker (MYW) linear equations and root-finding polynomials are established to estimate the diagonal phase difference.

[0114] (2) Since the diagonal array element spacing is greater than half of the carrier wavelength, ambiguity occurs in the diagonal phase difference estimation. Based on the linear combination relationship satisfied by the diagonal phase difference and the axial phase difference, a simple and effective diagonal phase difference ambiguity elimination scheme is proposed.

[0115] (3) The diagonal phase difference estimation on the FFRA in the XY sensor plane shows that the proposed estimation method is effective for complex signal sources such as independent and coherent signal sources. After eliminating the estimation blur, the RMSE of the diagonal phase difference slowly tends to the corresponding CRB. BRIEF DESCRIPTION OF THE DRAWINGS

[0116] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following briefly introduces the drawings used in the description of the embodiments of the present invention. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the contents of the embodiments of the present invention and these drawings without creative work.

[0117] Figure 1 Schematic diagram of FFRA on the XY sensor plane.

[0118] Figure 2 Schematic diagram of the FFRA diagonal ULA under study.

[0119] Figure 3 FFRA axial and diagonal phase difference estimation results for independent and coherent sources.

[0120] Figure 4 It is the estimation result of different diagonal differences of independent and coherent sources.

[0121] Figure 5 It is the search result of the diagonal difference of independent coherent sources of different ULA categories.

[0122] Figure 6 It is the curve of the RMS value of the diagonal difference between independent and coherent signal sources on the main diagonal and sub-diagonal as the signal-to-noise ratio changes.

[0123] Figure 7 is the RMS error of the diagonal phase difference of different ULAs. DETAILED DESCRIPTION

[0124] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It is to be understood that the specific embodiments described herein are only used to explain the present invention, rather than to limit the present invention. It should also be noted that, for ease of description, only parts related to the present invention, rather than all structures, are shown in the accompanying drawings.

[0125] Example 1

[0126] The FFRA diagonal ULA phase difference estimation and fuzzy resolution method based on the degenerate space ARMA model include the following:

[0127] (I) 2D array model:

[0128] (1) Without loss of generality, a 2D array is used. The array is composed of M×N isotropic sensor elements. The number of sensor elements parallel to the X and Y axes are N and M respectively. K far-field narrow-band stationary sources are incident on the 2D array. Their incident azimuth and elevation angles are φ respectively. k ,θ k, where k = 1, ..., K, the sensor array element located at the origin of the array coordinate is assumed to be a zero-phase array element, and the phase of the kth signal source propagating to the (m, n)th array element of the 2D array is assumed to be but for:

[0129]

[0130] Among them, m and n represent the spatial position index of the array element along the Y axis and X axis respectively. and are the phase differences of the kth source along the X-axis and Y-axis respectively, and the calculation formula is as follows:

[0131]

[0132] The scaling factor f and c are carrier frequency and propagation speed, Δd x , Δd y are the sensor array element spacing along the X and Y axis directions, For the defined parameters, the calculation formula is as follows:

[0133]

[0134] where θ k and φ k are the elevation angle and azimuth angle of the kth source, which are defined as the angle between the incident direction of the source and the Z axis and the angle between the projection of the incident direction on the XY array and the X axis respectively;

[0135] (2) For the axial ULAs in FFRA, the phase difference is equal to Δξ x and Δξ y , Δξ x and Δξ y denotes the phase difference of ULAs located on / parallel to the X-axis and Y-axis respectively, and for diagonal ULAs in FFRA, the corresponding phase difference Δξ D and Δξ D is Δξ x and Δξ y The linear combination of x , Δξ y and Δξ z A linear combination of , so it has a polynomial function form;

[0136] where Δξ D and Δξ D They represent the phase difference of ULAs on / parallel to the main diagonal and the phase difference of ULAs on / parallel to the secondary diagonal, Δξ zis the phase difference of the sensor array element along the Z axis; the specific description is as follows:

[0137] As a basic one-dimensional array structure, ULA consists of sensors with equal array element spacing on a straight line in space, and the phase associated with ULA only involves one variable array element spatial position index. In contrast, as shown in equation (1), the phase associated with a sensor of a two-dimensional FFRA usually has two array element spatial position indexes. When selecting sensors of a 2-D FFRA to be combined into a 1-D ULA, the requirement that the phase has only one variable array element spatial position index must also be followed. According to the change of the sensor index, ULAs (plural form of uniform linear arrays) in FFRA can basically be divided into the following two types:

[0138] Type 1: Only one array element's spatial position index increases or decreases linearly, and the other array element's spatial position index remains unchanged.

[0139] Type 1 refers to axial ULAs in FFRA, for example, X-axis ULAs with sensor coordinate positions from (1,1) to (1,N) and (1,1) to (M,1). 0 and Y 0 are two typical axial ULAs. In formula (2), the phase difference of any signal source along the X-axis and Y-axis is Δξ x and Δξ y ; (For convenience, the superscript k represents the phase difference of the kth source along the X-axis, without the superscript Δξ x The phase difference is valid for all sources; similarly, Δξ with a subscript x x represents the phase difference of ULAs located on / parallel to the X-axis, and Δξ without a subscript represents the phase difference applicable to all ULAs; the same representation rules also apply to other phases and phase differences);

[0140] Type 2: The spatial position indexes of two array elements change synchronously

[0141] Type 2 refers to ULAs located on / parallel to the main / minor diagonal (ULAs located on the main / minor diagonal, or ULAs parallel to the main / minor diagonal);

[0142] FFRA contains a large number of diagonal ULAs. For convenience, we first introduce the symbols used to describe diagonal ULAs. Symbol D i Used to describe ULAs located on / parallel to the main diagonal, where the integer i represents the sequence number of the ULA, i>0 and i<0 represent D i Located in D 0 above and below, where D 0 The ULA representing the main diagonal;

[0143] For D i , when M = N, its phase Given by:

[0144]

[0145] In the formula, Δξ D =Δξ x +Δξ y ,l is D i The spatial position of the sensor element, Indicated by D i The fixed phase caused by the spatial position of i The two-dimensional coordinates of the starting sensor are (i+1,1), and (1,|i|+1) when i<0; therefore, and l are given by:

[0146]

[0147] Δξ D Is minus What you get

[0148]

[0149] Δξ D For ULAD i From equations (4) and (5), we know that the ULAs located on / parallel to the main diagonal have the same phase difference Δξ D and different fixed phases; therefore, all ULAs located / parallel to the main diagonal, regardless of i ≥ 0 or i < 0, can be classified into the same ULA category; the phase difference is Δξ D Can be used as corresponding category markers;

[0150] Similarly, the ULAD on / parallel to the secondary diagonal i , for D i , when M = N, its phase Given by:

[0151]

[0152] In the formula, Δξ D For ULAD i The external signal source difference on D =Δξ x -Δξ y ,l is D i The spatial position of the sensor element, Because D iThe fixed phase caused by the spatial position, D i The two-dimensional coordinates of the starting sensor are (M, i+1) when i ≥ 0 and (M-|i|, 1) when i < 0, so and l are given by:

[0153]

[0154] Similarly, all ULAs located on / parallel to the secondary diagonal, regardless of i ≥ 0 or i < 0, can be classified into the same ULA category with a difference of Δξ D as the corresponding category marker;

[0155] From equations (4) and (8), it can be concluded that for FFRA axial ULAs, the corresponding phase difference is equal to Δξ x and Δξ y , and the corresponding phase differences with the diagonal ULAs of the FFRA are Δξ x and Δξ y Linear combinations of

[0156] In summary, in addition to the XY sensor plane, there are a large number of FFRA diagonal ULAs in the XZ and YZ sensor planes as well as the body diagonal ULAs of the three-dimensional elongated array.

[0157] Similar analysis shows that the phase difference along the surface and volume diagonal can be summarized as Δξ x , Δξ y and Δξ z A linear combination of z is the phase difference of the sensor array element along the Z axis, and its function form is as follows:

[0158] Δξ z =bΔd z cosθ, (11)

[0159] where Δd z is the array element spacing between sensors along the Z axis;

[0160] For convenience, Δξ x , Δξ y and Δξ z represents the axial phase difference, and Δξ D and Δξ D Represents the diagonal phase difference; since all diagonal phase differences of the MD sensor array are Δξ x , Δξ y and Δξ z A linear combination of , so it has a polynomial function form;

[0161] (II) Spatial spectrum model of FFRA diagonal ULAs and its diagonal phase difference estimation

[0162] Step 2.1, select the appropriate reference signal d′(t) in FFRA

[0163] First, using spatial differential operations on phase, the linear equations and root-finding polynomials for ULAs located / parallel to the main diagonal are theoretically derived. Although the functional form of the fixed phase is different, the sensor data for ULAs located / parallel to the main diagonal is given by the following equation:

[0164]

[0165] Among them, d l (t) is the data received by the sensors of the ULAs located / parallel to the main diagonal, and L are the fixed phase and number of sensors generated by the kth source, respectively, s k (t) and w l (t) are the signal of the kth source and the additive noise on the lth sensor, where k = 1,…,K; is the phase difference of the ULAs on / parallel to the main diagonal corresponding to the kth source, and is the phase difference of the ULAs located on / parallel to the main diagonal corresponding to the kth source along the X-axis and Y-axis respectively; According to equations (5) and (6), and L will change with each ULA change, only when s k When (t) has narrowband resolution characteristics, the complex exponential multiplication in equation (12) can achieve the corresponding time domain delay effect.

[0166] Step 2.2, construct and solve the noise-free correlation or variance MYW linear equation system of the selected diagonal ULA, and use the total least squares (TLS) method to give the solution of the MYW linear equation system; the specific contents are as follows:

[0167] Define the phase difference of the ULAs located on / parallel to the main diagonal corresponding to the kth source The source propagator v k for:

[0168]

[0169] First, the correlation sequence of the ULA sensor data under study is constructed. Assuming that the external signal source and sensor noise are stable, a phase difference is extracted by performing spatial difference operation on the phase of the same signal source in two adjacent elements of the correlation sequence. After repeating the spatial difference operation on all K phase differences, a linear equation group is derived and expressed as a matrix: (Because the phase difference is not explicitly hidden in the sensor array data, we derive the matrix equation (15) satisfied by the signal source phase difference (propagator) based on the characteristics of the uniform linear array. The signal source phase difference can be obtained by solving equation (15))

[0170] H D γ=r D (14)

[0171] Among them, H D , γ and r D are the coefficient matrix, the parameters to be determined, and the column vector of the constant term, respectively. D , γ and r D The dimensioning and construction of is the same as the dense rectangular array of azimuth-only ULA, where:

[0172] The results showed that H D is a standard block Hankel matrix:

[0173]

[0174] γ=[γ K γ K-1 … γ1] T ,(16)

[0175] r D =-[h K+1 h K+2 …h L ] T ,(17)

[0176] in,(·) T is the transposition operator. In formula (15), H D and r D Elements used in It is given by the following formula:

[0177] h l =R l -R wl =E{d l (t)d ′H (t)}-E{w l (t)d ′H (t)},l=1,…,L (18)

[0178] In formula (18): R lis the correlation function of the sensor data, R wl is the correlation function of the noise, d ′ (t) is the reference signal, which is used to form the correlation function under the assumption that the external source and sensor noise are stationary. and d ′H (t) is the orthogonal transpose of the reference signal. Generally speaking, in practice, the noise of the sensor array element is usually unknown. We hope that when solving the matrix equation, the noise variance should not appear in the matrix equation. When the noise and the external signal source are unrelated, we can obtain the matrix equation without the noise variance by selecting d′(t). In this way, the unknown parameters of the matrix equation only contain the signal source difference, and the problem becomes relatively simple. For example, if d l (t) is selected as d′(t). According to (18), the term after the minus sign is equal to the noise variance, and the noise variance will appear in the matrix equation. Of course, this problem is very easy to solve in a two-dimensional array. d′(t) selects the sensor array element signal of the non-ULA to be investigated;

[0179] Where R Sk Yes k (t), and R Sk =E{s k (t)d ′H (t)}; In addition, when the source signal and the sensor noise are uncorrelated and the sensor noise is a white Gaussian process with complex space and time, by selecting an appropriate d′(t) so that becomes zero, thus constructing MYW linear equations with h being 0: l =R l ;Therefore, it is possible to make an estimate without considering the sensor noise;

[0180] Step 2.3, for ULAsD located on / parallel to the main diagonal i , whether i ≥ 0 or i < 0, the unknown number of column vector γ All have the following explicit form:

[0181]

[0182] In order to determine the K source propagators Construct the following polynomial:

[0183] f(v)=v K +γ1v K-1 +…+γ k v K-k +…+γ K , (20)

[0184] Where ν represents the unknown independent variable of the source propagator;

[0185] In formula (19), Substituting into formula (20) we get

[0186] f(v)=(v-v1)(v-v2)…(vv K ), (twenty one)

[0187] It is easy to prove that the source propagator is the root of the polynomial f(v);

[0188] Taking the root of f(v), we get but Use the following formula to find:

[0189]

[0190] For ULAsD located / parallel to the minor diagonal i , the phase difference of the ULAs on the sub-diagonal line corresponding to the kth signal source is The source propagator v k Given by:

[0191]

[0192] Assuming that the external source signal and sensor noise are stable, a phase difference is extracted by performing spatial difference operation on the phase of the same source in two adjacent elements of the correlation sequence; after repeating the spatial difference operation on all K phase differences, a linear equation system is derived and expressed in a matrix as follows:

[0193] H D c =r D , (twenty four)

[0194] Among them, H D , γ and r D are the coefficient matrix, the parameters to be determined, and the column vector of the constant term, respectively. D , γ and r D The dimension setting and construction method of are the same as the corresponding items in formula (14);

[0195] H D and r D Elements in Given by the following formula

[0196] h l =E{d l (t)d ′H (t)}-E{w l (t)d ′H (t)},l=1,…,L. (25)

[0197] where d l (t) and w l (t) is ULAD i The lth sensor data and noise;

[0198] The elements in the column vector γ to be found With the following explicit expression:

[0199]

[0200] In order to determine the K source propagators Construct the following polynomial:

[0201] f(v)=v K +γ1v K-1 +…+γ k ν K-k +…+γ K , (27)

[0202] Taking the root of f(v), we get for Given by:

[0203]

[0204] When the source and sensor noise are uncorrelated and the sensor noise is Gaussian distributed in time and space, the conclusions of the above degraded space ARMA model still hold.

[0205] (III) Diagonal phase blur problem

[0206] DoA ambiguity, also known as the multi-value problem of DoA, has always been a key issue in array signal processing and is the main reason for the appearance of virtual sources in the estimation process. Since the phase difference is an explicit function of the angle, the phase difference ambiguity and DoA ambiguity are essentially the same. In order to avoid DoA ambiguity, it is usually required that the sensor array element spacing of the relevant ULA meet rule, that is, the array element spacing is set to no greater than Where λ is the carrier wavelength. The rule is proven to be effective for azimuth-only ULAs. Due to the existence of a large number of ULAs with different array element spacings, DoA ambiguity is widely encountered in FFRA. However, since diagonal ULAs are rarely used by existing estimation methods, DoA ambiguity has rarely been mentioned in the past.

[0207] 3.1, to better illustrate the ambiguity of phase difference, Δξ represents the phase difference applicable to all ULAs, assuming is the estimated value of the phase difference Δξ, where Δξ is obtained using the corresponding MYW linear equations and the source propagator v, and Δξ satisfy the following equation:

[0208]

[0209] Where ν is the source propagator of Δξ. As can be seen from equation (29), due to the introduction of a non-zero integer p, there are multiple estimated values ​​for the same Δξ, which is of blur;

[0210] 3.2, let Δξ in equation (29) be the phase difference Δξ of ULAs along the X axis x , replace Δξ in equation (2) x Substituting the explicit expression into equation (29), we get:

[0211]

[0212] where u c =cosφsinθ, set for u x The estimated value u x The value range is [-1,1], the value range of the azimuth angle φ is [0,π], and the value range of the elevation angle θ is λ is the carrier wavelength. In order to avoid Blurred, and u x The one-to-one mapping should be maintained in the interval [-1,1]; Theoretical analysis shows that, similar to the azimuth-only ULA, only when Δd x In line with tradition Rules, u x The one-to-one mapping is established, Δd x is the sensor array element spacing along the X direction;

[0213] 3.3, let Δξ in equation (29) be the phase difference Δξ of ULAs along the Y axis y , then similarly:

[0214]

[0215] where u y = sinφsinθ, set for u y To avoid The blur, Δd y Also need to meet Rule, Δd y is the sensor array element spacing along the Y-axis direction;

[0216] 3.4, because the diagonal phase difference is the monomial Δξ x and Δξ yLinear combination of ; For the convenience of discussion, first rewrite the diagonal difference into a monomial;

[0217] Let Δξ be ULAD i The external signal source on the D , we get Δξ D Estimated value of

[0218]

[0219] where the definition definition Δd D is the diagonal array element spacing, So:

[0220]

[0221] This shows that the monomial u in equation (34) D The value of is also in the range of [-1,1]; Substituting formula (34) into formula (33) yields:

[0222]

[0223] Which set for u D Therefore, The blur depends on the diagonal element spacing Δd D ; Similarly, without considering Δd x and Δd y For Δd D In the case of constraints, when Δd D conform to Rules, In the interval

[0224] There is no blur problem in [-1,1]; due to the diagonal array element spacing Δd D same, The same rules apply to ULAs located on / parallel to the secondary diagonal;

[0225] 3.5, Two cases of phase difference blur:

[0226] In the first case, let Δd D equal So and No ambiguity problem arises in the estimation; however, because and Axial element spacing Δd x and Δd y will inevitably be less than Therefore, this will cause axial misalignment and The resolution is reduced;

[0227] In the second case, let Δd x and Δd y are equal to but and With maximum resolution, but and Blurring may occur;

[0228] Therefore, the axial element spacing Δd x and Δd y It plays an important role in the trade-off between resolution and ambiguity of phase difference, which is a factor that needs to be fully considered in 2D DoA estimation based on FFRAs.

[0229] (IV) Solution to the diagonal phase blur problem:

[0230] The present invention takes the second case as an example to solve the fuzzy problem of diagonal phase difference. In actual estimation, and has its actual interval, here we assume [a xL ,a xU ]for The actual interval, a xL and a xU are the corresponding lower and upper bounds; similarly, [a yL ,a yU ]for The actual interval of; In the case of no ambiguity, for The interval is reduced from [-1,1] to [a xL ,a xU ],for The interval is reduced to [a yL ,a yU ];

[0231] Likewise, along the main diagonal There is also a practical interval [a DL ,a DU ], according to Δξ D With Δξ x and Δξ y The linear combination relationship of [a yL ,a yU ] and [a xL ,a xU ], the actual interval [a DL ,a DU ] is determined by the following formula:

[0232]

[0233] From formula (36), it can be concluded that, considering and After the actual interval of The interval is reduced from the original [-1,1] to [a DL ,a DU ]; due to [a yL ,a yU ] and [a xL ,a xU ] within and There is no blur, In [a DL ,a DU ] also does not have any ambiguity, so [a DL ,a DU The estimated value within the interval is judged as The true estimated value of is, and the estimated value outside this interval is judged as The false estimates of The fuzzy problem;

[0234] Through similar analysis, for the The actual interval is reduced from the original interval [-1,1] to [a D L ,a D U ], and meet the following conditions:

[0235]

[0236] Using [a D L ,a D U ], so it will belong to [[a D L ,a D U The estimated value within the interval is judged as The true estimated value of is, and the estimated value outside this interval is judged as spurious estimates; after distinguishing true and false estimates, solve fuzzy problem.

[0237] The array structure of FFRA provides a basis for eliminating the ambiguity of diagonal phase difference estimation. First, the diagonal phase difference and array element spacing are explicit functions of the axial phase difference and array element spacing, respectively. and When there is no ambiguity, Δξ can be used xand Δξ y The estimated solution Δξ D and Δξ D Based on the proposed estimation method, the present invention proposes a simple and effective solution, namely: by using Δξ x and Δξ y The estimated value of the diagonal aberration is derived and given, and the corresponding actual interval of diagonal aberration ambiguity is obtained, thereby solving the problem of diagonal aberration estimation ambiguity. The proposed elimination scheme is also applicable to various diagonal ULAs in FFRA on the XZ and YZ sensor planes.

[0238] Example 2

[0239] Several numerical examples are now presented to evaluate the effectiveness and performance of the proposed method.

[0240] exist Figure 1 and Figure 2 In the FFRA shown, the number of sensors along the X-axis and Y-axis is pre-specified as M = N = 10, and the axial array element spacing Δd x and Δd y for The number of external information sources K is known a priori, and here we assume K=3.

[0241] In this study, a complex-valued zero-mean Gaussian random process with the same power is used to simulate the external signal source and sensor noise, and their powers are and The signal-to-noise ratio can be expressed as All possible sensors of the selected ULAs are used as reference signals d′(t) to construct a noise-free variance MYW linear equation system that meets the requirements.

[0242] 1. Estimation of axial and diagonal phase difference

[0243] Figure 3 The incident light incident on the four ULAsX 0 , Y 0 , D 0 and D 0 The search results of the phase difference of independent and coherent sources on the graph are shown in Figure 2. The absolute value of the reciprocal of the corresponding root-finding polynomial is defined as the ordinate of the four sub-curves. The peak position of each curve is determined as the estimated value of the correlation phase difference. For the convenience of discussion, u is used to represent the phase difference of the four sub-curves. x 、u y 、u D and u D Instead of the phase difference Δξ x , Δξ y , Δξ D and Δξ D .

[0244] In an independent Monte Carlo experiment, due to the axial array element spacing exist Figure 3 Three amplitude peaks can be clearly found in the [-1,1] interval in (a) and (b), and the peak positions are and In contrast, Figure 3 There are four peaks in . Obviously, in addition to the three real peaks, there is also a false peak and an estimated value. Figure 3 in, u D The estimated value of is also blurred. The reason for the blur is that when the axial array element spacing Δd x and Δd y Set as When the diagonal array element spacing

[0245]

[0246] In order to compare the estimation results with different diagonal array element spacing, the ULAsD 0 and D 0 For example, Figure 4 The diagonal element spacing Δd is plotted D for Independent source of u D and u D Obviously, there are three peaks on the two estimated curves, and there is no blur. Figure 3 Compared with the peak in the figure, the estimated resolution is significantly reduced with the “blunted” peak, especially for ULAD 0 .

[0247] Also compared Figure 5 The estimation results of different ULAs belonging to the same category in . D , the ULAs used are D -2 , D 0 and D 2 , they have different fixed phases and numbers of sensors. D , the ULAs used are D -2 , D 0 and D 2 . Because the diagonal array element spacing Δd D equal So using all diagonal ULAs, the multi-value problem can be observed.

[0248] For each phase difference, almost identical search curves reflect that when the diagonal array element spacing exceeds When, for ULAs of the same category, the number of sensors and fixed phase have less impact on the estimation.

[0249] In addition, not only Figure 3 between independent and coherent sources in Figure 5 No differences in peak shape and position were found between ULAs belonging to the same category. The study shows that the proposed estimation method can be applied to various FFRA ULAs and is also effective for complex multipath scenarios. 2. Binary Elimination of Diagonal Aberration

[0250] According to the above analysis, when Δd x and Δd y equal When u along the axis direction is x 、u y , without causing ambiguity. Therefore, the estimated set and It can be reduced from [-1,1] to [a xL ,a xU ]and

[0251] [a yL ,a yU ].like Figure 3 As shown, [a xL ,a xU ]=[-0.354,0.483] and [a yL ,a uU ]=[0.171,0.847].

[0252] Therefore, u D The actual interval [a DL ,a DU ] and u D [a D L ,a D U ] can be derived from equations (36) and (37). After finding and selecting [a DL ,a DU ] and [a D L ,a D U ], and then eliminate the false estimates and By calculating, [a DL ,a DU ]=[-0.129,0.941] and

[0253] [a D L ,a D U ]=[-0.849,0.221]. Figure 3As shown in (c) and (d), u D and u D The estimated sets are (-0.481, 0.183, 0.330, 0.933) and (-0.683, 0.250, 0.096, 0.731), then Figure 3 The values ​​in -0.481 and Figure 4 The value 0.731 in is judged to be a false estimate. D equal When the derivation results are Figure 4 The results shown are consistent.

[0254] 3. Performance Analysis of the Proposed Method

[0255] The axial element spacing Δd x and Δd y equal In the case of FFRA diagonal ULAs, the estimation performance of the proposed method with respect to SNR is numerically analyzed after eliminating the corresponding estimation ambiguities.

[0256] The RMSE (root mean square error) of the estimated difference describes more subtle performance

[0257]

[0258] Where P is the number of Monte Carlo trials. Indicates the kth phase difference u in the pth Monte Carlo experiment k The estimated value of .

[0259] Using u D Two typical ULAsD 0 and u D D 0 , the RMSEs and CRBs (Cramer-Rao bounds) of the diagonal phase differences of independent and coherent sources are plotted against SNR. Figure 6 The number of snapshots in the four subgraphs is 200, 1000, and 10000 respectively. Figure 6 As shown, in general, whether it is an independent source or a coherent source, and The RMSEs of all sub-curves decrease monotonically and steadily with SNR, which shows that the proposed method is effective for various sources including independent sources and coherent sources. At the same time, in the four sub-curves, the RMSEs with different numbers of snapshots converge slowly to the corresponding CRB.

[0260] Taking independent sources as an example, the estimated root mean square error of different ULAs belonging to the same category is Figure 7 D in ULAs -2, D 0 and D 2 To estimate u D ,D -2 , D 0 and D 2 To estimate u D . Since the diagonal array element spacing Δd D Exceed For three ULAs with different numbers of sensors, The RMSE of Similar observations were also made on the RMSE of .

[0261] The conclusion is that after eliminating the estimation ambiguity, under the condition of large diagonal array element spacing, for the same category of ULA, the number of sensors has little effect on the estimation performance.

[0262] The above examples show that the proposed estimation method is applicable to independent and coherent sources, and the RMSE slowly converges to the corresponding CRB after eliminating the estimation ambiguity involved. The conclusions of the degraded space ARMA model and its estimation method are not only applicable to ULA of FFRA in different sensor planes and 3D cuboid sensor arrays, but also to ULA of dense rectangular arrays.

[0263] The preferred embodiments of the present invention are described in detail above in conjunction with the accompanying drawings. However, the protection scope of the present invention is not limited to the specific details in the above embodiments. Within the technical concept of the present invention, any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention. These simple variations all belong to the protection scope of the present invention.

[0264] It should also be noted that the various specific technical features described in the above specific embodiments can be combined in any suitable manner without contradiction. In order to avoid unnecessary repetition, the present invention will not further describe various possible combinations.

[0265] In addition, various embodiments of the present invention may be arbitrarily combined, and as long as they do not violate the concept of the present invention, they should also be regarded as the contents disclosed by the present invention.

Claims

1. A dense rectangular array diagonal ULA phase difference estimation method based on the degenerate space ARMA model, characterized in that: The specific contents include the following: (I) 2D array model: (1) Without loss of generality, a 2D array is used. The array is composed of M×N isotropic sensor elements. The number of sensor elements parallel to the X and Y axes are N and M respectively. K far-field narrow-band stationary sources are incident on the 2D array. Their incident azimuth and elevation angles are φ respectively. k ,θ k , where k = 1, ..., K, the sensor array element located at the origin of the array coordinate is assumed to be a zero-phase array element, and the phase of the k-th signal source propagating to the (m, n)-th array element of the 2D array is assumed to be but for: Among them, m and n represent the spatial position indices of the array element along the Y axis and X axis respectively. and are the phase differences of the kth source along the X-axis and Y-axis respectively, and the calculation formula is as follows: The scaling factor f and c are carrier frequency and propagation speed, Δd x , Δd y are the sensor array element spacing along the X and Y axis directions, For the defined parameters, the calculation formula is as follows: where θ k and φ k are the elevation and azimuth angles of the kth source; (2) For the axial ULAs in FFRA, the corresponding phase difference is equal to Δξ x and Δξ y , Δξ x and Δξ y denote the phase difference of ULAs located on / parallel to the X-axis and Y-axis respectively, while for diagonal ULAs in FFRA, the corresponding phase difference Δξ D and Δξ D is Δξ x and Δξ y The linear combination of x , Δξ y and Δξ z A linear combination of , so it has a polynomial function form; where Δξ D and Δξ D They represent the phase difference of ULAs on / parallel to the main diagonal and the phase difference of ULAs on / parallel to the secondary diagonal, Δξ z is the phase difference of the sensor element along the Z axis; (II) Spatial spectrum model of FFRA diagonal ULAs and its diagonal phase difference estimation Step 2.1, the sensor data for ULAs located / parallel to the main diagonal is given by: Among them, d l (t) is the data received by the sensors of the ULAs located / parallel to the main diagonal, and L are the fixed phase and number of sensors generated by the kth source, respectively, s k (t) and w l (t) are the signal of the kth source and the additive noise on the lth sensor, respectively, where k = 1, ..., K; is the phase difference of the ULAs on / parallel to the main diagonal corresponding to the kth source, and is the phase difference of the ULAs located on / parallel to the main diagonal corresponding to the kth source along the X-axis and Y-axis respectively; Step 2.2, construct and solve the noise-free correlation or variance MYW linear equation system of the selected diagonal ULA, and use the total least squares (TLS) method to give the solution of the MYW linear equation system; the specific contents are as follows: Define the phase difference of the ULAs located on / parallel to the main diagonal corresponding to the kth source The source propagator v k for: First, the correlation sequence of the ULA sensor data under study is constructed. Assuming that the external source and sensor noise are stable, a phase difference is extracted by performing spatial difference operation on the phase of the same source in two adjacent elements of the correlation sequence. After repeating the spatial difference operation on all K phase differences, a linear equation system is derived and expressed in a matrix as follows: H D γ=r D (14) Among them, H D , γ and r D are the coefficient matrix, the parameters to be determined, and the column vector of the constant term, respectively. D , γ and r D The dimensioning and construction of is the same as the dense rectangular array of azimuth-only ULA, where: γ=[γ K c K-1 … c1] T , (16) r D =-[h K+1 h K+2 … h L ] T , (17) in,(·) T is the transposition operator. In formula (15), H D and r D Elements used in It is given by the following formula: h l =R l -Rw l =E{d l (t)d′ H (t)}-E{w l (t)d′ H (t)},l=1,…,L (18) Where: R l is the correlation function of the sensor data, R wl is the correlation function of the noise, d′(t) is the reference signal, which is used to form a correlation function under the assumption of stationary external source and sensor noise. and d′ H (t) is the orthogonal transpose of the reference signal, R Sk Yes k (t), and R Sk =E{s k (t)d′ H (t)}; In addition, when the source signal and the sensor noise are uncorrelated and the sensor noise is a white Gaussian process with complex space and time, by selecting an appropriate d′(t) so that becomes zero, thus constructing MYW linear equations with h being 0: l =R l ;Therefore, it is possible to make an estimate without considering the sensor noise; Step 2.3, for ULAs D located on / parallel to the main diagonal:, for both i ≥ 0 and i < 0, the unknown number of column vector γ All have the following explicit form: In order to determine the K source propagators Construct the following polynomial: f(v)=v K +γ1v K-1 +…+c k v K-k +…+c K , (20) Where v represents the unknown independent variable of the source propagator; In formula (19), Substituting into formula (20) we get f(v)=(v-v1)(v-v2)…(vv K ), (21) It is easy to prove that the source propagator is the root of the polynomial f(v); Taking the root of f(v), we get but Use the following formula to find: For ULAs located on / parallel to the minor diagonal D i, the phase difference of the ULAs on / parallel to the sub-diagonal corresponding to the kth source Source propagator v k Given by: Assuming that the external source signal and sensor noise are stable, a phase difference is extracted by performing spatial difference operation on the phase of the same source in two adjacent elements of the correlation sequence; after repeating the spatial difference operation on all K phase differences, a linear equation system is derived and expressed in a matrix as follows: H Dγ ＝r D ， (24) Among them, H D , γ and r D are the coefficient matrix, the parameters to be determined, and the column vector of the constant term, respectively. D , γ and r D The dimension setting and construction method of are the same as the corresponding items in formula (14); H D and r D Elements in Given by the following formula h l =E{ d l (t)d′ H (t)}-E{ w l (t)d′ H (t)},l=1,…,L. (25) in d l (t) and w l (t) is ULA D i The lth sensor data and noise; Column vector to be found γ Elements in With the following explicit expression: In order to determine the K source propagators Construct the following polynomial: f( v )＝ v K + γ 1 v K-1 +…+ γ k v K-k +…+ γ K ， (27) For f( v ) to find the root, we get for Given by: (III) Diagonal phase blur problem 3.1, Δξ is the phase difference applicable to all ULAs, assuming is the estimated value of the phase difference Δξ, where and Δ ξ satisfies the following formula: Where v is the source propagator of Δξ. As can be seen from equation (29), due to the introduction of a non-zero integer p, there are multiple estimated values ​​for the same Δξ, which is of blur; 3.2, let Δξ be the phase difference Δξ of ULAs along the X-axis x , replace Δξ in equation (2) x Substituting the explicit expression into equation (29), we obtain: where u x =cosφsinθ, set for u x The estimated value u x The value range is [-1, 1], the value range of the azimuth angle φ is [0, π], and the value range of the elevation angle θ is λ is the carrier wavelength. In order to avoid Blurred and u x The one-to-one mapping should be maintained in the interval [-1, 1]; only when Δd x In line with tradition Rules, u x The one-to-one mapping is established; 3.3, let Δξ in equation (29) be the phase difference Δξ of ULAs along the Y axis y , then similarly: where u y = sinφsinθ, set for u y To avoid The blur, Δd y Also need to meet rule; 3.4, first rewrite the diagonal difference as a monomial; Let Δξ be ULA D i The external signal source on the D , we get Δξ D Estimated value of where the definition definition Δd D is the diagonal array element spacing, So: This shows that the monomial u in equation (34) D The value of is also in the range of [-1, 1]; Substituting formula (34) into formula (33) yields: Which set for u D The estimated value of The blur depends on the diagonal element spacing Δd D ; Similarly, without considering Δd x and Δd y For Δd D In the case of constraints, when Δd D conform to When the rules There is no ambiguity problem in the interval [-1, 1]; due to the diagonal array element spacing Δd D same, The same rules apply to ULAs located on / parallel to the secondary diagonal; 3.5, Phase difference blur: Let Δd x and Δd y are equal to but and With maximum resolution, but and Δξ D Blurring may occur. (IV) Solution to the diagonal phase blur problem: In actual estimation, and has its actual interval, here we assume [a xL , α xU ]for The actual interval of a xL and a xU are the corresponding lower and upper bounds; similarly, [a yL , a yU ]for The actual interval of The interval is reduced from [-1, 1] to [a xL , a xU ],for The interval is reduced to [a yL , a yU ]; Likewise, along the main diagonal There is also a practical interval [a DL , a DU ], according to Δξ D With Δξ x and Δξ y The linear combination relationship of [α yL , a yU ] and [a xL , a xU ], the actual interval [a DL , a DU ] is determined by the following formula: From formula (36), it can be concluded that, considering and After the actual interval of The interval is reduced from the original [-1, 1] to [a DL , a DU ]; due to [a yL , a yU ] and [a xL , a xU ] within and There is no ambiguity. In [a DL , a DU ] also does not have any ambiguity problem, so [a DL , a DU The estimated value within the interval is judged as The true estimated value of is, and the estimated value outside this interval is judged as The false estimates of The fuzzy problem; Through similar analysis, for the The actual interval is reduced from the original interval [-1, 1] to [a D L , α D U ] and meet the following conditions: Using [a D L , a D U ], so it will belong to [[a D L , a D U The estimated value within the interval is judged as The true estimated value of is, and the estimated value outside this interval is judged as spurious estimates; after distinguishing true and false estimates, solve fuzzy problem.

2. The dense rectangular array diagonal ULA phase difference estimation method based on the degenerate space ARMA model according to claim 1, characterized in that: The specific description of part (2) of part (a) is as follows: According to the changes in the sensor corner mark, ULAs in FFRA are divided into the following two types: Type 1 refers to axial ULAs in FFRA. In equation (2), the phase differences of any source along the X-axis and Y-axis are Δξ x and Δξ y ;; Type 2 refers to ULAs located on / parallel to the main / minor diagonal; D i Used to describe ULAs located on / parallel to the main diagonal, where the integer i represents the sequence number of the ULA, i>0 and i<0 represent D i Located in D 0 above and below, where D 0 The ULA representing the main diagonal; For D i , when M=N, its phase Given by: In the formula, Δξ D =Δξ x +Δξ y , l is D i The spatial position of the sensor element. Indicated by D i The fixed phase caused by the spatial position of i The two-dimensional coordinates of the starting sensor are (i+1, 1), and (1, |i|+1) when i < 0; therefore, and l are given by: Δξ D Is minus What you get Δξ D For ULA D i From equations (4) and (5), we can see that the ULAs located on / parallel to the main diagonal have the same phase difference Δξ D and different fixed phases; therefore, all ULAs located / parallel to the main diagonal, regardless of i ≥ 0 or i < 0, can be classified into the same ULA category; the phase difference Δξ D as the corresponding category marker; Similarly, the ULA on / parallel to the secondary diagonal D i ,for D i , when M=N, its phase Given by: In the formula, Δξ D For ULA D i The external signal source difference on D =Δξ x -Δξ y , l is D i The spatial position of the sensor element. Reason D i The fixed phase caused by the spatial position of D i The two-dimensional coordinates of the starting sensor are (M, i+1) when i≥0 and (M-|i|, 1) when i<0, so and l are given by: Similarly, all ULAs located on / parallel to the secondary diagonal, regardless of i ≥ 0 or i < 0, can be classified into the same ULA category with a difference of Δξ D as the corresponding category marker; From equations (4) and (8), it can be concluded that for FFRA axial ULAs, the corresponding phase difference is equal to Δξ x and Δξ y , and the corresponding phase difference with the diagonal ULAs of the FFRA is Δξ x and Δξ y Linear combinations of Similar analysis shows that the phase difference along the face and body diagonals can be summarized as Δξ x , Δξ y and Δξ z A linear combination of z is the phase difference of the sensor element along the z-axis, and its function form is as follows: Right z =bΔd z cosθ,(11) where Δd z is the array element spacing between sensors along the z-axis; For convenience, Δξ x , Δξ y and Δξ z represents the axial phase difference, and Δξ D and Δξ D Represents the diagonal phase difference; since all diagonal phase differences of the MD sensor array are Δξ x , Δξ y and Δξ z It is a linear combination of , so it has a polynomial function form.

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