Spatial power spectrum estimation method for small aperture linear arrays based on ellipsoidal semi-axis measurement

Abstract

This invention discloses a method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement, comprising: S1. acquiring a time-domain data matrix; S2. calculating a frequency-domain data matrix; S3. calculating a set of frequency indices; S4. decimating the frequency-domain data matrix; S5. determining a set of scanning directions; S6. calculating a time compensation vector; S7. calculating a phase compensation matrix; S8. performing phase compensation on the decimated frequency-domain data matrix; S9. constructing and solving a positive semi-definite programming problem; S10. calculating the ellipsoidal semi-axis measurement; and S11. calculating the spatial power spectrum based on the ellipsoidal semi-axis measurement. This method directly utilizes the frequency-domain data matrix to estimate the spatial power spectrum, eliminating the need to calculate the signal covariance matrix or perform eigenvalue decomposition. While reducing computational complexity, compared to traditional spatial power spectrum estimation methods based on conventional beamformers, it is more advantageous for estimating target directions when the number of array elements is small or the element spacing is much smaller than half the signal wavelength.

Description

Technical Field

[0001] This invention relates to the field of array signal processing technology, specifically, to a method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement. Background Technology

[0002] The applications of array signal processing involve many fields such as radar, communication, sonar, seismology, exploration, radio astronomy and biomedical engineering. Spatial power spectrum estimation is one of the key research contents of array signal processing. Specifically, spatial power spectrum estimation studies the ability of a processing system composed of a space multi-sensor array to accurately estimate various parameters of a spatial signal of interest. Its main purpose is to estimate the spatial parameters of the signal or the location of the signal source.

[0003] Based on whether they rely on array received data, existing array spatial power spectrum estimation methods can be divided into two categories: one is spatial power spectrum estimation methods independent of array received data, represented by those based on conventional beamformers (CBFs). The basic principle of these methods is to use a conventional beamformer to scan the region of interest, thereby achieving spatial power spectrum estimation. The other category is spatial power spectrum estimation methods related to array received data, represented by methods such as Minimum Variance Distortionless Response (MVDR) and Multiple Signal Classification (MUSIC). While these MVDR and MUSIC methods offer the advantage of high spatial resolution, practical applications often suffer from adverse factors such as array amplitude and phase errors, mutual coupling, element position errors, and near-field scattering. These factors drastically degrade the spatial power spectrum estimation performance of MVDR and MUSIC methods. On the other hand, although the spatial resolution of CBF-based spatial power spectrum estimation methods is weaker than that of MVDR and MUSIC methods, they exhibit better robustness to the aforementioned disadvantages in practical applications, and are therefore still widely used in practice. However, for small-aperture linear arrays, specifically those with a small number of elements or element spacing much smaller than half the signal wavelength, CBF-based spatial power spectrum estimation cannot obtain sharp spectral peaks in the target direction, which is detrimental to target direction estimation. Therefore, researching a spatial power spectrum estimation method based on conventional beamforming principles but also applicable to small-aperture linear arrays has significant practical value. Summary of the Invention

[0004] To effectively address the problem that conventional beamformer-based spatial power spectrum estimation methods cannot obtain sharp spectral peaks near the target direction when the number of array elements is small or the element spacing is much smaller than half the signal wavelength, this invention aims to provide a small aperture linear array spatial power spectrum estimation method based on ellipsoidal half-axis measurement. Under conditions such as a small number of array elements and element spacing much smaller than half the signal wavelength, the power spectrum peaks near the target direction obtained by the method of this invention are significantly sharper than those obtained by conventional beamformer-based spatial power spectrum estimation methods.

[0005] The basic idea of ​​this invention is as follows: First, the received signals of all sensor elements in the array are sampled and stored to obtain a time-domain data matrix, and a frequency-domain data matrix is ​​calculated based on the time-domain data matrix. Next, according to the type of received signal, i.e., single-tone signal, multi-tone signal, and broadband signal, a frequency index set is calculated, and the frequency-domain data matrix is ​​extracted using the frequency index set. Then, the determined direction-of-arrival intervals are sampled at equal intervals to obtain a set of scanning directions. After that, the scanning direction angles in the set of scanning directions are selected sequentially, the corresponding time compensation vector and phase compensation matrix are calculated, and phase compensation is performed on the extracted frequency-domain data matrix. Finally, a positive semi-definite programming problem is constructed and solved to calculate the ellipsoidal semi-axis metric and the spatial power spectrum based on the ellipsoidal semi-axis metric.

[0006] The technical solution of the present invention will be described in detail below:

[0007] A method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement includes the following steps:

[0008] Step S1: Obtain the time-domain data matrix: Sample and store the received signals of all sensor elements in the array to obtain the time-domain data matrix X;

[0009]

[0010] Where N represents the length of the signal received by a single array element, and M represents the number of sensor array elements contained in the array;

[0011] Step S2: Calculate the frequency domain data matrix: Perform a discrete Fourier transform on each column of the time domain data matrix X from step S1 to obtain the frequency domain data matrix Y;

[0012]

[0013] Among them, the data in the l-th column of the time-domain data matrix X is [x 1,l x 2,l …x N,l ] TLet f be a discrete sequence of length N obtained by sampling the received signal of the l-th sensor element, and let f be the sampling frequency. s (Unit: Hz);

[0014] Step S3: Calculate the frequency sequence set: Determine and calculate the frequency sequence set based on the type of signal received by the array;

[0015] Step S4: Extract the frequency domain data matrix: Extract the frequency domain data matrix obtained in step S2 using the set of frequency indices obtained in step S3.

[0016] Step S5: Determine the set of scanning directions: Denote the direction of arrival interval of the target signal as [θ]. L ,θ H ], where θ L θ represents the lower limit of the direction-of-arrival interval. H Let θ represent the upper limit of the direction-of-arrival interval; then for a linear array, θ L The value is -90 degrees, θ H The value is 90 degrees. The determined wave direction interval is sampled at equal intervals to obtain the set of scanning directions Θ.

[0017] Θ={θ k |θ k =θ L +k(θ H -θ L ) / S,k=0,1,2,…S}

[0018] The total number of sampling points contained in the set of scanning directions is S+1;

[0019] Step S6: Calculate the time compensation vector: Based on the scan direction set Θ obtained in step S5, sequentially select the scan direction angle θ from this set. k Given k = 0, 1, 2, ..., S, calculate the corresponding time compensation vector t. d ;

[0020] t d =[d 11 sin(θ k ) / c,d 21 sin(θ k ) / c,…,d M1 sin(θ k ) / c] T

[0021] Where, d i,1 Let i = 1, 2, ..., M, representing the distance between the i-th sensor element and the first sensor element, and d. 11 Typically, the value is zero, where c represents the speed at which the signal propagates in the medium;

[0022] Step S7: Calculate the phase compensation matrix: Based on the frequency index set k obtained in step S3, construct the frequency index vector k. f ;

[0023] k f =[e1-1,e2-1,e3-1,…,e r -1] T

[0024] The constructed frequency index vector and the time compensation vector t obtained in step S6 d Calculate the phase compensation matrix C;

[0025]

[0026] Step S8: Perform phase compensation on the extracted frequency domain data matrix: The frequency domain data matrix extracted in step S4... Perform a matrix dot product with the phase compensation matrix C obtained in step S7 to obtain

[0027]

[0028] Among them, symbols This represents the matrix dot product, i.e.

[0029] Step S9: Construct and solve the positive semidefinite programming problem: Decompose each element of matrix Q in step S8 into its real and imaginary parts to construct a real matrix Q. r Find a matrix Q that contains the matrix Q. r All column vectors q1, q2, ... q M Given an ellipsoid such that the sum of the squares of all its semi-axis is minimized, construct a positive semidefinite programming problem.

[0030] Step S10: Calculate the ellipsoidal semi-axis metric: The sum of the squares of all semi-axis lengths of the ellipsoid is called the ellipsoidal semi-axis metric. When the scanning direction is θ... k When the semi-axial length of the ellipsoid is measured as V(θ) k Its value is equal to the optimal matrix D obtained from the semidefinite programming problem in step S9. opt The trace, i.e., V(θ) k )=tr(D opt );

[0031] Step S11: Calculate the spatial power spectrum based on the ellipsoidal semi-axis metric: Based on the ellipsoidal semi-axis metric V(θ) obtained in step S10... k ), calculate the scanning direction as θ k Spatial power spectrum P(θ) k );

[0032]

[0033] Repeat steps S6-S11 to obtain the spatial power spectrum corresponding to all scanning directions based on the ellipsoidal semi-axis metric.

[0034] Specifically, in step S3, the frequency sequence set is determined and calculated based on the type of signal received by the array. The determination and calculation of the frequency sequence set are divided into the following three cases:

[0035] a. When the array receives a single-tone signal, let f (in Hz) be the analog frequency of the single-tone signal. Then, the method for calculating the frequency index set k is as follows:

[0036] k={n|Nf / f s <n<Nf / f s +2, where n is an integer}

[0037] b. When the array receives a multi-tone signal, i.e., the superposition of q single-tone signals of different frequencies, let the q frequency values ​​be f1, f2, ..., f... q (Unit: Hz), then the method for calculating the frequency index set k is as follows:

[0038] k = k1∪k2∪…∪k q

[0039] In the above formula, the frequency index set k represents all frequency index subsets k1, k2, ..., k q The union of the frequency index subset k i Calculate using the following formula:

[0040] k i ={n|Nf i / f s <n<Nf i / f s +2, n is an integer}, i = 1, 2, ..., q

[0041] c. When the array receives a wideband signal, let f be the lower frequency limit of the wideband signal. L (Unit: Hz), upper frequency limit: f H (Unit: Hz), then the method for calculating the frequency index set k is as follows:

[0042] k={n|Nf L / f s <n<Nf H / f s +2, where n is an integer}.

[0043] Furthermore, step S4, which involves using the frequency index set obtained in step S3 to extract the frequency domain data matrix obtained in step S2, specifically involves:

[0044] Let all elements contained in the frequency index set k be denoted as e1, e2, e3, ..., e r , where r represents the number of elements contained in the frequency index set k;

[0045] Extracting the frequency domain data matrix Y from the frequency index set yields...

[0046]

[0047] in, This represents the matrix obtained by extracting the frequency domain data matrix Y according to the frequency index set. It has r rows, which corresponds to the number of elements in the frequency index set k; and M columns, which corresponds to the number of sensor array elements.

[0048] Specifically, the direction of arrival interval [θ] of the target signal mentioned in step S5 L ,θ H During the calculation, conventional beamforming methods are used to coarsely estimate the direction of arrival of the target signal, thereby determining a smaller range of direction of arrival intervals to reduce computational complexity.

[0049] Furthermore, the total number of sampling points in the set of scanning directions mentioned in step S5 is S+1. The larger the selected S, the denser the sample points in the scanning direction, the weaker the picket fence effect in spatial power spectrum estimation, and the greater the computational load required for subsequent steps. Conversely, the smaller the selected S, the sparser the sample points in the scanning direction, the more significant the picket fence effect in spatial power spectrum estimation, and the smaller the computational load required for subsequent steps. Therefore, the specific value of the number of sampling points S can be selected according to the actual engineering situation and computing power.

[0050] Specifically, step S9 involves finding a matrix Q. r All column vectors q1, q2, ... q M Given an ellipsoid such that the sum of the squares of all its semi-axis is minimized, construct a positive semidefinite programming problem as follows:

[0051] First, the real matrix Q... r

[0052]

[0053] Denoted as a column vector Q r =[q1,q2,…q M ]; where q i Let i = 1, 2, ..., M, representing matrix Q. rThe i-th column;

[0054] Then, find a matrix Q that can contain the matrix. r All column vectors q1, q2, ... q M Given an ellipsoid such that the sum of the squares of all its semi-axis is minimized, the positive semidefinite programming problem is constructed as follows:

[0055]

[0056] Here, matrix D and vector d are the optimization variables of the semidefinite programming problem; matrix D is a symmetric matrix of dimension 2r×2r, which is related to the shape of the ellipsoid; vector d is a column vector of dimension 2r×1, which is related to the center position of the ellipsoid; the objective function tr(D) represents the trace of matrix D, and its size is equal to the trace containing all column vectors q1,q2,…q M The sum of squares of the semi-axis lengths of the ellipsoid; the i-th constraint condition i = 1, 2, ..., M, representing matrix It must be a positive semi-definite matrix;

[0057] By efficiently solving the constructed positive semidefinite programming problem, we obtain the optimal solution for matrix D and vector d, denoted as D0 and D1, respectively. opt and d opt .

[0058] Furthermore, the constructed semidefinite programming problem is solved effectively to obtain the optimal solution for matrix D and vector d. The optimal solution for matrix D and vector d is solved effectively using the interior point method.

[0059] Compared with the prior art, the beneficial effects of the present invention are:

[0060] 1. When the number of array elements is small or the spacing between array elements is much smaller than half the signal wavelength, the method of the present invention, compared with the spatial power spectrum estimation method based on conventional beamformers, can obtain sharper spectral peaks near the target direction, which is beneficial for estimating the target direction.

[0061] 2. The method of the present invention directly uses the frequency domain data matrix to estimate the spatial power spectrum during the calculation process, without calculating the signal covariance matrix or performing eigenvalue decomposition of the signal covariance matrix, which effectively reduces the computational complexity. Attached Figure Description

[0062] Figure 1 A flowchart of the spatial power spectrum estimation method for small aperture linear arrays based on ellipsoidal semi-axis measurement provided by the present invention;

[0063] Figure 2 This is a comparison of the spatial power spectrum of the method of the present invention and the conventional beamforming method in the case of a single-tone signal.

[0064] Figure 3 This is a comparison diagram of the spatial power spectrum of the method of the present invention and the conventional beamforming method under multi-tone signal conditions.

[0065] Figure 4 A comparison of the spatial power spectrum of the method of this invention and conventional beamforming methods under broadband signal conditions. Detailed Implementation

[0066] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0067] Example 1

[0068] like Figure 1 As shown in the figure, this embodiment presents a method for estimating the spatial power spectrum of a small aperture linear array based on ellipsoidal semi-axis measurement, including the following steps:

[0069] Step S1: Obtain the time-domain data matrix: Sample and store the received signals of all sensor elements in the array to obtain the time-domain data matrix X;

[0070]

[0071] Where N represents the length of the signal received by a single array element, and M represents the number of sensor array elements contained in the array;

[0072] Step S2: Calculate the frequency domain data matrix: Perform a discrete Fourier transform on each column of the time domain data matrix X from step S1 to obtain the frequency domain data matrix Y;

[0073]

[0074] Among them, the data in the l-th column of the time-domain data matrix X is [x 1,l x 2,l … x N,l ] T Let f be a discrete sequence of length N obtained by sampling the received signal of the l-th sensor element, and let f be the sampling frequency. s (Unit: Hz);

[0075] Step S3: Calculate the frequency index set: Based on the type of signal received by the array, determine and calculate the frequency index set. The determination and calculation of the frequency index set are divided into the following three cases:

[0076] a. When the array receives a single-tone signal, let f (in Hz) be the analog frequency of the single-tone signal. Then, the method for calculating the frequency index set k is as follows:

[0077] k={n|Nf / f s <n<Nf / f s +2, where n is an integer}

[0078] b. When the array receives a multi-tone signal, i.e., the superposition of q single-tone signals of different frequencies, let the q frequency values ​​be f1, f2, ..., f... q (Unit: Hz), then the method for calculating the frequency index set k is as follows:

[0079] k = k1∪k2∪…∪k q

[0080] In the above formula, the frequency index set k represents all frequency index subsets k1, k2, ..., k q The union of the frequency index subset k i Calculate using the following formula:

[0081] k i ={n|Nf i / f s <n<Nf i / f s +2, n is an integer}, i = 1, 2, ..., q

[0082] c. When the array receives a wideband signal, let f be the lower frequency limit of the wideband signal. L (Unit: Hz), upper frequency limit: f H (Unit: Hz), then the method for calculating the frequency index set k is as follows:

[0083] k={n|Nf L / f s <n<Nf H / f s +2, where n is an integer};

[0084] Step S4: Extract the frequency domain data matrix: Using the frequency index set obtained in step S3, extract the frequency domain data matrix obtained in step S2, and denote all elements contained in the frequency index set k as e1, e2, e3, ..., e r , where r represents the number of elements contained in the frequency index set k;

[0085] Extracting the frequency domain data matrix Y from the frequency index set yields...

[0086]

[0087] in, This represents the matrix obtained after extracting the frequency domain data matrix Y according to the frequency index set. It has r rows, corresponding to the number of elements in the frequency index set k; and M columns, corresponding to the number of sensor array elements.

[0088] Step S5: Determine the set of scanning directions: Denote the direction of arrival interval of the target signal as [θ]. L ,θ H ], where θ L θ represents the lower limit of the direction-of-arrival interval. H Let θ represent the upper limit of the direction-of-arrival interval; then for a linear array, θ L The value is -90 degrees, θ H The value is 90 degrees. In this embodiment, conventional beamforming methods are used to coarsely estimate the direction of arrival of the target signal during calculation, thereby determining a smaller range of direction of arrival intervals to reduce computational complexity.

[0089] By sampling the determined direction-of-arrival (DOA) intervals at equal intervals, the set of scanning directions Θ is obtained;

[0090] Θ={θ k |θ k =θ L +k(θ H -θ L ) / S,k=0,1,2,…S}

[0091] The total number of sampling points contained in the set of scanning directions is S+1;

[0092] It should be noted that for the total number of sampling points S, the larger the selected S, the denser the sample points in the scanning direction, the weaker the picket fence effect in spatial power spectrum estimation, and the greater the computational load required for subsequent steps; conversely, the smaller the selected S, the sparser the sample points in the scanning direction, the more significant the picket fence effect in spatial power spectrum estimation, and the smaller the computational load required for subsequent steps. Therefore, the specific value of the number of sampling points S can be selected according to the actual engineering situation and computing power.

[0093] Step S6: Calculate the time compensation vector: Based on the scan direction set Θ obtained in step S5, sequentially select the scan direction angle θ from this set. k Given k = 0, 1, 2, ..., S, calculate the corresponding time compensation vector t. d ;

[0094] t d =[d 11 sin(θ k ) / c,d 21 sin(θ k ) / c,…,d M1 sin(θk ) / c] T

[0095] Where, d i,1 Let i = 1, 2, ..., M, representing the distance between the i-th sensor element and the first sensor element, and d. 11 Typically, the value is zero, where c represents the speed at which the signal propagates in the medium;

[0096] Step S7: Calculate the phase compensation matrix: Based on the frequency index set k obtained in step S3, construct the frequency index vector k. f ;

[0097] k f =[e1-1,e2-1,e3-1,…,e r -1] T

[0098] The constructed frequency index vector and the time compensation vector t obtained in step S6 d Calculate the phase compensation matrix C;

[0099]

[0100] Step S8: Perform phase compensation on the extracted frequency domain data matrix: The frequency domain data matrix extracted in step S4... Perform a matrix dot product with the phase compensation matrix C obtained in step S7 to obtain

[0101]

[0102] Among them, symbols This represents the matrix dot product, i.e.

[0103] Step S9: Construct and solve the positive semidefinite programming problem: Decompose each element of matrix Q in step S8 into its real and imaginary parts to construct a real matrix Q. r , the real matrix Q r

[0104]

[0105] Denoted as a column vector Q r =[q1,q2,…q M ]; where q i Let i = 1, 2, ..., M, representing matrix Q. r The i-th column;

[0106] Then, find a matrix Q that can contain the matrix. r All column vectors q1, q2, ... q MGiven an ellipsoid such that the sum of the squares of all its semi-axis is minimized, the positive semidefinite programming problem is constructed as follows:

[0107]

[0108] Here, matrix D and vector d are the optimization variables of the semidefinite programming problem; matrix D is a symmetric matrix of dimension 2r×2r, which is related to the shape of the ellipsoid; vector d is a column vector of dimension 2r×1, which is related to the center position of the ellipsoid; the objective function tr(D) represents the trace of matrix D, and its size is equal to the trace containing all column vectors q1,q2,…q M The sum of squares of the semi-axis lengths of the ellipsoid; the i-th constraint condition i = 1, 2, ..., M, representing matrix It must be a positive semi-definite matrix;

[0109] By efficiently solving the constructed positive semidefinite programming problem, we obtain the optimal solution for matrix D and vector d, denoted as D0 and D1, respectively. opt and d opt In this embodiment, the optimal solutions for matrix D and vector d are obtained using the interior point method.

[0110] Step S10: Calculate the ellipsoidal semi-axis metric: The sum of the squares of all semi-axis lengths of the ellipsoid is called the ellipsoidal semi-axis metric. When the scanning direction is θ... k When the semi-axial length of the ellipsoid is measured as V(θ) k Its value is equal to the optimal matrix D obtained from the semidefinite programming problem in step S9. opt The trace, i.e., V(θ) k )=tr(D opt );

[0111] Step S11: Calculate the spatial power spectrum based on the ellipsoidal semi-axis metric: Based on the ellipsoidal semi-axis metric V(θ) obtained in step S10... k ), calculate the scanning direction as θ k Spatial power spectrum P(θ) k );

[0112]

[0113] Repeat steps S6-S11 to obtain the spatial power spectrum corresponding to all scanning directions based on the ellipsoidal semi-axis metric.

[0114] Example 2

[0115] A simulation experiment was conducted according to the method described in Example 1. The processing results were compared with the processing results obtained by the spatial power spectrum estimation method based on conventional beamformers. The specific process and comparison results are as follows.

[0116] The common basic parameter settings for each simulation experiment (single-tone signal, multi-tone signal, and broadband signal) are as follows:

[0117] The uniform linear array has an element spacing of 0.5 meters, a signal sampling frequency of 8000 Hz, and 2048 sampling points for each element. The received noise of each element is statistically independent and the signal-to-noise ratio is 20 dB. The plane wave signal generated by the target arrives at the uniform linear array in a true direction of 3 degrees, and the propagation speed of the plane wave signal is 340 m / s.

[0118] Comparison results

[0119] like Figure 2 The figure shown is a comparison of the spatial power spectrum processing results of the method of this invention and the conventional beamforming method when the target signal is a single-tone signal. For ease of comparison, Figure 2 The spatial power spectra of both the method of this invention and conventional beamforming methods have been normalized. Figure 2 It contains 4 subgraphs, among which Figure 2 (a) corresponds to the case where the number of array elements is 28 and the frequency of the single-tone signal is 340Hz. At this time, the array aperture is 14 meters and the array element spacing is equal to half a wavelength. Figure 2 (b) corresponds to the case where the number of array elements is 8 and the frequency of the single-tone signal is 340Hz. In this case, the array aperture is 4 meters and the array element spacing is equal to half a wavelength. Figure 2 (c) corresponds to the case where the number of array elements is 28 and the frequency of the single-tone signal is 113Hz. At this time, the array aperture is 14 meters and the array element spacing is approximately one-sixth of the wavelength. Figure 2 (d) corresponds to the case where the number of array elements is 8 and the frequency of the single-tone signal is 113Hz. In this case, the array aperture is 4 meters and the spacing between array elements is approximately one-sixth of the wavelength.

[0120] contrast Figure 2 (a)- Figure 2 (d) It can be observed that when the target signal is a single-tone signal, as the number of array elements decreases and the spacing between array elements decreases, the spatial power spectrum estimation method based on conventional beamformers can no longer obtain sharp spectral peaks near the target direction. However, even when the number of array elements is small or the spacing between array elements is much smaller than half the wavelength of the single-tone signal, the method of the present invention can still obtain very sharp spectral peaks near the target direction.

[0121] like Figure 3 The figure shown is a comparison of the spatial power spectrum processing results of the method of this invention and the conventional beamforming method when the target signal is a multi-tone signal. For ease of comparison, Figure 3 The spatial power spectra of both the method of this invention and conventional beamforming methods have been normalized. Figure 3 It contains 2 subgraphs, among which Figure 3 (a) corresponds to the case where the number of array elements is 28 and the frequencies of the multi-tone signals are 140Hz, 240Hz and 340Hz respectively. At this time, the array aperture is 14 meters and the wavelengths corresponding to each frequency of the multi-tone signal are 2.43 meters, 1.42 meters and 1 meter respectively. Figure 3 (b) corresponds to the case where the number of array elements is 8 and the frequencies of the multi-tone signals are 80Hz, 160Hz and 240Hz respectively. At this time, the array aperture is 4 meters and the wavelengths corresponding to each frequency of the multi-tone signal are 4.25 meters, 2.13 meters and 1.42 meters respectively.

[0122] contrast Figure 3 Figures (a) and (b) show that when the target signal is a multi-tone signal, and the number of array elements is small or the spacing between array elements is much smaller than the half wavelength corresponding to each frequency of the multi-tone signal, the method of the present invention can still obtain a sharper spectral peak near the target direction compared to conventional beamforming methods.

[0123] like Figure 4 The figure shown is a comparison of the spatial power spectrum processing results of the method of this invention and the conventional beamforming method when the target signal is a broadband signal. For ease of comparison, Figure 4 The spatial power spectra of both the method of this invention and conventional beamforming methods have been normalized. Figure 4 It contains 2 subgraphs, among which Figure 4 (a) corresponds to the case where the number of array elements is 13 and the broadband signal frequency band is [300Hz, 400Hz]. At this time, the array aperture is 6.5 meters and the wavelength corresponding to the center frequency of the broadband signal is 0.97 meters. Figure 4 (b) corresponds to the case where the number of array elements is 8 and the broadband signal frequency band is [150Hz, 200Hz]. In this case, the array aperture is 4 meters and the wavelength corresponding to the center frequency of the broadband signal is 1.94 meters.

[0124] contrast Figure 4 All results in (a)-(b) show that when the target signal is a broadband signal, and when the number of array elements is small or the spacing between array elements is much smaller than half the wavelength corresponding to the center frequency of the broadband signal, the method of the present invention can form a sharper spectral peak near the target direction compared with conventional beamforming methods.

[0125] In summary, the method of this invention achieves spatial power spectrum estimation by directly utilizing the frequency domain data matrix, without the need to calculate the signal covariance matrix or perform eigenvalue decomposition of the signal covariance matrix. While reducing computational complexity, compared with traditional spatial power spectrum estimation methods based on conventional beamformers, this method can obtain sharper spectral peaks near the target direction when the number of array elements is small or the element spacing is much smaller than half the signal wavelength, which is more conducive to the estimation of the target direction.

[0126] The preferred implementation of the present invention has been described in detail above, but the present invention is not limited to the described embodiments. Those skilled in the art can make various equivalent modifications or substitutions without departing from the spirit of the present invention, and these equivalent modifications or substitutions are all included within the scope defined by the claims of this application.

Claims

1. A method for small aperture linear array spatial power spectrum estimation based on ellipsoidal semi-axis metric, characterized in that, Includes the following steps: Step S1: Obtain the time-domain data matrix: Sample and store the received signals of all sensor elements in the array to obtain the time-domain data matrix X; ； Where N represents the length of the signal received by a single array element, and M represents the number of sensor array elements contained in the array; Step S2: Calculate the frequency domain data matrix: Perform a discrete Fourier transform on each column of the time domain data matrix X from step S1 to obtain the frequency domain data matrix Y; ； Among them, the data in the l-th column of the time-domain data matrix X is Let represent a discrete sequence of length N obtained by sampling the received signal of the l-th sensor element, with the sampling frequency denoted as . ; Step S3: Calculate the frequency sequence set: Determine and calculate the frequency sequence set based on the type of signal received by the array; Step S4: Extract the frequency domain data matrix: Extract the frequency domain data matrix obtained in step S2 using the set of frequency indices obtained in step S3. Step S5: Determine the set of scanning directions: The direction-of-arrival interval of the target signal is denoted as... ,in, This indicates the lower limit of the range in the direction of arrival. This represents the upper limit of the direction-of-arrival interval; then for a linear array, Values Spend, Values The scan direction set is obtained by sampling at equal intervals within a defined direction-of-arrival (DOA) range. ; ； The total number of sampling points contained in the scan direction set is: ; Step S6: Calculate the time compensation vector: Based on the scan direction set obtained in step S5 Select the scanning direction angles from the set in sequence. , Calculate the corresponding time compensation vector ; ； in, , , indicating the first The distance between each sensor element and the first sensor element Usually, it is taken as zero. Indicates the speed at which a signal travels through a medium; Step S7: Calculate the phase compensation matrix: Based on the frequency index set k obtained in step S3, denote all elements contained in the frequency index set k as follows: , , , , ,in Let k represent the number of elements in the frequency index set k, and construct the frequency index vector. ; ； The constructed frequency index vector and the time compensation vector obtained in step S6 Calculate the phase compensation matrix C; ； Step S8: Perform phase compensation on the extracted frequency domain data matrix: The frequency domain data matrix extracted in step S4... Perform a matrix dot product with the phase compensation matrix C obtained in step S7 to obtain... ； Among them, symbols This represents the matrix dot product, i.e. ; Step S9: Construct and solve the positive semidefinite programming problem: Decompose each element of matrix Q in step S8 into its real and imaginary parts to construct a real matrix. Find a matrix that can contain All column vectors Given an ellipsoid such that the sum of the squares of all its semi-axis is minimized, construct a positive semidefinite programming problem. Step S10: Calculate the ellipsoidal semi-axis metric: The sum of the squares of all semi-axis lengths of the ellipsoid is called the ellipsoidal semi-axis metric. Therefore, when the scanning direction is... When, the semi-axis of the ellipsoid is measured as Its value is equal to the optimal matrix obtained from the semidefinite programming problem in step S9. traces, that is ; Step S11: Calculate the spatial power spectrum based on the ellipsoidal semi-axis metric: Based on the ellipsoidal semi-axis metric obtained in step S10... The scanning direction is calculated as follows: Spatial power spectrum of time ; ； Repeat steps S6-S11 to obtain the spatial power spectrum corresponding to all scanning directions based on the ellipsoidal semi-axis metric.

2. The method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement according to claim 1, characterized in that, Step S3 describes determining and calculating the frequency index set based on the type of signal received by the array. The determination and calculation of the frequency index set can be divided into the following three cases: a. When the array receives a single-tone signal, let the analog frequency of the single-tone signal be denoted as . Then the frequency index set k is calculated as follows: ； b. When the array receives a multi-tone signal, i.e. The superposition of single-tone signals of different frequencies, denoted as... The frequency values ​​are respectively , , , Then the frequency index set k is calculated as follows: ； In the above formula, the frequency index set Represents all frequency index subsets , , , Union of frequencies and subsets Calculate using the following formula: , ； c. When the array receives a broadband signal, the lower limit of the broadband signal frequency is denoted as... The upper limit of frequency is Then the frequency index set k is calculated as follows: 。 3. The method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement according to claim 1, characterized in that, Step S4, which involves using the frequency index set obtained in step S3 to extract the frequency domain data matrix obtained in step S2, specifically involves: Let all elements contained in the frequency index set k be denoted as... , , , , ,in This indicates the number of elements contained in the frequency index set k; Extracting the frequency domain data matrix Y from the frequency index set yields... ； in, This represents the frequency domain data matrix based on the set of frequency indices. The matrix obtained after extraction has r rows, corresponding to the number of elements in the frequency index set k; and M columns, corresponding to the number of sensor array elements.

4. The method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement according to claim 1, characterized in that, The direction of arrival interval of the target signal in step S5 During the calculation, conventional beamforming methods are used to coarsely estimate the direction of arrival of the target signal, thereby determining a smaller range of direction of arrival intervals to reduce computational complexity.

5. The method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement according to claim 1, characterized in that, The total number of sampling points included in the scan direction set mentioned in step S5 is Selected The larger the value, the denser the sample points in the scanning direction, the weaker the picket fence effect in spatial power spectrum estimation, and the greater the computational load required for subsequent steps; conversely, the smaller the value, the denser the sample points in the scanning direction. The smaller the value, the sparser the sample points in the scanning direction, the more significant the picket fence effect in the spatial power spectrum estimation, and the less computation is required for subsequent steps.

6. The method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement according to claim 1, characterized in that, The step S9 involves finding a matrix that can contain... All column vectors Given an ellipsoid such that the sum of the squares of all its semi-axis is minimized, construct a positive semidefinite programming problem as follows: First, the real matrix ; ； Denote it as a column vector ;in, , , representing a matrix The List; Then, find a matrix that can contain the matrix. All column vectors Given an ellipsoid such that the sum of the squares of all its semi-axis is minimized, the positive semidefinite programming problem is constructed as follows: ； Among them, matrix sum vector These are the optimization variables in a semidefinite programming problem; matrices. It is a dimension of The symmetric matrix is ​​related to the shape of the ellipsoid; vector It is a dimension of The column vectors are related to the center position of the ellipsoid; the objective function Representation matrix The trace, whose size is equal to the total number of column vectors. The sum of squares of the semi-axis lengths of the ellipsoid; to efficiently solve the constructed positive semidefinite programming problem, obtaining information about the matrix. sum vector The optimal solutions are denoted as follows: and .

7. The method for estimating the spatial power spectrum of a small-aperture linear array based on ellipsoidal semi-axis measurement according to claim 6, characterized in that, The aforementioned method effectively solves the constructed positive semidefinite programming problem, yielding information about the matrix. sum vector The optimal solution, matrix sum vector The optimal solution is obtained by using the interior point method.

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