Multi-path signal underdetermined DOA estimation method based on multi-frequency data fusion under sparse array

Abstract

The invention discloses a multipath signal underdetermined DOA estimation method based on multi-frequency data fusion under a sparse array, and the method comprises the steps: receiving a broadband multipath signal through the sparse array, and building an array frequency domain receiving signal model after DFT; dividing a broadband frequency into a plurality of sub-bands, taking a center frequency point of each sub-band as a reference frequency point, transforming a direction matrix of different frequency points in each sub-band to the reference frequency point, and constructing a decoherent narrowband data model under each sub-band reference frequency point; calculating and vectorizing a received data covariance matrix after decoherence, and constructing a virtual array receiving model; a spatial spectrum function is obtained by applying one-way spatial smoothing and a MUSIC algorithm, and a DOA estimated value is obtained through spectrum peak search. According to the method, serious errors caused by inaccurate angle estimation in a traditional broadband focusing decoherence algorithm are effectively avoided, the array aperture and the spatial degree of freedom are remarkably expanded through virtualization operation, and DOA estimation under the underdetermined condition can be achieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of array signal processing, and particularly relates to an underdetermined DOA estimation method for multipath signals based on multi-frequency data fusion under a sparse array. Background Technique

[0002] Direction of Arrival (DOA) estimation, also known as spatial spectrum estimation, aims to accurately estimate the spatial angle information of target signal sources by processing the signals received by a sensor array. Currently, DOA estimation technology has been widely applied in fields such as radar, navigation, and sonar. In recent years, DOA estimation based on sparse arrays has become one of the research hotspots, and minimum redundancy arrays, nested arrays, coprime arrays, and their optimized structures have been proposed one after another. Compared with traditional uniform arrays, sparse arrays have excellent characteristics such as large aperture, high degrees of freedom, low mutual coupling rate, low redundancy, and low hardware cost. At the same time, to further improve the DOA estimation performance, scholars have developed various DOA estimation algorithms from different perspectives. Subspace algorithms such as Multiple Signal Classification (MUSIC) and Estimating Signal Parameters via Rotational Invariance Technique (ESPRIT) can achieve excellent spatial spectrum estimation under the ideal conditions of uncorrelated signals and a full-rank covariance matrix.

[0003] However, the real environment is complex and changeable, and signals often undergo multipath propagation due to phenomena such as scattering, reflection, and diffraction, resulting in angle expansion of the signals in space and forming coherent multipath signal clusters. At this time, the covariance matrix of the signals received by the array is no longer full-rank, the dimension of the signal subspace decreases, thereby destroying the orthogonality between the subspaces, and traditional orthogonality-based algorithms thus fail.

[0004] In broadband multipath signal DOA estimation, the Coherent Signal Subspace Method (CSSM) is the most classic algorithm. This algorithm proposes the idea of focusing. By constructing a focusing matrix, the signal subspaces at different frequency points are all focused to the reference frequency point. After focusing, averaging processing can compensate for the rank deficiency of the covariance matrix caused by signal coherence, and a narrowband DOA algorithm is applied for estimation based on the focused and smoothed covariance matrix.

[0005] By constructing different focusing matrices, scholars have proposed a large number of CSSM algorithms, such as the Rotate Signal Subspace (RSS) algorithm, the Signal Subspace Transformation (SST) algorithm, etc. However, there are still some problems with the CSSM de-coherence algorithms. First, in a sparse array, the CSSM algorithm needs to estimate the angle based on the physical array, which means that this process cannot utilize the advantage of breaking through the physical element limit after virtualizing the sparse array. Moreover, the construction of the focusing matrix heavily depends on the estimated angle information. Inaccurate estimated angles will lead to large estimation errors or even complete failure of this type of algorithm. Second, the selection of the reference frequency also affects the accuracy of DOA estimation. In addition, the CSSM algorithm uses the frequency-domain smoothing method to de-cohere, and the de-coherence effect is not ideal in scenarios with a small number of frequency points.

[0006] Therefore, it is of great significance to study the underdetermined DOA estimation method for wideband multipath signals in a sparse array. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to provide an underdetermined DOA estimation method for multipath signals based on multi-frequency data fusion under a sparse array in view of the above-mentioned deficiencies of the prior art. The method can not only de-cohere wideband signals in an actual multipath environment, but also make full use of the advantages of a sparse array to achieve DOA estimation under underdetermined conditions.

[0008] To achieve the above technical objectives, the technical solution adopted by the present invention is as follows:

[0009] An underdetermined DOA estimation method for multipath signals based on multi-frequency data fusion under a sparse array, comprising:

[0010] Step 1), receiving wideband multipath signals in the spatial domain through a sparse array, and establishing an array frequency-domain received signal model after performing DFT on the received signals;

[0011] Step 2), uniformly dividing the wideband frequencies in the array frequency-domain received signal model into multiple sub-bands, taking the center frequency point of each sub-band as the reference frequency point, transforming the direction matrices of different frequency points within each sub-band to the same reference frequency point through first-order Taylor expansion, and constructing a de-cohered narrowband data model at the reference frequency point of each sub-band by fusing the multi-frequency received data within each sub-band;

[0012] Step 3), calculating the covariance matrix of the de-cohered received data according to the narrowband data model, performing vectorization processing on the covariance matrix of the received data, and further constructing a virtual array received model corresponding to each reference frequency point;

[0013] Step 4), based on the virtual array reception model, first apply the one-way spatial smoothing technique and the MUSIC algorithm to obtain the spatial spectrum function, and then obtain the DOA estimation value through spectral peak search.

[0014] To optimize the above technical solution, the specific measures taken also include:

[0015] The array frequency-domain received signal model established in the above step 1) is:

[0016]

[0017] where X(f j ), A(f j , θ), S(f j ) and N(f j ) are the frequency-domain array received data, direction matrix, incoming wave signal and noise corresponding to the frequency point f j respectively; J is the total number of frequency points; (·) T represents the transpose operation; θ is the set of incident angles; X m (f j ) represents the frequency-domain received data of the m-th antenna element at the frequency point f j ; is the steering vector corresponding to the frequency point f j and the incident angle θ k , k = 1, 2,..., K, K is the number of multipath signals, q2, q M represent the positions of the 2nd and M-th elements; represents the k-th multipath signal at the frequency point f j , α k and ρ k represent the amplitude attenuation and time delay coefficients respectively, S1(f j ) is the DFT signal of the far-field broadband radiation source signal s1(t); N m (f j ) represents the frequency-domain data of the noise on the m-th antenna element at the frequency point f j , m = 1, 2,..., M.

[0018] In the above step 2), the broadband frequency in the array frequency-domain received signal model is evenly divided into multiple sub-bands, and with the center frequency point of each sub-band as the reference frequency point, the direction matrices at different frequency points within each sub-band are transformed to the same reference frequency point through first-order Taylor expansion, specifically:

[0019] The total of J frequency points in the array frequency-domain data model are evenly divided into P sub-bands, each sub-band contains G adjacent frequency points, where G > 2K, and the frequency points within the p-th sub-band are respectively expressed as The center frequency point is expressed as As the reference frequency point of the p-th sub-band;

[0020] Consider the direction matrix corresponding to G frequency points within the p-th sub-band. Taking the direction matrix at the center frequency point as a reference, denoted as Then, the direction matrix at the g-th frequency point within the p-th sub-band is transformed to the reference frequency point through a first-order Taylor expansion as follows:

[0021]

[0022] where represents the direction matrix corresponding to the reference frequency point of the p-th sub-band, c represents the propagation speed of electromagnetic waves in air, represents the value of the partial derivative matrix of the direction matrix A with respect to at the frequency point ;

[0023] The array received data at the g-th frequency point within the p-th sub-band is transformed into:

[0024]

[0025] where and represent the multipath signal data and noise data at the g-th frequency point within the p-th sub-band, respectively;

[0026] Let represent the extended direction matrix, represent the extended multipath signal frequency-domain data, then Equation (6) is expressed as:

[0027]

[0028] The above step 2) constructs a decorrelated narrowband data model at the reference frequency point of each sub-band by fusing the multi-frequency point received data within each sub-band, specifically:

[0029] Fuse the multi-frequency point received data corresponding to the p-th sub-band to obtain the decorrelated narrowband data model at the reference frequency point of the p-th sub-band:

[0030]

[0031] where represents the incident wave signal component corresponding to the sum of all received data within the sub-band, represents the noise component corresponding to the fusion of multi-frequency point data.

[0032] The above step 3) includes the following steps:

[0033] Step 31), calculate the covariance matrix of the received data after decorrelation according to the narrowband data model

[0034] Step 32), perform vectorization on to obtain:

[0035]

[0036] where, is the covariance matrix of the received data after vectorization, vec(·) represents the vectorization operation, represents the signal power vector, represents the virtual array steering matrix containing the positions of redundant array elements, (·) * represents taking the conjugate, the vectorized identity matrix I M represents the identity matrix of size M×M, represents the noise N p variance of; ⊙ represents the Khatri-Rao product.

[0037] Step 33), sort and remove redundancy to obtain the virtual array reception model corresponding to each reference frequency point:

[0038]

[0039] In the formula, is the virtual array steering matrix obtained by sorting and removing redundancy for all row vectors of ; I is the vector obtained after removing redundancy, with only the middle element being 1 and the rest of the elements approaching 0; represents the virtual array received data vector at the frequency point, and P is the number of subbands.

[0040] In the above step 4), the virtual array reception model is divided into H1 mutually overlapping subarrays in a one-way spatial sliding manner, and each subarray contains H2 = 2M2(M1 + 1) - H1 array elements. Among them, H1 > 2K, and the array element positions of the i-th subarray are:

[0041] {[M2(M1 + 1) - H2 + 1 - i]d1, [M2(M1 + 1) - H2 + 2 - i]d1,..., [M2(M1 + 1) - i]d1}(13)

[0042] Among them, M1 and M2 are the number of array elements of the first-level sub-array and the second-level sub-array in the sparse array (taking the two-level nested array as an example); d1 is the element spacing.

[0043] The received signal vector of the i-th sub-array is From the (2M2(M1 + 1) - H2 + 1 - i)-th row to the (2M2(M1 + 1) - i)-th row of, denoted as Its covariance matrix is:

[0044]

[0045] Among them, represents the received data vector of the virtual array at the frequency point; P represents the total number of sub-bands.

[0046] The one-way spatial smoothing covariance matrix is obtained by finding the average of the covariance matrices of H1 sub-arrays:

[0047]

[0048] Among them, represents the reference frequency point where the rank-deficient recovered one-way spatial smoothing covariance matrix is obtained.

[0049] For perform eigenvalue decomposition:

[0050]

[0051] Among them, is a diagonal matrix composed of the first K large eigenvalues; is composed of the eigenvectors corresponding to the first K large eigenvalues, representing the signal subspace; is a diagonal matrix composed of the remaining small eigenvalues; is composed of the remaining eigenvectors, representing the noise subspace.

[0052] In the above step 4), the MUSIC algorithm is used to calculate the spatial spectrum at each reference frequency point:

[0053]

[0054] The spatial spectrum function is obtained according to the P spatial spectra:

[0055]

[0056] Among them, represents the steering vector at the p-th reference frequency point after spatial smoothing.

[0057] In the above step 4), by traversing the value of the angle θ, for the spatial spectrum function Perform spectral peak search to use the θ values corresponding to the K maximum points as the DOA estimation values.

[0058] The present invention has the following beneficial effects:

[0059] The present invention first constructs a frequency-domain data model through discrete Fourier transform; subsequently, the broadband frequency is evenly divided into multiple sub-bands, each sub-band contains at least twice the number of frequency points of the DOA, and the center frequency point of each sub-band is used as the reference frequency point. According to the first-order Taylor expansion, the direction matrices of different frequency points within each sub-band can be transformed to the same reference frequency point. By fusing the multi-frequency point received data within each sub-band, a narrow-band data model after decoherence at the reference frequency point of each sub-band is constructed; then, by vectorizing the covariance matrix of each received data after decoherence, a virtual array reception model is further constructed; finally, the spatial spectrum at each reference frequency point is obtained using the spatial smoothing MUSIC algorithm, and the DOA estimation is achieved by calculating the average value of multiple spatial spectra. Compared with the CSSM (broadband focusing) type decoherence algorithm, the present invention does not require angle prediction based on a physical array, nor does it require constructing a focusing matrix for frequency focusing, thus effectively avoiding serious errors caused by inaccurate angle prediction. In addition, the virtualization operation significantly expands the array aperture and degrees of freedom, enabling DOA estimation under underdetermined conditions, and has important application value. Description of the Drawings

[0060] Figure 1 is a flowchart of the method of the present invention;

[0061] Figure 2 is a scene diagram of DOA estimation of multipath signals of the present invention;

[0062] Figure 3 is a comparison diagram of spatial spectrum estimation using the method of the present invention (Proposed) and the traditional rotating signal subspace algorithm (RSS). Detailed Embodiment

[0063] The following further describes the embodiments of the present invention in detail with reference to the drawings.

[0064] The method for underdetermined DOA estimation of multipath signals based on multi-frequency data fusion under a sparse array described in the present invention has a detailed process as Figure 1As shown in the figure, first, a wideband multipath signal is collected by a sparse array, and the received time-domain signal is subjected to a discrete Fourier transform (DFT) to construct a frequency-domain data model, where each frequency point corresponds to a narrowband model. Subsequently, the wideband frequency is evenly divided into multiple subbands, and each subband contains at least twice the number of frequency points of the DOA. Taking the center frequency point of each subband as the reference frequency point, according to the first-order Taylor expansion, the direction matrices of different frequency points within each subband can be transformed to the same reference frequency point. By fusing the multi-frequency point received data within each subband, a narrowband data model after decoherence at the reference frequency point of each subband is constructed. Then, by vectorizing each decohered received data covariance matrix, a virtual array reception model with extended spatial degrees of freedom is further constructed. Finally, the spatial spectrum at each reference frequency point is obtained using the spatial smoothing MUSIC algorithm, and the DOA estimation is achieved by calculating the average value of multiple spatial spectra. This method uses a sparse array (taking a second-level nested array as an example) to receive multipath signals and estimate the direction of arrival of multipath signals. The specific implementation is as follows:

[0065] Step 1), receive the wideband multipath signal in the airspace by a sparse array, and establish an array frequency-domain received signal model after performing DFT on the received signal;

[0066] In the embodiment, E(·) represents expectation, (·) H represents the conjugate transpose operation, (·) T represents the transpose operation, (·) * represents taking the conjugate, represents the Kronecker product, diag(a) represents the diagonal matrix composed of the elements in vector a, rank(·) represents the rank operation, vec(·) represents the vectorization operation, and the exponent i represents the imaginary unit.

[0067] The wideband multipath signal is received by a sparse array and a frequency-domain data model is established after DFT, where each frequency point corresponds to a narrowband model. The specific process of establishing the frequency-domain array received data model is as follows:

[0068] The second-level nested array is a classic sparse array structure. The method of the present invention uses a sparse array to collect airspace signals. Taking the second-level nested array as an example, as Figure 2 shown. The second-level nested array is composed of two uniform subarrays connected in sequence. The first-level subarray is a small-spacing uniform subarray with M1 array elements and an element spacing of d1, and the second-level subarray is a large-spacing uniform subarray with M2 array elements and an element spacing of (M1 + 1)d1. Therefore, the total number of physical array elements is M = M1 + M2, and the first array element is the reference array element.

[0069] There is currently a far-field broadband radiation source signal s1(t) with a frequency range of f0 to f1. Assuming that the radiation source generates K multipath signals after passing through the channel, which are completely coherent with each other and arrive at the second-level nested array at angles θ i (i = 1,..., K). To prevent angle ambiguity in DOA, the element spacing is set to d1 = c / (2f1), which is half the wavelength corresponding to the maximum frequency of the incident signal, where c represents the electromagnetic wave propagation speed. Therefore, the received signal at the m-th element can be expressed as

[0070]

[0071] where x m (t) (m = 1, 2,..., M) represents the expression of the received signal at the m-th antenna, and s1(t) represents the true radiation source signal. α k and ρ k represent the amplitude attenuation and time delay coefficient of the k-th multipath respectively. Then the k-th multipath signal is s k (t) = α k s1(t - ρ k ), with α1 = 1 and ρ1 = 0. τ m,k is the time delay difference of the k-th multipath signal arriving at the m-th element relative to the reference element, which can be expressed as τ m,k = q m sinθ k / c, where q m represents the position of the m-th element. n m (t) is the noise at the m-th element, which is usually assumed to be zero-mean additive white Gaussian noise, uncorrelated with the source, and uncorrelated between elements.

[0072] Different from narrowband signals, the envelope delay of broadband signals cannot be ignored, which results in that the broadband time-domain model cannot extract the steering vector like the narrowband model and cannot perform subsequent DOA estimation. Therefore, the Fourier transform is performed on both sides of Equation (1) to obtain the frequency-domain model:

[0073]

[0074] where X m (f j ), S1(f j ) and N m (f j ) correspond to the DFT of x m (t), s1(t) and n m (t) respectively.

[0075] According to the Nyquist sampling theorem, the sampling frequency should be at least greater than twice the maximum frequency of the signal. This means that many elements in the frequency-domain data of the received signal are close to zero. At the same time, to reduce complexity, only J adjacent frequency points within the signal frequency band are taken here, namely f1 to f J .

[0076] As can be seen from (2), at frequency f j , the frequency-domain received signal model of the array is:

[0077]

[0078] where X(f j ), A(f j , θ), S(f j ) and N(f j ) are the frequency-domain array received data, direction matrix, incoming wave signal and noise corresponding to the frequency point f j respectively; represents the k-th multipath signal at the frequency point f j , α k and ρ k represent the amplitude attenuation and time delay coefficient respectively; the steering vector a(f j , θ k ) is related not only to the incoming wave direction but also to the frequency of the broadband signal, and can be expressed as X m (f j ) represents the frequency-domain received data of the m-th antenna element at the frequency point f j , N m (f j ) represents the frequency-domain data of the noise on the m-th antenna element at the frequency point f j , m = 1, 2,..., M.

[0079] For the Fourier transform of the broadband array signal, the time-domain snapshots are transformed into frequency points, and each frequency point has single-snapshot data. Therefore, frequency-domain snapshots also need to be constructed in signal processing. The usual processing method is time-domain segmented transformation, that is, dividing the time domain into L non-overlapping segments (L >> K), and performing DFT on each segment respectively to obtain L independent segments of frequency-domain data. In this way, the frequency-domain broadband signal at each frequency point has L frequency-domain snapshots, making At this time, as can be seen from (3), each frequency point in the frequency-domain model of the array received signal corresponds to a narrowband data model, and A(f j , θ) represents the direction matrix corresponding to the frequency point f j .

[0080] Step 2), divide the wideband frequencies in the array frequency-domain received signal model evenly into multiple subbands. Taking the center frequency point of each subband as the reference frequency point, transform the direction matrices at different frequency points within each subband to the same reference frequency point through first-order Taylor expansion, and construct a narrowband data model after decoherence at the reference frequency point of each subband by fusing the multi-frequency received data within each subband;

[0081] In the embodiment, the wideband is evenly divided into multiple subbands, and each subband contains at least twice the number of DOA frequencies; taking the center frequency point of each subband as the reference frequency point, according to the first-order Taylor expansion, the direction matrices at different frequency points within each subband can be transformed to the same reference frequency point. By fusing the multi-frequency received data within each subband, a narrowband data model after decoherence at the reference frequency point of each subband is constructed as follows:

[0082] Step 21), divide the subbands, and according to the first-order Taylor expansion, transform the direction matrices at different frequency points within each subband to the reference frequency point:

[0083] As can be seen from Equation (3), S(f j ) in the data model is a set of coherent signals, and the virtualization method of the sparse array requires that the covariance matrix of the signals be a diagonal matrix, which means that there should be no correlation between the signals. Therefore, it is necessary to perform decoherence processing on the signals first to fully utilize the advantages of the sparse array.

[0084] In the frequency-domain data model, J frequency points within the signal bandwidth are selected. First, divide the J frequency points evenly into P subbands, and each subband contains G adjacent frequency points, where G > 2K. The frequency points within the p-th subband are respectively denoted as The center frequency point is denoted as

[0085] Considering the direction matrices corresponding to the G frequency points within the p-th subband, taking the direction matrix at the center frequency point as the reference, denoted as Then the direction matrix of the g-th frequency point within the p-th subband can be transformed to the reference frequency point through first-order Taylor expansion, that is

[0086]

[0087] where, represents the direction matrix corresponding to the reference frequency point of the p-th subband, represents the direction matrix corresponding to the reference frequency point of the p-th subband, c represents the propagation speed of electromagnetic waves in the air, represents the partial derivative matrix of the direction matrix A with respect to at the frequency point The value at, where the (m, k)-th element is expressed as

[0088]

[0089] At this time, approaches The first-order Taylor expansion (4) can approximate

[0090] Therefore, the array received data at the g-th frequency point can be transformed into

[0091]

[0092] where, and respectively represent the multipath signal data and noise data at the g-th frequency point in the p-th subband. Let the extended direction matrix In subsequent DOA estimation, plays a dominant role, and the extended multipath signal frequency-domain data Then equation (6) can be expressed as

[0093]

[0094] where, So is also a set of completely coherent data.

[0095] Step 22), next add the multi-frequency received data corresponding to the p-th subband to construct the array received data model after decorrelation at the reference frequency point of each subband:

[0096]

[0097] where, through multi-frequency data fusion, a row full-rank

[0098] represents the incident wave signal component corresponding to the sum of all received data within the subband, and the decorrelation operation is completed; is the data model after decorrelation at the p-th reference frequency point; represents the noise component corresponding to the multi-frequency data fusion.

[0099] Although is a set of completely coherent data, the data between different frequencies are independent of each other, that is are linearly independent, which means Therefore, the signal terms after multi-frequency data fusion are full-rank, and the decorrelation operation is completed. This method is similar to the CSSM algorithm, which is equivalent to decorrelation by frequency-domain smoothing. The difference is that the method of the present invention performs frequency-domain smoothing at the signal level and does not require estimating the angle based on the data received by the physical array, avoiding a serious decline in algorithm performance caused by inaccurate angle estimation during the construction of the focusing matrix. In addition, subsequent DOA estimation under underdetermined conditions can be directly achieved through the virtualization method.

[0100] Equation (8) is the narrowband data model after decorrelation at each sub-band reference frequency point. It should be noted that in practical applications, the frequency point spacing is usually much smaller than the electromagnetic wave propagation speed c, which results in the signal terms in the lower half having an amplitude much smaller than that in the upper half Therefore, in subsequent DOA estimation, the left half of the direction matrix corresponding to the signal component plays a dominant role.

[0101] Step 3), calculate the covariance matrix of the received data after decorrelation according to the narrowband data model, perform vectorization processing on the covariance matrix of the received data, and further construct a virtual array reception model corresponding to each reference frequency point;

[0102] In the embodiment, by vectorizing each covariance matrix of the received data after decorrelation, a virtual array reception model is further constructed, including:

[0103] Step 31), first, calculate the covariance matrix of the received data after decorrelation according to Equation (8)

[0104]

[0105] where represents the covariance matrix of the signal after decorrelation; represents each row corresponding to the power of the signal, represents the variance of the noise N p ; I M represents an identity matrix of size M×M.

[0106] Step 32), perform vectorization processing on to obtain

[0107]

[0108] where represents the signal power vector, represents the virtual array direction matrix containing the positions of redundant array elements,​

[0109] This two-level nested array can be stretched into a virtual uniform linear array with 2M2(M1 + 1) - 1 array elements, and the array element position D can be expressed as

[0110] D = {nd1, n = -[M2(M1 + 1) - 1],..., [M2(M1 + 1) - 1]} (11)

[0111] Step 33), according to Equation (11), After sorting and removing redundancy, the virtual array reception model corresponding to each reference frequency point is obtained:

[0112]

[0113] In the formula, is the virtual array direction matrix obtained by sorting all row vectors and removing redundancy; I in the noise term is the vector obtained after removing redundancy, with only the middle element being 1 and the remaining elements approaching 0; represents the virtual array received data vector at the frequency point. It can be seen from Equation (12) that the sparse array virtualization method adopted in the present invention significantly expands the array aperture and spatial degrees of freedom, and can achieve DOA estimation under underdetermined conditions.

[0114] Step 4), for the virtual array reception model, first obtain the spatial spectrum at each reference frequency point through one-way spatial smoothing and MUSIC algorithm, and then achieve DOA estimation by calculating the average value of multiple spatial spectra.

[0115] In the embodiment, since the virtual array reception model is single-snapshot data, it is necessary to use the spatial smoothing MUSIC algorithm to obtain the spatial spectrum at each reference frequency point, and achieve DOA estimation by calculating the average value of multiple spatial spectra, that is, first restore the rank of the array covariance matrix through spatial smoothing, and then use the MUSIC algorithm for DOA estimation, including:

[0116] Step 41), the total number of elements of the virtual array is 2M2(M1 + 1) - 1. Using the one-way spatial smoothing method, the virtual uniform array (virtual array reception model) is divided into H1 mutually overlapping sub-arrays in a sliding manner, and each sub-array contains H2 = 2M2(M1 + 1) - H1 array elements. Among them, H1 > 2K, and the element positions of the i-th sub-array are

[0117] {[M2(M1 + 1) - H2 + 1 - i]d1, [M2(M1 + 1) - H2 + 2 - i]d1,..., [M2(M1 + 1) - i]d1} (13)

[0118] ​Among them, M1 and M2 are the number of array elements of the first-level sub-arrays and the number of array elements of the second-level sub-arrays in the second-level nested array; d1 is the element spacing.

[0119] The received signal vector of this sub-array is from the (2M2(M1 + 1) - H2 + 1 - i)-th row to the (2M2(M1 + 1) - i)-th row, denoted as Its covariance matrix is

[0120]

[0121] According to Equation (14), the covariance matrices of H1 sub-arrays can be obtained, and the one-way spatial smoothing covariance matrix can be obtained by taking the average of the sub-array covariance matrices, that is

[0122]

[0123] Among them, H1 represents the number of sub-arrays divided by spatial smoothing, represents the reference frequency point The spatial smoothing covariance matrix whose rank deficiency is restored at the reference frequency point is obtained by taking the average of the sub-array covariance matrices, and its corresponding direction matrix can be expressed as Steering vector

[0124] As can be seen from Equation (8), the left half of the direction matrix plays a leading role in subsequent DOA estimation, and the signal energy is mainly concentrated in the column space. Therefore, in the eigenvalue decomposition of the covariance matrix, the eigenvectors corresponding to the first K larger eigenvalues are selected to span the signal subspace, and the eigenvectors corresponding to the remaining smaller eigenvalues span the noise subspace.

[0125] After is subjected to eigenvalue decomposition, it can be expressed as

[0126]

[0127] Among them, is a diagonal matrix composed of the first K large eigenvalues; is composed of the eigenvectors corresponding to the first K large eigenvalues, representing the signal subspace; is a diagonal matrix composed of the remaining small eigenvalues; is composed of the remaining eigenvectors, representing the noise subspace.

[0128] Step 42), next, use the MUSIC algorithm to calculate the spatial spectrum for the covariance matrix at each reference frequency point,

[0129]

[0130] Then the MUSIC spatial spectrum in the average sense is

[0131]

[0132] where \(P\) represents the total number of sub - frequency bands, represents the steering vector at the \(p\) - th reference frequency point after spatial smoothing. By traversing the value of the angle \(\theta\), a spectral peak search is performed on the spectral function defined by Equation (18), the \(\theta\) values corresponding to the \(K\) maximum points of

[0133] To verify the effectiveness and superiority of the algorithm described in the present invention, the following MATLAB simulation is carried out for verification,

[0134] As Figure 2 shown is the DOA estimation scenario diagram of the multi - path signal of the present invention. A broadband signal source in the airspace is incident on a two - level nested array of \(M1 + M2\) array elements after multi - path propagation. In the simulation, \(M1 = 2\), \(M2 = 3\), and the spacing between adjacent array elements is half of the wavelength corresponding to the maximum frequency of the broadband signal.

[0135] Figure 3 Shown is the comparison diagram of the spatial spectra of the two algorithms. The simulation studies the case when the number of DOAs exceeds the number of physical array elements. In the two - level nested array, the spatial spectrum estimation is respectively carried out using the method of the present invention (Proposed) and the Rotational Signal Subspace Transformation algorithm (RSS).

[0136] In the simulation, it is assumed that there is a signal source with a bandwidth of 10 MHz, which generates 6 multi - path signals after passing through the channel. These signals are completely coherent with each other and are incident on a 5 - element two - level nested array at \(-37^{\circ}\), \(-25^{\circ}\), \(-5^{\circ}\), \(12^{\circ}\), \(25^{\circ}\), \(40^{\circ}\) respectively. \(M1 = 2\), \(M2 = 3\), and the signal - to - noise ratio is 7 dB. The RSS algorithm is a classic CSSM - type algorithm. In this simulation, the algorithm first uses the Capon algorithm to process the received data of the physical array to obtain the angle pre - estimate value, then constructs a focusing matrix according to the pre - estimated angle, and focuses the direction matrices at all frequency points within the bandwidth to the same reference frequency point through the focusing transformation to obtain the decorrelated covariance matrix at the reference frequency point. Subsequently, the covariance matrix is vectorized to obtain the virtual array data model, and finally the DOA estimation can be obtained using the spatial smoothing MUSIC algorithm. The pre - estimation of the initial angle of this algorithm cannot utilize the virtual array model that breaks through the physical array element limit and is affected by the correlation between signals, which will lead to inaccurate pre - estimated angles. And the construction of the focusing matrix highly depends on the pre - estimated angle information, and inaccurate pre - estimated angles will lead to large estimation biases for this type of algorithm.

[0137] As Figure 3As shown, when the number of DOAs exceeds the number of physical array elements, serious errors occur in the spatial spectrum estimation of the RSS algorithm. On the contrary, the method of the present invention can effectively and accurately estimate the DOAs of multipath signals and can achieve DOA estimation under underdetermined conditions.

[0138] The above are only the preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions falling within the idea of the present invention belong to the protection scope of the present invention. It should be noted that for those of ordinary skill in the art, several improvements and refinements made without departing from the principle of the present invention should be regarded as within the protection scope of the present invention.

Claims

1. A method for underdetermined DOA estimation of multipath signals based on multi-frequency data fusion under a sparse array, characterized in that, Including: Step 1), receiving the spatial domain broadband multipath signal through a sparse array, performing DFT on the received signal, and establishing an array frequency domain received signal model; Step 2), evenly dividing the broadband frequency in the array frequency domain received signal model into multiple sub-bands, taking the center frequency point of each sub-band as the reference frequency point, transforming the direction matrices at different frequency points within each sub-band to the same reference frequency point through first-order Taylor expansion, and constructing a narrow-band data model after decoherence at the reference frequency point of each sub-band by fusing the multi-frequency point received data within each sub-band; Step 3), calculating the covariance matrix of the received data after decoherence according to the narrow-band data model and performing vectorization processing on the received data covariance matrix, and further constructing a virtual array received model corresponding to each reference frequency point; Step 4), based on the virtual array received model, first applying the one-way spatial smoothing technique and MUSIC algorithm to obtain the spatial spectrum function, and then obtaining the DOA estimation value through spectral peak search.

2. The method for underdetermined DOA estimation of multipath signals based on multi-frequency data fusion under a sparse array according to claim 1, wherein The array frequency domain received signal model established in the said Step 1) is: Among them, X(f j )、A(f j ,θ)、S(f j ) and N(f j ) are the received data of the frequency domain array, the direction matrix, the incoming wave signal and the noise corresponding to the frequency point f j respectively; J is the total number of frequency points; (·) T represents the transpose operation; θ is the set of incident angles; X m (f j ) represents the frequency domain received data of the m-th antenna element at the frequency point f j ; is the steering vector corresponding to the frequency point f j and the incident angle θ k , k = 1, 2,..., K, K is the number of multipath signals, q2, q M represent the positions of the 2nd and M-th array elements; represents the k-th multipath signal at the frequency point f j , α k and ρ k represent the amplitude attenuation and the time delay coefficient respectively, S1(f j ) is the DFT signal of the far-field broadband radiation source signal s1(t); N m (f j ) represents the frequency domain data of the noise on the m-th antenna element at the frequency point f j , m = 1, 2,..., M.

3. The method for underdetermined DOA estimation of multipath signals based on multi-frequency data fusion under a sparse array according to claim 2, wherein In Step 2), evenly dividing the broadband frequency in the array frequency domain received signal model into multiple sub-bands, taking the center frequency point of each sub-band as the reference frequency point, and transforming the direction matrices at different frequency points within each sub-band to the same reference frequency point through first-order Taylor expansion, specifically: Divide the total of J frequency points in the array frequency-domain data model evenly into P sub-bands, where each sub-band contains G adjacent frequency points, with G > 2K. The frequency points within the p-th sub-band are respectively expressed as The center frequency point is expressed as and serves as the reference frequency point for the p-th sub-band; Consider the direction matrix corresponding to G frequency points in the p-th sub-band. Taking the direction matrix at the center frequency point as a reference, denoted as Then, for the g-th frequency point in the p-th sub-band, the direction matrix is transformed to the reference frequency point through first-order Taylor expansion as follows: Among them, represents the reference frequency point of the p-th sub-band The corresponding direction matrix, c represents the propagation speed of electromagnetic waves in air, represents the direction matrix A with respect to The value of the partial derivative matrix at the frequency point ; Array received data at the g-th frequency point in the p-th subband is converted to: Among them, and respectively represent the multipath signal data and noise data at the g-th frequency point in the p-th subband; Let represent the extended direction matrix, represent the extended multipath signal frequency-domain data, then Equation (6) can be expressed as:

4. The underdetermined DOA estimation method for multipath signals based on multi-frequency data fusion under a sparse array according to claim 3, wherein, In Step 2), constructing a narrow-band data model after decoherence at the reference frequency point of each sub-band by fusing the multi-frequency point received data within each sub-band, specifically: Fuse the multi-frequency point received data corresponding to the p-th sub-band to obtain the reference frequency point of the p-th sub-band Narrowband data model after downcoherence: Among them, represents the wave signal component corresponding to the sum of all received data within the sub-band, represents the noise component corresponding to the multi-frequency data fusion.

5. The underdetermined DOA estimation method for multipath signals based on multi-frequency data fusion under a sparse array according to claim 1, wherein The said Step 3) includes the following steps: Step 31), calculate the covariance matrix of the received data after decorrelation according to the narrowband data model Step 32), for perform vectorization to obtain: Among them, is the received data covariance matrix after vectorization processing, and vec(·) represents the vectorization operation, represents the signal power vector, represents the virtual array direction matrix containing the positions of redundant array elements, (·) * represents taking the conjugate, and the vectorized identity matrix I M represents the identity matrix of size M×M, represents the noise N p variance; ⊙ represents the Khatri-Rao product; Step 33), After sorting and removing redundancy, obtain the virtual array reception model corresponding to each reference frequency point: In the formula, is the virtual array direction matrix obtained by sorting and removing redundancy for all row vectors; I is the vector obtained after removing redundancy, with only the middle element being 1 and the remaining elements approaching 0; represents the virtual array received data vector at the frequency point, and P is the number of sub - bands.

6. The method for underdetermined DOA estimation of multipath signals based on multi-frequency data fusion under a sparse array according to claim 1, wherein In the said Step 4), dividing the virtual array received model into H1 mutually overlapping sub-arrays in a one-way spatial sliding manner, each sub-array contains H2 = 2M2(M1 + 1) - H1 array elements, where H1 > 2K, and the array element positions of the i-th sub-array are: {[M2(M1 + 1) - H2 + 1 - i]d1, [M2(M1 + 1) - H2 + 2 - i]d1,..., [M2(M1 + 1) - i]d1}(13) where M1 and M2 are the number of array elements of the first-level sub-array and the second-level sub-array in the sparse array; d1 is the array element spacing; The received signal vector of the i-th subarray is from the (2M2(M1 + 1) - H2 + 1 - i)-th row to the (2M2(M1 + 1) - i)-th row, denoted as Its covariance matrix is: Among them, represents the virtual array received data vector at the frequency point; P represents the total number of sub-bands; Obtaining the one-way spatial smoothing covariance matrix by finding the average value of the covariance matrices of H1 sub-arrays; Among them, represents the reference frequency point where the rank deficiency of the one-way spatial smoothing covariance matrix is restored; P is the number of subbands; For perform eigenvalue decomposition: Among them, is a diagonal matrix composed of the top K largest eigenvalues; is composed of the eigenvectors corresponding to the top K largest eigenvalues, representing the signal subspace; is a diagonal matrix composed of the remaining small eigenvalues; is composed of the remaining eigenvectors, representing the noise subspace.

7. The method for underdetermined DOA estimation of multipath signals based on multi-frequency data fusion under a sparse array according to claim 6, characterized in that, In the said Step 4), calculating the spatial spectrum at each reference frequency point using the MUSIC algorithm; Obtaining the spatial spectrum function according to P spatial spectra; Among them, represents the steering vector at the p-th reference frequency point after spatial smoothing.

8. The underdetermined DOA estimation method for multipath signals based on multi-frequency data fusion under a sparse array according to claim 1, wherein In step 4), by traversing the values of the angle θ, the spatial spectrum function is searched for spectral peaks, and the θ values corresponding to the K maximum points of are used as the DOA estimation values.

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