A non-ambiguous search-free coprime array doa estimation method

Abstract

The application discloses a non-fuzzy search-free co-prime array DoA estimation method, and belongs to the field of array signal processing.The method is implemented as follows: firstly, the co-prime array is divided into two independent sub-arrays, a plurality of graphical parameters reflecting the relationship between the sub-arrays are preset according to antenna structure parameters; then, Root-MUSIC algorithm is completed in the two sub-arrays of the co-prime array respectively, and the estimation results of the two sub-arrays are corrected by means of the previously calculated graphical parameters and a threshold decision method; finally, the joint optimal estimation of the two sub-arrays is completed according to an adaptive weighting formula.Under the same number of array elements, the co-prime array can obtain a larger antenna aperture and higher estimation precision, and can reduce the difficulty of array design and the mutual coupling effect of the antenna.The phase ambiguity problem is eliminated by means of graphical analysis and geometric method, and the joint optimal estimation of the co-prime array is realized by means of an adaptive weighting scheme, so that the estimation precision can be improved while the ambiguity problem is avoided.

Description

Technical Field

[0001] This invention relates to target positioning and direction finding technology in the field of wireless communication, specifically to a fuzzy, search-free method for estimating the DoA of coprime arrays, belonging to the field of array signal processing. Background Technology

[0002] Target localization, as one of the core research directions in the field of communications, has been widely applied in many important application scenarios such as millimeter-wave radar, autonomous driving technology, and 5G space-division multiple access communication. These fields place extremely high demands on accurate target localization technology, and direction of arrival (DoA) estimation is a key component of target localization. This technology can accurately determine the direction of arrival of target signals, assisting communication systems in achieving environmental perception and maintaining stable communication.

[0003] With the continuous improvement of communication speeds and frequency bands, DoA estimation technology faces increasingly higher requirements for accuracy and real-time performance. Against this backdrop, coprime arrays have gradually become a popular research direction for solving high-precision DoA estimation problems due to their unique performance advantages. A coprime array consists of two uniform linear arrays that satisfy a coprime relationship, and its unique structural design does not need to adhere to the half-wavelength limitation of traditional antenna arrays. This characteristic not only significantly expands the array aperture but also effectively reduces the complexity of antenna design while minimizing the impact of mutual coupling effects on array performance. These advantages make coprime arrays demonstrate enormous application potential in the field of communications.

[0004] However, coprime arrays also face significant challenges in application. When traditional DoA estimation algorithms are directly applied to coprime arrays, severe phase ambiguity arises due to the increased spacing between the uniform linear arrays. This phase ambiguity renders the DoA estimation results completely invalid, severely impacting the reliability and accuracy of target localization. Researchers have conducted extensive research and proposed various improved algorithms to address the phase ambiguity problem in coprime arrays. However, existing solutions generally suffer from high computational complexity and insufficient real-time performance. Therefore, how to combat the phase ambiguity problem while fully utilizing the high degrees of freedom and high resolution of coprime arrays to achieve fast and reliable DoA estimation has become an important research topic in the field of communications. Summary of the Invention

[0005] To address the unreliable positioning results caused by phase ambiguity when using coprime arrays for target localization, this invention aims to provide an unambiguous, search-free DoA estimation method for coprime arrays. This method visualizes the phase ambiguity problem through a graphical approach, eliminates it using graphical analysis and geometric methods, and achieves joint optimal estimation of the coprime arrays using an adaptive weighting scheme. This improves estimation accuracy while avoiding ambiguity issues.

[0006] This invention discloses a fuzz-free, search-free method for estimating the DoA of coprime arrays, comprising the following steps:

[0007] Step 1: Construct a coprime array consisting of two uniform linear arrays. The two uniform linear arrays each possess... and The sensors correspond to different array element spacings. , .in For the signal wavelength, and The two linear arrays are coprime. They are arranged along the same dimension and share the first sensor element. The total number of sensors is... .

[0008] First, we analyze the mapping relationship between the estimated solution and the true value with fuzzy parameters under a given array structure. Let the relationship parameters of two uniform linear arrays be... ;definition Indicates the first The uniform linear array in the first The ambiguity coefficients in the angular domain are defined. Indicates the first Each angle range has a preset distance.

[0009] when season , Calculate according to formula (1) Positive critical angles corresponding to the angle domain The uniform linear array number corresponding to this critical angle .

[0010]

[0011] If the calculated critical angle does not meet the requirements Then let:

[0012]

[0013]

[0014]

[0015]

[0016] in At this time, we obtain The ambiguity coefficients of the two linear arrays and preset distance The critical angle at this point is calculated again according to equation (1). If not satisfied If the requirements are met, then the calculation steps of equations (2)-(5) are repeated again. Until this point... Corresponding critical angle satisfy The requirement at this time instruct The number of angle domains divided at time, which is further recorded as .

[0017] for The situation directly led to

[0018]

[0019]

[0020] in A set of parameters describing the relationship between two uniform linear arrays in a coprime array is obtained. .

[0021] Step 2: Receive using the coprime array constructed in Step 1 These are two uncorrelated far-field narrowband signals. Each signal has its own incident direction. Composition of angle set At this point, the middle of the coprime array... A uniform linear array Sampling data at time Represented as:

[0022]

[0023] in Corresponding to two uniform linear arrays, Represents independent source signal vectors. Indicates the first The sampled waveform of a signal. It is the first Gaussian white noise vector on a uniform linear array It is the coprime array of the first The direction matrix of a uniform linear array Indicates the first The signal corresponds to the first Direction vectors on a uniform linear array.

[0024] Step 3: Calculate based on the sampling data model established in Step 2. The covariance matrix of each of the two uniform linear arrays was captured by a sampling snapshot. And perform eigenvalue decomposition on the covariance matrix:

[0025]

[0026] The results of the eigenvalue decomposition are sorted in descending order of eigenvalues. and Corresponding to the first in the sorting feature values The diagonal matrix formed by the eigenvalues ​​and the subsequent... A diagonal matrix composed of eigenvalues, and and They are and The corresponding feature vector.

[0027] Definition of the first In a uniform linear array about polynomial function Among them, the polynomial coefficients intermediate parameters .calculate ,get We have eigenvalues, and we select the one that is closest to the unit circle. A complex root Then, calculate the first according to formula (10). The estimation of the i-th uniform linear array yields the following result for the i-th... The ambiguous spatial phase difference of the incident signals:

[0028]

[0029] in This indicates that the phase angle is extracted. Indicates calculation The remainder. According to equation (10), the estimated fuzzy spatial phase difference sets of the two uniform linear arrays can be calculated: and .

[0030] Step 4: Calculate the Cartesian product of the two sets of fuzzy spatial phase differences obtained in Step 3:

[0031]

[0032] Calculate the set according to equation (12) The points corresponding to the Cartesian products in the two-dimensional plane reach the origin and the slope is the relation parameter. The distance of the straight line.

[0033]

[0034] Compare the calculation result with the preset distance calculated in step 1. The matching process involves using a threshold method to extract valid values. This step can be represented as an optimization problem, as shown in equation (13):

[0035]

[0036] Threshold Set to:

[0037]

[0038] The solution set of the optimization problem in equation (13) corresponds to the legal matching results of the two uniform linear arrays. "Legal" indicates that these matches correspond to the same incident signal. Based on this solution set, the unambiguous estimated phase difference is calculated. The specific calculation process is as follows:

[0039]

[0040] in Indicates the first The signal at the first An unambiguous estimate after correction on a uniform linear array. These are the ambiguity coefficients of the coprime array obtained in step 1. Then, an unambiguous phase difference matching set is obtained:

[0041]

[0042] Step 5: Based on the matching set obtained in Step 4 and the relationship between the signal phase difference and the incident angle given by Equation (10), obtain the unambiguous angle matching set:

[0043]

[0044] The joint final estimate of the phase difference of the coprime array with respect to the input signal is calculated according to equation (18):

[0045]

[0046] This leads to the angle set. .in The adaptive weighting coefficients for joint estimation are expressed as follows:

[0047]

[0048] gather It is the calculation of the polynomial function in step 3. Received There are 1 characteristic roots. This is an intermediate parameter used to simplify the formula expression, and its calculation formula is given by (20):

[0049]

[0050] in Indicates taking the conjugate. This indicates taking the real part.

[0051] Step 6: Calculate the angle set obtained in Step 5. As the final estimation result output, the final estimation result is the unambiguous DoA estimation result, which is the unambiguous search-free coprime array DoA estimation.

[0052] Beneficial effects:

[0053] 1. Compared with traditional uniform linear arrays, the present invention discloses a fuzz-free, search-free coprime array DoA estimation method based on an expanded coprime array. With the same number of array elements, the coprime array can obtain a larger antenna aperture and higher estimation accuracy, and can reduce the array design difficulty and antenna mutual coupling effect.

[0054] 2. This invention discloses a fuzz-free, search-free DoA estimation method for coprime arrays. It achieves rapid elimination of the phase fuzziness problem unique to coprime arrays by leveraging the relationship parameters of the two uniform linear arrays constituting the coprime array. Compared to existing fuzziness elimination algorithms, this invention visualizes the phase fuzziness problem through a graphical method and eliminates it using graphical analysis and geometric methods. This achieves rapid elimination of the phase fuzziness problem in coprime arrays without the need for a search operation, resulting in lower computational complexity and thus improving the DoA estimation efficiency for coprime arrays.

[0055] 3. This invention discloses a fuzz-free, search-free DoA estimation method for coprime arrays. It achieves optimal estimation of two uniform linear arrays in a joint coprime array using adaptive weighting, and the resulting estimation is optimal in the sense of minimizing the mean square error. Compared to the mean-weighted scheme in traditional methods, the estimation performance of this invention is always better than or at least equal to that of the mean-weighted scheme. Attached Figure Description

[0056] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0057] Figure 1 This is a flowchart of a fuzz-free, search-free method for estimating the DoA of coprime arrays disclosed in this invention.

[0058] Figure 2 This is a schematic diagram of the unfolded coprime array structure used in this invention.

[0059] Figure 3 The present invention proposes a method based on and A graphical representation of the relationship between two coprime linear arrays.

[0060] Figure 4 To substitute the estimation results Figure 3 The matching result diagram shown in the graphical illustration.

[0061] Figure 5 This is a graph showing the relationship between the root mean square error of the method of this invention and other methods as a function of the input signal-to-noise ratio.

[0062] Figure 6 This graph shows the relationship between the root mean square error of the method of this invention and other methods as a function of the number of input snapshots.

[0063] Figure 7 The graph shows the relationship between the number of complex multiplications involved in estimating the mean square of a matrix using the method of this invention and other methods, and the matrix size. Detailed Implementation

[0064] The present invention will be further described and illustrated below with reference to the accompanying drawings and embodiments.

[0065] like Figure 1 As shown in the figure, the specific implementation steps of the unambiguous and search-free DoA estimation method for coprime arrays disclosed in this embodiment are as follows:

[0066] Step 1: Refer to the following Figure 2 The diagram shows an expanded coprime array structure. Two uniform linear arrays share the same array element at the origin, and the number of elements and the element spacing of the two arrays must satisfy certain coprime constraints. For example, let the number of elements be... and Two uniform linear arrays are arranged in opposite directions along the same dimension, and the spacing between their elements should satisfy the following condition: and .in The signal wavelength. Total number of array elements. .

[0067] The relationship between the estimated spatial phase difference and the true incident angle obtained from two coprime uniform linear arrays is transformed into a formula representing the x-axis and y-axis of a two-dimensional coordinate system:

[0068]

[0069] in Indicates the first The spatial phase difference estimated by a uniform linear array The direction sine represents the angle of incidence. Indicates calculation The remainder. A curve showing the relationship between the estimation results of two coprime uniform linear arrays can be plotted, as shown in the example. Figure 2 As shown in the figure. This figure contains... A straight line, and the number of coprime array angular domains in the invention is exactly 1. This corresponds to The angular domain of time. These straight lines have the same slope, corresponding to the relationship parameters of the two uniform linear arrays in the invention. The x-axis and y-axis of this two-dimensional coordinate system are estimated by the spatial phase difference obtained from two uniform linear arrays. This constitutes all possible estimates for two uniform linear arrays, and the drawn straight line corresponds to the valid result of combining the estimates of the two uniform linear arrays. Only when the estimated result lies on the drawn "valid straight line" is the corresponding estimated result truly meaningful. Furthermore, the calculation of the first... The array in the th... Fuzzy coefficients in the angular domain and This corresponds to the offset of the straight line on the x-axis and y-axis. The preset distance calculated in the invention description... Then the distance of each corresponding straight line relative to the straight line passing through the origin.

[0070] when season , Calculate according to equation (22) Positive critical angles corresponding to the angular domain The uniform linear array number corresponding to this critical angle .

[0071]

[0072] If the calculated critical angle does not meet the requirements Then let:

[0073]

[0074]

[0075]

[0076]

[0077] in Then, the critical angle at this point is calculated according to equation (22). If not satisfied If the requirements are met, then the calculation steps of equations (23)-(26) are repeated again. Until this point... Corresponding critical angle satisfy Requirements.

[0078] for The situation directly led to

[0079]

[0080]

[0081] in Then we need to describe the graphical parameters of the relationship between the two uniform linear arrays of the coprime array. This is also the first step of the "image-based method" in this invention. Based on this operation, image-based parameter calculations can be completed quickly and offline.

[0082] Step 2: Considering real-world scenarios where millimeter-wave radar or 5G base stations use coprime arrays for multi-target localization, the first step is to estimate the reliable direction of arrival for multiple targets. Assume there exist... There are signals with different incident directions, and the incident direction of each signal is... After frequency conversion, analog-to-digital conversion, and other operations, the coprime array can be obtained. A linear array Sampling data at time :

[0083]

[0084] in Corresponding to two uniform linear arrays, Represents independent source signal vectors. Indicates the first The sampled waveform of a signal. It is the first Gaussian white noise vector on a uniform linear array It is the coprime array of the first The orientation matrix of two uniform linear arrays, each with a different orientation vector:

[0085]

[0086]

[0087] Step 3: Calculate based on the sampled data model. The sampling covariance matrix of two uniform linear arrays captured by a single sampling snapshot:

[0088]

[0089] in This represents an estimate of the true value. Eigenvalue decomposition is performed on the covariance matrices of two uniform linear arrays, and the signal space and noise space are divided according to the number of incident signals, yielding:

[0090]

[0091] The results of the eigenvalue decomposition are sorted in descending order of eigenvalues. and Corresponding to the first in the sorting feature values The diagonal matrix formed by the eigenvalues ​​and the subsequent... A diagonal matrix composed of eigenvalues. and They are and The corresponding feature vector.

[0092] Then define about Polynomial functions:

[0093]

[0094] Where the polynomial coefficients intermediate parameters .calculate ,get We have eigenvalues, and we select the one that is closest to the unit circle. A complex root Then, calculate the first according to formula (35). The first uniform linear array is obtained by using the above-described root-finding algorithm (which is essentially the Root-MUSIC algorithm). Spatial phase difference corresponding to each incident signal:

[0095]

[0096] Step 4: Use the Root-MUSIC algorithm to obtain the set of spatial phase differences with ambiguity from the two uniform linear arrays. and By arbitrarily pairing the elements, we obtain their Cartesian product:

[0097]

[0098] set Elements in the drawing are drawn to Figure 3 In the two-dimensional coordinate system depicting the two uniform linear arrays, we obtain the following: Figure 4 The matching diagram of the estimation results is shown. As mentioned earlier, this two-dimensional coordinate system shows all possible estimation results for the coprime array, and the points on the multiple straight lines correspond to valid estimation results. Excluding the paired points outside the straight lines eliminates invalid results caused by phase ambiguity. Considering the presence of noise in the actual environment, a threshold method is used here to compare the calculated results with the preset distance calculated by equation (26). Perform matching and extract valid values. This process can be represented as an optimization problem:

[0099]

[0100] Threshold Set to:

[0101]

[0102] The solution set of the optimization problem in equation (37) corresponds to Figure 4 The legal pairings are clearly indicated; the remaining pairings are incorrect estimates due to phase ambiguity. A legal pairing corresponds to an unambiguous estimated phase difference, specifically expressed as:

[0103]

[0104] in Indicates the first The incident signal at the th th An unambiguous estimate after correction on a uniform linear array. It is the fuzzy coefficient that describes the uniform linear array constituting the coprime array, calculated by equations (24) and (25).

[0105] Step 5: Convert the calculated phase difference into angles and represent it as a set:

[0106]

[0107] Substituting this set into the construct of the joint optimal estimate:

[0108]

[0109] Obtain the set of unambiguous optimal estimated angles .in The adaptive weighting coefficients for joint estimation are expressed as follows:

[0110]

[0111] gather It is obtained by calculating the polynomial equation when the polynomial function (34) is equal to 0. There are 1 characteristic roots. This is an intermediate parameter used to simplify the formula expression; its specific calculation formula is given by (43):

[0112]

[0113] in Indicates taking the conjugate. This indicates taking the real part.

[0114] Step 6: Set The final estimation result obtained according to this scheme. Output set. This completes the unambiguous, search-free DoA estimation proposed in this invention.

[0115] To illustrate the implementation effects of the embodiments of the present invention, several comparative experiments were established. These experiments employed arrays with array elements of the numbers described in the examples above. and Two uniform linear arrays form a coprime array, receiving signals from... Three uncorrelated signal sources incident at three different angles. Number of repetitions. Compared with the proposed solution of this invention, the traditional DECOM scheme, the Root-MUSIC scheme used for a half-wavelength uniform linear array with the same number of array elements, and the estimated root mean square error (RMSE) for the array in four cases of CRLB. Figure 5 This demonstrates how the RMSE curve changes with the input signal-to-noise ratio (SNR) (calculating the number of snapshots in the covariance matrix). ). Figure 6 The above RMSE curve is shown as the number of snapshots used to calculate the covariance matrix. The variation (input signal-to-noise ratio SNR=10dB).

[0116] Similarly, this is to demonstrate the low computational complexity of the embodiments of the present invention. Figure 7 The curves showing the computational complexity of complex multiplication involved in the present invention and the traditional DECOM algorithm as a function of the total number of elements in the coprime array are illustrated. The changes.

[0117] As can be seen from the simulations above, this invention, based on a coprime array, achieves significantly better estimation performance than traditional ULA arrays. Furthermore, the estimation accuracy and computational complexity of this invention are significantly improved compared to traditional methods.

[0118] The specific examples described in this invention are merely illustrative and are not intended to limit the scope of this application. Those skilled in the art can make various modifications, additions, or similar substitutions to the described specific examples without departing from the invention or exceeding the scope defined by the appended claims.

Claims

1. A fuzz-free, search-free method for estimating the DoA of coprime arrays, characterized in that: Includes the following steps, Step 1: Construct a coprime array consisting of two uniform linear arrays; the two uniform linear arrays respectively possess and The sensors correspond to different array element spacings. , ;in For the signal wavelength, and They are coprime numbers; Let the relationship parameters of the two uniform linear arrays be... ;definition Indicates the first The uniform linear array in the first The ambiguity coefficients in the angular domain are defined. Indicates the first Each angle domain has a preset distance; when season , ; calculate according to formula (1) Positive critical angles corresponding to the angle domain The uniform linear array number corresponding to this critical angle ; If the calculated critical angle does not meet the requirements Then let: in Then, the critical angle at this time is calculated again according to equation (1). If not satisfied If the requirements are met, then the calculation steps of equations (2)-(5) are repeated again; until this point... Corresponding critical angle satisfy The requirements, and the The number of time-divided angle domains is recorded as follows ; for In this case, directly order: in This yields a set of parameters describing the relationship between two uniform linear arrays in a coprime array. ; Step 2: Receive using the coprime array constructed in Step 1 Several uncorrelated far-field narrowband signals; each signal's incident direction Composition of angle set At this time, the middle of the coprime array A uniform linear array Sampling data at time Represented as: in Corresponding to two uniform linear arrays, Represents independent source signal vectors. Indicates the first The sampled waveform of a signal. It is the first Gaussian white noise vector on a uniform linear array It is the coprime array of the first The direction matrix of a uniform linear array Indicates the first The signal corresponds to the first Direction vectors on a uniform linear array; Step 3: Calculate based on the sampling data model established in Step 2. The covariance matrix of each of the two uniform linear arrays was captured by a sampling snapshot. And perform eigenvalue decomposition on the covariance matrix: The results of the eigenvalue decomposition are sorted in descending order of eigenvalues. and Corresponding to the first in the sorting feature values The diagonal matrix formed by the eigenvalues ​​and the subsequent... A diagonal matrix composed of eigenvalues, and and They are and The corresponding feature vector; Definition of the first In a uniform linear array about polynomial function ;wherein the polynomial coefficients intermediate parameters ;calculate The closest to the unit circle A complex root Then, calculate the first according to formula (10). The estimation of the i-th uniform linear array yields the following result for the i-th... The ambiguous spatial phase difference of the incident signals: in This indicates that the phase angle is extracted. Indicates calculation The remainder; then the fuzzy spatial phase difference set of the two uniform linear arrays can be calculated: and ; Step 4: Calculate the Cartesian product of the two sets of fuzzy spatial phase differences obtained in Step 3: Calculate the set according to equation (12) The points corresponding to the Cartesian products in the equation are all points on the two-dimensional plane that pass through the origin and have a slope of . The straight-line distance; Eliminating illegal values ​​using the optimization problem shown in equation (13): Threshold Set to: Based on the solution set of equation (13), calculate the estimated phase difference for the unambiguous equation: in Indicates the first The signal at the first The unambiguous estimates are corrected on a uniform linear array; then the unambiguous phase difference matching set is obtained: Step 5: Based on the matching set obtained in Step 4 and the relationship between the signal phase difference and the incident angle given by Equation (10), obtain the unambiguous angle matching set: The joint final estimate of the phase difference of the coprime array with respect to the input signal is calculated according to equation (18): This leads to the angle set. ;in The adaptive weighting coefficients for joint estimation are expressed as follows: This is an intermediate parameter used to simplify the formula expression, and its calculation formula is given by (20): in Indicates taking the conjugate. Indicates taking the real part; Step 6: Calculate the angle set obtained in Step 5. As the final estimation result output, the final estimation result is the unambiguous DoA estimation result obtained.

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