Multi-path high-order QAM (Quadrature Amplitude Modulation) signal direct positioning method based on dual atom norm minimization

Abstract

The invention discloses a multipath high-order QAM (Quadrature Amplitude Modulation) signal direct positioning method based on dual atom norm minimization, which comprises the following steps of: firstly, constructing a distributed multi-sensor array positioning model in a multipath environment to obtain a received signal; secondly, calculating a simplified fourth-order cumulant of a received signal of each sensor array, and expanding the simplified fourth-order cumulant to obtain a virtual signal; then, respectively constructing positive semi-definite constraint problems, and solving to obtain dual vectors; and finally, combining all dual vectors to construct a cost function to directly search the position of the radiation source so as to obtain a position estimation value. The multipath high-order QAM signal direct positioning method based on dual atom norm minimization designed by the invention can effectively position a radiation source. In addition, the method fully utilizes the characteristics of the high-order QAM signal, the array aperture can be expanded, and more radiation sources can be estimated. The positioning performance of the method is better than that of a traditional spatial smoothing subspace data fusion direct positioning method and a discrete Fourier transform direct positioning method.

Description

Technical Field

[0001] The present invention relates to the technical field of wireless positioning, and particularly to a direct positioning method for multipath high-order QAM signals based on dual atomic norm minimization. Background Art

[0002] In modern communication systems, QAM (Quadrature Amplitude Modulation) signals play a crucial role. With the rapid development of communication technologies, the requirements for data transmission rate and spectral efficiency are becoming increasingly stringent. QAM, with its excellent ability to achieve high-speed data transmission within a limited bandwidth, has become the preferred modulation method for many communication standards. By modulating both the amplitude and phase of the carrier simultaneously, QAM can transmit multiple bits of information within a symbol period. For example, in high-order QAM modulations such as 64-QAM and 256-QAM, each symbol can carry 6 bits, 8 bits or even more information, greatly improving the spectral utilization rate. This characteristic enables QAM to be widely used in high-speed wireless communication systems such as 5G mobile communication and Wi-Fi.

[0003] Radio positioning technology plays a crucial role in many fields such as modern communication, navigation, monitoring, and security. Traditional radio positioning methods mainly include indirect positioning technologies based on ranging, angle of arrival (AoA), time of arrival (ToA), etc. The implementation of these technologies relies on the accurate measurement of signal propagation characteristics and subsequent calculation processes. However, in complex environments with significant multipath effects, signals will undergo reflection, scattering, and diffraction, etc., which significantly degrade the performance of the above indirect positioning methods and make it difficult to reliably guarantee the positioning accuracy.

[0004] To effectively address the challenges brought by multipath effects, direct positioning technology has attracted much attention in recent years. This technology directly extracts the signal source location information from the received signal without deriving the signal propagation characteristics through intermediate links. However, the existing direct positioning methods currently have many problems when dealing with multipath environments: on the one hand, these methods restore the target signal at the cost of sacrificing the array aperture, resulting in a reduction in the number of radiating sources that can be estimated; on the other hand, these methods often have difficulty fully exploiting the useful information in multipath high-order QAM signals, thus limiting the improvement of positioning accuracy.

[0005] Dual atomic norm minimization technology has shown great potential in the field of signal processing. This technology can efficiently utilize the sparsity and structured characteristics of signals to restore the target signal from the observed data, significantly enhancing the robustness and accuracy of signal processing. However, how to effectively integrate the dual atomic norm minimization technology in a multipath environment remains a difficulty and challenge in the current technological development process.

[0006] In view of this, the present invention proposes a direct positioning method for multipath high-order QAM signals based on dual atomic norm minimization, aiming to solve the above technical problems and improve the positioning accuracy and reliability of high-order QAM signals in a multipath environment. Summary of the Invention

[0007] The technical problem to be solved by the present invention is to provide a direct positioning method for multipath high-order QAM signals based on dual atomic norm minimization for the defects involved in the background technology.

[0008] The present invention adopts the following technical solutions to solve the above technical problems:

[0009] A direct positioning method for multipath high-order QAM signals based on dual atomic norm minimization, characterized by including the following steps:

[0010] Step 1), construct a distributed multi-sensor array positioning model in a multipath environment to obtain the received signal r l (t);

[0011] The distributed multi-sensor array positioning model is used to locate K far-field QAM radiation sources with unknown positions, and includes L sensor arrays with known positions. Each sensor array is equipped with an M-element uniform linear array arranged horizontally along the x-axis with a spacing of d; the centroid position vector of the l-th sensor array is expressed as representing its abscissa and ordinate respectively;

[0012] The position vector of the k-th radiation source is expressed as representing the abscissa of the k-th radiation source position, representing the ordinate of the k-th radiation source position; let the k-th radiation source reach the l-th sensor array after multiple reflections, including a line-of-sight path and V l,k different non-line-of-sight paths; the arrival angle of the line-of-sight path from the k-th radiation source to the l-th sensor array is expressed as wherein, when the logical expression is true, it takes the value of 1, otherwise it takes the value of 0, arctan(·) ∈ (-π / 2, π / 2), to eliminate the ambiguity of arctan(y / x) = arctan(-y / (-x)); the arrival angle of the v-th non-line-of-sight path from the k-th radiation source to the l-th sensor array is expressed as θ l,k,v , v ∈ <1, V l,k >, <1, V l,k > represents the set of integers greater than or equal to 1 and less than or equal to V l,k ;

[0013] The received signal of the l-th sensor array at time t is t ∈ <1, T>, where T is the number of snapshots; s k (t) represents the QAM signal with non-zero kurtosis emitted by the k-th radiation source at time t; n l (t) is the additive white Gaussian noise vector of the l-th sensor array, and its covariance is I M represents the M-dimensional identity matrix, represents the noise power; α l,k,v represents the fading coefficient of the v-th non-line-of-sight path from the k-th radiation source to the l-th sensor array; for a given θ, the i-th element of the steering vector a(θ) is where λ represents the signal carrier wavelength and j is the imaginary unit;

[0014] Step 2), calculate the reduced fourth-order cumulant z l of the received signal of each sensor array, and expand it to obtain the virtual signal

[0015] The reduced fourth-order cumulant of the received signal of the l-th sensor array is expressed as

[0016]

[0017] where, represents the mathematical expectation, e i represents the vector with the i-th element being 1 and the remaining elements being 0, n l 、r l 、s k are the reduced forms of n l (t), r l (t), s k (t) respectively;

[0018] Then expand it to obtain the virtual signal where:

[0019] The diagonal matrix Λ = diag{1, 1 / 2, …, 1 / M, 1 / (M - 1), …, 1};

[0020] The matrix 0 M×(M-1) represents the M×(M - 1)-dimensional zero matrix;

[0021] vec{·} represents the vectorization operator, Toep(z l ) represents constructing a Toeplitz matrix according to the vector z l ;

[0022] The Vandermonde matrix Bl The columns of are composed of 2M - 1 dimensional vectors

[0023] is the virtual transmitted signal;

[0024] Step 3), construct the semidefinite constraint problem respectively, and solve to obtain the dual vector q l ;

[0025] Step 3.1), use the atomic norm to sparsely represent Define an atomic set Then the arrival angle estimation of the radiation source is expressed as the following atomic norm minimization problem: where represents the overcomplete dictionary composed of the atomic set and represents the sparse vector;

[0026] Step 3.2), the dual problem of this atomic norm minimization problem is: where q l represents the dual vector, the superscript H represents the conjugate transpose, and ‖·‖ ∞ represents the infinity norm;

[0027] Step 3.3), represent the dual problem of this atomic norm minimization problem as the following semidefinite programming:

[0028] where H l is the Hermitian matrix;

[0029] Step 3.4), use the convex optimization toolbox to solve the semidefinite programming to obtain the dual vector q l ;

[0030] Step 4), jointly construct a cost function for all dual vectors, and search to obtain the radiation source position estimation;

[0031] Step 4.1), use the dual vector q l to construct a cost function where p represents the position vector, and θ l (p) represents the arrival angle of the line-of-sight path from p to the l-th sensor array, |·| represents the modulus of a complex number;

[0032] Step 4.2), perform a spectral peak search on this cost function, and the coordinates corresponding to the peak are the estimation results of the radiation source position.

[0033] The present invention adopts the above technical solution, and compared with the prior art, has the following technical effects:

[0034] The present invention can make full use of the sparsity and structural characteristics of signals, better recover the target signal from the observed data, expand the array aperture, suppress the multipath effect, increase the available degrees of freedom, and its position estimation accuracy is better than that of the traditional SSP-SDF direct positioning method and the DFT direct positioning method, and no additional parameter pairing process is required. Brief Description of the Drawings

[0035] Figure 1 is a flowchart of the present invention;

[0036] Figure 2 is a scene diagram of joint positioning of a distributed multi-sensor array in a multipath environment;

[0037] Figure 3 is a schematic diagram of the available degrees of freedom of the present invention and traditional positioning methods under different numbers of sensors;

[0038] Figure 4 is a schematic diagram of the root mean square error performance of the present invention and traditional positioning methods under different signal-to-noise ratios;

[0039] Figure 5 is a schematic diagram of the root mean square error performance of the present invention and traditional positioning methods under different numbers of snapshots. Detailed Embodiment

[0040] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings:

[0041] The present invention can be implemented in many different forms and should not be considered limited to the embodiments described herein. On the contrary, these embodiments are provided so that this disclosure is thorough and complete, and will fully convey the scope of the present invention to those skilled in the art. In the drawings, components are enlarged for clarity.

[0042] As Figure 1 shown, the present invention discloses a direct positioning method for multipath high-order QAM signals based on dual atomic norm minimization, which specifically includes the following steps:

[0043] Step 1), construct a positioning model for a distributed multi-sensor array in a multipath environment to obtain the received signal r l (t).

[0044] As Figure 2 shown, the distributed multi-sensor array positioning model is used to locate K far-field QAM radiation sources with unknown positions, and includes L sensor arrays with known positions. Each sensor array is equipped with an M-element uniform linear array arranged horizontally along the x-axis with a spacing of d. The position vector of the kth radiation source is expressed as where the superscript T represents the matrix transpose, denotes the abscissa of the position of the k-th radiation source, denotes the ordinate of the position of the k-th radiation source, and the vector of the center position of the l-th sensor array is expressed as and denote its abscissa and ordinate respectively. We assume that the k-th radiation source reaches the l-th sensor array after multiple reflections, including a Line-of-Sight (LoS) path and V l,k different Non-Line-of-Sight (NLoS) paths. The Angle-of-Arrival (AoA) of the LoS from the k-th radiation source to the l-th sensor array is expressed as

[0045]

[0046] where the logical expression takes the value of 1 when it is true and 0 otherwise, arctan(·) ∈ (-π / 2, π / 2), which is used to eliminate the ambiguity of arctan(y / x) = arctan(-y / (-x)).

[0047] The AoA of the v-th NLoS from the k-th radiation source to the l-th sensor array is expressed as θ l,k,v , v ∈ <1, V l,k >, where <1, V l,k > represents the set of integers greater than or equal to 1 and less than or equal to V l,k .

[0048] The received signal of the l-th sensor array at time t (t ∈ <1, T>, T is the number of snapshots) is

[0049]

[0050] where s k (t) represents the QAM signal with non-zero kurtosis transmitted by the k-th radiation source at time t, n l (t) is the additive Gaussian white noise vector of the l-th sensor array, and its covariance is I M represents the M-dimensional identity matrix, represents the noise power, α l,k,v represents the fading coefficient of the v-th non-line-of-sight path from the k-th radiation source to the l-th sensor array. For a given θ, the i-th element of the steering vector a(θ) has the following form:

[0051] where λ represents the signal carrier wavelength and j is the imaginary unit.

[0052] Step 2), calculate the reduced fourth-order cumulant z of the received signals of each sensor array l , and expand it to obtain a virtual signal

[0053] The reduced fourth-order cumulant of the received signal of the l-th sensor array can be expressed as

[0054]

[0055] where represents the mathematical expectation, e i represents a vector with the i-th element being 1 and the remaining elements being 0, n l , r l , s k are the reduced forms of n l (t(, r l (t), s k (t), respectively.

[0056] Then expand it to obtain a virtual signal where the diagonal matrix Λ = diag{1, 1 / 2, …, 1 / M, 1 / (M - 1), …, 1}, the matrix

[0057] 0 M×(M-1) represents an M×(M - 1) dimensional zero matrix, the superscript T represents the matrix transpose, vec{·} represents the vectorization operator, Toep(z l ) represents constructing a Toeplitz matrix according to the vector z l , the Vandermonde matrix B l has columns composed of 2M - 1 dimensional vectors For a given θ, the i-th element of b(θ) has the following form: can be regarded as the virtual transmitted signal.

[0058] Step 3), respectively construct a semi-definite constraint problem and solve to obtain the dual vector q l .

[0059] The virtual received signal obtained in Step 2 can be sparsely represented by the atomic norm. Define an atomic set Then the AoA estimation of the radiation source can be expressed as the following atomic norm minimization problem:

[0060]

[0061] wherein represents an overcomplete dictionary composed of an atomic set and represents a sparse vector.

[0062] The dual problem of the atomic norm minimization problem is:

[0063]

[0064] where q l represents the dual vector, the superscript H represents the conjugate transpose, and ‖·‖ ∞ represents the infinity norm.

[0065] The dual problem of the atomic norm minimization problem can be expressed as the following semidefinite programming:

[0066] where H l is a Hermitian matrix;

[0067] The above semidefinite programming can be solved using a convex optimization toolbox to obtain the dual vector q l .

[0068] Step 4), construct a cost function by combining all dual vectors, and search to obtain the radiation source position estimate.

[0069] Using the dual vector obtained in Step 3, construct the following cost function:

[0070]

[0071] where p represents the position vector, and θ l (p) represents the AoA of the LoS from p to the l-th sensor array, |·| represents the modulus of a complex number.

[0072] Perform a spectral peak search on this cost function, and the coordinates corresponding to the peak are the estimated result of the radiation source position.

[0073] In the case of the same number of array elements, the spatial degrees of freedom of the traditional SSP-SDF direct positioning method are less than M - 1, and generally, in order to ensure performance, the subarray length is taken as M / 2. In the case of the same number of array elements, the spatial degrees of freedom of the DFT direct positioning method are M - 1, and the spatial degrees of freedom obtained by the method of the present invention are 2M - 2, increasing the available degrees of freedom. Figure 3 is a schematic diagram of the available degrees of freedom of the method of the present invention and the traditional positioning method varying with the number of sensors. The simulation sets the spatial smoothing subarray length to M / 2. From Figure 3It can be seen that the method described in the present invention has more available degrees of freedom than the traditional SSP-SDF direct positioning method and the DFT direct positioning method.

[0074] The performance estimation criterion of the present invention is the root mean square error (RMSE), which is defined as:

[0075]

[0076] where M c is the number of Monte Carlo experiments, represents the estimated value of the position of the k-th radiation source in the i-th experiment, p k represents the actual value of the position of the k-th radiation source, and ‖·‖ represents the 2-norm. The signal-to-noise ratio (SNR) is defined as

[0077]

[0078] Figure 4 This is the performance curve graph of the root mean square error of the method described in the present invention, the traditional SSP-SDF direct positioning method, and the DFT direct positioning method varying with the signal-to-noise ratio. The simulation conditions are as follows: the positions of 2 radiation sources are [(50 m, 850 m), (840 m, 380 m)] respectively, and the positions of 4 sensor arrays are [(-870 m, -1170 m), (-270 m, -1070 m), (330 m, -970 m), (930 m, -870 m)] respectively. Each sensor array is equipped with a uniform linear array with 14 array elements. There is one LoS and 2 NLoS for each target arriving at the sensor array. The AoAs of the NLoS are θ 1,1,1 = 0.1865 radians, θ 1,1,2 = 0.6823 radians, θ 1,2,1 = -0.1691 radians, θ 1,2,2 = -0.5078 radians, θ 2,1,1 = 0.4069 radians, θ 2,1,2 = -0.0876 radians, θ 2,2,1 = -0.9305 radians, θ 2,2,2 = -0.4366 radians, θ 3,1,1 = 0.2140 radians, θ 3,1,2 = 0.9440 radians, θ 3,2,1 = -0.3537 radians, θ 3,2,2 = 0.0713 radians, θ 4,1,1 = 0.8067 radians, θ 4,1,2 = 0.1926 radians, θ 4,2,1 = 0.4577 radians, θ 4,2,2= -1.0653 radians, and the complex fading coefficients are |α 1,1,1 | = 0.85, |α 1,1,2 | = 0.76, |α 1,2,1 | = 0.75, |α 1,2,2 | = 0.66, |α 2,1,1 | = 0.86, |α 2,1,2 | = 0.89, |α 2,2,1 | = 0.76, |α 2,2,2 | = 0.79, |α 3,1,1 | = 0.84, |α 3,1,2 | = 0.89, |α 3,2,1 | = 0.74, |α 3,2,2 | = 0.79, |α 4,1,1 | = 0.74, |α 4,1,2 | = 0.82, |α 4,2,1 | = 0.64, |α 4,2,2 | = 0.72, the number of snapshots is 8000, and the global search range y ∈ {-1200 m, 1200 m}, the search step size is 1 m, the simulation is performed 500 times. In the comparison algorithm SSP-SDF direct positioning, the length of the spatial smoothing subarray is set to 7, the number of subarrays is 8, the radiation source signal is 256-QAM, the sampling rate is 10 kHz, and the symbol rate is 1 kHz. From Figure 4 It can be seen that the positioning accuracy of the present invention is always better than that of the traditional SSP-SDF direct positioning method and the DFT direct positioning method.

[0079] Figure 5 This is the performance curve graph of the root mean square error of the method of the present invention, the traditional SSP-SDF direct positioning method, and the DFT direct positioning method varying with the number of snapshots. The simulation conditions are set the same as Figure 4 and the signal-to-noise ratio is -8 dB. From Figure 5 It can be seen that the position estimation performance of the method proposed by the present invention is always better than that of the traditional SSP-SDF direct positioning method and the DFT direct positioning method.

[0080] In summary, from the analysis of the simulation effect diagram, it can be known that a direct positioning method for multipath high-order QAM signals based on dual atomic norm minimization proposed by the present invention can effectively locate the target. In addition, this method can make full use of the sparsity and structured characteristics of the signal, recover the target signal from the observation data without losing the array aperture, suppress the multipath effect, increase the available degrees of freedom, improve the positioning accuracy, and there is no need for an additional parameter pairing process. Its positioning accuracy is better than that of the traditional SSP-SDF direct positioning method and the DFT direct positioning method.

[0081] Those skilled in the art can understand that, unless otherwise defined, all terms used herein (including technical and scientific terms) have the same meaning as the general understanding of those of ordinary skill in the art to which the present invention pertains. It should also be understood that terms such as those defined in a general dictionary should be understood to have a meaning consistent with their meaning in the context of the prior art, and will not be interpreted in an idealized or overly formal sense unless defined as such here.

[0082] The specific embodiments described above further elaborate on the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above are only specific embodiments of the present invention and are not used to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A direct positioning method for multipath high-order QAM signals based on dual atomic norm minimization, characterized in that: The following steps are involved: Step 1) construct a distributed multi-sensor array positioning model in a multipath environment and obtain the received signal r l (t); The distributed multi-sensor array positioning model is used to locate K far-field QAM radiation sources with unknown positions, including L sensor arrays with known positions, each of which is equipped with an M-element uniform linear array arranged horizontally along the x-axis with a spacing of d; the center position vector of the l-th sensor array is expressed as Represent its horizontal and vertical coordinates respectively; The position vector of the kth radiation source is expressed as represents the horizontal coordinate of the kth radiation source position, Represents the ordinate of the kth radiation source position; let the kth radiation source reach the lth sensor array after multiple reflections, including a line-of-sight path and V l,k different non-line-of-sight paths; the arrival angle of the line-of-sight path from the kth radiation source to the lth sensor array is expressed as Among them, the logical expression The value is 1 when it is true, otherwise it is 0, arctan(·)∈(-π / 2,π / 2), To eliminate the ambiguity of arctan(y / x)=arctan(-y / (-x)); the arrival angle of the vth non-line-of-sight path from the kth radiation source to the lth sensor array is denoted by θ l,k,v ,v∈<1,V l,k >,<1,V l,k > indicates greater than or equal to 1 and less than or equal to V l,k The set of integers ; The received signal of the lth sensor array at time t is T is the number of snapshots; s k (t) represents the QAM signal with non-zero kurtosis emitted by the kth radiation source at time t; n l (t) is the additive white Gaussian noise vector of the lth sensor array, and its covariance is I M represents the M-dimensional identity matrix, represents the noise power; α l,k,v represents the fading coefficient of the vth non-line-of-sight path from the kth radiation source to the lth sensor array; the steering vector a(θ) has the ith element Where λ represents the signal carrier wavelength, and j is an imaginary unit; Step 2), calculate the simplified fourth-order cumulant z of the received signal of each sensor array l , and expand it to get the virtual signal The simplified fourth-order cumulant of the received signal of the lth sensor array is expressed as in, represents the mathematical expectation, e i represents a vector whose i-th element is 1 and the rest are 0, n l 、r l 、s k They are n l (t), r l (t), s k (t) in simplified form; Then expand it to get the virtual signal in: Diagonal matrix Λ=diag{1,1 / 2,…,1 / M,1 / (M-1),…,1}; matrix 0 M×(M-1) represents an M×(M-1)-dimensional zero matrix; vec{·} represents the vectorization operator, Toep(z l ) means that according to the vector z l Construct a Toeplitz matrix; Vandermonde matrix B l The columns consist of 2M-1 dimensional vectors Composition; for a given θ, the i-th element of b(θ) is It is a virtual transmission signal; Step 3), construct the semi-positive definite constraint problem respectively and solve it to obtain the dual vector q l ; Step 3.1), use atomic norm to sparsely represent Define an atomic set The arrival angle estimation of the radiation source is then expressed as the following atomic norm minimization problem: in, Represented by a collection of atoms An overcomplete dictionary, represents a sparse vector; Step 3.2), the dual problem of the atomic norm minimization problem is: Among them, q l represents the dual vector, the superscript H represents the conjugate transpose, ‖·‖ ∞ represents the infinite norm; Step 3.3), the dual problem of the atomic norm minimization problem is expressed as the following semi-positive definite programming: Among them, H l is a Hermitian matrix; Step 3.4), use the convex optimization toolbox to solve the semi-positive definite programming and obtain the dual vector q l ; Step 4), combine all dual vectors to construct a cost function and search to obtain the radiation source position estimate; Step 4.1), using the dual vector q l Constructing cost function Where p represents the position vector, θ l (p) represents the arrival angle of the line-of-sight path from p to the lth sensor array, |·| represents the modulus of a complex number; Step 4.2), perform a spectrum peak search on the cost function, and obtain the coordinates corresponding to the peak value, which is the estimated result of the radiation source position.

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