A non-circular signal direction finding method using a time-modulated array

Abstract

The present invention relates to a method for direction finding of non-circular signals using a time-modulated array, comprising: establishing a signal model for a time-modulated array receiving signal under non-circular signals; addressing the problem of non-uniform noise caused by time modulation by theoretically analyzing the time-modulated sequence and performing pre-whitening processing on the noise; combining the signal covariance matrix and the signal conjugate covariance matrix in the multidimensional domain of a dual quaternion, and constructing a direction finding fitness function using the orthogonality of the signal subspace; numerically analyzing the direction finding fitness function to decouple the signal direction parameter from the non-circular phase parameter, and constructing a dual quaternion-based non-circular signal DOA estimator using a time-modulated array under the guidance of the theoretical lower bound of the direction finding fitness function obtained by the analysis. The present invention solves the problem of direction finding of non-circular signals using a time-modulated array by increasing the array direction finding degrees of freedom and virtual array aperture based on the dual quaternion, thereby achieving improved direction finding accuracy and resolution.

Description

Technical Field

[0001] The present invention relates to a signal direction of arrival (DOA) estimation technology in the field of signal processing, and in particular to a time-modulated array non-circular signal direction finding method. Background Art

[0002] Array signal processing is an important branch of signal processing. As a major research area within array signal processing, signal direction of arrival estimation (also known as direction finding) is widely used in numerous fields, including communications, sonar, radar, exploration, and radio astronomy. Existing direction-finding methods face a significant trade-off between system implementation cost and performance. To achieve a better balance between performance and implementation cost, time-modulated arrays have been introduced into direction-finding systems to address this dilemma.

[0003] Currently, most research on direction finding using time-modulated arrays is based on the assumption that the incident signal is a complex circular signal. In actual mobile communication scenarios, many non-circular signals exist, such as amplitude modulation (AM), pulse amplitude modulation (PAM), and binary phase-shift keying (BPSK). By considering the characteristics of non-circular signals in the direction-finding system, the array aperture can be effectively expanded, thereby achieving better resolution. In recent years, researchers have proposed methods for implementing direction finding using the characteristics of non-circular signals within a quaternion or dual-quaternion framework. These methods leverage the stronger orthogonality between data in the hypercomplex domain (quaternions, dual-quaternions), effectively improving the accuracy and robustness of direction finding. Due to changes in the array structure, existing algorithms for direction finding using non-circular signals are no longer applicable to time-modulated arrays. Therefore, designing a non-circular signal direction finding method for time-modulated arrays has become a highly worthy research question. Summary of the Invention

[0004] Technical problems to be solved

[0005] To address the problem of existing direction-finding algorithms failing to effectively perform direction-finding in time-modulated arrays when the number of signal sources to be estimated exceeds the number of array elements, the present invention provides a non-circular signal direction-finding method for time-modulated arrays. This method achieves direction-finding capabilities when the number of signal sources exceeds the number of array elements, and significantly improves direction-finding accuracy and resolution compared to existing algorithms.

[0006] Technical Solution

[0007] A time-modulated array non-circular signal direction finding method, characterized by the following steps:

[0008] Step 1: Establish a time-modulated array receiving signal model under non-circular signals;

[0009] Step 2: To address the non-uniform noise problem caused by time modulation, pre-whiten the noise through theoretical analysis of the time modulation sequence;

[0010] Step 3: Combine the signal covariance matrix and the signal conjugate covariance matrix in the multidimensional domain of dual quaternion, and construct the direction finding fitness function using the orthogonality of the signal subspace;

[0011] Step 4: Numerical analysis is performed on the direction-finding fitness function to decouple the signal direction parameter from the non-circular phase parameter. Guided by the theoretical lower bound of the direction-finding fitness function obtained from the analysis, a dual quaternion-based non-circular signal DOA estimator using a time-modulated array is constructed.

[0012] A further technical solution of the present invention is as follows: The process of establishing the time modulation array receiving signal model under the non-circular signal described in step 1 is as follows:

[0013] Assume that a far-field uncorrelated narrowband non-circular signal with the same carrier frequency f0 is incident on a direction finding system based on a time-modulated array. Due to the influence of high-speed RF switching, the signal passing through the single-pole multi-throw switch can be written as

[0014]

[0015] where s k (t) represents the time from θ k The kth signal incident from the direction, β and d represent the wave number and array element spacing respectively, n m (t) is the variance σ 2 Zero-mean circular Gaussian white noise, U m (t) is the time modulation function on the mth array element, which can be expressed as

[0016]

[0017] Because U m (t) is time-periodic and satisfies the Dirichlet condition, so it can be decomposed into a Fourier series, given by where α m,q represents the Fourier coefficient, expressed as

[0018]

[0019] Where sinc(x)=sin(πx) / (πx);

[0020] Therefore, each incident signal will generate an infinite sideband signal whose center frequency is located at f0±qf after time modulation. p, q=1,2,...,+∞. Since it is impossible to use infinite sidebands in practice, we can only use the carrier frequency and ±Q sidebands. Given the complex Fourier coefficients and the expression of the modulated signal, the qth order (q=-Q,...,0,...,+Q) sideband signal can be expressed as

[0021]

[0022] Temporal modulation frequency f p Should satisfy f p The constraint of ≥2B is used to avoid frequency aliasing, where B refers to the signal bandwidth. Then, different sideband signals can be easily separated by digital bandpass filters. After passing through a single-channel receiver, all sideband signals are down-converted and sampled by the ADC relative to time t, which can be written as

[0023]

[0024] For convenience, the above expression can be rewritten in matrix format and given by

[0025] Y=B T [AS+N]

[0026] in(·) T represents the transposition operator, A=[a(θ1),a(θ2),...,a(θ K )] is the array manifold matrix, where k=1,...,K represents the steering vector of the incident signal from different directions; Y=[y -Q ,y -Q+1 ,...,y Q ] T is the received signal matrix signal composed of each sideband, S=[s1,s2,...,s K ] T Represents the incident signals from different sources, N=[n1,n2,...,n M ] T is the noise covariance matrix; By the Fourier coefficient α m,q Composition, called time modulation matrix;

[0027] Since the incident signal is assumed to be an arbitrary non-circular signal, we have

[0028]

[0029] where E{·} is the expectation operator, R S and denote the source covariance and source conjugate covariance matrices respectively; where ρ = diag{ρ1,ρ2,…,ρ K},0≤ρ k ≤1, diag{·} represents a diagonal matrix; ρ k is the kth non-circularity of the non-cyclic signal, is the kth non-circular phase caused by the communication channel; Due to the above non-circular characteristics, the covariance matrix and conjugate covariance matrix of the received signal of this direction finding system are given as follows R Y =E{YY H}=B T AR S (B T A) H +σ 2 B T (B T ) H

[0030]

[0031] A further technical solution of the present invention: The non-uniform noise pre-whitening process described in step 2 is as follows:

[0032] The noise covariance matrix after time modulation is no longer the identity matrix before time modulation, so Obviously M H =M; element β in row a and column b of M a,b It can be calculated as follows, where a, b = 1, 2, ..., 2Q + 1;

[0033]

[0034] Where sinc(x,y) means sinc(x,y)=[sin(πx)· sin(πy)] / (πx·πy). It is easy to find that only the term Related to the antenna index m, m=1,2,...,M; let k=ab, for the non-diagonal items of M, it is obvious that -2Q≤k≤2Q, k≠0; thus we can get the following relationship

[0035]

[0036] On the one hand, this shows that when 2Q+1≤M, the off-diagonal terms of M are equal to 0; on the other hand, the diagonal terms of M are expressed as β a,a , can be calculated using the following formula:

[0037]

[0038] Therefore, M is a diagonal matrix and can be written as

[0039] M=B T (BT ) H =diag{μ1,μ2,...,μ 2Q+1}

[0040] where μ a =β a,a , a=1,2,...,2Q+1; therefore, R N becomes R N =σ 2 ·diag{μ1,μ2,…,μ 2Q+1};

[0041] Correct the noise covariance matrix to the identity matrix through pre-whitening, and set C = M -1 / 2 is the non-uniform noise correction matrix, expressed as

[0042]

[0043] Obviously, C is a reversible diagonal matrix. The covariance matrix and conjugate covariance matrix of the received signal after non-uniform correction are expressed as follows:

[0044]

[0045]

[0046] where d k =CB T a(θ k ) is the steering vector after noise non-uniformity correction, corresponding to the kth incident signal, so that the noise is corrected to uniform noise.

[0047] A further technical solution of the present invention is as follows: Step 3 combines the signal covariance matrix and the signal conjugate covariance matrix in the multidimensional domain of the dual quaternion, which is expressed as follows

[0048]

[0049] set up Then R becomes the new quaternion combination covariance matrix. It is worth noting that Then it can be inferred

[0050]

[0051] Thus R can be rewritten as

[0052]

[0053] in is the dual quaternion steering vector;

[0054] The eigenvalue decomposition of the dual quaternion covariance matrix R is obtained by the following formula:

[0055]

[0056] in is the signal subspace, is the noise subspace, since and is the real eigenvalue of the diagonal matrix R consisting of the largest K real eigenvalues ​​and the smallest (2M-K);

[0057] By extending the subspace theory in the complex domain to the dual quaternion domain, it can be verified that the orthogonality between the signal subspace and the noise subspace still exists; then, due to the virtual steering vector The spanned space coincides with the signal subspace, so the corresponding lateral fitness function is constructed as follows

[0058]

[0059] in is a virtual dual quaternion steering vector corresponding to the possible signal incidence angles θ.

[0060] A further technical solution of the present invention is to perform numerical analysis on the direction finding fitness function in step 4, and guide the virtual dual quaternion vector Substitute the expression into the fitness function get

[0061]

[0062] in Obviously, μ ＊ (θ) = μ(θ); thus

[0063] μ(θ)=μ0(θ)+iIμ1(θ)+jIμ2(θ)+kIμ3(θ)

[0064] in Substitute the above formula into the fitness function get

[0065]

[0066] Fitness function Relative to non-circular phase variables Perform partial differential operation and set the deviation result to 0, and we get

[0067]

[0068]

[0069] in Using the above numerical relationship, the theoretical lower bound of the fitness function can be derived

[0070]

[0071] Therefore, the DOA of the incident signal can be obtained by solving the following formula:

[0072]

[0073] A computer system, characterized in that it includes: one or more processors, and a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the above-mentioned method.

[0074] A computer-readable storage medium is characterized by storing computer-executable instructions, which are used to implement the above method when executed.

[0075] Beneficial effects

[0076] The present invention provides a method for super-degree-of-freedom target direction finding using a non-circular signal using a dual quaternion time-modulated array. While greatly reducing the cost of implementing the direction-finding system by utilizing the time-modulated array, the method can achieve accurate direction finding when the number of signal sources is greater than the number of array elements, greatly improving the array's degrees of freedom and array virtual aperture, and significantly enhancing the direction-finding accuracy and resolution compared to existing algorithms. BRIEF DESCRIPTION OF THE DRAWINGS

[0077] The accompanying drawings are only for the purpose of illustrating particular embodiments and are not to be considered limiting of the present invention. Like reference symbols denote like parts throughout the drawings.

[0078] Figure 1 Schematic diagram of the structure of the single-channel time modulation array direction finding system designed for the present invention;

[0079] Figure 2 Schematic diagram of the principle of the method of the present invention;

[0080] Figure 3 This is the DOA estimation result of the underdetermined target with super degree of freedom of the present invention;

[0081] Figure 4 Schematic diagram of the relationship between the root mean square error and signal-to-noise ratio of DOA estimation in the present invention;

[0082] Figure 5 Schematic diagram of the relationship between the root mean square error of DOA estimation and the number of fast sampling rows in the present invention. DETAILED DESCRIPTION

[0083] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are only intended to illustrate the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0084] The non-circular signal direction finding system based on time modulation array constructed by the present invention is constructed as follows: Figure 1 As shown in the figure, the receiving antenna array of the system is a uniformly spaced time-modulated linear array with M elements and uniform static excitation. All array elements are connected and controlled by single-pole multi-throw (SPMT) switches, whose "on-off" time sequence is set by a field-programmable gate array (FPGA). A single-channel receiver is used to receive the signal passing through the SPMT switch and finally transmit the signal to a digital signal processor (DSP) for processing. The single-channel receiver consists of a low-noise amplifier (LNA), a mixer, a low-pass filter (LPF), an analog-to-digital converter (ADC), etc. The time modulation period T is set to p (time modulation frequency f p You can use f p =1 / T p Obtain) is evenly divided into M intervals, expressed as τ = T p / M. The SPMT switch connects to the mth array element in the mth time interval (m = 1, 2, ..., M). Therefore, the system only requires one processing channel, greatly reducing system complexity and implementation cost compared to traditional arrays.

[0085] In the present invention, integers, real numbers, complex numbers (with imaginary units ), quaternion and dual quaternion sets are respectively and express.

[0086] The dual quaternion a is represented as:

[0087] a=a0+ia1+ja2+ka3

[0088] =(a0) r +i(a1) r +j(a2) r +k(a3) r +I(a0) i +iI(a1) i +jI(a2) i +kI(a3) i

[0089] The definition of the "total conjugate" of a is:

[0090]

[0091] in(·) * Represents the general "complex conjugate" operation. Correspondingly, the dual quaternion matrix A can be defined as

[0092] A=A0+iA1+jA2+kA3

[0093] =(A0) r +i(A1) r +j(A2) r +k(A3) r +I(A0) i +iI(A1) i +jI(A2) i +kI(A3) i

[0094] Thus, the "total conjugate transpose" operation of the dual quaternion matrix A can be defined as

[0095]

[0096] in(·) H Represents the general "conjugate transpose" operation of a complex matrix.

[0097] like Figure 2 As shown, this embodiment provides a technology for realizing super-freedom underdetermined target direction finding by utilizing non-circular signal characteristics in a dual quaternion framework based on a time modulation array, which specifically includes the following processes:

[0098] Step 1: Establish a time-modulated array receiving signal model under non-circular signals:

[0099] Assume that a far-field uncorrelated narrowband non-circular signal with the same carrier frequency f0 is incident on a direction finding system based on a time-modulated array. Due to the influence of high-speed RF switching, the signal passing through the single-pole multi-throw switch can be written as

[0100]

[0101] where s k (t) represents the time from θ k The kth signal incident from the direction of the light. β and d represent the wave number (β = 2πf0 / c, c represents the speed of light) and the element spacing, respectively. m (t) is the variance σ 2 The zero-mean circular Gaussian white noise is statistically uncorrelated with the incident signal. m (t) is the time modulation function on the mth array element, which can be expressed as

[0102]

[0103] Because U m (t) is time-periodic and satisfies the Dirichlet condition, so it can be decomposed into a Fourier series, given by where α m,q represents the Fourier coefficient, expressed as

[0104]

[0105] Where sinc(x)=sin(πx) / (πx) (sinc(0) is defined as 1).

[0106] Therefore, each incident signal will generate an infinite sideband signal whose center frequency is located at f0±qf after time modulation. p (q=1,2,...,+∞). Since it is impossible to use infinite sidebands, we can only use the carrier frequency and ±Q sidebands. Given the complex Fourier coefficients and the expression of the modulated signal, the qth order (q=-Q,...,0,...,+Q) sideband signal can be expressed as

[0107]

[0108] Temporal modulation frequency f p Should satisfy f p The constraint of ≥2B is used to avoid frequency aliasing, where B refers to the signal bandwidth. The different sideband signals can then be easily separated by a digital bandpass filter. After passing through a single-channel receiver, all sideband signals are down-converted and sampled by the ADC relative to time t, which can be written as

[0109]

[0110] For convenience, the above expression can be rewritten in matrix format and given by

[0111] Y=B T [AS+N]

[0112] in(·) T Indicates the transposition operator. A=[a(θ1),a(θ2),...,a(θ K )] is the array manifold matrix, where k=1,...,K represents the steering vectors of incident signals from different directions. -Q ,y -Q+1 ,...,y Q ] T is the received signal matrix signal composed of each sideband, S=[s1,s2,...,s K ] TRepresents the incident signals from different sources, N=[n1,n2,...,n M ] T is the noise covariance matrix. By the Fourier coefficient α m,q The composition is called the time modulation matrix.

[0113] Since the incident signal is assumed to be an arbitrary non-circular signal, we have

[0114]

[0115] Where E{·} is the expectation operator. S and Denote the source covariance and source conjugate covariance matrix respectively. Where ρ=diag{ρ1,ρ2,…,ρ K},0≤ρ k ≤1, diag{·} represents a diagonal matrix. ρ k is the kth non-circularity of the non-cyclic signal, is the kth non-circular phase caused by the communication channel. Some special modulation signals with a non-circularity ratio ρ equal to 1 (such as BPSK, AM) are called linear signals, which are the focus of this invention. Due to the above non-circular characteristics, the covariance matrix and conjugate covariance matrix of the received signal of this direction finding system are given as follows:

[0116] R Y =E{YY H}=B T AR S (B T A) H +σ 2 B T (B T ) H

[0117]

[0118] Step 2: To address the non-uniform noise problem caused by time modulation, perform pre-whitening on the noise through theoretical analysis of the time modulation sequence:

[0119] The noise covariance matrix after time modulation is no longer the identity matrix before time modulation. Obviously M H = M. The element β in the ath row and bth column of M a,b (a,b=1,2,...,2Q+1) can be calculated as follows

[0120]

[0121] Where sinc(x,y) means sinc(x,y)=[sin(πx)·sin(πy)] / (πx·πy). It is easy to find that only the term Dependent on the antenna index m (m = 1, 2, ..., M). Let k = ab, for the non-diagonal terms of M, we have Thus, the following relationship can be obtained

[0122]

[0123] On the one hand, this shows that when 2Q+1≤M, the off-diagonal terms of M are equal to 0. On the other hand, the diagonal terms of M are expressed as β a,a , can be calculated using the following formula:

[0124]

[0125] Therefore, M is a diagonal matrix and can be written as

[0126] M=B T (B T ) H =diag{μ1,μ2,...,μ 2Q+1}

[0127] where μ a =β a,a (a=1,2,...,2Q+1). Therefore, R N becomes R N =σ 2 ·diag{μ1,μ2,...,μ 2Q+1}.

[0128] In summary, for the time series scheme adopted by the present invention, the noise after time modulation is still zero-mean Gaussian white noise, but the variance is non-uniform in different sidebands. Past related studies have shown that the presence of non-uniform noise will lead to a decrease in the performance of DOA estimation. In order to suppress the negative impact of non-uniform noise, we correct the noise covariance matrix to the unit matrix through pre-whitening. Let C = M -1 / 2 is the non-uniform noise correction matrix, expressed as

[0129]

[0130] Obviously, C is a reversible diagonal matrix. The covariance matrix and conjugate covariance matrix of the received signal after non-uniform correction are expressed as follows:

[0131]

[0132]

[0133] where dk =CB T a(θ k ) is the steering vector after noise non-uniformity correction, corresponding to the kth incident signal. Thus, the noise is corrected to uniform noise.

[0134] Step 3: Combine the signal covariance matrix and the signal conjugate covariance matrix in the multidimensional domain of dual quaternion, and construct the direction finding fitness function using the orthogonality of the signal subspace:

[0135] The signal covariance matrix and the pre-signal conjugate covariance matrix are combined in the multidimensional domain of dual quaternion, which is expressed as follows

[0136]

[0137] set up Then R becomes the new quaternion combination covariance matrix. It is worth noting that, Then it can be inferred

[0138]

[0139] Thus R can be rewritten as

[0140]

[0141] in is the dual quaternion steering vector.

[0142] The eigenvalue decomposition of the dual quaternion covariance matrix R is obtained by the following formula:

[0143]

[0144] in is the signal subspace, is the noise subspace. Since and It is the real eigenvalue of the diagonal matrix R consisting of the largest K real eigenvalues ​​and the smallest (2M-K).

[0145] By extending the subspace theory in the complex domain to the dual quaternion domain, we can verify that the orthogonality between the signal subspace and the noise subspace still exists. Then, since the virtual steering vector The spanned space coincides with the signal subspace, so the corresponding lateral fitness function is constructed as follows

[0146]

[0147] in is a virtual dual quaternion steering vector corresponding to the possible signal incidence angles θ.

[0148] Step 4: Numerical analysis is performed on the direction-finding fitness function to decouple the signal direction parameter from the non-circular phase parameter. Guided by the theoretical lower bound of the direction-finding fitness function obtained from the analysis, a dual quaternion-based non-circular signal DOA estimator using a time-modulated array is constructed:

[0149] Directing a virtual dual quaternion to a vector Substitute the expression into the fitness function get

[0150]

[0151] in Obviously, μ * (θ)=μ(θ). Thus

[0152] μ(θ)=μ0(θ)+iIμ1(θ)+jIμ2(θ)+kIμ3(θ)

[0153] in Substitute the above formula into the fitness function get

[0154]

[0155] Fitness function Relative to non-circular phase variables Perform partial differential operation and set the deviation result to 0, and we get

[0156]

[0157]

[0158] in Using the above numerical relationship, the theoretical lower bound of the fitness function can be derived

[0159]

[0160] Therefore, the DOA of the incident signal can be obtained by solving the following formula:

[0161]

[0162] The effectiveness of the present invention is verified by simulation experiments below.

[0163] In such Figure 1 The direction finding system framework shown adopts a time-modulated uniform linear array with M=5 elements (to ensure M=2Q+1, the corresponding maximum harmonic order is selected as Q=2). Figure 3The spatial spectrum performance of different direction-finding methods (based on time-modulated arrays) is presented for the cases of four non-circular signal inputs and eight non-circular signal inputs. The proposed method (TMAA-BN-MUSIC) leverages the non-circular nature of the signal to expand the array's degrees of freedom, enabling it to locate the direction of 2(M-1) signal sources, while other existing direction-finding methods based on time-modulated arrays can only achieve direction finding for (M-1) signal sources.

[0164] like Figure 4 and 5 As shown in the figure, simulations compare the DOA estimation accuracy of different methods under different signal-to-noise ratios (SNRs) and snapshot numbers. The simulations assume a time-modulated uniform linear array with M = 7 elements (to ensure M = 2Q + 1, the corresponding maximum harmonic order is selected as Q = 3), and three far-field narrowband uncorrelated BPSK signals are incident on the array from -9°, 0°, and 12°, respectively. It can be found that under different simulation conditions, the method proposed in this invention has significant performance advantages.

[0165] The above description is only a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with this technical field can easily think of various equivalent modifications or replacements within the technical scope disclosed in the present invention, and these modifications or replacements should all be included in the scope of protection of the present invention.

Claims

1. A time-modulated array non-circular signal direction finding method, characterized in that Here are the steps: Step 1: Establish a time-modulated array receiving signal model under non-circular signals; Step 2: To address the non-uniform noise problem caused by time modulation, we perform pre-whitening on the noise through theoretical analysis of the time modulation sequence to obtain the covariance matrix and conjugate covariance matrix of the received signal after non-uniform correction. Step 3: Combine the covariance matrix and conjugate covariance matrix of the received signal after non-uniform correction in the multidimensional domain of dual quaternion, and construct the direction finding fitness function using the orthogonality of the signal subspace; Step 4: Numerical analysis is performed on the direction-finding fitness function to decouple the signal direction parameter from the non-circular phase parameter. Guided by the theoretical lower bound of the direction-finding fitness function obtained from the analysis, a dual quaternion-based non-circular signal DOA estimator using a time-modulated array is constructed.

2. The time-modulated array non-circular signal direction finding method according to claim 1, characterized in that: The process of establishing the time modulation array receiving signal model under the non-circular signal described in step 1 is as follows: Assume that a far-field uncorrelated narrowband non-circular signal with the same carrier frequency f0 is incident on a direction finding system based on a time-modulated array. Due to the influence of high-speed RF switching, the signal passing through the single-pole multi-throw switch can be written as where s k (t) represents the time from θ k The kth signal incident from the direction, β and d represent the wave number and array element spacing respectively, n m (t) is the variance σ 2 Zero-mean circular Gaussian white noise, U m (t) is the time modulation function on the mth array element, which can be expressed as Because U m (t) is time-periodic and satisfies the Dirichlet condition, so it can be decomposed into a Fourier series, given by where α m,q represents the Fourier coefficient, expressed as Where sinc(x)=sin(πx) / (πx); Therefore, each incident signal will generate an infinite sideband signal whose center frequency is located at f0±qf after time modulation. p , q=1,2,...,+∞. Since it is impossible to use infinite sidebands in practice, we can only use the carrier frequency and ±Q sidebands. Given the complex Fourier coefficients and the expression of the modulated signal, the qth order (q=-Q,...,0,...,+Q) sideband signal can be expressed as Time modulation frequency f p Should satisfy f p The constraint of ≥2B is used to avoid frequency aliasing, where B refers to the signal bandwidth. Then, different sideband signals can be easily separated by digital bandpass filters. After passing through a single-channel receiver, all sideband signals are down-converted and sampled by the ADC relative to time t, which can be written as For convenience, the above expression can be rewritten in matrix format and given by Y=B T [AS+N] in(·) T represents the transposition operator, A=[a(θ1),a(θ2),...,a(θ K )] is the array manifold matrix, where Represents the steering vector of the incident signal from different directions; Y = [y -Q ,y -Q+1 ,...,y Q ] T is the received signal matrix signal composed of each sideband, S=[s1,s2,...,s K ] T Represents the incident signals from different sources, N=[n1,n2,...,n M ] T is the noise covariance matrix; By the Fourier coefficient α m,q Composition, called time modulation matrix; Since the incident signal is assumed to be an arbitrary non-circular signal, we have where E{·} is the expectation operator, R S and denote the source covariance and source conjugate covariance matrices respectively; where ρ = diag{ρ1,ρ2,…,ρ K },0≤ρ k ≤1, diag{·} represents a diagonal matrix; ρ k is the kth non-circularity of the non-cyclic signal, is the kth non-circular phase caused by the communication channel; Due to the above non-circular characteristics, the received signal covariance matrix and conjugate covariance matrix of this direction finding system are given as follows R Y =E{YY H }=B T AR S (B T A) H +σ 2 B T (B T ) H 3. The time-modulated array non-circular signal direction finding method according to claim 2, characterized in that: The non-uniform noise pre-whitening process described in step 2 is as follows: The noise covariance matrix after time modulation is no longer the identity matrix before time modulation, so Obviously M H =M; element β in row a and column b of M a,b It can be calculated as follows, where a, b = 1, 2, ..., 2Q + 1; Where sinc(x,y) means sinc(x,y)=[sin(πx)·sin(πy)] / (πx·πy). It is easy to find that only the term Related to the antenna index m, m=1,2,...,M; let k=ab, for the non-diagonal items of M, it is obvious that -2Q≤k≤2Q, k≠0; thus we can get the following relationship On the one hand, this shows that when 2Q+1≤M, the off-diagonal terms of M are equal to 0; on the other hand, the diagonal terms of M are expressed as β a,a , can be calculated using the following formula: Therefore, M is a diagonal matrix and can be written as M=B T (B T ) H =diag{μ1,μ2,...,μ 2Q+1 } where μ a =β a,a , a=1,2,…,2Q+1; therefore, R N becomes R N =σ 2 ·diag{μ1,μ2,...,μ 2Q+1 }; Correct the noise covariance matrix to the identity matrix through pre-whitening, and set C = M -1 / 2 is the non-uniform noise correction matrix, expressed as Obviously, C is a reversible diagonal matrix. The covariance matrix and conjugate covariance matrix of the received signal after non-uniform correction are expressed as follows: where d k =CB T a(θ k ) is the steering vector after noise non-uniformity correction, corresponding to the kth incident signal, so that the noise is corrected to uniform noise.

4. A time modulation array non-circular signal direction finding method according to claim 3, It is characterized in that: Step 3 combines the covariance matrix and the conjugate covariance matrix of the received signal after non-uniform correction in the multidimensional domain of dual quaternion, which is expressed as follows set up Then R becomes the new quaternion combination covariance matrix. It is worth noting that Then it can be inferred Thus R can be rewritten as in is the dual quaternion steering vector; The eigenvalue decomposition of the dual quaternion covariance matrix R is obtained by the following formula: in is the signal subspace, is the noise subspace, since and is the real eigenvalue of the diagonal matrix R consisting of the largest K real eigenvalues ​​and the smallest (2M-K); By extending the subspace theory in the complex domain to the dual quaternion domain, it can be verified that the orthogonality between the signal subspace and the noise subspace still exists; then, due to the virtual steering vector The spanned space coincides with the signal subspace, so the corresponding lateral fitness function is constructed as follows in is the virtual dual quaternion steering vector corresponding to the signal incident angle θ.

5. The time-modulated array non-circular signal direction finding method according to claim 4, characterized in that: In step 4, the direction finding fitness function is numerically analyzed to guide the virtual dual quaternion to the vector Substitute the expression into the fitness function get in Obviously, μ ＊ (θ) = μ(θ); thus μ(θ)=μ0(θ)+iIμ1(θ)+jIμ2(θ)+kIμ3(θ) in Substitute the above formula into the fitness function get Fitness function Relative to non-circular phase variables Perform partial differential operation and set the deviation result to 0, and we get in Using the above numerical relationship, the theoretical lower bound of the fitness function can be derived Therefore, the DOA of the incident signal can be obtained by solving the following formula:

6. A computer system, characterized in that include: One or more processors, a computer-readable storage medium for storing one or more programs, wherein when the one or more programs are executed by the one or more processors, the one or more processors implement the method according to any one of claims 1 to 5.

7. A computer-readable storage medium, characterized in that Computer-executable instructions are stored, and when the instructions are executed, they are used to implement the method according to any one of claims 1 to 5.

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