A radar complementary sparse frequency waveform sequence set design method based on CCM algorithm

Abstract

The application discloses a radar complementary sparse frequency waveform sequence set design method based on a complex circle flow surface algorithm, and specifically comprises the following steps: according to a monitored spectrum environment, delimiting a usable spectrum range and an unusable spectrum range; converting a weighted integral sidelobe level of a waveform sequence set into a quadratic function form with a waveform sequence set vector as a variable, constructing an objective function to describe the weighted integral sidelobe level of the waveform set; calculating a power spectrum of the waveform set, controlling a weighting coefficient according to an expected spectrum, performing weighted summation on the power spectrum, and converting a weighted summation expression of the power spectrum into a quadratic function form with the waveform sequence set vector as the variable; and performing weighted summation on a WISL objective function and an EFS objective function to construct a joint objective function. The CCM optimization algorithm is applied to the complementary sparse frequency waveform sequence set design problem for the first time, effectively solving the constant modulus constraint problem of the waveform sequence, and compared with the existing cyclic iteration algorithm and the interior point method.

Description

Technical Field

[0001] This invention belongs to the field of radar signal processing technology, and specifically relates to a design method for radar complementary sparse frequency waveform sequence sets based on the ComplexCircle Manifold (CCM) algorithm. Background Technology

[0002] Complementary waveform sequence sets possess ideal aperiodic autocorrelation properties, thus they are widely used in fields such as communications and radar, attracting considerable attention from researchers. Research on complementary waveform sequence sets can be traced back to 1949, when Golay first proposed the concept of a set of two mutually complementary waveform sequences, namely Golay complementary pairs. Complementary waveform sequence sets constructed mathematically satisfy the strict complementarity property, meaning the sum of the autocorrelation of the sequences in the set is an ideal impulse function. However, strictly complementary waveform sequence sets are strictly limited in sequence length and number, failing to meet the requirements for flexible waveform applications. Therefore, optimizing the design of waveform sequence sets to approximate the autocorrelation properties of complementary waveform sequence sets has gained widespread attention. This research approach overcomes the limitations of strictly complementary waveform sequence sets, offering greater design flexibility. For example, the paper (J.Song,P.Babu andDP Palomar. Sequence Set Design With Good Correlation Properties via Majorization-Minimization[J].IEEE Transactions on Signal Processing,2016,64(11):2866-2879.) designed a complementary waveform sequence set with low sidelobes using the Majorization-Minimization (MM) algorithm. In addition, this paper also used Fast Fourier Transform (FFT) for computation, which reduced the computational complexity of the algorithm.

[0003] On the other hand, with the rapid development of radio technology, spectrum resources are becoming increasingly scarce, and radar operating frequency bands are subject to interference from navigation, communication, and other devices operating in the same frequency band. Therefore, to avoid interfering frequency bands and ensure radar detection performance, sparse frequency waveform design has received widespread attention. Its main idea is: first, to divide the unusable frequency bands in the available spectrum into frequency stopbands; and second, to divide the available frequency bands into frequency passbands, the resulting spectrum is called the desired spectrum. Then, the radar waveform sequence is optimized to reduce the energy of the waveform sequence in the frequency stopbands, so that the spectrum of the waveform sequence is consistent with the desired spectrum. For example... Figure 1This is a spectrum diagram of a sparse frequency waveform. The red line represents the full frequency band; the shaded area represents the occupied frequency band, i.e., the frequency stopband, which includes [f1,f2]∪[f3,f4]∪[f5,f6]∪[f7,f8]∪[f9,f6] 10 Five frequency stopbands are shown; the black line represents the desired frequency spectrum. This waveform design method improves bandwidth utilization and the anti-jamming capability of the radar system. However, the design of sparse frequency waveform sequences also faces many challenges. The main difficulty is that, due to the sparsity of the waveform spectrum, direct stitching of waveforms with smaller bandwidths leads to a wide range of high sidelobes in the autocorrelation function of the sparse frequency waveforms. Therefore, it is essential to design a set of sparse frequency waveform sequences with lower sidelobes.

[0004] To ensure that sparse frequency waveform sequences have low sidelobes, an effective design approach is to design a set of sparse frequency waveform sequences, making the autocorrelation function of the waveform sequences in the set approximate the complementary characteristics. This set of sequences is called a complementary sparse frequency waveform sequence set. The literature (Xiang Li, Mai Chaoyun, Gan Junying. Design of sparse frequency radar waveforms using complementary codes [J]. Signal Processing, 2019, 35(8):1432-1438.) establishes two nonlinear optimization problems, taking power spectral density (PSD) and integrated sidelobe level (ISL) as objective functions and constraints respectively, in order to design a complementary sparse frequency waveform sequence set. To solve the established optimization problems, the paper first ignores the constant mode constraint of the waveform sequence and approximates other nonlinear constraints into linear constraints; then it uses the interior point method to solve the problem; finally, it directly extracts the phase of the solution to satisfy the constant mode constraint. Obviously, this solution method involves a series of approximations, and the solution obtained is a suboptimal solution to the original nonlinear optimization problem. Furthermore, this method can only suppress the autocorrelation sidelobes of complementary sparse frequency waveform sequence sets across the entire delay range, without considering suppression within a specific delay range; however, the latter often meets the needs of some special practical scenarios. The literature (S.Hong, YQFu, YTDong and Y.X.Zhang.The optimal design of radar complementary waveform sets with sparse frequency[C].IET International Radar Conference(IET IRC 2020),2020:897-901.) uses the weighted integral sidelobe level (Weighted ISL, WISL) of the waveform sequence set and the weighted sum between the mean square error of the waveform sequence set spectrum and the desired spectrum as the objective function, and employs a Cyclic Iterative Algorithm (CIA) to design radar complementary sparse frequency waveform sequence sets. However, in the iterative solution process, the solution of other auxiliary variables is introduced, and high-dimensional matrix square root operations are required, resulting in high algorithm complexity and low convergence performance.

[0005] Manifold optimization, also known as Riemannian optimization, is an optimization framework based on Riemannian manifolds proposed by Absil PA et al. in 2008 (Absil PA, Mahony R, Sepulchre R. Optimization Algorithms on Matrix Manifolds [M]. Princeton University Press, 2008.), and has received increasing attention and application in recent years. Manifold optimization provides a new perspective for solving and analyzing a special class of constrained optimization problems, namely manifold-constrained optimization problems. The Complex Circle Manifold (CCM) algorithm is a type of manifold optimization, specifically a gradient-based algorithm for complex circle manifold optimization. Essentially, it uses a gradient algorithm to solve for the objective function on a complex circle manifold constructed under constraints. Because the CCM algorithm can directly meet the constant modulus requirements of communication and radar transmission signals, it has been widely used in communication and radar.

[0006] Figure 2 For the CCM algorithm geometry, combined Figure 2 The CCM algorithm is described below:

[0007] The CCM algorithm is a gradient-based manifold optimization algorithm. Essentially, it solves for the objective function on a complex circular manifold under constant modulus constraints using a gradient algorithm. The method for solving the objective function is as follows:

[0008] Define S as a complex circle, defined as being ... In this paper, the search space for the constant modulus constrained optimization problem can be viewed as the product of MN complex circles, defined as...

[0009]

[0010] Among them, [x m ] n Represents vector x m The nth element.

[0011] The optimization problem to be solved is

[0012]

[0013] Here, f(x) is the objective function for optimization.

[0014] The CCM algorithm mainly consists of three steps: projection, descent, and retraction.

[0015] Projection: Optimizing the objective function on a complex circular manifold under constant modulus constraints requires finding the Riemann gradient of the objective function's descent. First, the objective function f(x) at the l-th iteration point x can be calculated. (l) ∈S MN The negative gradient in Euclidean space at a point, i.e. The objective function f(x) at the l-th iteration point x (l) ∈S MN The Riemann gradient of a point is in manifold S MN cross-section The projection operator can then be used to calculate the search gradient in Euclidean space. Projected onto the tangent plane From the above, we can obtain the objective function f(x) at the l-th iteration point x. (l) Riemann gradient on The calculation method is as follows:

[0016]

[0017] in, This represents the Hadama product.

[0018] Descent: According to step size β (l) and search direction The update is performed, resulting in new optimization variables:

[0019]

[0020] Retraction: The new variable obtained by updating through equation (54) Therefore, it needs to be mapped back to a circular manifold. That is, through a shrinkage operator, defined as...

[0021]

[0022] The literature (Alhujaili K, Monga V, Rangaswamy M. Transmit MIMO Radar Beampattern Design via Optimization on the Complex Circle Manifold[J].IEEE Transactions on Signal Processing, 2019, 67(13):3561-3575.) applies the CCM algorithm to MIMO radar beam design, and the literature (Alhujaili K, Monga V, Rangaswamy M. Correlation-Gradient-Descent: Efficient Optimization Methods for Unimodular Waveform Design with Desirable Correlation Properties[C]. 2020 IEEE International Radar Conference (RADAR). IEEE, 2020.) applies the CCM algorithm to the design of low autocorrelation and cross-correlation orthogonal waveforms for MIMO radar. However, existing literature has not optimized the design of complementary waveform sets and complementary sparse frequency waveform sets in traditional phased array radars.

[0023] To fill the aforementioned gaps, this invention is the first to utilize the CCM algorithm for the optimization design of complementary sparse frequency waveform sequence sets. This invention uses the autocorrelation and sparse spectral characteristics of the complementary sequence set as the objective function, and constant waveform modulus as the constraint, to establish a waveform optimization problem. The CCM algorithm is then used to solve this problem, and the required step size and matrix multiplication operations are accelerated to improve computational efficiency. Summary of the Invention

[0024] This invention provides a design method for radar complementary sparse frequency waveform sequence sets based on the CCM algorithm. This method can design waveform sequence sets with sparse frequency characteristics and complementary autocorrelation characteristics while satisfying spectral constraints. This invention is the first to apply the CCM algorithm to the design problem of complementary sparse frequency waveform sequence sets, effectively solving the constant mode constraint problem of waveform sequences. Compared with the interior-point method and CIA, it has lower autocorrelation sidelobes and faster convergence speed. Furthermore, the invented design method can also be extended to the design of complementary sparse frequency waveform sequence sets with partial autocorrelation sidelobe suppression, complementary waveform sequence sets with full autocorrelation sidelobe suppression without spectral constraints, and complementary waveform sequence sets with partial autocorrelation sidelobe suppression without spectral constraints, exhibiting wider adaptability.

[0025] This invention is achieved through the following technical solutions.

[0026] The present invention discloses a radar complementary sparse frequency waveform sequence set design method based on the CCM algorithm, comprising the following steps:

[0027] Step 1: Based on the monitored spectral environment, delineate the usable and unusable spectral ranges to determine the passband and stopband of the desired spectrum;

[0028] Determining the passband and stopband of the desired spectrum refers to identifying the clean, radar-detectable frequency band as the passband Ω. p The frequency bands occupied by navigation, communication and other radio equipment, as well as the frequency bands with strong interference and clutter, are defined as the stopband Ω. s Assume the frequency range of the waveform sequence spectrum is... Then there is

[0029] Step 2: Express the weighted integrated sidelobe level (WISL) of the waveform sequence set as a quadratic function with the waveform sequence set vector as the variable, and construct the objective function O. WISL (z) is used to describe the autocorrelation sidelobe level of the waveform set, where z is the waveform set vector;

[0030] Step 3: Calculate the power spectrum of the waveform set. Set weighting coefficients according to the desired spectrum, and perform a weighted summation of the power spectrum. Express the weighted summation of the power spectrum as a quadratic function with the waveform sequence set vector as the variable, thus constructing the Energy in the Frequency Stopband (EFS) objective function O. EFS (z) is used to describe the energy of the waveform set within the frequency stopband;

[0031] Step 4: Change the objective function O from Step 2. WISL (z) and the objective function O in step 3 EFS (z) Perform weighted summation to construct the joint objective function. Using the waveform sequence set as variables, a joint optimization problem is established under the constant modulus constraint of the waveform set;

[0032] Step 5: Solve the established optimization problem using the CCM algorithm. First, perform a series of equivalent transformations on the established optimization problem to facilitate the CCM algorithm's solution. These transformations, in sequence, include: ① Using relaxation methods, substituting some matrix variables obtained from the previous iteration into the next iteration as constant matrices, thus ensuring that the objective function in each iteration is a quadratic function of the optimization variables. ② In each iteration, utilizing the constant modulus of the waveform sequence set and the eigenvalue properties of the constant matrix in transformation ①, transform the optimization problem to ensure that the objective function is a convex function of the optimization variables. ③ Introducing a positive constant α into the objective function in transformation ② further ensures the convexity of the function in each iteration, thereby ensuring the convergence of the CCM algorithm. Then, for the transformed optimization problem, calculate the solution for the optimization variables according to the three steps of projection, descent, and retraction in the CCM algorithm to obtain the solution for the optimization variables at the current iteration number. The solution process involves several steps: The projection step calculates the Riemann gradient of the objective function, requiring the determination of the constant α; the descent step uses the Riemann gradient as the search direction and calculates the updated optimization variables using gradient descent; and the retraction step maps the gradient-descent-updated optimization variables onto a complex circle to satisfy the constant modulus constraint. Finally, an iterative solution is performed until the termination condition is met. To reduce computational complexity, a series of fast computation methods are proposed for eigenvalue solving, constant α calculation, and search step size calculation.

[0033] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0034] (1) The present invention takes the WISL and EFS of the waveform sequence set as optimization targets, and can design a constant mode sequence set with unlimited length and number. The sum of the autocorrelation functions of the waveform sequence set has low correlation sidelobes, which approximates the correlation properties of the complementary waveform sequence set. The waveform sequence set has a small spectral power within the set stopband.

[0035] (2) This invention is the first to utilize the CCM algorithm to design a radar complementary sparse frequency waveform sequence set, which can directly satisfy the constant modulus constraint of the waveform sequence in each iteration; at the same time, the iteration process selects the fastest direction for minimizing the objective function, ensuring the convergence of the algorithm. Compared with existing optimization algorithms, the CCM algorithm proposed in this invention has lower autocorrelation sidelobes, faster convergence speed and stronger stability.

[0036] (3) This invention accelerates the operations required by the CCM algorithm, such as step size, matrix multiplication and matrix eigenvalue solving, and makes full use of FFT operation to improve computational efficiency. In particular, the design of complementary sparse frequency waveform sequence sets with long sequences and multiple sequences has high engineering practical significance.

[0037] (4) The method proposed in this invention can be extended from the design of complementary sparse frequency waveform sequence sets to the design of complementary waveform sequence sets without spectral constraints.

[0038] (5) The method proposed in this invention can take into account the weighted case of autocorrelation sidelobe, and has a wider range of applicability compared with the interior point method. Attached Figure Description

[0039] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0040] Figure 1 This is a spectrum diagram of a sparse frequency waveform.

[0041] Figure 2 This is a geometric interpretation of the CCM algorithm used in this invention.

[0042] Figure 3 This is a flowchart of the method of the present invention.

[0043] Figure 4 The autocorrelation of the complementary sparse frequency waveform sequence set in simulation experiment 1 of this invention.

[0044] Figure 5 This is the complementary sparse frequency waveform sequence set PSD in simulation experiment 1 of this invention.

[0045] Figure 6 The algorithm runtime in simulation experiment 1 of this invention is the algorithm runtime under different convergence accuracies ε.

[0046] Figure 7 The autocorrelation of the complementary sparse waveform sequence set with partial sidelobe suppression in simulation experiment 2 of this invention.

[0047] Figure 8 This is the complementary sparse frequency waveform sequence set PSD for partial sidelobe suppression in simulation experiment 2 of this invention.

[0048] Figure 9 The autocorrelation of the complementary waveform sequence set in simulation experiment 3 of this invention.

[0049] Figure 10 This is the curve showing the change of the cost function with the number of iterations in simulation experiment 3 of this invention.

[0050] Figure 11 The autocorrelation of the complementary waveform sequence set of partial sidelobe suppression in simulation experiment 4 of this invention.

[0051] Figure 12 This is the time-varying curve of the cost function of the complementary waveform sequence set for partial sidelobe suppression in simulation experiment 4 of this invention. Detailed Implementation

[0052] The embodiments of the present invention will be described in detail below with reference to examples. However, those skilled in the art will understand that the following examples are only for illustrating the present invention and should not be regarded as limiting the scope of the present invention.

[0053] refer to Figure 3 The flowchart illustrates a radar complementary sparse frequency waveform sequence set design method based on the CCM algorithm provided by this invention, comprising the following steps:

[0054] Step 1: Based on the monitored spectral environment, delineate the usable and unusable spectral ranges to determine the passband and stopband of the desired spectrum;

[0055] Determining the passband and stopband of the desired spectrum refers to identifying the clean, radar-detectable frequency band as the passband Ω. p The frequency bands occupied by navigation, communication and other radio equipment, as well as the frequency bands with strong interference and clutter, are defined as the stopband Ω. s Assume the frequency range of the waveform sequence spectrum is... Then there is

[0056] Step 2: Express the weighted integral sidelobe level WISL of the waveform sequence set as a quadratic function with the waveform sequence set vector as the variable, and construct the objective function O. WISL (z) is used to describe the autocorrelation sidelobe level of the waveform set;

[0057] (2.1) Define a complex constant-mode waveform sequence set containing M waveform sequences, each with a length of N. Where z m =[[z m ]1,[z m ]2,…,[z m ] N ] T , |[z m ] n |=1,m=1,2,…,M,n=1,2,…,N. Construct a waveform sequence set vector from multiple waveform sequences. Where L = (2M-1)N.

[0058] (2.2) The aperiodic autocorrelation function of vector z is a waveform set within the time delay [-N+1, N-1]. The sum of the autocorrelation functions of each waveform is:

[0059]

[0060] Where r zz (k) represents the aperiodic autocorrelation function of vector z. m=1,2,…,M, k=0,1,…,N-1, For sequence z m The nonperiodic autocorrelation function.

[0061] (2.3) Therefore, the waveform set The sum of the autocorrelation functions of each waveform can be expressed as:

[0062]

[0063] Among them [z] n This represents the nth element of vector z. Let represent a matrix where all elements on the k-th diagonal are 1, and all other elements are 0, i.e.:

[0064]

[0065] (2.4) For the autocorrelation function r zz (k) Perform a weighted summation, and the real weighting coefficient w at the k-th delay point. k Must meet:

[0066]

[0067] The WISL objective function for the waveform sequence set can then be expressed as:

[0068]

[0069] Among them, matrix The specific form is:

[0070]

[0071] Ignore the constant term in equation (60) The WISL objective function O that can construct a set of waveform sequences WISL (z) is as follows:

[0072]

[0073] Step 3: Calculate the power spectrum of the waveform set. Set weighting coefficients according to the desired spectrum, and perform a weighted summation of the power spectrum. Express the weighted summation expression of the power spectrum as a quadratic function with the waveform sequence set vector as the variable, thus constructing the frequency stopband energy EFS objective function O. EFS (z) is used to describe the energy of the waveform set within the frequency stopband;

[0074] (3.1) Waveform sequence set The spectrum of the m-th waveform sequence It can be calculated using the following formula:

[0075]

[0076] in, Describing the Fourier transform matrix F H The first N elements in the h-th row; matrix F H Defined as

[0077]

[0078] (3.2) The m-th waveform sequence z m The spectrum f m The power spectrum of this waveform sequence can be calculated as follows:

[0079]

[0080] in This represents the Hadamard product. It's used to quickly select the m-th waveform sequence z from z. m Define the selection matrix S m =[0 N×(2m-2)N ,I N×N ,0 N×(L-(2m-1)N) ], thus having z m =S m z. Therefore, the power spectrum p in equation (65) m It can be further written as:

[0081]

[0082] (3.3) Weighted summation of the power spectrum of the waveform. To calculate the signal energy of the waveform sequence within the stopband, the weighting coefficients w... f (h) is set to:

[0083]

[0084] Among them, Ω s Represents the stopband frequency set, Ω p This represents the passband frequency set. Therefore, the energy O of the waveform set in the frequency stopband... EFS (z) can be represented as:

[0085]

[0086] Wherein, matrix P CSF Represented as:

[0087]

[0088] Step 4: Change the objective function O from Step 2. WISL (z) and the objective function O in step 3 EFS (z) Perform weighted summation to construct the joint objective function. Using the waveform sequence set as variables, a joint optimization problem is established under the constant modulus constraint of the waveform set.

[0089] The objective function O established in step 2 WISL (z) and the objective function O established in step 3 EFS (z) Perform a weighted summation to obtain the joint cost function. Where μ∈[0,1] is a weighting factor used to adjust the weight between the autocorrelation sidelobe characteristics and the sparse spectral characteristics of the waveform sequence set. Substituting equations (62) and (68) into... In the middle, there is in Therefore, taking the sequence set vector z as the variable, under the constant modulus constraint of the waveform sequence set, the following optimization problem can be established:

[0090]

[0091] By solving problem (70), the autocorrelation sidelobe level of the waveform sequence set and the energy of the waveform sequence set in the frequency stopband can be suppressed, thereby obtaining a complementary sparse frequency waveform sequence set with excellent performance.

[0092] Step 5: Solve the established optimization problem using the CCM algorithm. First, perform a series of equivalent transformations on the established optimization problem to facilitate the CCM algorithm's solution. These transformations, in sequence, include: ① Using relaxation methods, substituting some matrix variables obtained from the previous iteration into the next iteration as constant matrices, thus ensuring that the objective function in each iteration is a quadratic function of the optimization variables. ② In each iteration, utilizing the constant modulus of the waveform sequence set and the eigenvalue properties of the constant matrix in transformation ①, transform the optimization problem to ensure that the objective function is a convex function of the optimization variables. ③ Introducing a positive constant α into the objective function in transformation ② further ensures the convexity of the function in each iteration, thereby ensuring the convergence of the CCM algorithm. Then, for the transformed optimization problem, calculate the solution for the optimization variables according to the three steps of projection, descent, and retraction in the CCM algorithm to obtain the solution for the optimization variables at the current iteration number. The solution process involves several steps: Projection, which calculates the Riemann gradient of the objective function, requiring the determination of a constant α; Descent, which uses the Riemann gradient as the search direction and calculates the updated optimization variables using gradient descent; and Retraction, which maps the updated optimization variables onto a complex circle to satisfy the constant modulus constraint. Finally, an iterative solution is performed until the termination condition is met. To reduce computational complexity, a series of fast computation methods are proposed for eigenvalue solving, constant α calculation, and search step size calculation.

[0093] The specific solution process is as follows:

[0094] (5.1) The matrix contained in the objective function E(z) of problem (70) It is related to the optimization variable z, which is inconvenient to solve.

[0095] Therefore, by using the relaxation method, the matrix of the (l-1)th iteration is calculated during the l-th iteration of the CCM algorithm. Will Replace the solution in the l-th iteration At this point, during the l-th iteration of the solution... Transformed into a constant matrix independent of the optimization variable z. The optimization problem in the l-th iteration becomes:

[0096]

[0097] Where E=(1-μ)R+μP CSF ,

[0098] (5.2) To ensure the convexity of the objective function in problem (71) so that it can be solved using the CCM algorithm, matrix E needs to be a positive semi-definite matrix. Therefore, matrix E in the objective function of problem (71) is replaced with... Where λ min (R) is the smallest eigenvalue of matrix R, matrix It is an identity matrix. From equation (69), we know that matrix P... CSF It is a positive semi-definite matrix, therefore It is also a positive semi-definite matrix. Therefore, problem (71) can be transformed into the following optimization problem:

[0099]

[0100] st|[z m ] n |=1,m=1,…,M,n=1,…,N (72)

[0101] To obtain This leads to problem (72), which requires calculating λ. min (R). Directly solving for the eigenvalues ​​of matrix R results in high computational complexity due to its high dimensionality. Since the dimension of R is proportional to the length and number of sequences, this problem becomes more pronounced as the sequence length or number of sequences increases. To address this issue, a fast computation method is proposed, utilizing the following Lemma 1 to... min The calculation of (R) is transformed into the calculation of λ. min (R) Fast calculation of the upper bound.

[0102] Lemma 1: Define a complex vector of length 2N Depend on The elements in the matrix can be used to construct a Hermitian Toeplitz matrix T, as follows:

[0103]

[0104] Matrix T can also be quickly calculated using FFT, i.e. Where the FFT matrix F H The definition is shown in equation (64), F 1:N,: Let F be a submatrix consisting of the first N rows of matrix F. (Definition) The largest and smallest eigenvalues ​​of matrix T satisfy the following conditions:

[0105]

[0106]

[0107] in Represents vector The j-th element.

[0108] Obviously, the matrix R defined in equation (61) (i.e. ) is a Hermitian Toeplitz matrix, which can be quickly calculated from the complex vector c, where the complex vector c is:

[0109]

[0110]

[0111] Using Lemma 1, the smallest eigenvalue of matrix R satisfies:

[0112]

[0113] Therefore, we can obtain 0 ≤ R - λ min (R)I≤R-λ' min I, definition have:

[0114]

[0115] Will Replacement matrix R-λ in min (R)I, can be obtained Therefore, the optimization problem in equation (70) can be equivalent to:

[0116]

[0117] (5.3) To ensure the convergence of the iterative algorithm, it is necessary to guarantee that in each iteration, the objective function of problem (80) contains... It is a positive semi-definite matrix. Therefore, in each iteration, a positive constant α is introduced, further transforming problem (80) into the following equivalent problem:

[0118]

[0119] The value of α needs to be updated and calculated in each iteration.

[0120] (5.4) The optimization problem to be solved in the l-th iteration is the optimization problem in equation (81), which can be specifically written as:

[0121]

[0122] in, Based on the CCM algorithm, the solution can be obtained through a three-step process: Projection-Descent-Retraction, as detailed below:

[0123] Projection: Solve for the Riemann gradient of the objective function. Based on the objective function in equation (82), solve for the Euclidean negative gradient η at the l-th iteration. (l) :

[0124]

[0125] From η (l) The Riemann gradient of the objective function can be calculated as follows:

[0126]

[0127] Descent: The obtained Riemann gradient This represents the search direction for decreasing the objective function value. Based on the gradient descent concept, the optimization variables obtained in the l-th iteration can be written as:

[0128]

[0129] Where, β (l) Update the step size for the variable. The constant α in the Riemann gradient. (l) and step size β (l) It can be solved by Lemma 2.

[0130] Lemma 2: When using the CCM algorithm to solve the standard problem (82), in order to ensure the convergence of the algorithm, it is required that... It is a positive semi-definite matrix, and the constant α (l) and step size β (l) The following conditions must be met:

[0131]

[0132]

[0133] As can be seen from Lemma 2, to choose an appropriate α (l) and β (l) , needs to be calculated and For ease of explanation, here will be and Abbreviated as and Because the matrix dimension is large, directly performing eigenvalue decomposition to find the largest eigenvalue would significantly increase the computational complexity of the algorithm. Furthermore, because... The dimension of the problem is directly proportional to the length and number of sequences, and this problem becomes more pronounced as the sequence length or number of sequences increases. To address this problem, a fast computation method is proposed, utilizing Lemma 3 for rapid calculation. The upper bound of is then used to find the upper bound of . The upper boundary.

[0134] Lemma 3 (Weyl's Lemma): Given two Hermitian matrices and Sort the eigenvalues ​​of matrices A, B, and A+B in ascending order, and write them as follows: and Furthermore, the larger the value of i, the larger the eigenvalue it represents. The eigenvalues ​​of these three matrices satisfy the following relationship:

[0135] λ i (A+B)≤λ i+j (A)+λ N-j (B),i=1,2,…,N,j=0,1,…,Ni (88)

[0136] First, calculate From the above, we can see that And matrix sum matrix μP CSF Let there be two Hermitian matrices. Using Lemma 3, choosing i = N, j = 0, we can obtain:

[0137]

[0138] From equation (89), we can see that... The upper bound needs to be calculated separately. and λ max (P CSF The upper bound of ); specifically includes the following two steps:

[0139] (5.4.1) First, find The upper bound. By It can be known

[0140]

[0141] Where, λ' min It has already been calculated using equation (78). To find λ... max (R) can be derived using equation (74) in Lemma 1, similar to the derivation process in equations (76)-(78):

[0142]

[0143] According to equation (90), using equation (91), we have

[0144]

[0145] (5.4.2) Then, find λ. max (P CSF The upper bound of ).

[0146] Using the selection matrix S m=[0 N×2(m-1)N ,I N×N ,0 N×(L-(2m-1)N) The property of P in equation (69) can be used to... CSF Rewritten as:

[0147]

[0148] Where blkdiag{·} denotes a block diagonal matrix; matrix matrix Defined as:

[0149]

[0150] Among them, w f =[w f (1),w f (2),...,w f (2N)] T .

[0151] From equation (93), we can see that P CSF Let S be a diagonally partitioned matrix whose eigenvalues ​​are equal to the eigenvalues ​​of its inner block matrix S. If λ(S) represents the eigenvalues ​​of matrix S, then λ(P) CSF )=λ(S).

[0152] According to equation (94), the relationship between the eigenvalues ​​and eigenvectors of matrix S can be written as follows:

[0153]

[0154] Here, x is the eigenvector corresponding to its eigenvalue λ(S).

[0155] Multiply both sides of equation (95) by have:

[0156]

[0157] In equation (96) As a new feature vector, we can know That is, the eigenvalues ​​of matrix S and the matrix The eigenvalues ​​are the same. To solve... The following lemma 4 can be used.

[0158] Lemma 4: Given two positive semi-definite Hermitian matrices and Sort the eigenvalues ​​of matrices A, B, and AB in ascending order, and write them as follows: and Furthermore, the larger the value of i, the larger the eigenvalue it represents. The eigenvalues ​​of these three matrices satisfy the following relationship:

[0159] λ i (AB)≤λ N (B)λ i (A), i = 1, 2, ..., N (97)

[0160] Due to the matrix and diag(w f All are positive semi-definite Hermitian matrices. Let i = N. By Lemma 4, we know that:

[0161]

[0162] Obviously, there are Therefore, we can conclude that:

[0163]

[0164] From equation (99), we can know that λ max (P CSF The upper bound of ) is λ p =2N≥λ max (S).

[0165] (5.4.3) Finally, find The upper bound. Due to Further results can be obtained using this:

[0166]

[0167] Substituting equations (99) and (100) into equations (86) and (87), we obtain the constant α. (l) and step size β (l) The range of values ​​for α; based on this range, the constant α can be selected. (l) and step size β (l) for:

[0168]

[0169]

[0170] Among them, τ and γ can be arbitrarily chosen as non-negative constants that satisfy inequalities (86) and (87).

[0171] Retraction: The vector obtained according to equation (85) It does not satisfy the constant modulus constraint condition, that is It is not on the complex circular manifold formed by the constant modulus constraints in the joint optimization problem. To obtain the constant modulus waveform sequence, the vector is further processed using the following formula. Mapping back to the circular manifold:

[0172]

[0173] Furthermore, from equation (103) Waveform sets can be obtained The m-th sequence for:

[0174]

[0175] Depend on Reconstructable

[0176] After performing the three steps of Projection-Descent-Retraction, the l-th iteration is complete. Then, let l = l + 1 to begin the next iteration, and repeat this process until the iteration termination condition is met. The iteration termination condition can be set as ||z||. (l+1) -z (l) ||2≤ε or running time t≥T, etc.

[0177] Therefore, the present invention proposes a radar complementary sparse frequency waveform sequence set design method based on the CCM algorithm, the specific steps of which are shown in the table below:

[0178]

[0179] Simulation Experiment

[0180] The effects of the present invention can be further illustrated by the following specific examples:

[0181] The performance metrics for the waveform sequence set obtained from the simulation experiment are selected as follows:

[0182] To evaluate the effectiveness of the method of this invention and the good correlation and sparse frequency characteristics of the designed complementary sparse frequency waveform sequence set, two performance evaluation metrics for the waveform sequence set are defined: WISL and Average Passband Stopband Power Ratio (APSPR). Specifically, to ensure good correlation of the waveform sequence set, the WISL metric should be as small as possible; simultaneously, to achieve sparse frequency characteristics of the waveform sequence set, the APSPR metric should be as large as possible. WISL and APSPR can be calculated by the following formulas:

[0183]

[0184]

[0185] Among them, P p For the average passband power, P s This represents the average stopband power.

[0186] Simulation Experiment 1: Under the constraints of five normalized frequency stopbands [0.04,0.21], [0.23,0.25], [0.28,0.37], [0.39,0.49] and [0.52,0.56], a set of complementary sparse frequency waveform sequences with complete sidelobe suppression is designed using the CCM algorithm.

[0187] Simulation parameter settings:

[0188] The waveform sequence sets contain M=2 and M=3 sequences, respectively, with each sequence having a length of N=143. The autocorrelation sidelobe weighting is set as follows:

[0189]

[0190] The nonnegative constants τ = 0 and γ = 0.1. When the number of sequences is M = 2, the weighting coefficient is μ = 0.956, and the iteration stopping condition is that the error accuracy threshold satisfies ε = 10. -4 Or the total running time of the algorithm exceeds t = 3000s; when the number of sequences is M = 3, the weighting coefficient is μ = 0.955, and the iteration stopping condition is: the error accuracy threshold satisfies ε = 10. -4 Or the total running time of the algorithm exceeds t = 8000s.

[0191] Simulation results:

[0192] First, consider the case where the number of sequences is M=2, and compare the CCM algorithm of this invention with CIA (S.Hong, YQFu, YTDong and YXZhang. The optimal design of radar complementary waveformsets with sparse frequency[C]. IET International Radar Conference(IET IRC2020),2020:897-901.) and interior point method (Xiang Li, Mai Chaoyun, Gan Junying. Complementary code design of sparse frequency radar waveforms[J]. Signal Processing, 2019, 35(8):1432-1438.). The simulation results are as follows: Figure 4 , Figure 5 (a) and Figure 5 As shown in (b). Wherein Figure 4 Autocorrelation function for complementary sparse frequency waveform sequence sets designed for CCM algorithm, CIA and interior point method. Figure 4 In this invention, the CCM algorithm, CIA, and interior-point method can all design complementary sparse frequency waveform sequence sets with low sidelobes; and the WISL values ​​of the complementary sparse frequency waveform sequence sets designed by the CCM algorithm, CIA, and interior-point method are -3.83dB, -3.70dB, and -3.75dB, respectively. Figure 4 It can be seen that the WISL value obtained by the CCM algorithm proposed in this invention is the lowest. Figure 5 The PSD of two waveform sequences in a set of complementary sparse frequency waveform sequences optimized by three algorithms. Figure 5 (a) and Figure 5 (b) The gray shaded areas represent the five defined stopband regions. As shown in the figure, after optimization by the three algorithms, the waveform sequences exhibit lower energy within the defined frequency stopband regions. Specifically, the APSPRs designed by the CCM algorithm of this invention are 24.47 dB and 25.96 dB, the APSPRs of the two waveforms designed by the CIA algorithm are 24.19 dB and 23.64 dB, and the APSPRs of the two waveforms designed by the interior point method are 23.81 dB and 23.84 dB, respectively.

[0193] To provide a clearer comparison, this invention further considers the case where the number of sequences M=3, and records the WISL and APSPR values ​​of the complementary sparse frequency waveform sequence sets designed by the three algorithms for M=2 and M=3 in Table 1. In Table 1, when the number of sequences is M=2 and M=3, the CCM algorithm of this invention outperforms the CIA and interior-point methods in both autocorrelation sidelobe suppression and stopband energy suppression. Therefore, the complementary sparse frequency waveform sequence set obtained by the algorithm optimized in this paper has lower autocorrelation sidelobe characteristics and better sparse frequency characteristics, verifying the effectiveness of the design method.

[0194] Table 1. Performance Comparison of Complementary Sparse Frequency Waveform Sequence Sets for All Sidelobe Suppressions

[0195]

[0196] The convergence speed of the algorithms was also compared. Figure 6 The graph represents the time required to design complementary sparse frequency waveform sequence sets for the three algorithms at different convergence accuracies when M=2. The horizontal axis represents the convergence accuracy ε of the optimization variable. Since the interior-point method has a convergence accuracy ε=10... -5 Under certain conditions, the running time is too long; therefore, only the running time of the algorithm at the first five convergence accuracies is recorded. As shown in the figure, the running time of all three algorithms increases with the increase of the convergence accuracy of the optimization variable. However, at the same convergence accuracy, the time required for the CCM algorithm of this invention to design a complementary sparse frequency waveform sequence set is much shorter than that of CIA and the interior-point method. Therefore, the convergence speed of the algorithm of this invention in optimizing and designing a complementary sparse waveform sequence set is faster than that of CIA and the interior-point method.

[0197] Simulation Experiment 2: Under the constraints of five normalized frequency stopbands [0.04,0.21], [0.23,0.25], [0.28,0.37], [0.39,0.49] and [0.52,0.56], a complementary sparse frequency waveform sequence set with partial autocorrelation sidelobe suppression is designed using the CCM algorithm.

[0198] Simulation parameter settings:

[0199] The waveform sequence set contains M=2 and M=3 sequences, and each waveform sequence has a length of N=143. The autocorrelation sidelobe weighting is set as follows:

[0200]

[0201] Non-negative constants τ = 0, γ = 0.1. When M = 2, the weighting coefficient is μ = 0.5, and the iteration stopping condition is: the error accuracy threshold satisfies ε = 10. -4 Or the total running time of the algorithm exceeds t = 2000s; when M = 3, the weighting coefficient is μ = 0.6, and the iteration stopping condition is: the error accuracy threshold satisfies ε = 10. -4 Or the total running time of the algorithm exceeds t = 5000s.

[0202] Simulation results:

[0203] The interior-point method fails when partial autocorrelation sidelobe suppression is required. Considering the case where the number of sequences is M=2, and comparing the CCM algorithm of this invention with the CIA-optimized design of complementary sparse frequency waveform sequence sets, the simulation results are as follows: Figure 7 and Figure 8 As shown. Among them. Figure 7 Design autocorrelation functions for complementary sparse frequency waveform sequence sets for CCM algorithm and CIA. Figure 7 In this invention, both the CCM algorithm and CIA can design complementary sparse frequency waveform sequence sets with partial autocorrelation sidelobe suppression. Furthermore, the WISL of the complementary sparse frequency waveform sequence sets designed by the CCM algorithm and CIA in this invention are -34.28 dB and -16.87 dB, respectively, across the entire delay range. Figure 7 It can be seen that the WISL value of the CCM algorithm of this invention is lower. Figure 8 Optimize the PSD of two waveform sequences in the complementary sparse frequency waveform sequence set for CCM and CIA. Figure 8 (a) and Figure 8 (b) The gray shaded areas represent the five defined stopband regions. As shown in the figure, within the defined frequency stopband regions, the waveforms optimized by the algorithm exhibit lower energy. Specifically, the APSPRs designed by the CCM algorithm of this invention are 28.05 dB and 28.85 dB, while the APSPRs of the two waveforms designed by CIA are 19.43 dB and 18.52 dB, respectively. Figure 7 and Figure 8 A comparison of the WISL and APSPR of the waveforms designed by the two algorithms shows that the CCM algorithm proposed in this invention outperforms CIA in both WISL and APSPR.

[0204] Similarly, to more clearly demonstrate the advantages of the algorithm of this invention, this invention further considers the case where the number of sequences M=3. Table 2 records the WISL and APSPR values ​​of the complementary sparse frequency waveform sequence sets with partial autocorrelation sidelobe suppression designed by the two algorithms for M=2 and M=3. In Table 2, when the number of sequences is M=2 and M=3, the CCM algorithm of this invention outperforms CIA in both autocorrelation sidelobe suppression and stopband energy suppression capabilities. Therefore, it can be seen that the complementary sparse frequency waveform sequence set with partial autocorrelation sidelobe suppression obtained by the algorithm of this invention has lower autocorrelation sidelobe characteristics and better sparse frequency characteristics, verifying the effectiveness of the proposed method.

[0205] Table 2 compares the performance of complementary sparse frequency waveform sequence sets for partial autocorrelation sidelobe suppression.

[0206]

[0207] Simulation Experiment 3: Considering removing the spectral constraints, design a set of complementary waveform sequences with complete autocorrelation sidelobe suppression using the CCM algorithm.

[0208] Simulation parameter settings:

[0209] The number of sequences in the waveform sequence set is M = 2, and the weighting coefficient between the autocorrelation sidelobes and the stopband energy is μ = 0. According to equation (107), the autocorrelation sidelobes are weighted to suppress all sidelobes, with non-negative constants τ = 2N and γ = 0.1. The iteration stopping condition is: the error accuracy threshold of the optimization variables satisfies ε = 10⁻¹⁰. 4 Or the total running time of the algorithm exceeds t = 1000s.

[0210] Simulation results:

[0211] The CCM algorithm is used to solve this optimization problem, obtaining a set of complementary waveform sequences that satisfy the constraints. To highlight the superiority of the algorithm, it is compared with the CIA and MM algorithms. The simulation results are as follows. Figure 9 , Figure 10 As shown, where Figure 9 Design autocorrelation functions for complementary waveform sequence sets for three algorithms: CCM, CIA, and MM. Figure 9It can be seen that the autocorrelation sidelobes of the complementary waveform sequence sets designed by the CCM algorithm, CIA algorithm, and MM algorithm are approximately -80dB, -40dB, and -60dB, respectively. Furthermore, the WISL values ​​for all sidelobes suppressed are -44.01dB, -21.24dB, and -31.26dB, respectively. Therefore, compared to the CIA and MM algorithms, the complementary waveform sequence set designed by the CCM algorithm of this invention has lower autocorrelation sidelobes.

[0212] The convergence speed of the algorithms was also compared. Figure 10 The graph shows the cost function of the three algorithms as a function of the number of iterations. As can be seen, the cost function of the three algorithms decreases with increasing iteration count. At the same number of iterations, the cost function suppressed by the CCM algorithm is lower than that of the CIA and MM algorithms. This indicates that the method of this invention is more efficient in suppressing autocorrelation sidelobes and has a faster convergence speed.

[0213] Simulation Experiment 4: Considering removing the spectral constraints, design a complementary waveform sequence set with partial autocorrelation sidelobe suppression using the CCM algorithm.

[0214] Simulation parameter settings:

[0215] The waveform sequence set contains M = 2 sequences, and the weighting coefficient μ = 0 between the autocorrelation sidelobes and the stopband energy. The autocorrelation sidelobe weighting is set as follows:

[0216]

[0217] Non-negative constants τ = 2N, γ = 0.1, and the iteration stopping condition is: the algorithm error accuracy threshold satisfies ε = 10. -9 Or the total running time of the algorithm exceeds t = 10000s.

[0218] Simulation results:

[0219] The optimization problem is solved using the CCM algorithm, yielding a set of complementary waveform sequences with partially suppressed autocorrelation sidelobes that satisfy the constraints. The simulation results are as follows: Figure 11 , Figure 12 As shown, where Figure 11 Design autocorrelation functions for partial sidelobe suppression waveform sequence sets for three algorithms: CCM, CIA, and MM. Figure 11 It can be seen that the autocorrelation sidelobes of the complementary waveform sequence sets designed by the CCM algorithm, CIA algorithm, and MM algorithm are approximately -230dB, -140dB, and -170dB, respectively. Furthermore, their WISL values ​​within the sidelobe suppression range are -208.82dB, -127.33dB, and -146.96dB, respectively. Therefore, compared to the CIA and MM algorithms, the complementary waveform sequence set designed by the CCM algorithm of this invention has lower autocorrelation sidelobes.

[0220] The convergence speed of the algorithms was also compared. Figure 12 The graph shows the cost function of the three algorithms over time. As can be seen, the cost function of the three algorithms decreases with increasing time. However, within the same iteration time, the cost function suppressed by the CCM algorithm is significantly lower than that of the CIA and MM algorithms. This indicates that the method of this invention has a more efficient and faster convergence speed for autocorrelation sidelobe suppression.

Claims

1. A method for designing radar complementary sparse frequency waveform sequence sets based on the CCM algorithm, characterized in that, Includes the following steps: Step 1: Based on the monitored spectral environment, delineate the usable and unusable spectral ranges to determine the passband and stopband of the desired spectrum; Step 2: Express the weighted integral sidelobe level of the waveform sequence set as a quadratic function with the waveform sequence set vector as the variable, and construct the objective function. To describe the autocorrelation sidelobe level of the waveform set, where It is a waveform set vector; Step 3: Calculate the power spectrum of the waveform set. Set weighting coefficients according to the desired spectrum, and perform a weighted summation of the power spectrum. Express the weighted summation expression of the power spectrum as a quadratic function with the waveform sequence set vector as the variable, thereby constructing the frequency stopband energy objective function. To describe the energy of a waveform set within the frequency stopband; Step 4: Change the objective function from Step 2 and the objective function in step 3 We perform a weighted summation to construct the joint objective function. Using the waveform sequence set as variables, a joint optimization problem is established under the constant modulus constraint of the waveform set; Step 5: Solve the established optimization problem using the CCM algorithm; First, perform a series of equivalent transformations on the established optimization problem to facilitate the CCM algorithm's solution; In sequence, these transformations include: ① using relaxation methods to substitute some matrix variables obtained from the previous iteration into the next iteration as constant matrices, thereby ensuring that the objective function in each iteration is a quadratic function of the optimization variables; ② in each iteration, using the constant modulus of the waveform sequence set and the eigenvalue properties of the constant matrix in transformation ①, transforming the optimization problem to ensure that the objective function is a convex function of the optimization variables; ③ introducing a positive constant into the objective function in transformation ②. This further ensures the convexity of the function in each iteration, thereby guaranteeing the convergence of the CCM algorithm. Then, for the transformed optimization problem, the solution of the optimization variables at the current iteration number is calculated according to the three steps of projection, gradient, and shrinkage in the CCM algorithm. Among them, the projection step is to calculate the Riemann gradient of the objective function, and a constant needs to be determined to calculate this gradient. The value of is determined by the gradient step, which uses the Riemann gradient as the search direction and calculates the updated optimization variables according to the gradient descent method. The shrinking step maps the optimized variables updated by gradient descent onto the complex circle to satisfy the constant modulus constraint of the optimization variables. Finally, iterative loops are performed to solve the problem until the iteration termination condition is met. In this solution process, in order to reduce the computational complexity, a series of fast calculation methods are proposed for the eigenvalue solving, constant calculation, and search step size calculation processes.

2. The radar complementary sparse frequency waveform sequence set design method based on CCM algorithm according to claim 1, characterized in that, In step 1, determining the passband and stopband of the desired spectrum refers to identifying the clean, radar-detectable frequency band as the passband. The frequency bands occupied by navigation and communication radio equipment, as well as those with strong interference and clutter, are defined as stopbands. Assume the frequency range of the waveform sequence spectrum is... Then there is .

3. The radar complementary sparse frequency waveform sequence set design method based on CCM algorithm according to claim 1, characterized in that, In step 2, the objective function for constructing the waveform sequence set is... The specific steps are as follows: (2.1) Define a containing There are 1 waveform sequence, and the length of each waveform sequence is 1. Complex constant mode waveform sequence set ,in , Construct a waveform sequence set vector from multiple waveform sequences. ,in Represents vector The One element, ; (2.2) Vector The aperiodic autocorrelation function in time delay The inner part represents the sum of the autocorrelation functions of all waveforms in the waveform set, i.e.: (1) in Represents vector The nonperiodic autocorrelation function, , , , , is a sequence The nonperiodic autocorrelation function; (2.3) Thus, the waveform set The sum of the autocorrelation functions of each waveform can be expressed as: (2) in, Indicates the first A matrix in which all elements on the diagonal are 1 and all other elements are 0, i.e.: (3) (2.4) For the autocorrelation function Perform a weighted summation, on the... Real weighted coefficients at each delay point Must meet: (4) The WISL objective function for the waveform sequence set can then be expressed as: (5) Among them, matrix The specific form is as follows: (6) Ignore the constant term in equation (5) The WISL objective function that can be used to construct a set of waveform sequences as follows: (7)。 4. The radar complementary sparse frequency waveform sequence set design method based on CCM algorithm according to claim 3, characterized in that, In step 3, the objective function for constructing the waveform sequence set is... The specific steps are as follows: (3.1) Waveform sequence set The Middle Spectrum of a waveform sequence It can be calculated using the following formula: (8) in, Represents the Fourier transform matrix No. Before the trip elements, matrix Defined as: (9) (3.2) From the first Waveform sequence Spectrum The power spectrum of this waveform sequence can be calculated as follows: (10) in, Represents the Hadama product; for in Quickly select the first Waveform sequence Define the selection matrix Therefore, Therefore, the power spectrum in equation (10) It can be further written as: (11) (3.3) Perform a weighted summation of the power spectrum of the waveform; to calculate the signal energy of the waveform sequence within the stopband, the weighting coefficients are... Set to: (12) in, Represents the stopband frequency set. This represents the passband frequency set; therefore, the energy of the waveform set in the frequency stopband... It can be represented as: (13) in, (14)。 5. The radar complementary sparse frequency waveform sequence set design method based on CCM algorithm according to claim 4, characterized in that, Step 4 establishes a joint optimization problem, specifically as follows: The objective function established in step 2 and the objective function established in step 3 We perform a weighted summation to obtain the joint cost function. ,in The weighting factor is used to adjust the weight between the autocorrelation sidelobe characteristics and the sparse spectral characteristics of the waveform sequence set; substituting equations (7) and (13) into... In the middle, there is ,in Therefore, using sequence set vectors Let be the variable, and under the constant modulus constraint of the waveform sequence set, the following optimization problem can be established: (15) By solving problem (15), the autocorrelation sidelobe level of the waveform sequence set and the energy of the waveform sequence set in the frequency stopband can be suppressed, thereby obtaining a complementary sparse frequency waveform sequence set with excellent performance.

6. The radar complementary sparse frequency waveform sequence set design method based on CCM algorithm according to claim 5, characterized in that, In step 5, the CCM algorithm is used to solve the established optimization problem (15). The specific steps are as follows: (5.1) In the objective function of problem (15) The included matrices With optimization variables This is related to the fact that it is inconvenient to solve; therefore, the relaxation method is used in the CCM algorithm. In the iteration, the ... The matrix of the next iteration Replace the first The next iteration At this time, in the first During the second iteration of the solution Becomes with optimization variables Unrelated constant matrix ;No. The optimization problem in the next iteration becomes (16) in, , ; (5.2) To ensure the convexity of the objective function in problem (16) so that it can be solved using the CCM algorithm, the matrix needs to be made convex. It is a positive semi-definite matrix; therefore, the matrix in the objective function of problem (16) is... Replace with ,in For matrix The smallest eigenvalue of the matrix It is an identity matrix; as shown in equation (14), the matrix... It is a positive semi-definite matrix, therefore It is also a positive semi-definite matrix; therefore, problem (16) can be transformed into the following optimization problem: (17) To obtain And further, we arrive at problem (17), which requires calculation. If directly on the matrix Perform eigenvalue solving. A higher matrix dimension leads to higher computational complexity; because The dimension of the sequence is directly proportional to the length and number of sequences, and this problem becomes more pronounced as the sequence length or number of sequences increases. To address this problem, a fast computation method is proposed, which utilizes the following Lemma 1... The calculation is converted to a Fast calculation of the lower bound; Lemma 1: Define a length of Complex vectors ,Depend on The elements in can be used to construct a Hermitian Toplitz matrix. ,as follows: (18) matrix It can also be quickly calculated using FFT, i.e. The definition of the FFT matrix is ​​shown in equation (9). Representation matrix The former Submatrix formed by rows; definition Then the matrix The largest and smallest eigenvalues ​​satisfy the following conditions: (19) (20) in, Represents vector The One element; Obviously, the matrix defined in equation (6) (Right now ) is a Hermitian Toplitz matrix, which can be obtained from complex vectors. It is obtained quickly, where the complex vector for: (21) (22) Using Lemma 1, the matrix The smallest eigenvalue satisfies: (23) From this, we can obtain ,definition ,have: (24) Will Replacement matrix In ,available Therefore, the optimization problem in equation (17) can be equivalent to: (25) (5.3) To ensure the convergence of the iterative algorithm, it is necessary to guarantee that in each iteration, the objective function of problem (25) contains... It is always a positive semi-definite matrix; therefore, in each iteration, a positive constant is introduced. Problem (25) can be further transformed into the following equivalent problem: (26) in, The value of needs to be updated and calculated in each iteration; (5.4) Section The optimization problem to be solved in the next iteration is the optimization problem in equation (26), which can be specifically written as: (27) in, , Based on the CCM algorithm concept, the solution can be obtained through three steps: projection, gradient, and shrinkage, as detailed below: Projection: Solve for the Riemann gradient of the objective function; based on the objective function in equation (27), solve for the first... Negative gradient in Euclidean space at the next iteration : (28) Depend on The Riemann gradient of the objective function can be calculated as follows: (29) Gradient: The obtained Riemann gradient This represents the search direction for decreasing the objective function value; based on the idea of ​​gradient descent, the... The optimization variables obtained in the next iteration can be written as: (30) in, Update the step size for the variable; constants in problem (27) Step size in equation (30) It can be solved by Lemma 2; Lemma 2: When using the CCM algorithm to solve the standard problem (27), in order to ensure the convergence of the algorithm, it is required that... It is a positive semi-definite matrix, and its constant is... and step length The following conditions must be met: (31) (32) As can be seen from Lemma 2, to select an appropriate and , needs to be calculated and For ease of explanation, here we will... and Abbreviated as and Due to the matrix With a large dimensionality, directly performing eigenvalue decomposition to find the largest eigenvalue would significantly increase the computational complexity of the algorithm; furthermore, because The dimension of the sequence is directly proportional to the length and number of sequences, and this problem becomes more pronounced as the sequence length or number of sequences increases. To address this problem, a fast computation method is proposed, which utilizes Lemma 3 for rapid computation. The upper bound; then using Find the upper bound. The upper bound; Lemma 3 Weyl Lemma: Given two Hermitian matrices and , matrix , and The eigenvalues ​​are sorted in ascending order and written as , and ,and The larger the value, the larger the eigenvalue it represents; the eigenvalues ​​of these three matrices satisfy the following relationship: (33) First, calculate From the above, we can see that And matrix sum matrix Given two Hermitian matrices; using Lemma 3, choose... ,available: (34) From equation (34), we can see that... The upper bound needs to be calculated separately. and The upper bound; specifically includes the following two steps: (5.4.1) First, find The upper bound; by It can be known (35) in, It has already been calculated by equation (23); in order to find Using equation (19) in Lemma 1, similar to the derivation process in equations (21) to (23), we can obtain: (36) According to equation (35), using equation (36), we have (37) (5.4.2) Then, find The upper bound; using the selection matrix The properties of equation (14) can be used to... Rewritten as: (38) in, Represents a block diagonal matrix; matrix Defined as: (39) in, ; From equation (38), we can see that, It is a diagonal block matrix whose eigenvalues ​​are equal to the internal block matrix. eigenvalues; if using Representation matrix The eigenvalues ​​of then have ; According to equation (39), the matrix can be written as follows. The relationship between the eigenvalues ​​and eigenvectors is as follows: (40) in, Its eigenvalues The corresponding eigenvector; multiply both sides of equation (40) by ,have: (41) In equation (41) As a new feature vector, we can know , i.e., matrix Eigenvalues ​​and matrices The eigenvalues ​​are the same; to solve Lemma 4 can be used as follows; Lemma 4: Given two positive semi-definite Hermitian matrices and , matrix , and The eigenvalues ​​are sorted in ascending order and written as , and ,and The larger the value, the larger the eigenvalue it represents; the eigenvalues ​​of these three matrices satisfy the following relationship: (42) Due to the matrix and Both are positive semi-definite Hermitian matrices, let From Lemma 4, we know that: (43) Obviously, there are Therefore, we can conclude that: (44) From equation (44), we can know that... The upper bound is ; (5.4.3) Finally, find The upper bound; due to Using equations (37) and (44), we can obtain: (45) Then, using Furthermore, we can obtain: (46) Substituting equations (45) and (46) into equations (31) and (32), we obtain the constant. and step length The range of values ​​for ; based on this range, a constant can be selected. and step length for: (47) (48) in, , It can be arbitrarily chosen as a nonnegative constant that satisfies inequalities (31) and (32); Retraction: The vector obtained according to equation (30) It does not satisfy the constant modulus constraint condition, that is Not on the complex circular manifold formed by the constant modulus constraints in the joint optimization problem; to obtain the constant modulus waveform sequence, the vector is further expressed using the following formula. Mapping back to the circular manifold: (49) Furthermore, from equation (49) Waveform sets can be obtained The Middle Sequences for: (50) Depend on Reconstructable ; After performing the above three steps of projection, gradient, and shrinkage, the first step is complete. Solve in the next iteration; then, let The iteration continues in this manner, proceeding to the next iteration, until the iteration termination condition is met; the iteration termination condition can be set as follows: or running time .

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