A method for designing a jamming-resistant phase-coded waveform based on frequency-domain coordinate descent

Abstract

This invention discloses a phase-coded waveform design method for anti-suppression interference based on frequency domain coordinate descent. The method includes: constructing a time-domain optimization model of the phase-coded waveform based on the criterion of jointly minimizing the output power of the noise FM interference signal and the integral sidelobe level of the transmitted waveform; transforming the objective function in the time-domain optimization model from the time domain to the frequency domain to construct a frequency-domain optimization model of the phase-coded waveform; introducing the frequency domain coordinate descent method within the ADMM framework to obtain the ADMM-FCDM algorithm to solve the optimization model and obtain the transmitted waveform; and accelerating and simplifying the ADMM-FCDM algorithm using Fast Fourier Transform (FFT) technology to construct the ADMM-FFT-FCDM algorithm. This invention significantly improves the computational speed while ensuring radar anti-jamming and detection performance.

Description

Technical Field

[0001] This invention relates to the field of phase-coded waveform design technology, and specifically to a phase-coded waveform design method based on frequency domain coordinate descent to resist suppression interference. Background Technology

[0002] With the continuous development of electronic countermeasures technology, modern radar systems face severe challenges in their anti-jamming capabilities in complex electromagnetic environments. Frequency modulation (FM) noise jamming, as a typical form of suppression jamming, has advantages such as ease of implementation, wide applicability, and fast response speed, and is widely used in actual countermeasures.

[0003] Existing anti-interference measures for frequency modulation noise mainly fall into two categories: receiver signal processing and transmitter waveform design. However, receiver signal processing is a passive response strategy and fails to take advantage of the high degree of freedom in transmitter waveform design.

[0004] However, existing waveform design methods often rely on mismatch filtering or spectral sparsity, resulting in signal-to-noise ratio loss and limited applicability. To overcome these shortcomings, a new study establishes an objective function based on the joint optimization of the integral sidelobe level of the transmitted waveform and the output power of the interference after matched filtering. A power method-like iteration based on the alternating direction method of multipliers (ADMM-PMLI) is proposed for phase-coded waveform design, which improves the waveform's anti-interference capability and expands its application range to some extent. However, this method involves a large computational load and low computational efficiency during iterative solution, making it difficult to meet the real-time requirements of engineering applications. Summary of the Invention

[0005] The purpose of this invention is to propose a phase coding waveform design method for anti-suppression interference based on frequency domain coordinate descent. This method can effectively solve the problems of high computational complexity and difficulty in real-time processing of existing anti-noise FM interference phase coding waveform design algorithms. While ensuring radar anti-interference performance and detection performance, the computation speed is significantly improved.

[0006] To achieve the above-mentioned technical objectives, the technical solution adopted by the present invention is as follows:

[0007] A method for designing anti-suppression interference phase-coded waveforms based on frequency domain coordinate descent, the method comprising the following steps:

[0008] S1. Based on the criterion of jointly minimizing the output power of the noise FM interference signal and the integral sidelobe level of the transmitted waveform, a time-domain optimization model of the phase-coded waveform is constructed.

[0009] S2, transform the objective function in the time-domain optimization model from the time domain to the frequency domain, introduce constraints that connect the time and frequency domains, and construct a frequency-domain optimization model for the phase-coded waveform;

[0010] S3. Based on the characteristics of noise frequency modulation interference, the frequency domain optimization model is simplified by splitting the frequency domain sequence into amplitude and phase parts and transforming the frequency domain optimization model. The constraints are written into the augmented Lagrangian function to transform the frequency domain optimization model. The frequency domain coordinate descent method under the alternating direction multiplier method framework is used to solve the transformed frequency domain optimization model to obtain the transmission waveform.

[0011] Furthermore, in step S1, the time-domain optimization model of the phase-encoded waveform is as follows:

[0012]

[0013] st|s(n)|=1,n=0,…,N-1

[0014] In the formula, ξ is a weighting coefficient used to adjust the weights for interference suppression and sidelobe suppression; the integrated sidelobe level of the transmitted waveform is ISL = s H Q s s, the output power P of the noise FM interference signal after passing through the matched filter J =s H R J s; s=[s(0),s(1),…,s(N-1)] T This represents a phase-coded waveform sequence of length N. Let T be the covariance matrix of the transmitted waveform. h R is the transition matrix; J Let covariance be the interference signal. j i =[j i (0),j i (1),…,j i (N-1)] T Let E[] be the interference sample with a sequence length of N, where E[] represents the expected value; the superscript H represents the conjugate transpose, and the superscript T represents the transpose.

[0015] Step S2 further includes:

[0016] By introducing the constraint f = As in the time-frequency domain, the level change of the integral sidelobe of the transmitted waveform is transformed from the time domain to the frequency domain, and its frequency domain expression is as follows:

[0017]

[0018] Where ⊙ represents the Hadamard product operation, and s = [s(0), s(1), ..., s(N-1)] TLet f = As represent a phase-coded waveform sequence of length N, and let f = As represent the 2N-point discrete Fourier transform of s. It is a 2N×N DFT matrix. The superscript H indicates conjugate transpose, the superscript T indicates transpose, and the superscript * indicates complex conjugate operation;

[0019] The output power of the frequency-modulated noise interference signal is transformed to the frequency domain after passing through a matched filter. Its frequency domain expression is as follows:

[0020] P J =f H R f f

[0021] in, Let f be the frequency domain covariance matrix of the interference signal, E[] denote the mathematical expectation, and f J =A1j i 'For interference sample j i '=[j i (0),j i (1),…,j i (2N-1)] T 2N-point discrete Fourier transform sequence, Represents a 2N-point DFT matrix. It is a 2N-point discrete Fourier basis vector;

[0022] By transferring the time-domain optimization model to the frequency domain, we obtain the frequency-domain optimization model in the following form:

[0023]

[0024] stf = As

[0025] |s(n)|=1,n=0,…,N-1

[0026] In the formula, ξ is a weighting coefficient used to adjust the weights of interference suppression and sidelobe suppression.

[0027] Step S3 further includes:

[0028] Based on the characteristics of frequency-modulated noise interference, the optimization problem is simplified, and the simplified frequency domain optimization model is expressed as follows:

[0029]

[0030] stf = As

[0031] |s(n)|=1,n=0,…,N-1

[0032] Where W is a diagonal matrix, and its diagonal elements are formed by R. f The elements on the main diagonal constitute the composition;

[0033] Decompose f into two parts: amplitude and phase.

[0034]

[0035] in, Let f be the magnitude vector and v be the phase vector of f. v(p)=e jarg(f(p)) p = 0, ..., 2N-1;

[0036] The frequency domain optimization model is transformed using the decomposed constraints. The transformed frequency domain optimization model is as follows:

[0037]

[0038] |s(n)|=1,n=0,…,N-1

[0039] |v(p)|=1,p=0,…,2N-1

[0040] Where ||·||4 is the 4-norm of the vector.

[0041] Constraints By writing the augmented Lagrangian function, the optimization problem becomes:

[0042]

[0043] st|s(n)|=1,n=0,…,N-1

[0044] |v(p)|=1,p=0,…,2N-1

[0045] Where ρ is the penalty term of the constraint condition, and u represents the dual variable.

[0046] Furthermore, in step S3, the process of solving the transformed frequency domain optimization model using the frequency domain coordinate descent method within the framework of the alternating direction multiplier method includes the following steps:

[0047] A1, v (t) s (t) u (t) Treating it as a known quantity, update f using the following formula:

[0048]

[0049] in For the (t+1)th iteration value; w = diag(W), where diag(W) represents extracting the diagonal elements of matrix W and writing them as a vector;

[0050] A2, will s (t) u (t) Treating it as a known quantity, v is updated using the following formula:

[0051]

[0052] In the formula, exp() and arg() correspond to the complex exponentiation operation and the argument extraction function, respectively, and j is the imaginary unit;

[0053] A3, will v (t+1) u (t) Treating it as a known quantity, update it using the following formula:

[0054]

[0055] A4, will v (t+1) s (t+1) Treating it as a known quantity, u is updated using the following formula:

[0056]

[0057] A5. Repeat steps A1 to A4 until the number of iterations reaches the upper limit or the objective function satisfies the convergence condition.

[0058] A6 outputs the optimized parameters obtained after the iteration stops.

[0059] Furthermore, in step S3, FFT and IFFT are introduced to replace A and A in the update process. H The matrix-vector multiplication method is used to improve the frequency domain coordinate descent method under the alternating direction multiplier framework. The improved frequency domain coordinate descent method under the alternating direction multiplier framework is then used to solve the transformed frequency domain optimization model. Specifically, the following steps are included:

[0060] B1, v (t) s (t) u (t) Treat it as a known quantity, and update it using the following formula.

[0061]

[0062] in, For the t-th iteration value, d (t) For s (t) The sequence obtained by performing a 2N-point FFT;

[0063] B2, will s (t) u(t) Treating it as a known quantity, update it using the following formula:

[0064]

[0065] In the formula, exp() and arg() correspond to the complex exponentiation operation and the argument extraction function, respectively, and j is the imaginary unit;

[0066] B3, will v (t+1) u (t) Treating it as a known quantity, s is updated using the following formula:

[0067] s (t+1) =exp(jarg(g))

[0068] Where g is Perform a 2N-point IFFT and take the vector obtained from the first N points;

[0069] B4, will v (t+1) s (t+1) Treating it as a known quantity, u is updated using the following formula:

[0070]

[0071] Where, d (t+1) For s (t+1) The sequence obtained by performing a 2N-point FFT;

[0072] B5. Repeat steps B1 to B4 until the number of iterations reaches the upper limit or the objective function satisfies the convergence condition.

[0073] B6 outputs the optimized parameters obtained after the iteration stops.

[0074] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0075] The present invention provides an anti-suppression interference phase-coded waveform design method based on frequency domain coordinate descent. By transferring the optimization problem from the time domain to the frequency domain, the high-dimensional optimization problem is decomposed into multiple one-dimensional subproblems that can be processed in parallel. Furthermore, efficient updates are achieved by deriving closed-form solutions for the frequency domain sequence elements of the waveform, thereby significantly improving the computational efficiency of waveform design. Compared with existing methods, this invention, while ensuring radar anti-jamming and detection performance, has lower computational complexity and faster convergence speed, better meeting the real-time requirements of engineering applications. Attached Figure Description

[0076] Figure 1The diagram shows a comparison of the signal matched filtering results before and after optimization obtained from the simulation experiment; where (a) is the spectrum diagram, (b) is the matched filtering result before optimization, (c) is the matched filtering result after ADMM-PMLI optimization, and (d) is the matched filtering result after ADMM-FFT-FCDM optimization.

[0077] Figure 2 The diagram shows a comparison of the sidelobe optimization performance of the two algorithms; where (a) is the performance diagram before optimization, (b) is the performance diagram after ADMM-PMLI optimization, and (c) is the performance diagram after ADMM-FFT-FCDM optimization.

[0078] Figure 3 This diagram illustrates the optimized convergence performance of the ADMM-PMLI and ADMM-FFT-FCDM algorithms. Detailed Implementation

[0079] The embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0080] This invention discloses a method for designing noise-resistant FM interference phase-coded waveforms based on frequency-domain coordinate descent. The method includes: constructing a time-domain optimization model for the phase-coded waveform based on the criterion of jointly minimizing the output power of the noise FM interference signal and the integral sidelobe level of the transmitted waveform; transforming the objective function in the time-domain optimization model from the time domain to the frequency domain to construct a frequency-domain optimization model for the phase-coded waveform; introducing the frequency-domain coordinate descent method (FCDM) within the framework of the Alternating Direction Method of Multipliers (ADMM) to obtain the ADMM-FCDM algorithm for solving the optimization model and obtaining the transmitted waveform; and accelerating and simplifying the ADMM-FCDM algorithm by combining it with Fast Fourier Transform (FFT) technology to construct the ADMM-FFT-FCDM algorithm.

[0081] The noise-resistant FM interference phase coding waveform design method based on frequency domain coordinate descent of the present invention specifically includes:

[0082] S1 constructs a time-domain optimization model for the phase-coded waveform based on the criterion of jointly minimizing the output power of the noise-modulated interference signal and the integral sidelobe level of the transmitted waveform.

[0083] It may further include:

[0084] The integral sidelobe level of the transmitted waveform is:

[0085] ISL = sH Q s s (1)

[0086] Where s = [s(0), s(1), ..., s(N-1)] T This represents a phase-coded waveform sequence of length N, where |s(n)| = 1, n = 0, 1, 2, ..., N-1. Let T be the covariance matrix of the transmitted waveform. h Let be the transition matrix, defined as

[0087] The output power of the noise FM interference signal after passing through the matched filter is:

[0088] P J =s H R J s(2)

[0089] Among them, R J Let covariance be the interference signal. j i =[j i (0),j i (1),…,j i (N-1)] T These are interference samples with a sequence length of N.

[0090] Considering both the output power of the interference signal and the integral sidelobe level of the transmitted waveform, and introducing a weighting coefficient ξ, the optimization model that jointly minimizes the output power of the interference signal and the integral sidelobe level of the transmitted waveform can be expressed as:

[0091]

[0092] S2 transforms the objective function in the time-domain optimization model from the time domain to the frequency domain, and constructs a frequency-domain optimization model for the phase-coded waveform to reduce computational complexity and simplify the problem structure.

[0093] in,

[0094] The frequency domain expression of the integral sidelobe level of the transmitted waveform is as follows:

[0095]

[0096] Where ⊙ denotes the Hadamard product operation, and f=As denotes the 2N-point discrete Fourier transform of s. It is a 2N×N DFT matrix.

[0097] The frequency domain expression of the output power of the FM noise interference signal after passing through the matched filter is as follows:

[0098] PJ =f H R f f (5)

[0099] in, Let f be the frequency domain covariance matrix of the interference signal. J =A1j i 'For interference sample j i ' =[j i (0),j i (1),…,j i (2N-1)] T 2N-point discrete Fourier transform sequence, Represents a 2N-point DFT matrix. It is a 2N-point discrete Fourier basis vector.

[0100] The time-frequency domain conversion relationship of the output power of the frequency-modulated noise interference signal is proven as follows:

[0101]

[0102] Where z = [s(0), s(1), ..., s(N-1), 0, ..., 0] T , z∈C 2N×1 Let A1 and s be the zero-padding sequence of s. These are the 2N-point DFT and IDFT matrices, respectively. Based on the properties of the Discrete Fourier Transform and its inverse, we know...

[0103] Introducing the constraint f = As in the frequency domain to the frequency domain optimization model, the optimization model is then transferred to the frequency domain as follows:

[0104]

[0105] S3, within the framework of the Alternating Direction Method of Multipliers (ADMM), introduces the Frequency-domain Coordinate Descent Method (FCDM) to obtain the ADMM-FCDM algorithm, which is used to solve the optimization model and obtain the transmitted waveform.

[0106] First, the frequency domain optimization problem needs to be simplified and transformed to a certain extent in order to facilitate the solution of the optimization problem.

[0107] Since frequency-modulated noise interference is usually composed of random phase-modulated signals, the correlation between its different frequency components is weak. Therefore, the off-diagonal elements of its frequency domain covariance matrix can be approximated as zero. The proof is as follows:

[0108]

[0109] Based on the characteristics of frequency modulation noise interference, its autocorrelation function is only related to the time interval, and equation (9) can be rewritten as:

[0110]

[0111] Where Δt=nm, substituting m=n-Δt into equation (10) yields:

[0112]

[0113] When p1 = p2, R f (p1, p2) represents the elements on the diagonal of the frequency domain covariance matrix. Substituting p1 = p2 into equation (11) yields:

[0114]

[0115] When p1≠p2, R f (p1, p2) represent the off-diagonal elements of the frequency domain covariance matrix. Because the autocorrelation function of the frequency-modulated noise decays exponentially with Δt, R... f (p1,p2) are mainly composed of Therefore, equation (12) can be approximated as:

[0116]

[0117] Therefore, the diagonal matrix W is used to replace R. f The diagonal elements of W are composed of R f It is composed of the main diagonal elements.

[0118] According to Parseval's theorem, f can be obtained. H f = 2N 2 Then the objective function in equation (7) can be written as:

[0119]

[0120] Ignore the constant term in equation (13) and replace R with W. f Then equation (7) can be expressed as:

[0121]

[0122] By splitting f into amplitude and phase components, the frequency domain optimization model is transformed:

[0123]

[0124] in, Let f be the magnitude vector and v be the phase vector of f. v(p)=e jarg(f(p)) , p=0,…,2N-1.

[0125] Equation (14) can be updated to:

[0126]

[0127] Where ||·||4 is the 4-norm of the vector.

[0128] Equation (16) is a simplified quartic optimization problem under non-convex constraints. A common approach for this type of problem is to construct an augmented Lagrangian function with a penalty term for solving it. The alternating direction multiplier method is a commonly used algorithm for solving the augmented Lagrangian function. ADMM-FCDM is an optimization algorithm that combines frequency domain coordinate descent with the ADMM framework. Its solution process is as follows:

[0129] Constraints By writing the augmented Lagrangian function, the optimization problem can be transformed into:

[0130]

[0131] Where ρ is the penalty term for the constraint. Let ρ be... v (t) s (t) u (t) Let be the value after the t-th iteration. Solve this problem by following these steps:

[0132] A1, updated using the following formula At this time v (t) s (t) u (t) Given quantities, ignore those in the objective function. Irrelevant items, for The optimization problem can be expressed as:

[0133]

[0134] Ignoring the constant term, equation (18) can be simplified to:

[0135]

[0136] in:

[0137]

[0138] Decomposing equation (19) into 2N subproblems, the p-th subproblem can be expressed as:

[0139]

[0140] Where w = diag(W), diag(W) represents extracting the diagonal elements of matrix W and writing them as a vector. The objective function in equation (21) is then expressed as... This means, that is:

[0141]

[0142] Finding the solution to the optimization problem in equation (21) can be transformed into finding the zeros of the derivative of equation (22), that is:

[0143]

[0144] Equation (23) has a unique real solution, that is, the solution in equation (21) with respect to... The optimization problem has a global optimal solution.

[0145] The proof process is as follows:

[0146] For equation (23), the discriminant for determining the solution of the equation is:

[0147]

[0148] When Δ < 0, the equation has one real root and two conjugate complex roots.

[0149] When Δ>0, the equation has three distinct real roots.

[0150] When Δ = 0, the equation has multiple roots or two real roots (one simple root and one double root).

[0151] Because w p Since is a positive number, equation (23) has one real root and two conjugate complex roots. Taking the second derivative of equation (22) yields: It is always greater than 0. Therefore, the real solutions to equation (23) are those related to... The global optimal solution to the optimization problem.

[0152] The global optimal solution of equation (21) can be obtained from Cardan's formula as follows:

[0153]

[0154] By solving the 2N subproblems in parallel, we can obtain... The update expression is:

[0155]

[0156] A2, update v, at this point s (t) u (t)Given the variables, and ignoring terms in the objective function that are irrelevant to v, the optimization problem for v can be expressed as:

[0157]

[0158] Ignoring the constant term in the equation, decomposing equation (38) into 2N subproblems, the p-th subproblem can be written as:

[0159]

[0160] complex numbers and Treating vectors as vectors on the complex plane, the objective function in equation (28) can be expressed as:

[0161]

[0162] Where cos<·,·> represents the cosine of the angle between the vectors. Ignoring the constant part, equation (29) can be transformed into:

[0163]

[0164] The geometric meaning of the above equation is to find the vectors in the complex plane that make the vectors in the complex plane equal. and The angle v(p) with the smallest possible angle can be obtained as follows:

[0165]

[0166] From equation (31), the updated expression for v can be obtained as follows:

[0167]

[0168] A3, update s, at this time v (t+1) u (t) Given the variables s, and ignoring terms in the objective function that are irrelevant to s, the optimization problem for s can be expressed as:

[0169]

[0170] The solution method for v is the same, and the update expression for s can be obtained as follows:

[0171]

[0172] A4, update the Lagrange multiplier u, at this point v (t+1) s (t+1) The quantity is known.

[0173]

[0174] Repeat steps A1 to A4 until the preset maximum number of convergences is reached or the condition is met. The algorithm terminates when the time is right.

[0175] A5 outputs the optimization results obtained after the iteration stops, and the optimized design scheme of the signal is obtained based on the optimization results.

[0176] S4: The ADMM-FCDM algorithm is accelerated and simplified by combining the Fast Fourier Transform (FFT) technique to construct the ADMM-FFT-FCDM algorithm, which further improves the computational efficiency.

[0177] Its introduction of FFT and IFFT to replace the update process involves A and A H The matrix-vector multiplication is used to form the ADMM-FFT-FCDM algorithm, and the update process is as follows:

[0178] B1, updated using the following formula At this time v (t) s (t) u (t) Treat it as a known quantity:

[0179]

[0180] in, For the t-th iteration value, d (t) For s (t) The sequence obtained by performing a 2N-point FFT.

[0181] B2 updates v using the following formula, at which point... s (t) u (t) Treat it as a known quantity:

[0182]

[0183] B3, update s using the following formula, at this time... v (t+1) u (t) Treat it as a known quantity:

[0184] s (t+1) =exp(jarg(g)) (38)

[0185] Where g is Perform a 2N-point IFFT and take the vector obtained from the first N points.

[0186] B4, update u using the following formula, at this time v (t+1) s (t+1) Treat it as a known quantity:

[0187]

[0188] Where, d (t+1) For s (t+1) The sequence obtained by performing a 2N-point FFT.

[0189] In practical calculations, the above process can be implemented as follows:

[0190] Step 0: Generate a noise frequency-modulated interference signal and calculate its covariance matrix R. f Take out R f The elements on the diagonal are obtained as w; the emitted signal s is initialized. (0) and Lagrange multipliers u (0) , through s (0) Find the DFT sequence f of the initial transmitted signal. (0) Decompose its amplitude and phase to obtain the initial value. and v (0) Set the iteration count t = 0;

[0191] Step 1: For s (t) Perform a 2N-point FFT to obtain d (t) ,according to and renew

[0192] Step 2: According to Update v;

[0193] Step 3: [To] Perform an IFFT and take the first N points to obtain g, based on s. (t+1) =exp(jarg(g)) updates s;

[0194] Step 4: [Regarding s] (t+1) Perform a 2N-point FFT to obtain d. (t+1) ,according to Update u;

[0195] Step 5: Set t←t+1;

[0196] Step 6: Determine whether the obtained solution meets the convergence condition. If it does, complete the algorithm; otherwise, return to Step 2 for iteration.

[0197] Based on the specific implementation method, simulation experiments were conducted, and the simulation conditions for the radar transmission waveform are shown in Table 1.

[0198] Table 1 Simulation conditions for radar transmitted waveforms

[0199] Parameter name Parameter value Sequence length N 200 / 600 / 1000 Signal bandwidth / MHz 50 Interference bandwidth / MHz 50 / 75 / 100 weighting coefficients 200 Input signal-to-interference ratio (dB) 20

[0200] The normalized interference output power is defined as follows:

[0201]

[0202] The normalized transmit waveform average sidelobe expression is defined as follows:

[0203]

[0204] The peak sidelobe level ratio (PSLR) of the transmitted waveform is defined as follows:

[0205]

[0206] The sequence length is set to 1000, the interference bandwidth to 75MHz, and other simulation parameters are set as shown in Table 1. The initial transmission waveform uses a randomly generated phase-coded signal. Simulations are performed using the optimization design method of this invention, and the ADMM-FFT-FCDM algorithm of this invention is replaced with the ADMM-PMLI algorithm for comparison. Simulation results before and after optimization for both algorithms are attached. Figure 1 and Figure 2 As shown.

[0207] from Figure 1 It can be seen that before optimization, the noise frequency modulation interference significantly obscured the target signal, making it impossible to detect the target effectively. After optimization, however, the interference was suppressed to a certain extent, and the target could be effectively identified through constant false alarm rate (CFAR) detection, indicating that the optimized waveform has enhanced anti-interference capability.

[0208] Figure 1 (c) and (d) further show that the waveform optimized by the ADMM-FFT-FCDM algorithm described in this invention has a lower overall sidelobe level than the comparative algorithm ADMM-PMLI, and its peak sidelobe to main lobe ratio has a lower value.

[0209] from Figure 2 As can be seen, the waveform optimized by the ADMM-FFT-FCDM algorithm described in this invention has a certain degree of improvement in average sidelobe suppression performance and peak sidelobe ratio compared with the unoptimized waveform, and is superior to the comparative algorithm ADMM-PMLI in the above indicators.

[0210] To further compare the performance of the two algorithms under different parameters, the parameter settings are shown in Table 1. Let the reduction in interference output power after algorithm optimization be ΔP. out The decrease in the average integral sidelobe level of the transmitted waveform is _____. The reduction in peak sidelobe ratio of the transmitted waveform is ΔPSLR, and the average value is taken from 100 Monte Carlo experiments. Tables 2 and 3 show the performance of the two algorithms under different interference bandwidths and different waveform sequence lengths, respectively.

[0211] Table 2. Optimization performance of two waveform optimization algorithms under different interference bandwidths (N=1000)

[0212]

[0213] Table 3. Optimization performance of two waveform optimization algorithms under different waveform sequence lengths (B) J =50MHz)

[0214]

[0215] Table 2 illustrates the impact of interference bandwidth on algorithm performance. Experimental data shows that the optimization performance of both algorithms decreases with increasing interference bandwidth. This is because increasing the interference bandwidth expands its frequency domain coverage, enhances the interference effect, and consequently reduces algorithm performance.

[0216] Table 3 shows the impact of waveform sequence length on algorithm performance. Experimental data shows that the optimization performance of both algorithms improves with the increase of waveform sequence length. This is because increasing the waveform sequence length is equivalent to increasing the degree of freedom in waveform design, which can reduce the objective function value to a lower level.

[0217] However, under the same parameter conditions, the ADMM-FFT-FCDM algorithm proposed in this invention consistently outperforms the comparative algorithms in terms of interference suppression and sidelobe suppression. This advantage is mainly attributed to the algorithm transforming the optimization problem into the frequency domain and effectively avoiding the local optima problem present in the comparative algorithms by solving the closed-form solution of frequency domain elements in parallel.

[0218] Appendix Figure 3 The convergence results of the proposed ADMM-FFT-FCDM and the contrasting algorithm ADMM-PMLI are shown in Figure B. J =75MHz, N=1000), waveform parameters are set as shown in Table 1, and the convergence condition is set to...

[0219] Figure 3 The results show that the objective function values ​​of both algorithms decrease continuously with the increase of the number of iterations until they stabilize, verifying that both algorithms have good convergence. Furthermore, under the same conditions, the algorithm of this invention converges faster than the comparative algorithm, and the final convergence value is better, indicating that it has certain advantages in convergence efficiency and optimization performance.

[0220] Table 4 Comparison of convergence speed of the two algorithms under different parameters

[0221] ADMM-FFT-FCDM ADMM-PMLI <![CDATA[N=200,B J =50MHz]]> 0.012 7.21 <![CDATA[N=600,B J =50MHz]]> 0.043 240.89 <![CDATA[N=1000,B J =50MHz]]> 0.074 1176.83 <![CDATA[N=1000,B J =75MHz]]> 0.083 1268.44 <![CDATA[N=1000,B J =100MHz]]> 0.132 1590.12

[0222] Table 4 shows a comparison of the convergence speeds of the two algorithms under different parameters. As can be seen from Table 4, the ADMM-FFT-FCDM algorithm of this invention has a significantly higher computational efficiency than the ADMM-PMLI algorithm.

[0223] Furthermore, as the problem size increases, the runtime of the method described in this invention significantly outperforms the comparative algorithms, exhibiting an overall near-linear growth characteristic, while the runtime of the ADMM-PMLI algorithm shows an near-cubic growth trend. This result further verifies that the method described in this invention has superior computational efficiency and good engineering applicability when handling large-scale waveform optimization tasks.

[0224] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0225] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

Claims

1. A method for designing anti-suppression interference phase-coded waveforms based on frequency domain coordinate descent, characterized in that, The method includes the following steps: S1. Based on the criterion of jointly minimizing the output power of the noise FM interference signal and the integral sidelobe level of the transmitted waveform, a time-domain optimization model of the phase-coded waveform is constructed. S2, transform the objective function in the time-domain optimization model from the time domain to the frequency domain, introduce constraints that connect the time and frequency domains, and construct a frequency-domain optimization model for the phase-coded waveform; S3. Based on the characteristics of noise frequency modulation interference, the frequency domain optimization model is simplified by splitting the frequency domain sequence into amplitude and phase parts and transforming the frequency domain optimization model. The constraints are written into the augmented Lagrangian function to transform the frequency domain optimization model. The frequency domain coordinate descent method under the alternating direction multiplier method framework is used to solve the transformed frequency domain optimization model to obtain the transmission waveform.

2. The anti-suppression interference phase coding waveform design method based on frequency domain coordinate descent according to claim 1, characterized in that, In step S1, the time-domain optimization model of the phase-encoded waveform is as follows: ； In the formula, These are weighting coefficients used to adjust the weights for interference suppression and sidelobe suppression; Integral sidelobe level of the transmitted waveform The output power of the noise FM interference signal after passing through the matched filter ; Indicates length is Phase-coded waveform sequence, Let be the covariance matrix of the transmitted waveform. Here is the transition matrix; Let covariance be the interference signal. , Let E[] be the interference sample with a sequence length of N, where E[] represents the expected value; the superscript H represents the conjugate transpose, and the superscript T represents the transpose.

3. The anti-suppression interference phase coding waveform design method based on frequency domain coordinate descent according to claim 1, characterized in that, Step S2 further includes: Introducing constraints in the time and frequency domains The integral sidelobe level of the transmitted waveform is transformed from the time domain to the frequency domain, and its frequency domain expression is as follows: ； in, This represents the Hadamard product operation. Indicates length is Phase-coded waveform sequence, express The 2N-point discrete Fourier transform, for The DFT matrix, The superscript H indicates conjugate transpose, the superscript T indicates transpose, and the superscript * indicates complex conjugate operation. The output power of the frequency-modulated noise interference signal is transformed to the frequency domain after passing through a matched filter. Its frequency domain expression is as follows: ； in, Let E[] be the frequency domain covariance matrix of the interference signal, and let E[] represent the expected value. Interference sample 2N-point discrete Fourier transform sequence, Represents a 2N-point DFT matrix. It is a 2N-point discrete Fourier basis vector; By transferring the time-domain optimization model to the frequency domain, we obtain the frequency-domain optimization model in the following form: ； In the formula, The weighting coefficient is used to adjust the weights of interference suppression and sidelobe suppression.

4. The anti-suppression interference phase coding waveform design method based on frequency domain coordinate descent according to claim 3, characterized in that, Step S3 further includes: Based on the characteristics of frequency-modulated noise interference, the optimization problem is simplified, and the simplified frequency domain optimization model is expressed as follows: ； in, It is a diagonal matrix, whose diagonal elements are formed by... The elements on the main diagonal constitute the composition; Will It can be divided into two parts: amplitude and phase. ； in, for The magnitude vector, for phase vector, , , In the formula, arg() represents the argument extraction function, and j is the imaginary unit; The frequency domain optimization model is transformed using the decomposed constraints. The transformed frequency domain optimization model is as follows: ； in, The 4-norm of a vector. ; Constraints By writing the augmented Lagrangian function, the optimization problem becomes: ； in, The penalty term for the constraint condition, This represents the dual variable.

5. The anti-suppression interference phase coding waveform design method based on frequency domain coordinate descent according to claim 4, characterized in that, Step S3, the process of solving the transformed frequency domain optimization model using the frequency domain coordinate descent method under the alternating direction multiplier method framework, includes the following steps: A1, will Treat it as a known quantity, and update it using the following formula. : ； in For the (t+1)th iteration value; , , Indicates from matrix Extract the diagonal elements and write them as a vector; A2, will Treat it as a known quantity, and update it using the following formula. : ； In the formula, exp() represents complex exponentiation; A3, will Treating it as a known quantity, update it using the following formula: ； A4, will Treat it as a known quantity, and update it using the following formula. : ； A5, repeat steps A1 to A4 until the number of iterations reaches the upper limit or the objective function satisfies the convergence condition; A6 outputs the optimized parameters obtained after the iteration stops.

6. The anti-suppression interference phase coding waveform design method based on frequency domain coordinate descent according to claim 4, characterized in that, In step S3, FFT and IFFT are introduced to replace the updates involved in the process. and The matrix and vector multiplication method is used to improve the frequency domain coordinate descent method under the alternating direction multiplier method framework. The improved frequency domain coordinate descent method under the alternating direction multiplier method framework is used to solve the transformed frequency domain optimization model. Specifically, the following steps are included: B1, will Treat it as a known quantity, and update it using the following formula. : ； in, For the t-th iteration value, , for The sequence obtained by performing a 2N-point FFT; B2, will Treating it as a known quantity, update it using the following formula: ； In the formula, exp() represents complex exponentiation; B3, will Treat it as a known quantity, and update it using the following formula. : ； in, for Perform a 2N-point IFFT and take the vector obtained from the first N points; B4, will Treat it as a known quantity, and update it using the following formula. : ； in, for The sequence obtained by performing a 2N-point FFT; B5. Repeat steps B1 to B4 until the number of iterations reaches the upper limit or the objective function satisfies the convergence condition. B6 outputs the optimized parameters obtained after the iteration stops.

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