Low-interception waveform generation method based on cyclic spectrum approximate design

Abstract

The invention discloses a low-interception waveform generation method based on cyclic spectrum approximate design, which comprises the following steps: firstly, setting parameters, and selecting a proper cyclic spectrum estimation method; generating an optimization problem model, introducing auxiliary variables to realize conversion from a quartic coupling problem to a quadratic decoupling problem, and then entering an iteration step: solving a to-be-optimized signal vector, a spectrum frequency vector and a similarity constraint auxiliary vector; calculating and solving an optimal solution by using a CVX solver and a Matlab tool; updating the dual variables; performing statistics on the target function and various residual information; and continuously iterating through the iteration step, outputting an optimal anti-cyclostationary analysis radar low-interception waveform, and carrying out statistics on three-dimensional and two-dimensional section information of an obtained waveform cyclic spectrum. According to the method, violent inversion of a large-dimensional matrix is avoided, the optimal solution of the problem is solved, more ideal cyclic spectrum distribution is obtained, and low interception performance is achieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar waveform design, and particularly relates to the technology for generating low-intercept waveforms. Background Art

[0002] In the modern battlefield where the competition for the electromagnetic spectrum space is becoming increasingly intense, the design of low probability of intercept (LPI) waveforms has always been in a dynamic game of "spear and shield" with electronic reconnaissance technology. With the engineering applications of traditional low-intercept technologies (such as spread spectrum, frequency hopping, power control, etc.) becoming more mature, the second-generation detection theory based on cyclostationary analysis is gradually breaking through the limitations of traditional signal feature extraction. By exploiting the hidden periodic characteristics of signals (such as symbol rate, carrier offset, modulation period, etc.), it can effectively intercept complex modulated signals even under low signal-to-noise ratio and wideband conditions. This technological breakthrough makes the design of low-intercept waveforms face an urgent need for paradigm reconstruction - simply relying on traditional means such as spectral spreading and power hiding is difficult to cope with the threat of intelligent detection algorithms, and there is an urgent need to establish a new design framework of "cyclic feature blanking".

[0003] The cyclic spectrum is mainly used to analyze cyclostationary signals, which are common in communication and radar systems. It can reveal the periodic characteristics of signals and is helpful for signal detection, parameter estimation, and interference suppression. Cyclic spectrum analysis can be used to optimize the beamforming or interference suppression of antenna arrays. The core of cyclostationary analysis lies in extracting the periodic statistical features of signals, and the cyclic spectrum, as a high-order statistic, can capture more hidden non-linear coupling features in signals. However, this high-dimensional feature extraction brings a huge computational burden: cyclic spectrum analysis needs to calculate the statistical characteristics of signals in multiple dimensions such as time delay, frequency, and cyclic frequency, and its computational complexity increases exponentially with the signal length and dimension. For broadband signals or long-time observation signals, real-time calculation is almost infeasible, which poses extremely high computational efficiency requirements for the design of low-intercept waveforms in this field. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a low-intercept waveform generation method starting from reducing the sidelobes of the cyclic spectrum and approximating the cyclic features of stationary noise signals.

[0005] The technical solution adopted by the present invention to solve the above technical problem is a low-intercept waveform generation method based on cyclic spectrum approximation design, including the steps:

[0006] Steps for establishing the transmitted waveform model: Set the similarity constraint threshold, select the cyclic spectrum estimation method, and generate the smoothing calculation matrix; Based on the given smoothing calculation matrix, establish the transmitted waveform optimization model with the constraints of similarity constraint and energy constraint to suppress the sidelobes of the generated cyclic spectrum as the optimization objective; Introduce auxiliary variables into the transmitted waveform optimization model to obtain the decoupled and optimized transmitted waveform optimization model, and then transform it into an unconstrained optimization problem through the augmented Lagrangian function;

[0007] Iteration steps: Iteratively solve the unconstrained optimization problem, and update the optimized signal vector and introduce auxiliary variables through the CVX solver and Matlab tool for each iteration until the iteration ends and output the latest optimized signal vector as the final low-intercept waveform vector;

[0008] Waveform generation steps: Generate the low-intercept waveform radar signal using the low-intercept waveform vector.

[0009] Specifically, the penalty weight in the smoothing calculation matrix is set to a larger value at the sidelobes of the transmitted waveform and a smaller value at the main lobe of the transmitted waveform, so as to suppress the sidelobes of the generated cyclic spectrum in the transmitted waveform optimization model.

[0010] Specifically, the smoothing calculation matrix Π α,f is the smoothing calculation matrix with the cyclic frequency index of α and the spectral frequency index of f, ω α is the penalty weight at different cyclic frequency indices α, and Γ α,f is the intermediate matrix.

[0011] The transmitted waveform optimization model is expressed as:

[0012]

[0013] s.t.s H s = N

[0014] |s n -c n | 2 ≤θ 2 ,n = 1,2,…,N

[0015] where s is the transmitted waveform to be optimized, the cyclic frequency index of the cyclic spectrum is α, the spectral frequency index of the cyclic spectrum is f, Π α,f is the smoothing calculation matrix with the cyclic frequency index of α and the spectral frequency index of f, s n is the value of s at the nth sampling point, F N is the Fourier transform matrix with the discrete sampling points of N, c n is the value of c at the nth sampling point, θ is the similarity constraint threshold, and H is the conjugate transpose.

[0016] Specifically, the introduced auxiliary variables include a first auxiliary variable b representing the spectral frequency vector α,f = F N s and a second auxiliary variable h α,f = F N s, and a third auxiliary variable d = s - c representing the similarity constraint auxiliary vector.

[0017] The requirement for the waveform generated by the present invention is that, under the energy constraint and the similarity constraint to ensure good radar detection performance, when dealing with the low-intercept LPI radar signal transmitted by the third-party intercepting machine for cyclic stationary analysis, the radar transmitted signal can be submerged in Gaussian white noise. Currently, the third party can easily apply the cyclic detection theory to intercept the radar signal waveform. To address this situation, the present invention proposes that the transmitting party actively designs and generates the LPI waveform in response to the cyclic detection theory. During the process of designing and generating the waveform, considering that the model objective functions established for such problems are all non-convex, and due to the bispectral characteristics of the cyclic theory, the spectral frequency and the cyclic frequency are coupled with each other during the optimization process, directly optimizing the waveform will face challenges of high computational complexity. Therefore, a decoupled alternating direction multiplier method that can be used for low-intercept waveform design in the field of cyclic stationary theory is newly proposed to solve the optimization model, which can avoid the inverse operation of large-dimensional matrices, and utilize the independent characteristics after decoupling to obtain the LPI waveform through the parallel update of Matlab.

[0018] The beneficial effects of the present invention are as follows: By using the limitation of the similarity constraint, the detection performance of the LPI waveform is guaranteed, and at the same time, the sidelobes of the generated cyclic spectrum are suppressed, achieving the purpose of designing the LPI waveform to cope with the cyclic detection theory of modern intercepting machines. The radar signal based on the LPI waveform generated can resist the cyclic stationary analysis of the third party and ensure the security of radar communication; the designed LPI waveform shows good robustness, higher computational efficiency and scalability. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 FIG. is a flowchart of the decoupled ADMM algorithm for optimizing and designing the LPI waveform based on cyclic spectrum approximation according to the present invention.

[0020] Figure 2 FIG. is the simulation result of the cyclic spectral density under different similarity constraints (section f = 0).

[0021] Figure 3 FIG. is an enlarged view of the spectral line at α = 0.4375.

[0022] Figure 4 、 5 FIG. is a three-dimensional cyclic spectrum comparison diagram of the reference waveform c and the optimized waveform s * .

[0023] Figure 6For the comparison of cyclic spectra at different angles. Specifically, it includes (a) the top view of the reference waveform c; (b) the top view of the optimized waveform s * ; (c) the front view of the α cyclic frequency of the reference waveform c; (b) the optimized waveform s * 's front view of the α cyclic frequency. Detailed implementation manner

[0024] To better describe the embodiments of the present invention, the following definitions are first made:

[0025] Low probability of intercept waveform design: In a modern battlefield environment, the enemy usually uses high-sensitivity electronic reconnaissance equipment (such as enemy radars, communication monitoring equipment, jamming receivers, etc.) to capture and analyze signals. By adopting low probability of intercept waveform design, such as low signal power density, spectrum spreading, frequency hopping, etc., the probability of being detected by the enemy can be significantly reduced, thereby improving one's own survivability.

[0026] Cyclostationary analysis theory: It is a theoretical framework for analyzing non-stationary signals with implicit periodicity. Different from the traditional stationary signal analysis, although the cyclostationary signals do not satisfy the stationarity condition as a whole, their statistical characteristics (such as mean value, autocorrelation function, etc.) show periodic changes over time.

[0027] Cyclic spectrum: It is a high-order spectrum analysis tool in signal processing, used to study the non-linear and non-Gaussian characteristics of signals. It is a frequency-domain tool that can capture the periodicity and phase relationship of signals, and is often used to analyze periodic or repetitive signals, especially in a non-Gaussian noise environment. The cyclic spectrum shows both the frequency and the cyclic frequency of the signal, usually presented in the form of a two-dimensional graph, with the horizontal axis being the frequency of the signal, i.e., the spectral frequency; and the vertical axis being the cyclic frequency. The cyclostationary characteristics of the signal will be manifested as obvious peaks at specific cyclic frequencies.

[0028] Similarity constraint: The similarity constraint in radar waveform design refers to the restrictive conditions imposed on the similarity between the designed waveform and the reference waveform (or between waveform sets). Its purpose is to control the correlation or difference between waveforms while ensuring the performance of the radar system (such as resolution, anti-jamming ability, Doppler tolerance, etc.) to meet specific application requirements.

[0029] Energy constraint: The goal is to ensure that the designed radar waveform has sufficient power to achieve good detection performance, while not exceeding the power limit of the system or consuming excessive power. The energy usage efficiency of the radar system has an important impact on factors such as detection performance, concealment, and system cost.

[0030] ADMM algorithm: An effective algorithm for solving convex optimization problems, especially having significant advantages when dealing with large-scale problems and distributed optimization problems. The core idea of ADMM is to decompose a complex optimization problem into a series of relatively easy-to-solve sub-problems by introducing Lagrange multipliers and decomposition strategies, and then alternately optimize the solutions of these sub-problems to finally reach the global optimal solution. The ADMM algorithm is applicable to a class of convex optimization problems with relatively complex constraint conditions, such as sparse signal recovery, image processing, machine learning and other fields.

[0031] Lagrange multiplier method: A mathematical method for solving optimization problems with constraint conditions. It combines the constraint conditions with the objective function by introducing Lagrange multipliers, thus transforming the constrained optimization problem into an unconstrained optimization problem for solution. Specifically, it introduces Lagrange multipliers (also known as Karush-Kuhn-Tucker (KKT) multipliers) to construct a new Lagrangian function, which is a linear combination of the original objective function and all constraint conditions. Then, by taking the derivative of this Lagrangian function and setting the derivative to zero, the optimal solution can be obtained.

[0032] In an embodiment of the present invention, a decoupled ADMM algorithm for optimizing the design of LPI waveforms based on cyclic spectrum approximation includes the following steps:

[0033] Step 1, set the parameters required for the problem; including a reference waveform c with good detection performance, a similarity constraint threshold θ, the energy E of the transmitted signal, the number N of discrete sampling points, and the Fourier transform matrix F N ;

[0034] Step 2, select an appropriate cyclic spectrum estimation method. Since the direct frequency smoothing algorithm is obtained from the original definition and has better effects, the target problem is formed accordingly to generate a smoothing matrix Π α,f , the cyclic frequency index of the cyclic spectrum is α, and the spectral frequency index is f; Smoothing calculation matrix Π α,f contains information on the penalty weights ω at different cyclic frequency indices α α , and it is necessary to set the penalty weight ω α corresponding to the sidelobe of the generated waveform to be larger, and set the penalty weight ω α corresponding to the main lobe of the generated waveform to be smaller;

[0035] Step 3, based on the given smoothing calculation matrix Π α,f , use the constraints and suppression based on similarity constraints to generate an optimization model for the transmitted waveform with the purpose of optimizing the sidelobe of the cyclic spectrum;

[0036] Step 4, introduce auxiliary variables to realize the conversion from a quartic coupling problem to a quadratic decoupling problem;

[0037] Step 5: In each iteration process, use the CVX solver and Matlab tool to calculate the signal waveform s to be optimized obtained in this iteration, the first and second auxiliary variables b representing the spectral frequency vectors α,f , h α,f and the third auxiliary variable d representing the similarity constraint auxiliary vector, and update the dual variables corresponding to the original variables λ d , and complete a new round of iteration;

[0038] Step 6: Statistically analyze the objective function and various residual information;

[0039] Step 7: Through continuous iteration of Steps 5 to 6, output the optimal anti-cyclostationary analysis LPI waveform, and statistically analyze the three-dimensional and two-dimensional cross-section information of the cyclic spectrum of the obtained waveform.

[0040] As Figure 1 shown, the implementation steps of the embodiment are as follows:

[0041] Let s be the transmit waveform to be optimized, and c be a reference waveform with good detection performance. In the embodiment, c is set as an LFM signal, and Π α,f is a smoothing calculation matrix with a cyclic frequency index of α and a spectral frequency index of f, θ represents the similarity constraint threshold, E is the energy of the transmit signal, E = N, where N is the number of discrete sampling points, and F N is a Fourier transform matrix with N discrete sampling points.

[0042] Establish the following transmit waveform optimization model:

[0043]

[0044] s.t. s H s = N

[0045] |s n - c n | 2 ≤ θ 2 , n = 1, 2, …, N

[0046] where s n is the value of s at the nth sampling point, c n is the value of c at the nth sampling point, ω α is the penalty weight at different cyclic frequency indices α, H is the conjugate transpose, and the middle matrix Γ α,f is defined as follows:

[0047]

[0048] Among them, i and j respectively represent the i-th row and j-th column of the intermediate matrix, M is the number of samples participating in smoothing, k is the smoothing start index, k = mod(N + f - α - M / 2, N), and mod is the remainder function.

[0049] In the emission waveform optimization model, minimizing the objective function reflects the purpose of the model to suppress the sidelobes of the generated cyclic spectrum. After the smoothing calculation matrix Π α,f is given, due to the large penalty weight ω α at the sidelobes and the small penalty weight ω α corresponding to the main lobe of the waveform s, when generating the waveform s, the generated sidelobes need to be small enough to make the objective function value as small as possible, so as to achieve the effect of suppressing the sidelobes. The condition |s n -c n | 2 ≤θ 2 reflects the limitation based on the similarity constraint. s H s = E = N reflects the limitation based on the energy constraint.

[0050] It can be observed that the solution of the above optimization model is a quartic coupling with respect to s. Next, the optimization model is decoupled and optimized. The cyclic frequency index α and the spectral frequency index f are variable parameters, and F N s is a variable independent of the variable parameters α and f. For any α and f, the first auxiliary variable b α,f and the second auxiliary variable h α,f are added for the following transformation:

[0051] b α,f = F N s

[0052] h α,f = F N s

[0053] To improve the iteration efficiency, b α,f = F N s, h α,f = F N s changes so that there is a corresponding single variable for different α and f, thus realizing the decoupling of the calculation. After adding the auxiliary variables b α,f and h α,f , the quartic decoupling problem can be transformed into a quadratic decoupling problem, which is also the most crucial step in solving the optimization model. At the same time, in order to make the problem to be optimized conform to the convex optimization form, a third auxiliary variable d n is added for the value of d at the n-th sampling point. The optimization model is further transformed into:

[0054]

[0055] d = s - c

[0056] |d n | 2 ≤θ 2 , n = 1, 2, …, N

[0057] s H s = N

[0058] f represents for any α, f;

[0059] For this, the objective function in the optimization model is expressed as the augmented Lagrangian function as follows:

[0060]

[0061] ρ1, ρ2 and ρ3 are the first, second and third penalty term coefficients respectively, λ3 are the first, second and third dual variables respectively, and ‖‖ is the 2-norm.

[0062] Then, after the optimization problem is decoupled, the Alternating Direction Method of Multipliers (ADMM) algorithm is used to solve it. The ADMM algorithm is a commonly used optimization algorithm, and its idea is to transform the original problem into an equivalent form and then solve this equivalent problem using the variable alternating iteration method.

[0063] As Figure 1 shown in the flowchart of the decoupled ADMM algorithm for optimizing the design of LPI waveforms based on cyclic spectrum approximation of the present invention, it includes the following steps:

[0064] (1) Update the waveform s to be optimized. In this step, the iterative initial value can be used for the update. Solve the following sub-problem:

[0065]

[0066] s.t. (s (m) ) H s (m) = N

[0067] where the superscript (m) represents the previous iteration value or the m-th iteration value, and the superscript (m + 1) represents the current iteration value or the (m + 1)-th iteration value. The initial value of m is 0.

[0068] This problem is an optimization problem with an equality constraint. The closed-form solution can be obtained:

[0069]

[0070] where ψ is defined as:

[0071]

[0072] (2) Since b α,f and h α,f are essentially auxiliary variables independent of α and f, they can be considered independent of each other, avoiding double summation over α and f. At the same time, the Matlab tool can be used to accelerate the iterative process through operations.

[0073] Fix s (m+1) and update b α,f . Specifically, solve the following sub-problem:

[0074]

[0075] This sub-problem is an unconstrained quadratic extremum problem and can be solved using the ready-made solver CVX. The solution is:

[0076]

[0077] where

[0078]

[0079] E N is the N - order identity matrix, and the process saves the updated value of b α,f for this iteration.

[0080] (3) Update the auxiliary variable h α,f . h α,f and b α,f are symmetric. Update the value of h α,f in a similar way to step (2) as:

[0081]

[0082] Correspondingly,

[0083]

[0084] (4) Update the similarity - constraint auxiliary variable d. It is transformed into updating each component d n , and solve the following given by:

[0085]

[0086] s.t. |d n | 2 ≤ θ 2 , n = 1, 2, …, N

[0087] Each component has the following solution

[0088]

[0089] where

[0090]

[0091] (5) Update the dual variables corresponding to the original variables. The results of applying the gradient ascent method are as follows:

[0092]

[0093] (6) Determine whether another round of iteration is needed after this round and whether the judgment can end. If b α,f , h α,f and F N The residuals of s, the iterative residuals and dual residuals of d and s - c are small enough, and s meets the similarity constraint and energy constraint. Then the iteration can be ended, and at the same time, the numerical value of the objective function, various residual data, and running time at this time are output. If any of the above conditions is not met, then update m = m + 1 and return to step (2) to continue the iteration. Loop like this until the iteration end condition is satisfied.

[0094] The value of the similarity constraint θ affects the balance between detection performance and low intercept performance. According to Figures 2 to 6 The simulation results, it is not difficult to see that as θ increases, the cyclic frequency sidelobes of the cross-section diagram of the normalized value 0 (after normalization) of the spectral frequency index f show a suppression trend (as Figure 2 ), where when θ is 0.3, the normalized value of the cyclic frequency index α is 0.43750.4375, and the normalized value is the index divided by the number of waveform samples. The corresponding original sample cyclic frequency index is 28. At this time, the normalized cyclic spectrum value is optimized to 0.200524, reduced to half of the original waveform (as Figure 3 ), Figure 4 , Figure 5 , Figure 6 Compare the cyclic spectrum characteristics of the reference waveform c and the optimized waveform s * from different angles. The cyclic spectrum values outside the main lobe are basically controlled within 0.2, and can be well "submerged" in Gaussian white noise, thus realizing the LPI performance of anti-cyclic stationary analysis.

Claims

1. A method for generating a low-intercept waveform based on the design of cyclic spectrum approximation, characterized in that, Including the steps: Step of establishing the emission waveform model: Set the similarity constraint threshold, select the cyclic spectrum estimation method and generate the smoothing calculation matrix; Based on the given smoothing calculation matrix, with the similarity constraint and energy constraint as limitations, establish an emission waveform optimization model with the purpose of suppressing the sidelobes of the generated cyclic spectrum; Introduce auxiliary variables into the emission waveform optimization model to obtain the decoupled and optimized emission waveform optimization model, and then transform it into an unconstrained optimization problem through the augmented Lagrangian function; Iteration step: Iteratively solve the unconstrained optimization problem, and update the optimized signal vector and introduce auxiliary variables through the CVX solver and Matlab tool for each iteration until the iteration ends and output the latest optimized signal vector as the final low-intercept waveform vector; Waveform generation step: Generate a low-intercept waveform radar signal using the low-intercept waveform vector.

2. The method according to claim 1, wherein The penalty weight in the smoothing calculation matrix is set to a larger value at the sidelobes of the emission waveform and a smaller value at the main lobe of the emission waveform, so as to suppress the sidelobes of the generated cyclic spectrum in the emission waveform optimization model.

3. The method according to claim 2, wherein Smoothing calculation matrix Π α,f is the smoothing calculation matrix with the cyclic frequency index α and the spectral frequency index f. ω α is the penalty weight at different cyclic frequency indices α. The intermediate matrix Γ α,f Specifically: Where, i and j respectively represent the i-th row and j-th column of the intermediate matrix, M is the number of samples participating in the smoothing, k is the smoothing start index, k = mod(N + f - α - M / 2, N), mod is the remainder function, and N is the number of discrete sampling points.

4. The method according to claim 3, wherein The auxiliary variables include a first auxiliary variable and a second auxiliary variable representing the spectral frequency vector, and a third auxiliary variable representing the similarity constraint auxiliary vector.

5. The method according to claim 3, characterized in that, The emission waveform optimization model is expressed as: s.t.s H s = N |s n -c n | 2 ≤θ 2 , n = 1, 2, …, N Among them, s is the transmit waveform to be optimized, the cyclic frequency index of the cyclic spectrum is α, the spectral frequency index of the cyclic spectrum is f, and Π α,f is the smoothing calculation matrix with the cyclic frequency index α and the spectral frequency index f, and s n is the value of s at the nth sampling point, and c n is the value of c at the nth sampling point, θ is the similarity constraint threshold, and H is the conjugate transpose.

6. The method according to claim 4, wherein The specific unconstrained optimization problem is: Among them, is the objective function of the unconstrained optimization problem; where s is the transmit waveform to be optimized, the cyclic frequency index of the cyclic spectrum is α, the spectral frequency index of the cyclic spectrum is f, Π α,f is the smoothing calculation matrix with cyclic frequency index α and spectral frequency index f, b α,f is the first auxiliary variable, h α,f is the second auxiliary variable, F N is the Fourier transform matrix with N discrete sampling points, means that for any α, f, d is the third auxiliary variable, c is the reference waveform, θ is the similarity constraint threshold, d n is the value of d at the nth sampling point, ρ1, ρ2, and ρ3 are the first, second, and third penalty term coefficients respectively, λ3 are the first, second, and third dual variables respectively, ‖‖ is the 2-norm, H is the conjugate transpose.

7. The method according to claim 6, wherein The iterative solution of the unconstrained optimization problem is specifically to obtain the value of the transmit waveform \(s\) to be optimized in the current iteration through variable alternating iteration (m+1) ; fix \(s\) (m +1) to obtain the value of the first auxiliary variable \(b\) in the current iteration α,f (m+1) ; fix \(s\) (m+1) and \(b\) α,f (m+1) to obtain the value of the second auxiliary variable \(b\) in the current iteration α,f (m+1) ; fix \(s\) (m+1) and update the components of the third auxiliary variable \(d\) n (m+1) thereby obtaining the value of the third auxiliary variable \(d\) in the current iteration (m+1) ; then, according to \(s\) (m+1) , \(b\) α,f (m+1) , \(b\) α,f (m+1) and \(d\) (m+1) obtain the values of the first, second, and third dual variables in the current iteration and The superscript \((m + 1)\) represents the current iteration value, that is, the \((m + 1)\)-th iteration value.

8. The method according to claim 7, wherein The value s of the transmit waveform to be optimized in the current iteration (m+1) is calculated as follows: ψ is the intermediate value; Where, the superscript (m) represents the previous iteration value or the m-th iteration value, and the initial value of m is 0.

9. The method according to claim 8, wherein The first auxiliary variable value b of the current iteration α,f (m+1) , the second auxiliary variable value b α,f (m+1) and the third auxiliary variable value d (m+1) are calculated as follows: is the intermediate value; where, E N is an N-order identity matrix; is the intermediate value; d n (m+1) is the intermediate value; 10. The method according to claim 9, wherein The first, second, and third dual variable values of the current iteration and are calculated as follows:

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