Spectrum shaping method for non-contiguous spectrum signals with low sidelobes

Abstract

The application discloses a low-peak sidelobe non-continuous spectrum signal spectrum shaping method, and belongs to the field of radar waveform design.The present application aims at the problem that the existing spectrum shaping method does not consider the peak sidelobe and trade-off signal-to-interference-and-noise ratio, so that the spectrum template is relatively simple and has more defects.The application comprises the following steps: based on the power spectrum representation of the transmitted signal, the signal-to-interference-and-noise ratio and the weighted peak sidelobe level are obtained, and an exponential logarithmic smoothing function is used to transform the weighted peak sidelobe level to obtain the transformed weighted peak sidelobe level; the target function is weighted; the power spectrum constraint condition is set, and the spectrum shaping optimization problem is established in combination with the target function; the spectrum shaping optimization problem is solved to obtain the optimal power spectrum; the signal-to-interference-and-noise ratio and the peak sidelobe corresponding to the optimal power spectrum are calculated; and the spectrum template of the transmitted signal is obtained according to the relationship between the signal-to-interference-and-noise ratio and the peak sidelobe corresponding to the optimal power spectrum under different weighting coefficients, so that the spectrum shaping is realized.The application is used for spectrum shaping.

Description

Technical Field

[0001] This invention relates to a method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes, belonging to the field of radar waveform design. Background Technology

[0002] Radar is a system that detects targets by emitting and receiving electromagnetic waves. With technological advancements, the number of sensors has exploded, exacerbating the supply and demand problem of spectrum resources. Faced with dense electromagnetic interference on the same frequency, modern radar technology employs receiver interference cancellation and transmitter waveform design to mitigate the effects of interference. However, receiver interference cancellation often results in a loss of signal-to-noise ratio; therefore, transmitter waveform design has gained widespread attention and application.

[0003] To combat co-channel interference, the transmitted signal spectrum can be designed with a concave shape to avoid interfering frequency bands, improving the signal-to-interference-plus-noise ratio (SNR) while maintaining range resolution. However, setting a concave spectrum can raise the sidelobes of the transmitted signal's autocorrelation function, causing nearby weak targets to be blocked. Therefore, the autocorrelation sidelobe level and SNR need to be considered simultaneously in signal waveform design. Since the peak sidelobe represents the lower limit of the signal's correlation function performance in the sidelobe region, the peak sidelobe level is more appropriate for evaluating the blocking effect on weak targets. Researchers have designed transmitted signals by using the peak sidelobe level and SNR as objective functions or constraints for optimization problems, or by making the signal spectrum shape approximate a certain template to achieve performance similar to the template. With the continuous development of research, it has become necessary and urgent to comprehensively consider the SNR and the peak sidelobe level of the autocorrelation function to obtain a spectral template in environments with dense co-channel interference, and to analyze the trade-off between the two performance characteristics.

[0004] Existing technologies use a weighted average of the integral main lobe energy and integral side lobe energy of the signal autocorrelation function to comprehensively evaluate the range resolution and range side lobe level of the air-filtered signal. Since the signal power spectrum and the autocorrelation function are Fourier transforms of each other, a relationship can be established between them. The power spectrum can then be used to represent the integral main lobe energy and integral side lobe energy of the autocorrelation function, and an optimization problem can be established by weighting these two performance parameters. In practical radar transmission systems, the transmitted signal energy is constant. According to Passavar's theorem, energy constraints need to be imposed on the power spectrum. In addition, amplitude constraints also need to be imposed in the interference band. Since the Fourier transform is a linear transform, it is easy to see that the objective function is a convex function, and the constraints are all linear constraints, which are also convex constraints. Therefore, this optimization problem is a convex problem. By solving this problem, the optimal power spectrum is obtained, and the trade-off between range resolution and range side lobe level is analyzed.

[0005] However, the above techniques only consider the sidelobe level of the autocorrelation function, and the setting of the spectral concave depth is relatively simple. They do not deeply consider the signal-to-interference-plus-noise ratio (SIR) level and establish the relationship between the power spectrum and the SIR. At the same time, the above techniques only consider the integral sidelobe level and do not consider the peak sidelobe level.

[0006] Existing signal design methods use relatively simple 0 / 1 templates for their spectral templates, which cannot accurately measure the correlation performance of signals. Furthermore, the solution to the infinite norm problem of peak sidelobes in existing signal design methods is transformed into the p-norm, which has an inactive region. Therefore, the spectral templates used in existing signal design methods and the handling of peak sidelobes have significant defects and shortcomings. Existing spectrum shaping methods also do not consider the relationship between peak sidelobes and signal-to-interference-plus-noise ratio (SIR), making it impossible to conduct a trade-off analysis between the two. Summary of the Invention

[0007] To address the problem that existing spectrum shaping methods do not consider peak sidelobes and weigh the signal-to-interference-plus-noise ratio, resulting in relatively simple spectrum templates and significant drawbacks, this invention provides a spectrum shaping method for discontinuous spectrum signals with low peak sidelobes.

[0008] The present invention provides a method for spectral shaping of discontinuous spectrum signals with low peak sidelobes, comprising:

[0009] Step 1: Based on the power spectrum, represent the signal-to-interference-plus-noise ratio and weighted peak sidelobe level of the transmitted signal, and use an exponential logarithmic smoothing function to transform the weighted peak sidelobe level to obtain the transformed weighted peak sidelobe level;

[0010] Step 2: Weight the signal-to-interference-plus-noise ratio (SINR) based on the power spectrum representation and the converted weighted peak sidelobe level to obtain the objective function;

[0011] Step 3: Set constraints on the power spectrum and establish a spectrum shaping optimization problem based on the objective function;

[0012] Step 4: Determine that the spectrum shaping optimization problem has a unique optimal solution, and solve the spectrum shaping optimization problem to obtain the optimal power spectrum; calculate the signal-to-interference-plus-noise ratio and peak sidelobes corresponding to the optimal power spectrum;

[0013] Step 5: Adjust the weighting coefficients of the objective function, return to Step 2, obtain the optimal power spectrum under different weighting coefficients, and obtain the signal-to-interference-plus-noise ratio and peak sidelobe corresponding to the optimal power spectrum under different weighting coefficients;

[0014] Step 6: Obtain the spectrum template of the transmitted signal based on the relationship between the signal-to-interference-plus-noise ratio and the peak sidelobe corresponding to the optimal power spectrum under different weighting coefficients, thereby realizing spectrum shaping.

[0015] According to the low-peak-sidelobe discontinuous spectrum signal spectrum shaping method of the present invention, step one of obtaining the signal-to-interference-plus-noise ratio based on power spectrum representation includes:

[0016] Signal-to-interference-plus-noise ratio (SINR) of the transmitted signal obtained by matched filtering the radar echo. MF Represented as:

[0017]

[0018] In the formula This is the zero-padding form of the transmitted signal S, which includes N points: S = [s1,...,s...]. N ];

[0019] but Represented as:

[0020] R is the covariance matrix;

[0021] maxSINR MF The problem is transformed into

[0022] Assuming the total dimension K of the covariance matrix is ​​greater than 50, then the covariance matrix R is approximately:

[0023] R≈F H ΛF,

[0024] In the formula, F is the Fourier transform matrix, and Λ is the diagonal matrix of interference power spectral coefficients:

[0025]

[0026] but Represented as:

[0027]

[0028] Here are the coefficients of the interference power spectral density at the k-th frequency unit:

[0029]

[0030] Λ k,k These are the diagonal elements of the diagonal matrix Λ representing the interference power spectral coefficients;

[0031] p(k) is the power spectrum;

[0032] Then maxSINR MF The problem is transformed into the following optimization problem:

[0033] minf1(p) (3)

[0034] In the formula, f1(p) is the signal-to-interference-plus-noise ratio based on the power spectrum.

[0035]

[0036] According to the low-peak-side-lobes discontinuous spectrum signal spectrum shaping method of the present invention, step one, the method for obtaining the weighted peak-side-lobe level of the transmitted signal based on the power spectrum representation includes:

[0037] The weighted peak sidelobe level WPSL1 is defined as:

[0038] WPSL1 = max{w n |r n | 2}, n=[-N+1,-1]∪[1,N-1] (5)

[0039] In the formula w n r is the weighting coefficient of the autocorrelation sidelobe at point n. n The autocorrelation at point n;

[0040] The autocorrelation function r of the transmitted signal, expressed by the inverse Fourier transform based on the power spectrum, is:

[0041]

[0042] In the formula, P is the power spectrum vector: P = [p(1), p(2), ..., p(K)], K = 2N;

[0043]

[0044] Then, the weighted peak sidelobe level WPSL1 based on the power spectrum representation of formula (5) is further expressed as:

[0045]

[0046] in:

[0047]

[0048] By transforming the weighted peak sidelobe level WPSL1, we obtain the weighted peak sidelobe level WPSL2 based on the power spectrum representation:

[0049]

[0050] According to the low-peak-side-lobes discontinuous spectrum signal spectrum shaping method of the present invention, in step one, the method for transforming the infinite norm problem of the weighted peak-side-lobes level is as follows:

[0051] Define the exponential-logarithmic smoothing function f(z):

[0052]

[0053] In the formula, Q is the exponential logarithmic smoothing coefficient, z i Let be the i-th element in vector z.

[0054] When Q is greater than 2, the exponential-log smoothing function f(z) has no inactive region, satisfying:

[0055]

[0056] When Q is less than or equal to 2, the exponential logarithmic smoothing function defined by equation (11) is convex and Lipschitz continuous, then the weighted peak sidelobe level is transformed into a smoothed form expressed in terms of power spectrum:

[0057]

[0058] According to the low-peak-sidelobe discontinuous spectrum signal spectrum shaping method of the present invention, the objective function in step two is defined as f2(p):

[0059]

[0060] In the formula, λ is the weighting coefficient.

[0061] The beneficial effects of this invention are as follows: This invention proposes a discontinuous spectrum shaping method that combines peak sidelobes and signal-to-interference-plus-noise ratio (SINR) to overcome the deficiencies in existing technologies. It first establishes a relationship between SINR, peak sidelobes, and signal power spectrum, considers multiple criterion trade-offs, uses index weighting to construct the objective function, and constructs a multi-criteria, multi-constraint optimization problem. Through the derivation of the Hessian matrix and proof of its semi-definiteness, the convexity of this optimization problem is proven. Finally, the optimization problem is solved using a convex optimization toolbox. Simulation results verify that this invention can comprehensively consider SINR and peak sidelobe performance, and by changing the weighting coefficients, the coupling relationship between SINR and peak sidelobes can be obtained.

[0062] This invention establishes a weighted objective function by comprehensively considering the peak sidelobe level and signal-to-interference-plus-noise ratio (SNR) of the signal's autocorrelation function, and then constrains this function to create a spectrum shaping optimization problem. By fixing the weighting values, the optimal power spectrum is obtained by solving the optimization problem, which can serve as a template for subsequent signal design. Furthermore, the upper limits of peak sidelobe and SNR can be calculated. In addition, by adjusting the weighting values, this invention can obtain different power spectrum templates and their corresponding peak sidelobe and SNR performance. This allows for analysis of the trade-off between these two performance characteristics, thus enabling a better balance between them in transmit signal design. This effectively solves the problem of existing technologies being unable to simultaneously consider peak sidelobe performance and SNR performance for spectrum shaping and to balance these two aspects. Attached Figure Description

[0063] Figure 1 This is a flowchart of the method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes as described in this invention;

[0064] Figure 2 The output graph shows the relationship between λ and SINR.

[0065] Figure 3 The output graph shows the relationship between λ and WPSL.

[0066] Figure 4 This is a graph showing the relationship between SINR and WPSL as λ changes.

[0067] Figure 5 This is the output diagram of the optimal power spectrum shape when λ = 0.7962;

[0068] Figure 6 The output diagram shows the optimal shape of the autocorrelation function when λ = 0.7962.

[0069] Figure 7 This is an output graph showing the relationship between λ and SINR after changing the spectral indentation depth;

[0070] Figure 8 This is the output graph showing the relationship between λ and WPSL after changing the spectral indentation depth;

[0071] Figure 9 This is an output graph showing the relationship between SINR and WPSL as the spectral depth changes with λ. Detailed Implementation

[0072] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0073] It should be noted that, unless otherwise specified, the embodiments and features described in the present invention can be combined with each other.

[0074] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the scope of the invention.

[0075] Specific Implementation Method 1: Combination Figure 1 As shown, this invention provides a method for spectral shaping of discontinuous spectrum signals with low peak sidelobes, comprising:

[0076] Step 1: Based on the power spectrum, the signal-to-interference-plus-noise ratio (SINR) and weighted peak sidelobe level of the transmitted signal are represented, and the weighted peak sidelobe level is transformed using an exponential logarithmic smoothing function to obtain the transformed weighted peak sidelobe level.

[0077] Step 2: Weight the signal-to-interference-plus-noise ratio (SINR) based on the power spectrum representation and the converted weighted peak sidelobe level to obtain the objective function;

[0078] Step 3: Set constraints on the power spectrum and establish a spectrum shaping optimization problem based on the objective function;

[0079] Step 4: Determine that the spectrum shaping optimization problem has a unique optimal solution, and solve the spectrum shaping optimization problem to obtain the optimal power spectrum; calculate the signal-to-interference-plus-noise ratio and peak sidelobes corresponding to the optimal power spectrum;

[0080] Step 5: Adjust the weighting coefficients of the objective function, return to Step 2, obtain the optimal power spectrum under different weighting coefficients, and obtain the signal-to-interference-plus-noise ratio and peak sidelobe corresponding to the optimal power spectrum under different weighting coefficients;

[0081] Step 6: Obtain the spectrum template of the transmitted signal based on the relationship between the signal-to-interference-plus-noise ratio and the peak sidelobe corresponding to the optimal power spectrum under different weighting coefficients, thereby realizing spectrum shaping.

[0082] Furthermore, in step one, the method for obtaining the signal-to-interference-plus-noise ratio based on power spectrum representation includes:

[0083] Signal-to-interference-plus-noise ratio (SINR) of the transmitted signal obtained by matched filtering the radar echo. MF (Signal to Interference plus Noise Ratio, SINR) is represented as:

[0084]

[0085] In the formula This is the zero-padding form of the transmitted signal S, which includes N points: S = [s1,...,s...]. N ];

[0086] but Represented as:

[0087] R is the covariance matrix;

[0088] For a radar system, the total energy of its transmitted signal is a constant value; therefore, the factor affecting the SINR is its denominator. In detection, the higher the SINR, the higher the detection probability; therefore, the maximum SINR can be set. MF The problem is transformed into

[0089] according to Theorem: When the dimension K of the covariance matrix is ​​sufficiently large, for example, assuming the total dimension K of the covariance matrix is ​​greater than 50, then the covariance matrix R is approximately:

[0090] R≈F H ΛF,

[0091] In the formula, F is the Fourier transform matrix, and Λ is the diagonal matrix of interference power spectral coefficients:

[0092]

[0093] but Represented as:

[0094]

[0095] Here are the coefficients of the interference power spectral density at the k-th frequency unit:

[0096]

[0097] Λ k,k Λ represents the diagonal elements of the diagonal matrix Λ of the interference power spectral coefficients; p(k) represents the power spectrum.

[0098] Clearly, equation (2) is the weighted form of the power spectral density. It is obvious that λ k The value is non-zero within a finite set of stopband pointers, and zero at other positions. λ k The weights of the leakage power penalty function in each stopband determine the depth of the power spectrum dip in each stopband. In detection, a higher SINR corresponds to a higher detection probability; therefore, the max SINR is further... MF The problem is transformed into the following optimization problem:

[0099] min f1(p) (3)

[0100] In the formula, f1(p) is the signal-to-interference-plus-noise ratio based on the power spectrum.

[0101]

[0102] In step one, the method for obtaining the weighted peak sidelobe level of the transmitted signal based on the power spectrum representation includes:

[0103] Setting a concave pattern in the spectrum can raise the sidelobes of the transmitted signal's autocorrelation function, causing weak targets near strong targets to be obscured. Therefore, it is necessary to optimize the autocorrelation sidelobe level of the signal. The peak value of the autocorrelation function defines the highest value of the autocorrelation function in the sidelobe region, representing the lower limit of the autocorrelation function sidelobe performance. Therefore, the peak sidelobe level is used to measure sidelobe performance. Since different sidelobe regions have different focuses, the Weighted Peak Sidelobe Level (WPSL) is introduced to measure sidelobe performance. The Weighted Peak Sidelobe Level WPSL1 is defined as follows:

[0104] WPSL1 = max{w n |r n | 2}, n=[-N+1,-1]∪[1,N-1] (5)

[0105] In the formula w n r is the weighting coefficient of the autocorrelation sidelobe at point n. n The autocorrelation at point n;

[0106] The autocorrelation function r of the transmitted signal, expressed by the inverse Fourier transform based on the power spectrum, is:

[0107]

[0108] In the formula, P is the power spectrum vector: P = [p(1), p(2), ..., p(K)], K = 2N;

[0109]

[0110] Then, the weighted peak sidelobe level WPSL1 based on the power spectrum representation of formula (5) is further expressed as:

[0111]

[0112] in:

[0113]

[0114] By transforming the weighted peak sidelobe level WPSL1, we obtain the weighted peak sidelobe level WPSL2 based on the power spectrum representation:

[0115]

[0116] Since WPSL is an infinite norm problem, it is not conducive to solving and transforming the problem; in step one, the method for transforming the infinite norm problem of weighted peak sidelobe level is as follows:

[0117] Define the exponential-logarithmic smoothing function f(z):

[0118]

[0119] In the formula, Q is the exponential logarithmic smoothing coefficient, z i Let be the i-th element in vector z.

[0120] When Q is large, for example, Q is greater than 2, the exponential-log smoothing function f(z) has no inactive region, satisfying:

[0121]

[0122] In addition, when Q is less than or equal to 2, the exponential logarithmic smoothing function defined by equation (11) is convex and Lipschitz continuous, then the weighted peak sidelobe level is transformed into a smoothed form expressed in terms of power spectrum:

[0123]

[0124] Furthermore, radar waveforms typically need to meet multiple performance requirements during the design process, and these requirements are often coupled and contradictory. Therefore, it is necessary to weight these requirements to achieve a trade-off. To ensure that the radar waveform achieves good anti-jamming capability and weak target detection performance, both the signal-to-interference-plus-noise ratio (SINR) and the weighted peak sidelobe level need to be considered simultaneously. Therefore, a weighted form for SINR and weighted integral sidelobe level is defined.

[0125] In step two, the objective function is defined as f2(p):

[0126]

[0127] In the formula, λ is the weighting coefficient.

[0128] It is evident that the weighting function is a function of the power spectrum. Besides the objective function, the power spectrum should satisfy several constraints.

[0129] Furthermore, in step three, the power spectrum is constrained:

[0130] First, for energy constraints, the total energy of the transmitted signal is defined as 1. According to Passavar's theorem, we get:

[0131]

[0132] Meanwhile, all power spectrum values ​​are non-negative. Furthermore, to prevent some stopband positions in the power spectrum from becoming too high during optimization, an upper limit for the power spectrum at the stopband positions is set as follows:

[0133]

[0134] In the formula ε(Ω) u ) represents the stopband located at Ω u The upper limit of the power spectrum at p(Ω)u ) represents the location of Ω u The power spectrum at Ω = [Ω1,...,Ω] U Let ] be the set of stopband locations, u = 1, 2, 3, ..., U, where U is the total number of stopband locations;

[0135] Combining the objective function with the above constraints, we can establish a spectrum shaping optimization problem:

[0136]

[0137] In step four, the method for determining that the spectrum shaping optimization problem has a unique optimal solution includes obtaining the Hessian matrix of the function f:

[0138] To solve the optimization problem defined in equation (17), we first analyze the optimization problem. All constraints in the optimization problem are linear constraints, which are convex constraints; therefore, we analyze the convexity of the objective function. The first term of the objective function is a linear term and a convex function; therefore, we analyze the convexity of the second term.

[0139] For ease of formula writing, p(k) will be replaced with p in the following sections. k ;

[0140] To analyze the convexity of the second term in the optimization problem, let the function f be:

[0141]

[0142] Then we can analyze the properties of the Hessian matrix by examining the function f.

[0143] For the power spectrum p at the m-th position in the power spectrum vector P m Find the first-order partial derivative:

[0144]

[0145] make but:

[0146]

[0147] Then f with respect to the power spectrum p m The first derivative is expressed as:

[0148]

[0149] In f with respect to the power spectrum p m Based on the first derivative, and with respect to the power spectrum p at the l-th position l By taking the second-order partial derivative, we can obtain the Hessian matrix of the function f:

[0150]

[0151] In step four, the method for determining that the spectrum shaping optimization problem has a unique optimal solution also includes determining that the Hessian matrix of the function f is a positive semi-definite matrix:

[0152] The Hessian matrix of function f:

[0153]

[0154] Perform the following variable substitutions:

[0155]

[0156] Where m = 1, ..., 2N, n = 0, ..., 2N-1

[0157] but:

[0158]

[0159] Again To conduct the analysis, let:

[0160]

[0161] First, let's analyze matrix A. m,l , matrix A m,l Decomposed into a matrix sum matrix

[0162]

[0163] Easy-to-understand matrix All elements in the matrix are equal, its determinant is 0, and it is a positive semi-definite matrix.

[0164] Next, we will analyze the matrix.

[0165] Calculate matrix The determinant det(A) 1 )get:

[0166]

[0167] because:

[0168]

[0169] Therefore: det(A) 1 )=0(28)

[0170] Then matrix It is a positive semi-definite matrix, since Furthermore, matrix A m,l It is a positive semi-definite matrix;

[0171] Similarly, determine B.m,l and C m,l It is a positive semi-definite matrix:

[0172] matrix Divided into two parts in in Easy-to-understand matrix All elements in the matrix are equal, therefore the matrix is The determinant is 0, making it a semi-positive definite matrix. The following matrix... The determinant can also be proven to be 0, therefore the matrix It is a positive semi-definite matrix.

[0173] Finally, for matrix C m,l Perform the analysis. Matrix C m,l It can also be divided into two parts. in It can be seen from the expression Therefore matrix The determinant is 0, making it a semi-positive definite matrix. The determinant is 0, therefore matrix C m,l It is a positive semi-definite matrix.

[0174] Therefore, the Hessian matrix of function f is a positive semi-definite matrix after analysis. Since function f is a convex function, the problem is a convex problem. The spectrum shaping optimization problem represented by equation (17) has a unique optimal solution.

[0175] In step four, the method for obtaining the optimal power spectrum is as follows:

[0176] The optimal power spectrum P* is obtained by solving the spectrum shaping optimization problem using the optimization toolbox. Then, the corresponding signal-to-interference-plus-noise ratio and peak sidelobe are obtained by calculating using equations (1) and (5).

[0177] Finally, in step six, a two-dimensional Pareto curve is obtained based on the relationship between the signal-to-interference-plus-noise ratio (SIR) and peak sidelobes corresponding to the optimal power spectrum under different weighting coefficients. The trade-off between SIR and peak sidelobes is analyzed based on the two-dimensional Pareto curve. Then, weighting coefficients are selected according to the performance required by the radar system to determine the spectrum template.

[0178] In step four, the nature of the optimization problem was analyzed, revealing it to be a convex optimization problem. Therefore, given the stopband location and a determined weighting value λ, a unique optimal power spectral density P* and its corresponding SINR and WPSL can be obtained. Adjusting the weighting value λ yields different optimal power spectral densities P* and corresponding achievable upper limits for SINR and WPSL performance. Next, two-dimensional Pareto curves showing how SINR and WPSL change with the weighting value λ can be obtained, allowing analysis of the trade-off between the two. Based on the performance requirements of the radar system, a better weighting value can be selected for designing the transmitted signal. Specific implementation examples:

[0180] First, set the following simulation conditions: N is 256, the frequency stopband dip position is Ω=[0.0988,0.1469]∪[0.2593,0.2840]∪[0.6074,0.6938]∪[0.8185,0.8556], the weighting value λ of SINR and WPSL varies from 0.1 to 0.8, and the autocorrelation sidelobe weighting value w n =1, n=1,...2N-1, Interference intensity at the passband The interference intensity at the stopband is 0. Given a set of Gaussian distributed random numbers with a mean of 10 and a variance of 0.1, ε(Ω) u ),Ω u ∈Ω is set to -20dB. Simulation results show the SINR variation with λ as follows: Figure 2 As shown, the variation of WPSL with λ is as follows: Figure 3 As shown, the relationship between SINR and WPSL is as follows: Figure 4 As shown, it can be seen that as λ increases, the weight of the SINR term increases, while the weight of the WPSL term decreases. Ultimately, the optimal spectrum obtained through spectral shaping corresponds to a higher SINR and a lower WPSL. This indicates that a larger weight during the shaping process signifies greater emphasis on this performance, resulting in a better final optimized performance. Furthermore, it can be seen that... Figure 4 It can be seen from this that as λ changes, As the value changes from 0 to 100, SINR changes from 0.01 to +∞, while WPSL changes from -23.5dB to -24.5dB. It can be seen that due to the influence of the spectral dip on the signal autocorrelation, there is a design upper limit for WPSL after determining the location of the spectral dip, and the performance of WPSL is contradictory to SINR.

[0181] The optimization results for λ = 0.7962 are given below. The optimal power spectrum shape is as follows: Figure 5 As shown, the corresponding autocorrelation function is as follows: Figure 6 As shown, by Figure 5 It is evident that there are deep spectral dips in the designated unusable frequency bands, resulting in a high signal-to-interference-plus-noise ratio. Figure 6 It can be seen that the peak sidelobe of the autocorrelation function is -23.5532dB.

[0182] Finally, set ε(Ω) u ),Ω u With ∈Ω = -40dB and other conditions remaining constant, the simulation results show that SINR varies with λ as follows: Figure 7 As shown, the variation of WPSL with λ is as follows: Figure 8 As shown, the relationship between SINR and WPSL is as follows: Figure 9 As shown, it can be observed that the deeper the spectral dip, the more severe the damage to the autocorrelation function, and the worse the WPSL performance can be obtained, while the SINR trend remains basically unchanged.

[0183] While the invention has been described herein with reference to specific embodiments, it should be understood that these embodiments are merely examples of the principles and applications of the invention. Therefore, it should be understood that many modifications can be made to the exemplary embodiments, and other arrangements can be designed without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that different dependent claims and features described herein can be combined in ways different from those described in the original claims. It is also understood that features described in conjunction with individual embodiments can be used in other described embodiments.

Claims

1. A method for spectral shaping of discontinuous spectrum signals with low peak sidelobes, characterized in that... include, Step 1: Based on the power spectrum, represent the signal-to-interference-plus-noise ratio and weighted peak sidelobe level of the transmitted signal, and use an exponential logarithmic smoothing function to transform the weighted peak sidelobe level to obtain the transformed weighted peak sidelobe level; Step 2: Weight the signal-to-interference-plus-noise ratio (SINR) based on the power spectrum representation and the converted weighted peak sidelobe level to obtain the objective function; Step 3: Set constraints on the power spectrum and establish a spectrum shaping optimization problem based on the objective function; Step 4: Determine that the spectrum shaping optimization problem has a unique optimal solution, and solve the spectrum shaping optimization problem to obtain the optimal power spectrum; calculate the signal-to-interference-plus-noise ratio and peak sidelobes corresponding to the optimal power spectrum; Step 5: Adjust the weighting coefficients of the objective function, return to Step 2, obtain the optimal power spectrum under different weighting coefficients, and obtain the signal-to-interference-plus-noise ratio and peak sidelobe corresponding to the optimal power spectrum under different weighting coefficients; Step 6: Obtain the spectrum template of the transmitted signal based on the relationship between the signal-to-interference-plus-noise ratio and the peak sidelobe corresponding to the optimal power spectrum under different weighting coefficients, thereby realizing spectrum shaping.

2. The method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes according to claim 1, characterized in that, In step one, the methods for obtaining the signal-to-interference-plus-noise ratio based on power spectrum representation include: Signal-to-interference-plus-noise ratio (SINR) of the transmitted signal obtained by matched filtering of the radar echo. MF Represented as: In the formula This is the zero-padding form of the transmitted signal S, which includes N points: S = [s1,...,s...]. N ]; but Represented as: R is the covariance matrix; maxSINR MF The problem is transformed into Assuming the total dimension K of the covariance matrix is ​​greater than 50, then the covariance matrix R is approximately: R≈F H ΛF, In the formula, F is the Fourier transform matrix, and Λ is the diagonal matrix of interference power spectral coefficients: but Represented as: Here are the coefficients of the interference power spectral density at the k-th frequency unit: Λ k,k These are the diagonal elements of the diagonal matrix Λ representing the interference power spectral coefficients; p(k) is the power spectrum; Then maxSINR MF The problem is transformed into the following optimization problem: minf1(p) (3) In the formula, f1(p) is the signal-to-interference-plus-noise ratio based on the power spectrum. 。 3. The method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes according to claim 2, characterized in that, In step one, the method for obtaining the weighted peak sidelobe level of the transmitted signal based on the power spectrum representation includes: The weighted peak sidelobe level WPSL1 is defined as: WPSL1=max{w n |r n | 2 },n=[-N+1,-1]∪[1,N-1] (5) In the formula w n r is the weighting coefficient of the autocorrelation sidelobe at point n. n The autocorrelation at point n; The autocorrelation function r of the transmitted signal, expressed by the inverse Fourier transform based on the power spectrum, is: In the formula, P is the power spectrum vector: P = [p(1), p(2), ..., p(K)], K = 2N; Then, the weighted peak sidelobe level WPSL1 based on the power spectrum representation of formula (5) is further expressed as: in: By transforming the weighted peak sidelobe level WPSL1, we obtain the weighted peak sidelobe level WPSL2 based on the power spectrum representation: 。 4. The method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes according to claim 3, characterized in that, In step one, the method for transforming the infinite norm problem of the weighted peak sidelobe level is as follows: Define the exponential-logarithmic smoothing function f(z): In the formula, Q is the exponential logarithmic smoothing coefficient, z i Let be the i-th element in vector z. When Q is greater than 2, the exponential-log smoothing function f(z) has no inactive region, satisfying: When Q is less than or equal to 2, the exponential logarithmic smoothing function defined by equation (11) is convex and Lipschitz continuous, then the weighted peak sidelobe level is transformed into a smoothed form expressed in terms of power spectrum: 。 5. The method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes according to claim 4, characterized in that, In step two, the objective function is defined as f2(p): In the formula, λ is the weighting coefficient.

6. The method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes according to claim 5, characterized in that, In step three, the power spectrum is constrained: Define the total energy of the transmitted signal as 1, and according to Passavar's theorem: Set the upper limit of the power spectrum at the stopband position: In the formula ε(Ω) u ) represents the stopband located at Ω u The upper limit of the power spectrum at p(Ω) u ) represents the location of Ω u The power spectrum at Ω = [Ω1,...,Ω] U Let ] be the set of stopband locations, u = 1, 2, 3, ..., U, where U is the total number of stopband locations; Based on the objective function, establish the spectrum shaping optimization problem: 。 7. The method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes according to claim 6, characterized in that, In step four, the method for determining that the spectrum shaping optimization problem has a unique optimal solution includes obtaining the Hessian matrix of the function f: For ease of formula writing, p(k) will be replaced with p in the following sections. k ; To analyze the convexity of the second term in the optimization problem, let the function f be: For the power spectrum p at the m-th position in the power spectrum vector P m Find the first-order partial derivative: make but: Then f with respect to the power spectrum p m The first derivative is expressed as: In f with respect to power spectrum p m Based on the first derivative, and with respect to the power spectrum p at the l-th position l By taking the second-order partial derivative, we can obtain the Hessian matrix of the function f: 。 8. The method for shaping the spectrum of discontinuous spectrum signals with low peak sidelobes according to claim 7, characterized in that, In step four, the method for determining that the spectrum shaping optimization problem has a unique optimal solution also includes determining that the Hessian matrix of the function f is a positive semi-definite matrix: The Hessian matrix of function f: Perform the following variable substitutions: Where m = 1, ..., 2N, n = 0, ..., 2N-1 but: Again To conduct the analysis, let: Let matrix A m,l Decomposed into a matrix sum matrix matrix All elements in the matrix are equal, its determinant is 0, and it is a positive semi-definite matrix. Calculate matrix The determinant det(A) 1 )get: because: Therefore: det(A) 1 )=0(28) Then matrix It is a positive semi-definite matrix, and therefore matrix A m,l It is a positive semi-definite matrix; Similarly, determine B. m,l and C m,l It is a positive semi-definite matrix; Therefore, the Hessian matrix of function f is a positive semi-definite matrix, and function f is a convex function. The spectrum shaping optimization problem represented by equation (17) has a unique optimal solution.

9. The method for shaping the spectrum of a discontinuous spectrum signal with low peak sidelobes according to claim 8, characterized in that, In step four, the method for obtaining the optimal power spectrum is as follows: The optimal power spectrum P* is obtained by solving the spectrum shaping optimization problem using the optimization toolbox. Then, the corresponding signal-to-interference-plus-noise ratio and peak sidelobe are obtained by calculating using equations (1) and (5).

10. The method for shaping the spectrum of a discontinuous spectrum signal with low peak sidelobes according to claim 9, characterized in that, In step six, a two-dimensional Pareto curve is obtained based on the relationship between the signal-to-interference-plus-noise ratio (SIR) and peak sidelobes corresponding to the optimal power spectrum under different weighting coefficients. The trade-off between SIR and peak sidelobes is analyzed based on the two-dimensional Pareto curve. Then, weighting coefficients are selected and the spectrum template is determined according to the performance required by the radar system.

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