Design method of frequency hopping signal between pulses of discontinuous spectrum

Abstract

A method for designing a non-continuous spectrum inter-pulse frequency hopping signal belongs to the field of radar waveform design. The present invention addresses the problems of low efficiency and high computational complexity of the optimization process in existing non-continuous spectrum waveform designs. It includes obtaining the signal center autocorrelation function and the discretized center autocorrelation expression based on the signal time domain expression and echo processing process of the inter-pulse frequency hopping signal; using the weighted integral sidelobe of the discrete center autocorrelation as the objective function, and considering the signal's anti-co-frequency interference performance as a constraint condition to establish a signal design optimization problem; then, under the setting that the amplitude p(f) at the frequency offset f in the signal autocorrelation function is the optimal amplitude function, the initial value of the optimized frequency offset f is calculated, and the signal design optimization problem is converted into a frequency optimization problem with a differentiable objective function and a convex set constraint; solving the frequency optimization problem to obtain the final optimization result. The method of the present invention realizes the design of an inter-pulse frequency hopping signal.

Description

Technical Field

[0001] The invention relates to a method for designing a discontinuous spectrum inter-pulse frequency hopping signal, and belongs to the field of radar waveform design. Background Art

[0002] When radar is operating, the accuracy and resolution of its acquisition of information such as target position and velocity are proportional to the bandwidth of the transmitted signal. Traditional radars typically use a continuous frequency band to meet performance requirements. With the advancement of electronic information technology, radar has been widely used in fields such as ocean exploration and weather forecasting. Simultaneously, the use of various electronic systems, such as radio stations, jammers, and mobile communications, is also increasing. This has led to a growing demand for spectrum. However, due to the limited frequencies that can be generated and processed by current software and hardware, as well as security concerns associated with using high-frequency bands, available spectrum resources are limited, and spectrum congestion is becoming increasingly prominent.

[0003] Furthermore, the interplay of modern jamming and anti-jamming technologies has further complicated the radar's operating electromagnetic environment. Consequently, maintaining or improving radar performance has become a challenging issue. Among the currently considered solutions, waveform design leverages the various degrees of freedom of the transmit waveform to ensure radar performance. Discontinuous spectrum waveform design is one such approach. This approach improves the efficiency of existing spectrum resources by synthesizing a wide-bandwidth signal for transmission using discontinuous, quiet frequency bands, ensuring high range resolution. Furthermore, by notching the signal spectrum in the jamming frequency band, it mitigates co-channel interference, ensuring the radar's signal-to-interference-noise ratio (SIN) and improving its detection performance.

[0004] However, the discontinuity of the signal spectrum can lead to poor signal autocorrelation performance, that is, increased sidelobe levels. When using matched filtering for radar target detection, high sidelobes can lead to weak target masking or false target detection. To mitigate this adverse effect, the discontinuous spectrum signal can be optimized. Quantitative descriptions of autocorrelation sidelobe performance include the integrated sidelobe level and the peak sidelobe level. The integrated sidelobe level is the bi-norm of the signal autocorrelation function outside the mainlobe region, reflecting the average level of the signal sidelobes and is applicable to scenarios with uniformly distributed clutter.

[0005] Optimization techniques for discontinuous spectrum waveforms, which use signal frequency as a degree of freedom, typically involve highly non-convex optimization problems. Existing optimization techniques, such as random evolutionary algorithms like genetic algorithms, particle swarm optimization, and simulated annealing, are inefficient. Greedy downhill algorithms like gradient descent and the MM method only converge locally for non-convex optimization problems and require finding a good initial solution. Directly calculating the objective function gradient during the optimization process is computationally complex. In engineering applications, fast and efficient optimization methods for discontinuous spectrum signals are needed. Summary of the Invention

[0006] Aiming at the problems of low efficiency and high computational complexity of the optimization process in the existing discontinuous spectrum waveform design, the present invention provides a method for designing a discontinuous spectrum inter-pulse frequency hopping signal.

[0007] A method for designing a discontinuous spectrum inter-pulse frequency hopping signal of the present invention comprises:

[0008] A one-dimensional range image of the pulse signal is obtained based on the signal time domain expression of the inter-pulse frequency hopping signal and the echo processing process; a signal center autocorrelation function of the pulse signal is obtained from the one-dimensional range image, and the signal center autocorrelation function is discretized to obtain a discretized center autocorrelation expression;

[0009] The signal design optimization problem is established by taking the weighted integrated sidelobe of discrete center autocorrelation as the objective function and considering the signal's anti-co-channel interference performance as the constraint condition;

[0010] Based on the continuous distribution of the frequency offset of the pulse signal and the ability to modulate the amplitude of each pulse, a signal autocorrelation function expression is established with the amplitude of each pulse as a variable; under the setting that the amplitude p(f) at the frequency offset f in the signal autocorrelation function is the optimal amplitude function, an initial value of the optimized frequency offset f is calculated, and based on the initial value, the signal design optimization problem is converted into a frequency optimization problem with a differentiable objective function and a convex set constraint;

[0011] The frequency optimization problem is solved by using the non-uniform sampling Fourier transform accelerated gradient descent method, and the final optimization result is obtained to realize the design of the inter-pulse frequency hopping signal.

[0012] According to the method for designing a discontinuous spectrum inter-pulse frequency hopping signal of the present invention, the time domain expression of the inter-pulse frequency hopping signal is:

[0013]

[0014] Where s T (t) is the pulse signal in the time domain, t is time, N is the number of pulses, A n is the amplitude of the nth pulse signal, f n is the frequency deviation of the nth pulse signal, f c is the pulse signal carrier frequency, T r is the pulse signal period, φ n is the initial phase of the nth pulse signal, p(t) is the pulse signal intra-pulse modulation complex envelope;

[0015] A n =1, so that the pulse signal satisfies the constant modulus constraint;

[0016] The echo processing process includes:

[0017] After the pulse signal is reflected by the target, the received signal S r (t) is:

[0018]

[0019] Where α0 is the target reflection coefficient, τ0 is the target round-trip delay, and n0(t) is the echo noise;

[0020] Assume that the target reflection coefficient α0 does not change with frequency: α0 = 1;

[0021] The received signal S r (t) Sampling at the same distance position to obtain the discrete form s of the pulse signal r :

[0022]

[0023] Where s r (N) is the Nth discrete signal of the pulse signal, τ1 is the time delay corresponding to each pulse sampling point, τ is the time delay, g(τ) is the range ambiguity function of the pulse signal, and z(n) is the echo noise after matched filtering;

[0024] According to formula (3), the one-dimensional range image r(τ) of the pulse signal is obtained:

[0025]

[0026] According to the discontinuous spectrum inter-pulse frequency hopping signal design method of the present invention, the method for obtaining the discretized central autocorrelation expression is:

[0027] Let τ0 = 0, normalize formula (4), and let τ-τ0 = -υ, and obtain the signal center autocorrelation function χ(υ) of the pulse signal:

[0028]

[0029] Discretize formula (5) and get the discretized central autocorrelation χ(υ k ) expression:

[0030]

[0031] Where υ k is the kth sampling point delay, υ k =(k-1)Δ, k=1, 2, ..., K; K is the number of delay sampling points; Δ is the delay sampling interval.

[0032] According to the discontinuous spectrum inter-pulse frequency hopping signal design method of the present invention, the objective function F(f) is established as:

[0033]

[0034] Where WISL represents the weighted integrated sidelobe, υ main is the main lobe width, υ main =1 / B, B is the working bandwidth of frequency coding, w k is χ(υ k )’s weight;

[0035] If w k =0,υ k <υ main , the objective function F(f) is:

[0036]

[0037] When w k =1,υ k ≥υ main When , formula (8) is the integrated side lobe;

[0038] w k =10 -H(υk) / 10 ,υ k ≥υ main When χ(υ k ) has a specified upper envelope shape H(υ k );

[0039] Considering the signal's anti-co-channel interference performance, the constraints are as follows:

[0040]

[0041] Where M is the number of available passbands in the frequency band, l b is the upper boundary of the bth available passband, u b is the lower boundary of the bth available passband;

[0042] Combining formula (8) and formula (9), the signal design optimization problem is established;

[0043]

[0044] where f v =[f1,f2,...,f N ].

[0045] The beneficial effects of the present invention are as follows: the method of the present invention aims at the spectrum congestion problem faced by radar during operation, optimizes the integral sidelobe of the signal, and quickly designs the frequency hopping signal between discontinuous spectrum pulses. The designed signal has both anti-co-frequency interference performance and good autocorrelation performance (the integral sidelobe level is minimized).

[0046] The method of the present invention is based on the autocorrelation expression of the signal, and takes the autocorrelation sidelobe integral level as the objective function, considers the signal's anti-co-channel interference constraint, and establishes an optimization problem. The optimization variable is the frequency offset of each pulse of the signal, and the feasible space of its constraint conditions is non-convex and discontinuous. At the same time, the objective function is also non-convex with respect to the optimization variable. The method of the present invention transforms the optimization problem accordingly, and on the basis of obtaining a better iterative initial solution, the new constraint space of the optimization variable can be made a continuous interval. Finally, the gradient descent method is used to solve the new optimization problem, wherein, based on the derivation of the objective function gradient expression, a method is found that can use non-uniform sampling Fourier transform and inverse transform to accelerate gradient calculation. The better initial solution and the fast gradient calculation method improve the computational efficiency of the present invention.

[0047] Through simulation verification, the signal designed by the method of the present invention avoids the co-frequency interference frequency and has good integrated sidelobe and peak sidelobe levels, which can improve the performance of the radar in complex electromagnetic environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 1 is a flow chart of the method for designing a discontinuous spectrum inter-pulse frequency hopping signal according to the present invention;

[0049] Figure 2 This is a time-frequency diagram of the inter-pulse frequency hopping signal; T0 in the figure represents the pulse width;

[0050] Figure 3 The stop band distribution is Comparison diagram of the autocorrelation functions of the time-passband uniform stepped frequency signal and the iterative initial solution signal and the final optimized solution signal obtained by the method of the present invention;

[0051] Figure 4 The stop band distribution is Comparison diagram of the frequency distribution of the time-passband uniform stepped frequency signal and the method of the present invention;

[0052] Figure 5 The stop band distribution is Comparison diagram of the autocorrelation functions of the time-passband uniform stepped frequency signal and the iterative initial solution signal and the final optimized solution signal obtained by the method of the present invention;

[0053] Figure 6 The stop band distribution is Comparison diagram of the time-passband uniform stepped frequency signal and the frequency distribution obtained by the method of the present invention. DETAILED DESCRIPTION

[0054] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making any creative efforts shall fall within the scope of protection of the present invention.

[0055] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features therein may be combined with each other.

[0056] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but they are not intended to limit the present invention.

[0057] Specific implementation method 1. Combination Figure 1 and Figure 2 As shown, the present invention provides a method for designing a discontinuous spectrum inter-pulse frequency hopping signal, comprising:

[0058] A one-dimensional range image of the pulse signal is obtained based on the signal time domain expression of the inter-pulse frequency hopping signal and the echo processing process; a signal center autocorrelation function of the pulse signal is obtained from the one-dimensional range image, and the signal center autocorrelation function is discretized to obtain a discretized center autocorrelation expression;

[0059] The signal design optimization problem is established by taking the weighted integrated sidelobe of discrete center autocorrelation as the objective function and considering the signal anti-co-channel interference performance as the constraint condition;

[0060] Based on the continuous distribution of the frequency offset of the pulse signal and the ability to modulate the amplitude of each pulse, a signal autocorrelation function expression is established with the amplitude of each pulse as a variable; under the setting that the amplitude p(f) at the frequency offset f in the signal autocorrelation function is the optimal amplitude function, an initial value of the optimized frequency offset f is calculated, and based on the initial value, the signal design optimization problem is converted into a frequency optimization problem with a differentiable objective function and a convex set constraint;

[0061] The frequency optimization problem is solved by using the non-uniform sampling Fourier transform accelerated gradient descent method, and the final optimization result is obtained to realize the design of the inter-pulse frequency hopping signal.

[0062] Next, we first establish the mathematical model of the inter-pulse frequency hopping discontinuous spectrum signal and the mathematical expression of the optimization problem:

[0063] Assume that the pulse-to-pulse frequency hopping signal transmits N pulses, and the signal time domain expression of the pulse-to-pulse frequency hopping signal is:

[0064]

[0065] Where s T(t) is the pulse signal in the time domain, t is time, N is the number of pulses, A n is the amplitude of the nth pulse signal, f n is the frequency deviation of the nth pulse signal, f c is the pulse signal carrier frequency, T r is the pulse signal period, φ n is the initial phase of the nth pulse signal, p(t) is the pulse signal intra-pulse modulation complex envelope;

[0066] In this embodiment, A n =1, so that the pulse signal satisfies the constant modulus constraint;

[0067] The echo processing process includes:

[0068] After the pulse signal is reflected by the target, the received signal S r (t) is:

[0069]

[0070] Where α0 is the target reflection coefficient, τ0 is the target round-trip delay, and n0(t) is the echo noise;

[0071] Assume that the target reflection coefficient α0 does not change with frequency: α0 = 1;

[0072] The received signal is usually mixed and low-pass filtered to obtain the baseband signal, and then each pulse is passed through the matched filter. Finally, the A / D device is used to sample the same distance unit position at each pulse matched filter output to obtain the received signal S r (t) Sampling at the same distance position to obtain the discrete form s of the pulse signal r :

[0073]

[0074] Where s r (N) is the Nth discrete signal of the pulse signal, τ1 is the time delay corresponding to each pulse sampling point, τ is the time delay, g(τ) is the range ambiguity function of the pulse signal, and z(n) is the echo noise after matched filtering;

[0075] According to formula (3), the one-dimensional range image r(τ) of the pulse signal is obtained:

[0076]

[0077] In this embodiment, the method for obtaining the discretized central autocorrelation expression is:

[0078] In this embodiment, only the signal-related part of the range profile is considered, and the target delay does not affect the detection or resolution performance analysis of the range profile. Therefore, τ0 is set to 0, and τ1 is sampled according to the assumed target delay position. Its essence is the same as the consideration of τ in the above formula. Formula (4) is normalized and τ-τ0=-υ is set to obtain the signal center autocorrelation function χ(υ) of the pulse signal:

[0079]

[0080] Discretize formula (5) and get the discretized central autocorrelation χ(υ k ) expression:

[0081]

[0082] Where υ k is the kth sampling point delay, υ k =(k-1)Δ, k=1, 2, ..., K; K is the number of delay sampling points; Δ is the delay sampling interval.

[0083] Furthermore, the signal autocorrelation function sidelobe performance and the anti-co-channel interference performance are considered. When optimizing the waveform autocorrelation performance, the weighted integrated sidelobe is used as the objective function.

[0084] The objective function F(f) is established as:

[0085]

[0086] Where WISL represents the weighted integrated sidelobe, υ main is the main lobe width, υ main =1 / B, B is the working bandwidth of frequency coding, w k is χ(υ k )’s weight;

[0087] If w k =0,υ k <υ main , the objective function F(f) is:

[0088]

[0089] When w k =1,υ k ≥υ main When , formula (8) is the integrated side lobe;

[0090] When χ(υ k ) has a specified upper envelope shape H(υ k );

[0091] Considering the signal's anti-co-channel interference performance, the constraints are as follows:

[0092]

[0093] In formula (9), the set represents the available frequency band range with weak or no co-channel interference, where M is the number of available passbands in the frequency band, l b is the upper boundary of the bth available passband, u b is the lower boundary of the bth available passband;

[0094] Combining formula (8) and formula (9), the signal design optimization problem is established;

[0095]

[0096] where f v =[f1,f2,...,f N ].

[0097] Next, we will transform the optimization problem and iterate the initial solution acquisition process:

[0098] In formula (10), the variable feasible space is discontinuous and non-convex, and the objective function is also non-convex with respect to the optimization variables. Random evolutionary algorithms, such as genetic algorithms and particle swarm optimization, can be used to solve the problem, but these algorithms are inefficient. Therefore, this embodiment considers transforming the optimization problem to achieve a more efficient solution.

[0099] Previously, it was assumed that the pulse signal is continuously distributed in the frequency offset range [0, B] and the amplitude of each pulse can be modulated. When this restriction is removed, that is, the signal is continuously distributed in the frequency offset range [0, B] and the amplitude at the frequency offset f is p(f), then the signal autocorrelation function χ with each pulse amplitude as a variable is exp (υ k ) is:

[0100]

[0101] When p(f) takes a certain distribution, the weighted integral sidelobe level of formula (11) can be minimized, while satisfying that the amplitude value of the signal spectrum at the interference frequency point is 0. At this time, p(f) is the optimal amplitude function. In the actual solution of p(f), this embodiment first discretizes the amplitude p(f) to obtain the amplitude discrete vector p v :

[0102]

[0103] Where p L-1 is the Lth discretization amplitude, L is the total number of discretizations; δ represents the discrete sampling interval, which needs to be as small as possible to better reflect the continuous situation;

[0104] The unique solution of the amplitude p(f) is calculated by the following convex optimization problem as the optimal amplitude function:

[0105]

[0106] Where Φ is the set of unavailable frequency points.

[0107] The method to obtain the frequency optimization problem is:

[0108] Calculate the cumulative distribution function P(f) of the optimal amplitude function p(f):

[0109]

[0110] When applying discretization, just replace the integral with cumulative summation.

[0111] Divide the cumulative distribution function P(f) into N equal parts to obtain N+1 boundary points, then the frequency f of the qth boundary point is q 'satisfy:

[0112] P(f q ')=q / N,q=0,1,...,N(15);

[0113] Calculate the frequency offset f n Initial value of

[0114]

[0115] For the initial value in formula (16) that does not satisfy formula (9), Correction is performed, forcing it to be the nearest available passband edge frequency point to obtain the final initial value Arrange all final initial values ​​into an initial value vector

[0116]

[0117] The final initial value The passband range of the jth passband [lj,u j ], j∈{1,2,…,M} as the final initial value The optimization feasible space is:

[0118]

[0119] After the above processing, we get the frequency optimization problem where the objective function is differentiable and the constraint is a convex set:

[0120]

[0121] where l n ,u n The nth frequency offset f of the optimization variable is n The search upper and lower bounds.

[0122] The above process also obtains an initial solution. Since the initial solution is obtained based on the optimal amplitude distribution, it is close to the global optimal solution. In addition, the optimization variable search space in Equation (19) is continuous, and many existing optimization algorithms can be used. In this embodiment, the gradient descent method is used to solve the optimization problem, and the non-uniform sampling Fourier transform and inverse transform are used to accelerate the calculation of the gradient in the algorithm.

[0123] This embodiment proposes a non-uniformly sampled Fourier transform accelerated Gradient Descent method (NFT-GD) to solve the optimization problem.

[0124] In this embodiment, the method for solving the frequency optimization problem and obtaining the final optimization result is:

[0125] According to the initial value vector And the nth pulse signal frequency offset f determined in the frequency optimization problem n The boundary value of t is set, and the iteration pointer t = 0 at the initial moment, and the frequency offset of the tth iteration is calculated using the non-uniform sampling Fourier transform accelerated gradient descent method and gradient G t ;

[0126] Determine the adjustment step size η through linear search t ,make Reach a minimum value;

[0127] Update the frequency offset f for the t+1th iteration n t+1 =max(min(f n t -η t G t ,u n ),l n ) so that the obtained solution satisfies the constraints of formula (19);

[0128] Then adjust the iteration pointer to continue iterative calculation and update the frequency offset value until the specified termination criterion is met, and the final frequency offset f is obtained. n t+1 As f n The final optimization value of the frequency vector is obtained by all the final optimization values

[0129] The adjustment step length η is determined according to the following formula (20): t :

[0130] An inexact search based on interpolation to determine the adjustment step size η t , so that the search terminates with the strong Wolf condition:

[0131]

[0132] Wherein c1 and c2 are both constants, and 0<c1<c2<1.

[0133] From the perspective of the algorithm steps, the existing methods mainly spend time on calculating the objective function value and gradient value. This implementation derives the objective function gradient and finds a method for fast calculation. The specific derivation process is as follows:

[0134] Going further, the specific process of solving the frequency optimization problem and obtaining the final optimization result is as follows:

[0135] Frequency offset f n Derivative:

[0136]

[0137]

[0138] Where Im(·) represents the imaginary part; σ represents the impulse function. For σ(m1-n), when m1=n, the value of σ(m1-n) is 1, otherwise the value of σ(m1-n) is 0.

[0139] Substituting formula (22) into formula (21) yields:

[0140]

[0141] Where α k For intermediate variables:

[0142]

[0143] From formulas (23) and (24), we can see that α k The calculation of the summation part is performed by performing non-uniform sampling Fourier transform on the all-1 amplitude data, wherein the non-uniform sampling Fourier transform is non-uniform sampling in the data domain and uniform sampling in the transform domain;

[0144] Gradient G t The calculation is done by k The data is non-uniformly sampled and Fourier inverse transformed; all the partial derivatives of the frequency offset are arranged in sequence into vectors to obtain the gradient G t :

[0145]

[0146] Where T is the Fourier transform matrix:

[0147]

[0148] (·) H represents the conjugate transpose of a matrix;

[0149] α is the intermediate variable vector:

[0150] α=[α1,α2,...,α K ] T .

[0151] It can be seen from the calculation expression of the gradient that its main computational effort is mainly contained in the matrix multiplication operation of T, which can be quickly calculated through non-uniform sampling Fourier transform, thus making the entire algorithm run faster.

[0152] In this embodiment, the specified termination criteria are:

[0153]

[0154] Where ξ is the preset solution accuracy. Specific embodiment:

[0156] The effect of the method of the present invention is verified by simulation examples below:

[0157] The simulation sets up 2 groups, the stopband distribution is (single stopband), The inventive method was applied under dual-stopband conditions, using a uniformly stepped frequency signal across the passband for performance comparison. Specifically, the total passband bandwidth was divided into N frequency points, and the frequencies within the same passband were sampled at equal intervals. The simulation parameters are shown in Table 1.

[0158] Table 1 Simulation parameters

[0159]

[0160] Tables 2 and 3 respectively show the integrated sidelobe level and peak sidelobe level of the initial solution signal, the passband uniformly stepped frequency signal, and the final signal obtained after optimization during the design signal process under single stopband and double stopband conditions.

[0161] Table 2 Effect of the method of the present invention under single stop band

[0162]

[0163] Table 3 Effect of the method of the present invention under double stop band

[0164]

[0165] Combined with Table 2, 3 and Appendix Figures 3 to 6 It can be seen that the autocorrelation peak sidelobe level of a simple passband uniformly stepped frequency signal is higher than that of an ordinary stepped frequency signal. This is a negative impact caused by discontinuous use of the frequency band. However, the signal designed by the method of the present invention is Figure 4 、 Figure 6 It can be seen that it is not uniformly distributed within the passband. Its peak sidelobe and integrated sidelobe levels are much lower than those of the passband uniformly stepped frequency signal. The single stopband peak sidelobe is about 2dB lower, and the double stopband peak sidelobe is about 3dB lower. This shows that the method of the present invention is effective in reducing the degradation of signal autocorrelation performance caused by non-continuous frequency bands.

[0166] In addition, from Figures 3 to 6 As can be seen from Tables 2 and 3, the initial solution obtained by the method of the present invention is close to the optimal solution, which indicates that the subsequent iterative algorithm can converge quickly.

[0167] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the invention. It should be understood that many modifications may be made to the illustrative embodiments, and that other arrangements may be devised, without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in ways other than those described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be used in conjunction with other described embodiments.

Claims

1. A method for designing a discontinuous spectrum pulse frequency hopping signal, characterized in that: include: According to the time domain expression of the pulse frequency hopping signal and the echo processing process, the one-dimensional range image of the pulse signal is obtained; Obtaining a signal center autocorrelation function of the pulse signal from the one-dimensional range image, discretizing the signal center autocorrelation function to obtain a discretized center autocorrelation expression; The signal design optimization problem is established by taking the weighted integrated sidelobe of discrete center autocorrelation as the objective function and considering the signal anti-co-channel interference performance as the constraint condition; Based on the continuous distribution of the frequency offset of the pulse signal and the ability to modulate the amplitude of each pulse, a signal autocorrelation function expression is established with the amplitude of each pulse as a variable; under the setting that the amplitude p(f) at the frequency offset f in the signal autocorrelation function is the optimal amplitude function, an initial value of the optimized frequency offset f is calculated, and based on the initial value, the signal design optimization problem is converted into a frequency optimization problem with a differentiable objective function and a convex set constraint; The frequency optimization problem is solved by using the non-uniform sampling Fourier transform accelerated gradient descent method to obtain the final optimization result and realize the design of the inter-pulse frequency hopping signal. The signal time domain expression of the pulse frequency hopping signal is: （1）， In the formula is a pulse signal in the time domain, is the time, N is the number of pulses, is the amplitude of the nth pulse signal, is the frequency offset of the nth pulse signal, is the pulse signal carrier frequency, is the pulse signal period, is the initial phase of the nth pulse signal, It is the complex envelope of the pulse signal intra-pulse modulation; , so that the pulse signal satisfies the constant modulus constraint; The echo processing process includes: After the pulse signal is reflected by the target, the received signal is for: （2）， In the formula is the target reflection coefficient, is the target round-trip delay, is the echo noise; Assuming the target reflection coefficient Does not vary with frequency: ; To receive the signal Sampling at the same distance position to obtain the discrete form of the pulse signal : （3）， In the formula is the Nth discrete signal of the pulse signal, For each pulse sampling point corresponding to the delay, For delay, is the range ambiguity function of the pulse signal, is the echo noise after matched filtering; According to formula (3), the one-dimensional range image of the pulse signal is obtained : （4）； The method to obtain the discretized central autocorrelation expression is: make , normalize formula (4) and let , get the signal center autocorrelation function of the pulse signal : （5）； Discretize formula (5) to obtain the discretized central autocorrelation The expression: （6）， In the formula is the kth sampling point delay, , ; is the number of delay sampling points; is the delay sampling interval; Establishing the objective function for: （7）， In the formula represents the weighted integrated sidelobe, is the main lobe width, =1 / B, is the working bandwidth of the frequency coding, for The weight of like , the objective function for: （8）， when When , formula (8) is the integrated side lobe; hour, With specified upper envelope shape ; Considering the signal's anti-co-channel interference performance, the constraints are as follows: （9）， Where M is the number of available passbands in the frequency band, is the upper boundary of the bth available passband, is the lower boundary of the bth available passband; Combining formula (8) and formula (9), the signal design optimization problem is established; （10）， in ; Assume that the pulse signal is The internal frequency offset is continuously distributed, and the amplitude of each pulse can be modulated. Then the signal autocorrelation function with the amplitude of each pulse as a variable is The expression is: （11）； Amplitude Discretize to get the amplitude discrete vector : （12）， In the formula is the Lth discretization amplitude, L is the total number of discretizations; represents the discrete sampling interval; The amplitude is calculated by the following convex optimization problem The unique solution of , as the optimal amplitude function: （13）， In the formula is a set of unavailable frequencies; The method to obtain the frequency optimization problem is: Calculate the cumulative distribution function of the optimal amplitude function p(f) : （14）， Cumulative distribution function Divide into N equal parts, get N+1 boundary points, then the frequency of the qth boundary point is satisfy: （15）； Calculating frequency offset Initial value of : （16）； For the initial value in formula (16) that does not satisfy formula (9), Correction is performed, forcing it to be the nearest available passband edge frequency point to obtain the final initial value ; Arrange all final initial values ​​into an initial value vector : （17）； The final initial value The passband range of the jth passband , As the final initial value The optimization feasible space is: （18）； After the above processing, we get the frequency optimization problem where the objective function is differentiable and the constraint is a convex set: （19）。 2. The method for designing a discontinuous spectrum pulse-to-pulse frequency hopping signal according to claim 1, wherein: The method to solve the frequency optimization problem and obtain the final optimization result is: According to the initial value vector And the nth pulse signal frequency offset determined in the frequency optimization problem The boundary value of the frequency offset of the tth iteration is calculated using the non-uniform sampling Fourier transform accelerated gradient descent method. and gradient ; Determine the adjustment step size through linear search ,make Reach a minimum value; Update the frequency offset for the t+1th iteration ; Then adjust the iteration pointer to continue iterative calculation and update the frequency offset value until the specified termination criteria are met. As The final optimization value of the frequency vector is obtained by all the final optimization values .

3. The method for designing a discontinuous spectrum pulse-to-pulse frequency hopping signal according to claim 2, wherein: The adjustment step size is determined according to the following formula (20): : Interpolation-based method for determining the adjustment step size using an inexact search , so that the search terminates with the strong Wolf condition: （20）， In the formula and are constants, and .

4. The method for designing a discontinuous spectrum inter-pulse frequency hopping signal according to claim 3, wherein: The specific process of solving the frequency optimization problem and obtaining the final optimization result is as follows: Frequency offset Derivative: （21）， （22）， in Indicates taking the imaginary part; represents the impulse function, for ,when hour, The value is 1, otherwise The value is 0; Substituting formula (22) into formula (21) yields: （23）， In the formula For intermediate variables: （24）； The calculation of the summation part is performed by performing non-uniform sampling Fourier transform on the all-1 amplitude data, wherein the non-uniform sampling Fourier transform is non-uniform sampling in the data domain and uniform sampling in the transform domain; gradient The calculation is done by The data is non-uniformly sampled and Fourier inverse transformed; all the partial derivatives of the frequency offset are arranged in sequence into vectors to obtain the gradient : （25）， In the formula is the Fourier transform matrix: ， represents the conjugate transpose of a matrix; is the intermediate variable vector: 。 5. The method for designing a discontinuous spectrum inter-pulse frequency hopping signal according to claim 2, wherein: The specified termination criteria are: ， In the formula The preset solution accuracy.

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