Transmit-receive design method for robust detection of velocity-modulated targets in a cluttered background

Abstract

The application discloses a kind of for speed fuzzy target robust detection under clutter background transceiver design method, comprising the following steps: S1, intra-pulse inter-pulse parameter agile waveform component analysis;S2, intra-pulse inter-pulse waveform dependent airborne radar echo modeling;S3, the modeling of speed fuzzy target robust detection problem based on cognitive information;S4, transmission and Doppler filter bank robust joint design problem solution.The application proposes a kind of for speed fuzzy target robust detection under clutter background transceiver joint design method, based on block enhancement, principal component minimization and sequence convex approximation etc. Non-convex optimization method, efficient configuration intra-pulse inter-pulse waveform amplitude, phase and timing etc. Parameters and Doppler filter bank coefficient, maximize the worst signal-to-interference-and-noise ratio of speed fuzzy target, realize the robust detection of speed fuzzy target under clutter background.

Description

Technical Field

[0001] This invention belongs to the field of signal processing technology, specifically relating to a transceiver design method for robust detection of velocity-blurred targets in cluttered backgrounds. Background Technology

[0002] Pulse Doppler technology can suppress airborne radar clutter by utilizing the differences between airborne radar clutter and targets in space, time, and frequency. However, the pulse repetition frequency system of existing airborne radars has a technical bottleneck that makes it difficult to simultaneously address range ambiguity and velocity ambiguity between low and medium-high repetition frequencies. This results in velocity-ambiguous targets having difficulty acquiring velocity information and are prone to folding into strong clutter regions, causing serious missed detections.

[0003] Sufficiently high pulse repetition frequencies can avoid target velocity ambiguity, but at the same time reduce the maximum unambiguous range, leading to severe range ambiguity. On the other hand, multiple pulse train waveforms with coprime repetition frequencies can be transmitted, and velocity ambiguity can be resolved using the Chinese remainder theorem, but this wastes radar time and power resources to some extent, and the method has poor robustness in solving for the true velocity. With the development of arbitrary waveform generators and high-speed processing hardware, radars can utilize prior knowledge to adaptively adjust the transmitted waveform to improve radar detection performance in complex environments. Related research shows that pulse agility (such as carrier frequency, phase, timing, etc.) can achieve decoupling of range ambiguity and velocity ambiguity. For example, the paper "Unambiguous delay-doppler recovery from random phase coded pulses," IEEE Transactions on Signal Processing, 2021(69):4991–5004, uses compressed sensing-based methods to recover the target range unambiguously by randomly modulating the initial phase of the pulse and based on the sparse characteristics of the target in the range-Doppler plane. However, when the target velocity exceeds the maximum unambiguous velocity, the true velocity of the target cannot be obtained. Furthermore, when considering clutter environments, the sparse assumption is difficult to satisfy, often leading to algorithm failure. In addition, the paper "Range-Doppler reconstruction for frequencyagile and PRF-jittering radar[J]. IET" Radar, Sonar & Navigation, 2018, 12(3):348-352” uses inter-pulse repetition frequency and carrier frequency combined with agile waveforms to achieve unambiguous Doppler measurement. However, when the target velocity is large, there are coupled phase terms of Doppler frequency, time delay and carrier frequency, which increases the difficulty of waveform processing. In summary, the existing velocity-ambiguous target detection technology does not give full play to the intra-pulse and inter-pulse degrees of freedom of the waveform, has poor robustness, and is difficult to adapt to clutter environments. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a transceiver design method for robust detection of velocity-blurred targets in clutter backgrounds. Based on non-convex optimization methods such as block enhancement, principal component minimization, and sequence convex approximation, this invention optimizes the configuration of parameters such as amplitude, phase, and timing of intra-pulse and inter-pulse waveforms and the coefficients of the Doppler filter bank to maximize the worst signal-to-interference-plus-noise ratio of velocity-blurred targets and achieve robust detection of velocity-blurred targets in clutter backgrounds.

[0005] The objective of this invention is achieved through the following technical solution: a transceiver design method for robust detection of velocity-blurred targets against clutter backgrounds, comprising the following steps:

[0006] S1. Analysis of the constituent elements of the intra- and inter-pulse parameter agility waveform: Assuming that the intra- and inter-pulse parameter agility waveform consists of M pulses within a coherent processing time, it is represented as follows:

[0007]

[0008] Where t represents time, τ is the pulse width, and c m and t m Let be the complex modulation codeword and transmission time of the m-th pulse, respectively; rect(t) is the rectangular window function; and s(t) is the baseband signal within the pulse.

[0009] S2. Intra-pulse and inter-pulse waveform-dependent airborne radar echo modeling includes the following sub-steps:

[0010] S21. Velocity-Ambiguous Target Echo Modeling: Assuming there exists a point target moving at a constant velocity within the illumination range of the airborne radar's main lobe beam, which may have velocity ambiguity, and assuming no range ambiguity, the speed and time echoes of the point target are represented as follows:

[0011]

[0012] Where, α T k T and f T These are the amplitude of the target echo, the range cell it is located in, and the Doppler frequency, respectively, where s is the frequency f through which s(t) passes. s The sampled discrete intra-pulse modulation vector, where N is the number of discrete points, t and c represent the non-uniform pulse emission time vector and inter-pulse modulation vector, respectively, and a(t,f) T () is the slow-time steering vector, ⊙ and (·) T These are the Hadama product operator and the transpose operator, respectively.

[0013] S22. Airborne radar clutter time-lapse modeling: Considering the presence of 2K clutter echoes near the target. C +1 adjacent clutter units, where KC Let represent the number of adjacent clutter cells before and after the target. Assuming there is no range-ambiguous clutter, the clutter echo is:

[0014]

[0015] Among them, L C This indicates the number of clutter blocks above the ring at a distance. and They represent the kth c On the lth distance ring c Complex amplitude factor and Doppler frequency of each clutter block, It is a displacement matrix. It is the set of real numbers;

[0016] S23. Airborne Radar Fast-Slow Time Echo Modeling: Airborne Radar k-th... T The echo at each distance cell is represented as follows:

[0017]

[0018] in, This indicates that the mean is zero and the variance is... noise;

[0019] S3. Modeling a robust detection problem for velocity-fuzzy targets based on cognitive information, including the following sub-steps:

[0020] S31. Determination of the target's true distance and Doppler frequency range; Based on the perceived target distance and velocity information, the possible target distance and Doppler frequency range are assumed to be:

[0021] Ω={(r T ,f T )|r T ∈[r Tmin ,r Tmax ],f T ∈[f Tmin +ρf r ,f Tmax +ρf r ],ρ=1,2,…,P} (5)

[0022] Where, r Tmin and r Tmax These are the closest and furthest possible distances to the target, f. Tmin and f Tmax These are the minimum and maximum possible Doppler frequencies of the target, respectively, and ρ and P are the number of Doppler frequency ambiguities and the number of maximum Doppler frequency ambiguities, respectively.

[0023] Then, perform N on the possible target distance range. rDiscretization of points, performing N operations on the possible Doppler frequency range. f Discretization of the points yields discrete distance and Doppler frequency pairs.

[0024] S32, matched filter and Doppler filter bank are used for joint processing; radar fast and slow time echoes are processed together. The filter passes through a fast-time matched filter and a passband frequency of... filter bank After processing, the signal-to-interference-plus-noise ratio (SIN / N) of the (n,l)th output channel is expressed as:

[0025]

[0026] in, and Distance unit The speed of clutter, time echo, and noise at the location, (·) H and These represent the conjugate transpose operator and the expectation operator, respectively. is the distance unit corresponding to the target distance, and c is the speed of light; It is the round-up operator;

[0027] S33. Design of a robust target detection framework for velocity ambiguity: The maximum value of the output of all filters is compared with a preset threshold. If it is greater than the threshold, it is determined that there is a target at the corresponding range cell and Doppler cell, and the true range and velocity information of the target can be extracted; otherwise, there is no target in the region of interest.

[0028] Modeling of the joint design problem of S34 robust emission and Doppler filter bank;

[0029] The optimization criterion for worst signal-to-interference-plus-noise ratio is defined as follows:

[0030]

[0031] Considering both the energy constraints and peak-to-average power ratio (PAPR) constraints of the intra-pulse and inter-pulse agile waveforms, a robust optimization design problem based on maximizing the worst signal-to-interference-plus-noise ratio (SNR) of the intra-pulse and inter-pulse emission waveforms and the Doppler filter bank is established as follows:

[0032]

[0033] Among them, ||·|| 2 Let ||s|| denote the vector L2 norm. 2 ||c|| 2 =1 indicates an energy constraint; This represents the peak-to-average power ratio (PAPR) constraint, where γ is the PAPR modulation factor, and s k and cm Let s and c represent the k-th element and the m-th element, respectively.

[0034] S4. Solving the robust joint design problem of the transmitter and Doppler filter bank, including the following sub-steps:

[0035] S41. Based on the scale invariance of the objective function, the problem The equivalent representation is:

[0036]

[0037] S42. Problems based on block enhancement algorithms Solution framework construction: optimization through alternating iterations Improve worst-case signal-to-interference-plus-noise ratio The specific method is as follows: in each iteration, only one variable is changed, such as s, c, or Suppose the problem is solved after the (i-1)th iteration. The solution is Then, in the i-th iteration, the following problems need to be solved respectively:

[0038]

[0039]

[0040]

[0041] in, and These are the constraint sets of s and c at the i-th iteration, respectively;

[0042] S43, Doppler filter bank Solve the problem: In the objective function The problems are mutually separate. Based on the idea of ​​the minimum variance distortionless response method, the above problem is transformed into a minimax quadratic programming problem. Then, using the upper mirror diagram, it is further transformed into a second-order cone programming convex optimization problem. Finally, the filter bank is directly solved using the convex optimization toolbox. An enhanced solution;

[0043] S44, Solving for the intrapulse modulation vector s: [The problem is...] This is equivalently represented as a minimax quartic programming problem, and then the principal component minimization method is used to solve the problem. The fourth objective function in the algorithm is approximated as a linear function. Using the upper mirror image, it is further transformed into a second-order cone programming convex optimization problem. Finally, the convex optimization toolbox is used to solve iteratively. After the algorithm converges, an enhanced solution for the intrapulse modulation vector s is obtained.

[0044] S45. Solving the inter-pulse modulation vector c: Using the sequential convex approximation method, the minimum-maximum quadratic fractional programming problem is solved. Transform a minimax quadratic programming problem and, using the MM method and the mirror image of the problem, convert it into a second-order conic programming convex optimization problem. Finally, use the convex optimization toolbox to iteratively solve the problem and obtain a satisfactory solution for the inter-pulse modulation vector c.

[0045] The beneficial effects of this invention are: by modulating parameters such as amplitude, phase and timing of the transmitted waveform within and between pulses, and processing them at the receiving end using matched filters and Doppler filter banks, this invention can ensure that velocity-ambiguous targets falling within the region of interest have a good signal-to-interference-noise ratio, and is suitable for airborne radar warning-confirmation and tracking modes. Attached Figure Description

[0046] Figure 1 This is a flowchart of a waveform pulse multi-parameter cognitive design method for resisting airborne radar ambiguity clutter according to the present invention;

[0047] Figure 2 This is a graph showing the worst signal-to-interference-plus-noise ratio (SIR) output under different γ values ​​as a function of the number of iterations in an embodiment of the present invention.

[0048] Figure 3 For different N in the embodiments of the present invention f SINR(r) obtained by robust and non-robust designs under the given value T ,f T )result;

[0049] Figure 4 The minimum SINR(r) in the embodiments of the present invention T ,f T The graph shows the change in the number of blurring events with different maximum Doppler frequencies. Detailed Implementation

[0050] The technical solution of the present invention will be further described below with reference to the accompanying drawings.

[0051] like Figure 1 As shown, a transceiver design method for robust detection of velocity-blurred targets against clutter backgrounds according to the present invention includes the following steps:

[0052] S1. Analysis of the constituent elements of the intra- and inter-pulse parameter agile waveform: The amplitude, phase, and emission time of the intra- and inter-pulse parameter agile waveform can be arbitrarily modulated. Assuming that the intra- and inter-pulse parameter agile waveform consists of M pulses within a coherent processing time, it is represented as:

[0053]

[0054] Where t represents time, τ is the pulse width, and c m and t m Let be the complex modulation codeword and transmission time of the m-th pulse, respectively; rect(t) is the rectangular window function; and s(t) is the baseband signal within the pulse.

[0055] S2. Intra-pulse and inter-pulse waveform-dependent airborne radar echo modeling includes the following sub-steps:

[0056] S21. Velocity-Ambiguous Target Echo Modeling: Assuming there exists a point target moving at a constant velocity within the illumination range of the airborne radar's main lobe beam, which may have velocity ambiguity, and assuming no range ambiguity, the speed and time echoes of the point target are represented as follows:

[0057]

[0058] Where, α T k T and f T These are the amplitude of the target echo, the range cell it is located in, and the Doppler frequency, s = [s1, s2, ..., s]. N ] T It is s(t) passing through frequency f s The sampled discrete intra-pulse modulation vector, where N is the number of discrete points; t and c represent the non-uniform pulse emission time vector and inter-pulse modulation vector, respectively, t = [t1, t2, ..., tc]. M ] T and c = [c1, c2, ..., c M ] T ;a(t,f T ) is the slow-time steering vector. ⊙ and (·) T These are the Hadama product operator and the transpose operator, respectively.

[0059] The power of the target can be expressed as in, This represents the expectation operator. If the repetition frequency of all pulses is f... r ,but in is the Doppler frequency of the target blur, and P is the number of times the Doppler frequency blurs.

[0060] S22. Airborne radar clutter time-slow echo modeling: Considering the presence of -K near the target. C +k T ,…,k T ,…,K C +k T Total 2K C +1 adjacent clutter units, where K CLet represent the number of adjacent clutter cells before and after the target. Assuming there is no range-ambiguous clutter, the clutter echo is:

[0061]

[0062] Among them, L C This indicates the number of clutter blocks above the ring at a distance. and They represent the kth c On the lth distance ring c Complex amplitude factor and Doppler frequency of each clutter block; It is a displacement matrix, and its (p,q) elements are defined as follows:

[0063]

[0064] It is the set of real numbers; the k-th c On the lth distance ring c The power of each clutter block is

[0065] S23. Airborne Radar Fast-Slow Time Echo Modeling: Airborne Radar k-th... T The echo at each distance cell is represented as follows:

[0066]

[0067] in, This indicates that the mean is zero and the variance is... noise;

[0068] S3. Modeling a robust detection problem for velocity-fuzzy targets based on cognitive information, including the following sub-steps:

[0069] S31. The target's true range and Doppler frequency range are determined; based on the perceived target range and velocity information, i.e., the radar's detection area of ​​interest, are respectively r T ∈[r Tmin ,r Tmax ],f T ∈[f Tmin ,f Tmax ], where r Tmin and r Tmax These are the closest and furthest possible distances to the target, f. Tmin and f Tmax These are the target's possible minimum and maximum Doppler frequencies, respectively, but the target's actual Doppler frequency range may be f. T ∈[f Tmin +ρf r ,f Tmax +ρf r], ρ=1,2,…,P, where ρ and P are the number of Doppler frequency ambiguities and the number of the maximum Doppler frequency ambiguity, respectively. Therefore, assume the possible target distances and Doppler frequency ranges are:

[0070] Ω={(r T ,f T )|r T ∈[r Tmin ,r Tmax ],f T ∈[f Tmin +ρf r ,f Tmax +ρf r ],ρ=1,2,…,P} (18)

[0071] Ω represents the set of target range and Doppler frequency range, where r Tmin and r Tmax These are the closest and furthest possible distances to the target, f. Tmin and f Tmax These are the minimum and maximum possible Doppler frequencies of the target, respectively, and ρ and P are the number of Doppler frequency ambiguities and the number of maximum Doppler frequency ambiguities, respectively.

[0072] Then, perform N on the possible target distance range. r Discretization of points, performing N operations on the possible Doppler frequency range. f Discretization of the points yields discrete distance and Doppler frequency pairs.

[0073] S32, matched filter and Doppler filter bank are used for joint processing; radar fast and slow time echoes are processed together. The filter passes through a fast-time matched filter and a passband frequency of... filter bank If processed, the output of the (n,l)th path is represented as:

[0074]

[0075] The signal-to-interference-plus-noise ratio (SINR) of the (n,l)th output channel is expressed as:

[0076]

[0077] in, and Distance unit The speed of clutter, time echo, and noise at the location, (·) H and These represent the conjugate transpose operator and the expectation operator, respectively. is the distance unit corresponding to the target distance, and c is the speed of light; It is the round-up operator;

[0078] S33. Design of a robust target detection framework for velocity ambiguity: The maximum value of the output of all filters is compared with a preset threshold. If it is greater than the threshold, it is determined that there is a target at the corresponding range cell and Doppler cell, and the true range and velocity information of the target can be extracted; otherwise, there is no target in the region of interest.

[0079] Modeling of the joint design problem of S34 robust emission and Doppler filter bank;

[0080] The optimization criterion for worst signal-to-interference-plus-noise ratio is defined as follows:

[0081]

[0082] Considering both the energy constraints and peak-to-average power ratio (PAPR) constraints of the intra-pulse and inter-pulse agile waveforms, a robust optimization design problem based on maximizing the worst signal-to-interference-plus-noise ratio (SNR) of the intra-pulse and inter-pulse emission waveforms and the Doppler filter bank is established as follows:

[0083]

[0084] Among them, ||·|| 2 Let ||s|| denote the vector L2 norm. 2 ||c|| 2 =1 indicates an energy constraint condition; This represents the peak-to-average power ratio (PAPR) constraint, where γ is the PAPR modulation factor, and s k and c m Let s and c represent the k-th element and the m-th element, respectively.

[0085] S4. Solving the robust joint design problem of the transmitter and Doppler filter bank, including the following sub-steps:

[0086] S41. Based on the scale invariance of the objective function, the problem The equivalent representation is:

[0087]

[0088] S42. Problems based on block enhancement algorithms Solution framework construction: optimization through alternating iterations Improve worst-case signal-to-interference-plus-noise ratio The specific method is as follows: in each iteration, only one variable is changed, such as s, c, or Suppose the problem is solved after the (i-1)th iteration. The solution is Then, in the i-th iteration, the following problems need to be solved respectively:

[0089]

[0090]

[0091]

[0092] in, and These are the constraint sets for s and c in the i-th iteration, respectively, and are expressed as follows:

[0093]

[0094]

[0095] in, Represents the set of complex numbers.

[0096] S43, Doppler filter bank Solve the problem: In the objective function The problems are mutually separate. Based on the idea of ​​the minimum variance distortionless response method, the above problem is transformed into a minimax quadratic programming problem. Then, using the upper mirror diagram, it is further transformed into a second-order cone programming convex optimization problem, which can be expressed as:

[0097]

[0098] Where η is an auxiliary variable. It can be represented as:

[0099]

[0100] in,(·) * I represents the conjugate operator. M It is an M×M dimensional identity matrix.

[0101] Finally, the convex optimization toolbox is used to directly solve (29) to obtain the filter bank. An enhanced solution

[0102] S44, Solving for the intrapulse modulation vector s: [The problem is...] This is equivalent to a minimax quartic programming problem:

[0103]

[0104] in, It is a matrix ss H The vectorized form, It can be represented as:

[0105]

[0106] in, It is K 2 ×K 2 A dimensional unit matrix,

[0107]

[0108] Then, using the double principal component minimization (MM) method, the problem is solved. If the fourth-order objective function in the problem is approximated by a linear function, then the problem... The solution can be transformed into an iterative solution to the following problem: Define s (r) Let be the solution of the r-th iteration of s, and let . in I N It is an N×N dimensional identity matrix. It is a matrix If the largest eigenvalue is found, then the following problem needs to be solved in the (r+1)th iteration:

[0109]

[0110] in, It is a matrix The largest eigenvalue;

[0111]

[0112] Similarly, by using the upper mirror image, problem (34) is further transformed into a second-order cone programming convex optimization problem. Finally, the convex optimization toolbox is used to solve iteratively. After the algorithm converges, an enhanced solution si of the intrapulse modulation vector si is obtained. (i) ;

[0113] S45. Solving the inter-pulse modulation vector c: Using the sequential convex approximation method, the minimum-maximum quadratic fractional programming problem is solved. Transform the problem into a minimax quadratic programming problem;

[0114] definition It is the Cth The solution of the nth iteration, then the nth The problem to be solved in the next iteration can be expressed as:

[0115]

[0116] in,

[0117]

[0118]

[0119] also,

[0120]

[0121]

[0122] Problem (36) can be solved directly using the convex optimization toolbox, or it can be transformed into a second-order cone programming convex optimization problem using the MM method and the mirror image of the problem. Finally, the problem can be solved iteratively using the convex optimization toolbox to obtain a satisfactory solution c for the inter-pulse modulation vector c. (i) .

[0123] The technical effects of the present invention will be further verified through experiments below.

[0124] Assume the UAV platform flies at a constant speed of V = 20 m / s at an altitude of H = 8 km. The radar carrier frequency and bandwidth are f0 = 3 GHz and B = 2 MHz, respectively; the waveform pulse width and number of chips are τ = 16 μs and N = 32, respectively; the number of pulses is M = 50; and T... r =500μs. Unless otherwise specified, t is assigned a pre-designed non-uniform pulse emission time vector.

[0125] Set the noise power to Assume the target range and Doppler frequency range are r T ∈[23.7km,24.3km],f T ∈[-0.16+ρf r kHz, 0.16+ρf r kHz], ρ=1, f r =2kHz, the corresponding discrete distance and Doppler frequency are respectively and N r =9,N f =9. And set the target power to clutter power Set as, l c =1,…,L C Where K C =10, L C =181,

[0126] Define the maximum signal-to-noise ratio (SINR) of all channel outputs within the target range and Doppler frequency range. T ,f T As a variable reflecting the performance of target detection, it can be expressed as:

[0127]

[0128] Among them, (r T ,f T )∈Ω,

[0129]

[0130] Figure 2 The figure shows the worst-case signal-to-interference-plus-noise ratio (SNR) output by the method of this invention under different γ values ​​as a function of the number of iterations. As can be seen from the figure, the worst-case SNR obtained by the method of this invention continuously improves with the increase of the number of iterations, eventually monotonically converging to a stable value. Furthermore, the larger the γ value, the larger the final converged worst-case SNR. The results indicate that the method proposed in this invention can steadily improve the worst-case SNR of velocity-blurred targets by alternately optimizing the intra-pulse and inter-pulse waveforms and the Doppler filter bank, thereby ensuring robust detection of velocity-blurred targets against clutter backgrounds.

[0131] Figure 3 For different N f SINR(r) obtained by robust and non-robust designs under the given value T ,f T Results. Specifically, consider two design scenarios: considering only the target distance and the center value of the Doppler frequency range (r). T =24.0km,f T The non-robust design (=2kHz) and the robust design considering the target distance and Doppler frequency range are then used. The method proposed in this invention is then used to design intra-pulse and inter-pulse waveforms and Doppler filters (groups). The robust design considers two different N... f value. Figure 3 (a) and Figure 3 (b) gives N respectively f =9 and N f =17 Target detection performance of robust and non-robust designs. Observation shows that the non-robust design only produces high SINR(r) near the target's range of interest and the center of the Doppler frequency range. T ,f T SINR(r) values ​​at other distances and Doppler frequencies T ,f T The SINR (r) value is very low, while robust designs can achieve nearly the same SINR (r) across the target range of interest and the Doppler frequency range. T ,f T ) value, and N f The larger the value of SINR(r) T ,f T The closer the values ​​are to 10dB, the better the radar can adapt to scenarios where the target distance and Doppler information are inaccurate.

[0132] Figure 4 Minimum SINR(r)T ,f T The curves show the variation of the number of maximum Doppler blurs P with different maximum Doppler frequencies. It can be seen that as the number of maximum Doppler blurs of the target increases, the obtained signal-to-interference-plus-noise ratio (SIR) is lower, but not lower than 7dB. Compared with the non-robust design, the SIR is improved by nearly 24dB.

[0133] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A transceiver design method for robust detection of velocity-blurred targets against cluttered backgrounds, characterized in that, Includes the following steps: S1. Analysis of the constituent elements of the intra- and inter-pulse parameter agility waveform: Assuming that within one coherent processing time, the intra- and inter-pulse parameter agility waveform consists of M pulses, represented as follows: Where t represents time, τ is the pulse width, and c m and t m These are the complex modulation codeword and transmission time of the m-th pulse, respectively. It is a rectangular window function, s(tt) m This is the intrapulse baseband signal; S2. Intra-pulse and inter-pulse waveform-dependent airborne radar echo modeling includes the following sub-steps: S21. Velocity-Ambiguous Target Echo Modeling: Assuming there is a point target with velocity ambiguity moving at a constant speed within the illumination range of the airborne radar main lobe beam, and assuming no range ambiguity, the speed and time echoes of the point target are represented as follows: Where, α T k T and f T These are the amplitude of the target echo, the range cell it is located in, and the Doppler frequency, respectively, where s is the frequency f through which s(t) passes. s The sampled discrete intra-pulse modulation vector, where t and c represent the non-uniform pulse emission time vector and inter-pulse modulation vector, respectively, and a(t,f) T () is the slow-time steering vector, ⊙ and (·) T These are the Hadama product operator and the transpose operator, respectively. S22. Airborne radar clutter time-lapse modeling: Considering the presence of 2K clutter echoes near the target. C +1 adjacent clutter units, where K C Let represent the number of adjacent clutter cells before and after the target. Assuming there is no range-ambiguous clutter, the clutter echo is: Among them, L C This indicates the number of clutter blocks above the ring at a distance. and They represent the kth c On the lth distance ring c Complex amplitude factor and Doppler frequency of each clutter block, It is a displacement matrix. It is the set of real numbers, and N is the number of discrete points; S23. Airborne Radar Fast-Slow Time Echo Modeling: Airborne Radar k-th... T The echo at each distance cell is represented as follows: in, This indicates that the mean is zero and the variance is... noise; S3. Modeling a robust detection problem for velocity-fuzzy targets based on cognitive information, including the following sub-steps: S31. Determination of the target's true distance and Doppler frequency range; Based on the perceived target distance and velocity information, the possible target distance and Doppler frequency range are assumed to be: Ω={(r T ,f T )|r T ∈[r Tmin ,r Tmax ],f T ∈[f Tmin +ρf r ,f Tmax +ρf r ],ρ=1,2,…,P} (5) Where, r Tmin and r Tmax These are the closest and furthest possible distances to the target, f. Tmin and f Tmax These are the minimum and maximum possible Doppler frequencies of the target, respectively, and ρ and P are the number of Doppler frequency ambiguities and the number of maximum Doppler frequency ambiguities, respectively. Then, perform N on the possible target distance range. r Discretization of points, performing N operations on the possible Doppler frequency range. f Discretization of the points yields discrete distance and Doppler frequency pairs. S32, matched filter and Doppler filter bank are used for joint processing; radar fast and slow time echoes are processed together. The filter passes through a fast-time matched filter and a passband frequency of... filter bank After processing, the signal-to-interference-plus-noise ratio (SIN / N) of the (n,l)th output channel is expressed as: in, and Distance unit The speed of clutter, time echo, and noise at the location, (·) H and These represent the conjugate transpose operator and the expectation operator, respectively. is the distance unit corresponding to the target distance, and c is the speed of light; It is the round-up operator; S33. Design of a robust target detection framework for velocity ambiguity: Compare the maximum value of all filter outputs with a preset threshold. If the value is greater than the threshold, it is determined that a target exists at the corresponding range cell and Doppler cell, and the target's true range and velocity information are extracted; otherwise, no target exists in the region of interest. Modeling of the joint design problem of S34 robust emission and Doppler filter bank; The optimization criterion for worst signal-to-interference-plus-noise ratio is defined as follows: Considering both the energy constraints and peak-to-average power ratio (PAPR) constraints of the intra-pulse and inter-pulse agile waveforms, a robust optimization design problem based on maximizing the worst signal-to-interference-plus-noise ratio (SNR) of the intra-pulse and inter-pulse emission waveforms and the Doppler filter bank is established as follows: Among them, ||·|| 2 Let ||s|| denote the vector L2 norm. 2 ||c|| 2 =1 indicates an energy constraint; This represents the peak-to-average power ratio (PAPR) constraint, where γ is the PAPR modulation factor, and s k and c m Let these represent the k-th and m-th elements of s and c, respectively. S4. Solving the robust joint design problem of the transmitter and Doppler filter bank, including the following sub-steps: S41. Based on the scale invariance of the objective function, the problem The equivalent representation is: S42. Problems based on block enhancement algorithms Solution framework construction: optimization through alternating iterations Improve worst-case signal-to-interference-plus-noise ratio The specific method is as follows: in each iteration, only one variable is changed, such as s, c, or Suppose the problem is solved after the (i-1)th iteration. The solution is Then, in the i-th iteration, the following problems need to be solved respectively: in, and These are the constraint sets of s and c at the i-th iteration, respectively; S43, Doppler filter bank Solve the problem: In the objective function The problems are mutually separate. Based on the idea of ​​the minimum variance distortionless response method, the above problem is transformed into a minimax quadratic programming problem. Then, using the upper mirror diagram, it is further transformed into a second-order cone programming convex optimization problem. Finally, the filter bank is directly solved using the convex optimization toolbox. An enhanced solution; S44, Solving for the intrapulse modulation vector s: [The problem is...] This is equivalently represented as a minimax quartic programming problem, and then the principal component minimization method is used to solve the problem. The fourth objective function in the algorithm is approximated as a linear function. Using the upper mirror image, it is further transformed into a second-order cone programming convex optimization problem. Finally, the convex optimization toolbox is used to solve iteratively. After the algorithm converges, an enhanced solution for the intrapulse modulation vector s is obtained. S45. Solving the inter-pulse modulation vector c: Using the sequential convex approximation method, the minimum-maximum quadratic fractional programming problem is solved. Transform a minimax quadratic programming problem and, using the MM method and the mirror image of the problem, convert it into a second-order conic programming convex optimization problem. Finally, use the convex optimization toolbox to iteratively solve the problem and obtain a satisfactory solution for the inter-pulse modulation vector c.

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