Pseudo-random PRI airborne radar multi-frame track-before-detect method based on non-uniform FFT

Abstract

The invention discloses a pseudo-random PRI airborne radar multi-frame track-before-detect method based on non-uniform FFT. The method comprises the following steps: firstly, constructing an airborne radar echo model under a pseudo-random PRI framework; then, the target and clutter folding problem in the RD domain is solved under a non-uniform coherent processing framework; and finally, target detection is realized through a multi-frame DP-TBD algorithm. According to the method, radar echo data of a plurality of scanning periods are fused, incoherent accumulation of target energy is realized, and the target detection capability under the background of low signal-to-noise ratio and strong clutter is remarkably improved; meanwhile, the Doppler ambiguity problem caused by non-uniform sampling is effectively solved by using the non-uniform FFT technology, and the representation accuracy of the target signal in the frequency domain is improved.

Description

Technical Field

[0001] This invention belongs to the field of radar signal processing technology, and in particular relates to a pseudo-random PRI airborne radar multi-frame detection pre-tracking method based on non-uniform FFT. Background Technology

[0002] With the rapid development of modern radar technology, airborne radar systems have shown broad application potential in key areas such as situational awareness in complex environments, multi-target tracking, and airspace traffic management. Traditional airborne radars mostly use uniform pulse repetition intervals (UPRI) for signal modulation, which is simple to implement and has regular processing characteristics. However, in complex electromagnetic countermeasures environments, pseudo-random pulse repetition interval (PrPRI) modulation, due to its non-periodic characteristics, can significantly improve the low detectability and anti-deception jamming capabilities of radar systems, and has become a key research and application direction in recent years.

[0003] However, while the introduction of PrPRI enhances the system's stealth and anti-interference capabilities, it also brings numerous technical challenges, such as non-uniform sampling of echo data, ambiguity in Doppler frequency estimation, decreased target detection performance, and increased complexity in track correlation. These challenges place higher demands on existing signal processing algorithms, especially target detection and tracking models, necessitating the development of efficient processing methods adapted to non-uniform sampling characteristics to ensure stable system operation and accurate sensing capabilities in complex environments.

[0004] Current airborne radar systems generally employ a "detect-before-track" (DBT) processing architecture, which involves thresholding single-frame echo data, extracting target traces, and then performing subsequent tracking processing. While this method is simple to implement and computationally efficient under uniform pulse repetition intervals, it is susceptible to interference from range ambiguity and velocity ambiguity in environments with strong clutter or low signal-to-noise ratios, leading to increased target miss rates and false tracks. Especially when facing stealth targets, small targets, or densely packed (clustered) targets, the detection sensitivity and tracking stability of traditional DBT methods significantly decrease, making it difficult to meet the practical needs of modern complex environments.

[0005] The core advantage of pseudo-random pulse repetition interval (PrPRI) modulation lies in enhancing the unambiguity and anti-jamming capability of radar signals. However, this non-periodic modulation method also makes the pulse characteristics of the echo signal more difficult to predict, thereby increasing the complexity of target detection and tracking. Especially under non-uniform sampling conditions, the traditional "detect first, track later" (DBT) method struggles to fully extract cross-frame information, easily leading to problems such as missed target detections and increased false alarm rates.

[0006] In traditional airborne radar systems, UPRI technology is often hampered by clutter interference such as range ambiguity and velocity ambiguity, as well as the effects of low signal-to-noise ratio environments, leading to frequent target misses and an increase in false tracks. Its performance degrades significantly, especially in detecting stealth targets or densely packed formations. While PrPRI modulation technology demonstrates advantages in enhancing system anti-jamming capabilities and mitigating clutter ambiguity, its aperiodic structure results in high background clutter characteristics, posing a significant challenge to traditional DBT methods.

[0007] The DBT method relies on threshold decision based on single-frame echoes, making it difficult to achieve effective energy accumulation for weak target signals. Especially in low SNR environments, it is prone to target information loss and false track enhancement. Therefore, under PrPRI conditions, traditional methods struggle to balance target detection sensitivity with tracking stability. Summary of the Invention

[0008] The purpose of this invention is to address the problems mentioned in the background art by proposing a multi-frame pre-detection tracking method for airborne radar based on non-uniform FFT. Specifically, it addresses the difficulty of target detection and tracking in complex environments for airborne radar with pseudo-random pulse repetition intervals (PrPRI). This invention proposes a track-before-detection (TBD) method based on non-uniform FFT (NUFFT) and dynamic programming (DP). By fusing multi-frame scan data and combining it with non-uniform clutter suppression techniques, this method effectively improves target detection accuracy and tracking stability in low signal-to-noise ratio (SNR) and strong clutter environments, significantly enhancing the radar system's anti-jamming capability. This invention overcomes the performance bottleneck of traditional single-frame processing methods in strong clutter environments and has good engineering adaptability and widespread application value.

[0009] To achieve the objective of this invention, a pseudo-random PRI airborne radar multi-frame pre-detection tracking method based on non-uniform FFT is disclosed, comprising the following steps:

[0010] Step 1: Construct an airborne radar echo model within the pseudo-random PRI framework;

[0011] Step 2: Solve the target and clutter folding problem in the RD domain within the framework of non-uniform coherent processing;

[0012] Step 3: Implement target detection using the multi-frame DP-TBD algorithm.

[0013] Furthermore, step 1 specifically includes the following steps:

[0014] Step 1-1: To construct an airborne radar echo model suitable for the pseudo-random PRI system, we first start with array structure modeling, introducing a forward-looking rectangular planar phased array model; this lays the foundation for subsequent modeling of the directional characteristics of targets and clutter signals.

[0015] Steps 1-2: Based on the array model, a point target echo model based on the pseudo-random PRI system is further established. This model considers the range delay and Doppler characteristics when the target is within the main lobe, and combines the randomness of the pulse transmission time under non-uniform PRI to characterize the non-uniform distribution of the echo in the slow time dimension. This provides a theoretical basis for subsequent non-uniform sampling processing and multi-frame energy accumulation.

[0016] Steps 1-3: Based on the accurate modeling of point target echoes, the significant impact of clutter on radar system performance is further considered, and a clutter echo model suitable for pseudo-random PRI system is constructed.

[0017] Furthermore, step 1-1 specifically involves:

[0018] Suppose that the forward-looking rectangular planar array of the airborne radar consists of M elements, and the position coordinates of the elements are (x, y, y). m ,y m ), with wavelength λ, for pitch angle The spatial steering vector of the incident signal at azimuth angle θ is expressed as:

[0019]

[0020] in, Therefore, the radiation pattern of the above planar array can be written as:

[0021]

[0022] in, Indicates antenna beam direction The spatial filtering weighted vector at time;

[0023] Steps 1-2 are as follows:

[0024] Assume the carrier aircraft is cruising at speed v, and there is a point target P within its line of sight. At, where R represents the radial distance between the target and the carrier aircraft, Let θ and y be the elevation angle and azimuth angle, respectively; according to the basic radar equations, the echo amplitude of target P is expressed as:

[0025]

[0026] Among them, P tThe maximum power of the radar transmitted signal is represented by σ, the radar cross section (RCS) of the target is represented by σ, and the two-way path loss is represented by L. and These represent the beam gain in the directions of the transmitting and receiving antennas, respectively.

[0027] In the pseudo-random PRI system, each Coherent Processing Interval (CPI) contains N pulses, and the emission time of the nth pulse is:

[0028]

[0029] in, Let T be a random variable uniformly distributed within a given range, typically an integer multiple of the pulse width τ, thereby breaking the periodic structure of the pulse sequence and enhancing the system's anti-interference capability; if all T u They are equal, that is, T0 = T1 = ... = T N-1 =PRI, then it degenerates into the traditional uniform PRI system;

[0030] When the airborne radar transmits a linear frequency modulated (chirp) pulse, its baseband form is expressed as:

[0031]

[0032] in, f0, B, and τ are the carrier frequency, modulation bandwidth, and pulse width, respectively;

[0033] After down-conversion processing, the received signal of the echo from target P is expressed as:

[0034]

[0035] Where A is the echo amplitude, c represents the speed of light, and a s It is the spatial steering vector of the echo signal. The Doppler frequency at the scattering point P;

[0036] Steps 1-3 are as follows:

[0037] First, based on the radar system's range resolution ΔR = c / (2B), the visible clutter domain is divided into several equally spaced concentric range rings, each ring corresponding to a fixed radial range. Then, within each range ring, it is divided into multiple angular sectors according to the radar's angular resolution, thereby locating each clutter scattering unit in polar coordinate space. Each unit is regarded as an equivalent fixed scattering point, and its echo is modeled using a method similar to that used for point targets.

[0038] In the modeling process, each scattering unit is treated as an equivalent fixed scattering point, and its effective RCS is expressed as:

[0039]

[0040] The γ parameter is determined based on the terrain. R, ΔR and Δθ represent the ground rubbing angle, the radial distance between the clutter scattering unit and the carrier, the distance resolution and the angular resolution, respectively. Then, according to equation (3), the echo amplitude of the scattering point can be obtained.

[0041] The visible clutter domain is divided into L range rings, and each range ring is further divided into Q angular elements; then the echo signal of the q-th clutter element in the l-th range ring is modeled as follows:

[0042]

[0043] Among them, A l,q , R l,q and These represent the amplitude, spatial steering vector, radial distance, and Doppler frequency of the q-th clutter scattering unit on the l-th range ring, respectively.

[0044] After accumulating the clutter echo signals of all angular resolution cells in each range loop, the total clutter echo signal on that range loop is obtained.

[0045]

[0046] Furthermore, the clutter responses of all range loops are superimposed, and the overall clutter echo signal is expressed as Equation (9).

[0047] Furthermore, in step 2, starting from time alignment and data rearrangement, pulse compression and non-uniform FFT spectrum estimation are performed sequentially to construct a high-precision range-Doppler (RD) spectrum, providing input for subsequent target detection and tracking algorithms. This specifically includes the following steps:

[0048] Step 2-1: Rearrange echo data;

[0049] Step 2-2, Pulse Compression; Pulse compression preprocessing is complete. Next, the target echo energy needs to be focused in the range dimension to improve resolution and target signal-to-noise ratio.

[0050] Steps 2-3: Non-uniform FFT; After completing pulse compression and obtaining the distance matrix R pcSubsequently, in order to further extract the radial velocity information of the target, it is necessary to perform spectral analysis on the echo in the slow time dimension. Since the radar pulse adopts the pseudo-random PRI method, the echo data is in a non-uniform sampling form in the slow time dimension. Therefore, it is necessary to introduce non-uniform FFT to estimate the velocity of the target.

[0051] Furthermore, in step 2-1, the specific steps are as follows:

[0052] First, regarding the echo signal s r (t) Perform uniform sampling at M points within a CPI, with a sampling interval of τ. s Due to the slow, non-uniform sampling caused by PrPRI, the time position of the real target in each pulse echo is offset. To achieve subsequent pulse compression processing and effectively align the energy distribution of the real target, it is necessary to calculate the pulse transmission time [t1, t2, ..., t] based on the transmission time of each pulse. N For echo signal s r (t) is time-shifted;

[0053] The time-aligned data is rearranged into a matrix form. That is, the data is stacked in pulse order, where the nth row corresponds to the echo sample of the nth pulse; the matrix element R(n,m) is represented as:

[0054]

[0055] Where n = 1, 2, ..., N, [·] indicates rounding operation.

[0056] Furthermore, in step 2-2, the specific steps are as follows:

[0057] The pulse compression process is achieved by performing matched filtering on the data in each row of the echo matrix R; specifically, the echo signal corresponding to each pulse is convolved (or correlated) with the matched filter formed by the time-inverted complex conjugate form of the transmitted signal to compress the time width of the echo pulse and improve the distance resolution.

[0058] After pulse compression processing, the echo matrix R is converted into the range compression matrix T. pc Each row corresponds to a pulse distance output to the matched filter;

[0059] A single pulse signal is represented as:

[0060]

[0061] Where A represents the amplitude. B and τ are the modulation bandwidth and pulse width, respectively. The corresponding matched filter can be written as the following expression:

[0062]

[0063] Assume the discretized matched filter vector representation is s m The pulse compression process described above can be represented as follows:

[0064] T pc (n,:)=s m *R(n,:)(n=1,2,…,N) (13)

[0065] After pulse compression, the target signal in each row of matrix R is highlighted in the range dimension. To maintain consistency with the original echo matrix dimension and facilitate subsequent processing, the pulse compression result matrix T is... pc

[0066] Perform a truncation operation: extract the last M sample points from each row and construct a new distance matrix. This matrix serves as input data for subsequent Doppler processing or target detection, preserving the compressed target distance information;

[0067] After compressing the range dimension, the next step is to estimate the target's spectral characteristics in the slow time dimension in order to extract its radial velocity information.

[0068] Furthermore, in steps 2-3, the specific steps are as follows:

[0069] Before proceeding, the slow-time echo vector at the range gate where the target is located should be extracted first; let the range cell index corresponding to this range gate be m0, then the corresponding slow-time echo vector... Represented as:

[0070] g r =[R pc (1,m0),R pc (2,m0),…,R pc (N,m0)] T (14)

[0071] The echo vector g r This represents the slow time-domain response of the target at a fixed distance cell m0 under N consecutive pulses; subsequently, by performing non-uniform FFT processing on this vector, the velocity spectrum characteristics of the target are obtained, and then the Doppler frequency estimation is completed.

[0072] Furthermore, in this invention, because the radar system employs pseudo-random PRI modulation, the sampling points on the slow time axis are non-uniform. Therefore, to achieve accurate estimation of the target's radial velocity, a non-uniform FFT method is used to convert the non-uniformly sampled slow-time echo signal g... rThe signal is transformed to a uniform frequency domain to obtain the Doppler spectrum information of the target; this method is used to transform non-uniform time-domain sampled signals. Mapping to uniform frequency point k∈I N Calculate its Fourier coefficients:

[0073]

[0074] The specific algorithm flow is as follows:

[0075] Input: Non-uniform sampling points x j ∈[0,1), corresponding signal value f j The target frequency domain points are N, the oversampling factor is σ>1, and the window function is φ(·) (such as a Gaussian window).

[0076] Pre-calculation: Calculate the Fourier coefficients c of the window function φ k (φ); Construct the oversampled uniform grid index l∈I σN ;

[0077] Interpolate to a uniform grid:

[0078]

[0079] Where h is the grid spacing, φ is the Gaussian window (e.g., ... );

[0080] Uniform FFT:

[0081] Frequency domain clipping and correction:

[0082]

[0083] Output: Fourier coefficients in the uniform frequency domain

[0084] Therefore, after applying Doppler filtering to the pulse-compressed range data based on non-uniform FFT, range-Doppler (RD) spectrum data is obtained; its expression is as follows:

[0085]

[0086] Wherein, NUFFT represents the non-uniformly sampled slow time-domain signal R. pc (:,m) is mapped to a uniform frequency domain point f k Thus, the spectral response of the corresponding distance unit m in the Doppler frequency domain is obtained.

[0087] Compared with the traditional Non-Uniform Discrete Fourier Transform (NUDFT) method, NUFFT has a significant advantage in computational efficiency when processing large-scale data. It adopts a structure that combines interpolation and FFT, which can quickly approximate the NUDFT result and effectively alleviate the problems of spectral distortion and energy leakage caused by non-uniform sampling. It is especially suitable for radar signal processing scenarios under pseudo-random PRI conditions.

[0088] Furthermore, in step 3, the multi-frame TBD algorithm DP-TBD based on "Dynamic Programming (DP)" is adopted. It takes the range-Doppler RD observation data of K consecutive frames of radar as input, and combines the target's motion model and state transition characteristics to achieve joint detection and initial trajectory estimation without prior information about the existence of the target. This method is particularly suitable for the detection of weak targets, stealth targets and maneuvering targets, and can effectively improve the overall perception capability of the system in low SNR and strong clutter environments.

[0089] This invention uses the commonly used uniform motion model CV for modeling, and its state transition relationship is shown below:

[0090] x k =Fx k-1 +Γv k-1 (20)

[0091] in, This represents the target's two-dimensional position and velocity state in the k-th frame; Let be the Gaussian distributed process noise; F is the state transition matrix, modeling the target's motion inertia; Γ is the process noise transition matrix, describing the influence of random disturbances on the state; for a CV model with a sampling interval of Δt, we have:

[0092]

[0093] To complement the aforementioned motion model, a data representation of the target in the radar observation domain is established; it is assumed that the radar echo data plane is divided into N... x ×N y If there are range-azimuth resolution units, then the two-dimensional measurement data Z received by the airborne radar at time k is... k Represented in matrix form:

[0094]

[0095] Among them, Z k [i,j] represents the echo measurement value on the (i,j)th resolution unit at time k, N x and N y These represent the number of resolvable units in the distance and orientation dimensions, respectively.

[0096] Within each observation unit, to distinguish between the cases of "target presence" and "mere background noise," the following binary hypothesis testing model is established:

[0097]

[0098] Where H0 indicates that the current resolution unit has no target and is only noise; H1 indicates that the current resolution unit has a target signal. A is complex Gaussian white noise with zero mean; k Let θ be the amplitude of the target signal at time k; k The phase of the target signal at time k;

[0099] Since traditional detection-tracking (DBT) methods are difficult to handle weak signals or low SNR situations, this invention introduces a more robust "track-before-detect" strategy, TBD (Track-Before-Detect).

[0100] Within the TBD framework, the target detection and tracking problem is uniformly modeled as an optimization verification problem, with the following basic form:

[0101]

[0102] in, Z represents the detection statistics function (such as likelihood function, energy accumulation, etc.); K represents the number of TBD joint processing frames; 1:K ={Z1,Z2,…,Z K} represents the echo observation data from frame 1 to frame K; X 1:K ={x1,x2,…,x K} represents the target's actual trajectory within K frames; This represents the estimated trajectory obtained through optimization; γ is the preset detection threshold.

[0103] Given that this optimization problem has a clear temporal structure and independent subproblems, dynamic programming (DP) algorithm is used for efficient solution. Its core idea is to transform the target path search problem over the entire time series into a recursive solution of the optimal substructure at each stage, thereby effectively reducing computational complexity.

[0104] Furthermore, the basic process of the DP-TBD algorithm is as follows:

[0105] Initialization; when k=1, for all discrete states have:

[0106]

[0107] in, It is a discrete state space. Representing state The corresponding value function, It represents the detection statistics of a single frame, while Ψ(·) is used to record the state transition relationship between frames;

[0108] Iterative accumulation; when 2≤k≤K, for all discrete states By performing the above steps sequentially, we obtain the iterative expression for the value function:

[0109]

[0110] in, This represents the set of state transitions at time k-1. Based on the target's motion characteristics, states within this set can transition to state k at the next time step (i.e., time k).

[0111] Threshold decision: If the maximum value of the value function in the last frame exceeds the threshold γ, the target is determined to exist; otherwise, the target is declared not to exist, and the algorithm ends here.

[0112]

[0113] Track backtracking; when k = K-1, K-2, ..., 1, track backtracking is performed for states that exceed the threshold:

[0114]

[0115] The final estimated trajectory of the target is obtained as follows:

[0116] Compared with existing technologies, the significant advancements of this invention are: 1) It proposes a pseudo-random PRI clutter ambiguity suppression method based on non-uniform FFT: Utilizing the signal processing characteristics under non-uniform sampling conditions, non-uniform FFT technology is introduced to suppress range and velocity ambiguity clutter, thereby significantly expanding the low-clutter region in radar echo data and providing a clearer background environment for subsequent target detection; 2) It constructs a multi-frame pre-detection tracking method based on dynamic programming: By introducing a dynamic programming strategy, a state transition model is established, and non-coherent accumulation of target energy is performed using multi-frame echo data. This method adaptively adjusts the state transition range according to the target's motion characteristics across multiple frames, thereby enhancing the energy aggregation capability of weak targets and effectively improving tracking robustness and accuracy; 3) While enhancing anti-clutter capabilities, this invention also considers the detection performance of weak targets and the utilization efficiency of multi-frame data, making it suitable for airborne radar target perception and tracking tasks in modern complex environments.

[0117] To more clearly illustrate the functional characteristics and structural parameters of the present invention, further explanation is provided below in conjunction with the accompanying drawings and specific embodiments. Attached Figure Description

[0118] The accompanying drawings, which are included to provide a further understanding of the invention and form part of this application, illustrate exemplary embodiments of the invention and, together with their description, serve to explain the invention and do not constitute an undue limitation thereof. In the drawings:

[0119] Figure 1 This is a flowchart illustrating a pseudo-random PRI airborne radar multi-frame detection pre-tracking method based on non-uniform FFT;

[0120] Figure 2 This is a diagram of an airborne radar array;

[0121] Figure 3a It is a planar array emission pattern;

[0122] Figure 3b It is a planar array receiving pattern;

[0123] Figure 4a It is an RD plot based on uniform PRI;

[0124] Figure 4b It is an RD graph based on pseudo-random PRI;

[0125] Figure 5 It is the target's actual trajectory and estimated trajectory;

[0126] Figure 6a , Figure 6b , Figure 6c The following are schematic diagrams of the value function intensity when the frame number is 1, 3, and 5, respectively.

[0127] Figure 7 This is a comparison chart of target detection probabilities between the single-frame DBT algorithm and the multi-frame DP-TBD algorithm under different signal-to-noise ratios. Detailed Implementation

[0128] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. 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.

[0129] To address the problems existing in the prior art, this invention proposes a pseudo-random PRI multi-frame pre-detection tracking method based on non-uniform FFT. This method achieves incoherent accumulation of target energy by fusing radar echo data from multiple scanning cycles, significantly improving target detection capabilities in low signal-to-noise ratio and strong clutter environments. Simultaneously, the non-uniform FFT technique effectively addresses the Doppler ambiguity problem caused by non-uniform sampling, improving the accuracy of target signal representation in the frequency domain. Based on effectively mitigating clutter ambiguity and enhancing clutter suppression capabilities, this method fully exploits the temporal correlation between multiple frames of echo data to achieve cross-frame accumulation of target energy. By introducing non-uniform FFT technology to improve spectral resolution, it significantly enhances target detection probability and tracking stability, making it particularly suitable for radar systems operating in low-observable targets and complex combat environments.

[0130] Compared to traditional single-frame threshold decision-making methods, multi-frame pre-detection tracking methods can fully utilize the temporal motion correlation of targets, retain more target feature information, and effectively suppress background clutter interference. While obtaining stable detection results, it directly outputs the initial tracking information of the target, laying a solid foundation for subsequent accurate tracking and identification. This invention improves the robustness of radar systems in detecting weak targets and also provides a new path for the development of target tracking technology in complex environments.

[0131] This invention first constructs an airborne radar echo model under the pseudo-random PRI framework, then solves the target and clutter folding problem in the RD domain under the non-uniform coherent processing framework, and finally achieves target detection through the multi-frame DP-TBD algorithm.

[0132] The processing flow of this invention is as follows: Figure 1 As shown, the process mainly includes three steps: echo model construction, RD domain signal processing, and DP-TBD. The principles will be explained in detail below.

[0133] I. Constructing an airborne radar echo model within the pseudo-random PRI framework;

[0134] (I) Airborne Radar Echo Model Based on Pseudo-random PRI

[0135] To construct an airborne radar echo model suitable for the pseudo-random PRI system, this invention first starts with array structure modeling, introducing a forward-looking rectangular planar phased array model to lay the foundation for subsequent modeling of the directional characteristics of targets and clutter signals.

[0136] Assume an airborne radar forward-looking rectangular planar array (e.g.) Figure 2 (As shown) It consists of M array elements, and the coordinates of the array elements are (x, y, y). m ,y m ), with wavelength λ, for pitch angle The spatial steering vector of an incident signal at an azimuth angle θ can be expressed as:

[0137]

[0138] in Therefore, the radiation pattern of the above planar array (e.g.) Figure 3a , Figure 3b (As shown) can be written as:

[0139]

[0140] in Indicates antenna beam direction The spatial filtering weighted vector at time.

[0141] (II) Based on the array model, this invention further establishes a point target echo model based on the pseudo-random PRI system. This model considers the range delay and Doppler characteristics when the target is within the main lobe, and combines the randomness of the pulse transmission time under non-uniform PRI to characterize the non-uniform distribution of the echo in the slow time dimension, providing a theoretical basis for subsequent non-uniform sampling processing and multi-frame energy accumulation.

[0142] Specifically, assuming the aircraft is cruising at speed v, there exists a point target P within its line of sight. At, where R represents the radial distance between the target and the carrier aircraft, Let θ and y be the elevation angle and azimuth angle, respectively. According to the basic radar equations, the echo amplitude of target P can be expressed as:

[0143]

[0144] Where P t The maximum power of the radar transmitted signal is represented by σ, the radar cross section (RCS) of the target is represented by σ, and the two-way path loss is represented by L. and These represent the beam gain in the direction of the transmitting and receiving antennas, respectively.

[0145] In the pseudo-random PRI system, each Coherent Processing Interval (CPI) contains N pulses, and the emission time of the nth pulse is:

[0146]

[0147] in Let T be a random variable uniformly distributed within a given range, typically an integer multiple of the pulse width τ, thereby breaking the periodic structure of the pulse sequence and enhancing the system's anti-interference capability. uThey are equal, that is, T0 = T1 = ... = T N-1 =PRI, then it degenerates into the traditional uniform PRI system.

[0148] When the airborne radar transmits a linear frequency modulated (chirp) pulse, its baseband form can be expressed as:

[0149]

[0150] in f0, B, and τ represent the carrier frequency, modulation bandwidth, and pulse width, respectively.

[0151] After down-conversion processing, the received signal of the echo from target P can be expressed as:

[0152]

[0153] Where A is the echo amplitude, c represents the speed of light, and a s It is the spatial steering vector of the echo signal. Let P be the Doppler frequency of the scattering point.

[0154] (III) Based on the accurate modeling of point target echoes, this invention further considers the significant impact of clutter on radar system performance and constructs a clutter echo model suitable for pseudo-random PRI system.

[0155] This model first divides the visible clutter domain into several concentric range rings with equal spacing, based on the radar system's range resolution ΔR = c / (2B), with each ring corresponding to a fixed radial range. Then, within each range ring, it is divided into multiple angular sectors according to the radar's angular resolution, thereby locating each clutter scattering unit in polar coordinate space. Each unit is considered as an equivalent fixed scattering point, and its echo can be modeled using a method similar to that used for point targets.

[0156] In the modeling process, each scattering unit is treated as an equivalent fixed scattering point, and its effective RCS can be expressed as:

[0157]

[0158] The γ parameter is determined based on the terrain. R, ΔR and Δθ represent the ground rubbing angle, the radial distance between the clutter scattering unit and the carrier, the distance resolution and the angular resolution, respectively. The echo amplitude of the scattering point can be obtained according to equation (3).

[0159] The visible clutter domain is divided into L range rings, and each range ring is further divided into Q angular elements. The echo signal of the q-th clutter element in the l-th range ring can then be modeled as:

[0160]

[0161] Where A l,q , R l,q and These represent the amplitude, spatial steering vector, radial distance, and Doppler frequency of the q-th clutter scattering unit on the l-th range ring, respectively.

[0162] After summing the clutter echo signals of all angular resolution cells in each range loop, the total clutter echo signal for that range loop can be obtained. Furthermore, by superimposing the clutter responses of all range loops, the overall clutter echo signal can be expressed as:

[0163]

[0164] II. Solving the target and clutter folding problem in the RD domain within the framework of non-uniform coherence processing

[0165] To extract target range and velocity information, a series of non-uniform processing steps must be performed on the airborne radar echo signal under the pseudo-random PRI system. Due to the slow-time non-uniform sampling effect of random PRI, traditional uniform processing procedures are difficult to apply directly. Therefore, this invention starts with time alignment and data rearrangement, sequentially performing pulse compression and non-uniform FFT spectrum estimation to construct a high-precision range-Doppler (RD) spectrum, providing input for subsequent target detection and tracking algorithms.

[0166] (I) Echo Data Rearrangement

[0167] First, regarding the echo signal s r (t) Perform uniform sampling at M points within a CPI, with a sampling interval of τ. s Due to the slow, non-uniform sampling caused by PrPRI, the temporal position of the real target is offset in each pulse echo. To achieve subsequent pulse compression processing and effectively align the energy distribution of the real target, it is necessary to calculate the pulse position based on the transmission time [t1, t2, ..., t] of each pulse. N For echo signal s r (t) is time-shifted.

[0168] Time-aligned data can be rearranged into a matrix form. This is achieved by stacking the pulses in sequence, where the nth row corresponds to the echo sample of the nth pulse. The matrix element R(n,m) is represented as:

[0169]

[0170] Where n = 1, 2, ..., N, · indicates rounding operation.

[0171] (II) Pulse Compression

[0172] The pulse compression preprocessing has been completed. The next step is to focus the target echo energy in the range dimension to improve the resolution and target signal-to-noise ratio.

[0173] Pulse compression is achieved by performing matched filtering on the data in each row of the echo matrix R. Specifically, the echo signal corresponding to each pulse is convolved (or correlated) with a matched filter formed by the time-inverted complex conjugate form of the transmitted signal to compress the time width of the echo pulse and improve the range resolution.

[0174] After pulse compression processing, the echo matrix R is converted into the range compression matrix T. pc Each row corresponds to a pulse distance-oriented matched filter output.

[0175] A single pulse signal can be represented as:

[0176]

[0177] Where A represents the amplitude. B and τ are the modulation bandwidth and pulse width, respectively. The corresponding matched filter can be written as the following expression:

[0178]

[0179] Suppose that the discretized matched filter vector can be represented as s m Therefore, the above pulse compression process can be represented as:

[0180] T pc (n,:)=s m *R(n,:)(n=1,2,…,N) (13)

[0181] After pulse compression, the target signal in each row of matrix R is highlighted in the range dimension. To maintain consistency with the original echo matrix dimension and facilitate subsequent processing, the pulse compression result matrix T is... pc Perform a truncation operation: extract the last M sample points from each row and construct a new distance matrix. This matrix serves as input data for subsequent Doppler processing or target detection, preserving the compressed target distance information.

[0182] After compressing the range dimension, the next step is to estimate the target's spectral characteristics in the slow time dimension in order to extract its radial velocity information.

[0183] (III) Non-uniform FFT

[0184] At this point, pulse compression has been completed and the distance matrix R has been obtained. pc Subsequently, to further extract the target's radial velocity information, spectral analysis of the echo is required in the slow-time dimension. Since the radar pulses use a pseudo-random PRI method, the echo data exhibits non-uniform sampling in the slow-time dimension; therefore, a non-uniform FFT is needed to estimate the target's velocity.

[0185] Before proceeding, the slow-time echo vector at the range gate of the target should first be extracted. Let the range cell index corresponding to this range gate be m0, then the corresponding slow-time echo vector... It can be represented as:

[0186] g r =[R pc (1,m0),R pc (2,m0),…,R pc (N,m0)] T (14)

[0187] The echo vector g r This represents the slow time-domain response of the target at a fixed range cell m0 under N consecutive pulses. Subsequent non-uniform FFT processing of this vector yields the target's velocity spectrum characteristics, thus enabling Doppler frequency estimation.

[0188] In this invention, because the radar system employs pseudo-random PRI modulation, the sampling points on the slow time axis are non-uniform. Therefore, to achieve accurate estimation of the target's radial velocity, this invention uses a type 1 non-uniform FFT method to convert the non-uniformly sampled slow-time echo signal g... r The signal is converted to a uniform frequency domain to obtain the Doppler spectrum information of the target. This method is used to convert non-uniform time-domain sampled signals. Mapping to uniform frequency point k∈I N Calculate its Fourier coefficients:

[0189]

[0190] The specific algorithm flow is as follows:

[0191] Input: Non-uniform sampling points x j ∈[0,1), corresponding signal value f j ; Target frequency domain points N, oversampling factor σ>1, window function φ(·) (such as Gaussian window).

[0192] Pre-calculation: Calculate the Fourier coefficients c of the window function φ. k (φ); Construct the oversampled uniform grid index l∈I σN .

[0193] Interpolate to a uniform grid:

[0194]

[0195] Where h is the grid spacing, φ is the Gaussian window (e.g., ... ).

[0196] Uniform FFT:

[0197]

[0198] Frequency domain clipping and correction:

[0199]

[0200] Output: Fourier coefficients in the uniform frequency domain

[0201] Therefore, by performing Doppler filtering on the pulse-compressed range data based on non-uniform FFT, range-Doppler (RD) spectrum data can be obtained. Its expression is as follows:

[0202]

[0203] Wherein, NUFFT represents the non-uniformly sampled slow time-domain signal R. pc (:,m) is mapped to a uniform frequency domain point f k Thus, the spectral response of the corresponding distance unit m in the Doppler frequency domain is obtained.

[0204] Compared to the traditional Non-Uniform Discrete Fourier Transform (NUDFT) method, NUFFT has a significant computational efficiency advantage when processing large-scale data. Its structure, combining interpolation and FFT, can quickly approximate NUDFT results, effectively mitigating spectral distortion and energy leakage caused by non-uniform sampling. It is particularly suitable for radar signal processing scenarios under pseudo-random PRI conditions.

[0205] 3. Target detection is achieved through the multi-frame DP-TBD algorithm.

[0206] After constructing the RD domain range-Doppler spectrum, accurately detecting and stably tracking weak targets in a low signal-to-noise ratio background becomes a key issue in pseudo-random PRI radar systems. Since the traditional "detect-before-track" (DBT) strategy is susceptible to noise and clutter interference under single-frame threshold decision conditions, making it difficult to achieve stable extraction of weak signals in complex environments, this invention further introduces a "track-before-detect (TBD)" mechanism. This mechanism improves detection performance and track initialization accuracy through cross-frame joint processing.

[0207] To address this, this invention employs a multi-frame TBD algorithm (DP-TBD) based on Dynamic Programming (DP). Using K consecutive frames of range-Doppler (RD) observation data from radar as input, and combining the target's motion model and state transition characteristics, it achieves joint detection and initial trajectory estimation without prior information about the target's existence. This method is particularly suitable for detecting weak, stealthy, and maneuvering targets, effectively improving the system's overall perception capability in low SNR and strong clutter environments.

[0208] This invention uses a commonly used uniform motion model (CV) for modeling, and its state transition relationship is shown below:

[0209] x k =Fx k-1 +Γv k-1 (20)

[0210] in, This represents the target's two-dimensional position and velocity state in the k-th frame; Let be the Gaussian distributed process noise; F is the state transition matrix, modeling the target's motion inertia; Γ is the process noise transition matrix, describing the effect of random disturbances on the state. For a CV model with a sampling interval of Δt, we have:

[0211]

[0212] To support the aforementioned motion model, it is necessary to establish a data representation of the target within the radar observation domain. Assume the radar echo data plane is divided into N... x ×N y If there are range-azimuth resolution units, then the two-dimensional measurement data Z received by the airborne radar at time k is... k It can be represented in matrix form:

[0213]

[0214] Among them, Z k [i,j] represents the echo measurement value on the (i,j)th resolution unit at time k, N x and N y These represent the number of resolvable units in the distance and orientation dimensions, respectively.

[0215] Within each observation unit, to distinguish between the cases of "target existence" and "only background noise," the following binary hypothesis testing model can be established:

[0216]

[0217] Where H0 indicates that there is no target in the current resolution unit and only noise; H1 indicates that there is a target signal in the current resolution unit. A is complex Gaussian white noise with zero mean; k Let θ be the amplitude of the target signal at time k; k Let be the phase of the target signal at time k.

[0218] Since traditional detection-tracking (DBT) methods are difficult to handle weak signals or low SNR situations, this invention introduces a more robust "track-before-detect" (TBD) strategy.

[0219] Within the TBD framework, the target detection and tracking problem can be uniformly modeled as an optimization verification problem, with the following basic form:

[0220]

[0221] in, Z represents the detection statistics function (such as likelihood function, energy accumulation, etc.); K represents the number of TBD joint processing frames; 1:K ={Z1,Z2,…,Z K} represents the echo observation data from frame 1 to frame K; X 1:K ={x1,x2,…,x K} represents the target's actual trajectory within K frames; This represents the estimated trajectory obtained through optimization; γ is the preset detection threshold.

[0222] Given that this optimization problem exhibits a clear temporal structure and the independence of its subproblems, this invention employs a dynamic programming (DP) algorithm for efficient solution. Its core idea is to transform the target path search problem over the entire time series into a recursive solution of optimal substructures at each stage, thereby effectively reducing computational complexity.

[0223] Therefore, the TBD problem described above can be viewed as a multi-stage decision optimization problem. Below, we will present the basic flowchart of the DP-TBD algorithm:

[0224] Initialization. When k=1, for all discrete states... have:

[0225]

[0226] in, It is a discrete state space. Representing state The corresponding value function, Ψ(·) represents the detection statistics for a single frame, while Ψ(·) is used to record the state transition relationship between frames.

[0227] Iterative accumulation. When 2≤k≤K, for all discrete states By performing the above steps sequentially, the iterative expression of the value function can be obtained:

[0228]

[0229] in, This represents the set of state transitions at time k-1. Based on the target's motion characteristics, states within this set can transition to state k at the next time step (i.e., time k).

[0230] Threshold decision. If the maximum value of the value function in the last frame exceeds the threshold γ, the target is determined to exist; otherwise, the target is declared not to exist, and the algorithm ends.

[0231]

[0232] Track backtracking. When k = K-1, K-2, ..., 1, track backtracking is performed for states that exceed the threshold:

[0233]

[0234] The final estimated trajectory of the target is obtained as follows:

[0235] Example

[0236] This invention focuses on forward-looking airborne radar equipped with a rectangular planar array, modeling and analyzing its target detection and tracking problem in complex clutter environments. By constructing a simulated real-world scenario, the effectiveness of the proposed pseudo-random PRI technique based on non-uniform FFT and the multi-frame dynamic programming detection-before-track (DP-TBD) algorithm is verified, and key performance indicators are quantitatively evaluated.

[0237] In signal modeling, it is assumed that the echo signal received by the airborne radar under pseudo-random PRI consists of three parts: target echo, clutter echo, and thermal noise signal. Then the echo signal can be represented as:

[0238] s(t)=s target (t)+s clutter (t)+w(t) (33)

[0239] Among them, s target (t) represents the echo signal from the target, s clutter w(t) and w(t) are the echo signal of the clutter and the thermal noise signal, respectively.

[0240] First, the received radar echo signal s(t) is subjected to analog-to-digital conversion (A / D sampling), and then data rearrangement, pulse compression and non-uniform FFT processing are performed in sequence to finally obtain the range-Doppler data matrix as shown in Equation (19).

[0241] In the simulation experiments of this invention, the focus is no longer on the independent processing of single-frame echo data, but rather on the joint processing of data from multiple consecutive frames using a dynamic programming-based multi-frame detection-before-track (DP-TBD) algorithm. This method achieves incoherent signal accumulation by mining the motion correlation of the target across multiple frames, effectively enhancing the target echo intensity and suppressing background noise, significantly improving the target detection probability, and ultimately achieving accurate estimation of the target trajectory.

[0242] The simulation experiments involved in this invention were all completed on a laptop computer equipped with a 2.6GHz processor, 32GB of memory and an NVIDIA GeForce GTX 4050 dedicated graphics card. The software development and running environment was the MATLAB platform.

[0243] Experiment 1: To analyze the clutter distribution characteristics and range-velocity ambiguity effect of radar systems under different PRI modes, this experiment simulates the range-Doppler (RD) spectrum under uniform PRI and pseudo-random PRI conditions.

[0244] The experimental scenario assumes an aircraft cruising horizontally at a speed of 200 m / s at an altitude of 9 km. At a certain moment, the radar receiver's beam direction is set to an elevation angle of -9° and an azimuth angle of 0°. A target exists within the main lobe beam, with a radial distance of 20 km, a radial velocity of 450 m / s, a signal-to-noise ratio of 10 dB, and a clutter ratio of 30 dB. Other key parameters of the radar system are shown in Table 1.

[0245] Based on the parameters listed in Table 1, the maximum unambiguous distance in the uniform PRI mode is 7.5 km, and the maximum unambiguous speed is 333.3 m / s, which is significantly lower than the actual speed and distance of the target, indicating that there may be a serious distance-velocity ambiguity problem.

[0246] Figure 4a and Figure 4b The distribution maps of RD surface clutter under uniform PRI and pseudo-random PRI conditions are shown respectively. Figure 4a It can be observed that clutter energy is widely distributed across the entire RD plane, and the target signal is easily submerged in the strong clutter background, leading to missed or false detections of the target. This phenomenon is attributed to the range and velocity blurring effect of clutter under uniform PRI, where a large amount of strong clutter is folded into the visible area, forming strong interference.

[0247] In comparison, Figure 4bThe pseudo-random PRI mode exhibits significant advantages. This mode breaks the periodic superposition structure of clutter, retaining only the main lobe clutter region, significantly expanding the clutter-free area, and allowing the target signal to be clearly presented. Although the sidelobe level of the pseudo-random PRI is relatively high, clutter can be suppressed through multi-frame TBD accumulation, improving the detectability of weak targets.

[0248] Table 1 Radar Parameters

[0249]

[0250] Experiment 2: This experiment simulated and verified the proposed DP-TBD algorithm and analyzed its performance in depth. The simulation scenario was set as the detection and tracking of a single target by an airborne radar, with the target moving in an approximately uniform linear motion within a two-dimensional plane. The scan period was set to 1 second, the measurement plane size to 59×82 resolution cells, and the initial target state to be 90km away and 200m / s at a speed of 200m / s. The detection statistics were taken from the amplitude of the echo data. To evaluate the algorithm's performance, a target detection probability P was introduced. d As an evaluation metric, this probability is defined as the probability that the final accumulated value function exceeds the set detection threshold, and the error between the estimated position and the target's true position falls within the allowable range.

[0251] Figure 5 The comparison between the target's actual trajectory and estimated trajectory is shown when the number of accumulated frames is 5. The results demonstrate that the DP-TBD algorithm can accurately recover the target trajectory after multiple frame accumulations, achieving effective tracking. Furthermore, Figure 6a , Figure 6b , Figure 6c The distribution of the value function as the number of accumulated frames is shown under the conditions of SNR of 2dB and CNR of 30dB. Experiments reveal that when the number of frames is 1 (i.e., without cross-frame accumulation), the target signal is submerged in noise and difficult to identify. However, as the number of frames increases, the value function of the target's location gradually becomes more prominent in the background. When the number of frames reaches 5, a significant peak forms at the target's location, and the signal characteristics are clearly discernible. These results demonstrate that joint processing of multi-frame data can significantly enhance target energy and effectively improve the signal-to-noise ratio. The DP-TBD algorithm exhibits good target detection and tracking capabilities in low SNR scenarios.

[0252] Experiment 3: To further evaluate the target detection and tracking performance of the proposed multi-frame DP-TBD algorithm in a real-world environment, this experiment conducts empirical verification based on airborne radar echo data sampled from a real simulation scenario. The radar system parameters and aircraft parameters remain consistent with those in Experiment 1, and the target parameters are also set using the same methods as in Experiment 2. To simplify the data processing flow and focus on key detection areas, it is assumed that the target and main lobe clutter are strictly separated in space, and the processing range is limited to the region of interest.

[0253] During data processing, the inter-frame accumulation number was set to 3 frames and 5 frames, and the target detection probability P defined in Experiment 2 was continued to be used. d As a performance evaluation metric, to compare with traditional methods, the single-frame detection strategy OS-CFAR was used to perform target detection on the last frame of echo data, with the false alarm rate set to P. fa =e -5 The detection results were compared with the performance of the DP-TBD method under multi-frame joint processing.

[0254] Figure 7 The results show a comparison of target detection probabilities between the single-frame DBT method and the multi-frame DP-TBD algorithm under different signal-to-noise ratios (SNR). As can be seen from the figures, the DP-TBD algorithm achieves a detection performance improvement of approximately 2 dB compared to DBT even with K=3 frames. Furthermore, the detection probability continues to increase with the number of accumulated frames, significantly improving the detectability of weak targets in low SNR scenarios. The results demonstrate that the DP-TBD algorithm exhibits stronger robustness and adaptability in terms of target signal energy accumulation and noise suppression, making it particularly suitable for weak target detection tasks in complex clutter backgrounds.

[0255] In summary, the experiments verified that the airborne radar system based on pseudo-random PRI technology using non-uniform FFT combined with a multi-frame DP-TBD processing framework can effectively alleviate range-velocity ambiguity and improve target detection capabilities under realistic simulation conditions. Compared with the traditional single-frame DBT method, this method shows significant advantages in detection accuracy and tracking stability, demonstrating its good potential and broad prospects for practical application in complex environments.

[0256] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0257] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A pseudo-random PRI airborne radar multi-frame pre-detection tracking method based on non-uniform FFT, characterized in that, Includes the following steps: Step 1: Construct an airborne radar echo model within the pseudo-random PRI framework; Step 2: Solve the target and clutter folding problem in the RD domain within the framework of non-uniform coherent processing; Step 3: Implement target detection using the multi-frame DP-TBD algorithm.

2. The method for multi-frame pre-detection tracking of pseudo-random PRI airborne radar based on non-uniform FFT according to claim 1, characterized in that, Step 1 specifically includes the following steps: Step 1-1: To construct an airborne radar echo model suitable for the pseudo-random PRI system, we first start with array structure modeling and introduce a forward-looking rectangular planar phased array model. Step 1-2: Based on the array model, a point target echo model based on the pseudo-random PRI system is further established. This model considers the range delay and Doppler characteristics when the target is in the main lobe, and combines the randomness of the pulse transmission time under non-uniform PRI to characterize the non-uniform distribution of the echo in the slow time dimension. Steps 1-3: Based on the accurate modeling of point target echoes, the significant impact of clutter on radar system performance is further considered, and a clutter echo model suitable for pseudo-random PRI system is constructed.

3. The method for multi-frame pre-detection tracking of airborne radar based on non-uniform FFT according to claim 2, characterized in that, Step 1-1 is as follows: Suppose that the forward-looking rectangular planar array of the airborne radar consists of M elements, and the position coordinates of the elements are (x, y, y). m ,y m ), with wavelength λ, for pitch angle The spatial steering vector of the incident signal at azimuth angle θ is expressed as: in, Therefore, the radiation pattern of the above planar array can be written as: in, Indicates antenna beam direction The spatial filtering weighted vector at time; Steps 1-2 are as follows: Assume the carrier aircraft is cruising at speed v, and there is a point target P within its line of sight. At, where R represents the radial distance between the target and the carrier aircraft, Let θ and y be the elevation angle and azimuth angle, respectively; according to the basic radar equations, the echo amplitude of target P is expressed as: Among them, P t The maximum power of the radar transmitted signal is represented by σ, the radar cross section (RCS) of the target is represented by σ, and the two-way path loss is represented by L. and These represent the beam gain in the directions of the transmitting and receiving antennas, respectively. In the pseudo-random PRI system, assuming that each coherent processing interval (CPI) contains N pulses, the emission time of the nth pulse is: in, Let T be a random variable uniformly distributed within a given range, typically an integer multiple of the pulse width τ, thereby breaking the periodic structure of the pulse sequence and enhancing the system's anti-interference capability; if all T u They are equal, that is, T0 = T1 = ... = T N-1 =PRI, then it degenerates into the traditional uniform PRI system; When the airborne radar transmits a linear frequency modulated (chirp) pulse, its baseband form is expressed as: in, f0, B, and τ are the carrier frequency, modulation bandwidth, and pulse width, respectively; The received signal of the echo from target P after down-conversion processing is expressed as follows: Where A is the echo amplitude, c represents the speed of light, and a s It is the spatial steering vector of the echo signal. The Doppler frequency at the scattering point P; Steps 1-3 are as follows: First, based on the radar system's range resolution ΔR = c / (2B), the visible clutter domain is divided into several equally spaced concentric range rings, each ring corresponding to a fixed radial range. Then, within each range ring, it is divided into multiple angular sectors according to the radar's angular resolution, thereby locating each clutter scattering unit in polar coordinate space. Each unit is regarded as an equivalent fixed scattering point, and its echo is modeled using a method similar to that used for point targets. In the modeling process, each scattering unit is treated as an equivalent fixed scattering point, and its effective RCS is expressed as: The γ parameter is determined based on the terrain. R, ΔR and Δθ represent the ground rubbing angle, the radial distance between the clutter scattering unit and the carrier, the distance resolution and the angular resolution, respectively. Then, according to equation (3), the echo amplitude of the scattering point can be obtained. The visible clutter domain is divided into L range rings, and each range ring is further divided into Q angular elements; then the echo signal of the q-th clutter element in the l-th range ring is modeled as follows: Among them, A l,q , R l,q and These represent the amplitude, spatial steering vector, radial distance, and Doppler frequency of the q-th clutter scattering unit on the l-th range ring, respectively. After accumulating the clutter echo signals of all angular resolution cells in each range loop, the total clutter echo signal on that range loop is obtained. Furthermore, the clutter responses of all range loops are superimposed, and the overall clutter echo signal is expressed as Equation (9).

4. The method for multi-frame pre-detection tracking of airborne radar based on non-uniform FFT according to claim 1, characterized in that, In step 2, starting from time alignment and data rearrangement, pulse compression and non-uniform FFT spectrum estimation are performed sequentially to construct a high-precision range-Doppler RD spectrum, providing input for subsequent target detection and tracking algorithms. Specifically, this includes the following steps: Step 2-1: Rearrange echo data; Step 2-2, Pulse Compression; Pulse compression preprocessing is complete. Next, the target echo energy needs to be focused in the range dimension to improve resolution and target signal-to-noise ratio. Steps 2-3: Non-uniform FFT; After completing pulse compression and obtaining the distance matrix R pc Subsequently, in order to further extract the radial velocity information of the target, it is necessary to perform spectral analysis on the echo in the slow time dimension. Since the radar pulse adopts the pseudo-random PRI method, the echo data is in a non-uniform sampling form in the slow time dimension. Therefore, it is necessary to introduce non-uniform FFT to estimate the velocity of the target.

5. The method for multi-frame pre-detection tracking of pseudo-random PRI airborne radar based on non-uniform FFT according to claim 4, characterized in that, In step 2-1, the specific steps are as follows: First, regarding the echo signal s r (t) Perform uniform sampling at M points within a CPI, with a sampling interval of τ. s Due to the slow, non-uniform sampling caused by PrPRI, the time position of the real target in each pulse echo is offset. To achieve subsequent pulse compression processing and effectively align the energy distribution of the real target, it is necessary to calculate the pulse transmission time [t1, t2, ..., t] based on the transmission time of each pulse. N For echo signal s r (t) is time-shifted; The time-aligned data is rearranged into a matrix form. That is, the data is stacked in pulse order, where the nth row corresponds to the echo sample of the nth pulse; the matrix element R(n,m) is represented as: Where n = 1, 2, ..., N, [·] indicates rounding operation.

6. The method for multi-frame pre-detection tracking of airborne radar based on non-uniform FFT according to claim 4, characterized in that, In step 2-2, the specific steps are as follows: The pulse compression process is achieved by performing matched filtering on the data in each row of the echo matrix R; specifically, the echo signal corresponding to each pulse is convolved with a matched filter formed by the time-inverted complex conjugate form of the transmitted signal to compress the time width of the echo pulse and improve the distance resolution. After pulse compression processing, the echo matrix R is converted into the range compression matrix T. pc Each row corresponds to a pulse distance output to the matched filter; A single pulse signal is represented as: Where A represents the amplitude, B and τ are the modulation bandwidth and pulse width, respectively. The corresponding matched filter can be written as the following expression: Assume the discretized matched filter vector representation is s m The pulse compression process described above can be represented as follows: T pc (n,:)=s m *R(n,:)(n=1,2,…,N) (13 After pulse compression, the target signal in each row of matrix R is highlighted in the range dimension. To maintain consistency with the original echo matrix dimension and facilitate subsequent processing, the pulse compression result matrix T is... pc Perform a truncation operation: extract the last M sample points from each row and construct a new distance matrix. This matrix serves as input data for subsequent Doppler processing or target detection, preserving the compressed target distance information; After compressing the range dimension, the next step is to estimate the target's spectral characteristics in the slow time dimension in order to extract its radial velocity information.

7. The method for multi-frame pre-detection tracking of airborne radar based on non-uniform FFT according to claim 4, characterized in that, Steps 2-3 are detailed below: Before proceeding, the slow-time echo vector at the range gate where the target is located should be extracted first; let the range cell index corresponding to this range gate be m0, then the corresponding slow-time echo vector... Represented as: g r =[R pc (1,m0),R pc (2,m0),…,R pc (N,m0)] T (14) The echo vector g r This represents the slow time-domain response of the target at a fixed distance cell m0 under N consecutive pulses; subsequently, by performing non-uniform FFT processing on this vector, the velocity spectrum characteristics of the target are obtained, and then the Doppler frequency estimation is completed.

8. The method for multi-frame pre-detection tracking of pseudo-random PRI airborne radar based on non-uniform FFT according to claim 4, characterized in that, To achieve accurate estimation of the target's radial velocity, a non-uniform FFT method is employed to convert the non-uniformly sampled slow-time echo signal g... r The signal is converted to a uniform frequency domain to obtain the Doppler spectrum information of the target; this method is used to transform non-uniform time-domain sampled signals. Mapping to uniform frequency point k∈I N Calculate its Fourier coefficients: The specific algorithm flow is as follows: Input: Non-uniform sampling points x j ∈[0,1), corresponding signal value f j The target frequency domain points are N, the oversampling factor is σ>1, and the window function is φ(·) (such as a Gaussian window). Pre-calculation: Calculate the Fourier coefficients c of the window function φ. k (φ); Construct the oversampled uniform grid index l∈I σN ; Interpolate to a uniform grid: Where h is the grid spacing, and φ is the Gaussian window (e.g., ...). ); Uniform FFT: Frequency domain clipping and correction: Output: Fourier coefficients in the uniform frequency domain Therefore, after applying Doppler filtering to the pulse-compressed range data based on non-uniform FFT, range-Doppler RD spectrum data is obtained; its expression is as follows: Wherein, NUFFT represents the non-uniformly sampled slow time-domain signal R. pc (:,m) is mapped to a uniform frequency domain point f k Thus, the spectral response of the corresponding distance unit m in the Doppler frequency domain is obtained.

9. The method for multi-frame pre-detection tracking of airborne radar based on non-uniform FFT according to claim 1, characterized in that, In step 3, the multi-frame TBD algorithm DP-TBD based on "dynamic programming DP" is adopted. The range-Doppler RD observation data of K consecutive frames of radar are used as input. Combined with the target's motion model and state transition characteristics, joint detection and initial trajectory estimation are achieved without prior information about the existence of the target. The commonly used uniform motion model CV is selected for modeling, and its state transition relationship is shown below: x k =Fx k-1 +Γv k-1 (20) Among them, This represents the target's two-dimensional position and velocity state in the k-th frame; The process noise is Gaussian distributed; F is the state transition matrix, which models the target's motion inertia; Γ is the process noise transition matrix, describing the effect of random disturbances on the state; for a CV model with a sampling interval of Δt, we have: To complement the aforementioned motion model, a data representation of the target in the radar observation domain is established; it is assumed that the radar echo data plane is divided into N... x ×N y If there are range-azimuth resolution units, then the two-dimensional measurement data Z received by the airborne radar at time k is... k Represented in matrix form: Among them, Z k [i,j] represents the echo measurement value on the (i,j)th resolution unit at time k, N x and N y These represent the number of resolvable units in the distance and orientation dimensions, respectively. Within each observation unit, to distinguish between the cases of "target existence" and "only background noise," the following binary hypothesis testing model is established: Where H0 indicates that the current resolution unit has no target and is only noise; H1 indicates that the current resolution unit has a target signal. A is complex Gaussian white noise with zero mean; k Let θ be the amplitude of the target signal at time k; k The phase of the target signal at time k; Introducing a more robust "pre-detection tracking" strategy, TBD; Within the TBD framework, the target detection and tracking problem is uniformly modeled as an optimization verification problem, with the following basic form: in, Z represents the detection statistics function (such as likelihood function, energy accumulation, etc.); K represents the number of TBD joint processing frames; 1:K ={Z1,Z2,…,Z K } represents the echo observation data from frame 1 to frame K; X 1:K ={x1,x2,…,x K } represents the target's actual trajectory within K frames; This represents the estimated trajectory obtained through optimization; γ is the preset detection threshold. Given that this optimization problem has a clear temporal structure and independent subproblems, dynamic programming (DP) algorithm is used for efficient solution. Its core idea is to transform the target path search problem over the entire time series into a recursive solution of the optimal substructure at each stage, thereby effectively reducing computational complexity.

10. A multi-frame pre-detection tracking method for pseudo-random PRI airborne radar based on non-uniform FFT according to claim 9, characterized in that, The basic process of the DP-TBD algorithm is as follows: Initialization; when k=1, for all discrete states have: in, It is a discrete state space. Representing state The corresponding value function, It represents the detection statistics of a single frame, while Ψ(·) is used to record the state transition relationship between frames; Iterative accumulation; when 2≤k≤K, for all discrete states By performing the above steps sequentially, we obtain the iterative expression for the value function: in, This represents the set of state transitions at time k-1. Based on the target's motion characteristics, states within this set can transition to state k at the next time step (i.e., time k). Threshold decision: If the maximum value of the value function in the last frame exceeds the threshold γ, the target is determined to exist; otherwise, the target is declared not to exist, and the algorithm ends here. Track backtracking; when k = K-1, K-2, ..., 1, track backtracking is performed for states that exceed the threshold: The final estimated trajectory of the target is obtained as follows:

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