Unambiguous range extension for pulse‐Doppler radar via Poisson disk sampling

National Natural Science Foundation of China, Grant/Award Numbers: 61790551, 61925106; Institute for Guo Qiang, Tsinghua University Abstract Since the maximum unambiguous range in the traditional single‐channel pulse‐Doppler (PD) radar is restricted by the pulse repetition interval (PRI), there is a trade‐off between the maximum unambiguous range and the maximum unambiguous velocity. As a result, conventional coherent processing used in PD radar systems can hardly provide large unambiguous range and large unambiguous velocity simultaneously. We propose a method of unambiguous range extension for PD radar via Poisson disk sampling. The method enables the PD radar to detect farther targets unambiguously without sacrificing the maximum unambiguous velocity. The Poisson disk sampling is adopted in the slow time domain to ensure the interval length between any two adjacent transmitted pulses to be larger than a desired value, which can be set longer than the Nyquist sampling interval to extend the maximum unambiguous range. Then, the Iterative Soft Thresholding like (IST‐like) algorithm is utilized on the non‐uniformly under‐sampled data to recover the range‐Doppler spectrum accurately. Compared to some of existing methods of unambiguous range extension based on stochastic sampling, the proposed method has better velocity estimation accuracy. Simulations and experiments on real PD radar data demonstrate the effectiveness of the proposed method.


| INTRODUCTION
The range and velocity estimation of moving targets have been important issues since radar was invented. The range and velocity information in Pulse Doppler (PD) radar can be extracted by coherently processing the echo pulse train reflected by the targets [1,2]. There is a trade-off in pulse repetition frequency (PRF) selection between the maximum unambiguous range and the maximum unambiguous velocity in the traditional PD radar with a uniform PRF. That is, a lower PRF is required to extend the pulse repetition interval (PRI), thereby a larger maximum unambiguous range can be obtained. At the same time, according to the Nyquist sampling theorem, a higher PRF is required for a larger maximum unambiguous velocity.
To extend the maximum unambiguous range, the PRI needs to be enlarged so that the echoes reflected by farther targets can be received. However, the uniform undersampling method that directly reduce the PRF results in heavy Doppler aliasing. Fortunately, it has been investigated that stochastic sampling strategies have anti-aliasing capability at the sub-Nyquist rates [3][4][5]. Some work introduced the concept of stochastic sampling into the slow time domain by transmitting non-uniform under-sampling pulses. PRF staggering is an existing common method to solve the range and velocity ambiguity problem in coherent PD radar. Arbabian et al. [6] introduces several staggered PRF modes and evaluates their velocity estimation performance by the appropriate cost function. However, due to the high sidelobes in the spectrum of staggered PRF, it is difficult to determine the number of targets and accurately estimate the parameters of each target in the multi-target scenarios [7]. The development of compressed sensing makes it possible to reconstruct sparse signals from under-sampled data [8,9]. A target detection method for PD radar based on random selection from uniform pulses and compressed sensing is proposed in Ref. [10]. The method shows remarkable performance in the Radar cross-section (RCS) and velocity estimation at sub-Nyquist sampling rates, but it cannot provide sufficient maximum unambiguous range because there is no constraint on the minimum interval between adjacent pulses. In Ref. [11], a method of velocity ambiguity resolving for PD radar based on jittered sampling and compressed sensing is discussed. The Doppler spectrum reconstructed by this method contains several high sidelobes, which will affect the accuracy of velocity estimation.
Another stochastic sampling pattern, the Poisson disk sampling, can ensure the minimum interval between adjacent samples to be larger than a desired value, thus it has the potential to extend the maximum unambiguous range. It has been shown that the Poisson disk sampling is able to convert the structured spectrum aliasing caused by uniform undersampling to non-structured noise [12]. Due to the excellent anti-aliasing performance, the Poisson disk sampling has been used in computer graphics [13][14][15] and Magnetic Resonance Image (MRI) reconstruction [16].
An unambiguous range extension method for PD radar based on the Poisson disk sampling is proposed. We apply the Poisson disk sampling in the slow time domain of PD radar to enlarge the minimum interval between adjacent pulses and extend the maximum unambiguous range. A dictionary corresponding to the law of non-uniform pulse interval is constructed, and then the Iterative Soft Thresholding (IST) like (IST-like) algorithm is carried out on the non-uniformly under-sampled pulses to reconstruct the high-resolution range-Doppler spectrum. Compared with the traditional PD radar with a uniform PRF [2], the proposed method can provide larger maximum unambiguous range, and the range and velocity of long-range targets can be recovered simultaneously. Compared to some of existing methods of unambiguous range extension based on stochastic sampling [6,11], the proposed method demonstrates better Doppler spectrum anti-aliasing performance and velocity estimation accuracy. Simulations and experiments on real PD radar data are employed to validate the proposed method. It is demonstrated that the proposed method produces fewer false alarms and missed detections in the range-Doppler spectrum than other stochastic sampling methods.
The remainder of this study is organized as follows. Section 2 introduces several stochastic sampling patterns and compares their spectral anti-aliasing performance in unambiguous range extension. The proposed method based on the Poisson disk sampling and the IST-like algorithm is formulated in Section 3. In Section 4, the simulations and experimental results on real PD radar data are demonstrated. The conclusion is given in Section 5.

| POISSON DISK SAMPLING FOR UNAMBIGUOUS RANGE EXTENSION IN PULSE-DOPPLER RADAR
In this section, we review the trade-off between the maximum unambiguous range and the maximum unambiguous velocity and illustrate the principle of stochastic sampling patterns to achieve unambiguous range extension.

| Trade-off between unambiguous range and unambiguous velocity
In PD radar, the relationship between the radial maximum unambiguous velocity v rmax and the maximum Doppler shift f dmax caused by the target motion is v rmax ¼ f dmax λ=2, where λ is the wavelength of radar transmitted signal. It can be seen that fast-moving targets result in large Doppler shift. According to the Nyquist sampling theorem, PRF must be larger than the maximum Doppler shift to avoid the velocity ambiguity, that is, where κ is the oversampling ratio required in the pulse domain. As shown in Figure 1, the upper limit of the sampling window time is 1/PRF À T p , where T p is the pulse width of the transmitted signal. In order to ensure that the echo at the maximum unambiguous range can be collected, the maximum unambiguous range R max must satisfy [2]: where T R ¼1/PRF is the PRI length that meets the Nyquist sampling theorem. It is clear that there is an inherent trade-off issue on PRF selection of PD radar. That is, higher PRF is required for larger maximum unambiguous velocity, while lower PRF is required for larger maximum unambiguous range. Therefore, it is difficult for traditional single-channel PD radar to meet the requirements of large unambiguous velocity and large unambiguous range simultaneously. Fortunately, the compressed sensing theory [17] makes it possible to break through the limitation of Equation (1), and accordingly provides the possibility to extend the maximum unambiguous range.

| Several stochastic sampling patterns for unambiguous range extension
According to Equation (2), longer PRI is required for larger maximum unambiguous range. To make sure that the echo of all targets within the maximum unambiguous range can be received completely, the sampling patterns must guarantee that the minimum sampling interval T min between any two adjacent pulses follows: In order to evaluate the capability of stochastic sampling strategies in terms of unambiguous range extension, we define the range extension coefficient α in PD radar as: where T avg ¼ 1/PRF avg is the average pulse interval, and PRF avg is the slow time average sampling rate of non-uniform under-sampling patterns. A larger α, which means a longer minimum sampling interval T min at specific sampling rates, allows for a larger maximum unambiguous range R max according to Equation (3). The transmission power of PD radar is assumed to be large enough, and the influence of the Earth curvature on the detection range is negligible. For nonuniformly transformed pulse train, the maximum unambiguous range can be expressed as: The Poisson disk sampling can ensure the minimum interval between any two adjacent pulses to be larger than a desired value, which provides the potential for unambiguous range extension in PD radar. The arrangement of the Poisson disk sampling points is close to human retinal cells, and its excellent anti-aliasing performance has attracted extensive attention in the field of computer graphics [13][14][15]. In the Poisson disk sampling pattern as shown in Figure 2b, the sampling instant is defined as: where t m and t m þ 1 are the m-th and (m þ 1)-th sampling instants in the slow time domain, respectively, Δt min is the determined minimum sampling interval, and δ t is an exponentially distributed random variable with the expectation of 1/η t . It is obvious that the average sampling interval of the Poisson disk sampling is Δt min þ1/η t , therefore the range extension coefficientαin the Poisson disk sampling can be expressed as α¼T min /T avg ¼Δt min /(Δt min þ1/η t ). We apply the Poisson disk sampling in the slow time domain of PD radar, and extend the maximum unambiguous range by setting the minimum interval Δt min to be longer than the Nyquist sampling interval.
In the existing literature, the staggered sampling has been used in Doppler ambiguity resolution for PD radar [18]. Staggered radar overcomes the ambiguity problem by continuously varying the pulse interval. The performance of several staggered PRF methods in solving the Doppler ambiguity problem is compared in Ref. [6]. Among them, the sampling pattern of the best effect symmetrical-burst approach is shown in Figure 2c. The staggered PD radar transmits N pulses at one PRF, and then transmits the following N pulses at another PRF. The PRF changes every N pulses. A larger maximum unambiguous range can be obtained in the staggered sampling pattern by selecting a set of PRF lower than the requirements of Nyquist sampling theorem. It is obvious that the range extension coefficientαin staggered sampling can be expressed as α¼T min /T avg ¼PRF avg /PRF max , where PRF max is the largest value in the selected set of PRF.
Jittered sampling is another common stochastic sampling pattern which has been widely used in computer graphics [5]. The impact of sampler jitter on the spectrum performance is analyzed in detail in Ref. [19], and the Doppler ambiguity of MTI radar is resolved by means of the spectrum anti-aliasing performance of jittered sampling [11]. The jittered sampling instants are determined by randomly perturbing the regular samples. As shown in Figure 2d, the sampling sequence is defined as: where t m is the m-th sample. T d is the determined average sampling interval of jittered sampling. ζ is a random variable uniformly distributed on the interval [À βT d /2,βT d /2], and 0≤β≤1 determines the size of perturbation around the regular grids. The range extension coefficientαin the jittered sampling can be expressed as α¼T min /T avg ¼1À βT d /T d ¼1À β. The minimum interval between adjacent pulses can be set longer than a specific value by limiting the jitter range of sampling points, and the maximum unambiguous range can be extended by sub-Nyquist sampling in the slow time domain.

| The advantages of Poisson disk sampling on spectrum anti-aliasing performance
The maximum unambiguous range can be extended by enlarging the minimum pulse interval, but uniform undersampling will cause the well-known Doppler spectrum aliasing. It has been demonstrated that stochastic under-sampling can convert this kind of spectrum aliasing into non-structured noise and sidelobes [5]. Though the non-structured noise and sidelobes are consistently lower than the real Doppler shift caused by target motion, they have non-negligible influence on radar velocity estimation accuracy in real situation that contains external noise and clutter to be discussed later. In order to measure their intensity quantitively, we use the spectrum peak sidelobe ratio (PSLR) to compare the anti-aliasing performance of different stochastic sampling patterns, where the maximum sidelobe refers to the spectrum peak other than the real Doppler shift caused by target motion. Higher PSLR means better spectrum anti-aliasing performance, which is conducive to radar velocity estimation. To measure the number of stochastic samples more intuitively, we define the slow time domain under-sampling rate as M ¼ PRF N /PRF avg , where PRF N is the minimum uniform PRF that meets the requirement of Nyquist sampling theorem, and a larger value of M indicates fewer stochastic samples. Therefore, Equation (5) can be expressed in the form of Equation (9) That is to say, a larger maximum unambiguous range can be obtained by increasing the parameters α and M. To evaluate the influence of both parameters on the performance of magnitude spectrum at sub-Nyquist rates, the PSLRs of several stochastic sampling patterns under different values of α and M are illustrated in Figure 3. Each result is obtained by averaging 1000 Monte Carlo trials. Figure 3a shows the performance of PSLR versus the under-sampling rate M with α ¼ 0.7. We can see that with the increase of under-sampling rate, though the PSLR of the Poisson disk sampling slowly declines, it is consistently much higher than that of other stochastic sampling patterns. This indicates the excellent spectrum anti-aliasing performance of the Poisson disk sampling at high under-sampling rates. Figure 3b shows the performance of PSLR versus the range extension coefficient α with M ¼ 5. We can see that the increase of α has a greater negative impact on the spectrum PSLR of jittered sampling than Poisson disk sampling, and the Poisson disk sampling shows significant anti-aliasing performance even at highαvalues, which can provide large maximum unambiguous range. The PSLR of the Poisson disk sampling atα ¼ 0.8 is only 3 dB lower than that at α ¼ 0.1, while the PSLR of jittered sampling at α ¼ 0.8 is 15 dB lower than that at α ¼ 0.1. Besides, the PSLR of the Poisson disk sampling is better than that of the staggered sampling at all α values. The results demonstrate that the spectrum anti-aliasing performance of the Poisson disk sampling is significantly better than that of other stochastic sampling patterns at sub-Nyquist rates. Therefore, the Poisson disk sampling is utilized to extend the maximum unambiguous range of PD radar.  Figure 4, where each result is obtained by averaging 1000 Monte Carlo trials. The purpose of this experiment is the evaluate the maximum unambiguous range of PD radar. It can be seen that the spectrum PSLR of the Poisson disk sampling remains at a high level within a wide range of α and M, and shows insensitivity to the change of both parameters in most instances. Therefore, the Poisson disk sampling provides the potential to detect long-range high-speed moving targets unambiguously without sacrificing the velocity estimation performance.

| THE SPARSITY-DRIVEN METHOD FOR HIGH-RESOLUTION RANGE-DOPPLER SPECTRUM RECONSTRUCTION
In this section, a method for high-resolution range-Doppler spectrum reconstruction based on the combination of the Poisson disk sampling and the IST-like algorithm is proposed.

| Pulse-Doppler radar signal model
The transmitted chirp signal within a pulse can be expressed as: where τ donates the fast time, T p is the pulse width, ω stands for the chirp rate, f 0 is the radar carrier frequency, and rect(⋅) represents the unit rectangular function. Assume that there are Q moving targets with different velocities in a certain coarse range cell. After pulse compression by matching filter, the received signal in a range cell can be expressed as [20]: where t donates the slow time, σ q is the backscattering coefficient of the q-th target, T a is the duration time of the coherent processing interval (CPI), r q and v q represent the range and velocity of the q-th target, respectively, λ is the carrier wavelength, and N(t) stands for the received additive noise.
We discretize the interested Doppler frequency scope and range cell in the range-Doppler spectrum into K�L grids uniformly, K and L are the numbers of discrete cells in the Doppler and range domain, respectively. Then, the received signal in Equation (11) can be written in the following twodimensional discrete form: where k and l represent the cell indices of Doppler frequency and range, respectively, y(τ m ,t n ) stands for the m-th fast time sampling point at the n-th pulse of the original signal, and t n corresponds to the n-th non-uniform sampling instant in the slow time domain. N(τ m ,t n ) is the discretized additive noise. σ k, l denotes the complex-valued amplitude that need to be recovered at the k-th Doppler frequency cell and the l-th range cell of the range-Doppler spectrum. v k and r l are the velocity and range of the k-th Doppler frequency cell and the l-th range cell, respectively.

| Sparse driven model for high-resolution range-Doppler
Since the radar echo are non-uniformly under-sampled in the pulse domain for unambiguous range extension, we need to recover the unknown sparse signal from limited measurements. We use the CS theory to reconstruct the sparse range-Doppler spectrum.
Assume that the PD radar transmits P non-uniform interval pulses in the slow time domain, and the number of samples in the fast time domain is L. Then, the discretized echo y(τ m ,t n ) and the complex-valued amplitude σ k,l can be recorded in the matrix form of Y∈ℂ P�L and S∈ℂ K�L , respectively. Then, Equation (12) can be rewritten as:  where N is the noise matrix. The matrix Λ is a redundant dictionary that composed of K column vectors expressed as Λ¼[χ 1 ,χ 2 ,⋯,χ K ] where k ¼ 1,2,⋯,K and f k ¼(kÀ 1)/K⋅f s represents the k-th sample in Doppler frequency, f s is the sampling rate set to be larger than the maximum Doppler shift f dmax to avoid ambiguity. The non-uniform samples t 1 ,⋯,t P are obtained according to Equation (6).
There are only a small number of dominant targets in the observed sparse scene such as sea and air, and the scattering intensity of the targets are relatively high. Besides, the amplitude of non-structured noise in the Poisson disk sampling spectrum is much lower than the spectrum peaks that indicates moving targets. By exploiting sparsity, CS can effectively suppress both of the non-structured noise caused by stochastic sampling and the external noise and clutter [21]. Therefore, sparse recovery is utilized to reconstruct the high-resolution range-Doppler spectrum from non-uniform under-sampled signal. It is known that the sparse spectrum S can be recovered from the non-uniform under-sampled measurements Y under certain conditions [8] by solving the following unconstrained optimization problem, where ρ is the sparse regulation parameter andŜ is the reconstructed range-Doppler spectrum. ‖ • ‖ F is the Frobenius norm of matrix, and ‖ • ‖ 1 is the L 1 norm of matrix. Then, the range and velocity information can be extracted from the sparse recovered range-Doppler spectrumŜ for further detection.

| Spectrum reconstruction via IST-like algorithm
The IST algorithm [22] is an effective method for sparse reconstruction obtained by gradient derivation. We use the IST-like algorithm, which has the same form as the IST algorithm, to solve the optimization problem in Equation (15). The pseudo code of the IST-like algorithm for reconstructing the high-resolution range-Doppler spectrum from the Poisson disk sampled data is illustrated in Algorithm 1 ( Figure 5). First, we input the threshold parameter μ, the regularization parameter ρ, and the maximum iterations number I. Then, the range-Doppler spectrum, the residual matrix and the recovery error are initialized, and they update in each iteration. The range-Doppler spectrum cannot be acquired directly by the Fast Fourier Transform (FFT) since the radar echo are collected non-uniformly. Instead, we use the NUDFT matrix Ψ to transform the non-uniform under-sampled signal into the Doppler domain. The matrix Ψ¼[γ 1 ,γ 2 ,⋯,γ P ] is consist of P column vectors, where n ¼ 1, 2, ⋯, P. The residual matrix R i is transformed into the Doppler domain, and then added to the result of the previous iteration. After that, the threshold operation is performed to get the updated range-Doppler spectrum S i . For a vector z¼(z 1 ,z 2 ,⋯,z N ) T ∈ℂ N , the threshold operator Thre(⋅) in step 4 is defined by Ref. [23].
where soft (z,ρ) ¼ sign(z)max(|z|À ρ,0)is the soft-thresholding function, and sign(⋅) is the sign function. The nonuniform inverse discrete Fourier transform (NUIDFT) is then operated on the updated range-Doppler spectrum to obtain the approximate observation of the non-uniform radar signal. The NUIDFT matrix Λ has been defined above. Then, the residual matrix R i is updated by subtracting the approximate observation from the original measurements Y. The iteration carries out until the algorithm reaches the maximum iterations number I or the recovery error ξ i is smaller than the threshold parameter μ, and the algorithm outputs the sparse recovered high-resolution range-Doppler spectrumŜ¼S i .

| Analysis on computational complexity
Due to the non-uniform sampling of PD radar in the slow time domain, the regular FFT and IFFT cannot be directly applied to the transformation between the non-uniformly sampled F I G U R E 5 The pseudocode of the Iterative Soft Thresholding algorithm for range-Doppler spectrum reconstruction signal and the uniformly sampled spectrum. Instead, the NUDFT matrix Ψ and the NUIDFT matrix Λ are utilized in Algorithm 1. However, directly calculating the matrix multiplication consumes too much computing resources. As a fast method to calculate the spectrum of non-uniformly sampled signal, NUFFT has been widely used in radar imaging, MRI reconstruction and computational electromagnetics [24][25][26]. The key idea of NUFFT is to compute an oversampled FFT of the weighted form of the given signal, and then interpolate optimally onto the desired locations using small local neighbourhoods in the frequency domain. The details of several commonly used interpolation methods can be found in Ref. [27].
The range-Doppler spectrum of the proposed method is obtained through multiple iterations. Due to the uncertainty of the number of iterations, we first analyze the time complexity of single iteration. As mentioned above, there are P non-uniform sampled pulses in the slow time domain, and there are K and L discrete cells in the reconstructed Doppler and range domain, respectively. In each iteration of Algorithm 1, the computational complexity of directly calculating the two matrix multiplications is O(2PKL), which is very time-consuming. where I iter is the number of iterations. Generally, the values of P, K and L are not more than ten thousand, while J is at most tens and I iter is not more than 30. In the following section, we further investigate the running time of the proposed method, as the experimental results shown in Section 4.
As for the memory cost of the proposed method, we only need to storage the input, the output, and the interpolation coefficients. In brief, the memory cost is at the order of O(K).

| EXPERIMENTAL RESULTS
In this section, the performance of the proposed method is verified by experiments on the simulated and measured PD radar data.

| Simulations
In this simulation, twelve point targets with the same scattering intensity are distributed in the radar observation scene. Figure 6 shows the range-velocity distribution of the targets in the scene. The distance between the radar transmitter and the scene centre is 20 km, and the distance between the nearest and farthest targets is 10 km. The four columns of targets from left to right move away from and close to the radar at the radial velocities of À 18 m/s, À 6 m/s, 6 m/s and 18 m/s, respectively, where negative velocities indicate that the target moves away from radar. Some main parameters of the simulated PD radar system are shown in Table 1.
The PRF is set to be larger than the maximum Doppler shift caused by targets motion, so there is no aliasing in the range-Doppler spectrum. Figure 7a shows the range-Doppler spectrum reconstructed from the traditional uniform pulses. The simulated data are composed of 10,000 uniform pulses in the slow time domain. The maximum unambiguous range calculated by Equation (2) is R max ¼ 22.5 km, which indicates that only part of the targets can be detected. Due to the range ambiguity, it is unable to estimate the range and velocity of the targets at 25 km. In the framework of uniform pulses, PRF must be reduced to increase the maximum unambiguous range, which will simultaneously reduce the maximum unambiguous velocity and result in inaccurate velocity estimation.
As mentioned above, the stochastic sampling patterns can convert the structured spectrum aliasing caused by undersampling to non-structured noise and sidelobes, which can be suppressed to some extent by CS. This provides the possibility to increase the minimum pulse interval, thereby extending the maximum unambiguous range. Figure 7b-d compare the range-Doppler spectrum reconstructed by the IST-like algorithm under different stochastic sampling patterns. The number of sparse samples in the slow time domain is 5000, which means the under-sampling rate is M ¼ 2, and the range extension coefficient α is 0.75. Therefore, the maximum unambiguous range can be extended to 41.25 km according to Equation (9), which allows us to detect farther targets.
It can be seen in Figure 7b that there is relatively strong noise around the point targets in the range-Doppler spectrum of staggered sampling. And the spectrum of jittered sampling shown in Figure 7 has several high sidelobes near the point targets. Thus these two stochastic sampling patterns cannot achieve accurate velocity estimation. In comparison, the Poisson disk sampling pattern used in Figure 7d demonstrates excellent performance on the range and velocity estimation of each target. The spectral non-structured noise caused by non-uniform sampling can be well suppressed to obtain the range-Doppler spectrum that is consistent with the state of targets in Figure 6.
It has been shown in Ref. [28] that the compressed sensing theory has the advantage of enhancing resolution and reducing sidelobes. In Figure 8, we compare the profiles in the slow time direction of the same scattering point in Figure 7a and 7d. It can be seen that the proposed method has fewer sidelobes than the traditional uniform PRF method with matched filtering. At the same time, the Doppler resolution, which is indicated by the width of main lobe, remains unchanged. The simulation results show the effectiveness of the proposed method in unambiguous range extension and demonstrate the advantages of the Poisson disk sampling in velocity estimation accuracy over other two stochastic sampling patterns.

| Experimental results on real radar data
In our experiment, the raw data are collected by a high frequency PD radar. The carrier frequency of the radar transmitted signal is 4.1 MHz. The uniform PRF is 0.23 Hz and the number of pulses is 128. Figure 9a shows the range-Doppler spectrum reconstructed from the uniform pulses with PRF ¼ 0.23 Hz. The observed scene is sparse since there are only a few moving ships, which are denoted by the white areas in Figure 9. And there is some clutter and noise in the centre of the spectrum. Here range-Doppler spectrum reconstructed by the IST-like algorithm from the raw data is deemed as the benchmark, as shown in Figure 9b. Note that the raw data used in Figure 9b were sampled with a constant PRF of 0.23 Hz.
In order to investigate the velocity estimation performance of the Poisson disk sampling on the measured data, we first generate the non-uniform under-sampled data from the raw data. The raw data are firstly interpolated in the slow time domain to increase the sampling rate for 30 times. The interpolated data is then resampled according to the Poisson disk sampling pattern given in Equation (6). It is worth emphasizing that the average sampling rate in the slow time domain is set as PRF avg ¼ 0.046 Hz, which is five times under-sampled, and the range extension coefficientαis set to 0.5. Therefore, the under-sampling rate is M ¼ 5 and the maximum unambiguous range is extended to 2.5 times according to Equation (9). The sparse recovery result obtained by the proposed method with Poisson disk sampling is illustrated in Figure 10a. As a reference, the sparse reconstruction result of the random PRF selection method with the same under-sampling rate is given in Figure 10b.
To compare the performance of range-Doppler spectrum reconstruction between the proposed method and other existing sparse-driven methods, the radar raw data is resampled in the slow time domain with PRF avg ¼ 0.046 Hz and α ¼ 0.5 according to the staggered sampling pattern and jittered sampling pattern, respectively. Then, the IST-like algorithm is performed on the non-uniform under-sampled data to recover the sparse range-Doppler spectrum, and their reconstructed spectrums are shown in Figure 10c,d, respectively. The blue circles and the red squares in Figure 10 mark the false targets and missed detections, respectively.
We can see that though the maximum unambiguous range has been extended to the same scale, the accuracy of velocity estimation among different stochastic sampling patterns is significantly different. Compared to the baseline in Figure 9b, the spectrum of staggered sampling and jittered sampling both have some false alarms and missed detections. In comparison, the Poisson disk sampling can achieve accurate reconstruction of the range-Doppler spectrum. The range and velocity of all moving targets can be exactly estimated. The experimental results further demonstrate the superiority of the Poisson disk sampling in radar unambiguous detection and velocity estimation performance over other methods.
The running time results of different methods are shown in Table 2. All the experiments are performed on the same computer of a 6-core 4.3-GHz CPU with 32G memory. The size of the measured data is moderate since the sampling rate is relatively low, and it only takes several seconds for the proposed method to form the reconstructed range-Doppler spectrum. As for the high frequency PD radar studied, the integration time of a coherent processing interval is 9.28 min, which is much longer than the running time of the proposed method. The results illustrate the capability of real-time processing of the proposed method.

| The analysis on SNR loss
Compared with the traditional uniform PRF method with matched filtering, the stochastic under-sampling patterns result in SNR loss in the recovered range-Doppler spectrum. The SNR loss comes from the coherent integration of non-uniform samples in the slow time domain [29]. In order to extend the unambiguous range, the stochastic sampling patterns adopted in this study all work in under-sampling mode. As mentioned above, the stochastic sampling patterns can convert the structured spectrum aliasing caused by uniform undersampling to non-structured noise. The non-structured noise can be suppressed by the proposed IST-like sparse recovery method in Algorithm 1 to a large extent. However, there is still some noise that cannot be suppressed, which leads to the SNR loss compared to the traditional uniform PRF method with matched filtering, and the SNR loss may result in the wellknown false alarm in target detection after coherent integration.
To measure the SNR loss of different stochastic sampling patterns, the PD radar is assumed to work with the parameters in Table 1, and there is only a moving point-like target that is 20 km away from the radar in the observation scene. Figure 11a compares the SNR loss of several stochastic sampling patterns after sparse reconstruction with M ¼ 2 and α ¼ 0.75. We can see that the Poisson disk sampling pattern has lower SNR loss compared to other sampling patterns, especially when the target has a large velocity. In Figure 11b and Figure 11c, the radial velocity of the moving target is set to 50 m/s. Figure 11b compares the SNR loss of several  Figure 11c compares the SNR loss of several stochastic sampling patterns with different undersampling rate M at α ¼ 0.75. It can be seen that the SNR loss caused by Poisson disk sampling is significantly lower than that of jittered sampling and staggered sampling, which demonstrate the advantages of the proposed method in noise suppression. We propose a method for unambiguous range extension with PD radar based on the combination of the Poisson disk sampling and the IST-like algorithm. The Poisson disk sampling is capable of ensuring the minimum interval between any two adjacent pulses to be larger than a desired value. Therefore, the maximum unambiguous range of PD radar can be extended by adopting the Poisson disk sampling in the slow time domain at sub-Nyquist rates. The comparison in terms of PSLR shows the advantages of the Poisson disk sampling over other sampling patterns in spectrum anti-aliasing performance. Then, simulation results demonstrate that the Poisson disk sampling can detect farther targets than traditional uniform pulses with a constant PRF, with the same range of the unambiguous velocity estimation. The range-Doppler spectrum reconstructed by the IST-like algorithm illustrates that the Poisson disk sampling can improve the velocity estimation accuracy in comparison with other stochastic sampling methods. The experimental results on real pulse Doppler radar data further demonstrate the advantages of the proposed method over other sparse-driven methods in the range and velocity estimation of moving targets.