A Novel SV-PRI Strategy and Signal Processing Approach for High-Squint Spotlight SAR

Abstract

High-resolution and high-squint spaceborne spotlight synthetic aperture radar (SAR) has significant potential for extensive application in remote sensing, but its swath width effectiveness is constrained by a critical factor: severe range cell migration (RCM). To address this, pulse repetition interval (PRI) variation offers a practical scheme for raw data reception. However, the current designs for continuously varying PRI (CV-PRI) exhibit high complexity in engineering. In response to the issue, this paper proposes a novel strategy of stepwise varying PRI (SV-PRI), which demonstrates higher reconstruction accuracy compared with CV-PRI. Furthermore, confronting the azimuth non-uniform sampling characteristics induced by the PRI variation, this paper introduces a complete uniform reconstruction processing based on the azimuth partitioning methodology, which effectively alleviates the inherent contradiction between resolution and swath width. The processing flow, utilizing the temporal point remapping (TPR) concept, ensures the uniformity and coherence of dataset partitioning and reassembly in the context of the interpolation on non-uniform grids. Finally, according to the simulation results, the point target data, processed through the processing flow proposed in this study, have demonstrated effective focusing results.

1. Introduction

Synthetic aperture radar (SAR) is a powerful remote sensing technique, capable of providing high-resolution microwave images of the Earth’s surface regardless of weather conditions and sunlight illumination [1]. The growing demand for Earth observation necessitates improvements in radar technology to achieve more refined and informative imagery, making high-resolution imaging a critical criterion [2,3,4,5,6,7,8,9]. As a way to meet the requirement of a high-resolution azimuth, the spotlight mode has gained increasing prominence in recent years. In the spotlight mode, the spaceborne SAR system controls the direction of the radar antenna’s beam to continuously and extensively illuminate a specific observation area. This operational mode facilitates the acquisition of detailed terrain feature information and enables a more precise inversion of terrain parameters [2,8]. Based on the current in-orbit satellite data, spotlight SAR systems have demonstrated high-precision imaging capabilities with a sub-meter resolution, as exemplified by the prominent German TerraSAR-X satellite [3]. For specific regions of interest, the high-squint mode is imperative. The inclusion of a large angle significantly augments the acquisition of nuanced information regarding target characteristics and angle-dependent backscattering coefficients, enhancing the SAR system’s adaptability and observational capacity [9,10,11,12,13,14,15,16,17,18]. However, compared with the conventional mode, the imaging data reception in the high-squint spotlight SAR is constrained by severe range cell migration (RCM). This is a limitation arising from the inherent contradiction between resolution and swath width. Therefore, the acquisition of high-quality, focused images necessitates a trade-off, often resulting in reduced scanning width. In the case, the pulse repetition interval (PRI) variation, that is, PRI is dynamically adjusted in real time according to the changes in parameters such as slant range or angle from the reference target, emerges as an effective strategy for mitigating pulse interference caused by RCM [19].

In recent years, scholars have adopted the variable PRI technique. They proposed numerous methodologies involving continuously varying PRI (CV-PRI) to tackle blind zones resulting from fixed PRI sequences, as detailed in references [20,21,22,23,24,25,26]. However, due to the temporal design complexity, implementing CV-PRI sequences in engineering applications presents notable challenges. Theoretically, an alternative solution is to employ stepwise varying PRI (SV-PRI) sequences, which essentially strikes a balance between design complexity and data transmission rate.

The variation of PRI, equivalent to non-uniform sampling in an azimuth, introduces several challenges to subsequent imaging processing. Therefore, the processing of non-uniformly sampled data before imaging is a crucial aspect to consider. The Back-Projection Algorithm (BPA), as recommended in the reference [22], is well-suited for focusing non-uniformly sampled data in the azimuth-time domain. However, the principle of reconstructing a uniformly sampled signal opens up opportunities to employ both efficient and conventional frequency domain algorithms, as discussed in references [24,25,26]. These processing methods typically encompass two primary aspects: de-aliasing and interpolation techniques. The subsequent exposition will delve into these two aspects in a comprehensive manner.

Since the pulse repetition frequency (PRF) is lower than the Doppler bandwidth, the raw signal data are under-sampled and aliased, which implies that direct resampling will lead to unacceptable errors. Reference [13] advocates for de-ramping in two-step processing to remove aliasing and introduces a nonlinear shift processing method which removes the squint angle’s impact on focusing. Reference [9] proposes the extended two-step approach (TSA) to process the full-aperture and high-resolution data. However, reference [26] points out that even when utilizing the TSA, the spotlight SAR data, characterized by their high resolution, high squint angle, and large transmitted bandwidth, still suffer from severe aliasing. Furthermore, while the reference [26] recommends employing phase compensation based on the reference phase history [27], this method imposes an upper bound constraint on the swath width in the direction of the radar’s line of sight. This issue fundamentally stems from the inherent contradiction between resolution and swath width. Moreover, the significant Doppler effect, a consequence of the large transmission bandwidth, requires extensive zero-padding to maintain the effectiveness of the time-domain segmentation in the range direction. Undoubtedly, this method leads to a significant increase in data volume. Consequently, it becomes essential to explore alternative solutions that address the resolution-swath width contradiction.

In interpolation processing, the predominant methods include the non-uniform fast Fourier transform (NUFFT) and various interpolation algorithms. In reference [26], the analysis demonstrated that the weighted NUFFT is accurate and efficient in comparison with the best linear unbiased (BLU) interpolation and other NUFFT implementations. Furthermore, reference [28] proposes a novel Sinc-interpolation method based on the non-uniform discrete Fourier transform (NUDFT). This method eliminates ambiguous targets and offers focusing performance comparable to that of the NUDFT. Additionally, reference [12] describes performing baseband Lagrange interpolation to resample the data onto a uniform grid.

This study mainly analyzes the reception and processing of spotlight SAR raw data with a decimeter resolution (0.1 m) and large squint angle (40°). In terms of parameter design, the paper presents a specific strategic solution for the SV-PRI sequences with variable granularity, aimed at reducing engineering complexity. The phenomenon of spaceborne echoes spanning multiple pulse repetition intervals often triggers cascading effects in the design of timing sequences. Consequently, the sharp decrease characteristic of SV-PRI may result in the loss of echo data. To solve this issue, this study introduces constraints on abrupt reductions in SV-PRI, and theoretical deduction demonstrates that the proposed sequence effectively circumvents the problem. Regarding data processing, we propose a processing flow based on the azimuth partition to obtain uniformly sampled signal data, thereby alleviating the inherent contradiction between resolution and swath width. This study tackles the compatibility issues associated with azimuth data partitioning and reassembly in azimuth resampling, and introduces a novel partitioning approach based on temporal point remapping (TPR). The TPR strategy facilitates the effective reconstruction of a uniform grid, substantially simplifying the processing workflow.

The rest of this paper is organized as follows. Section 2 starts with a brief review of high-squint spotlight SAR imaging geometry, and then the analysis of the SV-PRI sequence design is described. Section 3 discusses the proposed processing of non-uniformly sampled signals. The experimental results of the proposed method are presented in Section 4. Discussion and remarks on possible further work are given finally in Section 5.

2. SV-PRI Sequences for High-Squint Spotlight SAR

2.1. Acquisition Geometry

As the orbital model is not the focal point of this paper, an equivalent simplified model of the satellite-ground geometry is constructed for the parameter design analysis. Under the assumptions of a linear satellite trajectory and a planar Earth surface, and ignoring Earth’s rotation, a Cartesian coordinate system is established for a high-resolution and high-squint spotlight SAR. This system takes the subsatellite point as the origin, where the y-axis aligns with the satellite’s velocity vector, the x-axis is orthogonal to the y-axis and extends along the ground range, and the z-axis is directed upward and perpendicular to the Earth’s surface. As illustrated in Figure 1, the imaging geometry considers an orbital altitude of H, an off-nadir angle of ${\alpha }_0$, an along-track velocity of $V_r$, and a range swath width of $W_r$. The reference target in the scene is denoted by the point P, and $R_0$ represents the minimum slant range to the radar platform in a positive side-looking configuration. The spotlight irradiation begins at time ${\eta }_s$ and ends at time ${\eta }_e$, with corresponding squint angles marked as ${\theta }_{max}$ and ${\theta }_{min}$. The central time‘s squint angle is ${\theta }_{sq}$, and the beam rotation angle is ${\theta }_{syn}$.

During data acquisition in spotlight SAR, the antenna’s direction and velocity are meticulously controlled to facilitate prolonged illumination of the observed area. Concurrently, using high-squint angles broadens the variation range in slant range between a point target and the radar platform throughout the synthetic aperture period, consequently increasing the magnitude of RCM. The simulation analysis, depicted in Figure 2, investigates the maximal migration distance of the point target in spaceborne spotlight SAR when the nearest slant range is 1200 km, in relation to changes in resolution and squint angle. In Figure 2a, the central squint angle of the spaceborne SAR is maintained at 5°, whereas in Figure 2b, the azimuth resolution is fixed at 0.1 m. The analysis reveals that when azimuth resolution is 0.1m, the RCM can surpass 18 km at small squint angles. Notably, when the squint angle is increased to 45°, the RCM can escalate to as much as 300 km. Such a high magnitude of RCM significantly restricts the imaging swath and hinders the system’s ability to achieve the required azimuth resolution. Therefore, to enable high-resolution imaging within the effective swath width, it is critical to consider strategies for reducing RCM.

2.2. Design of the SV-PRI Sequence

PRI variation offers a practical scheme to significantly reduce RCM in echo data. In this section, we will detail the principles of pulse transmission and reception corresponding to fixed PRI and varying PRI, and introduce a novel iterative computation strategy for the SV-PRI sequence.

In single-base spaceborne SAR, pulse transmission and echo reception typically occur within a single PRI. Figure 3 illustrates the pulse timing of three different sequence types. Assuming a constant length and opening time of the receive echo window, denoted by $T_w$ and $T_s$, the pulse duration is $T_p$. $PR{I}_s$ in Figure 3a signifies a fixed interval, $PR{I}_n$ in Figure 3b represents the nth PRI value, and $PR{I}_k$ in Figure 3c represents the PRI value for the kth period. Figure 3 demonstrates that data blind zone can be mitigated by aligning echo data from the reference target with the receive echo window, using varying PRI calculated from variable parameters like slant range or squint angle. For the CV-PRI sequence, the interval between the arrival of the reflected pulse and the transmission of the previous pulse is kept constant at $\Delta {\eta }_c$. The SV-PRI sequence aims to keep the nth interval $\Delta {\eta }_n\left(n=1,2,3,\dots ,N\right)$ within an optimal range, and N is the total number of transmitted pulses. These two PRI variations allow the radar system to continuously gather signal data from the target area without gaps. However, considering the design complexity of the CV-PRI sequence, the SV-PRI sequence is more advantageous. For the accuracy comparison in data processing between the two types of sequences, an in-depth analysis will be conducted in the subsequent simulations.

In the following, we will introduce a method for calculating the SV-PRI sequence. The discussion begins by focusing on the receive echo window. As illustrated in Figure 4, $T_{ws}$ represents the interval for safeguarding data, and $T_{echo}$ signifies the echo length, which can be estimated by

$$T_{echo}=\frac{2(R_{far}-R_{near})}{c}+T_p$$

(1)

where $R_{far}$ and $R_{near}$ are the slant ranges of the farthest and nearest point targets in the scene at the central moment. To accommodate the specific transceiver mechanism, the receive echo window must meet two primary conditions, which are given by

$$T_s>0$$

(2)

$$T_w\le PR{I}_{min}-T_p-2\ast T_s$$

(3)

Formula (2) stipulates that the window only activates after the transmitter is fully deactivated. Additionally, it follows from Formula (3) that the duration of reception must not exceed the minimum sampling interval $PR{I}_{min}$ in the variable PRI sequence.

Take the target at scene center as the reference target. First of all, the initial sampling time interval $PR{I}_s$ should be determined according to the acquisition geometric parameters and the beam Doppler bandwidth. Assume that the pulse transmission time vector is ${\eta }_u=[{\eta }_{u1},{\eta }_{u2},{\eta }_{u3},\dots ,{\eta }_{uN}]$, and ${\eta }_{un}$ means the start time of the nth transmitted pulse. The echo delay time of nth transmitted pulse, defined as ${\eta }_{dn}$, can be calculated using slant range models $R_{ref}\left({\eta }_{un}\right)$ with different accuracy. Based on start-stop model, ${\eta }_{dn}$ can be approximately equivalent to

$${\eta }_{dn}\approx \frac{2R_{ref}\left({\eta }_{un}\right)}{c}$$

(4)

where c is the speed of light. To meet the high-precision timing design requirements for high-resolution imaging, ${\eta }_{dn}$ can be updated according to [26]:

$${\eta }_{dn}=\frac{R_{ref}\left({\eta }_{un}\right)+R_{ref}({\eta }_{un}+{\eta }_{dn})}{c}$$

(5)

The echo delay time from the reference target at the central time is defined as

$${\eta }_{dc}=\frac{2R_0}{ccos{\theta }_{sq}}$$

(6)

Let M denote the number of complete $PR{I}_s$ spanned by the echo delay time, which is given by

$$M=⌊\frac{{\eta }_{dc}}{PR{I}_s}⌋$$

(7)

Then, as shown in Figure 3a, $\Delta {\eta }_n$ equals

$$\Delta {\eta }_n={\eta }_{dn}-M·PR{I}_s$$

(8)

With the fixed $T_w$, $T_s$, and $T_p$, as shown in Figure 3b, CV-PRI sequences can reduce migration in a finite receive echo window by controlling $\Delta {\eta }_n$ to approximate $\Delta {\eta }_c$, which can be fixed to

$$\Delta {\eta }_c=\frac{T_w}{2}+\frac{T_p}{2}+T_s$$

(9)

In the given acquisition geometry, the slant range of the reference target gradually decreases, resulting in a consistent leftward shift of the echo data within the window when employing a fixed PRI sequence. For the SV-PRI sequence, as shown in Figure 4, to maximize the utilization rate of the receive echo window, the first echo data of each period should be positioned on the right side as much as possible, which requires updating $\Delta {\eta }_c$; then,

$$\Delta {\eta }_c=T_p+T_s+\frac{T_w}{2}-\frac{T_{echo}}{2}-T_{ws}$$

(10)

$PR{I}_s$ and squint angle ${\theta }_{sq}$ in the azimuth central time determine the start time ${\eta }_{u1}$ of the first transmitting pulse in the synthetic aperture time. ${\eta }_{d1}$ can be calculated using Formulas (4) and (5), and then,

$$PR{I}_1=\frac{{\eta }_{d1}-\Delta {\eta }_c}{M}$$

(11)

To specify the period of the SV-PRI sequence, we define a granularity factor $\delta \left(\delta >0\right)$, and the SV-PRI sequence period is denoted by ${\delta }_M=\delta ·M\left({\delta }_M\ge 1\right)$, which requires positive integer quantification. As previously described, $\Delta {\eta }_n$ needs to be maintained within a certain range; that is,

$$T_p+T_s+T_{ws}<\Delta {\eta }_n<\Delta {\eta }_c$$

(12)

To achieve the goal of complete echo capture within this range, ${\delta }_M$ must meet the condition of

$$\left|{\eta }_{d1}-{\eta }_{d{\delta }_M}\right|<T_w-T_{echo}-2·T_{ws}$$

(13)

that is,

$$\frac{2\left|R_{ref}\left({\eta }_{u1}\right)-R_{ref}({\eta }_{u1}+({\delta }_M-1)·PR{I}_1)\right|}{c}<T_w-T_{echo}-2·T_{ws}$$

(14)

The number of periods in the SV-PRI sequence is

$$K=⌊\frac{N}{{\delta }_M}⌋$$

(15)

Using the iterative idea, subsequent $PR{I}_n$ and ${\eta }_{un}$ can be calculated until the entire sequence is obtained; that is,

$$PR{I}_n=\left\{\begin{array}{cc}({\eta }_{dn}-\Delta {\eta }_c)/\phantom{({\eta }_{dn}-\Delta {\eta }_c)M}M,& n=1,{\delta }_M+1,2{\delta }_M+1,\dots ,K·{\delta }_M+1\\ PR{I}_{n-1},& esle\end{array}\right.$$

(16)

$$\begin{array}{cc}{\eta }_{u(n+1)}={\eta }_{un}+PR{I}_n,& n=1,2,3,\dots ,N-1\end{array}$$

(17)

2.3. Analysis of Echo Data Loss

As mentioned in reference [24], due to the echo delay of M PRI before arrival in spaceborne SAR, there is a potential risk of echo data loss between two consecutive periods in the SV-PRI sequence. This paper conducts a thorough theoretical analysis to determine if the proposed SV-PRI sequence experiences this issue.

Assuming full capture of the last M pulse echoes in the kth period, the Formula (12) can be rewritten as

$$T_p+T_s+T_{ws}<{\eta }_{d((k+1)·{\delta }_M-M+l)}-(M-l+1)·PR{I}_k-(l-1)·PR{I}_{k+1}<\Delta {\eta }_c$$

(18)

where $k=1,2,3,\dots ,K-1$, and $1<l\le M$. Define the abrupt decrement value between two consecutive periods as $\Delta {\alpha }_k=PR{I}_k-PR{I}_{k+1}$, and the range of $\Delta {\alpha }_k$ is given by

$$\frac{M·PR{I}_k+(T_p+T_s+T_{ws})-{\eta }_{d((k+1)·{\delta }_M-M+l)}}{l-1}<\Delta {\alpha }_k<\frac{M·PR{I}_k+\Delta {\eta }_c-{\eta }_{d((k+1)·{\delta }_M-M+l)}}{l-1}$$

(19)

Formula (19) delineates the specific range for abrupt decreases in SV-PRI. According to the imaging geometry model constructed in this study, data loss is related to the constraints on the right side of Formula (19). As a result, this study focuses solely on the right-hand side constraint. From Formula (16), the values of $PR{I}_k$ and $PR{I}_{k+1}$ are

$$PR{I}_k=\frac{{\eta }_{d(k·{\delta }_M+1)}-\Delta {\eta }_c}{M}$$

(20)

$$PR{I}_{k+1}=\frac{{\eta }_{d((k+1)·{\delta }_M+1)}-\Delta {\eta }_c}{M}$$

(21)

Therefore, $\Delta {\alpha }_k$ can be updated to

$$\Delta {\alpha }_k=\frac{{\eta }_{d(k·{\delta }_M+1)}-{\eta }_{d((k+1)·{\delta }_M+1)}}{M}$$

(22)

And the right side of Formula (19) can be rewritten as

$$\frac{{\eta }_{d(k·{\delta }_M+1)}-{\eta }_{d((k+1)·{\delta }_M+1)}}{M}<\frac{{\eta }_{d(k·{\delta }_M+1)}-{\eta }_{d((k+1)·{\delta }_M-M+l)}}{l-1}$$

(23)

that is,

$$\delta ·\frac{{\eta }_{d(k·{\delta }_M+1)}-{\eta }_{d((k+1)·{\delta }_M+1)}}{\delta ·M}<\frac{{\delta }_M-M+l-1}{l-1}·\frac{{\eta }_{d(k·{\delta }_M+1)}-{\eta }_{d((k+1)·{\delta }_M-M+l)}}{{\delta }_M-M+l-1}$$

(24)

In Figure 1, the backward difference in slant range of the reference target presents a decreasing trend, so Formula (24) can be updated to

$$\delta \le 1+\frac{\delta -1}{l-1}·M$$

(25)

that is

$$(1-\frac{M}{l-1})·\delta \le 1-\frac{M}{l-1}$$

(26)

Formula (26) applies when $\delta >0$ and $1<l\le M$. Therefore, it can be inferred that the solution scheme proposed in this paper meets the necessary and sufficient conditions to prevent pulse echo loss, thereby confirming its effectiveness in preserving signal integrity within the SAR system.

3. Processing of Non-Uniformly Sampled Data

In this section, the processing of non-uniformly sampled data is discussed. As illustrated in Figure 5, the proposed flow primarily encompasses three key aspects: Doppler history recovery, azimuth partitioning and de-aliasing, and azimuth resampling and reassembly. The analysis will be sequentially conducted on these aspects.

3.1. Doppler History Recovery

For any single point target, the baseband signal in the range-frequency domain after non-uniform sampling and demodulation can be expressed as

$$S(f_{\tau },{\eta }_u)=W_r\left(f_{\tau }\right)exp\left\{-\mathrm{j}\frac{4\pi (f_0+f_{\tau })R\left({\eta }_u\right)}{c}\right\}exp\left\{-\mathrm{j}\frac{\pi f_{\tau }^2}{K_r}\right\}exp\left\{\mathrm{j}2\pi f_{\tau }\Delta \tau \right\}$$

(27)

where $f_{\tau }$ denotes the range frequency, $f_0$ denotes the carrier frequency, $W_r\left(f_{\tau }\right)$ denotes the range spectral envelope, and $K_r$ denotes the rate of frequency modulation in the chirp signal. The modification of PRI results in a temporal displacement $\Delta \tau$ during the initial phase of rapid time sampling. In this process, $\Delta \tau$ leads to the archiving and documentation of the 2D signal at a specific position, distinct from the range gate determined by the fixed PRI. To recover the Doppler history information, the consistent starting sampling can be compensated by multiplying the SV-PRI signal with the phase function $H_d(f_{\tau },{\eta }_u)$ in the range-frequency domain. The expression of $H_d(f_{\tau },{\eta }_u)$ is

$$H_d\left(f_{\tau },{\eta }_u\right)=exp\left\{-\mathrm{j}2\pi \left(f_{\tau }\Delta \tau \right)\right\}$$

(28)

However, this phase compensation results in significant signal folding in the range-time domain. In high-resolution and high-squint SAR systems, the conventional remediation, zero-padding, is no longer feasible. Consequently, the signal reconstruction methodology proposed in this paper is entirely based on the range-frequency domain, which effectively avoids the influence of this phenomenon.

3.2. TPR-Based Azimuth Partitioning and De-Aliasing

The partitioning approach presented in this paper primarily aims to surmount the limitations imposed by the range swath, thereby facilitating effective de-aliasing. As noted in the reference [26], the range swath along the radar’s line of sight is limited by

$$\Delta \mathrm{R}<\frac{\alpha -1}{\alpha ·PR{I}_{max}}\frac{R_{ref}\left({\eta }_{uc}\right)}{B_{dop,ref}}$$

(29)

where $\Delta R$ represents the maximum differential range of slant ranges in the direction of the radar’s line of sight at azimuth central moment ${\eta }_{uc}$, and $B_{dop,ref}$ denotes the Doppler bandwidth of the reference target. Considering the impact of the signal folding phenomenon mentioned earlier, dividing the range swath in the range-time domain to meet the condition given by Formula (29) is not an optimal solution. Essentially, Formula (29) exemplifies the contradiction between resolution and swath width. A more effective approach is to shift directly from range partitioning in the range-time domain to azimuth partitioning in the range-frequency domain. The entirety of the data in the azimuth time can be partitioned into multiple single-segment datasets, each acquired over a short azimuth time ${\eta }_u^{\left(k\right)}$ (k means the index of the kth dataset). In this way, the resultant residual bandwidth is compensated through an appropriate oversampling factor, and Formula (29) can be rewritten as

$$\frac{{\alpha }^{\left(k\right)}-1}{{\alpha }^{\left(k\right)}·PR{I}_{max}^{\left(k\right)}}\frac{R_{ref}\left({\eta }_{uc}^{\left(k\right)}\right)}{B_{dop,ref}^{\left(k\right)}}>\Delta R$$

(30)

The concept of azimuth partitioning in varying-PRI spotlight SAR, as previously introduced in the literature [29], is primarily aimed at conforming to the stepwise characteristics of the PRI sequence. This method undergoes two stages of interpolation. Initially, each segment of uniformly sampled data is zero-padded in the azimuth frequency domain to achieve a similar sampling rate. Then, after reassembling the datasets, a secondary, more precise interpolation is performed to establish a complete dataset with uniform grid points. These two interpolation processes cumulatively contribute to a considerable computational burden. To further simplify this processing, this paper introduces a data partitioning and reassembly strategy based on TPR. The strategy allows each data segment to undergo a single interpolation, resulting in a strictly uniform sampling signal after reassembly. More importantly, this method reduces the number of interpolations required, significantly decreasing the computational load. Next, the details of this method will be elaborated upon.

Firstly, by analyzing the non-uniform transmission pulse vector ${\eta }_u$ and the number of sampling points N, the corresponding uniform transmission pulse vector ${\eta }_n$ and uniform interval $PR{I}_{new}$ for the reconstructed uniform grid can be determined; that is,

$$PR{I}_{new}=\frac{{\eta }_{uN}-{\eta }_{u1}}{N-1}$$

(31)

$${\eta }_n=\left\{{\eta }_{ni}\left|{\eta }_{ni}={\eta }_{u1}+(i-1)·PR{I}_{new}\right.,i=1,2,\dots ,N\right\}$$

(32)

Determine the point number $N_b$ in each single-segment dataset that satisfies the condition of Formula (30), deriving the quantity of dataset K. The numbers of the kth non-uniformly and uniformly sampled points are denoted by $N_k$ and ${N_k}^{\prime }$. The kth non-uniformly and uniformly sampled data are calibrated as $P_k$ and ${P_k}^{\prime }$, where $k=1,2,\dots ,K$. Note that point number $N_K$ of datum $P_K$ is not strictly equal to $N_b$. Then, the single-segment pulse transmission time vector is given by

$$\begin{array}{cc}{\eta }_u^{\left(k\right)}=\left\{{\eta }_{ui}^{\left(k\right)}\left|i\in \left[(k-1)N_b+1,k{N}_b+1\right]\right.\right\},& k=1,\dots ,K-1\end{array}$$

(33)

$$\begin{array}{cc}{\eta }_u^{\left(k\right)}=\left\{{\eta }_{ui}^{\left(k\right)}\left|i\in \left[(k-1)N_b+1,N\right]\right.\right\},& k=K\end{array}$$

(34)

Define a set of temporal points $T_b$, which is given by

$$T_b=\left\{t_b^k\left|t_b^k={\eta }_{u(k-1)N_b+1}^{\left(k\right)},k=1,2,\dots ,K\right.\right\}$$

(35)

Define the remapping coefficient set $M_b$, which is given by

$$M_b=\left\{m_b^k\left|m_b^k=⌊\frac{t_b^k}{PR{I}_{new}}⌋,k=1,2,\dots K\right.\right\}$$

(36)

The mapped uniform transmission pulse vector is denoted by ${\eta }_n^{\left(k\right)}$; then,

$${\eta }_n^{\left(k\right)}=\begin{array}{cc}\left\{mPR{I}_{new}\left|m\in \left[m_b^k,m_b^{k+1}\right]\right.\right\},& k=1\end{array}$$

(37)

$${\eta }_n^{\left(k\right)}=\begin{array}{cc}\left\{mPR{I}_{new}\left|m\in \left[m_b^{k-1}+1,m_b^k\right]\right.\right\},& k=2,\dots ,K\end{array}$$

(38)

Following the computation of Formula (31) to Formula (38), the uniform grid ${\eta }_n^{\left(k\right)}$ is remapped from the non-uniform grid ${\eta }_u^{\left(k\right)}$. Then, the de-aliasing operation is performed via a multiplication with the phase function [29]:

$$Q_k(f_{\tau },{\eta }_u^{\left(k\right)})=exp\left\{\mathrm{j}\frac{4\pi (f_0+f_{\tau })R_{ref}\left({\eta }_u^{\left(k\right)}\right)}{c}\right\}$$

(39)

Figure 6 presents a concrete example of the TPR-based partitioning strategy, depicting a situation where the data are divided into three segments. From the illustration, it is evident that the start points of each segment on the non-uniform axis form a set of temporal points $T_b$. Utilizing the TPR approach, each start point on the non-uniform axis is mapped to the end point of the previous segment and the start point of the subsequent segment on the uniform axis. The interval between these two points is maintained at $PR{I}_{new}$, ensuring a completely uniform sampled grid after data reassembly. The TPR strategy is based on data correlation for remapping. It does not require setting the number of divided datasets to be the same as the number of changes in the SV-PRI sequence. Thus, the processing method is also applicable to SV-PRI with smaller $\delta$ and CV-PRI.

3.3. Azimuth Resampling and Reassembly

After the processing of azimuth partitioning and de-aliasing, the sampling intervals of the single-segment data $P_k$ all conform to the Nyquist criterion. For the data $P_k$, that is, $s\left({\eta }_u^{\left(k\right)}\right)$, an efficient and precise interpolation function can be utilized to reconstruct the values $s\left({\eta }_n^{\left(k\right)}\right)$ in a uniform grid ${\eta }_n^{\left(k\right)}$. This paper selects a weighted-Sinc interpolation suitable for non-uniformly resampled signals [28], with the function expressed as

$$\begin{array}{cc}s\left({\eta }_n^{\left(k\right)}\right)=\frac{1}{PR{I}_{new}}{\displaystyle \sum _{i=0}^{N_k-1}}s\left({\eta }_{ui}^{\left(k\right)}\right)\Delta {\eta }_{ui}^{\left(k\right)}sinc\left[\frac{1}{PR{I}_{new}}({\eta }_n^{\left(k\right)}-{\eta }_{ui}^{\left(k\right)})\right],& k=1,2,\dots ,K\end{array}$$

(40)

where $\Delta {\eta }_{ui}^{\left(k\right)}$ is the ith non-uniform sampling interval of datum $P_k$. Then, the resampled datasets should be multiplied with the function

$$Q_k^{\ast }(f_{\tau },{\eta }_n^{\left(k\right)})=exp\left\{-\mathrm{j}\frac{4\pi (f_0+f_{\tau })R_{ref}\left({\eta }_n^{\left(k\right)}\right)}{c}\right\}$$

(41)

After phase compensation, each dataset is reassembled based on its indices; that is,

$${\eta }_n=\left[{\eta }_n^{\left(1\right)},{\eta }_n^{\left(2\right)},{\eta }_n^{\left(3\right)},\dots ,{\eta }_n^{\left(K\right)}\right]$$

(42)

$$s\left({\eta }_n\right)=\left[s\left({\eta }_n^{\left(1\right)}\right),s\left({\eta }_n^{\left(2\right)}\right),s\left({\eta }_n^{\left(3\right)}\right),\dots ,s\left({\eta }_n^{\left(K\right)}\right)\right]$$

(43)

where $s\left({\eta }_n\right)$ represents the signals after uniform reconstruction in the range-frequency and azimuth-time domain. By utilizing azimuth fast Fourier transform (FFT), the 2D spectrum that is equivalent to uniformly sampled data can be acquired.

4. Simulation Results

In this section, simulated spaceborne SAR data are used to analyze the performance of the proposed method. The simulation focuses on the reference target and utilizes its performance as a benchmark to evaluate the effectiveness of the proposed SV-PRI sequences and the uniform signal reconstruction process. The data geometry acquisition model in the simulation is shown in Figure 1, and specific system parameters are summarized in

Table 1. SAR system parameters used in the simulation.

Parameter	Value
Orbit height	1000 km
Antenna length	12 m
Carrier frequency	9.6 GHz
Pulse bandwidth	3 GHz
Off-nadir angle	30°
Squint angle	40°
Cross-range resolution	0.1 m
Initialized PRF	1018.1 Hz

. All the simulation results in this paper are implemented in the MATLAB 2022b software.

4.1. Performance Analysis of the SV-PRI Sequence

4.1.1. Granularity Factor and Residual RCM

Firstly, we will discuss the inhibitory effects of the SV-PRI sequence on the RCM. The simulation provides two example results of SV-PRI sequences, as shown in Figure 7. Figure 7a,b illustrate the SV-PRI sequences with granularity factors of $\delta$ = 1 and $\delta$ = 10 and their residual RCM, respectively. From the left side of Figure 7 and with its magnified view, it is evident that these sequences exhibit distinct stepwise characteristics. Sequences with larger granularity factor $\delta$ possess a larger period of variation while also having lower temporal design complexity. The right side of Figure 7 indicates that both SV-PRI sequences reduce the residual RCM to within a controllable range. When the granularity factor $\delta$ = 10, the absolute value of the residual RCM is reduced to approximately 200 m. Meanwhile, when the granularity factor $\delta$ = 1, the absolute value of the residual RCM decreases to about 0.00025 m. The numerical results depicted in Figure 7 indicate that the finer the granularity factor of the sequence is, the stronger the inhibitory effect on RCM will be. Compared with the previously mentioned RCM of several hundred kilometers, the SV-PRI sequences proposed in this paper have a markedly significant effect on the suppression of RCM.

In summary, as the granularity factor increases, temporal design complexity is reduced, while the residual RCM escalates. In practical engineering, an appropriate granularity factor can be selected based on specific requirements, seeking a balance between temporal design complexity and residual RCM.

4.1.2. Resampling Relative Error

To evaluate the SV-PRI resampling performance, we define the relative error as the ratio of the error energy to the true signal energy [30], which is given by

$$\epsilon =\frac{{\displaystyle \sum _{p=1}^N}{\left|\stackrel{\wedge }{s}\left(p\right)-s\left(p\right)\right|}^2}{{\displaystyle \sum _{p=1}^N}{\left|s\left(p\right)\right|}^2}$$

(44)

In Formula (44), $\stackrel{\wedge }{s}(·)$ is the signal sample containing the interpolation error, and $s(·)$ is the true uniformly sampled signal. For comparison, this simulation selects the CV-PRI sequences in references [24,25], and the relative errors are as shown in

Table 2. Resampling errors with different types of PRI.

Types of PRI	Resampling Error (dB)
CV-PRI in  [24]	−33.73
CV-PRI in  [25]	−32.96
SV-PRI ($\delta$ = 1)	−42.77
SV-PRI ($\delta$ = 2)	−35.01
SV-PRI ($\delta$ = 5)	−34.98
SV-PRI ($\delta$ = 10)	−35.03

. The results demonstrate that, compared with the CV-PRI, the SV-PRI sequences’ errors are smaller. And it is particularly noteworthy that the SV-PRI with the period M exhibits superior reconstruction accuracy, which is reduced by about 9 dB. Obviously, the proposed SV-PRI sequences show advantageous reconstruction performance with higher precision.

4.2. Analysis of Uniform Signal Reconstruction Processing

In this section, the approaches described in references [28,29] have been simulated and contrasted with the method proposed in this study. To quantitatively assess the uniform reconstruction performance of this paper’s processing flow compared with the other literature, the relative error described in Section 4.1.2 continues to be selected as the evaluation metric.

For generality, we selected two SV-PRI sequences with the highest reconstruction accuracy and the lowest temporal design complexity for simulation and comparative analysis. The resampling errors in the uniform signal reconstruction methods are presented in

Table 3. Resampling errors in uniform signal reconstruction methods.

Method of Uniform Reconstruction	Resampling Error (dB)
	$\mathit{\delta }$= 1	$\mathit{\delta }$= 10
Method in [9]	2.97	2.98
Method in [29]	−42.76	−34.33
Proposed method	−42.77	−35.03

. The results indicate that the TSA method, combined with weighted-Sinc interpolation as referenced in [28], leads to severe deterioration errors. This phenomenon can be attributed to the fact that, as demonstrated in Figure 8a, the high-resolution, high-squint spotlight SAR data exhibit severe aliasing in the 2D spectrum even after TSA de-ramping. This issue is fundamentally rooted in the inherent contradiction between resolution and swath width. In Figure 8b, the TPR-based azimuth partitioning method alleviates this inherent contradiction, and as a result, the 2D spectrum of the reference target has been effectively compressed. Figure 8c demonstrates that the 2D spectrum achieves more accurate compression in the azimuth direction. Compared with the reference phase history method combined with the NUFFT in reference [29], the proposed method’s resampling error is similar when the granularity factor $\delta$ = 1. However, at a granularity factor of $\delta$ = 10, due to the NUFFT amplifying the periodic effects of the SV-PRI sequences, our method shows superior reconstruction accuracy.

4.3. Focusing Results

To analyze the impact of SV-PRI and the proposed method for uniform signal reconstruction on focused imaging, the simulation employs the BPA to focus on the reference point target. This study compares the focusing results of uniform signals with those of non-uniform signals processed using the proposed uniform reconstruction method.

Table 4. Focusing results of the reference target.

	PSLR (dB)	ISLR (dB)	IRW
Uniformly sampling	−13.2546	−10.0466	0.8696
Proposed method	−13.2549	−10.0468	0.8696

presents the simulation comparison results for the Peak Sidelobe Ratio (PSLR), Integrated Sidelobe Level Ratio (ISLR), and Impulse Response Width (IRW). The consistency between these results underscores that the imaging quality of the non-uniform signal, when processed with the method proposed in this study, is on par with that of the uniform signal. As illustrated in Figure 9, when processed according to the method outlined in this paper, the imaging of the reference point target achieves favorable focusing effects.

5. Discussion

In summary, the innovative work proposed in this paper is aimed at raw unfocused data for high-resolution images, primarily involving two major aspects: the reception and processing of the raw signal. For the significant challenges of severe RCM in high-resolution and high-squint spaceborne spotlight SAR, this paper proposes a novel strategy involving the variable granularity SV-PRI sequence, which ensures the reception of raw signal data and lays the foundation for high-resolution imaging. Compared with the CV-PRI sequence, the SV-PRI sequence shows higher resampling accuracy. Additionally, its stepwise feature minimizes the timing design complexity, thus simplifying engineering implementation. Varying-PRI technology resolves the issue of RCM, but leads to a non-uniformly sampled signal. To address the problem, this paper proposes a comprehensive processing workflow with TPR-based azimuth partitioning and weight-Sinc interpolation for uniform reconstruction. The results show that the proposed method exhibits superior performance.

This method has achieved commendable results, but it also possesses certain limitations. The corresponding data on the non-uniform and uniform grid are not of exactly the same length, which has a certain scale of temporal scaling. This characteristic imposes higher demands on subsequent resampling techniques.

It is noteworthy that the application of certain special mathematical sequences in SV-PRI sequences may enhance reconstruction accuracy and RCM suppression capability. In the future, we will focus on the potential application of special mathematical sequences in varying PRI techniques, such as coprime sequences, the Chinese Remainder Theorem, prime gap sequences, and so on. On the other hand, research will be primarily focused on developing interpolation and resampling algorithms with higher precision, expanding the application of the proposed method in high-resolution and high-squint spotlight SAR.

6. Conclusions

This paper introduces a novel strategy involving the variable granularity SV-PRI sequence to address the challenges of severe RCM in high-resolution, high-squint spaceborne spotlight SAR. This sequence significantly improves the reception of raw data, thereby establishing a solid foundation for achieving high-resolution imaging. In comparison with the conventional CV-PRI sequences, the SV-PRI methodology enhances resampling accuracy and reduces the complexity of timing design, facilitating simpler engineering implementations. Additionally, a comprehensive processing workflow that includes TPR-based azimuth partitioning and weighted-Sinc interpolation is introduced to resolve non-uniform sampling issues, which achieves notable results. In the future, more efforts will be concentrated on developing more precise interpolation and resampling algorithms to extend the application of this method in high-resolution and high-squint spotlight SAR scenarios.