Synthetic aperture radar (SAR) is an electromagnetic imaging system capable of acquiring two-dimensional (2D) images of the illuminated surface. Spotlight mode enhances azimuth resolution by steering the antenna beam to obtain wide Doppler bandwidth (BW) signals from the observed scene. According to the beam steering strategy, the spotlight mode is subdivided into staring and sliding modes. Staring mode steers the antenna beam to continuously illuminate a fixed scene, thereby enabling the acquisition of an extensive Doppler BW of the signal from the observed area [1, 2]. Sliding mode, a hybrid approach between the staring and strip-map modes, gradually slides the beam across the scene. It offers a broader image at the expense of reduced azimuth resolution compared to staring mode [3–6].

To achieve range resolution comparable to azimuth resolution in spotlight modes, wide-BW ramp signals are required. However, receiving the signals without distortion requires a high sampling frequency. Besides, storing and processing the large volume of sampled data imposes heavy demands on hardware and computational resources. The de-ramping technique mitigates these burdens by compressing the signal BW before sampling [1].

Accordingly, applying the de-ramping technique to signals acquired under spotlight mode geometry holds promise for generating high-resolution images with a reduced data volume. Despite this advantage, research on modeling and suitable image formation algorithms for the signal remains insufficient.

The frequency scaling algorithm (FSA) has emerged as an effective solution for processing de-ramped staring mode data [2, 7], as it inherently suppresses residual video phase, an inevitable byproduct of the de-ramping process [8]. In [2], the FSA is applied in combination with a sub-aperture method, which divides the synthetic aperture into multiple parts to limit the Doppler BW of each sub-aperture [9–13]. A real-time approach using phase linearization in each sub-aperture was proposed in [9]. A sub-aperture imaging method employing chirp scaling-dechirp to reduce the computational load in satellite SAR processing was proposed in [10]. A sub-aperture-based imaging method incorporating frequency-domain phase correction for highly squinted SAR geometry was proposed in [11]. A real-time multi-mode SAR imaging algorithm based on combining and scaling complex-valued sub-aperture images was proposed in [12]. A real-time spotlight SAR imaging method that fuses sub-aperture results within a modified range migration algorithm framework to reduce computational burden was proposed in [13].

Although sub-aperture-based techniques are indeed efficient in terms of computation load due to the segmented processing, their performance is susceptible to errors in the estimated effective velocity for each sub-aperture and is inconsistent depending on the number and size of the sub-apertures [14, 15]. Furthermore, with the recent advances in high-performance GPU-based parallel computing, the computational burden is becoming less critical.

To address the limitations of sub-aperture processing, the full-aperture method has become one of the primary frameworks for processing spotlight mode SAR data [15–23]. A two-step full-aperture processing approach for spotlight SAR, where range and azimuth focusing are performed sequentially without sub-aperture segmentation, was proposed in [16]. Extended two-step approaches applicable to airborne and spaceborne SAR, and squinted geometry were presented in [17, 18]. A non-sub-aperture processing method for sliding spotlight and terrain observation by progressive scans (TOPS) SAR data, which suppresses Doppler aliasing by selecting a pulse repetition frequency (PRF) slightly higher than the instantaneous Doppler BW, thereby eliminating the need for sub-aperture segmentation, was proposed in [15]. In addition, a full-aperture imaging algorithm tailored for highly squinted TOPS SAR data, which compensates for the time-varying Doppler centroid caused by progressive beam steering, was developed in [19]. A full-aperture focusing technique for spaceborne squinted sliding spotlight SAR was presented in [20], and an enhanced approach achieving very high resolution was proposed in [21]. An efficient full-aperture spotlight SAR processing technique using time-domain de-aliasing was proposed in [22]. Advanced processing of Gaofen-3 spaceborne dual-channel sliding spotlight SAR data was demonstrated in [23].

Although the full-aperture method has been extensively researched for spotlight SAR processing, its extension to de-ramped signal models has received relatively little attention. Accordingly, our recent work focused on integrating the FSA into a full-aperture processing framework [24]. In [24], a modified scaling factor and azimuth matched filter are derived based on a dechirped signal model. That is, unlike conventional works, this approach is specifically tailored to the dechirped data. However, [24] is designed exclusively for sliding mode only, limiting its applicability to the staring mode. In particular, the Doppler characteristics of staring mode SAR data differ significantly from those of sliding mode, making the processing strategy in [24] unsuitable for direct application.

The main contributions of this study are as follows:

The rest of this article is organized as follows. Section II provides a brief overview of the signal model of the de-ramped staring spotlight SAR data. The proposed method and its unique features compared to the conventional full-aperture FSA for the sliding mode are presented in Section III. At each processing step, signal characteristics, including verification via point target simulation, as well as relevant conditions for the convolution reference range, are analyzed. In Section IV, point target simulations and actual SAR raw data processing results are provided to verify the validity of the proposed method. In addition, a comparison with the conventional sub-aperture counterpart in terms of performance and computational load is provided. Finally, the conclusion is discussed in Section V.

This chapter overviews the 2D raw signal model of de-ramped staring mode data and its azimuth component properties. Fig. 1 shows the data acquisition geometry of staring mode.

t_a denotes the azimuth time of the SAR platform, T_I denotes the total synthetic aperture time, tr′ denotes the range time, θ_n denotes the nadir angle, θ_g denotes the grazing angle, v_r denotes the platform speed, t_c denotes the range time of the scene center, and tr,max′ and tr,min′ denote the maximum and minimum range times, respectively.

Based on the geometry, the SAR signal model of the received and de-ramped staring spotlight data for a point target P(t_o, r_o), within the observed scene, is in Eq. (1):

rect[·] denotes the rectangular function, λ denotes the radar wavelength, r(t_a) denotes the target range as a function of t_a, T_p denotes the transmitted pulse duration, r_c denotes the range of the scene center, and k_r denotes the pulse modulation rate. Note that the only difference compared to the de-ramped sliding spotlight SAR signal model is the azimuth envelope function [24]. By defining tr=tr′-tc where t_r denotes the relative range time from t_c, Eq. (1) is rewritten as follows:

With this substitution, the range phase correction step required in the conventional FSA is omitted, and the signal model becomes more concise [2, 24]. Assuming a zero squint and rectilinear geometry for brevity, r(t_a) is approximated as follows [25].

Accordingly, the azimuth part of Eq. (2) is expressed as follows:

where ka=-2vr2/(λro) denotes the Doppler rate of the target. Fig. 2 shows the azimuth time-frequency diagram (TFD). The azimuth signal extent, slope, and t-intercepts for each target in the graph are obtained from Eq. (4). The signal extent is obtained from the rect function. By differentiating the phase of Eq. (4) with respect to time, the related formulation between the azimuth time and frequency is derived.

As shown in Fig. 2, antenna beam steering in the staring mode leads to an increased overall Doppler BW that exceeds PRF. Therefore, proper pre-processing, such as azimuth convolution or synthetic aperture division, is essential in spotlight SAR processing to prevent aliasing in subsequent image formation algorithms.

Fig. 3(b) presents the overall workflow of the proposed method for processing the de-ramped staring mode data. Fig. 3(a) presents the traditional counterpart for sliding mode data. A key distinction revealed in Fig. 3(b) compared to Fig. 3(a) is that the azimuth post-processing is optional. The other is that the range phase correction step is omitted. This was made possible by redefining the range time as the relative range time from the scene center range time, as shown in Eq. (2). Besides, although the proposed workflow shares a general structure with the conventional one, there are differences in key parameters, which will be discussed in the rest of this chapter.

As a pre-processing step, azimuth convolution with zero-padding is employed to extend the PRF to exceed the total Doppler BW. For sliding mode, the convolution reference range of the azimuth convolution function is typically set to the range of the rotation center r_rot to minimize the azimuth signal extent [15, 20]. In contrast, since the scene center coincides with the rotation center in staring mode, a different convolution reference range r_conv > r_o (Case 1) is required. Though r_conv < r_o (Case 2) is not entirely infeasible, it is not recommended due to the impractical conditions it entails. Detailed conditions for r_conv are discussed later in this chapter. The convolution signal and the result are as follows:

where kconv=-2vr2/(λrconv) denotes Doppler rate of the convolution signal. In Eq. (7), the {·} term correspond to the initial multiplication in the pre-processing, which is as follows:

The following integration resembles a mathematical form of the Fourier transform. Therefore, adjusting the spacing Δt′ and number of azimuth samples P₁ via zero-padding to satisfy the condition of Eq. (9) allows the integral to be implemented by an fast Fourier transform (FFT) [26]. Δt denotes 1/PRF. The result is given in Eq. (10) and is derived by the principle of stationary phase (POSP), an approximation method of the FFT used for analyzing signal models [25]. The detailed POSP process is given in Appendix 1.

Z denotes a unique parameter of the proposed method, termed envelope shaping factor, as it determines the azimuth envelope, both the position and extent.

The convolution is completed by multiplying Eqs. (6) and (10).

where k_m = k_a · k_conv/(k_a – k_conv) denotes the modified Doppler rate. The azimuth TFDs for both cases are given in Fig. 4. The azimuth signal extent, slope, and t-intercepts of each target in the graph are obtained from Eq. (12). The signal extent is obtained from the rect function. By differentiating the phase of Eq. (12) with respect to time, the related formulation between the azimuth time and frequency is derived.

Unlike the sliding mode, the azimuth spatial positions of targets in the staring mode are not gathered at t_a = 0 and depend on their initial positions. The requirement T1′>T1 to avoid aliasing leads to the conditions rconv<2vr2TI/((B-PRF)·λ) for Case 1 and rconv>2vr2TI/((Bs-Ba+PRF)·λ) for Case 2. The simulation results for a single left-most point target to verify the conditions and the simulation parameters are presented in Fig. 5 and Table 1, respectively. The conditions imply that r_conv should be determined with the consideration of the data acquisition geometry, such as synthetic aperture length and range to the scene, as well as the radar hardware specifications, such as antenna beam-width and PRF. The conditions are summarized in Condition A in Table 2.

For the given parameters in Table 1, Condition A for Case 1 is r_conv < 928.9 km and for Case 2 is r_conv > 537.5 km. In Fig. 5(a), the condition is satisfied and therefore, no aliasing occurs, whereas in Fig. 5(b), the condition is violated, resulting in aliasing of the Doppler signal.

Considering the complete 2D signal model, Eq. (12) is reformulated as follows:

The sequential application of range FFT, IFFT, and azimuth FFT to Eq. (14) yields a suitable signal model for applying the FSA. The above three Fourier transforms are simplified as a single azimuth FFT operation in Fig. 3(b). We outline only the result of the successive FFT in Eq. (15). The full derivation process is given in Appendix 2.

β denotes the new scaling factor for staring mode, and src denotes the secondary range compression term, which are shown in Eq. (16):

The signal models for the staring and sliding modes diverge in several aspects at this stage. First, as shown in the azimuth envelope function in Eq. (15), while the Doppler BW in sliding mode is B_a/A, where A denotes the sliding factor, it is B_s in staring mode. This aligns with the fact that staring mode offers superior azimuth resolution compared to sliding mode. Second, while the Doppler centroid in sliding mode depends on the modified Doppler rate k_m, it is related to k_a in the staring mode. Lastly, although the notations for β and src are consistent for both modes, the change in r_conv results in a different value of k_m, and therefore, they are not equivalent.

Although the azimuth signals differ between the two modes, the similarities in range-related terms are obvious, as both are obtained through an identical de-ramping technique. Therefore, except for adjusting the scaling factor to suit staring mode, the FSA for staring mode also largely follows the conventional approach. Accordingly, the derivation is omitted in this chapter and instead provided in Appendix 3. The signal model and azimuth TFD after the FSA are given in Eq. (17) and Fig. 6, respectively. f_r denotes the range frequency.

The azimuth frequency extent and t-intercepts of the graph are obtained from the rect function of Eq. (17) and the rect function of its azimuth time domain signal, which is shown in Eq. (18).

Since the pre-acquired azimuth processing time extent T1′ already exceeds the azimuth signal time extent T, alias-free SAR imaging is possible at this stage. The requirement T1′>T leads to the condition rconv>2vr2T/(PRF·λ) for both cases. The conditions are summarized in Condition B-2 in Table 2. Generally, this condition is inherently satisfied so that high-quality SAR images are obtainable without the azimuth post-processing, which is essential in the sliding mode method.

On the other hand, a residual phase remains according to the initial azimuth position of the target, as shown in the phase term in Eq. (18). Therefore, if the reconstructed SAR image is intended for phase-sensitive applications such as SAR interferometry, post-processing used in the sliding mode method should be followed.

The azimuth post-processing follows the same steps as the conventional algorithm for sliding mode in Fig. 3(a) [24]. However, note that −k_a should be adopted instead of k_m in the convolution signal of the post-processing to remove the remaining phase. The azimuth TFD after the azimuth post-processing is given in Fig. 7. The azimuth processing BW becomes B2′=-kaT1′ after the post-processing. To ensure that it exceeds the total azimuth signal BW Bs=-kaTl,T1′ should be greater than T_I, which leads to the condition rconv>2vr2TI/(PRF·λ) for both cases. The conditions are summarized in Condition B-1 in Table 2. The simulation results for a left-most point target after the post-processing are given in Fig. 8. Condition B-1 for both cases is r_conv > 788.9 km. In Fig. 8(a), the condition is satisfied and therefore, no aliasing occurs, whereas in Fig. 8(b), the condition is violated, resulting in aliasing of the Doppler signal.

To prevent aliasing throughout the whole process, Conditions A and B-1 should be met. Condition B-1 is relaxed to a less stringent Condition B-2 when the objective is limited to SAR image formation without precise phase preserving.

The design checklist with a numeric worked example using the parameters in Table 1 is given in Table 3. As shown in Table 3, Case 2 is incompatible with Condition B-1, and therefore, there is no proper r_conv value for the whole process. Moreover, the choice of r_conv is still constrained even if the purpose is to satisfy the imaging-only condition since r_conv needs to be sufficiently separated from r_o to limit the variation of the modified Doppler rate across the scene. Therefore, Case 1 is highly recommended for r_conv selection.

Table 4 summarizes the key distinctions between the methods for sliding and staring modes.

T_sm denotes the synthetic aperture duration in strip-map mode. In addition to the factors listed in Table 4, since k_m is determined by r_conv, and β in turn depends on k_m, a difference in r_conv implies that both k_m and β are also different. Although the same full-aperture FSA framework is applied, its adaptations differ significantly between sliding and staring modes. This is because the data acquisition geometry and r_conv inherently affect several parameters, making it impractical to directly reuse the conventional approach.

This chapter presents the three types of point target simulation results. One is a multiple scatterer simulation with targets located at different positions, allowing assessment of the performance of the algorithm with respect to the convolution reference range. Another provides comparative results against the conventional sub-aperture FSA method in terms of performance and computational load. The other provides simulation results with and without the azimuth post-processing to confirm that the post-processing has no impact on SAR image quality. Moreover, to demonstrate the validity of the proposed method, experimental results using actual SAR raw data are provided, which also include a comparison with the conventional sub-aperture FSA.

First, we present the results of the multiple point scatterer simulation. The simulation parameters are the same as in Table 1, and r_conv values were selected as 900 and 600 km in accordance with the imaging-only mode. We conducted the simulation with nine-point targets uniformly distributed at 2 km intervals along both the range and azimuth axes. The SAR image results and the 2D impulse response functions (IRF) of three selected targets with different ranges are presented in Figs. 9(a)–9(b) and 10(a)–(f), respectively. Additionally, resolution and peak-to-side-lobe ratio (PSLR) values are provided in Table 5 to quantitatively assess the performance. When r_conv is set to 900 km, accurate and consistent compression performance is observed regardless of the range positions of targets. Whereas when r_conv is set to 600 km, significant degradation in azimuth compression performance is observed when the range positions of targets deviate from the scene center range, as shown in Fig. 10(d), 10(f), and Table 5. This is because of the proximity of the convolution reference range to the scene center. This induces a significant variation in modified Doppler rate across the range, causing the SAR focusing performance to degrade as the range of the target deviates from the scene center range.

Second, we simulated the same scenario with the conventional sub-aperture FSA. The SAR image results are presented in Figs. 9(c), 10(g)–10(i), and Table 5. There are also azimuth performance degradations when the range deviates from the scene center. These degradations occur because the azimuth space shift parameter required for azimuth compression in the conventional method becomes inaccurate as the range deviates from the scene center [2, 24].

Third, to confirm that post-processing has no impact on SAR image quality, results with and without post-processing are compared. With r_conv = 900 km, for a scenario with five point targets spaced 1 km apart along the azimuth axes, Figs. 11 and 12 show the SAR image and the 2D IRF results of the leftmost target, respectively. The wider scene extent in Fig. 11(b) compared to Fig. 11(a) is due to the absence of post-processing, which retains the original extent X1′=vrT1′, while Fig. 11(a) reduces to X = v_rT. The proposed method produces well-focused targets at their intended positions, with consistent compression performance regardless of whether the post-processing is applied or not.

Lastly, the actual SAR raw data processing results are provided. To the best of the authors’ knowledge, no publicly available raw data exist for the de-ramped staring mode. Therefore, the de-ramped sliding mode data were used instead. The sliding to staring conversion was performed using a Doppler band filtering technique that accounts for the imaging geometries of the two modes [27]. The SAR images produced by the proposed and the conventional methods are shown in Fig. 13. Both methods generate high-quality SAR images, and no noticeable difference is observed by visual inspection. These real SAR experiment results demonstrate the credibility of the proposed algorithm.

Since both methods consist of a large number of processing steps, directly comparing their computation complexity is tedious. Therefore, we provide a comparison based on the data size and the number of FFT operations, which are dominant contributors to computational load, along with the runtime measured for the representative scenes shown in Fig. 9(a) and 9(c). The numbers of range and azimuth pixels are assumed to be equal to N. Note that the FFT of length N requires 5N log₂ N FLOPs [25].

For the sub-aperture method, the number of sub-apertures is denoted as N_s1, and the number of pixels in each sub-aperture is denoted as N_s2. As shown in the block diagram of the conventional method [2], three range FFT/IFFT operations are required, while two azimuth FFT/IFFT operations are performed after the sub-aperture division, and one additional azimuth FFT/IFFT is required after the recombination. Therefore, the overall computational load of the sub-aperture method is as follows:

For the proposed full-aperture method, the number of pixels after zero-padding is denoted as N_f1, and the number of pixels after reducing the azimuth samples (in azimuth post-processing only) is denoted as N_f2. As shown in the block diagram of the proposed method, three range FFT/IFFT operations are required, while two azimuth FFT/IFFT operations are performed after zero-padding, and two additional azimuth FFT/IFFT operations are required after the size reduction. Therefore, the overall computational load of the full-aperture method is as follows:

Note that the last term is ignored in the imaging-only mode. For the representative scene, the runtime of the sub-aperture method was approximately 8.43 seconds, whereas the proposed full-aperture method required 8.20 seconds in the imaging-only mode and 11.248 seconds when the whole process was executed. In the imaging-only mode, the runtime is nearly identical to that of the sub-aperture method. This is because the time required for sub-aperture division, recombination, and the computations associated with phase preservation is saved.

This article proposed a processing method for de-ramped staring spotlight SAR data. Motivated by the limited research on the data, we conducted an integration of FSA, which is specialized for de-ramped signals, into the full-aperture approach, which is suitable for spotlight SAR processing. We focus specifically on the staring mode, revealing differences from the existing sliding mode method. A new parameter, termed the envelope shaping factor, was introduced to represent the azimuth-convoluted signal model, and the conditions for the convolution reference range to prevent aliasing were discussed. The conditions are closely associated with the data acquisition geometry and radar system parameters. The point target simulation proved that the performance of the algorithm varies according to the convolution reference range, and the azimuth post-processing is unnecessary in the staring mode method if SAR image generation is the only purpose. In addition, we provided a comparison with the conventional sub-aperture method in terms of performance and computational load. The results demonstrate that, unlike the sub-aperture approach, the proposed algorithm maintains consistent azimuth compression performance regardless of target range. Moreover, the credibility of the proposed method was further validated through experiments using actual SAR raw data.

Future studies will involve analysis of algorithm modifications required when the SAR platform deviates from a straight path or the synthetic aperture length is too long, where the conventional hyperbolic range equation model is no longer valid.