Early synthetic aperture radar (SAR) systems were primarily mounted on satellites. Thus, to cover long ranges between satellites and Earth’s surface, pulse radar was preferred. However, due to the increasingly small-scale observation areas and the emergence of SAR platforms such as automobiles and unmanned aerial vehicles, the demand for compact frequency-modulated continuous-wave (FMCW) radar with low-transmission power and cost-effective properties has increased. Accordingly, research on FMCW SAR signal processing algorithms has also increased [1–12].

SAR signal processing aims to determine the exact range and azimuth position of targets by compressing raw data in the range and azimuth direction. Azimuth resolution of SAR image greatly depends on the Doppler (azimuth) bandwidth of the signal. However, range cell migration (RCM) inevitably occurs during SAR data acquisition, resulting in azimuth resolution degradation due to the loss of the Doppler bandwidth. Therefore, before performing azimuth processing, it is necessary to align the azimuth phase information of a target to a single azimuth line to make the most of the Doppler bandwidth for the target. This correction process is called range cell migration correction (RCMC). Fig. 1 shows the bending of the energy trajectory of a single target according to RCM.

RCM mainly occurs because the range between the radar and the target varies for each pulse during the SAR integration time. This is the range migration that occurs between pulses. Moreover, since the platform is in motion, frequency shifts due to the relative velocity, and the range variation over the pulse duration may cause additional range migrations. This is the range migration caused by the intra-pulse motion of the SAR platform.

In pulse SAR, the pulse duration is short enough to be assumed that the radar platform is fixed during transmission and reception. Therefore, RCM originating from the latter is negligible in pulse SAR. This assumption is called the stop-and-go approximation (SAG). However, the pulse duration of FMCW radar is relatively long compared to pulse radar. Therefore, it is uncertain whether SAG is still applicable. If SAG cannot be applied, an additional RCMC process is required to compensate for the additional RCM caused by the intra-pulse motion of the SAR platform.

In this study, the range-Doppler algorithm (RDA) was used as a signal processing scheme because it is accurate, efficient, and the most widely used in SAR processing [13]. Fig. 2 shows block diagrams of RDA for FMCW SAR according to whether SAG is applied or not. As shown in Fig. 2(b), the additional RCMC process is omitted when SAG is applied. In general, RCMC involves an interpolation process that requires considerable computational loads [13]. Therefore, the processing time can be greatly reduced if SAG is applied to FMCW SAR.

In some studies, a platform motion during transmission and reception has been considered negligible in the FMCW SAR [2–4]. In other studies, SAG has been considered inapplicable to FMCW SAR [5–8]. Few of these studies have analytically determined whether SAG is applicable, and none have done so experimentally based on actual raw data [2–8]. Therefore, in this study, the conditions under which SAG can be applied to FMCW SAR were analyzed and experimentally confirmed through Ku-band FMCW SAR field tests.

This paper is organized as follows. In Chapter II, a review of the FMCW radar signal model is introduced, and the conditions for applying SAG in FMCW SAR are analytically derived. Chapter III presents the experimental results of Ku-band FMCW SAR field tests. It also presents comparisons of SAR quality parameters and processing times to confirm the applicability of SAG to FMCW SAR and its advantages. Finally, the conclusion of this paper is given in Chapter IV.

This chapter provides a review of the FMCW radar signal model to determine the range resolution. Linear frequency modulation and sawtooth sweep type are assumed. The transmission signal is

where f_c is the carrier frequency, T is the pulse repetition interval (PRI), t is the time variable within T, and K is the linear chirp rate. K is given as

where B is the sweep bandwidth of the transmitted signal. The reflected signal from the object with the range R is determined as

where τ is the signal round-trip time, and c is the speed of light. This reflected signal is mixed with a replica of the transmitted signal in the radar receiver. The beat signal, is given as

Since the only phase term related to time t is the second term, the beat frequency after applying a Fourier transform is calculated as

It can be seen that the beat frequency in the FMCW radar receiver is proportional to its distance from the object. The relation between the frequency resolution and T is

From Eqs. (5) and (6), the range resolution of the FMCW radar is

As mentioned in Chapter I, the pulse duration of FMCW radar is relatively long compared to pulse radar. Therefore, it should be determined whether SAG can be applied to FMCW SAR. Since the platform is in motion during the pulse duration, additional range migrations occur for two reasons. One reason is the frequency shift due to the relative velocity, and the other is the range variation over the pulse duration. For SAG to be applicable to FMCW SAR, these range migrations must be less than the range resolution of the radar.

First, the range migration, caused by the frequency shift due to the relative velocity, is discussed. A radar data acquisition geometry is shown in Fig. 3. For convenience, a zero squint case is assumed. L_s is the synthetic aperture length, θ_bw is the antenna beamwidth in the azimuth direction, P₀ is the position of closest approach, R₀ is the range of closest approach, and Δ_s is the sample spacing of the synthetic aperture signal. The sample target is continuously detected by the radar while it is in the beam illuminated area. P₁ is the position of the radar when the sample starts entering the beam illuminated area, and P₂ is the position of the radar when the sample exits the beam illuminated area. That is, the sample is detected while the radar is between P₁ and P₂. Since the distance between P₁ and P₂ is L_s, the positions of P₁ and P₂ can be specified as points separated by half the L_s from P₀. As shown in Fig. 3, L_s is determined from θ_bw and R₀ as

As the platform moves along the sensor path, the sample is detected by each pulse with a different Doppler frequency. The Doppler frequency shift caused by the relative velocity of the radar and the target is [13]:

where f_d is the Doppler frequency shift, V_s is the radar platform velocity, θ is the angle measured from boresight in the slant range plane, and λ is the radar wavelength. As represented in Eq. (5), in FMCW radar processing, the beat frequency is proportional to the range. Therefore, the Doppler frequency shift of the FMCW signal indicates range migration. By substituting Eq. (9) to Eq. (5), the range migration caused by the frequency shift due to the relative velocity can be determined as

where R_d is the range migration caused by the Doppler frequency shift and f is the radar frequency. Since the Doppler frequency shift is the largest when the platform is located at P₁ or P₂, the maximum value of R_d is determined as

Second, the range migration, caused by the range variation over the pulse duration, is discussed. Range migration due to range variation R_r can be defined as the difference between the range at the pulse start time and the range at the pulse end time (after one PRI). As illustrated in Fig. 3, the range between the radar and the sample when the radar is located at P₁ is R′ which is determined as

T_s is the SAR observation time. Since the platform is in motion, this range changes continuously over the pulse duration. After a PRI, the range between the radar and the sample becomes R″ which is determined as

Therefore, the range migration caused by the range variation during the pulse duration can be calculated as

This is the maximum value of R_r because when the radar moves to a position other than P₁, the range migration of the sample is less than R_r,max. Meanwhile, given that T_s consists of many PRIs, R_r,max is rewritten as

where N is the number of pulses that constitutes T_s. When the radar moves from P₁ to P₀ (approaching the sample), R_d has a positive value, meaning that range migration occurs in the away direction. R_r also has a positive value, but it means that the range migration occurs in the opposite direction. Therefore, the total range migration is

On the other hand, when the radar moves from P₀ to P₂ (away from the sample), R_d has a negative value, meaning that range migration occurs in the forward direction. R_r also has a negative value, but it means that range migration occurs in the away direction. Therefore, the total range migration is the same as in Eq. (16).

Thus far, the range resolution and the range migration due to the motion of the platform during the pulse have been defined. For SAG to be applicable to FMCW SAR, the total range migration R_m should be less than the range resolution of the radar ΔR. Thus, the following condition must be satisfied:

Fig. 4 shows the range migration (solid lines) as a function of frequency at different radar platform velocities and the range resolution (dashed lines) as a function of frequency bandwidth. For simplicity, a linear chirp rate of K = 2.5 THz/s and an antenna beamwidth of θ_bw= 34° are assumed. In addition, as will be discussed in Chapter III,R_r is not considered because its value is almost zero. On the other hand, for a fixed value of K, as T decreases, B also decreases, and ΔR degrades (i.e., widens). This is in line with the fact that SAG is valid for pulse radar with a short pulse duration. Notably, a value of V_s = 90 m/s is almost the fastest velocity available in FMCW SAR [1, 4, 7, 9–12]. This is because FMCW SAR is used only on aircraft or automobile platforms due to its detection range limitation

As shown in Fig. 4, range migration increases as the range frequency and radar platform velocity increases. Also, the wider the frequency bandwidth, the better (narrower) the range resolution. Therefore, the most difficult case to apply SAG to FMCW SAR is when the system has high-frequency, high platform velocity, and high range resolution properties. By comparing the expected range migration with the range resolution, it is possible to determine whether SAG is applicable to FMCW radar with certain system specifications. SAG can be applied to FMCW SAR in the Ku-band frequency region (12–18 GHz), even in the case of high ΔR (0.3 m) and high V_s (90 m/s) because the range migration in that region is considerably smaller than the range resolution. On the other hand, SAG is not applicable to FMCW SAR in the Ka-band frequency region (27–40 GHz) in the case of high ΔR (0.3 m) and high V_s (90 m/s) and can be applied only if V_s decreases or ΔR widens. Therefore, the system specifications should be considered to determine whether SAG is applicable to a given FMCW SAR system.

Chapter III presents range migration and range resolution calculations with actual Ku-band FMCW SAR system specifications. Also, it presents filed test demonstrating that SAG is applicable to Ku-band FMCW SAR even in tough cases.

The detailed specifications of the Ku-band FMCW SAR system used in the outdoor experiments are given in Table 1. The center frequency of the FMCW radar is in the Ku-band (14.25 GHz). The frequency bandwidth of the radar system is 500 MHz, which leads to a high ΔR (0.3 m). To avoid the limitations of a single scenario, experiments were performed at various radar platform velocities (82, 75, 63, and 45 m/s). In Fig. 5, since the expected range migration values are below the range resolution line, SAG is applicable to all cases. The applicability of SAG to a specific FMCW SAR system can be confirmed by directly calculating the additional range migration and the range resolution using the equations presented in Chapter II. The case of a radar platform velocity of 75 m/s is considered below.

In our system specification, using Eq. (11), the maximum range migration caused by the frequency shift due to the relative velocity R_d,max was 0.125 m, which was less than the range resolution ΔR. This value was consistent with the expected range migration value for the case of V_s = 75 m/s in Fig. 5. Moreover, since T_s consists of thousands of PRIs, R_r is almost zero and, therefore, negligible. In fact, using Eq. (15), the maximum range migration R_r,max caused by the range variation over the pulse duration was about 0.0044 m, which is considerably less than the range resolution. This means that SAG was applicable to our FMCW SAR system case, as shown by the calculation of Eq. (17).

To experimentally verify the applicability of SAG to FMCW SAR, SAR signal data were obtained from several test sites at various platform velocities. Appropriate SAR images appeared after processing SAG-applied RDA in the actual raw dataset. Fig. 6 shows the Ku-band FMCW SAR images. The SAR images match the aerial photographs of the test sites shown in Fig. 7, providing experimental evidence that our specific Ku-band FMCW SAR can work with SAG.

Moreover, an impulse response function (IRF) analysis was performed to provide a quantitative basis for the validity of SAG for Ku-band FMCW SAR. Essential SAR quality parameters, such as impulse response width (IRW) and peak sidelobe ratio (PSLR), were estimated from the IRF [13]. Fig. 8(a) shows a trihedral corner reflector (CR) installed at the field test site for the IRF analysis. Fig. 8(b) shows an aerial photograph of the CR test site. To determine whether SAG degraded SAR image quality, RDA without SAG (Fig. 2(a)) and with SAG (Fig. 2(b)) were performed and compared. Fig. 9(a) and (b) shows the results of the FMCW SAR images for each case. The CR clearly emerged inside the red circles shown in the images. The IRF of the CR in Fig. 9(a) is shown in Fig. 10(a–c). For visual clarity, an interpolation by a factor of 16 was implemented. Fig. 10(a) shows a normalized intensity of the IRF in 3D, and Fig. 10(b) and (c) shows one-dimensional profiles of the IRF in the range and azimuth direction, respectively. Likewise, the IRF of the CR in Fig. 9(b) is shown in Fig. 10(d–f). The peak of the signal was well represented in both the range and azimuth directions. The SAR quality parameters and processing times of RDA with and without SAG are summarized in Table 2. As shown in Fig. 10 and Table 2, applying SAG did not affect the IRW or PSLR values. This provides clear quantitative evidence that our specific Ku-band FMCW SAR can work with SAG. Furthermore, the processing time was significantly reduced (by about 36%) when SAG was applied. This suggests that SAG shortened the processing time without SAR quality parameter performance loss.

A major contribution of this study is its analytical and experimental approaches to determining the feasibility of SAG for FMCW SAR signal processing, which has been contested. No previous studies have verified the validity of SAG using practical raw data. Therefore, in this study, the conditions for applying SAG to FMCW SAR were analyzed and experimentally verified for Ku-band FMCW SAR. The results show that if SAG is applicable, the processing time is reduced without compromising the SAR image quality. Since the platform velocity used was at the aircraft level, the results may be generalizable to automobile applications as well. Moreover, since the operating frequency used in the experiment was in the Ku-band (14.25 GHz), SAG can also be applied to FMCW SAR using operating frequencies below the Ku-band if other parameters are similar. Future studies could investigate whether SAG is applicable to FMCW SAR using frequencies above the Ku-band, such as the K- or the Ka-band.