### I. Introduction

### II. Analysis of The Validity of Stop-and-go Approximation for FMCW SAR

*f*

*is the carrier frequency, T is the pulse repetition interval (PRI),*

_{c}*t*is the time variable within T, and

*K*is the linear chirp rate.

*K*is given as

*B*is the sweep bandwidth of the transmitted signal. The reflected signal from the object with the range

*R*is determined as

*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

*t*is the second term, the beat frequency after applying a Fourier transform is calculated as

*L*

*is the synthetic aperture length,*

_{s}*θ*

*is the antenna beamwidth in the azimuth direction,*

_{bw}*P*

_{0}is the position of closest approach,

*R*

_{0}is the range of closest approach, and Δ

*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.*

_{s}*P*

_{1}is the position of the radar when the sample starts entering the beam illuminated area, and

*P*

_{2}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

*P*

_{1}and

*P*

_{2}. Since the distance between

*P*

_{1}and

*P*

_{2}is

*L*

*, the positions of*

_{s}*P*

_{1}and

*P*

_{2}can be specified as points separated by half the

*L*

*from*

_{s}*P*

_{0}. As shown in Fig. 3,

*L*

*is determined from*

_{s}*θ*

*and*

_{bw}*R*

_{0}as

*f*

*is the Doppler frequency shift,*

_{d}*V*

*is the radar platform velocity,*

_{s}*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

*R*

*is the range migration caused by the Doppler frequency shift and*

_{d}*f*is the radar frequency. Since the Doppler frequency shift is the largest when the platform is located at

*P*

_{1}or

*P*

_{2}, the maximum value of

*R*

*is determined as*

_{d}*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*

_{r}*P*

_{1}is

*R′*which is determined as

*T*

*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*

_{s}*R″*which is determined as

*R*

*because when the radar moves to a position other than*

_{r}*P*

_{1}, the range migration of the sample is less than

*R*

_{r}_{,}

*. Meanwhile, given that*

_{max}*T*

*consists of many PRIs,*

_{s}*R*

_{r}_{,}

*is rewritten as*

_{max}*N*is the number of pulses that constitutes

*T*

*. When the radar moves from*

_{s}*P*

_{1}to

*P*

_{0}(approaching the sample),

*R*

*has a positive value, meaning that range migration occurs in the away direction.*

_{d}*R*

*also has a positive value, but it means that the range migration occurs in the opposite direction. Therefore, the total range migration is*

_{r}*P*

_{0}to

*P*

_{2}(away from the sample),

*R*

*has a negative value, meaning that range migration occurs in the forward direction.*

_{d}*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).*

_{r}*R*

*should be less than the range resolution of the radar Δ*

_{m}*R*. Thus, the following condition must be satisfied:

*K*= 2.5 THz/s and an antenna beamwidth of

*θ*

*= 34° are assumed. In addition, as will be discussed in Chapter III,*

_{bw}*R*

*is not considered because its value is almost zero. On the other hand, for a fixed value of*

_{r}*K*, as T decreases,

*B*also decreases, and Δ

*R*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*

*= 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*

_{s}*R*(0.3 m) and high

*V*

*(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 Δ*

_{s}*R*(0.3 m) and high

*V*

*(90 m/s) and can be applied only if*

_{s}*V*

*decreases or Δ*

_{s}*R*widens. Therefore, the system specifications should be considered to determine whether SAG is applicable to a given FMCW SAR system.

### III. Experimental Validation of the Feasibility of Stop-and-go Approximation in Ku-Band FMCW SAR Through Field Tests

*R*(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.

*R*

_{d}_{,}

*was 0.125 m, which was less than the range resolution Δ*

_{max}*R*. This value was consistent with the expected range migration value for the case of

*V*

*= 75 m/s in Fig. 5. Moreover, since*

_{s}*T*

*consists of thousands of PRIs,*

_{s}*R*

*is almost zero and, therefore, negligible. In fact, using Eq. (15), the maximum range migration*

_{r}*R*

_{r}_{,}

*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).*

_{max}