### I. Introduction

### II. The Ambiguity Performance of SAR Systems

### 1. Ambiguity-to-Signal Ratio

##### (1)

*G*

*and*

_{t}*G*

*are antenna transmitting and receiving power patterns, respectively;*

_{r}*σ*

^{0}is the backscattering coefficient;

*ρ*

*is the range ambiguous slant range;*

_{RA;}*θ*

*is the incidence angle;*

_{I}*θ*

*, related to the slant range, is the angle at which the range ambiguity signals occur;*

_{A}*m*is a nonzero integer, meaning the order that precedes or succeeds the reference pulse in terms of the slant range;

*θ*

*, related to the Doppler frequency, is the angle at which the azimuth ambiguity signals occur; and*

_{AZ}*PB*is the Doppler processing bandwidth. The integral with respect to

*PB*was calculated based on the relation between the Doppler frequency and the azimuth angle.

**is a nonzero integer, meaning the periodic order that is higher or lower than the Doppler centroid in terms of the azimuth frequency, and the other variables are the same as in Eq. (1). Generally, as the PRF of the SAR system increases, the RASR and AASR increases/decreases, respectively, and when the PRF decreases, the RASR and AASR decreases/increases, respectively [5]. In addition, as the antenna size increases, the ambiguity performance improves, but the resolution and swath width tend to be opposite. Therefore, it is necessary to select an optimal antenna size and system parameters to take this into account [15].**

*n*

*m***≠ 0**and

*n***≠ 0**[7]. The relation between each ambiguity component is shown in Fig. 1.

### 2. Polarimetric SAR Systems

*σ*

^{0}(

*θ*

*), is used in the ambiguity performance analysis. However, since the conventional quad-pol SAR system alternates horizontal (H) and vertical (V) polarizations, two polarization components must be used when calculating RASR for four images, HH, HV, VH, and VV. When calculating RASR, since the backscattering coefficient of cross-pol channels such as HV and VH is smaller than that of co-polarization (co-pol) channels such as HH and VV, the RASR performance of the cross-pol channel is severely deteriorated [2–4].*

_{I}

*S**), and can be transformed to signals obtained from conventional quad-pol SAR (*

_{HQ}

*S**) through the transformation relationship [18],*

_{CQ}^{^}above

*S*indicates after transformation. It is noticeable here that after the transformation, the RASR calculation of each polarization channel only considers its effect on that polarization [17]. In other words, the effect of the copol signal, which is relatively larger than the cross-pol signal due to the difference of the backscattering coefficient, disappears.

*S**that are combined to obtain a signal for*

_{HQ}

*S**, a phase difference occurs in the spectrum, and this does not completely cancel the unwanted components [2]. In the case of the HV signal as an example, according to Eq. (4)*

_{CQ}##### (6)

^{~}indicates a sampled signal and

*T*is the time interval between transmitted signals of the same polarization [2]. The Fourier transform of Eq. (6) yields the spectral relational equation in Eq. (7) (scale factors were ignored). Therefore, even if these two signals are summed, neither can cancel out, which can be expressed as shown in Fig. 2.

**is the transmitted polarization,**

*t***is the received polarization, and**

*r***is the orthogonal signal of**

*t̄***. For example, if**

*t***is H, then**

*t**t̄*is V. After this, the ambiguity performance analysis was performed based on Eq. (8).

##### (8)

### 3. Effects of Antenna Cross-Pol Pattern

*G*

*is the antenna co-pol pattern and*

_{co}*G*

*is the antenna cross-pol pattern. We considered all the effects of the antenna cross-pol pattern as their simple sum.*

_{cx}### III. Performance Analysis and Comparison of Spaceborne SAR System

### 1. Spaceborne SAR System Using an X-Band Parabolic Reflector Antenna

### 2. Analysis Results of the Ambiguity Performance

*m*or

*n*is always zero in Eq. (8). In this case, the effect of the antenna cross-pol pattern was not considered. In Case 2, the cross ambiguity component was added to Case 1, and the effect of the antenna cross-pol pattern was not considered in this case either. In other words, Case 2 is calculated from Eq. (8). In the third case, Case 3, the effect of the antenna cross-pol pattern was additionally considered on Case 2, which means that Eq. (8) should be modified by reflecting the technical description covered in Section II-3.