Precise Ambiguity Performance Evaluation for Spaceborne SAR with Diverse Waveforms

Abstract

The ambiguity suppression is a technical challenge for the present generation of spaceborne synthetic aperture radar (SAR) systems since this kind of suppression does not take the high spatial resolution and wide coverage into account simultaneously. The transmitting scheme based on the waveform diversity technique is a promising candidate for the conventional (one transmit, one receive channel) SAR systems and has been widely discussed, because it has almost no extra system costs and the ambiguity suppression performance is not closely related to pulse repetition frequency (PRF). However, the accurate method to evaluate the ratio of the intensities of the ambiguities to that of the signal is still a gap. To this end, starting from the precise signal model formulated in this paper, the ambiguity evaluation for spaceborne SAR with waveform diversity has been analyzed in detail. Particularly, the modified azimuth ambiguity-to-signal ratio (AASR) and range ambiguity-to-signal ratio (RASR) formulas are given for the single polarization SARs and quadrature-polarimetric (quad-pol) SARs, which contributes a lot, for the system designer, to precisely evaluating the ambiguity performance. Finally, detailed simulation experiments exploiting the system parameters of the LuTan (LT-1) system are carried out to corroborate the theoretical developments.

1. Introduction

Synthetic aperture radar (SAR), in recent years, can be called one of the most promising and attractive remote sensing technologies available for global comprehensive environmental monitoring, day-and-night and all-weather surveillance [1,2,3,4,5]. However, the present generation of spaceborne SAR systems cannot obtain high spatial resolution and wide coverage at the same time, and as a result, it causes the trade-off of the range and azimuth ambiguity [1,3,4]. Particularly, considering the fact that compared with the energies of the co-polarized channels, energies of cross-polarized channels returned are much weaker, quadrature-polarimetric (quad-pol) SAR systems, as a consequence, suffer frightful range and azimuth ambiguities [6]. Therefore, the ambiguity suppression has become an active and growing research field [7,8]. In [9], local azimuth ambiguity-to-signal ratio (LAASR) estimation and inpainting algorithm are employed to suppress the azimuth ambiguity. As introduced in [10], the range ambiguity energy can be suppressed by optimizing the elevation beam pattern. The designed two-way elevation beam pattern has relatively low sidelobe levels in the direction of strong range ambiguous echoes. Moreover, the azimuth ambiguity factor (AAF) is introduced and an accurate mathematical model is first constructed for azimuth ambiguity suppression [11]. Multiple transmit schemes in line with quasi-orthogonal waveforms [12,13,14,15] are envisioned for conventional (one transmit and one receive channel) SAR systems, where the ambiguity suppression principle behind the alternating transmit scheme exploits the low cross-correlation energy of the designed signal. These methods have many advantages, e.g., they do not need to increase channel numbers [16,17,18]. Additionally, different from the famous APC, namely, azimuth phase coding [19,20], this method is not related to pulse repetition frequency (PRF), and it has broad application prospects in spaceborne SAR systems.

The transmitting plan with waveform diversity was first introduced in [21]. In detail, it includes frequency diversity and code diversity. Considering the low spectral overlap of the transmitted waveforms with different center frequencies, the residual range ambiguity energy are quite small in the first transmitting scheme. Whereas, the incoherence of the transmitted waveform will cause frightful azimuth ambiguities [21]. Therefore, the ensuing studies are focused on the second scheme, e.g., [22,23,24,25].

It is worth remarking that this paper is not focused on the design of orthogonal waveforms, and we aim to give a precise evaluation of the azimuth ambiguity-to-signal ratio (AASR) and range ambiguity-to-signal ratio (RASR), which is very important for the design of spaceborne SAR systems. Up to now, the research on RASR and AASR has been widely reported in many publications [9,26,27,28]. The conventional AASR and RASR evaluation that we usually used, which is based on the received signal before range compression, was first detailed in [29]. Many AASR and RASR formulas have been studied for the transmitting scheme with waveform diversity; however, there are two very important points should be remarked: (1) the ambiguities are suppressed after range compression, thus, the AASR and RASR should be based on the signal after range compression; (2) the proposed AASR and RASR do not fit with the conventional definition. Therefore, this paper makes every effort to fill this gap to put forward more precise and general AASR and RASR assessment formulas accounting for the above two points.

The remainder of this paper is organized as follows. The conventional RASR and AASR evaluation is first reviewed in the Section 2. Next, in Section 3, the detailed signal model for the transmitting scheme with waveform diversity is presented. In Section 4, starting from the precise signal model, RASR and AASR equations without waveform diversity are first detailed, and then the revised RASR and AASR evaluation for waveform diversity is put forward. Additionally, in Section 5, the parameters of LuTan-1 (LT-1) mission (It is a spaceborne L-band SAR system, and it is an important part of the medium- and long-term development plans for China’s civil space infrastructure.) [30] are taken into account, and comparative simulation experiments are carried out to validate the developments in theories. In the end, we drawn the conclusions, which can be seen in Section 6.

2. Important Review for Ambiguity Evaluation

The conventional ambiguity-to-signal ratio (ASR) evaluation are based on the antenna power pattern. Considering a fixed Doppler frequency $f_0$ and a fixed range time delay ${\tau }_0$, the power of the ambiguity signal can be expressed as [31]

$$\begin{array}{c}\hfill P_{\mathrm{ambiguity}}=\sum _{\begin{array}{c}m,n=-\infty \\ m,n\ne 0\end{array}}^{\infty }{G}^2\left({\tau }_0+n/f_a,f_0+m{f}_a\right){\sigma }^0\left({\tau }_0+n/f_a,f_0+m{f}_a\right),\end{array}$$

(1)

where $m,n\in \mathbb{Z}$, $G^2(\tau ,f_{\eta })$ is the two-way antenna power pattern, ${\sigma }^0(\tau ,f_{\eta })$ denotes the backscattering reflectivity, and $f_a$ refers to the PRF. They are both the function of the range time delay $\tau$ and the Doppler frequency $f_{\eta }$. Additionally, the power of the desired signal can be represented as

$$\begin{array}{c}\hfill P_{\mathrm{desired}}=G^2\left({\tau }_0,f_0\right){\sigma }^0\left({\tau }_0,f_0\right).\end{array}$$

(2)

Therefore, starting from (1) and (2), the ASR can be written as the integral energy rate between the rang ambiguity signal and the desired signal, i.e., [31]

$$\begin{array}{c}\hfill \mathrm{ASR}\left(\tau \right)\cong \frac{{\displaystyle \sum _{\begin{array}{c}m,n=-\infty \\ m,n\ne 0\end{array}}^{\infty }{\int }_{-B_a}^{B_a}{G}^2\left(\tau +n/f_a,f_{\eta }+m{f}_a\right){\sigma }^0\left(\tau +n/f_a,f_{\eta }+m{f}_a\right)d{f}_{\eta }}}{G^2\left(\tau ,f_{\eta }\right){\sigma }^0\left(\tau ,f_{\eta }\right)},\end{array}$$

(3)

where $B_a$ denotes the processed Doppler bandwidth. However, the ASR evaluation based on the (3) is not convenient since the antenna power pattern must be represented as the range time delay and Doppler frequency [1]. Therefore, in this paper, the antenna power pattern is formulated as a function of the Doppler frequency $f_{\eta }$ and the look angle $\theta$, and the backscatter reflectivity is expressed as a function of the incidence angle $\varphi$ [6]. Furthermore, the ambiguity is divided into three types, which are referred to as the principal range ambiguity rate (i.e., $m=0$ in (3)), the principal azimuth ambiguity rate (i.e., $n=0$ in (3)), and the cross-term ambiguity rate (i.e., $m\ne 0,n\ne 0$ in (3)) [29]. Therefore, the principal range ambiguity rate (i.e., the RASR we generally used) can be expressed as

$$\begin{array}{c}\hfill \mathrm{RASR}\left({\varphi }_0\right)\cong \frac{{\sum }_{n=-\infty ,n\ne 0}^{n=\infty }\frac{{\sigma }^0\left({\varphi }_n\right){\int }_{-B_a}^{B_a}{G}^2({\theta }_n,f_{\eta })d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }^0\left({\varphi }_0\right){\int }_{-B_a}^{B_a}{G}^2({\theta }_0,f_{\eta })d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}},\end{array}$$

(4)

where ${\theta }_n$ and ${\varphi }_n$ (${\theta }_0$ and ${\varphi }_0$) are the look angle and incidence angle for the nth range ambiguity (desired scattered signal), respectively. $R_0$ and $R_n$ are the range for the desired signal and for the nth range ambiguity signal, respectively.

Additionally, the principal azimuth ambiguity rate (i.e., the AASR we generally used) can be expressed as

$$\begin{array}{c}\hfill \mathrm{AASR}\left({\theta }_0\right)\cong \frac{{\displaystyle \sum _{\begin{array}{c}m=-\infty \\ m\ne 0\end{array}}^{\infty }{\int }_{-B_a}^{B_a}{G}^2({\theta }_0,f_{\eta }+m{f}_a)d{f}_{\eta }}}{{\displaystyle {\int }_{-B_a}^{B_a}{G}^2({\theta }_0,f_{\eta })d{f}_{\eta }}}.\end{array}$$

(5)

Moreover, assuming the azimuth pattern in each evaluation is the same, (5) can be recast as

$$\begin{array}{c}\hfill \mathrm{AASR}\cong \frac{{\displaystyle \sum _{\begin{array}{c}m=-\infty \\ m\ne 0\end{array}}^{\infty }{\int }_{-B_a}^{B_a}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}}{{\displaystyle {\int }_{-B_a}^{B_a}{G}^2\left(f_{\eta }\right)d{f}_{\eta }}}.\end{array}$$

(6)

3. Signal Model

Based on the review presented in Section 2, some important observations are now necessary. First, the RASR and AASR that we generally used are actually referred to the principal range and azimuth ambiguity rate, respectively, and the cross-term ambiguity rate is ignored. Second, the above ASR equation derivation is unrelated to the transmitted signal. Therefore, to proceed further, considering the transmitted signal, the detailed signal model should be formulated for ASR evaluation.

First, considering a point-like target (the closest slant range is $R_0$, and the beam center offset time is ${\eta }_c$), the received desired baseband signal can be expressed as [1]

$$\begin{array}{c}\hfill s(t,\eta )=e^{j\xi \left({\varphi }_0\right)}\sqrt{{\sigma }^0\left({\varphi }_0\right)}\sqrt{\frac{G^2({\theta }_0,\beta (\eta -{\eta }_c))}{R^4\left(\eta \right)sin\left({\varphi }_0\right)}}{s}_t\left(t-\frac{2R\left(\eta \right)}{c}\right)exp\left(-j\frac{4\pi f_{c}R\left(\eta \right)}{c}\right),\end{array}$$

(7)

where ${\varphi }_0$ and ${\theta }_0$ are the incidence angle and look angle of the given target, respectively, $e^{j\xi \left({\varphi }_0\right)}\sqrt{{\sigma }^0\left({\varphi }_0\right)}$ is the backscatter reflectivity. It should be noted that $\varphi$ and $\theta$ are actually the function of azimuth time $\eta$, i.e., denoted by $\varphi \left(\eta \right)$ and $\theta \left(\eta \right)$. However, due to the assumption of narrow azimuth beam, the incidence angle and look angle can be assumed as constant values, i.e., $\varphi \left(\eta \right)=\varphi \left({\eta }_c\right)$ and $\theta \left(\eta \right)=\theta \left({\eta }_c\right)$, respectively. Additionally, t is the fast time, $s_t\left(t\right)$ is the transmitted baseband signal, $f_c$ is the carrier frequency, $\eta$ is the azimuth time referenced to the time of closest approach (i.e., $R_0$), and $R\left(\eta \right)$ is the instantaneous slant range (To simplify the analysis, the trajectory is assumed as a line.), as

$$\begin{array}{c}\hfill R\left(\eta \right)=\sqrt{R_0^2+v_r^2{\eta }^2},\end{array}$$

(8)

where $v_r$ is the equivalent velocity of spaceborne SAR. Moreover, $\beta (\eta -{\eta }_c)$ is the azimuth angle measured from boresight in the slant range plane (see Figure 1), which can be further expressed as

$$\begin{array}{c}\hfill \beta (\eta -{\eta }_c)={\theta }_{sq}\left(\eta \right)-{\theta }_{sq,c},\end{array}$$

(9)

where ${\theta }_{sq,c}$ is the squint angle of the beam center crossing time, i.e.,

$$\begin{array}{c}\hfill {\theta }_{sq,c}=-arctan\left(\frac{v_r{\eta }_c}{R_0}\right),\end{array}$$

(10)

and ${\theta }_{sq}\left(\eta \right)$ is the instantaneous squint angle, i.e.,

$$\begin{array}{c}\hfill {\theta }_{sq}=-arctan\left(\frac{v_r\eta }{R_0}\right),\end{array}$$

(11)

Thus, in the far field hypothesis, $\beta (\eta -{\eta }_c)$ can be recast as

$$\begin{array}{c}\hfill \beta (\eta -{\eta }_c)=-arctan\left(\frac{v_r\eta }{R_0}\right)-arctan\left(\frac{v_r{\eta }_c}{R_0}\right)\approx -arctan\left(\frac{v_r(\eta -{\eta }_c)}{R_0}\right),\end{array}$$

(12)

thus $G^2\left({\theta }_0,\beta (\eta -{\eta }_c)\right)$ can be denoted by $G^2({\theta }_0,\eta -{\eta }_c)$.

Furthermore, in a synthetic aperture time, $max\left[R\left(\eta \right)\right]-min\left[R\left(\eta \right)\right]\ll R_0$, (7) can be reformulated as

$$\begin{array}{c}\hfill s(t,\eta )\approx e^{j\xi \left({\varphi }_0\right)}\sqrt{{\sigma }^0\left({\varphi }_0\right)}\sqrt{\frac{G^2({\theta }_0,\eta -{\eta }_c)}{R_0^{4}sin\left({\varphi }_0\right)}}{s}_t\left(t-\frac{2R\left(\eta \right)}{c}\right)exp\left(-j\frac{4\pi f_{c}R\left(\eta \right)}{c}\right),\end{array}$$

(13)

Furthermore, since $R\left(\eta \right)\approx R_0+\frac{v_r^2{\eta }^2}{2R_0}$, (13) can be recast as

$$\begin{array}{c}\hfill s(t,\eta )\approx e^{j\xi \left({\varphi }_0\right)}\sqrt{{\sigma }^0\left({\varphi }_0\right)}\sqrt{\frac{G^2({\theta }_0,\eta -{\eta }_c)}{R_0^{4}sin\left({\varphi }_0\right)}}{s}_t\left(t-\frac{2R\left(\eta \right)}{c}\right)exp\left(-j\frac{4\pi R_0}{\lambda }\right)exp\left(-j\pi K_a{\eta }^2\right),\end{array}$$

(14)

where $\lambda =c/f_c$ is the wavelength (c is the velocity of light), and $K_a\approx \frac{2v_r^2}{\lambda R_0}$ is the azimuth frequency rate.

To proceed further, exploiting the Principle Of Stationary Phase (POSP) and employing the azimuth Fourier transform, (14) can be reformulated as

$$\begin{array}{c}\hfill s(t,f_{\eta })\approx e^{j\xi \left({\varphi }_0\right)}\sqrt{{\sigma }^0\left({\varphi }_0\right)}\sqrt{\frac{G^2({\theta }_0,f_{\eta }-f_{dc})}{R_0^{4}sin\left({\varphi }_0\right)}}\sqrt{\frac{1}{K_a}}{s}_t\left(t-\frac{2R_{rd}\left(\eta \right)}{c}\right)exp\left(j{\theta }_{rd}\left(f_{\eta }\right)\right),\end{array}$$

(15)

where $f_{dc}$ is the Doppler center frequency, which is a constant, and a similar derivation for the phase term of (15) can be found in Chapter 5.2.1 of [2]. This paper aims at giving the precise ASR evaluation equations, thus, the amplitude term, which is ignored in many studies, e.g., [2], is more important. Consequently, for the purpose of simplifying the steps of the analysis, from the view of mathematics, the analytic expressions of the terms $f_{dc}$, $R_{rd}\left(\eta \right)$, and ${\theta }_{rd}\left(f_{\eta }\right)$, which are useless for the following analysis, are ignored.

Next, employing Doppler frequency shift processing to compensate the shift of Doppler center frequency, (15) can be recast as

$$\begin{array}{c}\hfill s(t,f_{\eta })\approx e^{j\xi \left({\varphi }_0\right)}\sqrt{{\sigma }^0\left({\varphi }_0\right)}\sqrt{\frac{G^2({\theta }_0,f_{\eta })}{R_0^{3}sin\left({\varphi }_0\right)}}\sqrt{\frac{\lambda }{2v_r^2}}{s}_t\left(t-\frac{2R_{rd}\left(\eta \right)}{c}\right)exp\left(j{\theta }_{rd}\left(f_{\eta }\right)\right),\end{array}$$

(16)

4. Proposed RASR and AASR Evaluation Method

4.1. RASR and AASR without Waveform Diversity

In this sub-section, the conventional RASR and AASR, for the SAR without waveform diversity, after range compression is detailed.

First, starting from the detailed signal model given in (16), the nth range ambiguity can be represented as:

$$\begin{array}{c}\hfill s_n(t,f_{\eta })\approx e^{j\xi \left({\varphi }_n\right)}\sqrt{{\sigma }^0\left({\varphi }_n\right)}\sqrt{\frac{G^2({\theta }_n,f_{\eta })}{R_n^{3}sin\left({\varphi }_n\right)}}\sqrt{\frac{\lambda }{2v_r^2}}{s}_t\left(t-\frac{2R_{rd,n}\left(\eta \right)}{c}\right)exp\left(j{\theta }_{rd,n}\left(f_{\eta }\right)\right),\end{array}$$

(17)

where the subscript n denotes the corresponding variables for the nth range ambiguity.

Therefore, before range compression, the RASR for a point target at ${\varphi }_0$ can be represented as

$$\begin{array}{c}\hfill \mathrm{RASR}\left({\varphi }_0\right)\cong \frac{{\sum }_{n=-\infty ,n\ne 0}^{n=\infty }{\int }_{-B_a}^{B_a}{\int }_{-\infty }^{\infty }{\left|s_n(t,f_{\eta })\right|}^{2}dtd{f}_{\eta }}{{\int }_{-B_a}^{B_a}{\int }_{-\infty }^{\infty }{\left|s(t,f_{\eta })\right|}^{2}dtd{f}_{\eta }}=\frac{{\sum }_{n=-\infty ,n\ne 0}^{n=\infty }\frac{{\sigma }^0\left({\varphi }_n\right){\int }_{B_a}^{B_a}{G}^2({\theta }_n,f_{\eta })d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }^0\left({\varphi }_0\right){\int }_{B_a}^{B_a}{G}_r^2({\theta }_0,f_{\eta })d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}},\end{array}$$

(18)

which is consistent with the conventional RASR equation, i.e., (4). Furthermore, assuming the azimuth pattern in each evaluation is the same, the AASR before range compression can be computed as

$$\begin{array}{c}\hfill \mathrm{AASR}\cong \frac{{\displaystyle \sum _{\begin{array}{c}m=-\infty \\ m\ne 0\end{array}}^{\infty }{\int }_{-B_a}^{B_a}{\int }_{-\infty }^{\infty }{\left|s(t,f_{\eta }+m{f}_a)\right|}^{2}dtd{f}_{\eta }}}{{\int }_{-B_a}^{B_a}{\int }_{-\infty }^{\infty }{\left|s(t,f_{\eta })\right|}^{2}dtd{f}_{\eta }}=\frac{{\displaystyle \sum _{\begin{array}{c}m=-\infty \\ m\ne 0\end{array}}^{\infty }{\int }_{B_a}^{B_a}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}}{{\displaystyle {\int }_{B_a}^{B_a}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}}\end{array}$$

(19)

which is consistent with the conventional AASR equation, i.e., (6).

Next, focusing on the RASR after range compression. Assuming the peak output position is aligned after range compression (This paper aims at analyzing the energy; thus, the precise imaging position can be ignored), the expected signal and the received nth range ambiguity signal can be denoted as

$$\begin{array}{c}\hfill s^{\mathrm{rc}}(t,f_{\eta })\approx e^{j\xi \left({\varphi }_0\right)}\sqrt{{\sigma }^0\left({\varphi }_0\right)}\sqrt{\frac{G^2({\theta }_0,f_{\eta })}{R_0^{3}sin\left({\varphi }_0\right)}}\sqrt{\frac{\lambda }{2v_r^2}}p\left(t-\frac{2R_0}{c}\right)exp\left(j{\theta }_{rd}\left(f_{\eta }\right)\right),\end{array}$$

(20)

and

$$\begin{array}{c}\hfill s_n^{\mathrm{rc}}(t,f_{\eta })\approx e^{j\xi \left({\varphi }_n\right)}\sqrt{{\sigma }^0\left({\varphi }_n\right)}\sqrt{\frac{G^2({\theta }_n,f_{\eta })}{R_n^{3}sin\left({\varphi }_n\right)}}\sqrt{\frac{\lambda }{2v_r^2}}p\left(t-\frac{2R_n}{c}\right)exp\left(j{\theta }_{rd,n}\left(f_{\eta }\right)\right),\end{array}$$

(21)

where $p\left(t\right)$ is the matched filter output for the transmitted signal. Therefore, after range compression, the RASR for a point target at ${\varphi }_0$ can be represented as

$$\begin{array}{c}\hfill \mathrm{RASR}\left({\varphi }_0\right)\cong \frac{{\sum }_{n=-\infty ,n\ne 0}^{n=\infty }{\int }_{-B_a}^{B_a}{\left|s_n^{\mathrm{rc}}(t=2R_n/c,f_{\eta })\right|}^{2}d{f}_{\eta }}{{\int }_{-B_a}^{B_a}{\left|s^{\mathrm{rc}}(t=2R_0/c,f_{\eta })\right|}^{2}d{f}_{\eta }}=\frac{{\sum }_{n=-\infty ,n\ne 0}^{n=\infty }\frac{{\sigma }^0\left({\varphi }_n\right){\int }_{-B_a}^{B_a}{G}^2({\theta }_n,f_{\eta })d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }^0\left({\varphi }_0\right){\int }_{-B_a}^{B_a}{G}_r^2({\theta }_0,f_{\eta })d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}},\end{array}$$

(22)

which is also consistent with the (4).

Furthermore, the AASR after range compression can be computed as

$$\begin{array}{c}\hfill \mathrm{AASR}\cong \frac{{\displaystyle \sum _{\begin{array}{c}m=-\infty \\ m\ne 0\end{array}}^{\infty }}{\int }_{-B_a}^{B_a}{\left|s^{\mathrm{rc}}(t=2R_0/c,f_{\eta }+m{f}_a)\right|}^{2}d{f}_{\eta }}{{\int }_{-B_a}^{B_a}{\left|s^{\mathrm{rc}}(t=2R_0/c,f_{\eta })\right|}^{2}d{f}_{\eta }}=\frac{{\displaystyle \sum _{\begin{array}{c}m=-\infty \\ m\ne 0\end{array}}^{\infty }{\int }_{-B_a}^{B_a}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}}{{\displaystyle {\int }_{-B_a}^{B_a}{G}^2\left(f_{\eta }\right)d{f}_{\eta }}}\end{array}$$

(23)

which is also consistent with the (6). Based on the discussion of this section, the derived RASRs and AASRs before or after range compression are both consistent with the conventional ambiguity evaluation equations.

4.2. RASR and AASR with Waveform Diversity

In this sub-section, first, considering the diverse transmitted waveform in the single polarization SAR and quad-pol SAR and according to the same derivation method mentioned in Section 4.1, the conventional RASR and AASR after range compression is detailed. Then, the revised RASR and AASR is presented.

4.2.1. Single Polarization SAR with Waveform Diversity

Assuming two waveforms (denoted by $s_1\left(t\right)$ and $s_2\left(t\right)$) are alternately transmitted, the timing chart is shown in Figure 2. The waveform sequence of the odd-number ambiguities is different from the desired signal, resulting in the mismatched odd-number ambiguities after range compression. Furthermore, thanks to the low cross-correlation energy of the transmitted waveform, the range ambiguity energy can be suppressed [25]. Based on the above discussion, the conventional RASR in correspondence with the odd-number ambiguities with waveform diversity (after range compression) can be denoted as

$$\begin{array}{c}\hfill {\mathrm{RASR}}_{\mathrm{odd}}\left({\varphi }_0\right)\cong \frac{|p_{1,2}{\left(0\right)|}^2}{|p_{1,1}{\left(0\right)|}^2}\frac{{\sum }_{n=-\infty ,\mathrm{odd}}^{n=\infty }\frac{{\sigma }^0\left({\varphi }_n\right){\int }_{-B_a}^{B_a}{G}^2({\theta }_n,f_{\eta })d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }^0\left({\varphi }_0\right){\int }_{-B_a}^{B_a}{G}_r^2({\theta }_0,f_{\eta })d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}},\end{array}$$

(24)

where $p_{i,j}\left(t\right),i,j\in \{1,2\}$ denotes the matched output result between $s_i\left(t\right)$ and $s_j\left(t\right)$. Particularly, waveform diversity cannot affect the signal Doppler bandwidth. Moreover, it should be noted that (24) stems from (20)–(22) (setting $p\left(t\right)=p_{1,2}\left(t\right)$ for $s_n^{\mathrm{rc}}(t,f_{\eta })$, and $p\left(t\right)=p_{1,1}\left(t\right)$ for $s^{\mathrm{rc}}(t,f_{\eta })$ ). Furthermore, the waveform sequence of the even-number ambiguities is the same with that of the desired signal, resulting the RASR in correspondence of the even-number ambiguities with waveform diversity (after range compression) is the same with (22), i.e.,

$$\begin{array}{c}\hfill {\mathrm{RASR}}_{\mathrm{even}}\left({\varphi }_0\right)\cong \frac{{\sum }_{n=-\infty ,\mathrm{even},n\ne 0,}^{n=\infty }\frac{{\sigma }^0\left({\varphi }_n\right){\int }_{-B_a}^{B_a}{G}^2({\theta }_n,f_{\eta })d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }^0\left({\varphi }_0\right){\int }_{-B_a}^{B_a}{G}_r^2({\theta }_0,f_{\eta })d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}},\end{array}$$

(25)

Now, some important observations are necessary. In (24), only the range ambiguity energy at the peak location of the range compressed desired signal is considered. However, for the range ambiguities, the mismatched energy is dispersed along the range dimension. Considering the distributed target scenarios, the ambiguities at all range cells, in the current study, should be given more attention. Therefore, the conventional RASR assessment method is extremely imprecise for the waveform diversity mode, which also results in that many design schemes of quasi-orthogonal waveforms cannot reduce range ambiguity energy [22,23,24].

On the other hand, for the desired signal, the sidelobe energy is actually the noise energy, and using the peak energy denotes the desired signal energy is more reasonable. Therefore, the revised RASR for the single polarization SAR with waveform diversity is given by

$$\begin{array}{c}\hfill {\mathrm{RASR}}_{\mathrm{odd}}\left({\varphi }_0\right)\cong \frac{{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{|p_{1,1}{\left(0\right)|}^2}\frac{{\sum }_{n=-\infty ,\mathrm{odd}}^{n=\infty }\frac{{\sigma }^0\left({\varphi }_n\right){\int }_{-B_a}^{B_a}{G}^2({\theta }_n,f_{\eta })d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }^0\left({\varphi }_0\right){\int }_{-B_a}^{B_a}{G}_r^2({\theta }_0,f_{\eta })d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}\end{array}$$

(26)

Furthermore, when the transmitted waveform is a chirp signal, the evaluation results of RASR and AASR by the conventional equations and by the revised equations should be consistent. Additionally, the peak energies are equal for different waveforms with the same transmitting energy. Hence, a more reasonable revision is using the integral energy of a chirp waveform to replace the desired signal output, i.e.,

$$\begin{array}{c}\hfill {\mathrm{RASR}}_{\mathrm{odd}}\left({\varphi }_0\right)\cong \frac{{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}\frac{{\sum }_{n=-\infty ,\mathrm{odd}}^{n=\infty }\frac{{\sigma }^0\left({\varphi }_n\right){\int }_{-B_a}^{B_a}{G}^2({\theta }_n,f_{\eta })d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }^0\left({\varphi }_0\right){\int }_{-B_a}^{B_a}{G}_r^2({\theta }_0,f_{\eta })d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}},\end{array}$$

(27)

where $p_{\mathrm{chirp}}\left(t\right)$ denotes the matched output result of a standard chirp waveform. Moreover, azimuth ambiguities and the desired signal suffer from the same waveform, thus the AASR is the same with (19).

4.2.2. Quad-Pol SAR with Waveform Diversity

Considering the fact that compared with the energies of the co-polarized channels, energies of cross-polarized channels returned are much weaker, quad-pol SAR systems, as a consequence, suffer from severe range and azimuth ambiguities. The RASR and AASR, as for quad-pol SARs without waveform diversity, have been carefully studied in [6] (see Table IV), and the conventional RASR and AASR for quad-pol SARs with waveform diversity are provided in [25] (see Table V). However, complying with the same reasoning mentioned in Sub-Section 4.2.1, necessary revisions for RASR and AASR equations should be presented. First, for the RASR of the conventional quad-pol SAR with waveform diversity (the timing diagram is shown in Figure 3), the revised RASR equations are given by (the conventional RASR equations are given in Equation (31) of [25])

$$\begin{array}{c}\hfill {\mathrm{RASR}}_{\mathrm{HH}}\cong \frac{\frac{{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\sum }_{n=-\infty ,\mathrm{odd}}^{+\infty }\frac{{\sigma }_{\mathrm{HV}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{HH}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}+\frac{{\sum }_{n=-\infty ,\mathrm{even},\ne 0}^{+\infty }\frac{{\sigma }_{\mathrm{HH}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{HH}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}\hfill \\ \hfill {\mathrm{RASR}}_{\mathrm{HV}}\cong \frac{\frac{{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\sum }_{n=-\infty ,\mathrm{odd}}^{+\infty }\frac{{\sigma }_{\mathrm{HH}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{HV}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}+\frac{{\sum }_{n=-\infty ,\mathrm{even},\ne 0}^{+\infty }\frac{{\sigma }_{\mathrm{HV}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{HV}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}\hfill \\ \hfill {\mathrm{RASR}}_{\mathrm{VH}}\cong \frac{\frac{{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\sum }_{n=-\infty ,\mathrm{odd}}^{+\infty }\frac{{\sigma }_{\mathrm{VV}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{VH}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}+\frac{{\sum }_{n=-\infty ,\mathrm{even},\ne 0}^{+\infty }\frac{{\sigma }_{\mathrm{VH}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{VH}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}\hfill \\ \hfill {\mathrm{RASR}}_{\mathrm{VV}}\cong \frac{\frac{{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\sum }_{n=-\infty ,\mathrm{odd}}^{+\infty }\frac{{\sigma }_{\mathrm{VH}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{VV}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}+\frac{{\sum }_{n=-\infty ,\mathrm{even},\ne 0}^{+\infty }\frac{{\sigma }_{\mathrm{VV}}^0\left({\varphi }_n\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin\left({\varphi }_n\right)}}{\frac{{\sigma }_{\mathrm{VV}}^0\left({\varphi }_0\right)\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin\left({\varphi }_0\right)}}\hfill \end{array}$$

(28)

where the ${\sigma }_{pq}^0,p\in \{\mathrm{H},\mathrm{V}\}$, $q\in \{\mathrm{H},\mathrm{V}\}$ denotes the backscatter reflectivity of $pq$ polarimetric channel. Additionally, the AASR of the conventional quad-pol SARs still remains constant under diverse pulses, which is the same with (19).

Next, for the hybrid quad-pol SAR system with waveform diversity (the timing diagram is shown in Figure 4), the revised AASR equations are given by (the conventional AASR equations are given in Equation (45) of [25])

$$\begin{array}{cc}\hfill & \hfill {\mathrm{AASR}}_{\mathrm{HH}}\cong \frac{\frac{{\sigma }_{\mathrm{HV}}^0{\int }_T^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\sigma }_{\mathrm{HH}}^0{\int }_T^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{odd}\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }+{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{even},\ne 0\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}{{\int }_{-B_a/2}^{B_a/2}{G}^2\left(f_{\eta }\right)d{f}_{\eta }}\hfill \\ \hfill & \hfill {\mathrm{AASR}}_{\mathrm{HV}}\cong \frac{\frac{{\sigma }_{\mathrm{HH}}^0{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\sigma }_{\mathrm{HV}}^0{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{odd}\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }+{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{even},\ne 0\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}{{\int }_{-B_a/2}^{B_a/2}{G}^2\left(f_{\eta }\right)d{f}_{\eta }}\hfill \\ \hfill & \hfill {\mathrm{AASR}}_{\mathrm{VH}}\cong \frac{\frac{{\sigma }_{\mathrm{VV}}^0{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\sigma }_{\mathrm{VH}}^0{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{odd}\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }+{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{even},\ne 0\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}{{\int }_{-B_a/2}^{B_a/2}{G}^2\left(f_{\eta }\right)d{f}_{\eta }}\hfill \\ \hfill & \hfill {\mathrm{AASR}}_{\mathrm{VV}}\cong \frac{\frac{{\sigma }_{\mathrm{VH}}^0{\int }_{-T}^T{\left|p_{1,2}\left(t\right)\right|}^{2}dt}{{\sigma }_{\mathrm{VV}}^0{\int }_{-T}^T{\left|p_{\mathrm{chirp}}\left(t\right)\right|}^{2}dt}{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{odd}\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }+{\displaystyle \sum _{\begin{array}{c}k=-\infty \\ \mathrm{even},\ne 0\end{array}}^{\infty }}\underset{-B_a/2}{\overset{B_a/2}{\int }}{G}^2(f_{\eta }+m{f}_a)d{f}_{\eta }}{{\int }_{-B_a/2}^{B_a/2}{G}^2\left(f_{\eta }\right)d{f}_{\eta }}\hfill \end{array}$$

(29)

Additionally, the similar behaviors of the synthesized spectra of the hybrid and quad-pol SARs are worth remarking (as detailed in [25]), thus the revised AASR of $\pm \pi /4$ quad-pol SAR is the same with (29). Moreover, for the hybrid (or $\pm \pi /4$) quad-pol SAR, the range ambiguities and desired signal suffer from the same waveform and polarization characteristic, thus, the RASR is the same with Equation (46) of [25], i.e.,

$$\begin{array}{cc}\hfill {\mathrm{RASR}}_{pq}\cong & \hfill \frac{{\sum }_{n-\infty }^{+\infty }\frac{{\sigma }_{pq}^0\left({\varphi }_n\right){\int }_{-B_a/2}^{B_a/2}{G}^2\left({\theta }_n,f_{\eta }\right)d{f}_{\eta }}{R_n^{3}sin{\varphi }_n}}{\frac{{\sigma }_{pq}^0\left({\varphi }_0\right){\int }_{-B_a/2}^{B_a/2}{G}^2\left({\theta }_0,f_{\eta }\right)d{f}_{\eta }}{R_0^{3}sin{\varphi }_0}}.\hfill \\ \hfill & \hfill p\in \{H,V\},q\in \{H,V\}\hfill \end{array}$$

(30)

5. Simulation and Analysis

In this section, simulation experiments have been carried out to compare different assessment results by conventional and revised RASR and AASR equations. For this purpose, the system parameters of the LT-1 as detailed in

Table 1. Main simulation parameters of the LT-1 in quad-pol mode.

Parameters	Value
Orbit height	607 Km
Look angle access range	25.6–28.2 deg
SAR pulse width	80 $\mathsf{\mu }$s
Bandwidth	60 MHz
PRF	3320 Hz
Antenna type	Planar
Processed Doppler bandwidth $B_a$	1380 Hz
Backscatter model	Soil and Rock [32]

, are taken into account. Besides, it should be noted that this paper aims at giving a precise ambiguity performance evaluation method, and the design of orthogonal waveforms is not the research topic. To proceed further, three different quasi-orthogonal waveforms pairs, i.e., up-down chirp [22], Orthogonal Frequency Division Multiplexing (OFDM) chirp [24], and quasi-orthogonal NLFM waveforms [25], are exploited as the transmitted waveforms.

5.1. Point Target Simulation

First, the location of the point target at the non-ambiguous imaging area is shown in Figure 5. In the following, in order to carry out the assessment on the interferences in the first near range ambiguity induced by this point target, the focusing results of different transmission techniques are given in Figure 6. In this simulation, assume the energies of raw ambiguity signals are the same. Additionally, the obtained ambiguity energies involving a chirp signal are detailed in

Table 2. Measured ambiguity energy of a point target.

Quad-Pol Mode	Ambiguity	Tx Scheme	Ambiguity Energy
			Revised ${}^1$	Conventional ${}^2$
Conventional	Range	Up chirp waveform	0	0
		Up-down chirp waveform [22]	0	$-32.6$
		OFDM chirp waveforms [24]	$0.59$	$-11.98$
		quasi-orthogonal NLFM waveforms [25]	$-3.47$	$-42.14$
Hybrid	Azimuth	Up chirp waveforms	0	0
		Up-down chirp waveforms [22]	0	$-37.3$
		OFDM chirp waveforms [24]	$0.59$	$-11.98$
		quasi-orthogonal NLFM waveforms [25]	$-3.48$	$-42.14$

¹ The ambiguity energy in all range cell. ² The ambiguity energy in peak location.

, and the ambiguity energy of up chirp waveforms calculated by the conventional ambiguity evaluation method are set to 0 dB. As expected, the waveform diversity mode defocuses the ambiguous signals as shown in Figure 6b–d, and the energies are spread along the range.

As discussed before, for the conventional evaluation method, only the peak energy of the ambiguity signal is taken into account, and all three pairs of quasi-orthogonal waveforms can gain a good ambiguity suppression performance. However, due to the distributed target scene, the ambiguity energy will be accumulated, and the conventional ambiguity evaluation method for the quasi-orthogonal waveforms is inaccurate. To this end, the revised ambiguity performance evaluation method considers the ambiguity energy in all range cells. Therefore, as indicated in the Table 2, for the up chirp waveforms, the ambiguity energy calculated by the conventional and revised calculation methods are the same as 0 dB since the ambiguity signal of the up chirp waveforms is focused on the peak location. Additionally, the obtained total range ambiguity energies (i.e., the ambiguity energy in all range cell) of the up-down chirp pairs and the normal chirp are equal due to the same cross-correlation energy. Moreover, since the OFDM chirp waveforms increase the cross-correlation energy, the total ambiguity energy of OFDM chirp waveforms is larger than that of chirp waveforms about $0.59$ dB. Last but not least, compared with the normal chirp waveforms, the total ambiguity energies of the quasi-orthogonal NLFM waveforms, in this study, are reduced by approximately $3.47$ dB as compared with the standard chirp waveform.

Next, considering the implementation of the waveform diversity technique in the hybrid quad-pol SAR for azimuth ambiguity suppression, the focusing results of different transmission schemes for the first azimuth ambiguity region of a HV polarization channel are given in Figure 7. Figure 7a shows that the azimuth ambiguities of the chirp waveform are focused to some extent. Additionally, as shown in Figure 7b–d, the azimuth ambiguities of the up-down chirp waveforms, OFDM chirp waveforms, and quasi-orthogonal NLFM waveforms are both defocused, and the energies are both spread along the range dimension. The measured azimuth ambiguities relative to a chirp signal are listed in Table 2. Similar to the range ambiguity, if employing the conventional calculation method, the ambiguity energies of all quasi-orthogonal waveforms are significantly suppressed, which is unreasonable. However, according to the evaluation results by revised ASR calculation method, up-down chirp waveforms own a same ambiguity energy with that of chirp waveform due to the same cross-correlation energy. Besides, OFDM chirp waveforms gain a higher ambiguity energy due to the high cross-correlation energy. However, the ambiguity energy of the quasi-orthogonal NLFM waveforms is reduced by approximately $3.48$ dB as compared with the standard chirp waveform and up-down chirp waveforms because of the low cross-correlation energy, which is consistent with the theoretical analysis.

5.2. AASR and RASR

Furthermore, different RASR and AASR equations are employed for evaluating the ambiguity performance of the conventional quad-pol mode of LT-1 system, and the results are given in Figure 8. As shown in Figure 8a, if only one type of waveform (i.e., up chirp waveform) is employed, the conventional RASR is the same as the revised RASR, and the cross-polarized channels suffer from severe range ambiguities. Then, for the up-down chirp waveforms, the calculation results by the conventional RASR equation (see the Figure 8b, and focusing on the RASR at $26.5$ deg) indicate that they can suppress approximately $32.6$ dB range ambiguity energy for the HV and VH polarized channels, whereas they obviously cannot mitigate range ambiguities since their cross-correlation energy is actually not reduced with respect to the up chirp waveform.

On the other hand, according to the evaluation results by the revised RASR equations, up chirp and up-down chirp waveforms have the same RASR. Then, due to the higher cross-correlation energy of OFDM chirp waveforms, its RASR of the HV polarized channel is higher than that of chirp waveform about $0.59$ dB, as indicated in Figure 8c. Additionally, the calculation results by the revised RASR equation (given in Figure 8d) shows that quasi-orthogonal NLFM waveforms can suppress approximately $3.43$ dB compared with the up chirp waveform, which corroborates the point simulation results given in Table 2. Moreover, different polarized channels of conventional quad-pol only suffer from the same polarization azimuth ambiguities, thus, their AASRs are the same, as indicated in Figure 8e.

Finally, focused on the hybrid quad-pol mode, the RASRs and AASRs of different evaluation equations are given in Figure 9. Suppose only the up chirp waveform is used (see Figure 9b). In that case, the recovered cross-polarized channels suffer from severe azimuth ambiguities, and the conventional RASR is the same with the revised RASR. Then, as expected, the calculation results by conventional AASR equation (see Figure 9c) show that up-down chirp waveforms can greatly reduce the azimuth ambiguity energy of the cross-polarized channels, which is obviously unreasonable. On the other hand, the calculation results by the revised AASR equations illustrate that up-down chirp waveforms have the same azimuth ambiguity performance with the up chirp waveform due to the same cross-correlation energy. Additionally, the higher cross-correlation energy of OFDM chirp waveforms will increase the azimuth ambiguity energy, as shown in Figure 9d. Moreover, on account of the lower cross-correlation energy provided by the quasi-orthogonal NLFM waveforms, the azimuth ambiguity energy can be suppressed, as indicated in Figure 9e. Last but not least, the recovered signal in the conventional polarimetric basis only suffers from the same polarization range ambiguities; thus, the RASR is related with the transmitted waveform, and the corresponding RASR is shown in Figure 9a.

5.3. Distributed Scene Simulations

To better show the ambiguity performance of the transmitted waveform, distributed target experiments are further conducted, and the simulation parameters are the same with that of point target simulation experiments. Figure 10a,b are the images of the desired and the range ambiguity regions, respectively, and the range ambiguity suppression results of different waveforms are shown in Figure 10c–f. First, it can be clearly seen from Figure 10c–e that the range ambiguity energies of up-down chirp and OFDM chirp waveforms are not reduced compared with that of the up chirp waveform, whereas the range ambiguity energy of quasi-orthogonal NLFM waveforms is well suppressed due to the low cross-correlation energy. Furthermore, to verify the effectiveness and precision of the revised RASR equations, the azimuth profiles of range ambiguity energy (where the desired signal energy is set to 0 dB), in correspondence with the different transmitted waveforms, are illustrated in Figure 10e. The RASRs of up chirp waveform and up-chirp waveforms are approximately equal since they have the same cross-correlation energy, and the RASR of OFDM chirp waveforms is higher than that of the up chirp waveform. Additionally, the RASR of quasi-orthogonal NLFM waveforms is lower than that of up-chirp waveform because they own a lower cross-correlation energy. The most important is that all orthogonal waveforms cannot gain a significant ambiguity performance improvement compared with the single transmitted waveform (i.e., up chirp waveforms) since the ambiguity energy will be accumulated for distributed targets, and the range ambiguity evaluation results is close to the calculation results (shown in Figure 8) by the revised RASR equations.

Furthermore, the azimuth ambiguity simulation results of distributed targets are given in Figure 11. As expected, the azimuth ambiguity energies of up chirp waveforms and up-down chirp waveforms are approximately equal, whereas the ambiguity energy of OFDM chirp waveforms is higher. Additionally, because quasi-orthogonal NLFM waveforms can reduce the cross-correlation energy, they own the lowest ambiguity energy. Moreover, for a quantitative analysis to verify the effectiveness of the proposed AASR equations, the azimuth profiles of azimuth ambiguity energy are shown in Figure 11e, and the ambiguity energy evaluation results are also approximate to the AASR calculation results given in Figure 9.

6. Conclusions

Quasi-orthogonal waveforms, which have a low cross-correlation energy, can effectively suppress ambiguities in the SAR system by the waveform diversity technique, which has been widely discussed and verified in many open studies. Additionally, this method has almost no extra system costs and the ambiguity suppression performance is independent of PRF, thus it suits the PRF-limited spaceborne SAR system. However, the accurate method to evaluate the ambiguity performance of waveform diversity mode, i.e., RASR and AASR, is still a gap. Therefore, this paper formulates detailed signal model, and provides revised RASR and AASR formulas, which contributes a lot, for the system designer, to precisely evaluate the ambiguity performance, and the relevant equations are summarized in

Table 3. Equations for the conventional and revised RASR and AASR for the SAR with waveform diversity.

Mode	Conventional		Revised
	RASR	AASR	RASR	AASR
Single polarization	(24) & (25)	(6)	(25) & (27)	(6)
Conventional quad-pol	Equation (31) of [25]	(6)	(28)	(6)
Hybrid quad-pol	(30)	Equation (45) of [25]	(30)	(29)

. Besides, during our analysis, detailed simulation experiments exploiting the parameters of the LT-1 system are proceeded to strengthen the theoretical developments.