## 2 Range Ambiguity Analysis by Alternatively Up and Down Chirp Modulation  
As shown in Figure 1, range ambiguous signals are echoes arrived at the wrong time and scattered from undesired range zones. Considering a target (point A in Figure 1) in the main region with range history \(R(t)\) where \(t\) presents the azimuth time, the echoes from previous and latter transmitted pulses will arrive as ambiguous signals with the slant range history as [18]:  
\[R^{\prime}_{n}(t)=R(t)+\mathrm{c}/2\cdot n/prf(n=\pm 1,\pm 2,\pm 3\ldots) \tag{1}\]  
where \(c\) is the velocity of light, \(prf\) is the pulse repetition frequency, \(n\) is the ambiguous number (negative part presents previous pulses when positive part presents latter pulses). The signal echoes consist of mainlobe area parts and ambiguous area parts are shown below:  
\[\mathrm{s}(\tau,t)=\mathrm{s}_{main}(\tau,t)+\sum_{n}\mathrm{s}_{ambiguity }(\tau,t,n) \tag{2}\]  
where \(\tau\) is the range time. Supposing the system works by sending alternately up and down chirp signals with pulse length \(T_{p}\) and modulation rate \(k_{r}\), We can get the echo of ambiguous signals as:  
\[ \begin{array}{ll}\mathrm{s}_{ambiguity}(\tau,t,n)&=\sigma_{n}\cdot W_{a}(t, n)\cdot W_{r}[\tau-2R^{\prime}_{n}(t)/\mathrm{c}]\cdot\exp[-j4\pi/ \lambda\cdot R^{\prime}_{n}(t)]\\ &\cdot\exp\Bigl{\{}-(-1)^{|n|}j\pi k_{r}\cdot[\tau-2R^{\prime}_{n}(t)/\mathrm{ c}]^{2}\Bigr{\}}\end{array}  \tag{3}\]where \(\sigma\) is the backscattering intensity, \(\lambda\) is the wave length, \(W_{r}(\cdot)\) and \(W_{a}(\cdot)\) are the antenna patterns of range and azimuth direction. We can get the signals for main area as:  
\[ \begin{array}{ll}s_{main}(\tau,t)&=\sigma\cdot W_{a}(t)\cdot W_{r}[\tau-2R(t)/ c]\cdot\exp[-j4\pi/\lambda\cdot R(t)]\\ &\cdot\exp\left\{-j\pi k_{r}\cdot[\tau-2R(t)/c]^{2}\right\}\end{array}  \tag{4}\]  
We can see from (3) and (4) that the ambiguous signals with odd \(n\) show contrary modulation rate to the main signals, whereas those with even \(n\) show the same modulation rate. The filtering of an up chirp with a down chirp filter will result in defocusing phenomenon along range direction and the pulse compression result as follow [12]:  
\[s_{de}(\tau)=1/\sqrt{2}\cdot rect\left[\tau/2\cdot T_{p}\right]\cdot\exp\left( -j\pi k_{r}\tau^{2}/2\right) \tag{5}\]  
For comparison, the result of the correct filtering of the down chirp is [12]:  
\[s_{down}(\tau)=\exp(-\mathrm{j}\tau/4)\cdot\sqrt{k_{r}T_{p}^{2}}\sin c\left[ \pi k_{r}T_{p}\tau\right] \tag{6}\]  
where \(k_{r}T_{p}{}^{2}\) is the compression gain. We can see that the amplitude of mismatched up-chirp ambiguous signals (odd \(n\)) is lowered by \(1/\sqrt{2\cdot k_{r}T_{p}^{2}}\). The signal power of main area is unchangeable once the system antenna pattern and imaging targets are fixed, which leaves ambiguity suppression work to the reduction of ambiguous signal power. Due to the contrary modulation rate to the main signals with odd ambiguous number, the method of alternating up and down chirp modulation is provided with lower ambiguous signal power owing to the defocusing of mismatched range filter. Therefore, it can be widely adopted in the multi-polarization spaceborne SAR where a serious decline in the range-ambiguity-to-signal-ratio appears in cross polarization mode. Figure 1: This is a figure which shows the principle of range ambiguity. We can get the range-ambiguity-to-signal-ratio (RASR) as [18]:  
\[RASR=\frac{\sum\limits_{n=-\infty}^{\infty}\int_{-T_{s}/2}^{T_{s}/2}G^{2}[\eta(t,n )]\cdot\sigma(t,n)\sin[\eta(t,n)]/R_{n}^{\prime 3}(t)dt}{\int_{-T_{s}/2}^{T_{s}/2}G^{2}[\eta(t)] \cdot\sigma(t)\sin[\eta(t)]/R^{3}(t)dt} \tag{7}\]  
where \(\eta\) represents the incidence angle for a certain ambiguous number, \(G(\cdot)\) represents the azimuth antenna patterns for a certain incidence angle, \(T_{s}\) represents the data recording time.