**3.2 Raw echo algorithm of airborne stripmap SAR with trajectory error and attitude jitter under squint conditions**

This section presents the simulation algorithm of airborne SAR raw echo with trajectory offset and attitude jitter error under squint conditions.

#### *3.2.1 Simulation algorithm*

Suppose the received echo is as follows:

$$\begin{split} h(\mathbf{x}',r') &= \iint d\mathbf{x}d\mathbf{r} \chi(\mathbf{x},r) \cdot \exp\left\{-j\frac{4\pi}{\lambda}R - j\frac{4\pi}{\lambda}\frac{\Delta f/f}{c\tau}(r'-R)^2\right\} \\ &\times \operatorname{rect}\left[\frac{r'-R}{c\tau/2}\right] \cdot \mathcal{W}^2\left[\frac{\mathbf{x}'-\mathbf{x}-\delta\_{\mathbf{x}}}{X}\right] \end{split} \tag{32}$$

where *δ<sup>x</sup>* ¼ *r* þ *δrxr x*<sup>0</sup> , *<sup>x</sup>*,*r*, *<sup>ϕ</sup><sup>d</sup>* ð Þ ð Þ <sup>~</sup>*δaz <sup>x</sup>*<sup>0</sup> ð Þ is the antenna pointing error in azimuth direction. From [25] we have

$$\begin{aligned} &\delta \overline{r}\_{\mathbf{x}r}(\mathbf{x}', \mathbf{x}, r, \phi\_d) \approx d(\mathbf{x}') (\sin \theta(\mathbf{x}') \cos \phi\_d \cos \theta(\mathbf{x}, r) - \cos \theta(\mathbf{x}') \sin \theta(\mathbf{x}, r) \\ &+ \frac{1}{2} (\sin \phi\_d)^2 (\cos \theta(\mathbf{x}, r))^2 \sin \theta(\mathbf{x}') \cos \phi\_d \cos \theta(\mathbf{x}, r) + \frac{1}{2} \cos \theta(\mathbf{x}') \\ &\times \sin \theta(\mathbf{x}, r) (\sin \phi\_d)^2 (\cos \theta(\mathbf{x}, r))^2 \end{aligned} \tag{33}$$

Eq. (32) is a signal model for the exact time-domain simulation and can be used as the criterion to judge the validity of the algorithm proposed below.

When the beam pointing error is less than the beam width, the antenna pattern can be approximated as follows:

$$\begin{split} \mathcal{W}^2 \left( \frac{\mathbf{x}' - \mathbf{x} - \delta\_{\mathbf{x}'}}{X} \right) &\approx \mathcal{W}^2 \left( \frac{\mathbf{x}' - \mathbf{x}}{X} \right) - \mathcal{W}^{2(1)} \left( \frac{\mathbf{x}' - \mathbf{x}}{X} \right) (r + \delta \overline{r}\_{xr}(\mathbf{x}', \mathbf{x}, r, \phi\_d)) \delta\_{\mathbf{a}t} (\mathbf{x}') \\ &+ \frac{1}{2} \mathcal{W}^{2(2)} \left( \frac{\mathbf{x}' - \mathbf{x}}{X} \right) (r + \delta \overline{r}\_{xr}(\mathbf{x}', \mathbf{x}, r, \phi\_d))^2 \delta\_{\mathbf{a}x} \tag{34} \end{split} \tag{35}$$

where *<sup>W</sup>*<sup>2</sup>ð Þ *<sup>n</sup>* ð Þ� is the nth derivative of the antenna pattern. Accordingly, the echo can be decomposed as follows:

$$h(\mathbf{x}', r') \approx \hat{h}\_0(\mathbf{x}', r') + \hat{h}\_1(\mathbf{x}', r') + \hat{h}\_2(\mathbf{x}', r') \tag{35}$$

The above equation, ^ *h*<sup>0</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ only includes trajectory deviations, the latter two terms including trajectory offset and attitude jitter.

Assuming that the range Fourier transform of ^ *h*<sup>1</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ is ^ *h*<sup>1</sup> *x*<sup>0</sup> ð Þ , *η* , then

$$\hat{h}\_1(\mathbf{x}',\eta) = \exp\left[-j\left(\eta + \frac{4\pi}{\lambda}\right)\delta\overline{r}(\mathbf{x}',\phi\_d)\right]\overline{\hat{h}}\_1(\mathbf{x}',\eta) \tag{36}$$

The term <sup>~</sup>*δ*az *<sup>x</sup>*<sup>0</sup> ð Þ can be extracted from the integral, that is

$$
\overline{\hat{h}}\_1(\mathbf{x}',\eta) = \overline{\delta}\_{ax}(\mathbf{x}') \overline{\hat{h}}\_1(\mathbf{x}',\eta) \tag{37}
$$

$$\begin{split} \hat{h}\_{1}(\mathbf{x}',\eta) &= \text{rect}\left[\frac{\eta}{bc\tau}\right] \exp\left\{ \begin{array}{c} j\frac{\eta^{2}}{4b} \end{array} \right\} \Big| \int dx dr \chi(\mathbf{x},r) \exp\left\{ -j\left(\eta + \frac{4\pi}{\lambda}\right) \right\} \\ &\times (r + \Delta R(\mathbf{x}'-\mathbf{x},r) + \overline{\eta}(\mathbf{x}',r,\phi\_{d}) + \overline{\rho}(\mathbf{x}',\mathbf{x},r,\phi\_{d})) \\ &\times [-(r + \partial \overline{r}\_{\text{xr}}(\mathbf{x}',\mathbf{x},r,\phi\_{d}))] W^{2(1)}\left(\frac{\mathbf{x}'-\mathbf{x}}{X}\right) \end{split} \tag{38}$$

The azimuth Fourier transform of ^ h1 *x*<sup>0</sup> ð Þ , *η* is

$$\overline{\hat{\mathbf{h}}\_1}(\xi,\eta) = FT\left[\tilde{\delta}\_{ax}(\varkappa')\right] \otimes \tilde{\hat{h}}\_1(\xi,\eta) \tag{39}$$

where

$$\tilde{\hat{h}}\_1(\xi,\eta) = \int d\boldsymbol{r} \exp\left[-j\overline{\eta}\boldsymbol{r}\right] \int d\boldsymbol{l} \bar{\mathcal{G}}\_1(\xi-\boldsymbol{l},\eta,\boldsymbol{r}) \tilde{\hat{F}}\_1(\xi-\boldsymbol{l},\boldsymbol{l},\eta,\boldsymbol{r})\tag{40}$$

~ *<sup>F</sup>*^1ð Þ� is the Fourier transform of ~^*<sup>f</sup>* <sup>1</sup>ð Þ� , and

$$\begin{split} \tilde{\hat{f}}\_1(\mathbf{x}, \mathbf{x}', \boldsymbol{\eta}, \mathbf{r}) &= -[r + \delta \overline{r}\_{\mathbf{x}r}(\mathbf{x}', \mathbf{x}, r, \boldsymbol{\phi}\_d)] \boldsymbol{\upchi}(\mathbf{x}, \mathbf{r}) \\ &\times \exp\left\{ -j \left( \boldsymbol{\eta} + \frac{4\pi}{\lambda} \right) [\overline{\boldsymbol{\upmu}}(\mathbf{x}', r, \boldsymbol{\phi}\_d) + \overline{\boldsymbol{\upmu}}(\mathbf{x}', \mathbf{x}, r, \boldsymbol{\phi}\_d)] \right\} \end{split} \tag{41}$$

*Efficient Simulation of Airborne SAR Raw Data in Case of Motion Errors DOI: http://dx.doi.org/10.5772/intechopen.99378*

Similarly, a two-dimensional Fourier transform ^ *h*<sup>2</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ can be obtained. By performing a range FT of *h*<sup>2</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ and separating the factor exp �*jηδr x*<sup>0</sup> , *ϕ<sup>d</sup>* f g ð Þ , we get ^ *<sup>h</sup>*<sup>2</sup> *<sup>x</sup>*<sup>0</sup> ð Þ , *<sup>η</sup>* . Then, after azimuth FT of ^ *<sup>h</sup>*<sup>2</sup> *<sup>x</sup>*<sup>0</sup> ð Þ , *<sup>η</sup>* , we obtain: ^ *h*2ð Þ¼ *ξ*, *η*

$$\left[FT\_{\mathbf{x'}}\left[\tilde{\delta}^2\_{ax}(\mathbf{x'})\right]\otimes \tilde{\hat{h}}\_2(\xi,\eta), \text{ where}\right]$$

$$\tilde{\hat{h}}\_2(\xi,\eta) = \int d\boldsymbol{r} \, \exp\left[-j\overline{\eta}\boldsymbol{r}\right] \int d\boldsymbol{l} \, \tilde{\hat{G}}\_2(\xi-\boldsymbol{l},\eta,\boldsymbol{r}) \tilde{\hat{F}}\_2(\xi-\boldsymbol{l},\boldsymbol{l},\eta,\boldsymbol{r})\tag{42}$$

where <sup>~</sup> *<sup>F</sup>*^2ð Þ� is the two-dimensional FT of ~^*<sup>f</sup>* <sup>2</sup>ð Þ� . The difference between ~^*<sup>f</sup>* <sup>1</sup>ð Þ� and ~^*<sup>f</sup>* <sup>2</sup>ð Þ� is that the former contains a factor *<sup>r</sup>* <sup>þ</sup> *<sup>δ</sup>rxr <sup>x</sup>*<sup>0</sup> , *x*,*r*, *ϕ<sup>d</sup>* ½ � ð Þ while the latter contains a factor <sup>1</sup> <sup>2</sup> *r* þ *δrxr x*<sup>0</sup> , *<sup>x</sup>*,*r*, *<sup>ϕ</sup><sup>d</sup>* ½ � ð Þ <sup>2</sup> . If the two-dimensional spectrum of ~^ *h*1ð Þ *ξ*, *η* and ~^ *h*2ð Þ *ξ*, *η* is obtained from Eqs. (40) and (42), then the estimation of *h x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ can be obtained from **Figure 14**. However, due to the coupling of azimuth and distance between ~^*<sup>f</sup>* <sup>1</sup>ð Þ� and ~^*<sup>f</sup>* <sup>2</sup>ð Þ� , it is necessary to decouple them. Therefore, we adopt the following approach:

The amplitude approximation ~^*<sup>f</sup>* <sup>1</sup>ð Þ� and ~^*<sup>f</sup>* <sup>2</sup>ð Þ� satisfy the following conditions: *δrxr x*<sup>0</sup> , *x*,*r*, *ϕ<sup>d</sup>* ð Þ≈*δr x*<sup>0</sup> ,*r*, *ϕ<sup>d</sup>* ð Þ. Phase approximation needs to meet the following requirements:

$$\tilde{\hat{f}}\_1(\cdot) \approx (r + \delta \overline{r}\_{\text{sr}}(\mathbf{x}', r, \phi\_d)) \gamma(\mathbf{x}, r) \exp\left\{-j\frac{4\pi}{\lambda} \overline{\psi}(\mathbf{x}', r, \phi\_d)\right\} \tag{43}$$

$$\tilde{\hat{f}}\_2(\mathbf{x}, \mathbf{x}', \boldsymbol{\eta}, \mathbf{r}) \approx \frac{1}{2} (r + \delta \overline{r}\_r(\mathbf{x}', r, \phi\_d))^2 \boldsymbol{\eta}(\mathbf{x}, \mathbf{r}) \times \exp\left[ -j \frac{4\pi}{\lambda} \overline{\boldsymbol{\eta}}(\mathbf{x}', r, \phi\_d) \right] \tag{44}$$

$$|\overline{\rho}(\mathbf{x}', \mathbf{x}, r, \phi\_d)| < < \frac{\lambda}{4\pi} \tag{45}$$

**Figure 14.** *SAR raw data echo simulation diagram.*

$$|\overline{\wp}(\mathbf{x}', r, \phi\_d)| < < \frac{f}{\Delta f} \, \frac{\lambda}{2\pi} \tag{46}$$

then

$$\begin{split} \hat{\bar{h}}\_{1}(\xi,\eta) &= \int \exp\left(-j\overline{\eta}r\right) \left\{ d\!\!/\!G\_{1}(\xi-l,\eta,r)\Gamma(\xi-l,r)\!\!Q(l,\eta,r) \\ &= \int dr \, \exp\left(-j\overline{\eta}r\right) \left\{ \mathcal{Q}(\xi,\eta,r) \circledast\_{\xi} [\!\!G\_{1}(\xi,\eta,r)\Gamma(\xi,r)] \right\} \end{split} \tag{47}$$

where

$$Q(l,\eta,r) = FT\_{\mathcal{X}}\{\exp\left[-j\overline{\eta\nu}(\mathbf{x}',r,\phi\_d)\right]\}\tag{48}$$

$$\begin{split} G\_1(\xi,\eta,r) &\approx \text{rect}\left(\frac{\eta}{bc\tau}\right) \left[ W^2 \left(\frac{\xi-\xi\_d-\Delta\xi(r)}{2a}\right) - W^2 \left(\frac{\xi-\xi\_d}{2a}\right) \right] \\ &\times \exp\left(j\frac{\eta^2}{4b}\right) \exp\left[-j\left(\sqrt{\overline{\eta}^2-\xi^2}\cos\phi\_d-\overline{\eta}\right)r - j\xi r \sin\phi\_d \right] \end{split} \tag{49}$$

Similarly, we can get an estimate of ~^ *h*2ð Þ *ξ*, *η* , with the flow chart shown in **Figure 15**.

In the simulation, the attitude change needs to meet the following condition:

$$\delta\_{ax}(\mathbf{x'}) \ll \frac{X}{r + \delta \overline{r}\_r(\mathbf{x'}, r, \phi\_d)} < \frac{X}{r} = \frac{\lambda}{L} \tag{50}$$

Combined with conditions (45) and (46), we see that the algorithm is suitable for squint with medium trajectory offset error and antenna beam pointing error.

**Figure 15.** *The flowchart of medium trajectory error and attitude jitter.* *Efficient Simulation of Airborne SAR Raw Data in Case of Motion Errors DOI: http://dx.doi.org/10.5772/intechopen.99378*

The computational efficiency of the algorithm is analyzed below. Obviously, the computational complexity of the algorithm increases with the expansion order of the antenna pattern. Let *N* be the computational complexity with

$$N \approx nN\_a N\_r^2 \left( 3 + \log\_2 N\_a \right) \tag{51}$$

and *N*~ be the computational complexity of the time domain method. It turns out that the following ratio is obtained

$$\frac{\tilde{N}}{N} \approx \frac{N\_{sa}}{n\left(3 + \log\_2 N\_a\right)}\tag{52}$$

We observe from the above equation that the proposed algorithm has higher computational efficiency than the time-domain algorithm for the same order of antenna pattern expansion *n*.

#### *3.2.2 Simulation results*

In this section, the proposed algorithm is verified by comparing with the simulation results of time-domain algorithm. The trajectory offset error is shown in **Figure 16** and the antenna pointing error is given by Eq. (31). **Figure 17** shows the phase comparison results of azimuth and range directions, where **Figure 17(a)** is

**Figure 16.** *Moderate trajectory deviations [m] vs. the azimuth pixels.*

**Figure 17.** *SAR raw data phase comparison of the algorithm proposed in Section 3.2 and the time domain algorithm.*

the azimuth cut and **Figure 17(b)** is the range cut. It can be seen that the phase errors of the proposed algorithm and the time-domain algorithm are both minimal, proving the effectiveness of the proposed algorithm.
