**3. Raw echo algorithm of airborne stripmap SAR with trajectory error and attitude jitter**

Due to the air disturbance or the flight instability of the SAR platform, the platform trajectory deviation and attitude error are inevitably introduced into the sensor parameters. This section considers the raw echo algorithm of airborne stripmap SAR with trajectory error and attitude jitter.

### **3.1 Raw echo algorithm of airborne stripmap SAR with trajectory error and attitude jitter**

Firstly, the simulation algorithm of antenna beam pointing error is briefly introduced. In Ref. [22], a two-dimensional frequency-domain echo simulation algorithm is proposed to reflect the beam pointing error. The difference between the raw data echo, including antenna beam pointing error, and ideal echo is the azimuth envelope. When the amplitude of the antenna beam jitter is less than the beam width, the azimuth amplitude weighting function can be approximated by the Taylor expansion. From reference [22], When the beam pointing is sinusoidal jitter, a pair of echoes is erroneously produced.

Suppose the received echo is as follows:

$$\begin{split} h(\mathbf{x}',r') &= \iint d\mathbf{x}d\mathbf{r} \chi(\mathbf{x},r) \cdot \exp\left\{-j\frac{4\pi}{\lambda}R - j\frac{4\pi}{\lambda}\frac{\Delta f/f}{c\pi}(r'-R)^2\right\} \\ &\times \operatorname{rect}\left[\frac{r'-R}{c\pi/2}\right] \cdot \mathcal{W}^2\left[\frac{\mathbf{x}'-\mathbf{x}-\delta\_{\mathbf{x}}}{X}\right] \end{split} \tag{17}$$

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. When the amplitude of antenna beam jitter is less than the beam width, the azimuth amplitude weighting function is approximated by the second-order Taylor expansion:

$$\begin{split} W^2 \left( \frac{\mathbf{x}' - \mathbf{x} - \delta\_x}{X} \right) &\approx W^2 \left( \frac{\mathbf{x}' - \mathbf{x}}{X} \right) - W^{2(1)} \left( \frac{\mathbf{x}' - \mathbf{x}}{X} \right) (r + \delta r\_{\mathbf{x}r}(\mathbf{x}', \mathbf{x}, r)) \delta\_{\mathbf{u}\mathbf{x}}(\mathbf{x}') \\ &+ \frac{1}{2} W^{2(2)} \left( \frac{\mathbf{x}' - \mathbf{x}}{X} \right) (r + \delta r\_{\mathbf{x}r}(\mathbf{x}', \mathbf{x}, r))^2 \delta\_{\mathbf{u}\mathbf{x}}^2(\mathbf{x}') \end{split} \tag{18}$$

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

where *r* is the distance coordinate between the platform and the target point, and the original echo is decomposed into three parts:

$$h(\mathbf{x}', r') \approx h\_0(\mathbf{x}', r') + h\_1(\mathbf{x}', r') + h\_2(\mathbf{x}', r') \tag{19}$$

where *h*<sup>0</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ is equivalent to the original echo with trajectory deviations, which can be calculated by the traditional frequency-domain algorithm. The last two terms of Eq. (19) contain trajectory offset error and antenna attitude jitter error. *h*<sup>0</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ can be estimated by echo simulation algorithm, which only contains trajectory offset error. Therefore, the main task is to evaluate the two terms of *h*<sup>1</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ and *h*<sup>2</sup> *x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ.

In order to expand the range of distance effectiveness, we assume that

$$\frac{1}{r}\overline{f}\_1(\mathbf{x}, \mathbf{x}', \eta, r) \approx (r + \delta r\_r(\mathbf{x}', r)) \cdot \gamma(\mathbf{x}, r) \cdot \exp\left\{-j\frac{4\pi}{\lambda}\psi(\mathbf{x}', r)\right\} \tag{20}$$

which satisfies the following condition: *<sup>η</sup>* <sup>þ</sup> <sup>4</sup>*<sup>π</sup> λ* � �*<sup>φ</sup> <sup>x</sup>*<sup>0</sup> ð Þþ , *<sup>x</sup>*,*<sup>r</sup> ηψ <sup>x</sup>*<sup>0</sup> ð Þ ,*<sup>r</sup>* � � � � ≪ 1. The azimuth Fourier transform of *f* <sup>1</sup> can be expressed as: *F*1ð Þ¼ *χ*, *l*, *η*,*r* Γð Þ *χ*,*r Q*1ð Þ *l*,*r* . where Γð Þ� is the azimuth FT of *γ*ð Þ *x*,*r* .

$$Q\_1(l,r) = FT\_{x'} \left\{ (r + \delta r\_r(\mathbf{x'}, r)) \cdot \exp\left[ -j\frac{4\pi}{\lambda} \psi(\mathbf{x'}, r) \right] \right\} \tag{21}$$

Then *<sup>H</sup>*ð Þ¼ *<sup>ξ</sup>*, *<sup>η</sup>* <sup>Ð</sup> exp ½ � �*jη<sup>r</sup> <sup>Q</sup>*1ð Þ *<sup>ξ</sup>*,*<sup>r</sup>* <sup>⊗</sup> *ξ* ½ � *G*ð Þ� *ξ*, *η*,*r* Γð Þ *ξ*,*r* � �*dr*.

The system function *G*ð Þ� of SAR is decomposed into two parts

$$G(\xi,\eta,r) = G\_{A1}(\xi,r) \cdot G\_{B1}(\xi,\eta,r) \tag{22}$$

where the term.

*GA*1ð Þ¼ *ξ*,*r* exp �*j* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4*π λ* � �<sup>2</sup> � *<sup>ξ</sup>*<sup>2</sup> q � <sup>4</sup>*<sup>π</sup> λ* � �*<sup>r</sup>* � � describes the azimuth frequency modulation, including the change of focusing depth, and

$$G\_{B1}(\xi,\eta,r) = \text{rect}\left[\frac{\eta}{2bc\tau/2}\right] \cdot \left(\frac{PRF}{v}\right) \cdot \left[\mathcal{W}^2\left(\frac{\xi-\Delta\xi}{2\Omega\_r}\right) - \mathcal{W}^2\left(\frac{\xi}{2\Omega\_r}\right)\right] \cdot \exp\left\{\left.j\frac{\eta^2}{4b}\right\}\right) \tag{23}$$

$$\times \exp\left\{-j\left(\sqrt{\overline{\eta}^2-\xi^2}-\overline{\eta}\right)r\right\} \cdot \exp\left\{-j\left(\sqrt{\left(\frac{4\pi}{\lambda}\right)^2-\xi^2}-\frac{4\pi}{\lambda}\right)r\right\} \tag{25}$$

the migration effect of range element and the error of antenna beam jitter. Then, we have

$$\tilde{h}\_1(\xi,\eta) \approx \int \exp\left\{-j\overline{\eta}r\right\} \cdot G\_{\text{B1}}(\xi,\eta,r) \cdot \left\{Q\_1(\xi,r) \otimes\_{\xi} [G\_A(\xi,r) \cdot \Gamma(\xi,r)]\right\} dr \tag{24}$$

Similarly, let

$$\overline{f}\_2(\mathbf{x}, \mathbf{x}', \eta, r) \approx \frac{1}{2} (r + \delta r\_r(\mathbf{x}', r))^2 \cdot \gamma(\mathbf{x}, r) \cdot \exp\left\{-j\frac{4\pi}{\lambda} \psi(\mathbf{x}', r)\right\} \tag{25}$$

and

$$Q\_2(\xi, r) = FT\_\chi \left\{ \frac{1}{2} (r + \delta r\_r(\mathbf{x}', r))^2 \cdot \exp\left[ -j \frac{4\pi}{\lambda} \varphi(\mathbf{x}', r) \right] \right\} \tag{26}$$

$$\begin{split} G\_{\rm B2}(\xi,\eta,r) &= \text{rect}\left[\frac{\eta}{2bc\pi/2}\right] \cdot \left(\frac{PRF}{v}\right)^2 \cdot \left[W^2\left(\frac{\xi}{2\Omega\_\chi}\right) - 2W^2\left(\frac{\xi-\Delta\xi}{2\Omega\_\chi}\right)\right. \\ &+ W^2\left(\frac{\xi-2\Delta\xi}{2\Omega\_\chi}\right)\Big] \times \exp\left\{\begin{array}{c} j\frac{\eta^2}{4b}\right\} \cdot \exp\left\{-j\left(\sqrt{\overline{\eta}^2-\xi^2}-\overline{\eta}\right)r\right\} \\ &\times \exp\left\{\begin{array}{c} j\left(\sqrt{\left(\frac{4\pi}{\lambda}\right)^2-\xi^2}-\frac{4\pi}{\lambda}\right)r \end{array}\right\} \end{split} \tag{27}$$

we yield

$$\tilde{h}\_2(\xi,\eta) \approx \int \exp\left[-\tilde{\eta}\overline{\eta}r\right] \cdot G\_{\text{B2}}(\xi,\eta,r) \times \left\{Q\_2(\xi,r) \otimes\_{\xi} [G\_A(\xi,r)\Gamma(\xi,r)]\right\} dr \tag{28}$$

From Eqs. (24) and (28), we obtain the efficient computation of ~ *h*1ð Þ *ξ*, *η* and ~ *h*2ð Þ *ξ*, *η* . **Figure 9** summaries the computation flow chart of the procedures outlined above. Finally, *h x*<sup>0</sup> ,*r*<sup>0</sup> ð Þ can be evaluated from **Figure 10**. The effectiveness of the algorithm for antenna jitter is *<sup>δ</sup>az <sup>x</sup>*<sup>0</sup> ð Þ <sup>≪</sup> *<sup>X</sup> <sup>r</sup>*þ*δrr <sup>x</sup>*<sup>0</sup> ð Þ ,*<sup>r</sup>* <sup>&</sup>lt; *<sup>X</sup> <sup>r</sup>* <sup>¼</sup> *<sup>λ</sup> L*.

Note that the trajectory deviations should satisfy the following condition:

$$\left| \left( \eta + \frac{4\pi}{\lambda} \right) \wp(\mathbf{x}', \mathbf{x}, r) + \eta \wp(\mathbf{x}', r) \right| \ll \mathbf{1} \tag{29}$$

which ensures that the range of distance effectiveness of the algorithm.

**Figure 9.** *Flowchart of the simulation.*

#### *3.1.1 Computational complexity*

The complexity of algorithms given in Section 3.1 increases with the order of antenna pattern decomposition. Suppose *Na* and *Nr* are the azimuth and range pixel *Efficient Simulation of Airborne SAR Raw Data in Case of Motion Errors DOI: http://dx.doi.org/10.5772/intechopen.99378*

**Figure 10.** *Horizontal and vertical components of medium trajectory offset error.*

size of the original signal, *NaNr* is the computational complexity of <sup>γ</sup>ð Þ x, r . Let *<sup>N</sup>*<sup>~</sup> and *N* be the computational complexity of 3.1 part and medium trajectory offset error algorithm, then *<sup>N</sup>*<sup>~</sup> <sup>¼</sup> ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> *<sup>N</sup>*, where *<sup>N</sup>* <sup>¼</sup> *NxN*<sup>2</sup> *r*. Now we compare the computational complexity of part 3.1 and the exact time domain algorithm. It is known that the computational complexity of time domain is *<sup>N</sup>*^ <sup>¼</sup> *NxNrNtpNsa*, where *Ntp* and *Nsa* are the sizes of the emitted pulse and the synthetic aperture. Generally, *n* is less than *Nsa* and *Ntp* are the same size with *Nr*. Then the algorithm given in this part is more efficient than the time-domain algorithm. The calculation time-saving rate is

$$\frac{\hat{N}}{\bar{N}} \approx \frac{N\_{\text{sa}}}{n+1} \tag{30}$$

Therefore, the algorithm given in Section 3.1 can simulate the extended scene in a reasonable time.

### *3.1.2 Algorithm verification*

In this part, we will give some simulation results to verify the proposed algorithm. The simulation parameters are selected from the X-band SAR data in Ref.s [8, 9], and the main parameters are shown in **Table 3**. Note that the precise time domain simulation can be obtained from Eq. (17). In order to simplify, the algorithm given in part 3.1 is called algorithm A, and the time-domain algorithm is


**Table 3.** *Sensor simulation parameters.*

called algorithm B. Let a single point target be located in the middle of the scene (r = 5140 m), and the horizontal and vertical components are shown in **Figure 11**. Let the antenna pointing error be as follows:

$$\delta\_a(\mathbf{x'}) = \delta\_m \sin\left(\frac{2\pi}{\nu T\_b}\mathbf{x'} + \varrho\_0\right) \tag{31}$$

where *δ<sup>m</sup>* is the amplitude, *Tb* is the period of antenna jitter and *φ*<sup>0</sup> is the initial phase. Assumptions *<sup>δ</sup><sup>m</sup>* <sup>¼</sup> <sup>1</sup> 20 *λ <sup>L</sup>*, *Tb* <sup>¼</sup> <sup>1</sup> <sup>10</sup> *Ts* and *φ*<sup>0</sup> ¼ 0. **Figure 11** shows the horizontal and vertical components of the trajectory offset, and the antenna pointing error is Eq. (31). Now we consider the near range point (r = 4840 m) in the scene. The amplitude and phase comparison results of algorithms A and B are shown in **Figures 12** and **13**, respectively. Similar conclusion can be drawn that the amplitude error and phase error of algorithm A and B are very small, which verifies the effectiveness of algorithm A. **Table 4** shows the imaging quality evaluation results of algorithms A and B at near range (r = 4840 m), center range (r = 5140 m) and far range (r = 5440 m), including impulse response width (IRW), peak sidelobe ratio (PSLR) and integral sidelobe ratio (ISLR).

**Figure 11.**

*Amplitude comparison results of algorithm A and B (the blue line represents algorithm a and the red line represents algorithm B) (a) azimuth cut. (b) Range cut. The point target is located in the scene, and the close range R = 4840 meters.*

**Figure 12.**

*Phase comparison results of algorithm A and algorithm B. (a) Azimuth cut. (b) Range cut. The point target is located in the scene, and the close range R = 4840 meters.*

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

**Figure 13.** *Flowchart of medium trajectory error and attitude jitter at squint angle.*


**Table 4.**

*The focus performance of the point target response of algorithm A and B.*
