Simulation parameters.

## Abstract

In the simulation of SAR raw data, it is well-known that the frequency-domain algorithm is more efficient than a time-domain algorithm, making it is more suitable for extended scene simulation. However, the frequency-domain algorithm is perhaps better suited for ideal linear motion and requires some degrees of approximations to take the nonlinear motion effects. This chapter presents an efficient simulation approach based on hybrid time and frequency-domain algorithms under certain assumptions. The algorithm has high efficiency and is suitable for the simulation of extended scenes, which demands highly computational resources. The computational complexity of the proposed algorithm is analyzed, followed by numerical results to demonstrate the effectiveness and efficiency of the proposed approach.

### Keywords

- synthetic aperture radar (SAR)
- trajectory deviations
- antenna pointing errors
- raw data simulation
- Fourier transform (FT)

## 1. Introduction

There are two major approaches to the SAR raw echo data simulation: time-domain and frequency-domain methods. For airborne SAR, the sensor platform trajectory, due to nonideal factors such as atmospheric turbulence, deviates from the linear motion state in the synthetic aperture time, which seriously affects the imaging quality if no proper compensation is made [1]. The time-domain algorithm can accurately reflect the influence of various error factors, but the pitfall is that the computational efficiency is low [2, 3, 4]. On the other hand, the frequency-domain algorithm, based on the two-dimensional frequency-domain expression of the original echo signal, uses a fast FT to realize simulation [5, 6]. This kind of algorithm delivers high efficiency to a high degree. However, it is only for simulation under ideal flight conditions.

The fast simulation of SAR imaging signals has a long history of research. The main idea of the existing methods is to realize the SAR raw data simulation in the frequency-domain with the high efficiency of the fast FT [7]. In recent years, Franceschetti’s team in Italy has conducted a systematic study on the fast simulation algorithm of SAR raw data. A simulation method based on the two-dimensional frequency domain is proposed, which assumes slow platform motion with a narrow antenna beam [8]. Then, the algorithm is modified by introducing a one-dimensional azimuth Fourier transform and range time-domain integration [9]. Reference [10] improved the traditional 2-D frequency-domain method and extended it to the squint mode. However, these two algorithms only consider the case of the tracking error or the case of the squint mode and do not consider the case of the squint with the trajectory deviations. The time-frequency hybrid simulation methods were proposed to improve SAR raw echo data processing speed [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The original raw data is usually acquired in the squint situation due to the instability of the attitude. Furthermore, the change of antenna attitude will cause the change of beam direction, which leads to the degradation of image quality. The fast echo simulation algorithm for antenna attitude change has been proposed on the premise that the amplitude of linear antenna error is far less than the beam width [21, 22, 23]. For sinusoidal antenna azimuth attitude jitter, it will cause paired echoes. Similarly, sinusoidal antenna range attitude jitter will also cause paired echoes.

In this work, we present the fast echo simulation algorithm of trajectory offset and attitude jitter error. We first attempt to extend the Fourier domain approach to account for trajectory deviations at a non-zero squint angle. Then we consider the Fourier domain approach to account for trajectory deviations and antenna beam pointing errors, given rising more problematic for airborne SAR systems. The rest of the chapter is organized as follows: The algorithm for trajectory deviations is presented in Section 2. In Section 3, SAR signal simulations are carried out considering the trajectory deviations and antenna attitude variations. Finally, conclusions are drawn to close the chapter.

## 2. Raw echo algorithm of squint airborne strip SAR with trajectory offset error

In this section, an algorithm of raw echo with trajectory error in squint is proposed. The approach adopts the Acquisition Doppler (AD) [24] geometry instead of the standard cylindrical reference system (see Figure 1).

### 2.1 Raw echo algorithm of squint airborne strip SAR with trajectory offset error

The echo generation algorithm with trajectory error in squint is deduced by rotating the trajectory error along the line of sight. The motion error in the cone coordinate system is shown in Figure 2. The transformation relationship between a cylindrical coordinate system and a conical coordinate system is as follows:

where

where [25].

Then the trajectory deviations

where

Suppose that the radar transmits the following Linear frequency modulation (LFM) signal. After heterodyne, the echo signal obtained is

where

Furthermore, by separating the factor

where

If the following conditions are satisfied:

we can obtain the azimuth FT of

where

with

### 2.2 Computational complexity of the algorithm

Let us now consider the computational complexity of the algorithm in Figure 3. Suppose the data size is

Recalled that the required computation of time domain algorithm is as follows

where

It is evident that compared with the time domain simulation method, the proposed method has higher computational efficiency.

### 2.3 Algorithm simulation verification

In what follows, we use the SAR system parameters listed in Table 1 to verify the SAR raw data simulation algorithm of non-ideal track under squint condition. Figure 4 shows the track deviation between the actual track and the ideal track. For a point target located at the near range

Nominal height | 4000 m | Range pixel dimension | 3 m |
---|---|---|---|

Midrange coordinate | 5140 m | Chirp bandwidth | 45 MHz |

Wavelength | 3.14 cm | Chirp duration | 5 μs |

Pulse Repetition Frequency | 400 Hz | Velocity | 100 m/s |

Sampling Frequency | 50 MHz | Number of azimuth samples of the raw data | 972 |

Azimuth pixel dimension | 25 cm | Number of range samples of the raw data | 416 |

Simulation method | Azimuth direction | Range direction | ||||
---|---|---|---|---|---|---|

IRW (m) | PSLR (dB) | ISLR (dB) | IRW (m) | PSLR (dB) | ISLR (dB) | |

Time domain simulation algorithm | 0.2344 | −13.1035 | −9.6665 | 2.8125 | −13.2115 | −9.6759 |

Simulation algorithm of squint with trajectory offset | 0.2344 | −13.2322 | −9.6862 | 2.8125 | −13.2160 | −9.6871 |

Finally, the extended scene is considered. The final results are given in Figure 8, where Figure 8(a) shows the image without motion compensation, Figure 8(b) displays the results by a two-step MOCO algorithm, and Figure 8(c) is the image after the motion compensation [25]. Concerning the computational efficiency, it took only about 2 min for the simulation run, as shown in Figure 8(c), on an 8-GHz Intel Core i5 personal computer.

## 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:

where

where

where

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

which satisfies the following condition:

Then

The system function

where the term.

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

Similarly, let

and

we yield

From Eqs. (24) and (28), we obtain the efficient computation of

Note that the trajectory deviations should satisfy the following condition:

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

#### 3.1.1 Computational complexity

The complexity of algorithms given in Section 3.1 increases with the order of antenna pattern decomposition. Suppose

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 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:

Nominal height | 4000 m | Range pixel dimension | 3 m |
---|---|---|---|

Midrange coordinate | 5140 m | Chirp bandwidth | 45 MHz |

Wavelength | 3.14 cm | Chirp duration | 5 μs |

Pulse Repetition Frequency | 400 Hz | Azimuth antenna dimension | 1 m |

Sampling Frequency | 50 MHz | Number of azimuth samples in the raw data | 1941 |

Azimuth pixel dimension | 25 cm | Number of range samples in the raw signals | 830 |

where

Algorithm | r (m) | Azimuth direction | Range direction | ||||
---|---|---|---|---|---|---|---|

IRW (dB) | PSLR (dB) | ISLR (dB) | IRW (dB) | PSLR (dB) | ISLR (dB) | ||

A | 4840 | 0.2266 | −5.0346 | −3.9095 | −2.7188 | −6.8351 | −5.6503 |

5140 | 0.2227 | −3.4025 | −2.2344 | 2.8125 | −12.3214 | −9.5441 | |

5440 | 0.2188 | −2.4523 | −1.2622 | 2.7656 | −8.5 | −7.2288 | |

B | 4840 | 0.2266 | −5.3933 | −4.2801 | 2.7656 | −7.1473 | −6.0101 |

5140 | 0.2227 | −3.3611 | −2.1945 | 2.7656 | −12.8208 | −9.6497 | |

5440 | 0.2188 | −2.4399 | −1.2496 | 2.7656 | −8.0237 | −6.8483 |

### 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:

where

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:

where

The above equation,

Assuming that the range Fourier transform of

The term

The azimuth Fourier transform of

where

Similarly, a two-dimensional Fourier transform

where

The amplitude approximation

then

where

Similarly, we can get an estimate of

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

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.

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

and

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

#### 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 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.

## 4. Conclusion

In this chapter, we present a fast echo simulation algorithm of airborne striped SAR with trajectory error in squint condition, the fast echo simulation algorithm of airborne stripmap SAR with antenna attitude jitter and trajectory error, and the fast echo simulation algorithm of airborne stripmap SAR with attitude jitter and trajectory error in squint condition. From the comparison between the proposed hybrid-domain echo algorithm and the time-domain algorithm, it can be seen that the phase error slice is within a reasonable range, but the hybrid algorithm runs faster than the time-domain simulation by using FT, which is conducive to the simulation of large scenes.

## Acknowledgments

The author would like to thank Professor Kunshan Chen for his support in reviewing. This work is supported by Innovation Funds of China Aerospace Science and Technology (No. Y-Y-Y-FZLDTX-20).