Analysis of the High Frequency Vibration on Radar Imaging in the Terahertz Band

Abstract: High frequency vibration of the platform or the targets is an inevitable problem in the field of radar imaging, and it has significant effects on the image quality. A theoretical model of the high frequency vibration of the platform in the turntable imaging mode is established in this paper, and it shows that effects of the high frequency vibration in the terahertz band are more obvious than that in the microwave band. In addition, a 0.22 THz imaging radar system is introduced and imaging experiments on a corner reflector are carried out in this paper to verify the theory analysis.


Introduction
Terahertz (THz) waves usually refer to electromagnetic waves with frequencies between 0.1-10 THz.The terahertz band lies between the millimeter wave and infrared, which is a transitional band from electronics to photonics [1,2].With breakthroughs in the terahertz sources, signal detectors and other devices in recent years, the terahertz radar technology has developed rapidly, and the terahertz synthetic aperture radar (SAR) and inverse synthetic aperture radar (ISAR) imaging has been receiving more and more attention [3,4].The high frequency vibration of the platform often considered in the microwave band radar imaging is still problem in the terahertz band, and it may be even more obvious.The high frequency vibration in radar applications mainly includes: the high frequency vibration of the target while moving in the ISAR field, the high frequency vibration of the carrier aircraft and vehicles in the SAR field, or both.However, no matter in the SAR field or in the ISAR field, the essential effect of the high frequency vibration is a modulation of the echo phase, which will result in the deterioration of the imaging resolution.The high frequency vibration of the platform or the targets has been extensively studied and several suppression methods are discussed [5][6][7].However, researches on this problem in the terahertz band are insufficient.
A theoretical model of high frequency vibration of the platform in the turntable imaging is established in this paper, and its effects on imaging quality are analyzed in detail.After that, a 0.22 THz imaging radar system is adopted to verify the theory analysis.The paper is organized as follows: the motion model and the echo model of a scattering center in the turntable imaging mode are established in section II, and expressions of the range profile and the ISAR image are obtained through the theoretical deduction.In section III, a 0.22 THz imaging radar system is introduced and experiments on a corner reflector and a plane model are presented.The resolution deterioration induced by the high frequency vibration is analyzed and a phase autofocus algorithm is introduced to reduce the effects of the high frequency vibration.The conclusions are presented in section IV.

The echo model
The paper takes the turntable imaging mode as example to build the motion model and the echo model, and they are also applicable to SAR and ISAR mode due to the relativity of motions.It is generally known that SAR/ISAR achieves high range resolution by the large bandwidth of the transmitting signal, while the high azimuth resolution depends on the Doppler effect caused by a relative movement between radar and the target.Suppose there is a scattering center on the target located at The imaging radar mentioned in this paper utilizes frequency modulated continuous wave (FMCW) signal.Its bandwidth of the transmitting signal is B , the carrier frequency is c f ,the frequency-sweep period is p T , the chirp rate is γ .Then the transmitting signal can be expressed as: where t and m t represent the range fast-time and azimuth slowtime respectively.
Suppose the initial distance between the radar and the target is 0 R , and the vibration of the platform can be viewed as a simple harmonic motion.The actual distance considering the vibration is: where v a and v f are the amplitude and frequency of the vibration respectively.In general, it can be called high frequency vibration when the synthetic aperture time).The expression of the echo signal is: The transmitting and receiving signal possess large product of time-width and bandwidth, which requires high performance for hardware if sampling and processing directly according to Nyquist sampling theorem.Consequently, the dechirp method is often adopted, that is, mixing the echo signal with a reference signal, which usually refers to the echo signal of a target located at the reference distance ref R and it can be expressed as: The intermediate frequency signal after dechirp is : where

Resolution analysis
The range profile can be obtained by the Fourier transform of Equation ( 5) to the range fast-time: .The last two phase terms in Equation ( 6) are the residual video phase (RVP) term and the range skew term respectively, and they are easy to be compensated because the range profiles are Sinc functions with very narrow widths.After phase compensation, the Equation ( 6) can be rewritten as: According to characteristics of the Sinc function, it has a peak at , and its 3 dB width is 0.886 / p T .Consequently, the range resolution of the radar system is: It could be seen from the range profile in Equation ( 7) and the range resolution in Equation ( 8) that: vibration of the platform or the targets has no effect on the range resolution, but a position modulation of the range profiles, and the modulation amplitude and the modulation frequency just equal to the amplitude and frequency of the vibration.The range resolution of a microwave radar is commonly on the decimeter level, and vibrations on the level of millimeter or even micrometer are usually can be neglected.However, the terahertz radar can reach a range resolution of centimeter or even millimeter level due to the large bandwidth of the transmitting signal.In this situation, compensation or correction operations are often necessary if high quality imaging results are required.Considering that =, the Equation ( 7) can be written as: If no phase term associated with vibration existed, the ISAR image can be written as: where is the part unrelated to m t in Equation (9).The azimuth resolution is: If the phase term associated with vibration is considered, it can be decomposed as follow according to the Jacobi-Anger expansion.
( ) where ( ) n J ⋅ is the n-order Bessel function.The ISAR image can be expressed: As can be seen from the comparison between Equation (10) and Equation ( 13), the azimuth expression is no longer a Sinc function, but the sum of series of Sinc functions modulated by Bessel functions because of the influence of the vibration.The azimuth resolution in this situation will deteriorate, and the deterioration factor is proportional to the carrier frequency.So the deterioration factor in the terahertz band is much greater than that in the microwave band.Besides that, interference signals in azimuth will emerge.

The terahertz radar system
The imaging radar system adopted in this paper is based on the FMCW principle and has 221 GHz of carrier frequency with a synthetic bandwidth of 12.8 GHz, thereby realizing a 1.17 cm theoretical range resolution.The 0.22 THz radar system consists of five modules: the signal source, transmitting and receiving chains, cone-shaped horn antennas, the intermediate frequency (IF) module, and the data collection module.The terahertz signal is transmitted by a horn antenna after 16 times frequency multiplication of a Ku-band (13.45-14.25 GHz) sweeping generator in the transmitting chain, and the transmitting power is greater than 3 mW.The differential frequency in sweeping generators between the transmitting chain and the receiving chain is 60MHz.Through harmonic mixing, the received terahertz signal is down-converted to IF for super heterodyne reception.A photo of the o.22 THz radar system's transmitting/ receiving front-ends is shown in Figure 2.

Imaging experiments
The target is placed on a turntable at a distance of 4m from the radar system, and the reference distance is 4m as well.The angular velocity of the turntable is 30°/s.The synthetic aperture time m T is 0.1s, which realizes a 1.3 cm theoretical range resolution.The radar system and the targets are placed in an absorbing chamber during the experiments.The experiment scene is shown in Figure 3, and the imaging results of a corner reflector at different vibration situations are shown in Figure 4.For the above problem, the phase gradient autofocus (PGA) algorithm is adopted in this paper.PGA algorithm has been widely used in the SAR/ISAR field since it was proposed in 1989 due to its good robustness and fast convergence [8,9].The azimuth resolutions of a corner reflector vibrating at different vibration situations in Figure 4 after PGA process are shown in Figure 6, and it is obvious that the azimuth resolutions after PGA are close to that when the platform doesn't vibrate.
Furthermore, experiments on a plane model are carried out and the results are shown in Figure 7 and Figure 8.The platform vibrates on the micrometer level and millimeter level in Figure 7 and Figure 8 respectively.The results show that: for multiple scattering center targets, the performance of the PGA algorithm is very limit in the terahertz band and other high performance algorithms are a subject urgent to be studied presently.

Conclusion
The high frequency vibration is very common in the radar imaging field, and it effects more significantly in the terahertz band.In this paper, a theoretical model of the high frequency vibration of the platform in the turntable imaging mode was established, and the effects on imaging resolutions were analyzed in detail.In addition, a 0.22 THz imaging radar system is introduced and imaging experiments on a corner reflector and a plane model were presented to verify the theory analysis.Finally the PGA algorithm commonly used in the microwave radar was tried in the experiments.

Figure 3 .
Figure 3.The experiment scene

Figure 4 .Figure 5 .
Figure 4.The imaging results of a corner reflector: (a) is the imaging result when the platform doesn't vibrate.(b) is the imaging result when the platform vibrates on the micrometer level.(c) is the imaging result when the platform vibrates on the millimeter level.