# Motion Compensation for SAA FMCW Radar Based on Specific Switching Scheme
## 1 Introduction  
Switch antenna array (SAA) frequency modulated continuous wave (FMCW) radar has received extensive attention due to its low complexity architecture and small size. It has been widely used in scenes such as car collision avoidance and obstacle monitoring [1; 2]. Figure 1 shows the system architecture of the SAA FMCW radar. The transmitting frontend uses a single antenna, and the receiving frontend adopts an N-elements uniform linear array (ULA), which shares a receiving channel through a single-pole multithrow switch.  
In the conventional switching scheme, the receiving arrays are sequentially switched in order from 1 to \(N\)[3]. For the stationary target, the phase difference between adjacent elements is expressed as  
\[\varphi_{d}=\beta d\sin\theta \tag{1}\]  
which is the same as the multichannel antenna array, where \(\beta=2\pi/\lambda\) is the wave propagation constant, \(\lambda\) is the wavelength in free space, \(d\) is the spacing of ULA, and \(\theta\) is the direction-of-arrival (DoA). For the nonstationary target, the motion introduces an additional phase difference between adjacent elements, which is expressed as in the work by the authors of [4].  
\[\varphi_{v}=\beta 4 vT \tag{2}\]  
where \(v\) is the target's velocity, \(T\) is the up-chirp ramp duration, and the radar's pulse repetition interval (PRI) is \(2T\). The target moves with constant radial velocity and its tangential velocity can be neglected [5]. The phase distribution of SAA for the moving target is shown in Figure 2.  
The motion-induced phase \(\varphi_{v}\) will lead to erroneous DoA estimation [4]. Therefore, scholars have proposed some motion compensation methods to solve this problem. [PERSON] proposed a two-cycle switching scheme to achieve DoA estimation [3]. However, this method is only applicable to scenes where the target moves at a slower speed. Some random switching methods were proposed in works by the authors of [5; 6; 7] to realize motion compensation. However, these methods need to combine the target scattering characteristics and optimization algorithms to solve the optimal switching scheme, consuming time and computing resources, making application to scenarios with high real-time requirements difficult. [PERSON] [8] successfully estimated direction and velocity using two-dimensional MUSIC signal processing and [PERSON] [9] introduced the time-invariant steering vector (TISV) to achieve multiparameter (i.e., range, velocity, and direction) estimations. However, this approach suffers from an ambiguous velocity estimation problem. [PERSON] [10] proposed a motion-induced phase elimination method based on double-time switching scheme. Some scholars have proposed compensation methods for time division multiplexing (TDM) multiple-input-multiple-output (MIMO) radars. [PERSON] investigated a modified Doppler processing algorithm to achieve motion compensation [11]. The phase shift introduced by movement can be calculated using signals of redundant antenna elements [12; 13]. [PERSON] and [PERSON] [14] used the echo signal of the first antenna element to estimate a target's velocity, and then used this velocity to construct a term to compensate for the phase response of the moving target. Additionally, [PERSON] [15] found that the phase difference introduced by the target motion can be ignored when the transmit antenna was quickly switched at a single frequency. However, these methods cannot be applied to SAA radars.  
This paper proposes a specific switching scheme to compensate for the motion-induced phase for SAA FMCW radar. The receiving array is switched in the order of \(1,4,2,3,5,6,7,\ldots,N\)  
Figure 1: Block diagram of switch antenna array (SAA) frequency modulated continuous wave (FMCW) radar.  
Figure 2: Phase distribution for moving target.  
The motion-induced phase is extracted through a simple mathematical operation, and then it is used to compensate the phase of SAA. The proposed method only requires basic signal processing techniques and avoids complex optimization algorithms.