# A Waveform and Velocity Ambiguity Resolution Method for Corner Radar
## 2 Materials and Methods  
Presently, TDMA modulation dominates the automotive radar market; however, chip manufacturers are transitioning towards developing chips equipped with DDMA modulation, signaling the forthcoming prominence of DDMA [15] in the next generation. The widespread adoption of DDMA antennas notably enhances radar detection range. Channel separation in DDMA is facilitated through adjustments in the transmitter phase across different transmitter antennas via phase shifters. Notably, prior to azimuth estimation, channel separation is unnecessary, allowing range and velocity estimation under DDMA to reference the TDMA method. Moreover, the velocity ambiguity resolution method proposed in this paper remains effective for velocity estimation under DDMA. Under the assumption of radar stationarity and a target moving toward the radar at velocity v, Figure 1 illustrates the time-frequency relationship between the transmit and receive signals [16]. \(B\) represents the chirp transmit bandwidth, \(T_{c}\) denotes the chirp period,\(f_{0}\) signifies the center frequency, \(\tau_{0}\) represents the echo time delay, \(f_{d}\) denotes the Doppler frequency, and \(T_{s}\) stands for the sampling time length. The FM slope, denoted as \(\mu\), is expressed as the ratio of \(B\) to \(T_{u}\), where \(T_{u}\) signifies the rising edge time. The received and transmitted signals undergo mixing and filtering processes to yield the intermediate frequency (IF) signal  
Figure 1: FMCW radar transmitting and receiving signal model. \[f_{IF}=\frac{2\mu r}{c}-\frac{2 vf_{0}}{c}=\mu\tau_{0}-f_{d} \tag{1}\]  
The repetition frequency is denoted as PRF, where PRF equals 1 divided by \(T_{c}\). The input frequency for the FFT (Fast Fourier Transform) is \(F_{D}\). Consequently, the range of \(f_{d}\) spans from \(-F_{D}/2\) to \(F_{D}/2\). In scenarios where the object's velocity exceeds the permissible range, resulting in \(f_{d}\) falling outside this interval, velocity ambiguity arises. In such cases, the Doppler frequency is denoted as \(f_{D}\):  
\[f_{D}=f_{d}+k\cdot\frac{1}{T_{c}}=f_{d}+k\cdot F_{D},\ k\in Z,f_{d}\in\left[- \frac{F_{D}}{2},\frac{F_{D}}{2}\right) \tag{2}\]  
where \(k\) represents the ambiguous multiple. Additionally, there exists the following relationship between the radar's range resolution \(\Delta R\), maximum unambiguous range \(R_{max}\), velocity resolution \(\Delta V\), and maximum unambiguous velocity \(V_{max}\)[17]:  
\[\Delta R=\frac{c}{2B} \tag{3}\]  
\[R_{max}=\Delta R\cdot N_{FFT} \tag{4}\]  
\[\Delta V=\frac{\lambda}{2N_{dFFT}\cdot T_{c}} \tag{5}\]  
\[V_{max}=\frac{N_{dFFT}}{2}\cdot\Delta V=\frac{\lambda}{4T_{c}} \tag{6}\]  
\(N_{FFT}\) denotes the number of range FFT points, while \(N_{dFFT}\) represents the number of velocity FFT points. Equation (6) illustrates the positive correlation between \(V_{max}\) and \(\Delta V\). Typically, the waveform duration is initially determined based on the desired velocity resolution. However, it often fails to satisfy the requirements for maximum unambiguous velocity. For instance, when \(T_{c}=40\) us (\(f_{0}=76.5\) GHz), \(V_{max}\) is limited to 88 km/h, which may not align with the scene's demands, necessitating the restoration of true velocity. Moreover, \(R_{max}\) and \(\Delta R\) exhibit a positive correlation, with resolution often being sacrificed to detect targets at further distances, while higher resolution is prioritized for nearby targets. Consequently, a single waveform may not suffice for all range modes. The prevailing automotive radar systems currently encompass three modes: short-range radar (SRR), middle-range radar (MRR), and long-range radar (LRR). SRR typically operates within a range of less than 60 m, MRR covers approximately 100 m, and LRR spans distances generally exceeding 200 m [18]. The waveform configuration for multi-mode operation is depicted in Figure 2, where \"R&P\" denotes receiving and processing.