# Hand Trajectory Recognition by Radar with a Finite-State Machine and a Bi-LSTM
## 3 Methodology
### Simple Trajectory Gesture Recognition  
#### 3.1.1 Gesture Trajectory Extraction  
The gesture trajectory parameters include distance, azimuth, and elevation angle, representing the three-dimensional trajectory in the coordinate system during the gesture movement. The process of extracting gesture trajectory involves the measurement of target range and velocity, target detection, and estimation of target azimuth and elevation angles. The specific signal processing flowchart is shown in Figure 9.  
Firstly, 2D FFT processing is conducted on the radar IF echo signal to generate a range-velocity map. The specific procedure involves arranging the IF signals from multiple pulses into a two-dimensional data matrix and conducting FFT calculations along both the sampling point and pulse accumulation dimensions. In FMCW radar, according to the principle of range and velocity measurement, targets with different distances and velocities will be distinguished and displayed at different positions on the range-velocity map after 2D FFT processing. Figure 8: Gesture recognition processing flowchart. Secondly, target detection is performed on the range-velocity map to extract the hand target, while retaining the phase information of the corresponding position on the image for the subsequent calculation of the target's azimuth and elevation angles. The specific process involves using detection methods to filter out arm information and extracting hand information. This is because the obtained range-velocity map contains both hand and arm information, which may interfere with gesture trajectory extraction. In general, the amplitude of hand signals is greater than that of the arm signals. Therefore, OS-CFAR detection can be employed for hand target detection while filtering out arm targets. The steps of OS-CFAR involve performing one-dimensional CFAR detection along the distance direction on the range-velocity map, followed by extracting the CFAR detection results along the velocity dimension. If the radiation value in the distance dimension exceeds the predefined threshold, a hand target at that distance is identified, thus obtaining target distance and velocity information. In Figure 10, the range-velocity map displays two target peaks, with the peak of higher amplitude representing the hand target and the peak of lower amplitude representing the arm target. After OS-CFAR detection, only the information related to the hand target was retained in the range-velocity map, while the information concerning the arm target was filtered out. Afterward, estimate the azimuth and elevation of the hand target to obtain its azimuth and elevation information. The specific process involves using the Capon algorithm to process the complex information retained at the distance and velocity units corresponding to the target's location. The core of the Capon algorithm is filtering time-domain echo data, suppressing frequency components and noise unrelated to the target through adaptive filters. This allows the target signal frequency to pass through without distortion, facilitating the estimation of signal power at the target frequency. The signal power \(P_{\text{Capon}}(\theta)\) at angle \(\theta\) in Capon algorithm can be expressed as  
\[P_{\text{Capon}}(\theta)=\frac{1}{\alpha^{H}(\theta)\hat{R}_{x}^{-1}\alpha( \theta)} \tag{15}\]  
Figure 10: Hand target detection results. Figure 9: Diagram for extracting gesture trajectory parameters. where \(\alpha(\theta)\) represents the steering vector and \(\hat{R}_{x}\) represents the correlation matrix of the receiving signal. Then, by stepping at a certain angle, the observed angle is scanned and the power value at the corresponding angle is calculated. Subsequently, a peak search is conducted on the power spectrum, and the angle corresponding to the peak represents the desired target angle.