## 2 Linear and Rotational Doppler Effect on Hand Gesture Motion  
The common hand gestures include translational and rotational motion of the hand. The signals from air sonar with linear and rotational Doppler effect will be applied to our hand gesture recognition. Consider the constructed ultrasonic transmitter and receiver pair as an air sonar system fixed and located at the origin Q of the radar coordinate system (U, V, W) (Figure 1). The hand is described in the local coordinate (x, y, z) and has a translation and rotation with respect to the radar coordinate. A reference coordinate system (X, Y, Z) is introduced, which has the same translation as coordinate (x, y, z) but has no rotation with respect to the (U, V, W) coordinate. Suppose the hand has a translation velocity v with respect to the radar and an angular rotation velocity \(\omega\) represented in the reference coordinate as \(\omega=(\omega_{X},\omega_{Y},\omega_{Z})^{T}\). A point scatterer M on the hand, located at \(r_{0}=(X_{0},\,Y_{0},\,Z_{0})^{T}\), at time \(t=0\) will move to M\({}^{\prime}\) at time \(t\). This movement can be considered as a translation from M to M\({}^{\prime\prime}\) with velocity \(v\) and a rotation from M\({}^{\prime\prime}\) to M\({}^{\prime}\) with an angular velocity \(\omega\). At time \(t\), the range vector from the radar to the scatterer at M\({}^{\prime}\) becomes  
\[\overset{\rightharpoonup}{\text{OM}}^{\prime}=\overset{\rightharpoonup}{ \text{QO}}+\overset{\rightharpoonup}{\text{OM}}^{\prime}+\overset{\rightharpoonup}{ \text{OM}}^{\prime}=\overset{\rightharpoonup}{R_{0}}+\overset{\rightharpoonup}{v}t+ \overset{\rightharpoonup}{r}.\]  
The scalar range of \(\overset{\rightharpoonup}{\text{OM}}^{\prime}\) can be expressed as  
\[R_{t}=\parallel\overset{\rightharpoonup}{R_{0}}+\overset{\rightharpoonup}{v}t +\overset{\rightharpoonup}{r}\parallel.\]  
When the air sonar emits a continuous sinusoidal wave with a carrier frequency \(f_{0}\), the echo signal \(s(t)\) received from the scatterer on the hand at the position (\(x\), \(y\), \(z\)) is expressed as a function of \(R_{t}\). \[s(t)=\rho(x,y,z)\exp\biggl{\{}j\biggl{(}2\pi f\frac{2R_{t}}{c}\biggr{)} \biggr{\}}, \tag{1}\]  
where \(\rho(x,y,z)\) is the reflectivity function of the point scatterer M described in the target local coordinates (\(x\), \(y\), \(z\)), \(c\) is the speed of sound, and the phase of the signal is \(2\pi f(2R_{t}/c)\). The Doppler frequency shift by hand motion can be obtained by taking the time derivative of the phase as [23]  
\[f_{D}=\frac{1}{2\pi}\frac{d(2\pi f(\frac{2R_{t}}{c}))}{dt}=\frac{2f}{c}\frac{ dR_{t}}{dt}=\frac{2f}{c}(\overset{\rightharpoonup}{v}+\overset{\rightharpoonup}{ \omega}\times\overset{\rightharpoonup}{r})^{T}\overset{\rightharpoonup}{n_{m}}, \tag{2}\]  
where \(\overset{\rightharpoonup}{n_{m}}=(\overset{\rightharpoonup}{R_{0}}+\overset{ \rightharpoonup}{v}t+\overset{\rightharpoonup}{r})/R_{t}\). Figure 1: The coordinate of an air sonar and a target for micro Doppler analysis. Thus, the Doppler frequency shift included the effect of translation and rotation as well as the direction from the radar to the scatterer on the hand. Four hand gestures, namely, \"push\", \"wrist motion from flexion to extension\", \"pinch out\", and \"hand rotation\", were performed with the right hand for this study. (1) Push: As shown in Figure 2a, the push gesture involves only a translational motion of the palm. While the hand moved forward from an initial position to a stop position, the normal direction of the palm was toward the radar and the palm underwent the acceleration, near-constant velocity, and deceleration stages. The maximum Doppler frequency shift was contributed by the center of the palm because the velocity direction was normal to the direction of the radar to the center of the palm. The minimum Doppler frequency shift resulted from the corner of the moving palm.