# Training-Free Acoustic-Based Hand Gesture Tracking on Smart Speakers
## 2 Preliminaries  
Since the circular microphone array is employed in most commodity smart speakers, we first introduce the hand localization model based on received chirp signals at the circular microphone array. The chirp signal is widely employed in acoustic sensing [13], which can separate reflections from different positions. During the process of hand tracking, the smart speaker continuously transmits a sequence of chirp signals. The chirp signal is a sine wave whose frequency varies linearly over time. The transmitted signal can be represented as \[S_{T}(t)=cos\Big{(}2\pi f_{c}t+\pi kt^{2}\Big{)}, \tag{1}\]  
where \(f_{c}\) is the start frequency, \(k=\frac{\hbar}{T}\) is the sweep rate, \(B\) is the frequency bandwidth, and \(T\) is the chirp duration. After the transmitted signal is reflected from the hand, the received signal with delay and attenuation can be obtained, which can be represented as  
\[S_{R}(t)=acos\Big{(}2\pi f_{c}(t-\tau)+\pi k(t-\tau)^{2}\Big{)}, \tag{2}\]  
where \(a\) is the attenuation factor, and \(\tau\) is the Time-of-Flight (ToF) of the signal reflected by the hand. Then, the transmitted and received signals are processed to generate the mixed signal, which is a complex signal with the In-Phase component \(I(t)=S_{R}(t)\cdot S_{T}(t)\) and Quadrature component \(Q(t)=S_{R}(t)\cdot S_{T}^{*}(t)\). \(S_{T}^{*}(t)=sin\big{(}2\pi f_{c}t+\pi kt^{2}\big{)}\) is the 90\({}^{\circ}\) phase-shifted transmitted signal. By applying the formula \(cos\alpha\cdot cos\beta=\frac{1}{2}(\cos(\alpha+\beta)+\cos(\alpha-\beta))\) and a low-pass filter, the mixed signal can be combined as  
\[S_{M}(t)=I(t)+j\cdot Q(t)=\frac{1}{2}ae^{i2\pi\tau(kt+f_{c})}. \tag{3}\]  
We can analyze the TOF \(\tau\) to obtain the hand position information, i.e., the distance and angle. If the distance from the hand to the center of the circular array is \(d\), the TOF of the signal at this path is \(\frac{2d}{c}\), where \(2d\) is the round-trip distance, and \(c\) is the speed of sound. Suppose that there are \(M\) microphones distributing equally at the circumference as shown in Figure 1. The angle between the \(m^{th}\) microphone and the first microphone can be calculated by \(\varphi(m)=2\pi\cdot\frac{m-1}{M}\). Therefore, the TOF \(\tau(m)\) of the received signal at the \(m^{th}\) microphone can be calculated by  
\[\tau(m)=\frac{2d-rcos(\varphi(m)-\theta)}{c}, \tag{4}\]  
where \(r\) is the radius of the circle, and \(\theta\) is the incidence angle of the reflected signal. Then, we substitute \(\tau(m)\) into (3), and the mixed signal can be rewritten as  
\[\begin{split} S_{M}(t_{n},\ m)&=\frac{1}{2}ae^{i2 \pi(2d-rcos(\varphi(m)-\theta))(kt_{n}+f_{c})/c}\\ &=\frac{1}{2}ae^{i\phi(t_{n},\ m)}\end{split}, \tag{5}\]  
where \(t_{n}\) is the \(m^{th}\) sampling timestamp, and \(\phi(t_{n},\ m)\) is the phase of the mixed signal. Therefore, we can obtain distance \(d\) and angle \(\theta\) from the mixed signal to represent the hand position information.  
In practice, the signal is not only reflected by the hand but also by ambient objects, including static objects (e.g., walls and furniture) and dynamic objects (e.g., other humans). Thus, the mixed signal can be viewed as a superposition of reflections from \(L\) paths  
\[\begin{split} S(t_{n},\ m)&=\frac{1}{2}{\sum_{l=1 }^{L}a_{l}}e^{i2\pi(2d_{l}-rcos(\varphi(m)-\theta_{l}))(kt_{n}+f_{c})/c}\\ &=\frac{1}{2}{\sum_{l=1}^{L}a_{l}}e^{i\phi_{l}(t_{n},\ m)}\end{split}, \tag{6}\]  
where \(a_{l}\), \(d_{l}\), \(\theta_{l}\) and \(\phi_{l}(t_{