# Nonlinear Analysis of Built-in Sensor in Smart Device under the Condition of Voice Actuating
## 3 Nonlinear Characteristics Analysis  
Chaotic time series analysis is one of the most important methods in nonlinear dynamics theory, and phase space reconstruction is also the main goal which would be achieved. Usually, the basic target of phase space reconstruction is to demonstrate the inherent nonlinearity of a system by studying the chaotic attractor and characteristic parameters of a specific variable in the phase space. So, how to find the most accurate attractor to effectively reconstruct the phase space for this particular built-in acceleration system? Assuming that we obtain a univariate time series output signal of accelerometer through the observation or experiment as \(\{Xn\}=X_{1},X_{2},\ldots,X_{N}\) a state variable of the reconstructed phase space can be expressed as:  
\[X_{k}=\{X(k),X(k+\tau),\ldots,X(k+(m+1)\tau)\},\;k=1,2,3,\ldots,N \tag{1}\]  
Thus, an \(m\) dimensional state space \(Rm\) could be formed. Moreover, the mathematical model is to be established in the state space \(Rm\):  
\[Xn+1=G(Xn) \tag{2}\]  
In [9; 10], [PERSON] and [PERSON] proved that the reconstructed state space \(Rm\) has the same topological structure and geometric features with the original chaotic dynamic system, provided that the parameters \(m\) and \(\tau\) are properly selected. It also means that all these differential and topological invariants in the original chaotic dynamic system could be calculated in the reconstructed state  
Figure 4: Two chirp signals with different starting frequencies.  
Figure 5: Time-frequency diagram of response signals to two chirp signals.  
space. Furthermore, the dynamic system can be analyzed and predicted by establishing a specific mathematical model in the state space \(Rm\). It is clear that the correct determination of delay time \(\tau\) and embedding dimension \(m\) is the key for phase space reconstruction. In the modern chaos theory, it is generally believed that these two parameters are dependent on each other [11; 12]. Therefore, we could obtain the attractor of a built-in accelerometer system and complete phase space reconstruction just after the two parameters have been simultaneously determined.  
It can be known from the delay embedding theorem [13; 14] that the delay time \(\tau\) can be randomly selected if the time series is infinitely long, noiseless, and infinitely accurate. However, in a general practical application, the accelerometer output signal would be a time-limited sequence with noise pollution and measurement error. Thus, the delay time \(\tau\) can only be chosen depending on experience. Basically, the principle of it is to give \(Xn\) and \(Xn+\tau\) a certain degree of independence to process them as independent coordinates during the reconstruction of phase space. If \(\tau\) is small \(Xn\) and \(Xn+\tau\) would get too close to effectively differentiate in the phase space. Moreover, they are also very close to the \"diagonal\" in the reconstructed phase space, which means a large amount of redundant information would be carried by the attractor. Thus, the phase space would be disordered and mixed without rules. On the contrary, the attractor track would be projected in two completely opposite directions because of the minor relevancy between \(Xn\) and \(Xn+\tau\), which means that the actual evolution rule of the trajectory in the phase space cannot be reflected.  
Referencing to the C-C algorithm [15; 16], we define \(\tau_{w}=(m-1)\tau_{d}\) as the delay time window while \(\tau(\tau=t)\) is the delay time. So, the correlation integral can be represented as follows:  
\[C_{s}^{m}(m,N,r,t)=\frac{2}{M(M-1)}\sum_{1\leq i<j\leq M}\theta\{r-\|\,X_{i}-X _{j}\