# A Velocity-Aware Handover Trigger in Two-Tier Heterogeneous Networks
## Overall Handover Procedures  
### Allan Variance and RSS  
The Allan change was presented 50 years ago for investigating the security of frequency standards. Due to its metrological characteristics, it has been utilized as part of accelerometry, geophysics, geodesy, astrophysics, and even finance. For example, AVAR can segregate distinctive sorts of noise described by various power laws into spectral densities. Along these lines, the principal application of AVAR for time series analysis is in deciding the signal scatter level, and in spectral analysis it primarily aims to recognize the predominant noise type in a time series. Noise assessment in physical estimation time series is a critical measure of statistical attributes and overall quality. Among the best approaches to deal with investigating noise estimation (scatter) we have AVAR, which was initially introduced to measure how much the frequency of an oscillator changes from one period \(\tau\) to the next. Since then, AVAR has become an intense measurable technique for time series analysis, especially for the examination of geodetic and astronomical observations. To date, it has only been used for time arrangement investigation. Accordingly, in our paper we will apply AVAR to RSS to recognize any change in velocity. As shown by [18] AVAR has the capacity to recognize colored noise related to RSS and can assist in the identification and characterization of these components. The AVAR is a fluctuation for two-sample \(N_{m}=2\), where \(X(t)\) is the signal to be examined. Utilizing the IEEE-prescribed definition of frequency in view of the endpoint phases [19], the AVAR at various times \(t_{m}\) can be written as:  
\[\sigma_{\tau}^{2}=\frac{1}{2}E\{(\overline{X}(t_{m}+\tau)-\overline{X}(t_{m}) )^{2}\}=\frac{1}{2}E\{(\overline{X}_{m+1}-\overline{X}_{m})^{2}\} \tag{1}\]  
where  
\[\overline{X}_{m}=\frac{1}{\tau}\int_{t_{m}}^{t_{m}+\tau}X(t)dt \tag{2}\]  
and \(\overline{X}_{m+1}\) and \(\overline{X}_{m}\) are two successive integrations over a duration \(\tau\). Since the RSSs are considered observations points succeeding one another in time, this equation will be represented in the discrete case taking \((X_{j})_{j=1,..N}\) as a stochastic process whose realizations \(X_{j}\) are made at constant time intervals \(\tau_{0}\) (see Figure 1[20]) and \(m=\frac{\tau}{\tau_{0}}\). If we develop the empirical mean which is in fact an estimator of the expectation, the estimate of the variance of Allan used in this work is:  
\[\sigma^{2}(\tau)=\frac{1}{2*(N-2(\frac{\tau}{\tau_{0}}))}\sum_{(\overline{X}_ {k}+\frac{\tau}{\tau_{0}},\frac{\tau}{\tau_{0}}}-\overline{X}_{k,\frac{\tau}{ \tau_{0}}})^{2} \tag{3}\]  
Our study is based on the standard wireless RSS signal model given by:  
\[P_{y}(t)=PL_{0}-10\gamma\log(\frac{d}{d_{0}})+X_{g} \tag{4}\]  
where \(PL_{0}\) is the path loss at the reference distance \(d_{0}\), \(\gamma\) is the path loss exponent, and \(X_{g}\) is a normal (or Gaussian) random variable with zero mean reflecting the attenuation (in decibels) caused by flat fading. In our case we suppose that there is no fading, and therefore the distance is the only term varying with time. As presented in Figures 2 and 3 the results of the AVAR of the RSS sample can allow the detection of a change in velocity. In order to detect this variation, it was decided to calculate the angle \(\Theta_{t}\) between the two tangent lines for two adjacent intervals. The decision is target-based with \(\Delta\Theta=\Theta_{t_{1}}-\Theta_{t_{2}}\geq 1\%\), where \(\Theta_{t_{1}}\) and \(\Theta_{t_{2}}\) are two successive values of \(\Theta_{t}\). Figure 1: Principle of sub-sampling by Allan variance.