# Moment-Based Parameter Estimation for the \(\Gamma\)-Parameterized TWDP Model
## 2 Related Works  
The estimation of parameters that characterize fading channel is of practical importance in a variety of wireless scenarios. It includes not only delay insensitive channel characterization and link budget calculations, but also online adaptive coding/modulation for which the estimation of parameters must be both accurate and prompt [15]. In that sense, inaccurate estimation of channel parameters leads to a non-optimal channel capacity utilization, which might significantly affect highly mobile communication systems such as vehicular-to-vehicular and vehicular-to-infrastructure. Accordingly, different approaches in estimation of TWDP parameters are proposed to make a deal between their computational complexity and estimation accuracy. These approaches usually assume that the value of parameter \(\Omega\) can be directly estimated from the data set as a second moment [10], so the estimation problem has become focused on determination of values of \(K\) and \(\Delta\). Among the investigated approaches used to estimate parameters of \(\Delta\)-based parameterization, the distribution fitting approach is used for measurements performed in air-craft and buses at 2.4 GHz [3], while the maximum likelihood procedure (ML) is used for measurements performed at 60 GHz in the indoor environment [11] and in vehicular-to-vehicular propagation scenarios [10]. However, it is shown that both approaches are computationally very complex and inappropriate for online applications. Accordingly, the moment-based approach is considered in [4; 7; 12], as a compromise between estimator's complexity and its accuracy. Thereat, in [7], estimators are derived only as conditional expressions which can not be used for practical estimations in which both parameters are unknown. To overcome the issue, the exact joint estimators of TWDP parameters are derived in [4] as computationally simple expressions. However, for certain combinations of parameters \(K\) and \(\Delta\), the estimator of parameter \(\Delta\), \(\hat{\Delta}\), derived in [4], provides physically unsubstantiated results, which can be demonstrated by applying ([4], Equation (16)) on Monte Carlo simulated samples. Accordingly, for each combination of \(K\) and \(\Delta\), where \(K=\{1,3,10,30\}\) and \(\Delta=\{0,0.1,0.2,\ldots,1\}\), 500 realizations of the TWDP process with \(N=10^{4}\) i.i.d. samples are generated. The estimate of parameter \(\Delta\) from \(j\)-th realization, \(\hat{\Delta}_{j}\), is obtained by ([4], Equation (16)) and, for each combination of \(K\) and \(\Delta\), sample mean value \(\hat{\Delta}_{mean}\) is calculated as \((1/500)\sum_{j=1}^{500}\hat{\Delta}_{j}\). The results are presented in Figure 1.  
From Figure 1, it can be observed that, for large values of \(\Delta\) (i.e., for \(\Delta\approx 1\)), \(\Delta\) estimates may exceed 1 regardless of the value of parameter \(K\). However, according to definition of \(\Delta\):  
\[(1-\Delta)=\frac{(V_{1}-V_{2})^{2}}{V_{1}^{2}+V_{2}^{2}} \tag{5}\]  
\[(1-\Delta)\geq 0,\text{ for }(V_{1},V_{2})\in\mathbb{R}^{2} \tag{6}\]  
the parameter has to be lower or equal to one (\(\Delta\leq 1\)) for any \((V_{1},V_{2})\in\mathbb{R}^{2}\). Accordingly, estimates of \(\Delta\) greater than 1 are physically unsubstantiated and can not be used to gain any insight into the relation between \(V_{1}\) and \(V_{2}\). Presented results related to sample means, when \(\Delta\) is in the vicinity of 1, are also in contrast with underlying physical mechanisms. Namely, as magnitudes of two dominant waves become approximately the same (\(V_{1}\approx V_{2}\)), i.e., when \(\Delta\approx 1\), it gets harder to discern the two rays [16]. However, Figure 1 shows that error in estimation of \(\Delta\) (expressed by the difference between \(\hat{\Delta}_{mean}\) and \(\Delta\) and the dispersion of estimated values) in the vicinity of 1 is smaller than those obtained for moderate values of \(\Delta\).