# Moment-Based Parameter Estimation for the \(\Gamma\)-Parameterized TWDP Model
## 1 Introduction  
Small-scale fading severely degrades performance of wireless communication systems [1]. Thus, in order to design highly reliable and efficient transceivers for communication in deep fading conditions, it is of profound significance to have an accurate and tractable fading model [2, 3]. Traditionally, the signal affected by small-scale fading in nonline-of-sight (NLOS) environments has been modeled as a sum of many diffuse components and described by Rayleigh distribution. Withal, the Rician fading model has been used for the description of received signal variations in better-than-Rayleigh channel conditions, since, except for many diffuse components, it also assumes the presence of one dominant specular component [4]. However, in some multipath sparse channels, traditional fading models fall short of accurately describing small-scale variations of the received complex envelope [4], which makes them nonviable for accurate modeling of all propagation conditions [5]. The aforementioned is especially pronounced in mmWave 5G communication networks equipped with directional antennas or arrays [6] as well as in wireless sensor networks deployed in cavity environments [3], where measurements have shown that the signal in multipath sparse propagation conditions may, in some cases, experience fading with characteristics worse than those in Rayleigh channels [4]. In such scenarios, the TWDP fading model--which assumes the presence of two specular components instead of just one considered by Rician model--appears to be appropriate for modeling small-scale fading effects [7]. In this regard, it is shown that TWDP distribution describes small-scale fading more accurately than conventional distributions in static sensor networks with their nodes placed within cavity environments, such as aircrafts and buses [3] and abroad a large transport helicopter [8]. It is also shown that its PDF shape significantly coincides with empirical results obtained for train-to-infrastructure wireless communication [9], vehicle-to-vehicle 60 GHz urban communication [10], and mmWave communication in indoor environments [11, 12]. Thereat, the TWDP model itself represents generalization of Rayleigh and Rician fading models and can be used for modeling both better-than-Rayleigh and worse-than-Rayleigh fading conditions. It is based on the assumption that the waves arriving at the receiver can be observed as a sum of two specular components with constant magnitudes \(V_{1}\) and \(V_{2}\) and uniformly distributed phases, plus many diffuse components treated as a complex zero-mean Gaussian random process with average power \(2\sigma^{2}\). As such, the model is conventionally parameterized by \(K\) and \(\Delta\), defined as [13]:  
\[K=\frac{V_{1}^{2}+V_{2}^{2}}{2\sigma^{2}} \tag{1}\]  
\[\Delta=\frac{2V_{1}V_{2}}{V_{1}^{2}+V_{2}^{2}} \tag{2}\]  
where parameter \(K\) (\(K\geq 0\)) characterizes the ratio of the average power of specular components to the average power of remaining diffuse components (like the Rician parameter \(K_{Ricc}=V_{1}^{2}/\left(2\sigma^{2}\right)\)) and parameter \(\Delta\) (\(0\leq\Delta\leq 1\)) characterizes the relation between magnitudes of specular components. Consequently, the average received signal power \(\Omega\) is equal to \(V_{1}^{2}+V_{2}^{2}+2\sigma^{2}\). However, as it is elaborated in [14], the definition of parameter \(\Delta\) is not in accordance with the model's underlying physical mechanisms. Namely, according to the model's definition, specular components have constant magnitudes and are propagating in a linear medium. Consequently, \(V_{2}\) can be nothing but the linear combination of \(V_{1}\), wherefore the function which characterizes the ratio between \(V_{1}\) and \(V_{2}\) has to be linear [14]. However, \(\Delta\)-based parameterization introduces a nonlinear relation between magnitudes of specular components, since \(V_{2}=V_{1}(1-\sqrt{(1-\Delta^{2})})/\Delta\).