Wireless Fading Channel Models: from Classical to Stochastic Differential Equations (original) (raw)

Stochastic Control 300 (DPSD). Aulin [11] presented a methodology to find the Doppler power spectrum by computing the Fourier transform of the autocorrelation function of the channel impulse response with respect to time. A different approach, leading to the same Doppler power spectrum relation was presented by Gans [13]. These STF models suggest various distributions for the received signal amplitude such as Rayleigh, Rician, or Nakagami. Models based on autoregressive and moving averages (AR) are proposed in [14, 15]. However, these models assume that the channel state is completely observable, which in reality is not the case due to additive noise, and requires long observation intervals. First order Markov models for Raleigh fading have been proposed in [16, 17], and the usefulness of a finite-state Markov channel model is argued in [18]. Mobile-to-mobile (or ad hoc) wireless networks comprise nodes that freely and dynamically self-organize into arbitrary and/or temporary network topology without any fixed infrastructure support [19]. They require direct communication between a mobile transmitter and a mobile receiver over a wireless medium. Such mobile-to-mobile communication systems differ from the conventional cellular systems, where one terminal, the base station, is stationary, and only the mobile station is moving. As a consequence, the statistical properties of mobile-to-mobile links are different from cellular ones [20, 21]. Copious ad hoc networking research exists on layers in the open system interconnection (OSI) model above the physical layer. However, neglecting the physical layer while modeling wireless environment is error prone and should be considered more carefully [22]. The experimental results in [23] show that the factors at the physical layer not only affect the absolute performance of a protocol, but because their impact on different protocols is nonuniform, it can even change the relative ranking among protocols for the same scenario. The importance of the physical layer is demonstrated in [24] by evaluating the Medium Access Control (MAC) performance. Most of the research conducted on wireless channel modeling, such as [1-4, 25, 26], deals mainly with deterministic wireless channel models. In these models, the speeds of the nodes are assumed to be constant and the statistical characteristics of the received signal are assumed to be fixed with time. But in reality, the propagation environment varies continuously due to mobility of the nodes at variable speeds and movement of objects or scatter across transmitters and receivers resulting in appearance or disappearance of existing paths from one instant to the next. As a result, the current models that assume fixed statistics are unable to capture and track complex time variations in the propagation environment. These time variations compel us to introduce more advanced dynamical models based on stochastic differential equations (SDEs), in order to capture higher order dynamics of the wireless channels. The random variables characterizing the instantaneous power in static (deterministic) channel models are generalized to dynamical (stochastic) models including random processes with time-varying statistics [27-31]. The advantage of using SDE methods is due to computational simplicity simply because estimation and identification can be performed recursively and in real time. Parts of the results appearing in this chapter were presented in [27-31]. This chapter is organized as follows. In Section 2, the general time-varying (TV) wireless channel impulse response is introduced. The TV stochastic LTF, STF, and ad hoc wireless channel models are discussed in Sections 3, 4, and 5, respectively. Link performance for cellular and ad hoc channels is presented in Section 6. Finally, Section 7 provides the conclusion.