Fundamentals of Time-Varying Communication Channels

Wireless communication systems, i.e., systems transmitting information via electromagnetic (radio) or acoustic (sound) waves, have become ubiquitous. In many of these systems, the transmitter or the receiver is mobile. Even if both link ends are static, scatterers — i.e., objects that reflect, scatter, or diffract the propagating waves — may move with significant velocities. These situations give rise to time variations of the wireless channel due to the Doppler effect. Non-ideal local oscillators are another source of temporal channel variations, even in the case of wireline channels. Because of their practical relevance, linear time-varying (LTV) channels have attracted considerable interest in the fields of signal processing, communications, propagation, information theory, and mathematics. In their most general form, LTV channels are also referred to as time-frequency (TF) dispersive or doubly dispersive, as well as TF selective or doubly selective. In this chapter, we discuss the fundamentals of wireless channels from a signal processing and communications perspective. In contrast to existing textbooks (e.g., [Jak74, Par92, VB03, Mol05]), our focus will be on LTV channels. Many of the theoretical foundations of LTV channels were laid in the fifties and sixties of the twentieth century. In 1950, Zadeh proposed a “system function” that characterizes an LTV system in a joint TF domain [Zad50]. Driven by increasing interest in ionospheric channels, Kailath complemented Zadeh’s work by introducing a dual system function, discussing sampling models, and addressing measurement issues [Kai59, Kai62]. A related discussion focusing on the concept of duality (an important notion in TF analysis) was provided by Gersho [Ger63]. In a seminal paper on random LTV channels [Bel63], Bello introduced the assumption of wide-sense stationary uncorrelated scattering (WSSUS), which has been used almost universally since. The estimation of channel statistics was addressed by Gallager [Gal64] and a few years later by Gaarder [Gaa68]. A fairly comprehensive coverage of the modeling of and communication over random LTV channels was pro-