Thomas L. Marzetta

Models and Preliminaries

This chapter introduces the basic signal and channel models to be used throughout the book. We use standard complex baseband representations of all signals and noise, with the implicit assumption that all signals are eventually modulated onto a carrier with frequency f c and wavelength λ = c/f c , where c is the speed of light. Also, unless stated explicitly, all Gaussian random variables are complex-valued and circularly symmetric; see Appendix A for a treatment of such variables. Single-Antenna Transmitter and Single-Antenna Receiver The wireless channel takes an input signal x ( t ), emitted by a transmit antenna, and yields an output signal y ( t ), observed at a receive antenna. The relation between x ( t ) and y ( t ) is linear, owing to the linearity of Maxwells equations. However, this relation generally is timevarying, since the transmitter, receiver, and other objects in the propagation environment may move relative to one another. Coherence Time The time during which the channel can be reasonably well viewed as time-invariant is called the coherence time and denoted by T c (measured in seconds). To relate T c to the characteristics of the physical propagation environment, we consider a simple two-path propagation model where a transmit antenna emits a signal x ( t ) that reaches the receiver both directly via a LoS path, and via a single specular reflection; see Figure 2.1(a). If both paths have unit strength, and the bandwidth of x ( t ) is small enough that time-delays can be approximated as phase shifts, then by the superposition principle the received signal is where d 1 and d 2 are the propagation path lengths defined in Figure 2.1(a). Suppose, for the sake of argument, that when the receiver is located as shown in Figure 2.1(a), d 1 / λ and d 2 / λ are integers. Then the two paths add up constructively and y ( t ) = 2 x ( t ). Next, if the receiver is displaced d meters to the right, so that we have the situation in Figure 2.1(b), the received signal will instead be The two paths add up destructively if the cosine in (2.2) is equal to zero. As shown in Figure 2.2(a), this occurs periodically for displacements d that are spaced λ /2 meters apart. The channel may be considered time-invariant as long as the receiver does not move farther than this distance, λ /2.