**Wireless channels have been the subject of a large body of research,** including both analytic and empirical modeling; see [61, 151, 178] and references therein. Consider, for instance, the wireless network depicted on the left in Figure 3.5. The signals transmitted by the devices are bandlimited and can, by Nyquist sampling theory, be represented by sequences of discrete-time symbols. A commonly studied class of problems is based on an AWGN model where every output sample at nodes 2 and 3 can be written as respectively, where X1,X2, Y2,Y3,Z2,Z3 are complex random variables, huv and duv are the respective channel gains and distances between nodes u and v, and a is an attenuation exponent (see Section 2.2.1).

**Fig. 3.5 A wireless network and its graph.**

This wireless network is naturally represented by the graph shown on the right in Figure 3.5. The idea of this graph is that every node u has one channel input Xu and one channel output YU. Of course, if a node only receives or transmits then one can ignore the former or latter variable. We draw a directed edge from node u to node v if Xu contributes to Yv. The graph now permits broadcasting (X1 contributes to both Y2 and Y3) but this causes interference (X1 and X2 interfere at node 3).

**Thus wireless channels differ** in many ways from wireline channels (see Table 1.1). Additionally, there are significant time variations due to node mobility and the propagation environment. The channel changes are considered to be either slow or fast depending on many factors, for example the device velocity, bandwidth, delay constraints, and electromagnetic wave scattering and absorption. Slow channel variations might occur when the device has low velocity (laptops), the bandwidth is large so that individual symbols are short, there are tight delay constraints (voice traffic), or there is line-of-sight communication. In fact, signal strength can fluctuate rapidly even at low device velocity due to multipath propagation; this spatial phenomenon is called fading. Rapid channel variations occur when the device has high velocity (planes, trains), the bandwidth is small so that individual symbols are long, there are relaxed delay constraints (data traffic), or there is rich scattering causing multipath.

**Stated more succinctly,** a channel (or edge) is called slow fading if all the encoded symbols of each data packet traversing this channel encounter only one channel realization (often represented by a multiplicative gain, see (3.3) below). The channel is called fast fading if the encoded symbols of each data packet encounter many channel realizations. Of course, there are intermediate situations where the channel is neither slow nor fast fading. However, for simplicity we will consider only the two extreme cases.

**We next model channel time variations in more detail**. Suppose we use the channel n times, and we index the edge variables at time i, i = 1,2,…,n, as X1,i, Y2,i, h12,j, and so forth (see (3.3)). We might wish to choose the sequences {huv,i}™=1 by using electromagnetic wave propagation equations for specified geographies, and for specified device trajectories and velocities. This approach is, however, sensitive to the choice of system variables and is too complex to be useful for many scattering environments encountered in practice. Instead, we admit uncertainty and model the sequences {huv,i}™=1 as realizations of integer time stochastic processes {Huv,i}™=1. Several types of processes have been considered in the literature, and each has its own peculiarities. We opt for simple classes of models since our focus will be on cooperative strategies rather than channel characterization. These models are characterized by the marginal distributions of the Huv and the temporal correlation of the Huv,j.

**With respect to the distribution of the Huv,** we consider two models. First, the Rayleigh fading channel model assumes the signal at an antenna consists of a large number of independent randomly phased unresolvable multipath components. In this case, the Huv are independent, complex, Gaussian, zero-mean, unit variance random variables with independent real and imaginary parts with variance 1/2. We further assume that all the Huv are independent of X1,X2,Z2,Z3. In this case, the channel capacity of the individual link (u,v) does not depend on the channel phase, and it is sometimes convenient to follow [22] and use the simplified real-valued signals in which we ignore the quadrature signal components (but one should keep the underlying complex model in mind).

**In this case,** the channel gain Huv,i is a real-valued exponential random variable with expected value E[Huv,i] = 1. The actual data rates in the complex baseband channel will be double those derived under the real-valued model. This same real-valued model may also be applied to the OFDM channel model as the capacity of each subchannel is insensitive to the subchannel phase.

**An instructive second model is the uniform**-phase fading channel in which Huv = and the $uv are independent and uniform over the interval [0,2n). This model is unrealistic for wireless environments, and may seem peculiar since the capacity of the individual link (u, v) does not depend on the phase of Huv. However, as we shall see in topic 4, this model has the didactic advantages of giving simple capacity expressions and important insight for wireless networks in which multiple transmitters can derive advantages by phase aligning signals at a receiver. We remark that the name "fading" is perhaps inappropriate here because there are no signal strength variations. However, we feel that the suggestive label compensates adequately for the loss in precision.

**With respect to temporal correlation**, as mentioned above we focus on two classes of models that correspond to extreme forms of fast and slow fading. More precisely, in the fast fading model, the channel gains huv,j, i = 1,2,…,n, are chosen independently with the distributions PHuv (■). Our second model class describing slow fading has the huv ,1 drawn randomly before transmission for all (u, v), and we set huv,i = huv,1 for i = 2,…,n. In this slow fading case, the reliable communication rates one can achieve are therefore random variables, and one is usually interested in characterizing their probability distributions. This model is often referred to as block-fading or quasistatic, and it is a realistic model for shadow fading, or for systems employing time-division multiaccess (TDMA) or orthogonal frequency division multiplexing (OFDM) with TDMA symbols.

**An appropriate definition of capacity depends** on both the marginal distributions and the temporal correlations of the link gains. In addition, the capacity depends on how the transmitter and receiver observe the channel state. In the next section, we interpret the incomplete knowledge of the wireless channel state process as a constraint on a wireless device. We then return in Section 3.4 to examine capacity metrics in terms of wireless channel models and constraints on wireless devices.