In this section, we present a basic model for the wireless channel and show its performance to be inferior to that of the wireline channel.

The traditional wireline channel is modeled by the equation

(1.1)

where x

is the channel input. x is a complex number, referred to as symbol, representing two bits of information, i.e., it can take up to four different values according to the mapping (in general, x may be chosen to represent more than two bits of information):

(1.2)

We assume that all the above possible realizations of x are equally probable.

n is the channel noise, accounting for the thermal noise induced by different parts of the receiver. n is modeled as a zero mean, complex Gaussian random variable with variance σ2 per dimension, i.e., the real part of n and the imaginary part of n are zero mean, statistically independent Gaussian random variables with variance σ2. Note that

(1.3)

"E" and "*" denote statistical expectation and complex conjugate, respectively. The channel signal-to-noise-ratio (SNR) is given by

(1.4)

The receiver observes the channel's output y and decides which symbol, out of the four possible ones, was sent. The receiver tries to produce decisions with the best possible reliability. We measure reliability through error probability. Let x(y) be the receiver decision. The error probability, denoted here by P_r{ε}, is the probability that x(y) is different than x,

(1.5)

What receiver should we be using in order to achieve minimal error probability? The optimal receiver [2] decides that symbol x was sent, if the Euclidian distance between x and y is the smallest among all possible distances (total of four in our case). The optimal receiver, referred to as the maximum likelihood (ML) receiver, achieves error probability [2] satisfying

(1.6)

Throughout the paper, we provide upper bounds on the error probability. However, these bounds are tight for the mid-to-high SNR range [2], which is the interesting SNR range when targeting reliable, high-throughput communication. As we can see, for the wireline channel, the error probability decreases exponentially fast with the SNR. Does this excellent behavior remain intact when transmitting over wireless channels? Unfortunately, the answer is no. The fading, as shown next, dramatically increases the error probability.

The wireless channel model is similar to the wireline channel model, but with the input amplitude modified to account for the fading. The wireless channel is modeled with the equation

(1.7)

where h represents the fading. We consider an environment in which there is no line of sight (NLOS) between the transmitter and the receiver. For this kind of environment, h is modeled as a zero mean, complex Gaussian random variable with variance 0.5 per dimension. Suppose for a moment, that the fading h is a fixed deterministic number rather than a random variable. In that case, the channel SNR would be given by

(1.8)

and, as for the wireline channel, the error probability satisfies

(1.9)

Let

(1.10)

then

(1.11)

The above result accounts for the error probability for a given realization of the fading h. In order to obtain the error probability P_r{ε} we must average the conditioned error probability (1.11) with respect to the probability low of z

(1.12)

f(z) is the probability density function of z. Since h is Gaussian distributed, z is the squared root of the squared sum of two independent Gaussian random variables, which means [2] z is Rayleigh distributed

(1.13)

Averaging (1.11) with respect to the Rayleigh distribution we have

(1.14)

It should be clear now how severe the damage caused by the fading is: instead of having an error probability that decreases exponentially fast with the SNR (1.6), we have an error probability which is only inversely proportional to the SNR. In the next section, we show how by using multiple antennas the situation can be rectified to some extent.