B.1 Some Useful Probability Distributions

Consider that h is a complex Gaussian variable, i.e. a circularly complex variable with zero-mean and variance ? ². This means that both the real and imaginary parts of h are zero-mean Gaussian variables of variance ? ². In this case, s ? h follows a Rayleigh distribution

(B.1)

with the following properties:

(B.2)

(B.3)

The CDF of s is given by

(B.4)

The random variable y ? h ² = s ² follows a X ² distribution (with two degrees of freedom)

(B.5)

Note that for small ?, we have that P[ y < ?] ? ?.

Let us now consider n i.i.d. zero-mean complex Gaussian variables h ₁ , ,h _n with variance ? ². Defining , the moment generating function of u is given by

(B.6)

and the distribution of u reads as

(B.7)

This distribution is known as the ? ² distribution with 2 n degrees of freedom (the case n = 1 reduces to (B.5)). The corresponding CDF is given by

(B.8)

(B.9)

(B.10)

Finally, let us now analyze the case when h ₁ , ,h _n are non-zero mean. Assume that the real and imaginary parts of h _k are Gaussian variables of mean ? _k and variance ? ². In this situation, follows...