7.1 Random Variables Related to Mobile-Radio Signals

A signal s ₀( t) received at the mobile unit can be expressed:

(7.1)

The long-term-fading factor m( x) is extracted from s ₀( t) and the resultant can be expressed:

(7.2)

where r ₀( t) and ?( t) are the envelope and phase terms, respectively. The characteristics of r ₀( t) have been discussed in Chap. 6. The phase and time derivative of , is the random FM that is described in this chapter.

Assume that a _j( t) is the jth wave arrival, then s( t) represents the sum of all wave arrivals, as has been shown in Eq. (6.10):

(7.3)

where X ₁ and Y ₁ are as defined in Eqs. (6.11) and (6.12), respectively.

Hence ? ₁( t) can be defined:

(7.4)

The terms X ₁ and Y ₁ of the signal s( t) are two independent Gaussian variables with zero mean and a variance of ? ². This means that:

(7.5)

and

(7.6)

But when two signals s ₁ = s( t) and s ₂ = s( t + ?), expressed

(7.7)

and

(7.8)

are correlated, then E[ X ₁ X ₂] and E[ X ₁ Y ₂] are not necessarily zero. Consequently, it is first necessary to find the covariance...