On the signal-to-interference ratio of CDMA systems in wireless communications

Let {sij: i,.j=1,.2,..} consist of i.i.d. random variables in . with Es11=0, E|s11|2=1. For each positive integer N, let sk=sk(N)=(s1k,.s2k,..,.sNk)T, 1.k.K, with K=K(N) and K/N.c>0 as N... Assume for fixed positive integer L, for each N and k.K, .k=(.k(1),..,..k(L))T is random, independent of the sij, and the empirical distribution of (.1,..,..K), with probability one converging weakly to a probability distribution H on .L. Let .k=.k(N)=(.k(1)skT,..,..k(L)skT)T and set C=C(N)=(1/N).k=2K .k .k*. Let .2>0 be arbitrary. Then define SIR1=(1/N).1*(C+.2I).1 .1, which represents the best signal-to-interference ratio for user 1 with respect to the other K.1 users in a direct-sequence code-division multiple-access system in wireless communications. In this paper it is proven that, with probability 1, SIR1 tends, as N.., to the limit ..,.'=1L..1(.).1(.')a.,.', where A=(a.,.') is nonrandom, Hermitian positive definite, and is the unique matrix of such type satisfying A=(cE...1+..A.+.2IL).1, where ...L has distribution H. The result generalizes those previously derived under more restricted assumptions.