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 C with Es11 = 0, E|s11| = 1. For each positive integer N let sk = sk(N) = (s1k, s2k, . . . , sNk) , 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 αk = (αk(1), . . . , αk(L)) is random, independent of the sij , and the empirical distribution of (α1, . . . , αK), with probability one converges weakly to a probability distribution H on C. Let βk = βk(N) = (αk(1)sk , . . . , αk(L)sk ) and set C = C(N) = (1/N) ∑K k=2 βkβ ∗ k . Let σ 2 > 0 be arbitrary. Then with probability one SIR1 ≡ limN→∞(1/N)β∗ 1(C + σ2I)−1β1 = ∑L `,`′=1 ᾱ1(`)α1(` ′)a`,`′ where A = (a`,`′) is nonrandom, Hermitian positive definite, and is the unique matrix of such type satisfying A = ( cE αα ∗ 1+α∗Aα + σ IL )−1 where α ∈ C has distribution H. The quantity SIR1 is used as a model for the best signal-to-interference ratio for one user with respect to other K−1 users, for K large in a direct-sequence code-division multiple-access system in wireless communications. The result generalizes those previously derived but under more restricted assumptions, allowing for the analysis of user location with respect to the antennas in arbitrary settings.