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

Abstract

Let {s_ij: i, j=1, 2, …} consist of i.i.d. random variables in ℂ with $\mathsf{E}s_{11}=0$, $\mathsf{E}|s_{11}|^{2}=1$. For each positive integer N, let s_k=s_k(N)=(s_1k, s_2k, …, s_Nk)^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, α_k=(α_k(1), …, α_k(L))^T is random, independent of the s_ij, and the empirical distribution of (α₁, …, α_K), with probability one converging weakly to a probability distribution H on ℂ^L. Let β_k=β_k(N)=(α_k(1)s_k^T, …, α_k(L)s_k^T)^T and set C=C(N)=(1/N)∑_k=2^K β_k β_k^*. Let σ²>0 be arbitrary. Then define SIR₁=(1/N)β₁^*(C+σ²I)⁻¹ β₁, 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 ∑_ℓ,ℓ'=1^Lα̅₁(ℓ)α₁(ℓ')a_ℓ,ℓ', where A=(a_ℓ,ℓ') is nonrandom, Hermitian positive definite, and is the unique matrix of such type satisfying $A=\bigl(c\,\mathsf{E}\frac{\mathbf{\alpha}\mathbf{\alpha}^{*}}{1+\mathbf{\alpha}^{*}A\mathbf{\alpha}}+\sigma^{2}I_{L}\bigr)^{-1}$, where α∈ℂ^L has distribution H. The result generalizes those previously derived under more restricted assumptions.