The Annals of Applied Probability

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

Z. D. Bai and Jack W. Silverstein

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Let {sij: 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 sk=sk(N)=(s1k, s2k, …, sNk)T, 1≤kK, with K=K(N) and K/Nc>0 as N→∞. Assume for fixed positive integer L, for each N and kK, αk=(αk(1), …, αk(L))T is random, independent of the sij, and the empirical distribution of (α1, …, αK), with probability one converging weakly to a probability distribution H on ℂL. Let βk=βk(N)=(αk(1)skT, …, αk(L)skT)T and set C=C(N)=(1/N)∑k=2K βk βk*. 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'=1Lα̅1()α1(ℓ')a,ℓ', where A=(a,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∈ℂL has distribution H. The result generalizes those previously derived under more restricted assumptions.

Article information

Ann. Appl. Probab., Volume 17, Number 1 (2007), 81-101.

First available in Project Euclid: 13 February 2007

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Zentralblatt MATH identifier

Primary: 15A52 60F15: Strong theorems
Secondary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 94A05: Communication theory [See also 60G35, 90B18] 94A15: Information theory, general [See also 62B10, 81P94]

Code-division multiple access (CDMA) systems random matrix empirical distributions


Bai, Z. D.; Silverstein, Jack W. On the signal-to-interference ratio of CDMA systems in wireless communications. Ann. Appl. Probab. 17 (2007), no. 1, 81--101. doi:10.1214/105051606000000637.

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