Asymptotic Spectral Distribution of Crosscorrelation Matrix in Asynchronous CDMA

Asymptotic spectral distribution (ASD) of the crosscorrelation matrix is investigated for a large-system asynchronous direct sequence-code division multiple access (DS-CDMA) with random spreading sequences. The crosscorrelation matrix is formed by an infinite input symbol length. Two levels of asynchronism are considered. One is symbol-asynchronous but chip-synchronous (called chip-synchronous for short), and the other is chipasynchronous. The results are applicable to arbitrary chip waveforms. The existence of a nonrandom ASD is proven by moment convergence theorem, where the focus is on the existence of asymptotic eigenvalue moments (AEM) of the crosscorrelation matrix of all orders. A combinatorics approach based on noncrossing partition of set partition theory is adopted for AEM computation. It is shown that, in chip-synchronous CDMA, AEM are irrelevant to realizations of asynchronous delays and the adopted chip waveform, and AEM are equal to those of a symbolsynchronous CDMA system. The ASD converges almost surely to Marc̆enko-Pastur law. In chip-asynchronous CDMA, AEM are relevant to the shape of chip waveform. Whether AEM are relevant to realizations of asynchronous delays depends on chip waveform bandwidth. The empirical spectral distribution converges almost surely to an ASD, provided that a general constraint is satisfied by the chip waveform.