Asymptotic Analysis of Reduced-Rank Chip-Level MMSE Equalizers in the Downlink of CDMA Systems

In this paper, we address the performance of downlink CDMA receivers that consist in a reduced rank Wiener chip-rate equalizer followed by a despreading operation. In particular, we tackle the question of whether a performance close to the optimum can be achieved for small values of the rank. To answer this question, it is standard to consider the output signal-to-interference plus noise ratio (SINR), and to study its convergence speed versus the rank of the receiver. Unfortunately, this task is difficult due to the fact that the SINR expressions depend on the spreading codes allocated to the various users in a rather complicated way. To be able to obtain positive results, we assume that the spreading factor and the number of users tend to infinity while their ratio remains finite. As in the third-generation UMTS system, the spreading codes we consider coincide with orthogonal Walsh-Hadamard codes scrambled by an independent identically distributed sequence. In this context, we show that the SINR of each reduced-rank receiver converges toward a deterministic limit which depends only of the rank of the receiver, and not of the spreading codes given to the various users. Using some previous results, we prove that the asymptotic SINRs converge exponentially to the SINR of the plain Wiener receiver when the rank of the receiver increases. We obtain the corresponding convergence rate, and exhibit the parameters that influence the convergence speed. We finally compare our asymptotic performance expressions with the results of numerical simulations. We observe a good agreement for spreading factors as low as 16