Asymptotic analysis of reduced rank chip-level MMSE equalizers in the downlink of CDMA systems

We study the performance of downlink CDMA receivers consisting of a reduced rank Wiener chip rate equalizer followed by despreading. For this, it is standard to consider the output signal to interference plus noise ratio (SINR), and to study its convergence speed versus the order of the receiver. Unfortunately, this is a difficult task because the SINR expressions depend on the spreading codes allocated to the various users in a rather complicated way. In order to be able to obtain positive results, we follow the classical approach used for the first time in D. Tse et al. (1999), and assume that the spreading factor TV and the number of users K converge to +∞ at the same rate. The spreading codes are supposed to coincide with Walsh Hadamard codes scrambled by an independent identically distributed sequence. In this context, we show that the SINR of each order n reduced rank receiver converges toward a deterministic limit 0n independent of the spreading codes. In order to address the performance of the receiver versus n, we thus study the convergence speed of β_n when n→+∞, a simpler problem. For this, we use the results of P. Loubaton et al. (2003) based on the theory of orthogonal polynomials for the power moment problem. We obtain the convergence rate of β_n, and exhibit the parameters influencing the convergence speed. We finally compare our asymptotic expressions with the performances of the receivers obtained by numerical simulations. For K/N=1/2, we observe a good agreement as soon as N≥128.