Asymptotic analysis of reduced rank downlink CDMA Wiener receivers

In this paper, we study the performance of reduced rank Wiener filters in the context of downlink CDMA systems corrupted by a frequency selective channel. For this, we consider the output signal to interference plus noise ratio (SINR), and 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 [12]: the code matrix is modelled as the realization of a certain random matrix, and the behavior of the SINRs is studied when the spreading factor N and the number of users K converges to +∞ in such a way that K over N → +α. As the code matrices used in the downlink of CDMA systems are very often orthogonal, we model the code matrix allocated to the various users as a realization of a Haar distributed random unitary matrix. In this context, we show that the SINR of each order n reduced rank receiver converge toward a deterministic limit β_n independent of the spreading codes. In order to study the performance of the receiver versus n, we therefore study the convergence speed of β_n when n → +∞, a simpler problem. For this, we use the results of [8] 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.