On the asymptotic performance of the decorrelator

We derive the asymptotic signal-to-interference ratio (SIR) of the decorrelator in the large system limit, both for the case in which the number of users exceeds the spreading gain and for the case in which the number of users is less than the spreading gain. We show that, contrary to what is claimed by Tse and Hardy (see ibid., vol.45, p.641-57, Mar. 1999) and by Tse and Zeitouni (see ibid., vol.46, p.171-188, Jan. 2000) when the number of users exceeds the spreading gain and the decorrelator is defined in terms of the Moore-Penrose pseudoinverse, the SIR does not converge to zero.