Optimal sequences for CDMA under colored noise: A schur-saddle function property

We consider direct sequence code division multiple access (DS-CDMA), modeling interference from users communicating with neighboring base stations by additive colored noise. We consider two types of receiver structures: first we consider the information-theoretically optimal receiver and use the sum capacity of the channel as our performance measure. Second, we consider the linear minimum mean square error (LMMSE) receiver and use the signal-to-interference ratio (SIR) of the estimate of the symbol transmitted as our performance measure. Our main result is a constructive characterization of the possible performance in both these scenarios. A central contribution of this characterization is the derivation of a qualitative feature of the optimal performance measure in both the scenarios studied. We show that the sum capacity is a saddle function: it is convex in the additive noise covariances and concave in the user received powers. In the linear receiver case, we show that the minimum average power required to meet a set of target performance requirements of the users is a saddle function: it is convex in the additive noise covariances and concave in the set of performance requirements.