Signal and detector randomization for multiuser and multichannel communication systems

Randomization can be considered as a possible approach to enhance error performance of communication systems subject to average power constraints. In the first part of this dissertation, we consider downlink of a multiuser communications system subject to an average power constraint, where randomization is employed at the transmitter and receiver sides by modeling signal levels as random variables (stochastic signals) and employing different sets of detectors via time-sharing (detector randomization), respectively. In the second part, we consider single-user systems, where we assume that there exist multiple channels between the transmitter and receiver with arbitrary noise distributions over each of them and only one of the channels can be employed for transmission at any given time. In this case, randomization is performed by choosing the channel in use according to some probability mass function and employing stochastic signaling at the transmitter. First, the jointly optimal power control with signal constellation randomization is proposed for the downlink of a multiuser communications system. Unlike a conventional system in which a fixed signal constellation is employed for all the bits of a user (for given channel conditions and noise power), power control with signal constellation randomization involves randomization/time-sharing among different signal constellations for each user. A formulation is obtained for the problem of optimal power control with signal constellation randomization, and it is shown that the optimal solution can be represented by a randomization of (K + 1) or fewer distinct signal constellations for each user, where K denotes the number of users. In addition to the original nonconvex formulation, an approximate solution based on convex relaxation is derived. Then, detailed performance analysis is presented when the receivers employ symmetric signaling and sign detectors. Specifically, the maximum asymptotical improvement ratio is shown to be equal to the number of users, and the conditions under which the maximum and minimum asymptotical improvement ratios are achieved are derived. In the literature, it is known that employing different detectors with corresponding deterministic signals via time-sharing may enhance error performance of communications systems subject to average power constraints. Motivated by this result, as a second approach, we study optimal detector randomization for the downlink of a multiuser communications system. A formulation is provided to obtain optimal signal amplitudes, detectors, and detector randomization factors. It is shown that the solution of this joint optimization problem can be calculated in two steps, resulting in significant reduction in computational complexity. It is proved that the optimal solution is achieved via randomization among at most min{K, Nd} detector sets, where K is the number of users and Nd is the number of detectors at each receiver. Lower and upper bounds are derived on the performance of optimal detector randomization, and it is proved that the optimal detector randomization approach can reduce the worst-case average probability of error of the optimal approach that employs a single detector for each user by up to K times. Various sufficient conditions are obtained for the improvability and nonimprovability via detector randomization. In the special case of equal crosscorrelations and noise powers, a simple solution is developed for the optimal detector randomization problem, and necessary and sufficient conditions are presented for the uniqueness of that solution. Next, a single-user M−ary communication system is considered in which the transmitter and the receiver are connected via multiple additive (possibly nonGaussian) noise channels, any one of which can be utilized for a given symbol transmission. Contrary to deterministic signaling (i.e., employing a fixed constellation), a stochastic signaling approach is adopted by treating the signal values transmitted for each information symbol over each channel as random variables. In particular, the joint optimization of the channel switching (i.e., time-sharing among different channels) strategy, stochastic signals, and decision rules at the receiver is performed in order to minimize the average probability of error under an average transmit power constraint. It is proved that the solution to this problem involves either one of the following: (i) deterministic signaling over a single channel, (ii) randomizing (time-sharing) between two different signal constellations over a single channel, or (iii) switching (time-sharing) between two channels with deterministic signaling over each channel. For all cases, the optimal strategies are shown to employ corresponding maximum a posteriori probability (MAP) decision rules at the receiver.