Distributed Quantization of Order Statistics with Applications to CSI Feedback Matthew Pugh, Student Member, IEEE, and Bhaskar D. Rao, Fellow, IEEE Department of Electrical and Computer Engineering University of California, San Diego La Jolla, CA, 92093-0407, USA E-mail: {mopugh, brao}@ucsd.edu Abstract Feedback of channel state information (CSI) in wireless systems is essential in order to exploit multi-user diversity and achieve the highest possible performace. When each spatially distributed user in the wireless system is assumed to have i.i.d. scalar CSI values, the optimal fixed-rate and entropy-constrained point density functions are established in the high-resolution regime for the quantization of the CSI feedback to a centralized scheduler under the mean square error (MSE) criterion. The spatially distributed nature of the users leads to a distributed functional scalar quantization approach for the optimal high resolution point densities of the CSI feedback. Under a mild absolute moment criterion, it is shown that with a greedy schedul- ing algorithm at the centralized scheduler, the optimal fixed-rate point density for each user corresponds to a point density associated with the maximal order statistic distribution. This result is generalized to monotonic functions of arbitrary order statistics. Optimal point densities under entropy-constrained quantization for the CSI are established under mild conditions on the distribution function of the CSI metric. I. I NTRODUCTION In many wireless systems, resource allocation and/or scheduling decisions are made based on feedback of channel state information (CSI). In a cellular broadcast channel, cell phones provide CSI feedback to the base station and the base station computes a function of the fed back CSI to determine the appropriate resource allocations or scheduling. For example, in [1], each user in the broadcast channel feeds back their measured signal-to- noise plus interference ratio (SINR) on each random transmit beam, and the base station schedules the user with the highest SINR for each transmit beam. In this case, the SINR represents the CSI metric, and the scheduling function is the max(·) function, i.e. greedy scheduling. For a comprehensive overview of the role of feedback in wireless system, see [2]. A question that naturally arises in these situations is how to appropriately quantize the CSI for feedback. If the statistics of the CSI are known, then one could naively design a quantizer based on the distribution of the CSI metric. This, however, is sub-optimal if the function that determines the resource allocation or scheduling is known a priori. The problem of interest is finding the optimal high resolution point densities for quantization of the CSI when the users are spatially distributed and a function of the feedback is to be computed at a central controller. This problem is a specific case of the work on distributed functional scalar quantization (DFSQ) established in [3]. This contribution will consider the case where the scalar CSI at each spatially dis- tributed user is i.i.d. and the distribution is known. The main result of this work is to show that the optimal (in the MSE sense) high-resolution point density for quantization of the CSI at each user is based on the distribution of the order statistics for the fixed-rate