A multiple transmit antenna, single receive antenna (per receiver) downlink channel with limited channel feedback is considered. Given a constraint on the total system-wide channel feedback, the following question is considered: is it preferable to get low-rate feedback from a large number of receivers or to receive high-rate/high-quality feedback from a smaller number of (randomly selected) receivers? Acquiring feedback from many users allows multi-user diversity to be exploited, while high-rate feedback allows for very precise selection of beamforming directions. It is shown that systems in which a limited number of users feedback high-rate channel information significantly outperform low-rate/many user systems. While capacity increases only double logarithmically with the number of users, the marginal benefit of channel feedback is very significant up to the point where the CSI is essentially perfect.
Multiple antenna broadcast channels have been the subject of a tremendous amount of research since the seminal work of Caire and Shamai showed the sum-rate optimality of dirtypaper precoding (DPC) with Gaussian inputs [1]. If the transmitter is equipped with M antennas, then multi-user MIMO techniques (such as DPC or sub-optimal but low-complexity linear precoding) that allow simultaneous transmission to multiple users over the same time-frequency resource can achieve a multiplexing gain of M (as long as there are M or more receivers) even if each receiver has only one antenna. In contrast, orthogonal techniques (such as TDMA) that only serve one user achieve a multiplexing gain of only one.
Since the multiple antenna broadcast channel is a very natural model for many-to-one communication (e.g., a single cell in a cellular system), this line of work has been of great interest to both academia and industry. The multiple antenna broadcast channel with limited channel feedback has been of particular interest over the past few years because this accurately models the practical scenario where each receiver feeds back (imperfect) channel information to the transmitter. In a frequency-division duplexed system (or a time-division duplex system without accurate channel reciprocity) channel feedback is generally the only mechanism by which the transmitter can obtain channel state information (CSI). In the single receive antenna setting, most proposed feedback strategies either directly or indirectly involve each receiver quantizing its M -dimensional channel vector to the closest of a set of quantization vectors; finer quantization corresponds to a larger set of quantization vectors and thus higher rate channel feedback.
Within the literature on the MIMO broadcast with limited feedback, there has been a dichotomy between the extremes of systems with a small number of receivers (on the order of the number of transmit antennas) versus systems with an extremely large number of receivers.
• Finite systems have been shown to be extremely sensitive to the accuracy of the CSIT, and thus require highrate feedback. This has been shown from a fundamental information theoretic perspective [2], as well as in terms of particular transmit strategies. In particular, zero-forcing beamforming has been shown to require CSIT quality that scales proportional to SNR [3][5]. • Large systems have been shown to be able to operate near capacity with extremely low-rate channel feedback in the asymptotic limit as the number of users is taken to infinity. In particular, random beamforming (RBF) [6] can operate with only log 2 M bits of feedback per user (plus one real number). The performance of this technique in the asymptotic limit is quite amazing: not only does the ratio of random beamforming throughput to perfect CSIT capacity converge to one as the number of users is taken to infinity, but the difference between these quantities actually has been shown to converge to zero [7].
Finite systems require high-rate feedback because imperfect CSIT leads to multi-user interference that cannot be resolved at each receiver. In order to prevent such a system from becoming interference-limited, the CSIT must be very accurate; in terms of channel quantization, this corresponds to using a very rich quantization codebook that allows the direction of each receiver’s channel vector to be very accurately quantized. In large systems, on the other hand, multi-user diversity is exploited to allow the system to operate with extremely low levels of feedback. The RBF strategy involves a quantization codebook consisting of only M orthonormal vectors (e.g., the elementary basis vectors). If such a codebook is used with a small user population, each user’s quantization will likely be quite poor due to the limited size of the quantization codebook. However, as the number of users increases, it becomes more and more likely that at least some of the users have channel vectors that lie very close to one of the M quantization vectors. This effect allows the system to get by with very low rate feedback. Although the RBF throughput does converge in the strong absolute sense to the perfect CSIT capacity, convergence is extremely slow, even for systems with a small number of transmit antennas.
Motivated by the apparent dichotomy between finite and asymptotically large MIMO broadcast systems with limited channel feedback, in this paper we ask the following simple question: Is it preferable to have a system with a large number of receivers and low-rate feedback from each receiver (thereby exploiting multi-user diversity), or to have a system with a smaller number of receivers with high-rate feedback from each receiver (thereby exploiting the benefits of accurate CSIT)?
In order to fairly compare these systems, we equalize the total number of channel feedback bits (across users). Assuming that a total of T feedback bits are used, we compare the following:
• Random beamf
