172 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 2, FEBRUARY 2013 Performance of Conjugate and Zero-Forcing Beamforming in Large-Scale Antenna Systems Hong Yang and Thomas L. Marzetta Abstract—Large-Scale Antenna Systems (LSAS) is a form of multi-user MIMO technology in which unprecedented numbers of antennas serve a signiﬁcantly smaller number of autonomous terminals. We compare the two most prominent linear pre- coders, conjugate beamforming and zero-forcing, with respect to net spectral-efﬁciency and radiated energy-efﬁciency in a simpliﬁed single-cell scenario where propagation is governed by independent Rayleigh fading, and where channel-state informa- tion (CSI) acquisition and data transmission are both performed during a short coherence interval. An effective-noise analysis of the pre-coded forward channel yields explicit lower bounds on net capacity which account for CSI acquisition overhead and errors as well as the sub-optimality of the pre-coders. In turn the bounds generate trade-off curves between radiated energy-efﬁciency and net spectral-efﬁciency. For high spectral- efﬁciency and low energy-efﬁciency zero-forcing outperforms conjugate beamforming, while at low spectral-efﬁciency and high energy-efﬁciency the opposite holds. Surprisingly, in an optimized system, the total LSAS-critical computational burden of conjugate beamforming may be greater than that of zero- forcing. Conjugate beamforming may still be preferable to zero- forcing because of its greater robustness, and because conjugate beamforming lends itself to a de-centralized architecture and de-centralized signal processing. Index Terms—Large-scale antenna system, capacity, energy efﬁciency, spectral efﬁciency, spatial multiplexing, beamforming, pre-coder, computational burden I. I NTRODUCTION L ARGE-SCALE Antenna Systems (LSAS) bring huge spectral-efﬁciency and radiated energy-efﬁciency gains compared with 4G wireless technologies [4], [5], [6], [8], [11]. On the forward link a critical operation is pre-coding, i.e. the mapping of the message-bearing symbols intended for K terminals into the M (typically M >> K) signals which the service-antennas transmit. One advantage of LSAS is that the preponderance of service-antennas over terminals can make linear pre-coding perform nearly as well as nonlinear pre- coding as exempliﬁed by sphere-encoded modulation [1], [7], or the optimal dirty-paper coding [10]. The contribution of this paper is to compare, for a simpliﬁed but informative scenario, the relative spectral-efﬁciency and radiated energy-efﬁciency and the computational burden of the two most prominent forms of linear pre-coding: conjugate beamforming and zero- forcing. Manuscript received January 16, 2012; revised May 31, 2012 and Septem- ber 5, 2012. As this paper was co-authored by a guest editor of this issue, the review of this manuscript was coordinated by Senior Editor Wayne Stark. H. Yang and T. L. Marzetta are with Mathematics of Networks and Communications Research Department, Bell Laborataries, Alcatel-Lucent, Murray Hill, NJ 07974 USA (e-mail: h.yang@research.bell-labs.com and tlm@research.bell-labs.com). Digital Object Identiﬁer 10.1109/JSAC.2013.130206. Linear pre-coding involves multiplying the K dimensional vector of QAM symbols intended for the terminals by an M by K pre-coding matrix to obtain the M dimensional vector of signals that are actually transmitted. For conjugate beamforming, the pre-coding matrix is proportional to the conjugate of the estimated channel matrix, and for zero-forcing the pre-coding matrix is proportional to the pseudo-inverse of the estimated channel matrix. In the time-domain, conjugate beamforming is called “time-reversal beamforming” and it is equivalent to convolving each QAM sequence with its respective conjugated and time-reversed impulse-response es- timate, and summing over the K correlations. The reverse link counterpart to conjugate beamforming is matched-ﬁltering. The scenario is a single cell containing an array of M service-antennas and K autonomous single-antenna terminals. The propagation comprises independent Rayleigh fading, and a-priori nobody knows the instantiation of the channel. During a speciﬁed coherence interval the terminals transmit orthogo- nal pilot sequences on the reverse link, each service antenna estimates the reverse link channels to the terminals which, via time-division duplex (TDD) reciprocity, are also estimates for the forward-link channels, and during the remaining portion of the coherence interval message-bearing symbols are trans- mitted by the service-antennas through a linear pre-coding operation. The signal that each terminal receives is interpreted as the desired message-bearing signal, times a known gain, plus uncorrelated additive effective noise. This provides rigorous capacity lower-bounds which account for the noisy channel estimates and the sub-optimality of the linear pre-coding. We evaluate energy-efﬁciency relative to that of a conventional single-antenna, forward-pilot scheme. In turn we obtain plots of relative energy-efﬁciency versus net spectral efﬁciency, such that for each point on the curve, the design parameters (radiated power, number of terminals served, and duration of reverse link pilots) are chosen optimally. For the two linear pre-coding schemes we evaluate the computational burden with respect to the LSAS-critical signal processing steps, including fast Fourier transforms (FFT’s), correlation of received pilot signals with pilot sequences, computing the pre-coding matrix (for zero-forcing), and implementing the pre-coding. We ﬁnd that for an operating point which yields high spectral-efﬁciency and low energy-efﬁciency, zero-forcing out- performs conjugate beamforming, while the converse holds for high energy-efﬁciency and low spectral-efﬁciency. When the operating point is set such that the relative energy-efﬁciency is equal to 1/M we typically obtain very large spectral- 0733-8716/13/$31.00 c  2013 IEEE