We consider a downlink cellular network where multi-antenna base stations (BSs) transmit data to single-antenna users by using one of two linear precoding methods with limited feedback: 1) maximum ratio transmission (MRT) for serving a single user or 2) zero forcing (ZF) for serving multiple users. The BS and user locations are drawn from a Poisson point process, allowing expressions for the signal-to-interference coverage probability and the ergodic spectral efficiency to be derived as a function of system parameters, such as the number of BS antennas and feedback bits, and the pathloss exponent. We find a tight lower bound on the optimum number of feedback bits to maximize the net spectral efficiency, which captures the overall system gain by considering both of downlink and uplink spectral efficiency using limited feedback. Our main finding is that, when using MRT, the optimum number of feedback bits scales linearly with the number of antennas, and logarithmically with the channel coherence time. When using ZF, the feedback scales in the same ways as MRT, but also linearly with the pathloss exponent. The derived results provide system-level insights into the preferred channel codebook size by averaging the effects of short-term fading and long-term pathloss.
@cite_37: In multiple antenna wireless systems, beamforming is a simple technique for guarding against the negative effects of fading. Unfortunately, beamforming requires the transmitter to have knowledge of the forward-link channel which is often not available a priori. One way of overcoming this problem is to design the beamforming vector using a limited number of feedback bits sent from the receiver to the transmitter. In limited feedback beamforming, the beamforming vector is restricted to lie in a codebook that is known to both the transmitter and receiver. Random vector quantization (RVQ) is a simple approach to codebook design that generates the vectors independently from a uniform distribution on the complex unit sphere. This correspondence presents performance analysis results for RVQ limited feedback beamforming
@cite_9: Feedback of quantized channel state information (CSI), called limited feedback, enables transmit beamforming in multiple-input-multiple-output (MIMO) wireless systems with a small amount of overhead. Due to its efficiency, beamforming with limited feedback has been adopted in several wireless communication standards. Prior work on limited feedback commonly adopts the block fading channel model where temporal correlation in wireless channels is neglected. In this paper, we consider temporally correlated channels and design single-user transmit beamforming with limited feedback. Analytical results concerning CSI feedback are derived by modeling quantized CSI as a first-order finite-state Markov chain. These results include the information rate of the CSI quantizer output, the bit rate a CSI feedback channel is required to support, and the effect of feedback delay on throughput. In particular, based on the theory of Markov chain convergence rate, feedback delay is proved to reduce the throughput gain due to CSI feedback at least exponentially. Furthermore, an algorithm is proposed for CSI feedback compression in time. Combining the results in this work leads to a new method for designing limited feedback beamforming as demonstrated by a design example.
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@cite_23: Multiple-input-multiple-output (MIMO) communication systems can provide large capacity gains over traditional single-input-single-output (SISO) systems and are expected to be a core technology of next generation wireless systems. Often, these capacity gains are achievable only with some form of adaptive transmission. In this paper, we study the capacity loss (defined as the rate loss in bits s Hz) of the MIMO wireless system when the covariance matrix of the transmitted signal vector is designed using a low rate feedback channel. For the MIMO channel, we find a bound on the ergodic capacity loss when random codebooks, generated from the uniform distribution on the complex unit sphere, are used to convey the second order statistics of the transmitted signal from the receiver to the transmitter. In this case, we find a closed-form expression for the ergodic capacity loss as a function of the number of bits fed back at each channel realization. These results show that the capacity loss decreases at least as O(2 sup -B (2MMt-2) ) where B is the number of feedback bits, M sub t is the number of transmit antennas, and M=min M sub r ,M sub t where M sub r is the number of receive antennas. In the high SNR regime, we present a new bound on the capacity loss that is tighter than the previously derived bound and show that the capacity loss decreases exponentially as a function of the number of feedback bits.
@cite_25: Transmit beamforming and receive combining are simple methods for exploiting the significant diversity that is available in multiple-input multiple-output (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector; both are hard to realize. In this article, a quantized maximum signal-to-noise ratio (SNR) beamforming technique is proposed where the receiver only sends the label of the best beamforming vector in a predetermined codebook to the transmitter. By using the distribution of the optimal beamforming vector in independent and identically distributed Rayleigh fading matrix channels, the codebook design problem is solved and related to the problem of Grassmannian line packing. The proposed design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the codebook size are derived to guarantee full diversity order. Results on the density of Grassmannian line packings are derived and used to develop bounds on the codebook size given a capacity or SNR loss. Monte Carlo simulations are presented that compare the probability of error for different quantization strategies.
@cite_17: We study a multiple-antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any finite set of beamformers. The universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel. Using the bound, it is shown that finite information systems approach the perfect information case as (t-1)2 sup -B t-1 , where B is the number of feedback bits and t is the number of transmit antennas. The geometrical bounding technique, used in the proof of the lower bound, also leads to a design criterion for good beamformers, whose outage performance approaches the lower bound. The design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook, and is equivalent to the problem of designing unitary space-time codes under certain conditions. Finally, we show that good beamformers are good packings of two-dimensional subspaces in a 2t-dimensional real Grassmannian manifold with chordal distance as the metric.
There has been extensive prior work on the downlink transmission rate as a function of a finite CSI feedback rate. In toy setups where a base station (BS) serves a single user in a single cell by ignoring ICI, downlink transmission rates were characterized as a function of the codebook size and the SNR. For example, in @cite_25 @cite_29 , a channel codebook design method for a single-user transmission was proposed by using Grassmannian line packing. By employing random vector quantization (RVQ), the rate loss of a point-to-point MIMO system caused by using finite rate feedback was characterized in @cite_23 . In @cite_37 , by applying RVQ, the performance of conjugate beamforming was analyzed in a MISO system. In @cite_9 , the performance of the finite rate CSI feedback was characterized, assuming temporally correlated channel and a principle of designing a channel codebook for this particular condition was proposed. The main limitation of @cite_25 @cite_17 @cite_19 @cite_23 @cite_37 is that it assumed only a single pair of a BS and a user, in which important features of a cellular network, e.g., ICI are missing.