We consider the quantization of a transmit beamforming vector in multiantenna channels and of a signature vector in code division multiple access (CDMA) systems. Assuming perfect channel knowledge, the receiver selects for a transmitter the vector that maximizes the performance from a random vector quantization (RVQ) codebook, which consists of independent isotropically distributed unit-norm vectors. The quantized vector is then relayed to the transmitter via a rate-limited feedback channel. The RVQ codebook requires an exhaustive search to locate the selected entry. To reduce the search complexity, we apply generalized Lloyd or k-dimensional (kd)-tree algorithms to organize RVQ entries into a tree. In examples shown, the search complexity of tree-structured (TS) RVQ can be a few orders of magnitude less than that of the unstructured RVQ for the same performance. We also derive the performance approximation for TS-RVQ in a large system limit, which predicts the performance of a moderate-size system very well.Index Terms-Signature quantization, tree-structured codebook, CDMA, MIMO, random vector quantization, generalized Lloyd algorithm, kd tree.

We consider multiple-input multiple output (MIMO) non-orthogonal multiple access (NOMA) downlink channels with zeroforcing beamforming transmission. The base station has multiple transmit antennas while all mobile devices have single receive antenna. With a limited feedback rate, channel direction information (CDI) needs to be quantized and fed back from mobile users to a base station. Thus, the accuracy of the quantized channel state information at the transmitter (CSIT) will depend on the feedback rate. To increase spectral efficiency, 2 active users in some clusters share the same beamforming vector and thus, will interfere fully with each other. Given a total-feedback rate, we analyze the feedback allocation for clusters with single active user and 2 active users. The objective is to either minimize the maximum outage probability or maximize the minimum rate among all active users in a cell. We show that the proposed feedback allocation is close to the optimum. Some numerical examples show that the resulting rate performance with the proposed feedback allocation is increased by 100% over that with the uniform feedback allocation. Index Terms-Non-orthogonal multiple access (NOMA), multiple-input multiple-output (MIMO), channel state information at transmitter (CSIT), random vector quantization (RVQ), zeroforcing, beamforming, feedback.

A receiver with perfect channel state information (CSI) in a point-to-point multiple-input multiple-output (MIMO) channel can compute the transmit beamforming vector that maximizes the transmission rate. For frequency-division duplex, a transmitter is not able to estimate CSI directly and has to obtain a quantized transmit beamforming vector from the receiver via a rate-limited feedback channel. We assume that time evolution of MIMO channels is modeled as a Gauss-Markov process parameterized by a temporal-correlation coefficient. Since feedback rate is usually low, we assume rank-one transmit beamforming or transmission with single data stream. For given feedback rate, we analyze the optimal feedback interval that maximizes the average received power of the systems with two transmit or two receive antennas. For other system sizes, the optimal feedback interval is approximated by maximizing the rate difference in a large system limit. Numerical results show that the large system approximation can predict the optimal interval for finitesize system quite accurately. Numerical results also show that quantizing transmit beamforming with the optimal feedback interval gives larger rate than the existing Kalman-filter scheme does by as much as 10% and than feeding back for every block does by 44% when the number of feedback bits is small.Index Terms-MIMO, transmit beamforming, temporally correlated channels, Gauss-Markov process, finite-rate feedback, random vector quantization (RVQ), feedback interval.

Abstract-Assuming perfect channel state information (CSI), the receiver in a point-to-point multiantenna channel can compute the optimal transmit beamforming vector that maximizes channel capacity. The transmitter, which is not able to estimate the CSI, obtains the quantized transmit beamforming vector via a rate-limited feedback channel. We assume that time evolution of both MIMO and MISO channels can be modeled as the first-order autoregressive process parameterized by a temporalcorrelation coefficient. For a limited number of feedback bits, we would like to find out how often the feedback update should take place. Applying a large system limit and random vector quantization (RVQ), we derive the integer optimization problem, which determines the optimal feedback interval that maximizes the average capacity. The analytical results show that the optimal feedback interval depends on the temporal correlation coefficient, available feedback, and the number of transmit and receive antennas.

With finite-rate feedback, we propose two feedback methods for transmit beamforming in a point-to-point MISO-OFDM channel. For the first method, a receiver with perfect channel information, quantizes and feeds back the optimal transmit beamforming vectors of a few selected subcarriers, which are equally spaced. Based on those quantized vectors, the transmitter applies either constant, linear, or higher-order interpolation with the remaining beamforming vectors. With constant interpolation, we derive the approximate sum achievable rate and the optimal cluster size that maximizes the approximate rate. For linear interpolation, we derive a closed-form expression for the phase rotation by utilizing the correlation between OFDM subcarriers. We also propose a higher-order interpolation that requires more than two quantized vectors to interpolate transmit beamformers, and is based on existing channel estimation methods. Numerical results show that interpolation with the optimized cluster size can perform significantly better than that with an arbitrary cluster size. For the second proposed method, a channel impulse response is quantized with a uniform scalar quantizer. With channel quantization, we also derive the approximate sum achievable rate. We show that switching between the two methods for different feedback-rate requirements can perform better than the existing schemes.Comment: To appear in IEEE Transactions on Communication