We consider power-domain multiple-input multiple-output (MIMO) nonorthogonal multiple access (NOMA) down-links in which a multiple-antenna base station applies zeroforcing beamforming to transmit data to single-antenna users. Two users whose channel correlation coefficient exceeds a threshold are paired to share the same transmit beam. We derive an approximate of the expected signal-to-interference plus noise ratio (SINR) for each user, which depends on transmit power, the correlation threshold, and the number of quantization bits for channel direction information (CDI). A max-min SINR problem is formulated as a geometric program. We show that a near-optimal SINR can be achieved by optimizing only two CDI rates: one for unpaired users and the other for user pairs. A large SINR gain can be achieved by optimizing a pairing threshold when a total CDI rate is high.

In downlink, a base station (BS) with multiple transmit antennas applies zeroforcing beamforming to transmit to single-antenna mobile users in a cell. We propose the schemes that optimize transmit power and the number of bits for channel direction information (CDI) for all users to achieve the max-min signal-to-interference plus noise ratio (SINR) fairness. The optimal allocation can be obtained by a geometric program for both non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA). For NOMA, 2 users with highly correlated channels are paired and share the same transmit beamforming. In some small total-CDI rate regimes, we show that NOMA can outperform OMA by as much as 3 dB. The performance gain over OMA increases when the correlation-coefficient threshold for user pairing is set higher. To reduce computational complexity, we propose to allocate transmit power and CDI rate to groups of multiple users instead of individual users. The user grouping scheme is based on K-means over the user SINR. We also propose a progressive filling scheme that performs close to the optimum, but can reduce the computation time by almost 3 orders of magnitude in some numerical examples.

We consider non-orthogonal multiple access (NOMA) downlink channels with zeroforcing beamforming transmission. With limited feedback rate, channel direction information (CDI) needs to be quantized and fed back from mobile users to a base station. To increase spectral efficiency, two active users in some clusters share the same beamforming vector and thus, will interfere fully with each other. We derive the approximate achievable rates and outage probability for users in 1-user and 2-user clusters. With the approximate outage probability derived as a function of feedback rate, we find the optimal feedback allocation for both types of cluster that minimizes the maximum outage probability for all users. Numerical results show that significant gain over uniform feedback allocation is achieved in moderate-to-large SNR regimes.

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 finite-size 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 then feeding back for every block does by 44% when the number of feedback bits is small.

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.

We propose two feedback methods for transmit beamforming in a point-to-point multiple-antenna OFDM channel. For the first method, a receiver with channel information quantizes and feeds back the optimal transmit beamforming vectors of a few selected subcarriers that are equally spaced. Based on those quantized vectors, the transmitter linearly interpolates the remaining beamforming vectors with different phase rotation whose expression is explicitly shown. For the second proposed method, a channel impulse response is quantized with a uniform scalar quantizer. At the transmitter, channel frequency response can be reconstructed from the quantized impulse response and the optimal beamforming vectors can then be computed. We show that switching between the two methods for different feedback-rate requirement outperforms existing methods in the literature.

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 temporal-correlation 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.

In this work, we analyze the performance of a signature quantization scheme for reverse-link Direct Sequence (DS)- Code Division Multiple Access (CDMA). Assuming perfect estimates of the channel and interference covariance, the receiver selects the signature that minimizes interference power or maximizes signal-to-interference plus noise ratio (SINR) for a desired user from a signature codebook. The codebook index corresponding to the optimal signature is then relayed to the user with a finite number of bits via a feedback channel. Here we are interested in the performance of a Random Vector Quantization (RVQ) codebook, which contains independent isotropically distributed vectors. Assuming arbitrary transmit power allocation, we consider additive white Gaussian noise (AWGN) channel first with no fading and subsequently, with multipath fading. We derive the corresponding SINR in a large system limit at the output of matched filter and linear minimum mean squared error (MMSE) receiver. Numerical examples show that the derived large system results give a good approximation to the performance of finite-size system and that the MMSE receiver achieves close to a single-user performance with only one feedback bit per signature element.

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.

We examine the capacity of beamforming over a single-user, multiantenna link taking into account the overhead due to channel estimation and limited feedback of channel state information. Multi-input-single-output (MISO) and multi-input-multi-output (MIMO) channels are considered subject to block Rayleigh fading. Each coherence block contains L symbols, and is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a minimum mean squared error estimate of the channel matrix. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2Bi.i.d. random vectors, and sends the corresponding B bits back to the transmitter. We derive bounds on the beamforming capacity for MISO and MIMO channels and characterize the optimal (rate-maximizing) training and feedback overhead (T and B) as L and the number of transmit antennas Ntboth become large. The optimal Ntis limited by the coherence time, and increases as L/logL. For the MISO channel the optimal T/L and B/L (fractional overhead due to training and feedback) are asymptotically the same, and tend to zero at the rate 1/log Nt. For the MIMO channel the optimal feedback overhead B/L tends to zero faster (as 1/log2Nt).

For a point-to-point multi-input multi-output (MIMO) wireless channel, we propose a feedback scheme, which consists of transmit-antenna selection algorithm and beamforming quantization. A feedback, which is relayed from a receiver to a transmitter via a feedback channel, specifies a set of active transmit antennas and associated beamforming vector, which contains transmit antenna coefficients. Assuming perfect channel knowledge, the receiver selects the set of transmit antennas that maximizes the largest eigenvalue of a channel covariance matrix and then, chooses the beamforming vector that maximizes the capacity, from a random vector quantization (RVQ) codebook. Entries in the RVQ codebook are independent isotropically distributed and was previously shown to perform close to the optimum. We derive

Given a multiple-input multiple-output (MIMO) channel, feedback from the receiver can be used to specify a transmit precoding matrix, which selectively activates the strongest channel modes. Here we analyze the performance ofrandomvectorquantization(RVQ), in which the precoding matrix is selected from a random codebook containing independent, isotropically distributed entries. We assume that channel elements are independent and identically distributed (i.i.d.) and known to the receiver, which relays the optimal (rate-maximizing) precoder codebook index to the transmitter usingBbits. We first derive the large system capacity of beamforming (rank-one precoding matrix) as a function ofB, where large system refers to the limit asBand the number of transmit and receive antennas all go to infinity with fixed ratios. RVQ for beamforming is asymptotically optimal, i.e., no other quantization scheme can achieve a larger asymptotic rate. We subsequently consider a precoding matrix with arbitrary rank, and approximate the asymptotic RVQ performance with optimal and linear receivers (matched filter and minimum mean squared error (MMSE)). Numerical examples show that these approximations accurately predict the performance of finite-size systems of interest. Given a target spectral efficiency, numerical examples show that the amount of feedback required by the linear MMSE receiver is only slightly more than that required by the optimal receiver, whereas the matched filter can require significantly more feedback.

Signature quantization for reverse-link direct sequence (DS)-code division multiple access (CDMA) is considered. A receiver assumed to have perfect estimates of channel and interference covariance, selects the signature that maximizes signal-to-interference plus noise ratio (SINR) for a desired user from a signature codebook. The codebook index corresponding to the optimal signature is relayed to the user with finite number of bits via a feedback channel. Previously, it was shown that a random vector quantization (RVQ) codebook, which contains independent isotropically distributed vectors, is optimal (i.e., maximizes SINR) in a large system limit in which number of interfering users, processing gain, and feedback bits tend to infinity with fixed ratios. Here we derive exact expressions for a large system SINR for the user whose signature is selected from RVQ codebook. We assume that the receiver is linear minimum mean squared error (MMSE) and consider both ideal and multipath fading channels.

We examine the capacity of beamforming over a block Rayleigh fading multi-input/multi-output (MIMO) channel with finite training for channel estimation and limited feedback. A fixed-length packet is assumed, which is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a minimum mean squared error (MMSE) estimate of the channel matrix. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2Bi.i.d. random vectors, and relays the corresponding B-bit index back to the transmitter. We derive bounds on the large system capacity, i.e., as the number of transmit antennas Ntrarr infin and receive antennas Nrrarr infin with fixed ratio Nt/Nr. The bounds are used to show that the optimal T, which maximizes the capacity, increases as Nt/ log Nt, whereas the optimal B increases as Nt/log2Nt.

This work studies the performance of multiuser detection and decoding in the downlink of cellular systems based on orthogonal frequency-division multiplexing (OFDM). Of particular interest is a worst case scenario where the desired user is at the cell boundary and subject to an equally strong interferer from a neighboring cell. A flexible iterative turbo decision-feedback equalizer (DFE) is proposed and studied numerically, which exhibits good error performance and resilience to fading, requires moderate training and low complexity, and accommodates multiple antennas easily. The scheme performs well without knowledge of the pilots of interfering signals from other cells, which is typically unavailable in practice, while such knowledge may improve the performance significantly. Furthermore, in the absence of knowledge of pilots for the out-of-cell interference, it is found that estimating the filter coefficients of the DFE directly is superior to deriving the coefficients from estimates of the instantaneous channel gains.

We examine the capacity of beamforming over a multi-input/single-output block Rayleigh fading channel with finite training for channel estimation and limited feedback. A fixed-length packet is assumed, which is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a minimum mean squared error (MMSE) estimate of the channel vector. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2Bi.i.d. random vectors, and relays the corresponding B bits back to the transmitter. We derive bounds on the capacity and show that for a large number of transmit antennas Nt, the optimal T and B, which maximize the bounds, are approximately equal and both increase as Nt/logNt. We conclude that with limited training and feedback, the optimal number of antennas to activate also increases as Nt/logNt

We study the performance of joint signature-receiver optimization for direct-sequence code-division multiple access (DS-CDMA) with limited feedback. The receiver for a particular user selects the signature from a signature codebook, and relays the corresponding B index bits to the transmitter over a noiseless channel. We study the performance of a random vector quantization (RVQ) scheme in which the codebook entries are independent and isotropically distributed. Assuming the interfering signatures are independent, and have independent and identically distributed (i.i.d.) elements, we evaluate the received signal-to-interference plus noise ratio (SINR) in the large system limit as the number of users, processing gain, and feedback bits B all tend to infinity with fixed ratios. This SINR is evaluated for both the matched filter and linear minimum mean-squared error (MMSE) receivers. Furthermore, we show that this large system SINR is the maximum that can be achieved over any sequence of codebooks. Numerical results show that with the MMSE receiver, one feedback bit per signature coefficient achieves close to single-user performance. We also consider a less complex and suboptimal reduced-rank signature optimization scheme in which the user's signature is constrained to lie in a lower dimensional subspace. The optimal subspace coefficients are scalar-quantized and relayed to the transmitter. The large system performance of the quantized reduced-rank scheme can be approximated, and numerical results show that it performs in the vicinity of the RVQ bound. Finally, we extend our analysis to the scenario in which a subset of users optimize their signatures in the presence of random interference.

We study the capacity of a single-user channel with multiple antennas and limited feedback. The receiver has perfect channel knowledge, and can relay B bits, which specify a beamforming vector, to the transmitter. We show that a random vector quantization scheme is asymptotically optimal, and give a simple expression for the associated capacity.