Quantization Methods for Equal Gain Transmission With Finite Rate Feedback

We consider the design and analysis of quantizers for equal gain transmission (EGT) systems with finite rate feedback-based communication in flat-fading multiple input single output (MISO) systems. EGT is a beamforming technique that maximizes the MISO channel capacity when there is an equal power-per-antenna constraint at the transmitter, and requires the feedback of t-1 phase angles, when there are t antennas at the transmitter. In this paper, we contrast two popular approaches for quantizing the phase angles: vector quantization (VQ) and scalar quantization (SQ). On the VQ side, using the capacity loss with respect to EGT with perfect channel information at transmitter as performance metric, we develop a criterion for designing the beamforming codebook for quantized EGT (Q-EGT). We also propose an iterative algorithm based on the well-known generalized Lloyd algorithm, for computing the beamforming vector codebook. On the analytical side, we study the performance of Q-EGT and derive closed-form expressions for the performance in terms of capacity loss and outage probability in the case of i.i.d. Rayleigh flat-fading channels. On the SQ side, assuming uniform scalar quantization and i.i.d. Rayleigh flat-fading channels, we derive the high-resolution performance of quantized EGT and contrast the performance with that of VQ. We find that although both VQ and SQ achieve the same rate of convergence (to the capacity with perfect feedback) as the number of feedback bits B increases, there exists a fixed gap between the two     

Transmit beamforming in multiple-antenna systems with finite rate feedback: a VQ-based approach

This paper investigates quantization methods for feeding back the channel information through a low-rate feedback channel in the context of multiple-input single-output (MISO) systems. We propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization (VQ) design algorithm. The criterion is based on maximizing the mean-squared weighted inner product (MSwIP) between the optimum and the quantized beamforming vector. The performance of systems with quantized beamforming is analyzed for the independent fading case. This requires finding the density of the squared inner product between the optimum and the quantized beamforming vector, which is obtained by considering a simple approximation of the quantization cell. The approximate density function is used to lower-bound the capacity loss due to quantization, the outage probability, and the bit error probability. The resulting expressions provide insight into the dependence of the performance of transmit beamforming MISO systems on the number of transmit antennas and feedback rate. Computer simulations support the analytical results and indicate that the lower bounds are quite tight.     

Feedback gain in multiple antenna systems

Multiple antenna transmission and reception have been shown to significantly increase the achievable data rates of wireless systems. However, most of the existing analysis assumes perfect or no channel information at the receiver and transmitter. The performance gap between these extreme channel assumptions is large and most practical systems lie in between. Therefore, it is important to analyze multiple antenna systems in the presence of partial channel information. We upper bound the outage probability performance of multiple antenna systems with preamble-based channel estimation and quantized feedback. We design causal feedback and power control schemes to minimize this upper bound on outage probability. We consider the following practical issues in our analysis and design: (1) the channel information is imperfect both at the receiver and at the transmitter and (2) part of the total available resources for the system need to be used for estimation and feedback. Our results demonstrate that for block fading channels, sending a periodic preamble and causally receiving channel state information via a feedback channel can lead to substantial gains in the outage performance over any nonfeedback scheme. Most of the gains achieved by perfect feedback can be achieved by very few bits of feedback. Furthermore, it is demonstrated that these outage probability gains can be translated into improvements in frame error rate performance of systems using space-time codes. Thus, implementing a power control, even at the cost of reduced spectral resources for the forward channel is beneficial for block fading channels     

On the Optimality of Beamforming with Quantized Feedback

The ergodic capacity of a fading vector channel with multiple transmit antennas and a single receive antenna is explored. Perfect channel information is assumed to be available at the receiver while the transmitter has only partial knowledge of the direction of the user's channel vector based on quantized feedback. We present necessary and sufficient conditions for the optimality of beamforming in such systems. The conditions are applicable to all quantized feedback scenarios regardless of the channel distribution, number of transmit antennas, number of quantization vectors or transmit power. The optimality conditions are closely related to the iteration conditions of the Lloyd algorithm, revealing an interesting link between the optimality of beamforming and the optimality of the vector quantizers. Using the conditions, we prove the capacity optimality of beamforming for several quantized feedback scenarios such as the antenna-selection scheme. We also point out examples of quantized feedback scenarios where beamforming is not optimal. We find that for the independent identically distributed Rayleigh fading channel with more than a single bit of quantized feedback, there is no capacity benefit from increasing the number of antennas beyond the number of quantization vectors. Extensions of the necessary and sufficient optimality condition to the multiple-input multiple-output case are also provided.     

Limited feedback unitary precoding for spatial multiplexing systems

Multiple-input multiple-output (MIMO) wireless systems use antenna arrays at both the transmitter and receiver to provide communication links with substantial diversity and capacity. Spatial multiplexing is a common space-time modulation technique for MIMO communication systems where independent information streams are sent over different transmit antennas. Unfortunately, spatial multiplexing is sensitive to ill-conditioning of the channel matrix. Precoding can improve the resilience of spatial multiplexing at the expense of full channel knowledge at the transmitter-which is often not realistic. This correspondence proposes a quantized precoding system where the optimal precoder is chosen from a finite codebook known to both receiver and transmitter. The index of the optimal precoder is conveyed from the receiver to the transmitter over a low-delay feedback link. Criteria are presented for selecting the optimal precoding matrix based on the error rate and mutual information for different receiver designs. Codebook design criteria are proposed for each selection criterion by minimizing a bound on the average distortion assuming a Rayleigh-fading matrix channel. The design criteria are shown to be equivalent to packing subspaces in the Grassmann manifold using the projection two-norm and Fubini-Study distances. Simulation results show that the proposed system outperforms antenna subset selection and performs close to optimal unitary precoding with a minimal amount of feedback.     

Performance analysis of MIMO beamforming with imperfect feedback

The quality of channel state information at the transmitter (CSIT) is critical to MIMO beamforming systems. However, in practical wireless systems, CSIT suffers from imperfections originating from quantization effects, feedback error and feedback delay. In this paper, we study the impact of feedback error and delay on the symbol error rate of MIMO beamforming systems with finite rate feedback. The feedback channel is modeled as a uniform symmetric channel. We derive an symbol error rate upper bound that is tight for a good beamformer. We also quantify the diversity gain and array gain loss due to the feedback error and delay. The codebook design method that is applicable to the beamforming systems with error or delay feedback is discussed. Both analytical and simulation results show that feedback error and delay will make the system behave badly at high signal‐to‐noise ratios. Copyright © 2012 John Wiley & Sons, Ltd.     

Optimal bandwidth allocation for the data and feedback channels in MISO-FDD systems

In frequency-division duplex (FDD) systems, channel-state information (CSI) is estimated by the receiver and then fed back to the transmitter through a feedback link, which inevitably requires additional bandwidth and power. In this letter, we jointly study optimal bandwidth allocation between the data channel, modeled as a flat-fading multiple-input single-output (MISO) channel, and the feedback channel for maximum average throughput in the data channel using a beamforming scheme. We consider two models of the partial CSI at the transmitter (CSIT): the noisy CSIT, modeled as jointly Gaussian with the actual channel state, and the quantized CSIT. In the first model, we use distortion-rate theory to relate the CSIT accuracy to the feedback-link bandwidth. In the second model, we derive a lower bound on the achievable rate of the data channel based on the ensemble of a set of random quantization codebooks. We show that in the MISO flat-fading channel case, beamforming based on feedback CSI can achieve an average rate larger than the capacity without CSIT under a wide range of mobility conditions.     

Systematic Codebook Designs for Quantized Beamforming in Correlated MIMO Channels

The full diversity gain provided by a multi-antenna channel can be achieved by transmit beamforming and receive combining. This requires the knowledge of channel state information (CSI) at the transmitter which is difficult to obtain in practice. Quantized beamforming where fixed codebooks known at both the transmitter and the receiver are used to quantize the CSI has been proposed to solve this problem. Most recent works focus attention on limited feedback codebook design for the uncorrelated Rayleigh fading channel. Such designs are sub-optimal when used in correlated channels. In this paper, we propose systematic codebook design for correlated channels when channel statistical information is known at the transmitter. This design is motivated by studying the performance of pure statistical beamforming in correlated channels and is implemented by maps that can rotate and scale spherical caps on the Grassmannian manifold. Based on this study, we show that even statistical beamforming is near-optimal if the transmitter covariance matrix is ill-conditioned and receiver covariance matrix is well-conditioned. This leads to a partitioning of the transmit and receive covariance spaces based on their conditioning with variable feedback requirements to achieve an operational performance level in the different partitions. When channel statistics are difficult to obtain at the transmitter, we propose a universal codebook design (also implemented by the rotation-scaling maps) that is robust to channel statistics. Numerical studies show that even few bits of feedback, when applied with our designs, lead to near perfect CSI performance in a variety of correlated channel conditions.     

Interpolation based transmit beamforming for MIMO-OFDM with limited feedback

Transmit beamforming and receive combining are simple methods for exploiting spatial diversity in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) system. Optimal beamforming requires channel state information in the form of the beamforming vectors for each OFDM subcarrier. This paper proposes a limited feedback architecture that combines beamforming vector quantization and smart vector interpolation. In the proposed system, the receiver sends a fraction of information about the optimal beamforming vectors to the transmitter and the transmitter computes the beamforming vectors for all subcarriers through interpolation. A new spherical interpolator is developed that exploits parameters for phase rotation to satisfy the phase invariance and unit norm properties of the transmitted beamforming vectors. The beamforming vectors and phase parameters are quantized at the receiver and the quantized information is provided to the transmitter. The proposed quantization system provides only a moderate increase in complexity versus over comparable approaches. Numerical simulations show that the proposed scheme performs better than existing diversity techniques with the same feedback data rate.     

The Optimality of Transmit Beamforming: A Unified View

The optimality of transmit beamforming for a multiple antenna system with partial/limited feedback is investigated and a single general necessary and sufficient condition for beamforming to achieve ergodic capacity is derived. The condition obtained is universal - applicable to all partial/limited feedback scenarios in all ergodic fading channel distributions regardless of the number of transmit/receive antennas or transmit power. Using the universal condition we explore the optimality of beamforming for the quantized mean feedback scheme, which unifies previous results for the separate cases of mean feedback and quantized feedback. Numerical results are provided to complement the analysis     

On the Information Rate of MIMO Systems With Finite Rate Channel State Feedback Using Beamforming and Power On/Off Strategy

It is well known that multiple-input multiple-output (MIMO) systems have high spectral efficiency, especially when channel state information at the transmitter (CSIT) is available. In many practical systems, it is reasonable to assume that the CSIT is obtained by a limited (i.e., finite rate) feedback and is therefore imperfect. We consider the design problem of how to use the limited feedback resource to maximize the achievable information rate. In particular, we develop a low complexity power on/off strategy with beamforming (or Grassmann precoding), and analytically characterize its performance. Given the eigenvalue decomposition of the covariance matrix of the transmitted signal, refer to the eigenvectors as beams, and to the corresponding eigenvalues as the beam's power. A power on/off strategy means that a beam is either turned on with a constant power, or turned off. We will first assume that the beams match the channel perfectly and show that the ratio between the optimal number of beams turned on and the number of antennas converges to a constant when the numbers of transmit and receive antennas approach infinity proportionally. This motivates our power on/off strategy where the number of beams turned on is independent of channel realizations but is a function of the signal-to-noise ratio (SNR). When the feedback rate is finite, beamforming cannot be perfect, and we characterize the effect of imperfect beamforming by quantization bounds on the Grassmann manifold. By combining the results for power on/off and beamforming, a good approximation to the achievable information rate is derived. Simulations show that the proposed strategy is near optimal and the performance approximation is accurate for all experimented SNRs.     

Performance Analysis of Quantized Beamforming MIMO Systems

Multiple-input multiple-output (MIMO) wireless systems can achieve significant diversity and array gain by using single-stream transmit beamforming and receive combining. A MIMO beamforming system with feedback using a codebook based quantization of the beamforming vector allows practical implementation of such a strategy in a single-user scenario. The performance of this system in uncorrelated Rayleigh flat fading channels is studied from the point-of-view of signal-to-noise ratio (SNR) and outage probability. In this paper, lower bounds are derived on the expected SNR loss and the outage probability of systems that have a single receive antenna or two transmit antennas. For arbitrary transmit and receive antennas, approximations for the SNR loss and outage are derived. In particular, the SNR loss in a quantized MIMO beamforming system is characterized as a function of the number of quantization bits and the number of transmit and receive antennas. The analytical expressions are proved to be tight with asymptotically large feedback rate. Simulations show that the bounds and approximations are tight even at low feedback rates, thereby providing a benchmark for feedback system design     

Limited Feedback Beamforming Over Temporally-Correlated Channels

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.     

Necessary and sufficient conditions for full diversity order in correlated Rayleigh fading beamforming and combining systems

Transmit beamforming and receive combining are low complexity, linear techniques that make use of the spatial diversity advantage provided by transmitters and/or receivers employing multiple antennas. There has been a growing interest in designing beamforming schemes for frequency division duplexing systems that use a limited amount of feedback from the receiver to the transmitter. This limited feedback conveys a beamforming vector chosen from a finite set known to both the transmitter and receiver. These techniques often use a set of beamforming vectors where the probability of error expression can not be easily formulated or bounded. It is of utmost importance to guarantee that the sets of beamforming and combining vectors are chosen such that full diversity order is achieved. For this reason, necessary and sufficient conditions on the sets of possible beamformers and combiners are derived that guarantee full diversity order in correlated Rayleigh fading.     

On the Capacity and Design of Limited Feedback Multiuser MIMO Uplinks

The theory of multiple-input–multiple-output (MIMO) technology has been well developed to increase fading channel capacity over single-input–single-output (SISO) systems. This capacity gain can often be leveraged by utilizing channel state information at the transmitter and the receiver. Users make use of this channel state information for transmit signal adaptation. In this correspondence, we derive the capacity region for the MIMO multiple access channel (MIMO MAC) when partial channel state information is available at the transmitters, where we assume a synchronous MIMO multiuser uplink. The partial channel state information feedback has a cardinality constraint and is fed back from the basestation to the users using a limited rate feedback channel. Using this feedback information, we propose a finite codebook design method to maximize the sum rate. In this correspondence, the codebook is a set of transmit signal covariance matrices. We also derive the capacity region and codebook design methods in the case that the covariance matrix is rank one (i.e., beamforming). This is motivated by the fact that beamforming is optimal in certain conditions. The simulation results show that when the number of feedback bits increases, the capacity also increases. Even with a small number of feedback bits, the performance of the proposed system is close to an optimal solution with the full feedback.      

Quantized beamforming with channel prediction - transactions letters

This paper investigates a quantized beamforming system with feedback delay. A linear channel predictor is used to cope with the feedback delay. We derive an upper bound on the symbol error rate (SER) of phaseshift keying (PSK) signal. Based on the bound, we design a predictor that provides good error performance. We also demonstrate that the beamformer design methods developed in a delay-free scenario are applicable to the system with feedback delay.     

Index Assignment for Beamforming with Limited-Rate Imperfect Feedback

Beamforming in most multiple-antenna systems often requires channel state information (CSI) at the transmitter through feedback. In practice, CSI must be quantized into a finite set of vectors and feedback only sends the index representing the desired vector. In addition to quantization error of the channel coefficients, feedback errors, which lead to incorrect beam- forming vectors to be applied at the transmitter, also degrade beamforming performance. We present an index-assignment algorithm that minimizes the impact of feedback errors. The proposed algorithm requires exhaustive search to find the best mapping. When the codebook size is large, the complexity of the algorithm becomes prohibitive. We thus propose a group-based index assignment (GIA) that has a low computational load while, still performing better than random index assignments.     

Limited feedback-based block diagonalization for the MIMO broadcast channel

Block diagonalization is a linear preceding technique for the multiple antenna broadcast (downlink) channel that involves transmission of multiple data streams to each receiver such that no multi-user interference is experienced at any of the receivers. This low-complexity scheme operates only a few dB away from capacity but requires very accurate channel knowledge at the transmitter. We consider a limited feedback system where each receiver knows its channel perfectly, but the transmitter is only provided with a finite number of channel feedback bits from each receiver. Using a random quantization argument, we quantify the throughput loss due to imperfect channel knowledge as a function of the feedback level. The quality of channel knowledge must improve proportional to the SNR in order to prevent interference-limitations, and we show that scaling the number of feedback bits linearly with the system SNR is sufficient to maintain a bounded rate loss. Finally, we compare our quantization strategy to an analog feedback scheme and show the superiority of quantized feedback.     

Achievable Rates with Imperfect Transmitter Side Information Using a Broadcast Transmission Strategy

rdquoWe investigate the performance of the broadcast approach for various fading distributions, which correspond to different models of partial transmit channel state information (CSI). The first model considered is the quantized limited feedback. In this model, the receiver can send as feedback only a finite number of bits describing the fading gain. We derive the optimal power allocation for the broadcast approach for the quantized feedback model. For a Rayleigh fading channel, numerical results here show that if the feedback word can be longer than one bit, the broadcasting gain becomes negligible, due to diminished channel uncertainty. The second partial transmit CSI model is a stochastic Gaussian model with mean and variance information, which is commonly used for modeling the channel estimation error. In a single-input single-output (SISO) channel, this model also corresponds to the Ricean fading distribution, for which we derive maximal achievable broadcasting rates. We further consider a multiple-input single-output (MISO) channel, and derive the optimal power allocation strategy in a broadcast approach. Numerical results here show that uniform power allocation is preferable over beamforming power allocation in the region where broadcasting gain over single level coding is non-negligible.     

Capacity of correlated MIMO Rayleigh channels

Abstract-This paper presents some exact results on the capacity of multiple-input-multiple-output (MIMO) channels subject to correlated Rayleigh fading when perfect channel state information (CSI) is known at the receiver. The authors focus on the semicorrelated scenario in which correlation exists either at the transmitter or at the receiver., They consider two cases: 1) the transmitter does not have any CSI and as such allocates power equally among transmitter antennas and 2) the transmitter only knows the statistical distribution of the channel. The first case derives the moment generating function (MGF) of the mutual information (MI) and then deduces from this MGF the mean MI. The authors also study the cumulative distribution function (CDF) of the MI, which can serve as an upper bound to the outage probability under the capacity versus outage formulation when the channel is nonergodic. The second case studies the capacity achieved by optimum power-loading and beamforming schemes based on covariance feedback. Numerical results illustrate that the full capacity of MIMO systems can be preserved even for relatively high values of correlation coefficients.