Analysis of Multiple Antenna Systems With Finite-Rate Channel Information Feedback Over Spatially Correlated Fading Channels

This paper employs a high resolution quantization framework to study the effects of finite-rate quantization of the channel state information (CSI) on the performance of MISO systems over correlated fading channels. The contributions of this paper are twofold. First, as an application of the general distortion analysis, tight lower bounds on the capacity loss of correlated MISO systems due to the finite-rate channel quantization are provided. Closed-form expressions for the capacity loss in high-signal-to-noise ratio (SNR) and low-SNR regimes are also provided, and their analysis reveals that the capacity loss of correlated MISO channels is related to that of i.i.d. fading channels by a simple multiplicative factor which is given by the ratio of the geometric mean to the arithmetic mean of the eigenvalues of the channel covariance matrix. Second, this paper extends the general asymptotic distortion analysis to the important practical problem of suboptimal quantizers resulting from mismatches in the distortion functions, source statistics, and quantization criteria. As a specific application, two types of mismatched MISO CSI quantizers are investigated: quantizers whose codebooks are designed with minimum mean square error (MMSE) criterion but the distortion measure is the ergodic capacity loss (i.e., mismatched design criterion), and quantizers with codebook designed with a mismatched channel covariance matrix (i.e., mismatched statistics). Bounds on the channel capacity loss of the mismatched codebooks are provided and compared to that of the optimal quantizers. Finally, numerical and simulation results are presented and they confirm the tightness of theoretical distortion bounds.     

MIMO Transmit Beamforming Under Uniform Elemental Power Constraint

We consider multi-input multi-output (MIMO) transmit beamforming under the uniform elemental power constraint. This is a nonconvex optimization problem, and it is usually difficult to find the optimal transmit beamformer. First, we show that for the multi-input single-output (MISO) case, the optimal solution has a closed-form expression. Then we propose a cyclic algorithm for the MIMO case which uses the closed-form MISO optimal solution iteratively. The cyclic algorithm has a low computational complexity and is locally convergent under mild conditions. Moreover, we consider finite-rate feedback methods needed for transmit beamforming. We propose a simple scalar quantization method, as well as a novel vector quantization method. For the latter method, the codebook is constructed under the uniform elemental power constraint and the method is referred as VQ-UEP. We analyze VQ-UEP performance for the MISO case. Specifically, we obtain an approximate expression for the average degradation of the receive signal-to-noise ratio (SNR) caused by VQ-UEP. Numerical examples are provided to demonstrate the effectiveness of our proposed transmit beamformer designs and the finite-rate feedback techniques.     

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.     

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     

A performance analysis framework for limited feedback beamforming in correlated fading

Limited or finite rate, feedback is an efficient way to implement beamforming in multiple antenna systems using frequency division duplexing. Unfortunately, closed-form performance analysis of limited feedback beamforming has not been investigated. This paper provides an analytical framework for the correlated limited feedback beamforming problem by treating selection of the beamforming vector from the codebook as a multibranch selection problem.     

Design and analysis of transmit-beamforming based on limited-rate feedback

This paper deals with design and performance analysis of transmit beamformers for multiple-input multiple-output (MIMO) systems based on bandwidth-limited information that is fed back from the receiver to the transmitter. By casting the design of transmit beamforming based on limited-rate feedback as an equivalent sphere vector quantization (SVQ) problem, multiantenna beamformed transmissions through independent and identically distributed (i.i.d.) Rayleigh fading channels are first considered. The rate-distortion function of the vector source is upper-bounded, and the operational rate-distortion performance achieved by the generalized Lloyd's algorithm is lower-bounded. Although different in nature, the two bounds yield asymptotically equivalent performance analysis results. The average signal-to-noise ratio (SNR) performance is also quantified. Finally, beamformer codebook designs are studied for correlated Rayleigh fading channels, and a low-complexity codebook design that achieves near-optimal performance is derived.     

On the performance of random vector quantization limited feedback beamforming in a MISO system

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     

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     

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.     

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.     

Grassmannian beamforming for MIMO amplify-and-forward relaying

We consider the problem of beamforming codebook design for limited feedback half-duplex multiple-input multiple output (MIMO) amplify-and-forward (AF) relay system. In the first part of the paper, the direct link between the source and the destination is ignored. Assuming perfect channel state information (CSI), we show that the source and the relay should map their signals to the dominant right singular vectors of the source-relay and relay-destination channels. For the limited feedback scenario, we prove the appropriateness of Grassmannian codebooks as the source and relay beamforming codebooks based on the distributions of the optimal source and relay beamforming vectors. In the second part of the paper, the direct link is considered in the problem model. Assuming perfect CSI, we derive the optimization problem that identifies the optimal source beamforming vector and show that the solution to this problem is uniformly distributed on the unit sphere for independent and identically distributed (i.i.d) Rayleigh channels. For the limited feedback scenario, we justify the appropriateness of Grassmannian codebooks for quantizing the optimal source beamforming vector based on its distribution. Finally, a modified quantization scheme is presented, which introduces a negligible penalty in the system performance but significantly reduces the required number of feedback bits.     

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.     

Optimal pruning with applications to tree-structured source codingand modeling

An algorithm introduced by L. Breiman et al. (1984) in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved. These include variable-rate and minimum-entropy tree-structured vector quantization, minimum expected cost decision trees, variable-order Markov modeling, optimum bit allocation, and computer graphics and image processing using quadtrees. A concentration on the first of these and a detailed analysis of variable-rate tree-structured vector quantization are provided. It is found that variable-rate tree-structured vector quantization outperforms not only the fixed-rate variety but also full-search vector quantization. The successive approximation character of variable-rate tree-structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder. This has applications to the problem of buffer overflow     

Channel Adaptive Quantization for Limited Feedback MIMO Beamforming Systems

Multiple-input multiple-output (MIMO) wireless systems can achieve significant diversity and array gain by using transmit beamforming and receive combining techniques. In the absence of full channel knowledge at the transmitter, the transmit beamforming vector can be quantized at the receiver and sent to the transmitter using a low-rate feedback channel. In the literature, quantization algorithms for the beamforming vector are designed and optimized for a particular channel distribution, commonly the uncorrelated Rayleigh distribution. When the channel is not uncorrelated Rayleigh, however, these quantization strategies result in a degradation of the receive signal-to-noise ratio (SNR). In this paper, switched codebook quantization is proposed where the codebook is dynamically chosen based on the channel distribution. The codebook adaptation enables the quantization to exploit the spatial and temporal correlation inherent in the channel. The convergence properties of the codebook selection algorithm are studied assuming a block-stationary model for the channel. In the case of a nonstationary channel, it is shown using simulations that the selected codebook tracks the distribution of the channel resulting in improvements in SNR. Simulation results show that in the case of correlated channels, the SNR performance of the link can be significantly improved by adaptation, compared with nonadaptive quantization strategies designed for uncorrelated Rayleigh-fading channels     

Transmit beamforming for space-frequency coded MIMO-OFDM systems with spatial correlation feedback

This paper addresses the problem of joint optimization of transmit beamforming and space-frequency (SF) coding for MIMO-OFDM systems with spatial correlation feedback in broadband communications. This problem is challenging in the sense that the transmitter should be designed to beamform across multiple eigenspaces associated with the multipath environment simultaneously. With arbitrary transmit spatial correlation, the performance analysis for SF-coded MIMO-OFDM systems with beamforming is provided, and a general optimization problem for the beamforming design is formulated. Three suboptimal approaches to design the beamformer based on the derived design criteria are proposed: i) eigenvalue selection scheme; ii) eigenspace selection scheme; and iii) per-subcarrier approach based on decoding at each subcarrier. The proposed schemes take into account the multiple eigenspace information associated with the multipath-delay channel. Improvement in the performance over SF coding without beamforming is shown through simulations in terms of bit error rate. The eigenvalue selection scheme provides the best performance among the proposed algorithms. This scheme locates the subspace associated with the largest eigenvalues in the eigenspace of the covariance matrices. With the eigenvalue selection scheme, the performance improvement is about 3 dB over the SF coding without beamforming for highly correlated channels as shown in our simulations.     

Unitary space vector quantization codebook design for spatial correlated limited feedback MIMO system

In the transmitting, beamforming, and receiving combing （TBRC） MIMO system, a codebook based feedback strategy is usually used to provide the transmitter with the beamforming vector. The adopted codebook affects the system performance considerably. Therefore, the codebook design is a key technology in the TBRC MIMO system. In this article, the unitary space vector quantization （USVQ） codebook design criterion is proposed to design optimal codebooks for various spatial correlated MIMO channels. And the unitary space K-mean （USK） codebook generating algorithm is provided to generate the USVQ codebooks. Simulations show that the capacities of the feedback based TBRC systems using USVQ codebooks are very close to those of the ideal cases.     

Recursive and Trellis-Based Feedback Reduction for MIMO-OFDM with Rate-Limited Feedback

We investigate an adaptive MIMO-OFDM system with a feedback link that can only convey a finite number of bits. We consider three different transmitter configurations: i) beamforming applied per OFDM subcarrier, ii) precoded spatial multiplexing applied per subcarrier, and iii) precoded orthogonal space time block coding applied per subcarrier. Depending on the channel realization, the receiver selects the optimal beamforming vector or precoding matrix from a finite-size codebook on each subcarrier, and informs the transmitter through finite-rate feedback. Exploiting the fact that the channel responses across OFDM subcarriers are correlated, we propose two methods to reduce the amount of feedback. One is recursive feedback encoding that selects the optimal beamforming/precoding choices sequentially across the subcarriers, and adopts a smaller-size time-varying codebook per subcarrier depending on prior decisions. The other is trellis-based feedback encoding that selects the optimal decisions for all subcarriers at once along a trellis structure via the Viterbi algorithm. Our methods are applicable to different transmitter configurations in a unified fashion. Simulation results demonstrate that the trellis-based approach outperforms the recursive method as well as an existing interpolation-based alternative at high signal-to-noise-ratio, as the latter suffers from "diversity loss"     

Enhanced Multi-mode Transmission by User Scheduling in MISO Broadcast Channels with Finite-Rate Feedback

This paper investigates the user selection strategy in multiuser downlinks using zero-forcing beamforming (ZFBF) and finite-rate feedback (FRF). In order to mitigate the interference-limited effect in ZFBF-FRF systems, we propose an efficient user scheduling scheme combined with adaptive transmission mode selection strategy. In this scheme, each user first evaluates its preferred transmission mode and the corresponding achievable rate according to a derived tight lower bound of the rate. Given such information on the users through feedback, the BS then determines the global transmission mode of the system and selects users for simultaneous transmission. Asymptotic analysis shows that limited feedback beamforming to a single user is the best choice at both low and high SNR regimes. Simulation results demonstrate individual effects of different system parameters on the sum rate performance by the proposed scheme.     

Error performance of transmit beamforming with delayed and limited feedback

Feedback delay can severely affect the performance of transmit beamforming (TB) and the analytical quantification of the performance degradation has attracted much research interest recently. In this letter, we study the effect of delayed and limited-rate codebook index feedback on the error rate performance of TB systems over Rayleigh fading channels. We derive closed-form expressions for the moment generating function (MGF) and the probability density function (PDF) of the receiver output signal-to-noise ratio (SNR) including the effects of outdated and finite-rate feedback and further provide accurate analytical error rate expressions, which are verified by simulation results. The coding gain gap between the full-rate and limited-rate feedback and the coding gain advantage of multiple transmit antennas to the single antenna are analyzed, for limited and delayed feedback. These results are simple and concise and provide new analytical insight into the achievable diversity and combining gains and the loss caused by feedback delay for different system parameters and modulation formats.     

Quantized feedback information in orthogonal space-time block coding

This work considers how the presence of quantized channel information obtained from a feedback link may be utilized for determining a transmit weighting matrix that improves the performance of a predetermined orthogonal space-time block (OSTB) code. To reduce the effects of feedback delay, quantization errors and feedback channel bit errors, methods based on vector quantization for noisy channels are used in the design of the feedback link. The resulting transmission scheme and feedback link take the imperfect nature of the channel information into account while combining the benefits of conventional beamforming with those provided by OSTB coding.