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.     

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.     

Tight Upper Bound on the Outage Probability of QSTBC

Recently, the statistical properties of the equivalent channel representation of a multiple-input-multiple output (MIMO) system employing rate one quasi-orthogonal space-time block codes (QSTBC) for n _T=2ⁿ, nges2 transmit antennas was characterized. It was shown, that these QSTBC were capable of achieving a significant fraction of the outage mutual information of a MIMO system. In this letter, based on the Minkowski's determinant inequality and the Meijer's G-function we derive an upper bound for the fraction of outage probability achieved with QSTBC for the general case of n_T=2 ⁿ, nges2 transmit antennas. Simulations results show that this bound gets tight for all signal-to-noise-ratios (SNR) values by increasing the number of receive antennas     

Transmitter optimization and performance gain for multiple-input single-output systems with finite-rate direction feedback

Consider finite-rate channel-direction feedback in a system with multiple transmit but single receive antennas. We investigate how the transmitter should be optimized for symbol error rate with finite-rate feedback, and how the symbol error rate and outage probability improve as a function of the number of feedback bits. It is found that when the number of feedback directions is equal to or larger than the number of transmit antennas, transmit beamforming is optimal. Otherwise, the antennas should be divided into two groups, where antenna selection is used in the first group to choose the strongest channel, and equal power allocation is used in the second group. At high signal to noise ratio (SNR), the optimal power allocation between these two antenna groups is proportional to the number of antennas in each group. Based on high SNR analysis, we quantify the power gain of each feedback bit. It is shown that the incremental gain increases initially and diminishes when the number of feedback bits surpasses the logarithm (base 2) of the number of transmit antennas.     

Spatial multiplexing with transmit antenna and constellation selection for correlated MIMO fading channels

We consider spatial multiplexing systems in correlated multiple-input multiple-output (MIMO) fading channels with equal power allocated to each transmit antenna. Under this constraint, the number and subset of transmit antennas together with the transmit symbol constellations are determined assuming knowledge of the channel correlation matrices. We first consider a fixed data rate system and vary the number of transmit antennas and constellation such that the minimum margin in the signal-to-noise ratio (SNR) is maximized for linear and Vertical Bell Laboratories Layered Space-Time (V-BLAST) receivers. We also derive transmit antenna and constellation selection criteria for a successive interference cancellation receiver (SCR) with a fixed detection order and a variable number of bits transmitted on each substream. Compared with a system using all available antennas, performance results show significant gains using a subset of transmit antennas, even for independent fading channels. Finally, we select a subset of transmit antennas to maximize data rate given a minimum SNR margin. A lower bound on the maximum outage data rate is derived. The maximum outage data rate of the SCR receiver is seen to be close to the outage channel capacity.     

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     

Performance Analysis of Multiuser Diversity in MIMO Systems with Antenna Selection

In this paper, a framework is presented to analyze the performance of multiuser diversity (MUD) in multiuser point-to-multipoint (PMP) MIMO systems with antenna selection. Based on this framework, the tight closed-form expressions of outage capacity and average symbol error rate are derived for the multiuser transmit antenna selection with maximal-ratio combining (TAS/MRC) system, by which we show how and with what characteristics antenna selection gains, MIMO antenna configurations and fading gains impact on the system performance, with an emphasis on the study of multiuser diversity influence. From both theoretical and simulation results, our study shows that in multiuser PMP TAS/MRC systems an diversity order equals to the product of the number of transmit antennas, number of receive antennas and number of users can be achieved; what's more, users plays a key role in the system performance and can be viewed as equivalent 'virtual" transmit antennas, which is the source of the multiuser diversity inherent exists in the multiuser system. This kind of diversity can be efficiently extracted in the design of multiantenna systems.     

Optimal Successive Group Decoders for MIMO Multiple-Access Channels

We consider a slow-fading narrowband multiple-input multiple-output (MIMO) multiple-access channel (MAC) in which multiple users, each equipped with multiple transmit antennas, communicate to a receiver equipped with multiple receive antennas. The users are unaware of the channel state information (CSI) whereas the receiver has perfect CSI and employs a successive group decoder (SGD). We obtain achievable outage probabilities for the case where an outage must be declared simultaneously for all users (common outage) as well as the case where outages can be declared individually for each user (individual outage). We then derive the optimum successive group decoder (OSGD) that simultaneously minimizes the common outage probability and the individual outage probability of each user, over all SGDs of permissible decoding complexity. For each channel realization, the OSGD is also shown to maximize the error exponent of the decodable set of users. An adaptive SGD is derived which not only retains the outage optimality of the OSGD but also minimizes the expected decoding complexity. Asymptotically tight (in the limit of high signal-to-noise ratio (SNR)) affine approximations are then obtained for the weighted sum common and individual outage capacities and the symmetric outage capacitiy yielded by the OSGD. Limiting expressions for the relevant capacities as the number of users and the number of receive antennas approach infinity are also obtained and it is shown that the OSGD yields symmetric capacity gains commensurate with the decoding complexity allowed. Simulation results with practical low-density parity-check (LDPC) outer codes show that the OSGD offers significantly improved performance at low decoding complexity.      

Outage probability of multi-hop MIMO relaying with transmit antenna selection and ideal relay gain over rayleigh fading channels

We present a study on the outage probability of multi-hop wireless communication systems with multiple-input multiple-output (MIMO) link based on the transmit antenna selection and the maximal-ratio combining (MRC) at the receiver. A nonregenerative system (NS) is investigated with an ideal amplifying gain. MIMO channels are assumed in uncorrelated Rayleigh fading.We derive a moment generating function (MGF) of the reciprocal of the end-to-end signal-to-noise ratio (SNR) and obtain a closed-form approximation on the outage probability through the numerical inversion of a Laplace transform. Numerical results show that the presented outage is exactly matched with the outage probability when assuming the ideal relay gain. For more practical gains, the result is shown to be a lowerbound that gets tight at high average SNR as well as for a small number of hops and/or of antennas. We also compare the outage probabilities of nonregenerative MIMO relaying with a regenerative counterpart for multiple hops.     

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.     

General Capacity Bounds for Spatially Correlated Rician MIMO Channels

This paper considers the capacity of spatially correlated Rician multiple-input multiple-output (MIMO) channels. We consider the general case with double-sided correlation and arbitrary rank channel means. We derive tight upper and lower bounds on the ergodic capacity. In the particular cases when the numbers of transmit and receive antennas are equal, or when the correlation is single sided, we derive more specific bounds which are computationally efficient. The bounds are shown to reduce to known results in cases of independent and identically distributed (i.i.d.) and correlated Rayleigh MIMO channels. We also analyze the outage characteristics of the correlated Rician MIMO channels at high signal-to-noise ratio (SNR). We derive the mean and variance of the mutual information and show that it is well approximated by a Gaussian distribution. Finally, we present numerical results which show the effect of the antenna configuration, correlation level (angle spreads), Rician$K$-factor, and the geometry of the dominant Rician paths.     

Coordinated beamforming with limited feedback in the MIMO broadcast channel

In this paper, we propose a new joint optimization of linear transmit beamforming and receive combining vectors for the multiple-input multiple-output (MIMO) broadcast channel. We consider the transmission of a single information stream to two users with two or more receive antennas. Unlike past work in which iterative computation is required to design the beamformers, we derive specific formulations for the transmit beamformers for two active users via a power iteration and a generalized eigen analysis. To enable practical implementation, a new limited feedback algorithm is proposed that exploits the structure of the algorithm to avoid full channel quantization. The feedback overhead of the proposed algorithm is independent of the number of receive antennas. Monte Carlo simulations are used to evaluate the bit error rate and the sum rate performances of the proposed algorithm. Simulation results show that the proposed method performs close to the sum capacity of the MIMO broadcast channel even with limited feedback.     

On the information throughput and optimized power allocation for MIMO wireless systems with imperfect channel estimation

In this paper, we focus on the throughput analysis, outage evaluation and optimized power allocation for Multiple-Input Multiple-Output (MIMO) pilot-based wireless systems subject to short-term constraints on the radiated power and equipped with a feedback-path for communicating back to the transmitter the imperfect MIMO channel estimates available at the receiver. The case of the ergodic throughput for Gaussian distributed input signals is analyzed, and the conditions for the (asymptotical) achievement of the Shannon capacity are pointed out. The main contributions of this work may be so summarized. First, we develop closed-form analytical expressions for the computation of the ergodic information throughput conveyed by the considered MIMO system for the case of ideal feedback link. Second, we present an iterative algorithm for the optimized power allocation over the transmit antennas that explicitly accounts for the imperfect MIMO channel estimates available at the receiver. Third, after relaxing the assumption of ideal feedback link, we test the sensitivity of the proposed power allocation algorithm on errors possibly introduced by the feedback channel, and then, we numerically evaluate the resulting throughput loss. Finally, we develop closed-form upper and lower bounds on the outage probability that are asymptotically tight.     

On outage and error rate analysis of the ordered V-BLAST

Outage and error rate performance of the ordered BLAST with more than 2 transmit antennas is evaluated for i.i.d. Rayleigh fading channels. A number of lower and upper bounds on the 1st step outage probability at any SNR are derived, which are further used to obtain accurate approximations to average block and total error rates. For m Tx antennas, the effect of the optimal ordering at the first step is an m-fold SNR gain. As m increases to infinity, the BLER decreases to zero, which is a manifestation of the space-time autocoding effect in the V-BLAST. While the sub-optimal ordering (based on the before-projection SNR) suffers a few dB SNR penalty compared to the optimal one, it has a lower computational complexity and a 3 dB SNR gain compared to the unordered V-BLAST and can be an attractive solution for low-complexity/low-energy systems. Uncoded D-BLAST exhibits the same outage and error rate performance as that of the V-BLAST. An SNR penalty of the linear receiver interfaces compared to the BLAST is also analytically evaluated.     

Capacity of a Multiple-Antenna Fading Channel With a Quantized Precoding Matrix

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 of random vector quantization (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 using $B$ bits. We first derive the large system capacity of beamforming (rank-one precoding matrix) as a function of $B$, where large system refers to the limit as $B$ and 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.      

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.     

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 Noncoherent MIMO Channels in the Wideband Regime: Capacity and Reliability

We consider a multiple-input multiple-output (MIMO) wideband Rayleigh block-fading channel where the channel state is unknown to both the transmitter and the receiver and there is only an average power constraint on the input. We compute the capacity and analyze its dependence on coherence length, number of antennas and receive signal-to-noise ratio (SNR) per degree of freedom. We establish conditions on the coherence length and number of antennas for the noncoherent channel to have a "near-coherent" performance in the wideband regime. We also propose a signaling scheme that is near-capacity achieving in this regime. We compute the error probability for this wideband noncoherent MIMO channel and study its dependence on SNR, number of transmit and receive antennas and coherence length. We show that error probability decays inversely with coherence length and exponentially with the product of the number of transmit and receive antennas. Moreover, channel outage dominates error probability in the wideband regime. We also show that the critical as well as cutoff rates are much smaller than channel capacity in this regime     

Characterizing the statistical properties of mutual information in MIMO channels

We consider Gaussian multiple-input multiple-output (MIMO) frequency-selective spatially correlated fading channels, assuming that the channel is unknown at the transmitter and perfectly known at the receiver. For Gaussian codebooks, using results from multivariate statistics, we derive an analytical expression for a tight lower bound on the ergodic capacity of such channels at any signal-to-noise ratio (SNR). We show that our bound is tighter than previously reported analytical lower bounds, and we proceed to analytically quantify the impact of spatial fading correlation on ergodic capacity. Based on a closed-form approximation of the variance of mutual information in correlated flat-fading MIMO channels, we provide insights into the multiplexing-diversity tradeoff for Gaussian code books. Furthermore, for a given total number of antennas, we consider the problem of finding the optimal (ergodic capacity maximizing) number of transmit and receive antennas, and we reveal the SNR-dependent nature of the maximization strategy. Finally, we present numerical results and comparisons between our capacity bounds and previously reported bounds.     

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