Flexible codebook design for limited feedback downlink systems via smooth optimization on the Grassmannian manifold

The design procedure of hybrid beamforming is investigated considering multi-user MIMO communication systems taking into account the number of independent data streams for each user. Hybrid precoding and combining are utilized at the transmitter and receiver sides of millimeter-wave multi-user massive MIMO. The obtained results indicate the trade-off in performance based on the ratio of the number of antennas to the number of data streams at the transmitter as well as the receiver. The optimization of the number of parallel data streams per user allows the increase of transmitted data rate while keeping the levels of error vector magnitude (EVM) at acceptable levels.

The multiuser MIMO (MU-MIMO) systems tend to have inter-user interference in practical systems with limited feedback. Practical MU-MIMO systems with limited feedback can be divided into two classes in general. One is zero-forcing beamforming (ZFBF), and the other is per user unitary rate control (PU <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> RC) which is an extended version of random beamforming (RBF). To improve performance, ZFBF needs more feedback bits, and PU <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> RC needs a larger number of users. We propose a new hybrid MU-MISO system which has the advantages of the two MU-MIMO schemes simultaneously, and analyze the sum-rate performance and the quantization error of the hybrid scheme. The analysis shows how the number of feedback bits, the number of users, and the signal power affect the sum-rate. The throughput scaling laws are also derived in the high and the medium SNR regimes. Simulation results show that the proposed hybrid MU-MISO system performs better than either of the two MU-MIMO systems with the same number of feedback bits. Another advantage is that it has lower complexity for codeword search since it has a hierarchical codebook structure.

Communication systems with low-resolution analog-to-digital-converters (ADCs) can exploit channel state information at the transmitter and receiver. This paper presents codebook designs and performance analyses for limited feedback MIMO systems with finite-bit ADCs. A point-to-point single-user channel is firstly considered. When the received signal is sliced by 1-bit ADCs, the absolute phase at the receiver is important to align the phase of the received signals. A new codebook design for beamforming, which separately quantizes the channel direction and the residual phase, is therefore proposed. For the multi-bit case where the optimal transmission method is unknown, suboptimal Gaussian signaling and eigenvector beamforming is assumed to obtain a lower bound of the achievable rate. It is found that to limit the rate loss, more feedback bits are needed in the medium SNR regime than the low and high SNR regimes, which is quite different from the conventional infinite-bit ADC case. Second, a multi-user system where a multiple-antenna transmitter sends signals to multiple single-antenna receivers with finite-bit ADCs is considered. Based on the derived performance loss due to finite-bit ADCs and finite-bit CSI feedback, the number of bits per feedback should increase linearly with the ADC resolution in order to restrict the rate loss.

In Multi-user MIMO system, traditional user scheduling methods are usually aiming at maximizing the sum-rate capacity of the target system. Such methods select the users which have the best channel quality to service but ignore users' satisfaction. In this paper, we proposed a user satisfaction based judgment criteria, and designed a novel scheduling method in order to ensure users' satisfaction. We apply zero-forcing beam forming (ZFBF) precoding into downlink transmission, and assume perfect knowledge of CSI at the base station. Simulation results show that the algorithm achieved a higher system average user satisfaction degree while suffered a little performance loss compared with the classical SUS-ZFBF strategy.

In this paper, we consider practical multiuser multiple-input multiple-output (MU-MIMO) systems where each user has multiple receive antennas and limited channel feedback is available at the base station (BS). The joint combiner and precoding scheme for downlink transmission is proposed, in which we consider the joint use of the receive antenna combiner at each user and precoding at the BS. In the proposed scheme, each user independently chooses the codeword as its preferred-beam vector from the predefined codebook, which maximizes the effective channel gain after the receive antenna combiner. Utilizing the received feedback from users, the BS can easily construct the precoding matrix by zero-forcing beamforming (ZFBF) criterion. In addition, a user selection algorithm is provided to select served users, when the number of users exceeds the number of transmit antennas. Numerical results show that the proposed scheme improves the sum-rate performance of MU-MIMO downlink systems.

Limited feedback techniques, that employ codebook-based quantisation, are used in coordinated multi-point (CoMP) transmission to convey the channel direction information between the mobile user and each base station in the cooperative set and also among the cooperating base stations. However, unlike single-cell multiple-user multiple input multiple output (MU-MIMO) systems, CoMP systems face additional challenges in this codebook design. Among these challenges are the huge feedback overhead and the need for dynamic codebook size. These problems are solved when CoMP uses the so-called per-cell codebook, in which the channel to each cell is quantised separately. However, this results in a further problem, that is having random phase values between the quantised per-cell channels of the same user. In this study, the phase ambiguity problem is explained and its effect on performance degradation is quantified. A new quantisation technique is proposed where the quantisation of each of the real and the imaginary parts is performed independently using the same real codebook. This novel solution prevents the phase ambiguity problem instead of trying to mitigate its effects, in addition to simplifying the codebook design. The proposed quantisation technique outperforms two recently introduced solutions in the literature. The first solution is to quantise phase ambiguity as part of channel state information (CSI), whereas the other avoids phase ambiguity by an iterative selection procedure. Mathematical validations and simulations are provided to verify these findings.

Multimode precoding, where the number of independent data-streams is adapted optimally, can be used to maximize the achievable throughput in multi-antenna communication systems. Motivated by standardization efforts embraced by the industry, the focus of this work is on systematic precoder design with realistic assumptions on the spatial correlation, channel state information (CSI) at the transmitter and the receiver, and implementation complexity. For spatial correlation of the channel matrix, we assume a general channel model, based on physical principles, that has been verified by many recent measurement campaigns. We also assume a coherent receiver and knowledge of the spatial statistics at the transmitter along with the presence of an ideal, low-rate feedback link from the receiver to the transmitter. The reverse link is used for codebook-index feedback and the goal of this work is to construct precoder codebooks, adaptable in response to the statistical information, such that the achievable throughput is significantly enhanced over that of a fixed, non-adaptive, i.i.d. codebook design. We illustrate how a codebook of semiunitary precoder matrices localized around some fixed center on the Grassmann manifold can be skewed in response to the spatial correlation via low-complexity maps that can rotate and scale submanifolds on the Grassmann manifold. The skewed codebook in combination with a lowcomplexity statistical power allocation scheme is then shown to bridge the gap in performance between a perfect CSI benchmark and an i.i.d. codebook design.

Joint precoding selection diversity with limited feedback

Stochastic geometry provides a powerful analytical framework for evaluating interference-limited cellular networks with randomly deployed base stations (BSs). While prior studies have examined limited channel state information at the transmitter (CSIT) and low-resolution analog-to-digital converters (ADCs) separately, their joint impact in multi-user multiple-input multiple-output (MIMO) systems remains largely unexplored. This paper investigates a downlink cellular network in which BSs are distributed according to a homogeneous Poisson point process (PPP), employing zero-forcing beamforming (ZFBF) with limited feedback, and receivers are equipped with one-bit ADCs. We derive a tractable approximation for the achievable spectral efficiency that explicitly accounts for both the quantization error from limited feedback and the receiver distortion caused by coarse ADCs. Based on this approximation, we determine the optimal feedback rate that maximizes the net spectral efficiency. Our analysis reveals that the optimal number of feedback bits scales logarithmically with the channel coherence time but its absolute value decreases due to coarse quantization. Simulation results validate the accuracy of the proposed approximation and confirm the predicted scaling behavior, demonstrating its effectiveness for interference-limited multi-user MIMO networks.

Downlink multi-user transmission techniques that allow several users to be served simultaneously in frequency and time have been recognized as a promising means to increase system capacity. However they require accurate channel state information (CSI) at the base station (BS) such that appropriate signal processing can be performed to separate multiple users in the space domain. In practice however, CSI cannot be perfectly known to the BS due to the capacity-limited feedback channel in FDD systems for instance. In this letter we investigate multiuser multiple-input single-output (MU-MISO) transmission with limited feedback. In particular we identify the useful information that the BS needs to deal with inter-user interference and a codebook design is derived accordingly. In addition, an adaptive threshold-based feedback approach is proposed, where only users with good feedback quality are allowed to do feedback. It is shown that the adaptive scheme can effectively improve the system performance given a fixed number of feedback bits.

We treat the problem of quantization codebook design for MIMO precoding schemes. We propose a design criterion based on the capacity measure which is different from the maximum mutual-minimum-distance conventionally used to design the codebook. The latter criterion is derived from SNR maximization which is not necessarily the capacity- optimal quantization strategy. While the capacity expression does not directly define a valid distance metric for the design, a bound on the capacity expression can be used to formulate the quantization codebook design. We introduce a new distance metric on the Grassmanian manifold which is used to design the optimal precoder codebook. While the original formulation can be used for the (space) correlated channel model, we propose a differential codebook design in order to track changes in the channel statistics when the channel is correlated in time (or in frequency, e.g., across different tones in OFDM systems). Simulation results has shown considerable improvement through the proposed codebook design both for single user (SU-) and multiple user (MU-) MIMO systems.

In this paper, a singular value decomposition based signature matrix inversion (SVD-SI) scheme is proposed for downlink in multiuser MIMO systems, which is more efficient and able to reduce the feedback overhead by limited feedback. The wireless channels are decomposed into several eigenmodes with the right singular vectors as the spatial signatures, which are quantized and fed back to the base station. Then, the operating users are selected according to their signatures, and the multiuser interference is eliminated by the signature matrix inversion scheme. Finally, we characterize the performance of SVD-SI under limited feedback, and give the recommended feedback rate under different scenarios. Numerical results show that the proposed scheme achieves significant throughput improvement while reducing the feedback overhead.

Precoding codebook design for limited feedback MIMO systems is known to reduce to a discretization problem on a Grassmann manifold. The case of two-antenna beamforming is special in that it is equivalent to quantizing the real sphere. The isometry between the Grassmannian G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℂ</sup> and the real sphere <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> shows that discretization problems in the Grassmannian G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℂ</sup> are directly solved by corresponding spherical codes. Notably, the Grassmannian line packing problem in ℂ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , namely maximizing the minimum distance, is equivalent to the Tammes problem on the real sphere, so that optimum spherical packings give optimum Grassmannian packings. Moreover, a simple isomorphism between G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℂ</sup> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> enables to analytically derive simple codebooks in closed-form having low implementation complexity. Using the simple geometry of some of these codebooks, we derive closed-form expressions of the probability density function of the relative SNR loss due to limited feedback. We also investigate codebooks based on other spherical arrangements, such as solutions maximizing the harmonic mean of the mutual distances among the codewords, which is known as the Thomson problem. We find that in some special cases, Grassmannian codebooks based on these other spherical arrangements outperform codebooks from Grassmannian packing.

ABSTRACTIn this paper, several multi‐user scheduling algorithms with limited channel feedback are proposed for multi‐antenna downland channels. With little channel quality information (CQI) and channel direction information (CDI) at multi‐antenna base station, multi‐user scheduling algorithms are carefully designed to maximise the potential system throughput while reducing the feedback load. We first propose a scheduling algorithm that requires a quantised CDI and a threshold CQI. Then we propose a scheduling that needs limited feedback of a threshold CDI and a perfect CQI. Both scheduling algorithms are studied in terms of the achievable throughput and the feedback load. Analytical results show that the proposed scheduling with limited channel feedback has a small performance degradation compared with the scheduling with unlimited channel feedback when the thresholds for user scheduling are judiciously designed. Moreover, under a per user total bit rate constraint of the feedback channel, we propose a scheduling with the quantisation of both CQI and CDI. It is demonstrated that allocating 1 bit to the CQI with an optimal threshold and the remaining bits to the CDI achieves the near optimal sum‐rate performance. Copyright © 2011 John Wiley &amp; Sons, Ltd.

The distributed antenna system is a promising technique to enhance system performance in next-generation broadband wireless mobile communications. The application of linear precoding with space time codeword based on channel state information can further enhance the link quality and reduce the receiver complexity in DASs. However, designing the optimal codebook for DAS to convey the precoding information remains an open issue. This article provides a tutorial addressing various issues of DASs with limited feedback, focusing on the challenges in its codebook design. A suboptimal codebook design is then proposed, suitable for DAS. Furthermore, an effective feedback reduction strategy without performance loss is presented by exploiting the channel correlation property. Numerical results are given to demonstrate the effectiveness of the proposed scheme.