In recent years, the unmanned aerial vehicle (UAV) communications have become an important part of the space-air-ground integrated network. Unfortunately, the high mobility, as well as perturbation, of UAV poses a great challenge in aligning narrow high-gain beams between the UAV and base station (BS). To tackle this challenging issue, we propose efficient beam prediction and tracking solutions from the perspective of control in this paper. First of all, for an important and typical flight mode in practice (i.e., the mission flight mode - to assign a series of targets in advance and fly from one target to the next one in turn), we study in depth the underlying control principle and reveal important properties and relationships between beam direction and controlled variables. Then, to exploit the properties and relationships revealed, we propose an efficient learning-based beam prediction and tracking solution. Specifically, we develop an efficient learning model, together with offline training and online inference algorithms. To further reduce the computational complexity, we distinguish two kinds of beam offsets and prove an important property of the mission flight mode, i.e., a multicopter almost keeps fixed attitude and velocity in most part of a flight process, based on which an efficient algorithm is designed. Comprehensive experiment results from open-source software, hardware and real UAV confirm the effectiveness of our control-assisted approach.

In unmanned aerial vehicle (UAV) communications, beam tracking faces significant challenges due to persistent variations in mobile users’ positions and flight attitudes. Fixed tracking periods result in either unnecessary or insufficient overhead depending on whether the UAV’s relative motion is static or dynamic. To address these challenges, this paper proposes a two-level beam tracking scheme enabled by data-driven beam prediction. In the upper layer, a time-series prediction method is used to adaptively adjust the tracking interval by forecasting confidence intervals of future communication quality. In the lower layer, a threshold-based probing beam selection method is implemented, which selects probing beams whose conditional likelihoods exceeding a predefined threshold. Simulation results demonstrate that the total beam training overhead drop 43.37 %, while the average outage probability decreases from 6.229 % to 0.095 %.

Because of the flexible scheduling, improved reliability, enhanced capacity over much wider range, the unmanned aerial vehicle (UAV) communications have become an important part of the space-air-ground integrated network. However, the high mobility and perturbation of UAV impose a challenge on aligning narrow beams between the UAV and another node, such as the base station (BS). Although the position and attitude of UAV have been exploited to develop beam tracking algorithms, they belong to current or past information, which often provide limited performance improvement in the high-mobility scenario. To tackle this challenging issue, we, for the first time, identify a kind of important but ready-made information - the command or control sequence (CCS) provided by the flight control system (FCS). We explain in detail that CCS provides real and direct (rather than estimated) future information for beam prediction. Then, we propose an efficient learning-based algorithm to exploit the information. In particular, we prove theoretically that the convolutional neural network (CNN) is an appropriate choice of the network structure within the nonlinear prediction model. Experiment results from practical real UAVs confirm the effectiveness and superiority of our proposal.

The sixth-generation (6G) wireless network will bring major advances in ubiquitous connectivity, support massive device deployments, and open up new opportunities for diverse and decentralized Internet of things (IoT) applications. However, it also raises inherent trust issues, as IoT devices controlled by various independent entities cannot fully trust each other, thereby complicating secure and efficient random access control. To address this challenge, existing research turns to blockchain technology as a promising secure and trusted solution. This paper aims to explore and analyze blockchain-enhanced random access management from a game-theoretic perspective. We first investigate the well-known rogue’s dilemma during random access process, where selfish devices exploit access protocols and reduce system fairness. Next, we construct a two-device mixed-strategy game to illustrate the strategic interaction between honest and selfish IoT devices. To capture realistic behaviors under bounded rationality, we extend the analysis to a multi-device scenario with evolutionary game theory and replicator dynamics. Numerical results show that unregulated selfish behavior severely reduces IoT network efficiency. In contrast, blockchain-based penalties effectively discourage malicious actions and promote stable and efficient access equilibria.

In this paper, we introduce third-party intelligent reflecting surfaces (TIRSs) into the physical layer security aware wireless communication system, where a central legitimate transmitter is designed to transmit secret signals to a group of legitimate receivers in the presence of the threat from an active eavesdropper (EV). Due to the channel reshaping ability of TIRSs, they are able to not only help legitimate pairs (LPs) enhance the secure transmission rate but also assist EV in improving the eavesdropping performance. Furthermore, with the potential selfishness, TIRSs may dynamically choose to ally with LPs or EV in exchange for potential benefits (e.g., payoffs). This leads to complex dynamic ally-adversary relationships among LPs, EV, and TIRSs under unpredictable wireless channel conditions. To address this issue, we formulate a repeated coalition formation game (RCFG) with dynamic decision-making to model the long-term strategic interactions among LPs, EV, and TIRSs. In particular, we theoretically analyze the existence of Nash equilibrium in the formulated RCFG, and then propose a switch operations-based coalition selection along with a deep reinforcement learning (DRL)-based approach for obtaining such an equilibrium. Simulations examine the feasibility of the proposed approach and show its superiority over counterparts.

Extremely large-scale multiple-input multiple-output (XL-MIMO) communication systems introduce a new communication paradigm called near-field communications, which identifies users’ location within the near-field (Fresnel’s region). In the near-field, beams can be steered in the angle and distance dimensions, resulting in an enormous codebook and a prolonged two-dimensional beam alignment (BA) process. To keep a low BA overhead while achieving low BA error, in this paper, we design a novel hierarchical codebook and a BA scheme for near-field XL-MIMO systems. Specifically, we first propose a novel spatial partition where the angle-offset effect is revealed and leveraged to improve the beam gain inside the coverage area. Based on the partition, we design distance-coarse and focusing beams. Distance-coarse beams are leveraged to construct the high level of the codebook for angle dimension alignment. In contrast, focusing beams construct the last level codebook for distance dimension alignment. Corresponding to the proposed codebook structure, our BA scheme is a tree search consisting of two stages: the angle aligning stage and the distance aligning stage. Next, we formulate the desired codebook design problem as difference convex optimization problems, where three beam design guidelines are considered to minimize the BA error rate raised by the near-field angle-offset effect. After that, the proposed optimization problem is solved by the constrained concave-convex procedure. Numerical simulations verify the angle offset effect and our designed near-field beam. Furthermore, we show that our BA scheme only utilizes one percent of overhead but achieves a lower BA error rate than exhaustive searching.

Beam prediction and tracking (BPT) are key technology for millimeter wave communications. Typical techniques include Kalman filtering (KF) and Gaussian process (GP) regression. However, KF requires explicit system dynamics, which is difficult to obtain for complicated scenarios. In contrast, thanks to the data-driven manner, GP regression circumvents this challenging, which, however, suffers from prohibitive computational complexity. To tackle this issue, we propose a novel hybrid model and data driven approach, referred to as data-induced intelligent Kalman filtering (DIIKF). DIIKF learns the system dynamics via the data-driven manner, which can enjoy the advantages of both KF and GP while overcoming their drawbacks. In view that the system dynamics is available, we further propose long-term prediction and design an efficient algorithm. Simulation results show that our method approaches the optimal oracle solution (in terms of effective achievable rate), with the linear complexity order.

By sharing the same hardware platform, spectral resource as well as transmit waveform, dual-functional radar-communication (DFRC) based integrated sensing and communication (ISAC) framework has been envisioned as a key technology for future wireless networks. Most DFRC beamforming works focus on block-level precoding, which fails to exploit constructive interference. To tackle this issue, we propose symbol-level joint radar sensing and communication beamforming algorithms in this paper. First, we formulate the problem of joint radar-communication beamforming based on symbol-level precoding (SLP) by incorporating constructive interference into SLP, so as to improve the energy efficiency. To address the formulated problem, we tailor a highly parallelizable iterative algorithm, which is shown to converge to stationary points. To achieve better performance, we further propose an efficient recursive optimization algorithm. In particular, the recursive algorithm monotonously improves the performance of interest as the recursive procedure proceeds.

The current multiple-input multiple-output (MIMO) systems are still mainly implemented based on the centralized architecture, which thus has to process a huge amount of base-band data. In particular, the central processing unit (CPU) needs a large bus bandwidth to accommodate the prohibitive baseband data transmission, which hinders effective system implementation, especially for the massive MIMO systems. Moreover, the centralized scheme lacks flexibility and scalability when facing varying sizes of antenna arrays and diverse applications. This paper proposes an efficient decentralized bidirectional-chain (DBC) equalizer architecture. The advantages of the DBC architecture are two-fold. First, it can reduce the data traffic transmitted from the antennas to the processing unit by categorizing them into clusters, each of which is equipped with a local processing unit (LPU). Second, it can reduce the time delay by updating all clusters in parallel. To sufficiently exploit the proposed DBC architecture, we further propose efficient parallel iterative algorithms. The DBC-based parallelizable iterative algorithms achieve the state-of-the-art performance in terms of convergence rate and bit error rate. Finally, simulation results are provided to confirm the effectiveness and superiority of our proposal.

Beam training and tracking (BTT) are key technologies for millimeter wave communications. However, since the effectiveness of BTT methods heavily depends on wireless environments, complexity and randomness of practical environments severely limit the application scope of many BTT algorithms and even invalidate them. To tackle this issue, from the perspective of stochastic process (SP), in this paper we propose to model beam directions as a SP and address the problem of BTT via process inference. The benefit of the SP design methodology is that environment priors and uncertainties can be naturally taken into account (e.g., to encode them into SP distribution) to improve prediction efficiencies (e.g., accuracy and robustness). We take the Gaussian process (GP) as an example to elaborate on the design methodology and propose novel learning methods to optimize the prediction models. In particular, beam training subset is optimized based on derived posterior distribution. The GP-based SP methodology enjoys two advantages. First, good performance can be achieved even for small data, which is very appealing in dynamic communication scenarios. Second, in contrast to most BTT algorithms that only predict a single beam, our algorithms output an optimizable beam subset, which enables a flexible tradeoff between training overhead and desired performance. Simulation results show the superiority of our approach.

A recently proposed unified precoding and pilot design optimization (UPPiDO) framework offers a reduction in both training and feedback overhead of acquiring channel state information (CSI) and an enhancement in robustness (to CSI uncertainties) at the expense of a more computationally demanding precoding optimization. To address this increased complexity, in this paper we first propose an unfolding-friendly iterative algorithm, which can efficiently address a family of non-convex and non-smooth problems. Then, we develop an efficient approach to unfold the iterative algorithm designed. Besides being applicable to important and typical iterative optimization algorithms, a pivotal advantage of the proposed unfolding approach is that the trainable parameters are scalars (rather than matrices). This, in turn, reduces the number of training samples required and makes it suitable for rapidly fluctuating wireless environments. We apply the algorithm unfolding (AU) techniques developed to our UPPiDO-based symbol-level precoding and block-level precoding. Our complexity analysis indicates that the computational complexity is scalable both with the numbers of served users and antennas. Our simulation results demonstrate that the number of outer iterations (or layers) required is about 1/3 of that of the original iterative algorithms.

Optimal pilot design to acquire channel state information (CSI) is of critical importance for FDD downlink massive MIMO systems, and is still an open problem. To tackle this issue, in this paper we propose a two-stage precoding approach based on reduced CSI (rCSI-TSP) design framework and an efficient algorithm, whose core is to obtain an optimal precoder while also sparsifying physical CSI (pCSI), so as to save on CSI estimation. The advantages of the rCSI-TSP framework are three-fold. First, the framework enables to simultaneously extract and exploit statistical and instantaneous CSI. Second, it guarantees the most needed rCSI can be obtained and thus avoids performance loss due to heuristic pilot design. Third, we tailor an efficient online deep-learning based method for the TSP framework, which paves the way for practical applications. As an example, we apply the framework to the multi-user symbol-level precoding (SLP) and verify performance improvements.

Symbol detection (SD) plays an important role in a digital communication system. However, most SD algorithms require channel state information (CSI), which is often difficult to estimate accurately. As a consequence, it is challenging for these SD algorithms to approach the performance of the maximum likelihood detection (MLD) algorithm. To address this issue, we employ both semi-supervised learning and ensemble learning to design a flexible parallelizable approach in this paper. First, we prove theoretically that the proposed algorithms can arbitrarily approach the performance of the MLD algorithm with perfect CSI. Second, to enable parallel implementation and also enhance design flexibility, we further propose a parallelizable approach for multi-output systems. Finally, comprehensive simulation results are provided to demonstrate the effectiveness and superiority of the designed algorithms. In particular, the proposed algorithms approach the performance of the MLD algorithm with perfect CSI, and outperform it when the CSI is imperfect. Interestingly, a detector constructed with received signals from only two receiving antennas (less than the size of the whole receiving antenna array) can also provide good detection performance.

Large-scale antenna array techniques are key enablers for modern wireless communication systems. Channel state information (CSI) is indispensable for large-scale multi-antenna systems, but is challenging to obtain. To tackle this issue, in this paper we propose a unified precoding and pilot design framework, that allows minimal and precoding-sensitive modified CSI (mCSI) to be collected. This results in a significant reduction in the CSI overheads and complexity compared to classical physical CSI (pCSI) estimation. Based on this unified framework, we further propose an intelligent pilot (IP) approach that senses and selects the mCSI to be collected. The IP approach utilizes a compressive sensing formulation to attach sensing and selection of significant mCSI to precoding optimization. We apply the above techniques to the multi-user frequency division duplexing (FDD) downlink as an example. Our study shows that the advantages of the IP approach are three-fold. First, in contrast to the pCSI, precoding-sensitive information is only captured, which reduces the training and feedback overheads. Second, the precoders are optimized directly based on the mCSI, which avoids recovering the pCSI of high-dimension. Third, since the mCSI of reduced dimension is utilized, the scale of the problem to optimize the precoder is also reduced and thus it is much easier to solve.

The use of large-scale antenna arrays poses great difficulties in obtaining perfect channel state information (CSI) in multi-antenna communication systems, which is essential for precoding optimization. To tackle this issue, in this paper we propose a probabilistic-learning based approach (PLA), aiming at alleviating the requirement of perfect CSI. The rationale is that the existing precoding algorithms that output a single precoder are often overconfident in their abilities and the obtained CSI. To avoid overconfidence, we incorporate the idea of regularization in machine learning (ML) into precoding models, so as to limit representative abilities of the precoding models. Compared to the state-of-the-art robust precoding designs, an important advantage of PLA is that CSI uncertainty models are not required. As a specific application of PLA, we design an efficient robust symbol-level hybrid precoding algorithm for the millimeter wave system and confirm the effectiveness of PLA via simulations.

Millimeter wave (mmwave) communications, envisaged for the next generation wireless networks, rely on large antenna arrays and very narrow, high-gain beams. This poses significant challenges to beam alignment between transmitter and receiver, which has attracted considerable research attention. Even when alignment is achieved, the link is subject to beam drift (BD). BD, caused by non-ideal features inherent in practical beams and rapidly changing environments, is referred to as the phenomenon that the center of main-lobe of the used beam deviates from the real dominant channel direction, which further deteriorates the system’s performance. To mitigate the BD effect, in this paper we first theoretically analyze the BD effect on the performance of outage probability as well as effective achievable rate, which takes practical factors (e.g., the rate of change of the environment, beam width, transmit power) into account. Then, different from conventional practice, we propose a novel design philosophy where multi-resolution beams with varying beam widths are used for data transmission while narrow beams are employed for beam training. Finally, we design an efficient learning based algorithm which can adaptively choose an appropriate beam width according to the environment. Simulation results demonstrate the effectiveness and superiority of our proposals.

Millimeter wave communications, equipped with large-scale antenna arrays, are able to provide Gbps data rates by exploring abundant spectrum resources. However, the use of a large number of antennas along with narrow beams causes a large overhead in obtaining channel state information (CSI) via beam training, especially for fast-changing channels. To reduce beam training overhead, in this paper we develop an interactive learning design paradigm (ILDP) that makes full use of domain knowledge of wireless communications (WCs) and adaptive learning ability of machine learning (ML). Specifically, the ILDP is fulfilled via deep reinforcement learning (DRL), which yields DRL-ILDP, and consists of communication model (CM) module and adaptive learning (AL) module, which work in an interactive manner. Then, we exploit the DRL-ILDP to design efficient beam training algorithms for both multi-user and user-centric cooperative communications. The proposed DRL-ILDP based algorithms enjoy three folds of advantages. Firstly, ILDP takes full advantages of the existing WC models and methods. Secondly, ILDP integrates powerful ML elements, which facilitates extracting interested statistical and probabilistic information from environments. Thirdly, via the interaction between the CM and AL modules, the algorithms are able to collect samples and extract information in real-time and sufficiently adapt to the ever-changing environments. Simulation results demonstrate the effectiveness and superiority of the designed algorithms.

Millimeter wave (mmwave) communications have attracted increasing attention thanks to the abundant spectrum resource. The short wave-length of mmwave signals facilitates exploiting large antenna arrays to achieve large array gains and combat the large path-loss. However, the use of large antenna arrays along with narrow beams leads to a large overhead in beam training for obtaining channel state information, especially in dynamic environments. To reduce the overhead of beam training, in this paper we formulate the problem of beam alignment and tracking (BA/T) as a stochastic bandit problem. In particular, to sense the change of the environments, the actions are designed based on the offset of successive beam indexes (i.e., beam index difference), which measures the rate of change of the envir-onments. Then, we propose two efficient BA/T algorithms based on the stochastic bandit learning. To reveal useful insights, the performance of effective achievable rate is further analyzed for the proposed BA/T algorithms. The analytical results show that the algorithms can sense the change of the environments and adjust beam training strategies intelligently. In addition, they do not require any priori knowledge of dynamic channel modeling, and thus are applicable to a variety of complicated scenarios. Simulation results demonstrate the effectiveness and superiority of the proposed algorithms.

The use of the millimeter wave (mmwave) spectrum for next generation mobile communication systems has gained significant attention recently. Large antenna arrays along with beamforming techniques are required to combat the large path-loss at mmwave frequencies. However, the existing beam designs often cause a large peak to average power ratio, and thus require power-inefficient power amplifiers (PAs). In this paper, we propose power-efficient beam design methods that facilitate the use of power-efficient PAs. Specifically, we design digital and hybrid analog-digital mmwave beams that possess a per-antenna constant envelope (PACE) and thus are highly power-efficient. Meanwhile, we also minimize the ripples in the mainlobe and sidelobe of the beams and consider both infinite and finite resolution phase shifters. To this end, we first propose an efficient feasible point search method to provide a feasible solution for the considered difficult beam design problem. Then, a novel hybrid analog-digital mapping algorithm is developed to map a designed digital beam to a hybrid analog-digital beam. To achieve better performance, we propose an improved hybrid analog-digital beam design method employing further optimization based on the feasible point. The proposed method is applicable to both infinite-resolution and finite-resolution phase shifters. Comprehensive simulation results are provided to demonstrate the effectiveness and superiority of the proposed beam designs.

Millimeter wave (mmwave) communication has attracted increasing attention owing to its abundant spectrum resource. The short wave-length of mmwave signals facilitates exploiting large antenna arrays to achieve large array gains and combat large path-loss. However, the use of large antenna arrays and narrow beams leads to a large overhead in beam training for obtaining channel state information, especially in dynamic environments. To reduce the overhead of beam training, in this paper we propose an environment sensing based beam training algorithm via deep reinforcement learning. The proposed algorithm can sense the change of the environment and learn required latent probability information from the environment, and intelligently trains beams with a low overhead. In addition, the proposed algorithm does not require any priori knowledge of dynamic channel modeling, and thus is applicable to a variety of complicated scenarios. Simulation results demonstrate the effectiveness and superiority of the proposed intelligent beam training algorithm.

The hardware implementation of large-scale multi-antenna systems requires power-efficient power amplifiers (PAs). However, the existing beamforming designs often cause a large peak-to-average power ratio and have to rely on power-inefficient PAs. In this paper, we propose a unified power-efficient beamforming design framework, which incorporates per-antenna constant envelope constraints to improve the power efficiency. We further propose an efficient algorithm named "sequential outer approximation procedure" (SOAP) to search a feasible point. Power-efficient design for beam pattern synthesis is developed based on SOAP.

Millimeter wave (mmwave) communication operating in the band of 30-300 GHz is promising to provide Gbps data rates owing to its abundant spectrum resource, and has attracted increasing attention. Cooperative transmission, by converting undesired interferences into useful signals, is able to further improve performance of mmwave systems. In this paper, we propose a novel cooperative transmission scheme for mmwave communication networks, where each mobile user is cooperatively served by multiple access points (APs) that use hybrid precoders. Our goal is to maximize the system energy efficiency, which leverages on a joint design of the hybrid precoders of all APs. The formulated problem is a difficult nonlinear fractional programming subject to unit modulus constraints. We propose an efficient algorithm by incorporating penalty decomposition and block coordinate descent methods. Numerical results are provided to confirm the effectiveness of the proposed algorithm and reveal some important insights.

Millimeter wave (mmwave) communication has attracted increasing attention owing to its abundant spectrum resource. The short wavelength at mmwave frequencies facilitates placing a large number of antennas in a small space, and the mmwave channels are likely to be sparse in the directions. These two new features promise enhanced security by directional precoding. To explore this potential, we investigate the design of directional hybrid digital and analog precoding for the multiuser mmwave communication system with multiple eavesdroppers. Particularly, we consider two cost-efficient sub-connected hybrid architectures, i.e., multi-subarray architecture and switched-phased-array architecture, and optimize the hybrid precoding under per-antenna constant envelope (CE) constraints. The goal of our design is to guarantee the receive quality of the legitimate users while minimizing the power leaked to the eavesdroppers, so as to realize a directional transmission for a general mmwave channel. The resulting problems are very challenging due to the nonlinear CE constraints and binary integer constraint from antenna selection. To address them, we leverage exact penalty function methods to find efficient solutions to the CE hybrid directional precoding. Our analysis shows that the proposed algorithm is able to converge to a stationary point under some mild conditions. Simulation results are finally provided to confirm the effectiveness of the proposed schemes and their superiority over the existing schemes under both single-path and multi-path mmwave channels.

Large-scale antenna arrays which provide high beamforming gain are commonly used to combat the serious path-loss in millimeter wave (mmwave) systems. Traditionally, the beamforming is completely implemented at the baseband or the digital domain, which, however, causes high hardware cost and power consumption. The hybrid precoding method which can effectively avoids these problems is, therefore, more attractive. However, the design of hybrid precoder is challenging due to the constant modules of phase shifters. To solve this problem, we propose a novel hybrid precoding approach from the perspective of geometric construction in this paper. The new method can significantly reduce the number of RF chains meanwhile still achieve an almost optimal performance in terms of sum rate. The proposed algorithm is compared with the popular orthogonal matching pursuit algorithm (OMP) via numerical simulations, and shows that our proposal can increase the system spectral efficiency with reduced computational complexity.

This paper studies beam training design for a codebook- based beamforming millimeter wave (mmwave) system where multiple antenna arrays are employed and each array is capable of beamforming independently. To reduce the training overhead and the complexity of subsequent beam direction search, we propose a cooperative multi- subarray beam training method. Specifically, from the perspective of excluding noneffective beam direction combinations and thus reducing search space, method and criterion of beam superposition are proposed to construct a wide beam from multiple narrow beams corresponding to multiple subarrays. Then, a cooperative multisubarray beam training scheme is proposed based on the proposed criterion. Finally, simulation results show that the proposed scheme achieves a spectral efficiency close to that of the optimal exhaustive search scheme, while has greatly reduced training overhead and computational complexity.

This paper exploits the potential of large antenna arrays to develop a secure millimeter wave (mmwave) transmission scheme. To reduce the peak-to-average power ratio (PAPR), the idea of constant envelope precoding (CEP) is introduced to improve the power efficiency of power amplifiers. In the CEP scheme, only phase variation of each antenna is used to form the desired signal at the target receiver, while the sum power of noise-free signals received by all eavesdroppers is minimized for secure transmission. A nonconvex optimization problem is formulated with equality and unit modulus constraints. To tackle the nonconvex constraints, the augmented Lagrangian penalty method is employed to address the challenging problem. An efficient iterative algorithm is further proposed to tackle the problem of precoder design. Simulation results confirm the effectiveness and superiority of the proposed CEP secure transmission scheme.

Owing to abundant spectrum resources, millimeter wave (mmwave) communication promises to provide Gbps data rates, which, however, may be restricted by large path-loss. Thus, antenna arrays are commonly used along with beam alignment (BA) as an important step to achieve the array gain. Efficient BA relies on the beam training codebook design. In this paper, we propose a new hierarchical codebook to achieve uniform BA performance with low overhead. To better elaborate on the design principle, a single-path channel model is considered first to frame the proposal. The codebook design is formulated as an optimization problem, where the ripple in the main/side lobes is constrained such that each training beam is close to the ideal one with a flat magnitude response and a narrow transition band. Then, we propose an efficient algorithm to find such a beam training codebook. Furthermore, we derive closed-form expressions of the BA misalignment probability or error rate of the proposed beam training codebook. Our results reveal that using the proposed codebook, the error rate of tree-search-based BA exponentially decreases with the SNR for a given channel, and linearly decreases in the log-log coordinate axis for a fading channel. We further propose a power allocation scheme used in different training stages to further improve the BA performance. Finally, the proposed framework is extended to the more complex case of multi-path channels. Numerical results confirm the effectiveness of the proposed training codebook and power allocation scheme as well as the accuracy of the performance analysis.