Second-order Channel Statistics Research Articles

Joint Spatio-Temporal Precoding for Practical Non-Stationary Wireless Channels

The high mobility, density and multi-path evident in modern wireless systems makes the channel highly non-stationary. This causes temporal variation in the channel distribution that leads to the existence of time-varying joint interference across multiple degrees of freedom (DoF, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g</i> ., users, antennas, frequency and symbols), which renders conventional precoding sub-optimal in practice. In this work, we derive a High-Order Generalization of Mercer’s Theorem (HOGMT), which decomposes the multi-user non-stationary channel into two (dual) sets of jointly orthogonal subchannels (eigenfunctions), that result in the other set when one set is transmitted through the channel. This duality and joint orthogonality of eigenfuntions ensure transmission over independently flat-fading subchannels. Consequently, transmitting these eigenfunctions with optimally derived coefficients eventually mitigates any interference across its degrees of freedoms and forms the foundation of the proposed joint spatio-temporal precoding. The transferred dual eigenfuntions and coefficients directly reconstruct the data symbols at the receiver upon demodulation, thereby significantly reducing its computational burden, by alleviating the need for any complementary post-coding. Additionally, the eigenfunctions decomposed from the time-frequency delay-Doppler channel kernel are paramount to extracting the second-order channel statistics, and therefore completely characterize the underlying channel. We evaluate this using a realistic non-stationary channel framework built in Matlab and show that our precoding achieves ⩾4 orders of reduction in BER at SNR⩾15dB in OFDM systems for higher-order modulations and less complexity compared to the state-of-the-art precoding.

Downlink Channel Estimation for FDD Massive MIMO Using Conditional Generative Adversarial Networks

For implementation of massive multiple-input multiple-output (MIMO) cellular systems in frequency division duplex (FDD) mode, accurate estimation of downlink channel state information (CSI) is necessary, but full radio channel reciprocity between the uplink and downlink does not exist in that mode. Existing work on estimating downlink CSI in FDD massive MIMO systems has considered such approaches as angle-of-arrival reciprocity, compressive sensing, using second-order channel statistics (particularly the channel covariance matrix (CCM)), and machine learning using deep neural networks (DNNs). Typical DNN-based approaches are unsuitable for this problem because DNNs require large datasets, thousands of training epochs, and are susceptible to environmental variations. To overcome these shortcomings, we develop a conditional generative adversarial network (CGAN) approach to uplink-to-downlink mapping of both CCMs and CSI. To apply this method, we convert the uplink and downlink CCMs/CSI to images and employ CGAN techniques previously applied to image translation. The normalized mean square error performance of the proposed CGAN is evaluated for several array sizes for both CCM and CSI mapping. For uplink-to-downlink CSI mapping, we also examine the spectral efficiency performance of our CGAN-based method, as well as the impact of pilot reuse; both simulated and measured CSI data are considered. Our results demonstrate performance improvement over existing algorithms.

The Internet of Bodies: A Systematic Survey on Propagation Characterization and Channel Modeling

The Internet of Bodies (IoBs) is an imminent extension to the vast Internet of Things domain, where interconnected devices (e.g., worn, implanted, embedded, swallowed, etc.) are located in-on-and-around the human body form a network. Thus, the IoB can enable a myriad of services and applications for a wide range of sectors, including medicine, safety, security, wellness, entertainment, to name but a few. Especially, considering the recent health and economic crisis caused by the novel coronavirus pandemic, also known as COVID-19, the IoB can revolutionize today’s public health and safety infrastructure. Nonetheless, reaping the full benefit of IoB is still subject to addressing related risks, concerns, and challenges. Hence, this survey first outlines the IoB requirements and related communication and networking standards. Considering the lossy and heterogeneous dielectric properties of the human body, one of the major technical challenges is characterizing the behavior of the communication links in-on-and-around the human body. Therefore, this article presents a systematic survey of channel modeling issues for various link types of human body communication (HBC) channels below 100 MHz, the narrowband (NB) channels between 400 and 2.5 GHz, and ultrawideband (UWB) channels from 3 to 10 GHz. After explaining bio-electromagnetics attributes of the human body, physical, and numerical body phantoms are presented along with electromagnetic propagation tool models. Then, the first-order and the second-order channel statistics for NB and UWB channels are covered with a special emphasis on body posture, mobility, and antenna effects. For capacitively, galvanically, and magnetically coupled HBC channels, four different channel modeling methods (i.e., analytical, numerical, circuit, and empirical) are investigated, and electrode effects are discussed. Finally, interested readers are provided with open research challenges and potential future research directions.

Intelligent Reflecting Surfaces: Sum-Rate Optimization Based on Statistical Position Information

In this paper, we consider a multi-user multiple-input multiple-output (MIMO) system aided by multiple intelligent reflecting surfaces (IRSs) that are deployed to increase the coverage and, possibly, the rank of the channel. We propose an optimization algorithm to configure the IRSs, which is aimed at maximizing the network sum-rate by exploiting only the statistical characterization of the locations of the mobile users. As a consequence, the proposed approach does not require the estimation of either instantaneous channel state information (CSI) or second-order channel statistics for IRS optimization, thus significantly relaxing (or even avoiding) the need of frequently reconfiguring the IRSs, which constitutes one of the most critical issues in IRS-assisted systems. Numerical results confirm the validity of the proposed approach. It is shown, in particular, that IRS-assisted wireless systems that are optimized based on statistical position information still provide large performance gains as compared to the baseline scenarios in which no IRSs are deployed.

Pilot and data power allocation for spatially correlated massive MIMO systems with imperfect CSI

This paper investigates pilot and data power allocation for a multi-cell spatially correlated massive multiple-input multiple-output (MIMO) system with imperfect channel state information (CSI). By exploiting the availability of statistical CSI, the uplink closed-form spectral efficiency (SE) expressions are derived for both full covariance knowledge case and no prior statistical information case. In order to enhance the accuracy of channel estimation, novel pilot power allocation schemes are firstly designed by minimizing the sum normalized mean square error (NMSE) of all users. Then, based on the optimal pilot power, data power allocation schemes are proposed to maximize the sum SE of system. For solving these non-convex problems, an iterative optimization algorithm is developed to obtain the local optimal solution by exploiting second-order channel statistics of users. Compared with the traditional method, numerical results show that the proposed pilot and data power allocation schemes can achieve higher SE for spatially correlated massive MIMO system.

Average Contiguous Duration (ACD)-Based Quantization for Secret Key Generation in Generalized Gamma Fading Channels

The wireless channel-based Secret Key Generation (SKG) algorithms aim at securing the wireless link against unauthorized eavesdropping by exploiting the channel’s randomness for generating matching secret keys at the legitimate nodes for message encryption/decryption. To counter differences in hardware and noise conditions at the legitimate nodes, which can lead to key mismatch, the SKG algorithms typically include the intermediate steps of sampling, quantization, information reconciliation, and privacy amplification. These steps collectively aim to improve the performance trade-offs between Key Generation Rate (KGR), Key Agreement Probability (KAP), and Secret Key Randomness (SKR) properties. This paper derives a closed-form expression for the Average Contiguous Duration (ACD) of Generalized Gamma (GG) fading wireless channels. The ACD is a recently introduced novel quantifier for characterizing the second-order statistics of fading channels, which includes Average Fade Duration (AFD) as its special case. The proposed GG fading ACD expression is shown to include, as its special cases, the ACD for commonly observed fading distributions such as Gamma, Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> , and Rayleigh. By exploiting the derived GG ACD expression, a multi-level quantization scheme for SKG is proposed that determines suitable quantization intervals for identical likelihood of an equal number of consecutive channel samples falling in each quantization interval. A comprehensive comparative analysis of the proposed ACD-based quantization for SKG is conducted in relation to conventional Uniform Quantization (UQ) and Cumulative Distribution Function (CDF)-based Non-Uniform Quantization (NUQ) schemes. The presented numerical results confirm the superior performance trade-off between KGR and KAP offered by the proposed ACD-based quantization in relation to that offered by UQ and CDF-based NUQ.

Nested Hybrid Cylindrical Array Design and DoA Estimation for Massive IoT Networks

Reducing cost and power consumption while maintaining high network access capability is a key physical-layer requirement of massive Internet of Things (mIoT) networks. Deploying a hybrid array is a cost- and energy-efficient way to meet the requirement, but would penalize system degree of freedom (DoF) and channel estimation accuracy. This is because signals from multiple antennas are combined by a radio frequency (RF) network of the hybrid array. This article presents a novel hybrid uniform circular cylindrical array (UCyA) for mIoT networks. We design a nested hybrid beamforming structure based on sparse array techniques and propose the corresponding channel estimation method based on the second-order channel statistics. As a result, only a small number of RF chains are required to preserve the DoF of the UCyA. We also propose a new tensor-based two-dimensional (2-D) direction-of-arrival (DoA) estimation algorithm tailored for the proposed hybrid array. The algorithm suppresses the noise components in all tensor modes and operates on the signal data model directly, hence improving estimation accuracy with an affordable computational complexity. Corroborated by a Cramér-Rao lower bound (CRLB) analysis, simulation results show that the proposed hybrid UCyA array and the DoA estimation algorithm can accurately estimate the 2-D DoAs of a large number of IoT devices.

Capacity Scaling of Massive MIMO in Strong Spatial Correlation Regimes

This paper investigates the capacity scaling of multicell massive MIMO systems in the presence of spatially correlated fading. In particular, we focus on the strong spatial correlation regimes where the covariance matrix of each user channel vector has a rank that scales sublinearly with the number of base station antennas, as the latter grows to infinity. We also consider the case where the covariance eigenvectors corresponding to the non-zero eigenvalues span randomly selected subspaces. For this channel model, referred to as the "random sparse angular support" model, we characterize the asymptotic capacity scaling law in the limit of large number of antennas. To achieve the asymptotic capacity results, statistical spatial despreading based on the second-order channel statistics plays a pivotal role in terms of pilot decontamination and interference suppression. A remarkable result is that even when the number of users scales linearly with base station antennas, a linear growth of the capacity with respect to the number of antennas is achievable under the sparse angular support model. We note that the achievable rate lower bound based on massive MIMO "channel hardening", widely used in the massive MIMO literature, yields rather loose results in the strong spatial correlation regimes and may significantly underestimate the achievable rate of massive MIMO. This work therefore considers an alternative bounding technique which is better suited to the strong correlation regimes. In fading channels with sparse angular support, it is further shown that spatial despreading (spreading) in uplink (downlink) has a more prominent impact on the performance of massive MIMO than channel hardening.

Adaptive Pilot Allocation for Multi-cell Massive MIMO Systems

In the upcoming 5G system, the massive MIMO technology will perform an important role in significantly improving the system performance. To sufficiently exploit the potential of massive MIMO, the transceiver should perform appropriate accurate channel estimation, including carefully designed quantities of pilot resource and its allocation among different users. In this letter, a low-complexity adaptive pilot allocation algorithm using the second-order channel statistics of all users is proposed, which helps in finding a good balance between the allocated pilot length and pilot training interference. Numerical results indicate that the proposed algorithm can significantly improve the net spectrum efficiency in multi-cell massive MIMO systems.

Sherman-Morrison Formula Aided Adaptive Channel Estimation for Underwater Visible Light Communication With Fractionally-Sampled OFDM

In this paper, we investigate the channel estimation (CE) problem in an underwater visible light communication (UVLC) system invoking fractionally-sampled optical orthogonal frequency division multiplexing (FS-OOFDM). In practical UVLC scenarios, the communication links inevitably suffer from many stochastic channel effects including multi-path dispersion, scattering, turbulence, etc., and/or from the mobility of the transceiver, therefore resulting in a time-varying, location-dependent non-stationary propagation environment. Naturally, compared with the indoor visible light communication (VLC) scenario with a typical assumption being the time-flat channel models, it becomes a notable challenge for designing a low-complexity adaptive CE in the much more complicated UVLC scenarios. To solve this problem, we derive a class of Bayesian CE algorithms referred to as the Sherman-Morrison formula (SMF) based CE (SMF-CE), by exploiting the property of rank-one structure of the second-order channel statistics in the delay domain. Furthermore, an adaptive version of SMF-CE (ASMF-CE) can be obtained through updating the imperfect a priori knowledge of the channel and the noise's statistics. Simulation results demonstrate the superior performances of the proposed algorithms in comparison to existing methods, while maintaining a reduced computational complexity in comparison to the conventional linear minimum mean square error (LMMSE) scheme.

Doppler Shift Characterization of Wideband Mobile Radio Channels

The prevailing approach for characterizing the Doppler shift (DS) of mobile radio channels assumes the transmission of an unmodulated carrier. This consideration is valid for the analysis of narrowband channels, but its pertinence is questionable in regards to the modeling of wideband channels. In this correspondence, we redefine the DS from a time-frequency analysis perspective that does not depend on the aforementioned assumption. We systematically demonstrate that the DS can be characterized by the instantaneous frequency of the channel transfer function. This generic definition makes evident a fundamental aspect of the DS that is seldom acknowledged, namely, the DS is a frequency-varying quantity. We show that the second-order statistics of wideband mobile radio channels are non-stationary due to the DS's frequency variations. In addition, we present numerical results of a case study showing that such non-stationarities can cause significant system performance degradations.

Average SEP-Optimal Precoding for Correlated Massive MIMO With ZF Detection: An Asymptotic Analysis

This paper investigates the symbol error probability~(SEP) of point-to-point massive multiple-input multiple-output (MIMO) systems using equally likely PAM, PSK, and square QAM signallings in the presence of transmitter correlation. The receiver has perfect knowledge of the channel coefficients, while the transmitter only knows first- and second-order channel statistics. With a zero-forcing~(ZF) detector implemented at the receiver side, we design and derive closed-form expressions of the optimal precoders at the transmitter that minimizes the average SEP over channel statistics for various modulation schemes. We then unveil some nice structures on the resulting minimum average SEP expressions, which naturally motivate us to explore the use of two useful mathematical tools to systematically study their asymptotic behaviors. The first tool is the Szeg\"o's theorem on large Hermitian Toeplitz matrices and the second tool is the well-known limit: $\lim_{x\to\infty}(1+1/x)^x=e$. The application of these two tools enables us to attain very simple expressions of the SEP limits as the number of the transmitter antennas goes to infinity. A major advantage of our asymptotic analysis is that the asymptotic SEP converges to the true SEP when the number of antennas is moderately large. As such, the obtained expressions can serve as effective SEP approximations for massive MIMO systems even when the number of antennas is not very large. For the widely used exponential correlation model, we derive closed-form expressions for the SEP limits of both optimally precoded and uniformly precoded systems. Extensive simulations are provided to demonstrate the effectiveness of our asymptotic analysis and compare the performance limit of optimally precoded and uniformly precoded systems.

Multipath Cluster Fading Statistics and Modeling in Millimeter-Wave Radio Channels

The second-order statistics of indoor directional channels are investigated using millimeter-wave (mmWave) band (30.4–37.1 GHz) ultrawideband (UWB) channel measurements. Considering two main mmWave system assumptions (high bandwidth and high beamforming gain), this paper aims to investigate the validity of the Rayleigh–Rice fading models for the cluster fading envelope. The results from the mmWave band study are compared to an already well-studied lower frequency FCC band (3.4–10.1 GHz). During the measurements, only selective objects (emulated multipath clusters in the propagation channel) are illuminated in a small lecture room. The experiments show that for both UWB channels, the complex received (Rx) signal is a circularly symmetric non-Gaussian random variable with highly correlated inphase (I) and quadrature (Q) components. These properties demonstrate that the intracluster multipath components (MPCs) structure is sparse. Consequently, modeling the cluster fading envelope with Rayleigh–Rice distribution is not realistic. Therefore, the sum-of-cisoids principle is used for intracluster multipath modeling which inherently considers a correlation between I and Q components. It has been established that a reasonably good approximation of the cluster fading envelope can be obtained with $N=3-6$ equal amplitude cisoids. However, we remark that Rayleigh–Rice models may become realistic cluster fading envelopes for narrowband mmWave systems.

Characterization of Fading Statistics of mmWave (28 GHz and 38 GHz) Outdoor and Indoor Radio Propagation Channels

Extension of usable frequency spectrum from microwave to millimeter-wave (mmWave) is one of the key research directions in addressing the capacity demands of emerging 5th-generation communication networks. This paper presents a thorough analysis on the azimuthal multipath shape factors and second-order fading statistics (SOFS) of outdoor and indoor mmWave radio propagation channels. The well-established analytical relationship of plain angular statistics of a radio propagation channel with the channel’s fading statistics is used to study the channel’s fading characteristics. The plain angle-of-arrival measurement results available in the open literature for four different outdoor radio propagation scenarios at 38 GHz, as well as nine different indoor radio propagation scenarios at 28 GHz and 38 GHz bands, are extracted by using different graphical data interpretation techniques. The considered quantifiers for energy dispersion in angular domain and SOFS are true standard-deviation, angular spread, angular constriction, and direction of maximum fading; and spatial coherence distance, spatial auto-covariance, average fade duration, and level-crossing-rate; respectively. This study focuses on the angular spread analysis only in the azimuth plane. The conducted analysis on angular spread and SOFS is of high significance in designing modulation schemes, equalization schemes, antenna-beams, channel estimation, error-correction techniques, and interleaving algorithms; for mmWave outdoor and indoor radio propagation environments.

Pilot decontamination techniques based on beam‐domain channel characteristics in millimetre‐wave sparse channels

The upcoming 5G system is expected to utilise large-scale antenna arrays to increase the users' data rate and system capacity. With a large number of antennas, channel estimation becomes a very critical design issue due to the adverse effects of pilot contamination. Some problems in the existing channel estimation methods studied in the literature include: (i) the complexity of signal processing on the spatial domain is very high. (ii) Channels are quite frequency selective in the spatial domain. (iii) It is not easy to obtain the second-order statistics of channels. In this study, the channel estimation issue is investigated using the signal processing in the beam domain. It is due to the salient channel sparsity at millimetre-wave frequencies that only a few beam channels need to be estimated. The authors exploit these beam channels to propose some pilot decontamination techniques: basic pilot allocation, beam-set non-overlapping allocation, intra-cell interference mitigation allocation and maximum beam power based allocation (MBPBA). Both large pilot reuse factor and beam-domain power control are also presented to supplement the proposed techniques. Simulation results show that the best method is MBPBA in terms of overhead and normalised square error.

An Adaptive LMMSE Interpolation Algorithm for Channel Estimation in LTE-Advanced Systems

The traditional LMMSE interpolation algorithm assumes that the terminal knows the channel power delay profile (PDP), which is not consistent with the actual situation. This paper proposes an improved LMMSE interpolation algorithm based on RMS estimation. The time-delay parameters of channel first are estimated by pilots, then LMMSE interpolation process in frequency domain is completed by LUT (looking up table) for selection of corresponding frequency domain interpolation matrix. This solves the difficulty in acquiring the second-order channel statistics in the traditional LMMSE interpolation algorithm, and avoiding the real-time inversion process of matrices. Simulation results demonstrate that this method can reduce the complexity and show a good performance of BER and MSE, thus it is more suitable for practical application.

Multiuser Millimeter Wave Beamforming Strategies With Quantized and Statistical CSIT

To alleviate the high cost of hardware in mm-wave systems, hybrid analog/digital precoding is typically employed. In the conventional two-stage feedback scheme, the analog beamformer is determined by beam search and feedback to maximize the desired signal power of each user. The digital precoder is designed based on quantization and feedback of effective channel to mitigate multiuser interference. Alternatively, we propose a one-stage feedback scheme, which effectively reduces the complexity of the signalling and feedback procedure. Specifically, the second-order channel statistics are leveraged to design digital precoder for interference mitigation while all feedback overhead is reserved for precise analog beamforming. Under a fixed total feedback constraint, we investigate the conditions under which the one-stage feedback scheme outperforms the conventional two-stage counterpart. Moreover, a rate splitting (RS) transmission strategy is introduced to further tackle the multiuser interference and enhance the rate performance. Consider: 1) RS precoded by the one-stage feedback scheme and 2) conventional transmission strategy precoded by the two-stage scheme with the same first-stage feedback as 1) and also certain amount of extra second-stage feedback. We show that 1) can achieve a sum rate comparable to that of 2). Hence, RS enables remarkable saving in the second-stage training and feedback overhead.

Saturation Power-Based Simple Energy Efficiency Maximization Schemes for MISO Broadcast Channel Systems

In this paper, we investigate an energy efficiency (EE) maximization problem in multiple input single output broadcast channels. The optimization problem in this system model is difficult to solve in general, since it is in non-convex fractional form. Hence, conventional algorithms have addressed the problem in an iterative manner for each channel realization, which leads to high computational complexity. To tackle this complexity issue, we propose a new simple method by utilizing the fact that EE maximization becomes identical to spectral efficiency (SE) maximization for the region of the power below a certain transmit power termed as saturation power . In order to calculate the saturation power, we first introduce upper and lower bounds of the EE performance by adopting a maximal ratio transmission beamforming strategy. Then, we propose an efficient way to compute the saturation power for the EE maximization problem. Once we determine the saturation power in advance, we can transform the EE maximization problem into a simplified sub-optimal EE problem, which can be solved by the SE maximization schemes with low complexity. The derived saturation power is parameterized by employing random matrix theory, which relies only on the second-order channel statistics. Hence, this approach needs much lower computational complexity compared with a conventional scheme, which requires instantaneous channel state information. Numerical results validate that the proposed algorithm achieves near optimal EE performance with significantly reduced complexity.

On the use of the channel second-order statistics in MMSE receivers for time- and frequency-selective MIMO transmission systems

Equalization of unknown frequency- and time-selective multiple input multiple output (MIMO) channels is often carried out by means of decision feedback receivers. These consist of a channel estimator and a linear filter (for the estimation of the transmitted symbols), interconnected by a feedback loop through a symbol-wise threshold detector. The linear filter is often a minimum mean square error (MMSE) filter, and its mathematical expression involves second-order statistics (SOS) of the channel, which are usually ignored by simply assuming that the channel is a known (deterministic) parameter given by an estimate thereof. This appears to be suboptimal and in this work we investigate the kind of performance gains that can be expected when the MMSE equalizer is obtained using SOS of the channel process. As a result, we demonstrate that improvements of several dBs in the signal-to-noise ratio needed to achieve a prescribed symbol error rate are possible.

Multiuser hybrid analogue/digital beamforming for massive multiple‐input–multiple‐output

The authors propose a generalised multi-layer architecture for the phase shifter network (PSN) in hybrid analogue/digital beamforming (BF), which provides flexible trade-offs between the performance and implementation complexity. By changing the layer number, the authors can adjust the complexity of the PSN and the achievable performance. A two-stage BF algorithm is adopted, in which the analogue and digital BF matrices are designed separately. Analogue BF is adaptive to the second-order channel statistics. A simple iterative algorithm is proposed to maximise an upper bound of the average rate of the effective channel after analogue BF. Digital BF is adaptive to the effective channel for multiuser interference cancellation. Both ideal phase shifters and discrete phase shifters with limited phase and amplitude resolutions are considered in the algorithm design. Theoretical analysis and simulation are used to demonstrate the good performance and flexibility of the proposed technique. In a 64-antenna system, it can use 16 radio-frequency chains to perform very closely to full digital BF with a fully-connected PSN, or achieve 70–80% performance with a simple PSN. It is also robust against the limited phase shifter resolution. It enables us to select the proper performance and implementation complexity according to the requirement and capability of real systems.