Closed-form Sum Research Articles

On the Performance of Fully-Connected and Sub-Connected Hybrid Beamforming System

When the wireless communication system evolves to massive multiple-input multiple-output (MIMO) system, the system becomes more and more complex. Hybrid beamforming architecture provides excellent trade-off between the system performance and the complexity for the massive MIMO system. Hybrid beamforming architecture can be classified as fully-connected (FC) and sub-connected (SC) architecture. Existing studies on the system performance comparison of the two architectures are limited. In this paper, we investigate the performance of the FC and SC architecture in the multi-user multiple-input single-output (MU-MISO) system. We derive the closed-form sum rate of two architectures under the same total transmission power constraint. From the results, we conclude that the system performance of FC architecture is always better than that of SC architecture. When the number of users is small or the signal noise ratio (SNR) is low, the performance difference between the FC and SC architecture is small. Our analysis can serve as a guideline of choosing hybrid beamforming architectures.

Euler and the Cubic Basel Problem

In 1734, Leonhard Euler summed the infinite series of reciprocals of the squares, thereby solving a challenge known as the “Basel problem.” He later extended his method to find closed-form sums for the reciprocals of 4th, 6th, and other even powers. But those techniques did not yield a value for the sum of the reciprocals of the cubes. Here, we show how Euler tried to evaluate this series by transforming it into the sum of a strange constant and an even stranger integral.

Rate Analysis of Cell-Free Massive MIMO-NOMA With Three Linear Precoders

Although the hybrid of cell-free (CF) massive multiple-input multiple-output (MIMO) and non-orthogonal multiple access (NOMA) promises massive spectral efficiency gains, the type of precoders employed at the access points (APs) impacts the gains. In this paper, we thus comprehensively evaluate the system performance with maximum ratio transmission (MRT), full-pilot zero-forcing (fpZF) and modified regularized ZF (mRZF) precoders. We derive their closed-form sum rate expressions by considering Rayleigh fading channels, the effects of intra-cluster pilot contamination, inter-cluster interference, and imperfect successive interference cancellation (SIC). Our results reveal that this system supports significantly more users simultaneously at the same coherence interval compared to its OMA equivalent. However, intra-cluster pilot contamination and imperfect SIC degrade the system performance when the number of users is low. Moreover, with perfect SIC, mRZF and fpZF significantly outperform MRT. Also, we show that this system with either mRZF or fpZF precoding outperforms OMA systems with MRT. The analytical findings are verified by numerical results.

Underlaid Spectrum Sharing for Cell-Free Massive MIMO-NOMA

We investigate the achievable rate of a hybrid cell-free (CF) massive multiple-input multiple-output (MIMO) non-orthogonal multiple-access (NOMA) underlaid below a primary massive MIMO. We propose a low complexity sub-optimal user-clustering method and derive the closed-form sum rate expression for Rayleigh fading channels by considering the effects of intra-cluster pilot contamination, inter-cluster interference, imperfect successive interference cancellation (SIC) and statistical downlink channel state information (CSI) at secondary users (SUs). Our results reveal that NOMA based underlay CF massive MIMO-NOMA can exploit the scarce spectrum bands more efficiently than its counterpart orthogonal multiple-access (OMA).

A treaty of symmetric function: An approach in deriving general formulation for sums of power for an arbitrary arithmetic progression Part 1

Sum of Power had gathered interest of many classical mathematicians for more than two thousand years ago. The quests of finding sum of power or discrete sum of numerical power can be traced back from the time of Archimedes in third BC then to Faulhaber in the sixteen centuries. Until today there is no closed form sums of power formulation for an arithmetic progression has been found. Many mathematicians were involved in this research and many approaches have been introduced but none is found to be conclusive. The generalized equation for sums of power discovered in this research has been compared to Faulhaber’s sums of power for integers and it is found that this new generalized equation can be used for both integers and arithmetic progression, thus offering a new frontier in studying symmetric function, Fermat’s last theorem, Riemman’s Zeta function etc.

Sum Rate Analysis for Multi-Cell Random Beamforming With Heterogeneous Schedule Probabilities

In this letter, the closed form sum rate for the random beamforming is analyzed over the downlink multiple-input single-output and multiple input multiple-output interfering broadcast channel with finite users. The analysis approach is based on the limiting throughput distribution theorem, and can be applied for the multi-cell interference scenarios with heterogeneous signal to interference-plus-noise ratio (SINR) distributions and user schedule probabilities. It is proved that both the exact and the asymptotic sum rates achieved from SINR/signal-to-interference ratio analysis will converge to the Gumbel type distribution. The effectiveness of the proposed analytical framework is confirmed by simulation results.

Efficient Low-Resolution ADC Relaying for Multiuser Massive MIMO System

In this paper, we investigate a multiuser massive MIMO amplify-and-forward relay uplink, where low-resolution analog-to-digital convertors (ADCs) are used at the relay station (RS). Both the base station (BS) and the RS are equipped with large numbers, i.e., $\boldsymbol {N_B}$ and $\boldsymbol {N_R}$ , respectively, of antennas. For both perfect and imperfect channel state information (CSI), we derive closed-form sum rate expressions with respect to finite $\boldsymbol {N_B}$ and $\boldsymbol {N_R}$ . Under some mild assumptions, we demonstrate that sum rates with perfect and imperfect CSI, respectively, increase with $\boldsymbol {\alpha }$ and $\boldsymbol {\alpha ^2}$ both logarithmically, where $\boldsymbol {\alpha }$ is linear quantization gain of ADCs at the RS. We show that low-resolution, e.g., 2-3 bits, ADCs only cause limited sum rate loss when $\boldsymbol {N_B}$ and $\boldsymbol {N_R}$ are relatively large compared to the number of users, i.e., $\boldsymbol {K}$ . Based on the obtained sum rate expressions, we further discover the generalized power scaling laws that guarantee a nonvanishing sum rate, as $\boldsymbol {N_B}$ and $\boldsymbol {N_R}$ grow to infinity, where we find that low-resolution ADCs at the RS do not bring substantial impact on power scaling laws. Numerical results verify the correctness of our theoretical analysis and consequently justify the large-scale antenna arrays at the RS in a low-cost and energy efficient manner, yet still achieving considerable sum rate performance.

Degrees of Freedom in Cached MIMO Interference Networks With Asynchronous User Requests

In this paper, we study the average sum DoF of cached interference networks over random user requests. First, we derive a closed-form average sum DoF upper bound of the cached interference channels over all possible caching and transmission strategies. Then, we propose an online cache content placement algorithm to maximize the achievable average sum DoF without explicit knowledge of the popularity of the content files. We show that the gap between the DoF upper bound and an achievable DoF is small for some typical scenarios. Moreover, we quantify the DoF gain due to caching with respect to some important system parameters. We show that large DoF gain over interference network without cache is possible even if the number of files $L\rightarrow \infty$ . Both simulation and analysis indicate that the proposed scheme has significant gains over various baselines.

On certain series involving Laguerre polynomials

In this paper, we find closed-form sums for the series ∑n=1∞(α2)nn1n!1F1(−n,γ,x2), where α>0. This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H=−d2dx2+Bx2+Ax2+λxα, where 0≤x<∞, α,λ>0, A≥0.

Ergodic sum rate analysis and efficient power allocation for a massive MIMO two‐way relay network

The authors study the transmit power allocation (PA) problem for a network of two multi-antenna terminals (one of which is a massive multiple-input and multiple-output (MIMO) terminal) and a two-way, amplify-and-forward relay. The relay is limited to a single antenna. Using perfect channel state information, the terminals employ beamforming with maximum-ratio-transmission and maximum-ratio-combining for transmission and reception, respectively. The authors investigate two practical problems, namely; (i) maximising the sum rate subject to a total power constraint (ii) maximising the sum rate when one of the terminals must exceed a target signal-to-noise ratio (SNR). For the first case, the authors derive the closed-form optimal PA and for the second, the authors derive a sub-optimal PA. In both cases, the resulting sum rates are a function of instantaneous channel gains. Thus by averaging over the Nakagami-m distribution and exploiting the weak law of large numbers, the authors derive the closed-form ergodic sum rates. Finally, the simulation results validate the theoretical analysis and show the sum-rate improvements over uniform PA. For example, to achieve 4 bit/s/Hz, a uniform allocation needs 1 dB more than the authors’ optimal allocation. When one of the SNRs must exceed a target value, the gap between the authors’ sub-optimal PA and random PA increases to 2 dB.

Rate Maximization for Downlink Multiuser Visible Light Communications

We are interested in optimizing transmit beamforming for a visible light communication multiuser system such that the achievable sum rate is maximized. We first derive a closed-form sum rate expression via information-theoretic tools. Then, by further including light-emitting diode optical power constraints, we formulate the rate maximization problem in a mathematical optimization form. To address this difficult non-convex problem, we establish an iterative algorithm that exploits sequential parametric convex approximation. Compared with the existing zero-forcing beamforming strategies, the proposed approach does not restrict the co-channel interference to be zero, and thus achieves a higher sum rate, as is validated by simulations.

Sum-Rate and Power Scaling of Massive MIMO Systems With Channel Aging

This paper investigates the achievable sum-rate of massive multiple-input multiple-output (MIMO) systems in the presence of channel aging. For the uplink, by assuming that the base station (BS) deploys maximum ratio combining (MRC) or zero-forcing (ZF) receivers, we present tight closed-form lower bounds on the achievable sum-rate for both receivers with aged channel state information (CSI). In addition, the benefit of implementing channel prediction methods on the sum-rate is examined, and closed-form sum rate lower bounds are derived. Moreover, the impact of channel aging and channel prediction on the power scaling law is characterized. Extension to the downlink scenario and multi-cell scenario are also considered. It is found that, for a system with/without channel prediction, the transmit power of each user can be scaled down at most by $1/\sqrt{M}$ (where $M$ is the number of BS antennas), which indicates that aged CSI does not degrade the power scaling law, and channel prediction does not enhance the power scaling law; instead, these phenomena affect the achievable sum-rate by degrading or enhancing the effective signal to interference and noise ratio, respectively.

Symbolic methods for the evaluation of sum rules of Bessel functions

The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.

Random Beamforming with Heterogeneous Users and Selective Feedback: Individual Sum Rate and Individual Scaling Laws

This paper investigates three open problems in random beamforming based communication systems: the scheduling policy with heterogeneous users, the closed form sum rate, and the randomness of multiuser diversity with selective feedback. By employing the cumulative distribution function based scheduling policy, we guarantee fairness among users as well as obtain multiuser diversity gain in the heterogeneous scenario. Under this scheduling framework, the individual sum rate, namely the average rate for a given user multiplied by the number of users, is of interest and analyzed under different feedback schemes. Firstly, under the full feedback scheme, we derive the closed form individual sum rate by employing a decomposition of the probability density function of the selected user's signal-to-interference-plus-noise ratio. This technique is employed to further obtain a closed form rate approximation with selective feedback in the spatial dimension. The analysis is also extended to random beamforming in a wideband OFDMA system with additional selective feedback in the spectral dimension wherein only the best beams for the best-L resource blocks are fed back. We utilize extreme value theory to examine the randomness of multiuser diversity incurred by selective feedback. Finally, by leveraging the tail equivalence method, the multiplicative effect of selective feedback and random observations is observed to establish the individual rate scaling.

Closed Form Sum Rate of Random Beamforming

In this letter, we carry out analysis to derive the closed form sum rate for a general multi-antenna random beamforming system with an arbitrary but finite number of users. A simple and analytical useful expansion with general utility is first developed to decompose and interpret the probability density function of the selected user's signal-to-interference-plus-noise ratio. By using this decomposition, along with standard integration techniques, the closed form result for the sum rate is obtained. The established result explains the interplay between the system parameters for arbitrary but finite number of users.

Robust user pairing algorithm under channel estimation errors for uplink virtual multiple-input multiple-output systems

Employing user pairing schemes in uplink virtual multiple-input multiple-output (V-MIMO) systems can significantly improve system capacity owing to multi-user diversity. Most of the existing user pairing algorithms were developed under the assumption that a scheduler always possesses the perfect channel state information (CSI). In this paper, taking into account the effects of channel estimation error (CEE), the authors propose a robust user pairing algorithm based on a closed-form average sum rate under CEE. In addition, the authors further propose an adaptive user selection scheme as an effort to maximise the sum rates owing to the rate constraint under CEE. The gain of the proposed method is shown in simulation results.

Optimal Category Pricing with Endogenous Store Traffic

We propose a dynamic programming framework for retailers of frequently purchased consumer goods in which the prices affect both the profit per visit in the current period and the number of visitors (i.e., store traffic) in future periods. We show that optimal category prices in the infinite-horizon problem also maximize the closed form sum of a geometric series, allowing us to derive meaningful analytical results. Modeling the linkage between category prices and future store traffic fundamentally changes optimal pricing policy. Optimal pricing must balance current profits against future traffic; under general conditions, optimal long-run prices are uniformly lower across all categories than those that maximize current profits. This result explains the empirical generalization that category demand in grocery stores is inelastic. Parameterizing profit per visit and store traffic reveals that, as future traffic becomes more sensitive to price, retailers should increasingly lower current prices and sacrifice current profits. We also determine how the burden of drawing future traffic to the store should be distributed across categories; this is the foundation for a new taxonomy of category roles.

Energy bounds for a class of singular potentials and some related series

Perturbation expansions up to third order for the generalized spiked harmonic oscillator Hamiltonians H = -d^2/dx^2+ x^2 + A/x^2 + lambda/x^alpha, A >= 0, 2gamma > alpha, gamma=1+(1/2)sqrt(1+4A), and small values of the coupling lambda > 0, are developed. Upper and lower bounds for the eigenvalues are computed by means of the procedure of Burrows et al [J. Phys. A: Math. Gen. 20, 889-897 (1987)] for assessing the accuracy of a truncated perturbation expansion. Closed-form sums for some related perturbation double infinite series then immediately follow as a result of this investigation.

Closed-form sums for some perturbation series involving hypergeometric functions

Infinite series of the type Sum{n=1,infinity}(alpha/2)_n_2F_1(-n, b; gamma; y)/(n n!) are investigated. Closed-form sums are obtained for alpha a positive integer alpha=1,2,3, ... The limiting case of b --> infinity, after y is replaced with x^2/b, leads to Sum{n=1,infinity}(alpha/2)_n_1F_1(-n; gamma; x^2)/(n n!). This type of series appears in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -(d/dx)^2 + Bx^2 + A/x^2 + lambda/x^alpha, x >=0, alpha, lambda > 0, A >= 0. These results have immediate applications to perturbation series for the energy and wave function of the spiked harmonic oscillator Hamiltonian H = -(d/dx)^2 + Bx^2 + lambda/x^alpha, x >= 0, alpha, lambda > 0.

Closed-form sums for some perturbation series involving associated Laguerre polynomials

Infinite series sum_{n=1}^infty {(alpha/2)_n / (n n!)}_1F_1(-n, gamma, x^2), where_1F_1(-n, gamma, x^2)={n!_(gamma)_n}L_n^(gamma-1)(x^2), appear in the first-order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator Hamiltonian H = -d^2/dx^2 + B x^2 + A/x^2 + lambda/x^alpha 0 <= x < infty, alpha, lambda > 0, A >= 0. It is proved that the series is convergent for all x > 0 and 2 gamma > alpha, where gamma = 1 + (1/2)sqrt(1+4A). Closed-form sums are presented for these series for the cases alpha = 2, 4, and 6. A general formula for finding the sum for alpha/2 = 2 + m, m = 0,1,2, ..., in terms of associated Laguerre polynomials, is also provided.