Approximate Ergodic Capacity Research Articles

Outage probability and ergodic capacity analysis of satellite–terrestrial NOMA system with mixed RF/mmWave relaying

This paper investigates the performance of hybrid radio frequency/millimeter wave (RF/mmWave) satellite–terrestrial relay network with non-orthogonal multiple access (NOMA) scheme. In particular, the satellite network operates at the Ka-band while the mmWave band is adopted at decode-and-forward (DF) relay–destination link. The blockage effect in mmWave communication is considered to deduce the ordered probability density function (PDF) of the user equipment (UE) distance. However, the randomness of the UEs’ location and the random beamforming design cause the difficulty of the performance analysis. Herein, we provide a closed-form approximation for the outage probability. Moreover, the system diversity order is derived from the asymptotic expressions of the outage probability at high signal-to-noise ratio (SNR) slopes. Besides, approximate ergodic capacity performances of the multi-UEs are evaluated with numerical integration. Simulations are applied to demonstrate the accuracy of the theoretical analysis and show that the performance of the ordered selection NOMA scheme outperforms the orthogonal multiple access scheme and random selection scheme.

Performance Analysis of NOMA-Based Hybrid Satellite-Terrestrial Relay System Using mmWave Technology

This paper investigates the performance of NOMA-based hybrid Satellite-Terrestrial relays system (HSTR) using the millimeter wave (mmWave) technology. Furthermore, the relays are equipped with multiple antennas and utilize the amplify and forward (AF) protocol to forward the satellite’s superimposed information to multiple destinations. Then, the rain coefficient is considered as the fading factor of the mmWave band to choose the best relay. We considered the shadowed-Rician fading and Nakagami-m fading for satellite links and terrestrial links respectively, and in addition, we evaluated the shadowing effect for satellite links with two modes of: frequent heavy shadowing (FHS) and average shadowing (AS). With these suggestions, the closed-form outage probability (OP) and approximate ergodic capacity (EC) are derived to evaluate the efficiency of the proposed system. Next contribution of the research is an asymptotic analysis for the OP, which is derived in order to gain additional insight into important system parameters. Finally, the theoretical derivation is validated through simulation results and analyzed the impact of significant parameters. These results demonstrate NOMA’s superiority to the traditional orthogonal multiple access (OMA) method in the proposed system.

Performance of Dual-Hop Relaying for THz-RF Wireless Link Over Asymmetrical $\alpha$-$\mu$ Fading

Terahertz (THz) frequency bands can be promising for data transmissions between the core network and access points (AP) for next-generation wireless systems. In this paper, we analyze the performance of a dual-hop THz-RF wireless system where an AP facilitates data transmission between a core network and user equipment (UE). We consider a generalized model for the end-to-end channel with independent and not identically distributed (i.ni.d.) fading model for THz and RF links using the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> distribution, the THz link with pointing errors, and asymmetrical relay position. We derive a closed-form expression of the cumulative distribution function (CDF) of the end-to-end signal to noise ratio (SNR) for the THz-RF link, which is valid for real-valued <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> for a generalized performance analysis over THz fading channels. Using the derived CDF, we analyze the performance of the THz-RF relayed system using decode-and-forward (DF) protocol by deriving closed-form expressions of outage probability, moments of SNR, average BER, and a tight approximation of ergodic capacity in terms of system parameters. We develop asymptotic analysis on the ergodic capacity, outage probability, and average BER in the high SNR region and derive the diversity order of the system. We also analyze the considered system with an i.i.d. model and develop simplified performance to provide insight on the system behavior analytically under various practically relevant scenarios. Simulation and numerical analysis show a significant effect of fading parameters of the THz link and a nominal effect of normalized beam-width on the performance of the relay-assisted THz-RF system.

On the Ergodic Capacity of mmWave Systems Under Finite-Dimensional Channels

Due to the underlying sparse structure of the mmWave channels, which indeed makes the exact closed-form capacity expressions inherently hard to derive, there has been less research on the ergodic capacity of mmWave systems. To overcome this problem, by means of the majorization theory, this paper analyzes the ergodic capacity of point-to-point mmWave communication systems under finite-dimensional channel model. In particular, we derive several closed-form ergodic capacity approximations, which exhibit excellent tightness in spite of whether the steering matrices are singular or not. Then, several Jensen’s approximations and bounds of the ergodic capacity are also derived. The results indicate that the ergodic capacity seems to increase logarithmically with the number of antennas, the transmit SNR per antenna, and the eigenvalues of the steering matrix products. Besides, the DFT matrices can effectively characterize the spatial directions of mmWave channels when the number of antennas grows large. After that, high-SNR ergodic capacity, high-SNR slope, and power offset are also analyzed. It indicates that for a finite-dimensional channel, the maximum multiplexing gain increases with the number of paths instead of the number of antennas in Rayleigh channels. Numerical simulations are performed to validate the results.

Capacity of Backscatter Communication Systems With Tag Selection

In backscatter communication (BackCom) systems, the system performance is susceptible to the double fading channels. In this paper, we investigate the capacity performance of BackCom systems via multiple tags, where the tag corresponding to the maximum signal-to-noise ratio (SNR) is selected to combat the double fading. Although the exact ergodic capacity does not admit a closed-form expression, we derive its approximation by exploiting the extreme value theory, which shows that the capacity scales as $\log \log N$ with $N$ denoting the number of tags. Simulation results confirm the tightness of the approximate ergodic capacity, and the correctness of the capacity scaling law.

Simple and Accurate Low SNR Ergodic Capacity Approximations

Some of the existing ergodic capacity approximations for the low signal-to-noise ratio (SNR) region may lack accuracy. To overcome this, we derive two simple yet accurate Pade approximations for the low-SNR ergodic capacity. These approximations utilize the channel moments, which need not be updated for each distinct SNR value. The moments can be derived from the probability density function or from the moment generating function. For instance, we derive the general expressions for the moments of multiple input single output and multiple input multiple output channels. Numerical results demonstrate the superior accuracy of our approximations over some of the existing approximations. Moreover, we demonstrate how our approximations can be efficiently used to analyze several types of wireless links.

The Performance of Wireless Powered MIMO Relaying With Energy Beamforming

This paper analyzes the performance of energy-constrained dual-hop amplify-and-forward relaying systems with multi-antenna nodes in the presence of multiple co-channel interferers at the destination. To maximize the overall signal-to-interference-plus-noise ratio, as well as the harvested energy so as to mitigate the severe effects of fading and enable long-distance wireless power transfer, hop-by-hop information and energy beamforming is proposed where the transmitted signal is steered along the strongest eigenmode of each hop. The wirelessly powered relay scavenges energy from the source information radio frequency signal through energy beamforming, where both the time-switching receiver and power-splitting receiver are considered, and then uses the harvested energy to forward the source message to the destination. To this end, tight lower and upper bound expressions for the outage probability and ergodic capacity are presented in closed form. These are employed to investigate the throughput of the delay-constrained and delay-tolerant transmission modes. In addition, the asymptotic high signal-to-noise ratio outage probability and ergodic capacity approximations are derived, where the achievable diversity order is also presented. Numerical results sustained by Monte Carlo simulations show the tightness of the proposed analytical expressions. The impact of various parameters, such as energy harvesting time, power-splitting ratio, source transmit power, and the number of antennas on the system throughput, is also considered.

Closed-Form Average SNR and Ergodic Capacity Approximations for Best Relay Selection

This work analyses the average signal-to-noise ratio (SNR) and ergodic capacity of large-scale relay networks with best relay selection (BRS). First, extreme value theory (EVT) is used to obtain an implicit expression for the asymptotic cumulative distribution function (cdf) of the received SNR when the number of relays is high. Then, via high power approximations, closed-form expressions for the cdf and probability density function (pdf) of the received SNR are achieved, from which closed-form expressions of the average received SNR and ergodic capacity are derived. Insights on the array gain and ergodic capacity behavior of BRS can be observed from the derived results. Simulations show that the derived approximations are tight, even for not-so-large relay networks.

Ergodic Capacity Analysis of Decode-and-Forward Relay-Assisted FSO Systems Over Alpha–Mu Fading Channels Considering Pointing Errors

The ergodic capacity of decode-and-forward (DF) relay-assisted free-space optical (FSO) communication systems when line of sight is available is analyzed over gamma–gamma fading channels with pointing errors. A novel closed-form approximate ergodic capacity expression is obtained in terms of the H-Fox function for a three-way FSO communications system when the $\alpha{-} \mu$ distribution to efficiently approximate the probability density function (PDF) of the sum of gamma–gamma with pointing-error variates is considered. Moreover, we present novel asymptotic expressions at high signal-to-noise ratio (SNR) , as well as low SNR for the ergodic capacity of DF relay-assisted FSO systems. The main contribution in this paper lies in an in-depth analysis about the impact of pointing errors on the ergodic capacity for cooperative FSO systems. In order to maintain the same performance in terms of capacity, it is corroborated that the presence of pointing errors requires an increase in SNR, which is related to the fraction of the collected power at the receive aperture, i.e., $A_{0} $ . Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results.

Approximate ergodic capacity of multiuser massive multiple input multiple output in a Rayleigh fading uplink channel with variance profile

The effect of large-scale fading variations over massive multiple input multiple output (MIMO) antenna array on sum-capacity of massive MIMO system has not been thoroughly investigated in the literature. This study considers a multiuser massive MIMO system with a Rayleigh fading channel that takes into account large-scale fading variations over base station antenna array. Two scenarios are investigated: distributed antenna base station and linear array base station. In both scenarios, the authors obtain an approximate closed-form expressions for uplink ergodic sum-capacity for both low and high signal-to-noise ratio (SNR) regimes. It is shown that the proposed closed-form expressions are accurate in a wide range of practical SNRs; for example, the approximation error is below 10% for SNRs lower than −10 dB and higher than 17 dB for low-SNR and high-SNR closed-form expressions, respectively. It is also shown that the proposed closed-form expressions are not only valid for massive MIMO scenario but also for a system with a few number of base station antennas. The numerical results show that the proposed closed-form expressions have much less computational complexity than previous works.

On the Performance Analysis of Hybrid ARQ With Incremental Redundancy and With Code Combining Over Free-Space Optical Channels With Pointing Errors

In this paper, we investigate the performance of hybrid automatic repeat request (HARQ) with incremental redundancy (IR) and with code combining (CC) from an information-theoretic perspective over a point-to-point free-space optical (FSO) system. First, we introduce new closed-form expressions for the probability density function, the cumulative distribution function, the moment generating function, and the moments of an FSO link modeled by the Gamma fading channel subject to pointing errors and using intensity modulation with direct detection technique at the receiver. Based on these formulas, we derive exact results for the average bit error rate and the capacity in terms of Meijer's G functions. Moreover, we present asymptotic expressions by utilizing the Meijer's G function expansion and using the moments method, too, for the ergodic capacity approximations. Then, we provide novel analytical expressions for the outage probability, the average number of transmissions, and the average transmission rate for HARQ with IR, assuming a maximum number of rounds for the HARQ protocol. Besides, we offer asymptotic expressions for these results in terms of simple elementary functions. Additionally, we compare the performance of HARQ with IR and HARQ with CC. Our analysis demonstrates that HARQ with IR outperforms HARQ with CC.

Approximate Ergodic Capacity of a Class of Fading Two-User Two-Hop Networks

The fading AWGN two-user two-hop network is considered where the channel coefficients are independent and identically distributed (i.i.d.) according to a continuous distribution and vary over time. For a broad class of channel distributions, the ergodic sum capacity is characterized to within a constant number of bits/second/hertz, independent of the signal-to-noise ratio. The achievability follows from the analysis of an interference neutralization scheme where the relays are partitioned into M pairs, and interference is neutralized separately by each pair of relays. When M = 1, the proposed ergodic interference neutralization characterizes the ergodic sum capacity to within 4 bits/sec/Hz for i.i.d. uniform phase fading and approximately 4.7 bits/sec/Hz for i.i.d. Rayleigh fading. It is further shown that this gap can be tightened to 4 log π-4 bits/sec/Hz (approximately 2.6) for i.i.d. uniform phase fading and 4-4 log(3π/8) bits/sec/Hz (approximately 3.1) for i.i.d. Rayleigh fading in the limit of large M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> .

Performance Analysis of OSTBC for Partial Relay Selection with Correlated Antennas over Nakagami-m Fading

This letter considers the performance for partial relay selection of a dual-hop relay system with correlated antennas at each node. Orthogonal space time block coding (OSTBC) is employed at the source and the best relay. The source communicates with the destination through a relay with the maximum signal-to-noise-ratio (SNR) in the first hop. We assume non-identically distributed Nakagami-m fading channels. The closed form expressions of outage probability, the symbol error rate (SER), and the generalized higher moments of instantaneous end-to-end signal-to-noise-ratio (SNR) are provided. A tight approximation of ergodic capacity is evaluated and an asymptotic approximation is provided to obtain the diversity order. Analytical results are validated with Monte-Carlo simulations using Mathematica and MATLAB programs.

Impact of Antenna Correlation on a New Dual-Hop MIMO AF Relaying Model

A novel system model is proposed for the dual-hop multiple-input multiple-output amplify-andforward relay networks, and the impact of antenna correlation on the performance is studied. For a semiarbitrary correlated source-relay channel and an arbitrary correlated relay-destination channel, the complementary cumulative distribution function (CCDF) and the moment-generating function (MGF) approximations of the end-to-end signal-to-noise ratio (SNR) are derived. The outage probability, the average symbol error rate (SER), and the ergodic capacity approximations are also derived. Two special cases are treated explicitly: (1) dual-antenna relay and multiple-antenna destination and (2) uncorrelated antennas at the relay and correlated antennas at the destination. For the first case, the CCDF, the MGF and the average SER of an upper bound of the end-to-end SNR are derived in closed-form. For the second case, the CCDF, the MGF, the average SER, and the moments of SNR are derived in closed-form; as well, the high SNR approximations for the outage probability and the average SER are derived, and the diversity gain and coding gain are developed. Extensive numerical results and Monte Carlo simulation results are presented to verify the analytical results and to quantify the detrimental impact of antenna correlations on the system performance.

Linear Precoding for High-K-Factor Channels Exploiting Channel Mean and Covariance Information

In this paper, we consider the high-K-factor Ricean channel, where the channel mean is of arbitrary rank, and the fading channel components are correlated. We propose two suboptimal linear precoding schemes based on the knowledge of the channel mean and covariance at the transmitter. The first precoder is derived from an approximate ergodic capacity expression for the high-K-factor Ricean channels, whereas the second is derived from Jensen's inequality. We compare the capacity obtained to the true ergodic capacity found by using a numerical convex optimization algorithm. We further investigate the impact of the fading channel covariance, the rank of the channel mean, and the K-factor on the precoding schemes and the achievable ergodic capacity limit.

Rate of Channel Hardening of Antenna Selection Diversity Schemes and Its Implication on Scheduling

For a multiple-antenna system, we find a simple and accurate expression for the asymptotic distribution of the antenna selection gain when the transmitter selects the transmit antenna with the strongest channel. We use this to estimate the underlying channel capacity distributions and obtain the approximate ergodic capacity. This estimate is compared with upper and lower bounds. This analysis demonstrates that unlike multiple-input/multiple-output (MIMO) systems, the channel for antenna selection systems hardens at a slower rate, and thus a significant multiuser scheduling gain can exist-Theta (1/logm) for channel selection as opposed to Theta (1/radicm) for MIMO, where m is the number of transmit antennas.