Expression For Moment Generating Function Research Articles

Performance analysis of energy harvesting‐enabled relay networks inκ‐μfading channels

AbstractIn this paper, we investigate the performance of an energy harvesting (EH)‐enabled multiple relay network operating over generalized‐fading channels. Our approach involves utilizing a partial relay selection strategy and employing an amplify and forward relay protocol for efficient EH from RF signals. Additionally, we integrate a selection combining scheme at the receiver to combine both direct and relaying path signals. In our research, we first obtain the exact closed‐form expression of the cumulative distribution function (CDF) for the system under investigation. Subsequently, we derive expressions for the outage probability (OP) and the moment generating function (MGF) using the derived CDF expression. Furthermore, with the assistance of a CDF‐based approach, we derive closed‐form expressions for the average symbol error rate (ASER) for coherent quadrature amplitude modulation (QAM) schemes, including rectangular QAM (RQAM) and hexagonal QAM (HQAM). We also analyze the ASER expression for non‐coherent frequency shift keying (NCFSK), leveraging the derived MGF expression. To gain valuable insights into the performance of EH‐enabled multiple relay networks, we assess the asymptotic expression of the OP. The outcomes indicate how the behavior of the system is affected by various factors such as time switching ratio, channel fading components, number of relays, and the distance between nodes. To corroborate the accuracy of the analytical findings, Monte Carlo simulations are executed.

Joint Distribution of Ages of Information in Networks

We study a general setting of status updating systems in which a set of source nodes provide status updates about some physical process(es) to a set of monitors. The freshness of information available at each monitor is quantified in terms of the Age of Information (AoI), and the vector of AoI processes at the monitors (or equivalently the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">age vector</i> ) models the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">continuous</i> state of the system. While the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">marginal</i> distributional properties of each AoI process have been studied for a variety of settings using the stochastic hybrid system (SHS) approach, we lack a counterpart of this approach to systematically study their <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">joint</i> distributional properties. Developing such a framework is the main contribution of this paper. In particular, we model the discrete state of the system as a finite-state continuous-time Markov chain, and describe the coupled evolution of the continuous and discrete states of the system by a piecewise linear SHS with linear reset maps. Using the notion of tensors, we first derive first-order linear differential equations for the temporal evolution of both the joint moments and the joint moment generating function (MGF) for an arbitrary set of age processes. We then characterize the conditions under which the derived differential equations are asymptotically stable. The generality of our framework is demonstrated by recovering several existing results as special cases. Finally, we apply our framework to derive closed-form expressions of the stationary joint MGF in a multi-source updating system under non-preemptive and source-agnostic/ source-aware preemptive in service queueing disciplines.

Novel MGF expressions of [formula omitted] composite fading channels with applications

In this paper, we provide novel expressions of moment generating function (MGF) for Fisher–Snedocor F composite fading channels. Precisely, we derive the exact MGF and its approximated expression to incorporate the heavy as well as the light shadowed conditions. With the help of these expressions, it is shown that the MGF of the F distribution turn to be valid for the entire range of the composite fading parameters yielding finite results for each value of the average signal to noise ratio. The significance of these expressions is further explained by simplifying the key integrals which are used in the error computation of widely used efficient digital modulation schemes. The analytical framework is also validated with the help of Monte-Carlo simulations.

Error Probability Analysis and Power Allocation for Interference Exploitation Over Rayleigh Fading Channels

This paper considers the performance analysis of constructive interference (CI) precoding technique in multi-user multiple-input single-output (MU-MISO) systems with a finite constellation phase-shift keying (PSK) input alphabet. Firstly, analytical expressions for the moment generating function (MGF) and the average of the received signal-to-noise-ratio (SNR) are derived. Then, based on the derived MGF expression the average symbol error probability (SEP) for the CI precoder with PSK signaling is calculated. In this regard, new exact and very accurate asymptotic approximation for the average SEP are provided. Building on the new performance analysis, different power allocation schemes are considered to enhance the achieved SEP. In the first scheme, power allocation based on minimizing the sum symbol error probabilities (Min-Sum) is studied, while in the second scheme the power allocation based on minimizing the maximum SEP (Min-Max) is investigated. Furthermore, new analytical expressions of the throughput and power efficiency of the CI precoding in MU-MISO systems are also derived. The numerical results in this work demonstrate that, the CI precoding outperforms the conventional interference suppression precoding techniques with an up to 20 dB gain in the transmit SNR in terms of SEP, and up to 15 dB gain in the transmit SNR in terms of the throughput. In addition, the SEP-based power allocation schemes provide additional up to 13 dB gains in the transmit SNR compared to the conventional equal power allocation scheme.

MGF-based Analysis of Maximum Ratio Combining Receiver over Fisher-Snedecor Composite Fading Channel

Fisher Snedecor composite fading model is the combination of Nakagami-m and inverse Nakagami-m distribution. The Nakagami-m is used to characterize the small scale fading, whereas shadowing is modeled by inverse Nakagami-m distribution. In this paper, the closed-form expression for moment generating function (MGF) of instantaneous signal to noise ratio (SNR) over independent identically distributed (i.i.d) Fisher-Snedecor composite fading channel using maximum ratio combining (MRC) diversity technique is derived. By using newly derived MGF expression, we derive the closed-form expressions of average bit error rate (ABER) or average symbol error rate (ASER) for different binary and multilevel modulation schemes. The expressions for average channel capacity (ACC) under two adaptive transmission protocols like optimum rate adaption (ORA) and channel inversion with fixed rate (CIFR) are also derived using proposed MGF. Further, the numerical results of newly derived expression are presented and compared with the results of Rayleigh and Nakagami-m distribution which is the special case of Fisher Snedecor composite fading model.

MGF‐based analysis of κ‐μ/gamma composite fading model for indoor off body communication

AbstractThis paper presents the moment generating function (MGF)–based analysis of κ‐μ/gamma composite fading model, which is suitable for the indoor off‐body communication channel. In this paper, first of all, we derive an analytical expression for MGF of instantaneous signal‐to‐noise ratio. By using the derived MGF, the expressions of average symbol error rate (ASER)/average bit error rate (ABER) for binary and M‐ary modulation scheme are derived. Using the MGF approach, we also derive the expression of channel capacity for optimum rate adaption and channel inversion with fixed rate policy. The derived expressions for ABER/ASER and channel capacity are presented in the infinite series form. We further present the numerical results of error rate and channel capacity for different values of κ, μ, α, and β parameters. The result of ABER and channel capacity shows good agreement with the previously reported results as special cases that verify the derived expressions.

Imperfect CSI Based Multi-Way Satellite Relaying

The problem of multi-way satellite relaying is discussed in this letter. A new transmission method is proposed to improve the performance of the considered system. Further, it is well known that for detecting the transmitted data in a multi-way relaying system, global channel state information (CSI) is required at each earth station (ES). Due to nonreciprocal channels and high value of latency, it is very difficult to estimate the channel gains at respective ESs. A method of training is proposed to estimate the channel gains. Detection of data is performed by replacing exact CSI by estimated CSI in decision metric. Analytical results of the proposed system are obtained by using moment generating function (MGF) approach. An approximate closed-form expression of MGF is obtained by ignoring higher order noise terms. The symbol error rate and diversity order of the proposed system are derived by using MGF expression.

Imperfect CSI‐based large MIMO systems

The authors address the problem of channel estimation and data detection in a large multiple-input multiple-output (MIMO) systems in Rayleigh fading environment. A method of channel estimation and data detection is proposed in large MIMO systems. The channel matrix is decomposed in N d /2 × N s /2 channel matrices; where N d and N s are the number of antennas at the destination and source, respectively. The pilot signals are clubbed with data symbols encoded in N d /2 × N s /2 orthogonal space-time block codes (OSTBC), and then by using the properties of null matrix, channel estimation is performed. A decoder of OSTBC data is derived by assuming the perfect channel state information (CSI) at the destination; then perfect CSI is replaced by estimated CSI in the decision metric. The expression for the moment generating function (MGF) of the received signal-to-noise ratio at the destination node is derived for the proposed scheme. Using the MGF expressions, they obtain the analytical symbol error rate for the proposed channel estimation-based decoders for general OSTBCs with M-ary phase-shift keying modulation symbols.

On the Performance of RF-FSO System Over Rayleigh and Kappa-Mu/Inverse Gaussian Fading Environment

In this paper, we investigate a decode-and-forward cooperative communication based radio frequency (RF)-free space optical (FSO) system. An asymmetric environment is considered where source to relay channel is subjected to Rayleigh fading and relay to destination path is assumed to follow $\kappa $ - $\mu $ /Inverse Gaussian (IG) distribution. The analytical framework proposed is based on a versatile $\kappa $ - $\mu $ /IG model which includes one-sided Gaussian/IG, Rice (Nakagami- $n$ )/IG, Nakagami- $m$ /IG and Rayleigh/IG distributions as special cases. The expressions for moment generating function (MGF) and its special case are derived for this mixed RF and FSO network. The MGF is further utilized to derive the expression of end-to-end ergodic channel capacity for the RF-FSO communication system. Thereafter, end-to-end bit error rate is determined for BPSK, QPSK, DPSK and NCFSK modulation schemes. In addition, we have also derived analytical and asymptotic expressions of MGF and outage probability of the proposed RF-FSO system. Furthermore, we have also investigated the diversity gain and coding gain of the considered RF-FSO system.

Symbol error rate of QO‐STBC based decode‐and‐forward cooperative communication system over generalised – and – fading channels

In this study, the authors derive the closed form expressions for symbol error rate (SER) of a quasi-orthogonal space-time block code (QO-STBC) based multi-relay cooperative communication system using M-ary phase-shift keying (M-PSK) modulation over generalised – fading channel. For the same system over – fading channel, due to the unobtainability of the closed form SER expressions, the analysis is restricted to binary-PSK and the bit error rate (BER) has been derived. The authors employ decode-and-forward protocol at the relays and maximal ratio combiner at the destination. For the ease of analysis, the authors consider a three relay cooperative network and the SER has been derived which could be extended to any number of relays or QO-STBCs. First, the expressions of SER were expressed in terms of moment generating function (MGF) and finally, the available closed form expression for MGF of – and – channel distributions are used to obtain the final expression of theoretical BER/SER. The validity of the obtained BER/SER expressions for a three relay cooperative system are verified through Monte–Carlo simulations by varying the channel parameters of – and – fading channels.

Cooperative IDMA systems with regenerative relays over Weibull fading channels: outage probability and error analysis

Cooperative communications has recently been introduced into multiple-access networks to combat the signal attenuation caused by fading and therefore improve the network performance. This study proposes the outage probability and error analysis of cooperative interleave-division multiple-access (IDMA) networks with regenerative relays over Weibull fading channels. The main contribution of this study relies on deriving a moment-generating function (MGF) of the total signal-to-noise ratio with the help of Pade approximation for IDMA with regenerative relaying. The derived MGF expression is employed to evaluate the outage probability of the considered system. Moreover, a closed-form asymptotic expression for the error probability of the considered system is derived and then comprehensive computer simulations have been performed to validate the accuracy of the authors’ analytical results. It is shown that results obtained by the proposed MGF and asymptotic error expressions are in good agreement with their corresponding simulation results.

Average symbol error rate and outage probability of DS-CDMA systems with AF relaying over asymmetric fading channels

In this paper, approximate outage probability (OP) and average symbol error rate (SER) of cooperative direct-sequence code-division multiple access (DS-CDMA) systems with amplify-and-forward (AF) relaying are evaluated over asymmetric fading channels. Practically, the channels between source (S), relay (R) and destination (D) nodes in a DS-CDMA system can be subject to different fading due to the nature of wireless medium. Motivated by this reality, the S–D path is assumed to be Nakagami-m distributed which is a general fading model for different types of channel conditions while S–R and R–D paths are considered to experience Rician fading in order to include line of sight conditions. First, closed-form solutions are obtained for both the cumulative distribution function and the probability density function of S–R–D paths. Then, a closed-form total moment generating function (MGF) of the considered system is derived. The lower bound of the OP is calculated with the help of the inverse Laplace transform of the derived MGF expression while average SER is analyzed by evaluating the integral over the derived MGF. Finally, an asymptotic MGF expression is proposed for high signal-to-noise ratio (SNR) analysis of AF DS-CDMA systems. The simulation results are provided to verify the correctness of the analytical derivations. It is shown that the approximate OP and average SER results are in well agreement with the simulations and the asymptotic results are tight from medium to high SNR regime.

Channel Estimation and Detection in Satellite Communication Systems

The problem of channel estimation and detection in shadowed Rician (SR) fading land mobile satellite (LMS) channels is studied. It is assumed that the satellite contains a single antenna and that the earth station has multiple antennas. In the proposed detector, joint channel estimation and detection of symbols is performed. It is shown by simulation and analysis that joint detection performs better than the existing decoupled detection. In this paper, we derive closed-form expressions of the moment-generating function (MGF) of the received signal-to-noise ratio (SNR), bit error rate (BER), analytical diversity order, and capacity for the considered detector in correlated SR fading LMS links. Next, we consider independent and identically distributed (i.i.d.) SR fading LMS channels and derive approximate expressions of MGF, symbol error rate (SER), diversity order, and capacity of the satellite system with estimated channel gains for existing decoupled and proposed joint detectors. It is shown from the analysis and simulation that the channel estimation error does not affect the diversity order but reduces the capacity of the considered system.

OSTBC Transmission in Large MIMO Systems

We address the problem of orthogonal space-time block code (OSTBC) transmission in a large multiple-input multiple-output (MIMO) system in Rayleigh fading environment. Because there are hundreds of antennas at the transmitter, a very large dimension OSTBC is required in the case of massive MIMO system. Full rate OSTBC is available only for two transmit antenna-based system with complex symbols. Recently, an implementation of OSTBC for 32 transmit and 32 receive antennas is presented for real symbols. In this letter, we present a scheme of OSTBC transmission for practical large MIMO systems by embedding the OSTBCs of small dimensions with good rate. At the receiving end, the decoding of the OSTBCs is performed by utilizing the concept of null space. It is shown by simulation and analysis that the code rate can be improved at the cost of degraded performance; however, the performance of large MIMO system is not an issue because of the availability of very large antennas at the transmitter and receiver. The expression for the moment generating function (MGF) of the received signal-to-noise ratio at the receiver node is derived for the proposed OSTBC-based scheme. Using the MGF expression, we obtain the analytical symbol error rate for the proposed scheme with M-ary phase-shift keying modulation symbols.

Efficient Analysis of the Ergodic Capacity of Cooperative Non-Regenerative Relay Networks Over Generalized Fading Channels

In this paper, we proposed a novel efficient method of analyzing the ergodic channel capacity of the cooperative amplify-and-forward (CAF) relay system. This is accomplished by employing a very tight approximate moment generating function (MGF) of end-to-end signal-to-noise ratio of 2-hop multi-relay system, which is In this paper, we proposed a novel efficient method of analyzing the ergodic channel capacity of the cooperative amplify-and-forward (CAF) relay system. This is accomplished by employing a very tight approximate moment applicable to myriad of fading environments including mixed and composite fading channels. Three distinct adaptive source transmission policies were considered in our analysis namely: (i) constant power with optimal rate adaptation (ORA); (ii) optimal joint power and rate adaptation (OPRA); and (iii) fixed rate with truncated channel inversion (TCIFR). The proposed frame work based on the novel approximate MGF method is sufficiently general to encapsulate all types of fading environments (especially for the analysis of the mixed fading case)and provides significant advantage to model wireless system for mixed and composite fading channel. In addition to simplifying computation complexity of ergodic capacity for CAF relaying schemes treated in literature, we also derive closed form expressions for the above three adaptive source transmission policies under Nakagami-m fading with i.n.d statistics. The accuracy of our proposed method has been validated with existing MGF expressions that are readily available for specific fading environments in terms of bounds, and via Monte Carlo simulations.

Beamforming and combining based scheme over κ − μ shadowed fading satellite channels

In this study, the authors address the problem of beamforming and combining based amplify-and-forward relaying in a satellite communication system in which two earth stations (ESs) with multiple antennas communicate via a single antenna based satellite relay. The transparent satellite is used to relay the received signals from one ES to other ES. Both ESs are assumed to be geographically separated by a very large distance, therefore, no direct path exists in between the ESs. The ESs-satellite links are assumed to follow the κ − μ shadowed fading. A closed form expression of moment generating function (MGF) of the instantaneous received signal-to-noise ratio at destination ES is derived. An expression for the symbol error rate of the considered system is obtained by using the expression of MGF. The analytical diversity order of the system is also obtained by using an asymptotic property of the Meijer-G function. It is shown by simulation and analysis that the diversity order of the considered system is min(L1μ1, L2μ2), where L1 and L2 are the number of antennas at the ES-1 and ES-2, respectively.

Exact SINR Statistics in the Presence of Heterogeneous Interferers

We derive new results for the higher order moments of signal-to-interference-plus-noise ratio (SINR) in the presence of an arbitrary Poisson point process (PPP)-based heterogeneous interference field. The analysis leverages on a moment-generating-function (MGF) methodology, which only requires the statistics of intended signal and aggregate interference, thus eliminating the need for the exact distribution of SINR. We extend the existing results on interference statistics by deriving a generalized closed-form expression of the interference MGF considering Nakagami-m fading channels with exclusion region. In certain special cases, explicit expressions for the averages of different functions of SINR are found, which also lead to closed-form solutions for the probability distributions of aggregate interference reciprocal and signal-to-interference ratio. We prove that in such cases the effect of total PPP-based interference power on useful transmission is mathematically equivalent to the severe fluctuations from a one-sided Gaussian fading channel. As an application example, the proposed methodology is used together with stochastic geometry theory to characterize the average SINR and rate in heterogeneous cellular networks. The validity of our analytical derivations is confirmed via Monte Carlo simulations for various system settings. We show that with cellular network densification there exists a tradeoff between the average SINR and rate performance.

SER performance of cooperative systems with the Nth best path selection in generalized-K channels

We investigate performance of the both variable and fixed-gain amplify-and-forward cooperative communication systems over composite fading and shadowing channels modeled by independent but not necessarily identically distributed generalized-K distribution. We consider two different Nth best path selection schemes: the Nth best relay and link selection. In the Nth best relay selection (RS) scheme, signals received from the Nth best relay and the direct path are combined with maximal-ratio combining and decoded at the destination. In the Nth best link selection (LS) scheme, the direct path is included to the selection set and only the signal received from the selected path is decoded at the destination. New symbol error rate (SER) expressions are derived analytically by using the moment generating function (MGF) approach with Padé approximation and the cumulative distribution function approach. The MGF and SER expressions are in closed-form for the Nth best RS and LS systems, respectively. The derived SER expressions are almost exact at any signal-to-noise ratio. We also present the asymptotic diversity order of the system. The analytical results are verified by Monte-Carlo simulations. Since the generalized-K channel model is a quite general channel model, the existing symbol error probability results given in the literature for cooperative systems with path selection over conventional Rayleigh or Nakagami-m fading channels can be obtained as special cases of our results.

Performance Analysis of Two-Way AF MIMO Relaying of OSTBCs With Imperfect Channel Gains

In this paper, we consider the relaying of orthogonal space-time block codes (OSTBCs) in a two-way amplify-and-forward (AF) multiple-input-multiple-output (MIMO) relay system with estimated channel state information (CSI). A simple four-phase protocol is used for training and OSTBC data transmission. Decoding of OSTBC data at a user terminal is performed by replacing the exact CSI by the estimated CSI in a maximum-likelihood (ML) decoder. Tight approximations for the moment-generating function (MGF) of the received signal-to-noise ratio (SNR) at a user is derived under Rayleigh fading by ignoring the higher order noise terms. Analytical average error performance of the considered cooperative scheme is derived by using the MGF expression. Moreover, the analytical diversity order of the considered scheme is also obtained for certain system configurations. It is shown by simulations and analysis that the channel estimation does not affect the diversity order of the OSTBC-based two-way AF MIMO relay system.

Two-Way Mobile Satellite Relaying: A Beamforming and Combining Based Approach

In this letter, we propose a beamforming and combining scheme for a two-way amplify-and-forward protocol-based communication between two multi-antenna earth stations, via a single antenna satellite relay. Under the shadowed-Rician fading, an approximate expression for the symbol error rate of the scheme is obtained based on the moment generating function (MGF) of the instantaneous received signal-to-noise ratio. The proposed expression of MGF contains finite number of summations of the Meijer-G function. By using an asymptotic property of the Meijer-G function, the analytical diversity order of the proposed scheme is also obtained. It is shown by simulations that the considered two-way satellite relaying scheme can overcome the shadowing with improved diversity by increasing the number of antennas at the earth stations.