On the first- and second-order statistics of the capacity of N*Nakagami-m channels for applications in cooperative networks
© Rafiq et al; licensee Springer. 2012
Received: 1 July 2011
Accepted: 20 January 2012
Published: 20 January 2012
This article deals with the derivation and analysis of the statistical properties of the instantaneous channel capacitya of N*Nakagami-m channels, which has been recently introduced as a suitable stochastic model for multihop fading channels. We have derived exact analytical expressions for the probability density function (PDF), cumulative distribution function (CDF), level-crossing rate (LCR), and average duration of fades (ADF) of the instantaneous channel capacity of N*Nakagami-m channels. For large number of hops, we have studied the first-order statistics of the instantaneous channel capacity by assuming that the fading amplitude of the channel can approximately be modeled as a lognormal process. Furthermore, an accurate closed-form approximation has been derived for the LCR of the instantaneous channel capacity. The results are studied for different values of the number of hops as well as for different values of the Nakagami parameters, controlling the severity of fading in different links of the multihop communication system. The results show that an increase in the number of hops or the severity of fading decreases the mean channel capacity, while the ADF of the instantaneous channel capacity increases. Moreover, an increase in the severity of fading or the number of hops decreases the LCR of the instantaneous channel capacity of N*Nakagami-m channels at higher levels. The converse statement is true for lower levels. The presented results provide an insight into the influence of the number of hops and the severity of fading on the instantaneous channel capacity, and hence they are very useful for the design and performance analysis of multihop communication systems.
Multihop communication systems fall under the category of cooperative diversity systems, in which the intermediate wireless network nodes assist each other by relaying the information from the source mobile station (SMS) to the destination mobile station (DMS) [1–3]. This kind of communication scheme promises an increased network coverage, enhanced mobility, and improved system performance. It has applications in wireless local area networks (WLANs) , cellular networks , ad-hoc networks [6, 7], and hybrid networks . Based on the amount of signal processing used for relaying the received signal, the relays can generally be classified into two types, namely amplify-and-forward (or non-regenerative) relays [9, 10] and decode-and-forward (or regenerative) relays [9, 11]. The relay nodes in multihop communication systems can further be categorized into channel state information (CSI) assisted relays , which employ the CSI to calculate the relay gains and blind relays with fixed relay gains .
In order to characterize the fading in the end-to-end link between the SMS and the DMS in a multihop communication system with N hops, the authors in  have proposed the N*Nakagami-m channel model, assuming that the fading in each link between the wireless nodes can be modeled by a Nakagami-m process. The second-order statistical properties of multihop Rayleigh fading channels have been studied in , while for dualhop Nakagami-m channels, the second-order statistics of the received signal envelope has been analyzed in . Moreover, the performance analysis of multihop communication systems for different kinds of relaying can be found in [10, 13, 17] and the multiple references therein.
The statistical properties of the instantaneous capacity of different multiple-input multiple-output (MIMO) channels have been studied in several articles. For example, by assuming that the instantaneous channel capacity is a random variable, the PDF and the statistical moments of the instantaneous channel capacity have been derived in . Moreover, by describing the instantaneous channel capacity as a discrete-time or a continuous-time stochastic process, the LCR and ADF of the instantaneous channel capacity have been studied in . Furthermore, analytical expressions for the PDF, CDF, LCR, and ADF of the continuous-time instantaneous capacity of MIMO channels by using orthogonal space-time block codes have been derived in . The temporal behavior of the instantaneous channel capacity can be studied with the help of the LCR and ADF of the channel capacity. The LCR of the instantaneous channel capacity describes the average rate of up-crossings (or down-crossings) of the instantaneous channel capacity through a certain threshold level. The ADF of the instantaneous channel capacity denotes the average duration of time over which the instantaneous channel capacity is below a given level [20, 21]. In the literature, the analysis of the LCR and ADF has mostly been carried out for the received signal envelope, which provides useful information regarding the statistics of burst errors occurring in fading channels . However, in , the channel capacity for systems employing multiple antennas has been proposed as a more pragmatic performance merit than the received signal envelope. Therein, the authors have used the LCR of the instantaneous channel capacity to improve the system performance. Hence, it is important to study the LCR and ADF in addition to the PDF and CDF of the instantaneous channel capacity in order to meet the increasing demand for high data rates in mobile communication systemsb. In , the authors analyzed the statistical properties of the instantaneous capacity of dualhop Rice channels employing amplify-and-forward based blind relays. An extension of the work in  to the case of dualhop Nakagami-m channels has been presented in . The ergodic capacity of generalized multihop fading channels has been studied in . Though a lot of artilces have been published in the literature dealing with the performance and analysis of multihop communication systems, the statistical properties of the instantaneous capacity of N*Nakagami-m channels have not been investigated so far. The aim of this article is to fill in this gap of information.
In this article, the statistical properties of the instantaneous capacityc of N*Nakagami-m channels are analyzed. For example, we have derived exact analytical expressions for the PDF, CDF, LCR, and ADF of the channel capacity. The mean channel capacity (or the ergodic capacity) can be obtained from the PDF of the channel capacity , while the CDF of the channel capacity is helpful for the derivation of the outage capacity . Both the mean channel capacity and outage capacity have widely been used in the literature due to their importance for the system design. The mean channel capacity is the ensemble average of the information rate over all realizations of the channel capacity . The outage capacity is defined as the maximum information rate that can be transmitted over a channel with an outage probability corresponding to the probability that the transmission cannot be decoded with an arbitrarily small error probability . In general, the mean channel capacity is less complicated to study analytically than the outage capacity . Although the mean channel capacity and outage capacity are important quantities that describe the channel, they do not give any insight into the dynamic behavior of the channel capacity. For example, the outage capacity does not provide any information regarding the spread of the outage intervals or the rate of occurrence of these outage durations in the time domain. In , it has been demonstrated that the temporal behavior of the channel capacity is very useful for the improvement of the overall network performance.
The rest of the article is organized as follows. In Section 2, we briefly describe the N*Nakagami-m channel model and some of its statistical properties. Section 3 presents the statistical properties of the capacity of N*Nakagami-m channels. A study on the first-order statical properties of the channel capacity for a large number of hops N is presented in Section 4. The analysis of the obtained results is carried out in Section 5. The concluding remarks are finally stated in Section 6.
2 The N*Nakagami-m channel model
where each of the processes follows the Nakagami-m distribution with parameters m n and . To gain an insight into the relationship between the relay gains G n and the instantaneous signal-to-noise ratio (SNR) γ(t) at the DMS, one can see the results presented in [, Equations (1)-(3)]. Therein, it can easily be observed that increasing the relay gains G n increases the instantaneous SNR at the DMS for any arbitrary fixed values of the noise variances at the relays. However, at any instant of time t, the value of γ(t) is always less than or equal to γl(t), representing the instantaneous SNR at the first mobile relay In other words, as the value of G n increases, the value of γ(t) approaches γl(t) for any value of t. It is worth mentioning that in general, the total noise at the DMS can be represented as a sum of products. Specifically, it is a sum of N terms, where except for one (which is the noise component of the final hop), all the other (N - 1) terms can be expressed as a product of the corresponding hop's noise component and the channel gains of all the pervious hops . However, we have assumed that each product term has Gaussian distribution and is independent from the others. Hence, the sum is also assumed to be Gaussian distributed, making the AWGN assumption valid at the DMS. In the following, for the sake of simplicity, we will assume a fixed noise power N0 at the DMS. Hence, the instantaneous SNR at the DMS is given by γ(t) = P S (t)/N0. Here, P S (t) denotes the instantaneous signal power at the DMS and is expressed as .
Here, and represent the maximum Doppler frequencies of the SMS and DMS, respectively, while denotes the maximum Doppler frequency of the n th mobile relay MR n (n = 1, 2, ..., N - 1). It should be mentioned that the expression obtained in (7b) is only valid under isotropic scattering conditions [36, 37].
3 Statistical properties of the capacity of N*Nakagami-m channels
Unfortunately, for N*Nakagami-m channels, closed-form analytical expressions for the mean channel capacity, variance of the channel capacity, and the outage capacity given by (10), (11), and (13), respectively, are very difficult to obtain. Nevertheless, these results can be obtained numerically, as will be presented in Section 5.
where F C (r) and N C (r) are given by (4) and (8), respectively
4 Asymptotic analysis
respectively. In the next section, it will be shown by simulations that the approximations obtained in (23)-(26) perform well even for a small number of hops N.
5 Numerical results
where μ n,l (t) (l = 1, 2, ..., 2m n ; n = 1, 2, ..., N) are the underlying independent and identically distributed (i.i.d.) Gaussian processes, and m n is the parameter of the Nakagami-m distribution associated with the n th link of the multihop communication systems. The Gaussian processes μ n,l (t), each with zero mean and variances , were simulated using the sum-of-sinusoids model . The model parameters were computed using the generalized method of exact Doppler spread (GMEDS1) . The number of sinusoids for the generation of Gaussian processes μ n,l (t) was chosen to be 20. The parameter Ω n was chosen to be equal to 2(m n σ0)2, the values of the maximum Doppler frequencies were set to be equal to 125 Hz, and the quantity γ s was equal to 15 dB. The parameters Gn-1(n = 1, 2, ..., N) and σ0 were chosen to be unity. The simulation time for the channel realizations was set set to be 250 s with sampling duration of 50 μ s. Finally, using (3), (8), and (27), the simulation results for the statistical properties of the channel capacity were foundd. For analytical illustrations, the Meijer's G-function as well as the multifold integrals can be numerically evaluated using the existing built-in functions of the numerical computation tools, such as MATLAB or MATHEMATICA.
In this article, we have presented a statistical analysis of the capacity of N*Nakagami-m channels. Specifically, we have studied the influence of the severity of fading and the number of hops on the PDF, CDF, LCR, and ADF of the channel capacity. We have derived an accurate closed-form approximation for the LCR of the channel capacity. For a large number of hops N, we have investigated the suitability of the assumption that the N*Nakagami fading distribution can be approximated by the lognormal distribution. The findings of this article show that an increase in the number of hops N or the severity of fading decreases the mean channel capacity, while it results in an increase in the ADF of the channel capacity. Moreover, at higher levels r, the LCR N C (r) of the capacity of N*Nakagami-m channels decreases with an increase in severity of fading or the number of hops N. However, the converse statement is true for lower levels r. Furthermore, the variance of the channel capacity decreases by increasing the number of hops, while increase in the severity of fading has an opposite influence on the variance of the channel capacity. It is also observed that increasing the relay gains increases the received SNR at the DMS, however the received SNR at the DMS is always less than or equal to the SNR at the first mobile relay MR1. The analytical results are verified by simulations, whereby a very good fitting is observed.
Finally, by substituting (30), (30), (31), and (17b) in (28), we obtain the approximate closed-form expression for the LCR N C (r) of the channel capacity C(t) given by (16).
bThe scope of this article is limited only to the derivation and analysis of the statistical properties of the instantaneous channel capacity. However, a detailed discussion regarding the use of statistical properties of the channel capacity for the improvement of the system performance can be found in, e.g., [23, 38, 39] and the references therein.
dFor further details, the interested reader is referred to , where MATLAB source codes are provided for simulating different channel realizations as well as the corresponding statistical properties (such as the PDF, CDF, LCR, and ADF) for a variety of propagation scenarios.
The contribution of Dr. G. Rafiq and Prof. M. Pätzold in this article was partially supported by the Research Council of Norway (NFR) through the project 176773/S10 entitled "Optimized Heterogeneous Multiuser MIMO Networks-OptiMO".
The contribution of Dr. B. O. Hogstad was supported in part by the Basque Government through the MIMONET project (PC2009-27B), and by the Spanish Ministry of Science and Innovation through the projects COSIMA (TEC2010-19545-C04-02) and COMONSENS (CSD2008-00010).
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