A simple approach to evaluate the ergodic capacity and outage probability of correlated Rayleigh diversity channels with unequal signal-to-noise ratios
© Moinuddin and Naseem; licensee Springer. 2013
Received: 6 May 2012
Accepted: 10 January 2013
Published: 30 January 2013
In this article, we propose a novel method to derive exact closed-form ergodic capacity and outage probability expressions for correlated Rayleigh fading channels with receive diversity. Unlike the existing works, the proposed method employ a simple approach for the capacity and outage analysis for receiver diversity channels operating at different signal-to-noise ratios depicted in the diagonal elements of matrix Ω. With x being the channel gain vector, random variable of the form Y(a)=a + x∗Ω x is considered. Novelty of the work resides in the fact that the distribution of Y(a) is accurately determined by employing Fourier representation of unit step function followed by complex integration in a straight forward way. The ergodic channel capacity is thus calculated by using the first-order moment, , while the outage probability for a certain threshold γ0is evaluated using . Extensive experiments have been conducted demonstrating the accuracy of the proposed approach.
KeywordsErgodic channel capacity Rayleigh fading Receive diversity
Ergodic channel capacity and outage probability are two important parameters to be considered for the design of a given communication system . As such the Shannon capacity formula, initially derived for Gaussian environment, gives an upper bound for maximum transmission rate . Capacity however depends on the nature of a particular channel environment. Consequently, a number of investigations have been conducted for various fading channels. Efforts have been focused to derive closed-form expressions for exact/estimated capacity [3–5].
It is well known that diversity schemes enhance the system capacity by proper utilization of random variation in a multipath wireless channel. However, the capacity evaluation and the outage analysis of diversity schemes becomes complicated. Several works have attempted to study the capacity and outage analysis of diversity channels [6, 7]. Unfortunately, the results of these works are mostly (1) approximate using some assumptions, (2) limited for some specific scenarios, and/or (3) do not result in closed form expressions. For instance, a common practice is to assume independence across the multipath channels [6, 8]. In , for example, a number of closed-form expressions for channel capacity of independent multipath Rayleigh fading channels have been presented. While the assumption of independence across the channels seems appropriate enough, there are situations when such a premise is practically inadequate. Assumption of independence among diversity branches fails and there exist a correlation among them when there is insufficient antenna spacing. Consequently, the correlation between the multipath channels results in degradation of the overall performance . In [7, 10], capacity with correlated fading channels have been addressed for Rayleigh and Nakagami channels, respectively. Another commonly accepted practice is to approximate a weighted sum of chi-square variables by a single one with different degrees of freedom and an adequate scaling factor. The average capacity of correlated diversity Rician channels, for instance, is derived in  using this approximation.
In , closed-form expressions for the capacity of correlated Nakagami-m fading channels is derived. In particular, the following scenarios are considered: (i) Dual-branch maximal ratio combining, (ii) equal gain combining, (iii) selection combining, and (iv) switch and stay combining. The approach relies on the confluent hypergeometric function which results in an infinite series. The series is approximated by truncation and upper bound on the truncation error is calculated. The analysis in  can therefore be regarded as an approximate solution due to series truncation. Moreover, the analysis is limited to a specific scenario of equal correlation among the diversity branches.
In , information capacity of the random signature MIMO–CDMA system is calculated. Primarily, the distribution of eigenvalues of the covariance of channel signature matrix is employed. The results, however, are limited for the scenarios of unsaturated and over saturated systems. Moreover, the methodology cannot be employed for any other MIMO system without CDMA architecture.
The analysis in  gives closed-form expressions for the single-user capacity for Rayleigh fading channel for the MRC diversity system. Different adaptive transmission techniques are considered assuming multiple uncorrelated branches with equal average SNR. In real scenarios however, assumption of independent fading is not always true. For instance, small-size terminals with space antenna diversity may have insufficient antenna spacing to obtain independent fading in each branch. Thus, this study is limited to a specific scenario, incorporating equal average branch SNRs and equal correlation among the diversity branches.
The study  addresses the scenario of equal average branch SNRs and arbitrary correlation between branches under three adaptive policies: (i) Optimal power and rate adaptation, (ii) constant power with optimal rate adaptation, and (iii) channel inversion with fixed rate. The approach takes into account the probability distribution function (PDF) for the sum of individual branch average SNRs (i.e., , where n is the number of diversity branches).In this study, we present a novel approach of capacity and outage probability analysis of correlated diversity Rayleigh fading channels. In contrast to the above approaches, we aim to derive the PDF of a random variable of the form Y(a)=a + x ∗Ω x where x ∗is Hermitian transpose of x . Primarily, the PDF is calculated by incorporating the integration limits as a unit step function. Fourier representation of the unit step function is further used to facilitate the complex integration. Consequently, the expressions for the capacity and outage probability are evaluated using the derived PDF of Y(a). Analytical results are validated through extensive experiments.
A novel, exact, and simpler method for the capacity analysis of the correlated diversity Rayleigh channels is presented.
For the purpose of unified analysis, the PDF of a generalized random variable of the form Y(a)=a + x ∗Ω x is derived using Fourier representation of the unit step function.
Exact closed-form expression for the ergodic capacity of correlated diversity Rayleigh channels is evaluated for any degree of channel correlation and unequal SNRs.
Exact closed form expressions for the outage probability for certain threshold γ 0is evaluated.
The remainder of this article is organized as follows: Section 2 presents the problem formulation followed by the proposed approach in Section 3. Experiments are presented in Section 4 and the article is concluded in Section 5.
Ergodic capacity and outage probability of correlated diversity rayleigh fading channels
Ergodic capacity is defined as E C where E is the expectation operator showing ensemble average of a random variable.
The proposed approach
Thus, our approach relies on evaluating the PDF and the moment of the random variable of the quadratic form where D is a diagonal matrix. Next, we present the derivation for the PDF of random variable Y(a).
The PDF of Y(a)
Evaluation of ergodic capacity
Evaluation of outage probability
The experiments were aimed to investigate: (1) effect of correlation coefficient (γ c ) on the channel capacity; (2) variation in channel capacity with respect to the change in diversity order n; (3) effect of the SNR on the channel capacity; (4) effect of the SNR on the Outage Probability; (5) effect of correlation coefficient on the outage probability; and (6) agreement of simulation and analytical results for all sets of experiments.
Comparison of simulation and analytical results for the case study γ c = 0.2
Diversity order (n)
Capacity comparison with respect to variation in the correlation coefficient with n = 16
Capacity comparison for simulation and analytical experiments with n =8
A novel and exact expression for average capacity of correlated diversity Rayleigh fading channels is presented. The proposed approach relies on exact evaluation of the CDF of random variable of the form 1 + x∗Ω x. This is essentially achieved by using Fourier representation of the unit step function followed by complex integration. The main contribution of the proposed research is the exact analysis in a simpler way which avoids approximations and sophisticated expressions. Extensive experiments have been conducted to investigate the accuracy of the proposed approach. Analytical and simulation results are found to be in agreement for all sets of experiments.
aFor any matrix D, the quadratic form is defined as .
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