# A simple approach to evaluate the ergodic capacity and outage probability of correlated Rayleigh diversity channels with unequal signal-to-noise ratios

- Muhammad Moinuddin
^{1}and - Imran Naseem
^{2}Email author

**2013**:20

https://doi.org/10.1186/1687-1499-2013-20

© Moinuddin and Naseem; licensee Springer. 2013

**Received: **6 May 2012

**Accepted: **10 January 2013

**Published: **30 January 2013

## Abstract

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, $E\left[\underset{2}{log}\right(Y\left(1\right)\left)\right]$, while the outage probability for a certain threshold *γ*_{0}is evaluated using $\underset{0}{\overset{{\gamma}_{0}}{\int}}{f}_{Y\left(0\right)}\left(y\right)\mathit{\text{dy}}$. Extensive experiments have been conducted demonstrating the accuracy of the proposed approach.

### Keywords

Ergodic channel capacity Rayleigh fading Receive diversity## Introduction

Ergodic channel capacity and outage probability are two important parameters to be considered for the design of a given communication system [1]. As such the Shannon capacity formula, initially derived for Gaussian environment, gives an upper bound for maximum transmission rate [2]. 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 [9], 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 [10]. 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 [5] using this approximation.

- (a)
In [7], 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 [7] 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.

- (b)
In [8], 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.

- (c)
The analysis in [9] 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.

- (d)
The study [10] 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., $\gamma =\sum _{i=1}^{n}{\gamma}_{i}$, 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**[11]. 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)
A novel, exact, and simpler method for the capacity analysis of the correlated diversity Rayleigh channels is presented.

- (b)
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. - (c)
Exact closed-form expression for the ergodic capacity of correlated diversity Rayleigh channels is evaluated for any degree of channel correlation and unequal SNRs.

- (d)
Exact closed form expressions for the outage probability for certain threshold

*γ*_{0}is 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

*n*, let

**x**be the

*n*×1 vector of channel gains linking the transmitter with

*n*receive antennas. Let Ω be a diagonal matrix of order

*n*×

*n*such that the diagonal element

*ρ*

_{ i };

*i*=1,2,…,

*n*represents SNR of the

*i*th channel. The capacity of a diversity channel is given by [6]

Ergodic capacity is defined as *E* *C* where *E* is the expectation operator showing ensemble average of a random variable.

**x**is therefore colored circular complex Gaussian vector with zero mean and some covariance matrix say

**R**, that is,

^{a}as

*P*

_{out}for a threshold

*γ*

_{0}is defined as [12]

*f*

_{ γ }(

*γ*) represents the PDF of received SNR

*γ*. The received SNR

*γ*, for the maximum ratio combining scheme in Rayleigh fading channels, is given by

*γ*

_{ i }is the instantaneous SNR of the

*i*th receive diversity branch for Rayleigh channel given by

*γ*at the receiver’s output can be expressed as

## The proposed approach

*a*=1, that is,

*a*=0, that is,

Thus, our approach relies on evaluating the PDF and the moment of the random variable of the quadratic form $Y\left(a\right)=a+\left|\right|\mathbf{x}|{|}_{\mathbf{D}}^{2}$ where **D** is a diagonal matrix. Next, we present the derivation for the PDF of random variable *Y*(*a*).

### The PDF of *Y*(*a*)

*Y*defined in (8) can be rewritten in quadratic form as

**x**

_{ w }is the whitened version of

**x**, i.e., ${\mathbf{x}}_{w}\sim \mathcal{C}\mathcal{N}\phantom{\rule{1em}{0ex}}(\mathbf{0},\mathbf{I})$, where

**I**is the identity matrix. As a result, the random variable

*Y*(

*a*) can be setup as follows

*Y*(

*a*) is given as

*Y*(

*a*) becomes

*Y*becomes

**R**=

**Q**

*Λ*

**Q**

^{∗}where

**Q**is a matrix of eigenvectors of matrix

**R**and

**Λ**is a diagonal matrix with

*λ*

_{ i }=

**Λ**(

*i*

*i*) corresponding to the

*i*th eigenvalue of

**R**. Thus, by applying this eigenvalue decomposition and using a change of variable ${\stackrel{~}{\mathbf{x}}}_{w}={\mathbf{Q}}^{\ast}{\mathbf{x}}_{w}$, the inner integral in (19) can be set up as

*y*is obtained by differentiating Equation (20)

**R**, fraction in Equation (21) can be decomposed as

*ν*)>0 and Re(

*μ*)>0. Thus, PDF of

*Y*(

*a*) is found to be

### Evaluation of ergodic capacity

*Y*(

*a*) with

*a*=1 from (26) into (9). Thus,

*E*

_{ i }(.) is the

*Exponential Integral*function defined as [13]:

### Evaluation of outage probability

*Y*(

*a*) with

*a*=0 from (26) into (11). Thus, it is found that

## Experimental results

*n*, the channel correlation matrix

**R**with correlation coefficient

*γ*

_{ c }is given by

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.

*n*. The results are shown in Figure 1. Two extreme cases of

*γ*

_{ c }=0.2 and

*γ*

_{ c }=0.8 are considered. For all the experiments we design the matrix Ω such that

*ρ*

_{ i }=

*i*dB;

*i*=1,2,…,

*n*. For a comprehensive comparison, performance curves are obtained for Equation (1) using Monte Carlo method, each simulated value

*E*[

*C*] is the result of averaging over 4,000 independent computer trials.

*γ*

_{ c }=0.2 and

*γ*

_{ c }=0.8, the curves for analytical and simulation values almost overlap each other. A detailed comparison for the case study of

*γ*

_{ c }=0.2 is dilated in Table 1. For all values of

*n*, the relative error between the simulated and analytical values is quite low, as such the minimum error of 0.0007 is reported for the system with 19 receive antennas.

**Comparison of simulation and analytical results for the case study**
γ
_{
c
}
** = 0.2**

Diversity order (n) | Simulation | Analytical | Relative error |
---|---|---|---|

2 | 1.7523 | 1.7579 | 0.0056 |

3 | 2.4006 | 2.3768 | 0.0238 |

4 | 2.9253 | 2.9104 | 0.0150 |

5 | 3.3848 | 3.3883 | 0.0035 |

6 | 3.8236 | 3.8283 | 0.0046 |

7 | 4.2489 | 4.2415 | 0.0074 |

8 | 4.6411 | 4.6354 | 0.0057 |

9 | 5.0210 | 5.0151 | 0.0059 |

10 | 5.3878 | 5.3840 | 0.0038 |

11 | 5.7345 | 5.7448 | 0.0103 |

12 | 6.0926 | 6.0994 | 0.0067 |

13 | 6.4468 | 6.4491 | 0.0023 |

14 | 6.8037 | 6.7951 | 0.0086 |

15 | 7.1281 | 7.1382 | 0.0101 |

16 | 7.4709 | 7.4790 | 0.0080 |

17 | 7.8066 | 7.8179 | 0.0113 |

18 | 8.1453 | 8.1555 | 0.0102 |

19 | 8.4927 | 8.4920 | 0.0007 |

20 | 8.8326 | 8.8275 | 0.0051 |

*γ*

_{ c }, for different diversity orders. Results are shown in Figure 2, various diversity orders of 2, 4, 8, and 16 were considered. The performance curves obtained across the whole range

*γ*

_{ c }=0.1−0.9 clearly show a degradation with the increase of the correlation coefficient. The analytical values are quite similar to their simulated counterparts. Table 2 shows results for the case with

*n*=16. As such the minimum relative error of 0.0012 is reported for the case

*γ*

_{ c }=0.6.

**Capacity comparison with respect to variation in the correlation coefficient with**
n
** = 16**

γ
| Simulation | Analytical | Relative error |
---|---|---|---|

0.1 | 2.8994 | 2.9153 | 0.0159 |

0.2 | 2.9061 | 2.9104 | 0.0043 |

0.3 | 2.9095 | 2.9019 | 0.0077 |

0.4 | 2.8791 | 2.8891 | 0.0099 |

0.5 | 2.8798 | 2.8709 | 0.0089 |

0.6 | 2.847 | 2.8458 | 0.0012 |

0.7 | 2.8006 | 2.8107 | 0.0101 |

0.8 | 2.7752 | 2.7610 | 0.0142 |

0.9 | 2.689 | 2.6867 | 0.0023 |

*γ*

_{ c }=0.8 and results are illustrated in Figure 3. Experiments were conducted with various diversity orders of 4, 8, and 16, as expected the channel capacity increases with the increase in the SNR. Again, the analytical results are found to be quite similar to the simulation values, results for the case

*n*=8 are dilated in Table 3. As such no error is reported for the case of 9-dB SNR where analytical result is in exact agreement with the simulated value.

**Capacity comparison for simulation and analytical experiments with**
n
**=8**

SNR (dB) | Simulation | Analytical | Relative error |
---|---|---|---|

1 | 3.2437 | 3.2523 | 0.0086 |

2 | 3.5371 | 3.5479 | 0.0108 |

3 | 3.9269 | 3.8502 | 0.0768 |

4 | 4.1549 | 4.1579 | 0.0030 |

5 | 4.4301 | 4.4703 | 0.0402 |

6 | 4.7859 | 4.7865 | 0.0006 |

7 | 5.1043 | 5.1058 | 0.0015 |

8 | 5.3680 | 5.4276 | 0.0597 |

9 | 5.7515 | 5.7515 | 0.0000 |

10 | 6.1104 | 6.0770 | 0.0334 |

*ρ*

_{ i }) is investigated. The outage probability is calculated with

*γ*

_{ c }=0.5 for different values of SNRs on a wide rang of threshold

*γ*

_{0}. Four different values of SNR, namely 5, 10, 15, and 20 dB, are used in the experiment and the results are reported in Figure 4. It can be depicted from the figure that the outage probability at a certain threshold value decreases by increasing the SNR.

*γ*

_{ c }on the outage probability with equal branch SNRs of 20 dB is investigated. For a comprehensive analysis, three different values of the correlation coefficient, i.e., 0.01, 0.5, and 0.99, are chosen constituting three different experiments. The outage probability is calculated for a wide rang of threshold

*γ*

_{0}. The results depicted in Figure 5 show that the outage probability is maximum for

*γ*

_{ c }=0.01 before the value of

*γ*

_{0}approaches 580 (approximately). Beyond this point the outage probability is maximum for

*γ*

_{ c }=0.99.

## Conclusion

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.

## Endnote

^{a}For any matrix **D**, the quadratic form $\left|\right|\mathbf{v}|{|}_{\mathbf{D}}^{2}$ is defined as $\left|\right|\mathbf{v}|{|}_{\mathbf{D}}^{2}\triangleq {\mathbf{v}}^{\ast}\mathbf{D}\mathbf{v}$.

## Declarations

## Authors’ Affiliations

## References

- Sagias NC, Tombras GS, Karagiannidis GK: New results for the Shannon channel capacity in generalized fading channels.
*IEEE Commun. Lett*2005, 9(2):97-99. 10.1109/LCOMM.2005.02031View ArticleGoogle Scholar - Costa DB, Yacoub MD: Average channel capacity for generalized fading scenarios.
*IEEE Commun. Lett*2007, 11(12):949-951.View ArticleGoogle Scholar - Lee WCY: Estimate of channel capacity in Rayleigh fading environment.
*IEEE Trans. Veh. Technol*1990, 39(3):187-189. 10.1109/25.130999View ArticleGoogle Scholar - Günther CG: Comments on “Estimate of channel capacity in Rayleigh fading environment”.
*IEEE Trans. Veh. Technol*1996, 45(2):401-403.View ArticleGoogle Scholar - Zhang QT, Liu DP: A simple capacity formula for correlated diversity Rician fading channels.
*IEEE Commun. Lett*2002, 6(11):481-483.View ArticleGoogle Scholar - Telatar E: Capacity of multi-antenna Gaussian channels. AT&T Labs, BL0 112 170-950 615-07TM, 1995Google Scholar
- Khatalin S, Fonseka JP: Capacity of correlated Nakagami-m fading channels with diversity combining techniques.
*IEEE Trans. Veh. Technol*2006, 55: 142-150. 10.1109/TVT.2005.861206View ArticleGoogle Scholar - Rapajic PB, Popescu D: Information capacity of a random signature multiple-input multiple-output channel.
*IEEE Trans. Commun*2000, 48(8):1245-1248. 10.1109/26.864159MATHView ArticleGoogle Scholar - Alouini M, Goldsmith A: Capacity of Rayleigh fading channels under different adaptive transmission and diversity combined techniques.
*IEEE Trans. Veh. Technol*1999, 48(4):1165-1181. 10.1109/25.775366View ArticleGoogle Scholar - Malik RK, Win MZ, Shao JW, Alouini M, Goldsmith A: Channel capacity of adaptive transmission with maximal ratio combining in correlated Rayleigh fading channel.
*IEEE Trans. Wirel. Commun*2004, 3(4):1124-1133. 10.1109/TWC.2004.830823View ArticleGoogle Scholar - Al-Naffouri TY, Moinuddin M: Exact performance analysis of the ε-NLMS algorithm for colored circular Gaussian inputs.
*IEEE Trans. Signal Process*2010, 58(10):5080-5090.MathSciNetView ArticleGoogle Scholar - Proakis J:
*Digital Communications*. McGraw-Hill, New York; 2000.Google Scholar - Gradshteyn IS, Ryzhik IM:
*Table of Integral, Series and Products, Corrected and Enlarged Edition*. Academic Press, Inc., New York; 1980.Google Scholar

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