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Performance Analysis of Two-Hop OSTBC Transmission over Rayleigh Fading Channels

Open Access
Research Article

Abstract

A two-hop amplify and forward (AF) relay system is considered where source and destination are each equipped with multiple antennas while the relay has a single antenna. Orthogonal space-time block coding (OSTBC) is employed at the source. New exact expressions for outage probability in Rayleigh fading as well as symbol error rate (SER) expressions for a variety of modulation schemes are derived. The diversity order of the system is evaluated. Monte Carlo simulations demonstrate the accuracy of the analyses presented. Results that can be extended to relay systems with a direct source-destination link are also highlighted. To put the results in context, the two-hop system performance is then compared to that of a MIMO point-to-point system. Finally, the new analysis is applied to evaluate two-hop system performance as a function of relay location.

Keywords

Outage Probability Diversity Order MIMO System Rayleigh Fading Channel Single Antenna 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Through exploiting spatial diversity, it is well known that MIMO technology can improve the reliability of wireless communication links [1]. Orthogonal space-time block coding (OSTBC) is a key component of MIMO systems that has attracted tremendous attention. First, OSTBC does not require complicated feedback links to provide channel state information at the transmitter (CSIT). Second, OSTBC methods enable maximum likelihood detection to be performed with low computational complexity [2]. As a result of its practicality, OSTBC has been incorporated into emerging MIMO standards [3].

While MIMO systems offer significant physical layer performance enhancements, a significant problem in initial wireless network deployments is obtaining adequate coverage. The concept of relaying signals through intermediate nodes has been shown to be effective at extending the coverage of networks in a power-efficient manner. In addition, very simple relaying systems have been shown to increase diversity through node collaboration. As a result, the provision for relaying has recently been adopted into recent standards [4]. This paper investigates the effect of simple relaying on MIMO system performance.

Previously, end-to-end performance of two-hop relay systems was studied in [5, 6, 7], including outage probability and average bit error rate (BER) in a variety of fading environments. However, in [5, 6, 7] all assume a single antenna at both source and destination. Recently, a two-hop amplify and forward (AF) relay system in which the source and destination are both equipped with multiple antennas while the relay has a single antenna appears in [8, 9]. In [8], an OSTBC strategy is employed at the source, and end-to-end average bit error rate (BER) was investigated. However, the method in [8] is only suitable for systems with the same numbers of antennas at the source and destination. Moreover, an exact expression for outage probability was not given, and the diversity order of the system was not evaluated analytically. In [10], system performance including outage probability and average SER is determined for the special case of multiple antennas at the source and a single antenna at the destination. Although the method used in [10] has been often used in the literature, it cannot be easily generalized to the case of multiple antennas at the destination.

In this paper, the same system model is assumed as in [8, 9]. First, exact expressions for system outage probability with both exact and ideal relay gain are derived for arbitrary antenna configurations at the source and destination. Exact average SER expressions for different modulation schemes are then derived by calculating the probability density function (PDF) and moment generation method (MGF). Generalizations of results to systems that include a direct link are also briefly indicated where applicable. The diversity order of the system is also evaluated. Monte Carlo simulations confirm the analytical results, compare performance between the proposed relaying system and a MIMO point-to-point system as well as evaluate the two-hop system performance as a function of relay location.

2. System Model

A two-hop relay system is considered where there are Open image in new window antennas at the source, Open image in new window antennas at the destination, and a single antenna at the relay, as shown in Figure 1. To make the relay as simple as possible, an AF relaying protocol is employed. It is also assumed that a direct communication link between source and destination is not available, as is reasonable in the case where the communication link between source and destination is in a deep fading state and/or the separation distance between them is large. In addition, half-duplex transmission is assumed; that is, the relay cannot transmit and receive simultaneously in the same time slot or frequency band.
Figure 1

Two-hop relay system model.

OSTBC transmission containing Open image in new window symbols Open image in new window and block length of Open image in new window is utilized at the source to achieve space diversity. During the first time slot, the Open image in new window received vector signal at the relay can be written as
where Open image in new window denotes a Open image in new window OSTBC transmission matrix, Open image in new window denotes the Open image in new window Rayleigh fading complex channel gain vector from the source to the relay, and Open image in new window is the Open image in new window independent and identically distributed (i.i.d) complex Gaussian noise vector at the relay. During the second time slot, the Open image in new window received signal at the destination can be written as
where Open image in new window denotes the Open image in new window Rayleigh fading channel gain vector from the relay to the destination, Open image in new window is the Open image in new window i.i.d noise matrix at the destination where Open image in new window denotes the noise at the Open image in new window th receive antenna during the Open image in new window th symbol period, and Open image in new window denotes the Open image in new window signal vector sent by the relay where Open image in new window is a relay gain. As categorized in the literature, relay gains may be fixed or variable. In this paper, variable relay gain is considered. Relay gains can be further classified as exact or ideal. We denote Open image in new window as the exact relay gain, where Open image in new window and Open image in new window are average power constraints at the source and relay, respectively. If we ignore the noise at the relay, Open image in new window which is denoted as the ideal relay gain [5] and is amenable to mathematical manipulation. As in most of the literature, the ideal relay gain is used to compute exact average SERs of the proposed system in this paper. Later, it will be observed from simulations that the ideal relay gain provides a tight lower bound on outage probability and average SER in the case of medium-to-high SNR. Substituting Open image in new window into (2) leads to the received signal at the destination given by

Using maximum likelihood (ML) detection of OSTBCs for the case of spatially colored noise given in [11], the received SNR is obtained as follows.

Theorem 1.

Using the exact relay gain, the received SNR of two-hop AF OSTBC transmission is given by
When the ideal relay gain is utilized, the received SNR is given by

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window denotes the OSTBC code rate.

Proof.

It can be observed from (3) that the noise at the destination is temporally white but spatially colored; that is, the columns of noise matrix Open image in new window are independent and Gaussian with covariance matrix Open image in new window . Also, note that system (3) under consideration is equivalent to a conventional MIMO system with an effective channel gain matrix Open image in new window . We first whiten the colored noise and then employ the ML method given in [11, 12] to decode each symbol in the OSTBC. The noise-whitening process is given by
where Open image in new window , Open image in new window and Open image in new window . After ML detection, the MIMO system in (6) can be transformed into the following Open image in new window parallel and independent single-input and single-output (SISO) systems:
where Open image in new window and Open image in new window denotes the Frobenius norm of Open image in new window . Following similar arguments to [11], the received SNR for each symbol can be written as
where Open image in new window denotes trace of a matrix, Open image in new window is inverse of matrix Open image in new window , and Open image in new window denotes average power of a symbol where Open image in new window is assumed for OSTBCs. By substituting for Open image in new window and Open image in new window defined above, (8) can be written as
Using the Matrix Inversion Lemma Open image in new window [13], (9) can be written as
Using Open image in new window , (10) can be simplified to

Using the fact that Open image in new window and Open image in new window in OSTBCs, substituting the exact relay gain into (11) yields (4), and substituting the ideal relay gain into (11) yields (5).

Remark 2.

Using maximal ratio combining (MRC) at the destination, Theorem 1 can be generalized to the case of relay systems with a direct link by including a term corresponding to the SNR of the direct link that increases the SNR by Open image in new window , where Open image in new window is the Open image in new window Rayleigh fading complex channel gain matrix from source to destination and Open image in new window denotes the received noise variance during the source-to-destination time slot. It is assumed that relay-to-destination communication occurs in a separate second time slot, that is, half-duplex communication.

3. Outage Probability

In this section, analytical expressions for outage probability are derived.

Theorem 3.

For the exact relay gain, the outage probability of a two-hop AF relay system when an OSTBC strategy is employed at the source is given by
where Open image in new window in which Open image in new window denotes outage capacity and Open image in new window denotes the modified Bessel function of second kind and order Open image in new window . When the ideal relay gain is utilized, the outage probability is given by

Proof.

As Open image in new window and Open image in new window are complex Gaussian distributed, where Open image in new window and Open image in new window are elements of Open image in new window and Open image in new window , respectively, it is readily found that Open image in new window and Open image in new window are exponentially distributed. Setting rate parameters of Open image in new window and Open image in new window to be equal to Open image in new window and Open image in new window , respectively, the cumulative distribution function (CDF) and PDF of Open image in new window and Open image in new window are, respectively, [14]
The CDF of Open image in new window and Open image in new window , in terms of the constant parameter Open image in new window , is given by
where Open image in new window denotes the CDF of Open image in new window with the exact relay gain and Open image in new window denotes the CDF of Open image in new window with the ideal relay gain. Setting Open image in new window and substituting (14) and (17) into (18) yield
Moving the constant terms outside the integral and applying the binomial expansion yield

Using [15, Equation (3.324)], substituting Open image in new window into (20), and through straightforward mathematical manipulations (12) with Open image in new window and (13) with Open image in new window are yielded.

4. Exact Average SER Expressions for Different Modulation Schemes

In this section, exact SER expressions for different modulation schemes are derived assuming an ideal relay gain.

(i)First, we consider modulation schemes that have conditional SER Open image in new window [16], for example, BPSK, BFSK, and MPAM.

The average SER is obtained by integrating the conditional SER over the PDF of Open image in new window [14]
Using integration by parts yields
where Open image in new window denotes the derivative of Open image in new window and Open image in new window denotes the CDF of Open image in new window for the ideal relay gain. Substituting (13) with Open image in new window into (22) yields
Using integrals of combinations of Bessel functions, exponentials, and powers, for example, [15, Equation (3.381.4)] and, [15, Equation (6.621.3)], it can be shown that the following expression for average SER can be obtained from (23):

where Open image in new window is the Gauss hypergeometric function, Open image in new window for binary phase-shift keying (BPSK), Open image in new window for binary frequency-shift keying (BFSK), and Open image in new window for M-ary pulse amplitude modulation (MPAM).

(ii)Next, we consider the modulation schemes of MPSK and MQAM.

When MPSK or MQAM is employed, we derive exact SER expressions using the well-known MGF-SER relationships given in [17]. For MPSK and MQAM, the MGF-SER relationship can be written, respectively, as

The moment generation function (MGF) of Open image in new window is given by the following theorem.

Theorem 4.

Before proving Theorem 4, the probability density function (PDF) of Open image in new window is first presented in the following lemma.

Lemma 5.

Proof of Lemma 5.

Differentiating (13), where Open image in new window with respect to Open image in new window and using the expression for the modified Bessel function derivative in [15, Equation (8.486.12)]yield (28).

Proof of Theorem 4.

From Lemma 5, taking the Laplace transform of (28) and using [15, Equation (6.621.3)] yields (27).

Finally, from Theorem 4, substituting MGF (27) for the ideal relay gain into (25) and (26), respectively, we obtain exact average SER expressions for MPSK and MQAM.

Remark 6.

The SER expressions for MPSK and MQAM can straightforwardly be generalized to a system with a direct link. The MGF would be multiplied by the factor Open image in new window , where Open image in new window is the rate parameter of the exponentially distributed source-to-destination gain Open image in new window .

5. Diversity Order Analysis

The diversity order of the system can be determined directly by the definition

Theorem 7.

The diversity order of a two-hop AF relay system is Open image in new window when an OSTBC strategy is employed at the source.

Proof.

Since the diversity order of the system describes performance at asymptotically high SNR, the ideal relay gain assumption is used. We begin by determining the lower bound on diversity order of the system. Setting Open image in new window , then when Open image in new window , Open image in new window , it can be claimed that Open image in new window by ([18], Lemma Open image in new window ). Substituting (15) and (17) into the previous expression yields
According to the appendix in [19], the following expression can be obtained when Open image in new window :
Combining (29), (30), and (31), we obtain
An upper bound on diversity order can be obtained as follows:
Combining (29), (31), and (33), we obtain

Combining (32) and (34), we conclude that the diversity order is Open image in new window .

6. Numerical Results and Conclusions

In the following Monte Carlo simulations, without loss of generality, we assume equal transmit SNR at the source and destination: Open image in new window and outage capacity Open image in new window . In Figures 24, the variances of Rayleigh fading channels Open image in new window from the source antennas to the relay and the variance of Rayleigh fading channels Open image in new window from the relay to the destination antennas are set to 0.8 and 0.9, respectively. Also, OSTBCs with the highest code rate and minimum decoding delay are employed as given in [2]. Figure 2, showing outage probability for different antenna configurations, reveals that Monte Carlo simulations agree closely with the analysis predicted by Theorem 3. Also, the ideal relay gain provides a tight lower bound on outage probability even in the medium SNR regime. It is clear that the diversity order, observable by the slope of the outage probability curve, cannot be improved through simply adding transmit antennas only or receive antennas only for the case where an equal number of antennas is installed at the source and destination. This is as expected from Theorem 7, where the diversity order of the system is shown to be equal to Open image in new window .
Figure 2

Outage probability of the system with different numbers of antennas.

Figure 3

Average SER of the system with different modulation schemes.

Figure 4

Average SER of the BPSK system with different antenna configurations.

Figure 3 compares Monte Carlo simulations and analytical results for modulations with conditional SER Open image in new window including BPSK, BFSK, and 4 PAM as well as with QPSK and 16 QAM. Clearly, the simulations very closely match the analyses. Again, the ideal relay gain provides a tight lower bound on average SER even in the case of medium SNR.

Figure 4 shows average SER with BPSK modulation for different antenna configurations. Here, the diversity order can be observed as the slope of the average SER curve. In accordance with Theorem 7, it is observed that the diversity order cannot be improved through simply adding transmit or receive antennas for the case of equal numbers of source and destination antennas.

To assess the impact of relaying, Figure 5 compares the proposed relaying system with an MIMO point-to-point system. The relay is located between the source and destination. The normalized distance between the source and destination is assumed to be unity, so Open image in new window . The path loss exponent is set to 4, and it is assumed that shadowing effects are the same for the source-relay and relay-destination links, with a standard deviation Open image in new window , a value typically assumed in urban cellular environments. For a fair comparison, the transmit SNR as well as the OSTBC transmit strategy is assumed to be identical for both systems.
Figure 5

Average SER comparison between the relaying system and MIMO point-to-point system.

Specifically, the relay is assumed to be placed halfway between the source and destination in Figure 5. Figure 5 shows how system performance trades off between the relaying and MIMO point-to-point systems. As expected, the results clearly show that the diversity order of the MIMO point-to-point system, which is known to be Open image in new window , is larger than that of the relaying system for the same antenna configuration at both source and destination. In other words, as transmit SNR increases, the MIMO point-to-point system has an advantage over the relaying system. However, it can also be observed that the relaying system outperforms the MIMO point-to-point system in lower SNR regimes: for the case of Open image in new window , the relay system outperforms point-to-point MIMO systems at transmit SNRs below 6 dB. Adding a transmit antenna to the MIMO system lowers this threshold to -1 dB, while at the same time, the code rate for OSTBCs becomes 3/4. On the other hand, adding a receive antenna to the MIMO system lowers this threshold to -4 dB. Of course, the MIMO point-to-point system would be able to achieve similar system performance gain by adding transmit or receive antennas, but this would increase complexity.

Figure 6 shows average SER of the system for different source-relay distances with a fixed transmit SNR. It can be observed from the figure that performance is best when the relay is placed halfway between the source and destination. Although not shown here, it is found that a similar trend holds across a wide range of transmit SNR values.
Figure 6

Average system SER versus source-relay distance. Shown are transmit SNRs of 10 and 6 dB.

7. Conclusions

The performance of a MIMO system using OSTBC transmission that encounters a simple relay in a two-hop AF configuration has been analyzed, including outage probability, SER and diversity order. These results extend those found in [8, 10]. Monte Carlo simulations are found to agree closely with the analyses. In fact, the numerical results indicate that tight lower bounds are obtainable using the ideal relay gain approximation, even at SNRs as low as 5–8 dB. In addition, Monte Carlo simulations also compare system performance between the proposed relaying system and a MIMO point-point system and assess performance as a function of location of the relay between the source and destination.

Notes

Acknowlegment

This paper was supported in part by the Natural Sciences and Engineering Research Council of Canada Discovery (Grant no. 41731) and the Natural Science Foundation of Guangdong Province, China (Grant no. 10451009001004407).

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Copyright information

© Guangping Li et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Faculty of Information EngineeringGuangdong University of TechnologyGuangzhouChina
  2. 2.Department of Electrical and Computer EngineeringQueen's UniversityKingstonCanada
  3. 3.Department of Electronic and Communications EngineeringSun Yat-sen UniversityGuangzhouChina

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