# Distributed Space-Time Block Code over Mixed Rayleigh and Rician Frequency-Selective Fading Channels

- Quoc-Tuan Vien
^{1, 2}, - Le-Nam Tran
^{1}and - Een-Kee Hong
^{1}Email author

**2010**:385872

**DOI: **10.1155/2010/385872

© Quoc-Tuan Vien et al. 2010

**Received: **28 October 2009

**Accepted: **9 June 2010

**Published: **1 July 2010

## Abstract

This paper proposes a new distributed space-time block code (DSTBC) over frequency-selective fading channels for two-hop amplify and forward relay networks, consisting of a source node ( ), two relay nodes ( and ), and a destination node ( ). The proposed DSTBC is designed to achieve maximal spatial diversity gain and decoupling detection of data blocks with a low-complexity receiver. To achieve these two goals, uses zero-sequence padding, and relay nodes precode the received signals with a proper precoding matrix. The pairwise error probability (PEP) analysis is provided to investigate the achievable diversity gain of the proposed DSTBC for a general channel model in which one hop is modeled by Rayleigh fading and the other by Rician fading. This mixed Rayleigh-Rician channel model allows us to analyze two typical scenarios where are in the neighborhood of either or .

## 1. Introduction

The reliability of wireless communications over fading channels can be greatly improved by the use of diversity schemes. For multiple-input multiple-output (MIMO) systems, transmitted diversity can be realized in the form of space-time block codes (STBCs) [1, 2]. With low-complexity maximum-likelihood (ML) decodability and high achievable diversity gain, STBCs are widely used for wireless communications. Generally, the conventional STBCs were designed for the colocated antennas, and thus are easily deployed at the base station to improve the performance of the downlink transmission. Nevertheless, the realization of STBCs is impractical in the uplink transmission due to the constraints on size and hardware complexity in mobile handsets. Fortunately, mobile users can cooperate to form a virtual multiple-antenna system, which is now known as cooperative diversity [3, 4].

The distributed space-time block codes can be viewed as the distributed implementation of conventional STBCs for cooperative communications. Originally, the DSTBCs were proposed for flat-fading channels [5–7]. The problem of DSTBC in frequency-selective fading channels was investigated in [8] with decode-and-forward (DF) relaying, and in [9] with amplify-and-forward (AF) relaying. However, these DSTBCs were devised for relay networks where there exists one active relay node and a direct communication link between the source and the final destination.

In this paper, we design a new DSTBC for two-hop relay networks [6, 7] over frequency-selective fading channels with AF protocol, where there are two active relay nodes. The proposed DSTBC operates as follows: in the first time slot, the source ( ) sends two blocks of information data to two relays ( and ). What is remarkable in our proposed DSTBC is that one of the two relays precodes its received signals that will be sent to the destination ( ) in the next time slot. The precoding matrix is designed such that each relay conveys a distinct column of the block Alamouti scheme (see, e.g., [1, 10, 11]). Our main contributions in this paper are summarized in brief as follows.

- (i)
With our proposed DSTBC, the data rate of 1/2 is achieved, which is proved to be the maximum data rate for two-hop relaying networks. As we can see later, the extension of [9] to two relays results in a rate of 1/3.

- (ii)
We propose the precoding matrix at the relays such that the decoupling detection of two data blocks in both time and frequency domains is possible at . The PEP analysis is carried out with ML detection in time domain, and numerical results are obtained with minimum mean square error (MMSE) receiver in frequency domain.

- (iii)
We study the achievable diversity gain of the proposed DSTBC for the general scenario where the relays are located near the source or the destination, that is, one of the two hops ( to or to ) is line-of-sight (LOS) transmission, while the other is nonline-of-sight (NLOS) transmission. Accordingly, the considered channel model is a mix of Rayleigh and Rician fading.

The theoretical results prove that our proposed scheme achieves the spatial diversity order of , where and are the channel memory lengths for the links from to and from to , respectively. The analysis also shows that the -factor of Rician fading in the LOS component provides a coding gain to the PEP performance. It means that as the -factor increases, a better performance is observed.

The rest of this paper is organized as follows In Section 2, we describe the system model of the proposed DSTBC and the proof of decoupling capability in time domain and frequency domain. Performance analysis is presented in Section 3. We present the numerical results in Section 4, and Section 5 concludes this paper.

Notation 1.

Bold lower and upper case letters represent vectors and matrices, respectively; , , and denote transpose, complex conjugate, and Hermitian transpose operations, respectively; and denote an identity matrix and an all-zero matrix of size ; denotes the expectation; denotes the Euclidean norm of a vector; stands for a Fast Fourier Transform (FFT) matrix of size ; and represent the links from the source ( ) to the th relay ( ) and from to the destination ( ), respectively.

## 2. System Model and the Proposed DSTBC

In the first time slot, the source serially transmits two data blocks to two relays. In the next time slot, one relay only amplifies and forwards its received signals, while the other precodes its received data blocks by a precoding matrix before transmitting to the destination as illustrated in Figure 1. The idea behind our design is that precoding in is designed to send the second column of the block Alamouti's scheme (see, e.g., [1, 11]) to . This enables the decoupling detection of two data blocks at the destination and achieves a rate of 1/2. To achieve the same goal in the considered scenario, the source with repetition code in [9], which is devised for one-relay system, must send two columns during two time slots. Thus, the rate of this scheme is reduced to 1/3. Recall that the maximum achievable data rate of an -relay repetition-coding network is [12]. Clearly, our design can achieve a higher data rate transmission.

where is the average energy of link; is the circulant channel matrix of link; is the white Gaussian noise vector at the th relay with each entry having zeromean and variance of per dimension. For any circulant matrix , its entry is written as .

where is the average energy of link; is the normalized received signal; is the circulant matrix, denoting the channel ; is white Gaussian noise vector at the destination with each entry having zeromean and variance of .

where and for .

where is the Gaussian noise vector resulting from the space-time decoupling.

Since the general ML detection or standard equalization techniques based on (13) require complex receiver, we introduce the decoupling in the frequency domain that allows for low-complexity equalizers [13].

In (17), we omit the dependency of subscript because the detection of two data blocks is based on the same model. Typical frequency domain equalizers in [13] can be applied to the outputs of decoupling process.

## 3. Performance Analysis

In this section, we derive the PEP expression of the proposed DSTBC model over frequency-selective fading channels based on the joint channel model (11), where the links and experience Rician fading and Rayleigh fading, respectively. For the case when and are Rayleigh fading and Rician fading, the PEP expression can be similarly obtained with some interchanged parameters.

In (22), (23), and for the rest of the paper, the subscript stands for or for sake of generality because the links and can be treated in the same way. To derive the PEP, we differentiate three cases based on the relation of and because of the different characteristics of fading and .

Case 1 ( ).

where is the Nakagami- or Rician fading parameter, and represents the Gamma function defined by for any positive integer .

Case 2 ( ).

Case 3 ( ).

where the incomplete Gamma function is defined by .

Generalizing the three above cases with (31), (40) and (46), we can conclude that the diversity gain of our proposed DSTBC is by extracting the exponential terms of . Although the Rician fading parameter does not produce any diversity gain, it acts as a coding gain, and thus can improve the PEP.

We observe that the diversity order of this case is + . Additionally, PEP reduces to zero when the Rician parameter goes to infinity. That explains the positive effect of LOS component on the quality of received signals.

## 4. Numerical Results

As aforementioned, the PEP analysis is carried out based on (11). Moreover, the derivation of the PEP is accomplished under the assumption of an ML detection scheme at the receiver. However, such an ML scheme for the system model in our paper requires high complexity. Instead, we use MMSE frequency-domain linear equalizer to verify the analytical results. Since MMSE receiver is a suboptimal solution to the data detection problem, there is always a gap between the performance of ML and MMSE schemes. Note that the slopes of the performance curve of ML and MMSE receivers are similar at high SNR [16]. Consequently, the diversity gain can also be verified based on the performance of MMSE receiver.

In this section, we evaluate the BER performance of the proposed DSTBC via Monte Carlo simulation to justify the analysis of the achievable diversity gain. By observing the slope of these BER curves, we can confirm the validity of analysis since BER is proportional to PEP [16]. Each data block consists of 64 symbols including the zero sequence and information-carrying data which is modulated by QPSK with Gray mapping. We assume that the receiver has perfect channel state information.

Case 1.

(black curve and square marker); the diversity order is calculated by .

Case 2.

(red curve and round marker); the diversity order is .

Case 3.

(green curve and upper-triangular marker); the diversity order is .

The simulation results indicate that the slopes of the curves for Case 2 and Case 3 are steeper than that for Case 1 since Case 1 achieves smaller diversity gain. We can also see that the curves for Case 2 and Case 3 have the same slope, which can also be shown using the diversity order expressions we derived. These facts confirm our conclusion of the achievable diversity gain in the analysis. Another observation is for the same value of and (e.g., red curve and green curve), the better performance is achieved as the number of paths from the relays to destination increases. However the slopes of these BER curves are identical at high SNR since they have the same diversity gain of .

## 5. Conclusion

A DSTBC scheme for two-hop cooperative systems over frequency-selective fading channels has been proposed. The proposed DSTBC can achieve half-data rate transmission, spatial diversity order of . Furthermore, the data detection of the proposed DSTBC can be decoupled in frequency domain for a low-complexity receiver. The main idea in our design is that each relay conveys a distinct column of the conventional STBC developed for colocated antennas. We prove that the data rate of the proposed DSTBC is higher than that of repetition code which has the maximum data rate of 1/3 for a two-relay system. The design of DSTBC in our paper is inspired by the idea of linear dispersion codes where each column of the code matrix is a linear summation of transmit data and their conjugations [17]. In this paper, we analyze the PEP with a mixed Rayleigh and Rician frequency-selective fading channel model. The analysis shows that the LOS component effectively improves the error rate performance at the destination. Simulation results over different channel parameters and different -factors were provided to verify the theoretical analysis.

## Declarations

### Acknowledgments

This paper was partly supported by the IT R&D program of MKE/KEIT (KI001814, Game Theoretic Approach for Cross-layer Design in Wireless Communications) and MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency) (NIPA-2010-(C1090-1011-0001)).

## Authors’ Affiliations

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