Open Access

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

EURASIP Journal on Wireless Communications and Networking20102010:385872

DOI: 10.1155/2010/385872

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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq1_HTML.gif ), two relay nodes ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq3_HTML.gif ), and a destination node ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq4_HTML.gif ). 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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq5_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq6_HTML.gif are in the neighborhood of either https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq7_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq8_HTML.gif .

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 [57]. 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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq9_HTML.gif ) sends two blocks of information data to two relays ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq11_HTML.gif ). 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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq12_HTML.gif ) 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.

  1. (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.

     
  2. (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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq13_HTML.gif . 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.

     
  3. (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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq14_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq15_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq16_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq17_HTML.gif ) 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq18_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq20_HTML.gif are the channel memory lengths for the links from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq21_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq22_HTML.gif and from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq23_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq24_HTML.gif , respectively. The analysis also shows that the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq25_HTML.gif -factor of Rician fading in the LOS component provides a coding gain to the PEP performance. It means that as the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq26_HTML.gif -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; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq27_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq28_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq29_HTML.gif denote transpose, complex conjugate, and Hermitian transpose operations, respectively; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq31_HTML.gif denote an identity matrix and an all-zero matrix of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq32_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq33_HTML.gif denotes the expectation; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq34_HTML.gif denotes the Euclidean norm of a vector; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq35_HTML.gif stands for a Fast Fourier Transform (FFT) matrix of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq36_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq38_HTML.gif represent the links from the source ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq39_HTML.gif ) to the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq40_HTML.gif th relay ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq41_HTML.gif ) and from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq42_HTML.gif to the destination ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq43_HTML.gif ), respectively.

2. System Model and the Proposed DSTBC

We consider a four-node wireless relay network shown in Figure 1, where the source terminal cannot communicate directly with the intended destination. The data transmission from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq44_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq45_HTML.gif is completed via two-hop protocol [6, 7] with the assistance of two relays https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq47_HTML.gif . The frequency-selective channel from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq48_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq49_HTML.gif is characterized by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq50_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq51_HTML.gif is the channel memory order. Two transmitted data blocks https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq52_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq53_HTML.gif of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq54_HTML.gif , shown in Figure 1, are created by padding a zero sequence of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq55_HTML.gif to two information data blocks https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq56_HTML.gif , of length https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq57_HTML.gif . To achieve the decoupling property of data detection, the length of the zero sequence must satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq58_HTML.gif [9]. This condition makes circulant channel matrices from source to relays and relays to destination.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Fig1_HTML.jpg
Figure 1

System model and data structures.

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq59_HTML.gif is designed to send the second column of the block Alamouti's scheme (see, e.g., [1, 11]) to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq60_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq61_HTML.gif -relay repetition-coding network is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq62_HTML.gif [12]. Clearly, our design can achieve a higher data rate transmission.

We now proceed to prove that our proposed DSTBC can decouple the detection of two data blocks. Throughout this paper, the superscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq63_HTML.gif denotes the relay index, while the subscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq64_HTML.gif refers to data block index. The received signal at the relay is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ1_HTML.gif
(1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq65_HTML.gif is the average energy of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq66_HTML.gif link; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq67_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq68_HTML.gif circulant channel matrix of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq69_HTML.gif link; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq70_HTML.gif is the white Gaussian noise vector at the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq71_HTML.gif th relay with each entry having zeromean and variance of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq72_HTML.gif per dimension. For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq73_HTML.gif circulant matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq74_HTML.gif , its https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq75_HTML.gif entry is written as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq76_HTML.gif .

At https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq77_HTML.gif , the received data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq78_HTML.gif is conjugated, followed by the precoding operation which is denoted by precoding matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq79_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ2_HTML.gif
(2)
where the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq80_HTML.gif is designed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ3_HTML.gif
(3)
In (3), the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq81_HTML.gif of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq82_HTML.gif and the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq83_HTML.gif of size https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq84_HTML.gif have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq85_HTML.gif element given, respectively, by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ4_HTML.gif
(4)
We choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq86_HTML.gif to ensure that, after precoding, at least last https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq87_HTML.gif samples of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq89_HTML.gif are all zeros to make the circulant channel matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq90_HTML.gif Before transmitting the signals to the destination, the relay https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq91_HTML.gif normalizes its signals by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq92_HTML.gif to have unit average energy. The received signals at the destination are written by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ5_HTML.gif
(5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq93_HTML.gif is the average energy of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq94_HTML.gif link; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq95_HTML.gif is the normalized received signal; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq96_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq97_HTML.gif circulant matrix, denoting the channel https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq98_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq99_HTML.gif is white Gaussian noise vector at the destination with each entry having zeromean and variance of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq100_HTML.gif .

Using (1), we can rewrite (5) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ6_HTML.gif
(6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq101_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq102_HTML.gif include the Gaussian noise of relays and destination. It is common to normalize the noise variance in (6) to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq103_HTML.gif , which results in
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ7_HTML.gif
(7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ8_HTML.gif
(8)
The values of normalization factors in (7) and (8) are defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ9_HTML.gif
(9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq105_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq106_HTML.gif .

By conjugating, and multiplying both sides of (8) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq107_HTML.gif , and noting that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq108_HTML.gif for any circulant matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq109_HTML.gif , we can rewrite (8) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ10_HTML.gif
(10)
For mathematical convenience, we group (7) and (10) in vector-matrix form as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ11_HTML.gif
(11)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ12_HTML.gif
(12)
Let us denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq110_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq111_HTML.gif + https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq112_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq113_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq114_HTML.gif for any circulant matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq115_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq116_HTML.gif = https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq117_HTML.gif is a block-diagonal matrix. By multiplying both sides of (11) with the unitary matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq118_HTML.gif , we can decouple the detection of two data blocks. That means two data blocks can be detected independently, rather than joint detection, without any loss of gain, which is based on the following model
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ13_HTML.gif
(13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq119_HTML.gif 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].

We notice that any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq120_HTML.gif circulant matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq121_HTML.gif can be diagonalized as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq122_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq123_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq124_HTML.gif diagonal matrix whose diagonal elements are the DFT of the first column of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq125_HTML.gif . Taking the DFT of both sides of (11) results in
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ14_HTML.gif
(14)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq127_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ15_HTML.gif
(15)
Let us denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq128_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq129_HTML.gif is a block-diagonal matrix. By multiplying both sides of (14) with the unitary matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq130_HTML.gif , we can decouple the detection of each data block in frequency domain as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ16_HTML.gif
(16)
Since the matrix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq131_HTML.gif is diagonal, (16) can be decomposed into a set of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq132_HTML.gif scalar equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ17_HTML.gif
(17)

In (17), we omit the dependency of subscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq133_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq135_HTML.gif experience Rician fading and Rayleigh fading, respectively. For the case when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq137_HTML.gif are Rayleigh fading and Rician fading, the PEP expression can be similarly obtained with some interchanged parameters.

Defining the decoded codeword vector as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq138_HTML.gif and the Euclidean distance between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq139_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq140_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq141_HTML.gif , the conditional PEP under fading channels is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ18_HTML.gif
(18)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq142_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq143_HTML.gif function. By applying Chernoff bound to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq144_HTML.gif function, this PEP is upper bounded by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ19_HTML.gif
(19)
The Euclidean distance in (19) is calculated by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ20_HTML.gif
(20)
Using the fact in [9], we can approximate (20) to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ21_HTML.gif
(21)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq145_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq146_HTML.gif . We note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq148_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq149_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq150_HTML.gif denotes the eigenvalue of codeword difference matrix, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq151_HTML.gif is zero-mean complex Gaussian vectors with unit variance. In (21), we mean that the distance can be approximated by one of four possible forms. Each component of the summations of the right hand side of (21) can be expressed by one of the two following factors
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ22_HTML.gif
(22)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ23_HTML.gif
(23)

In (22), (23), and for the rest of the paper, the subscript https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq152_HTML.gif stands for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq153_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq154_HTML.gif for sake of generality because the links https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq156_HTML.gif can be treated in the same way. To derive the PEP, we differentiate three cases based on the relation of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq158_HTML.gif because of the different characteristics of fading https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq160_HTML.gif .

Case 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq161_HTML.gif ).

We consider (22) and define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq162_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq164_HTML.gif . Applying Chernoff bound, the PEP corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq165_HTML.gif is upper bounded by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq166_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq167_HTML.gif denotes the moment-generating function. If we consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq168_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq169_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq170_HTML.gif will be corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq171_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq172_HTML.gif ). https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq173_HTML.gif can be evaluated as [14]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ24_HTML.gif
(24)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq174_HTML.gif is the probability density function. Since the fading channels https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq175_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq176_HTML.gif are frequency-selective Rician and Rayleigh fading, respectively,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ25_HTML.gif
(25)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ26_HTML.gif
(26)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq177_HTML.gif is the Nakagami- https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq178_HTML.gif or Rician fading parameter, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq179_HTML.gif represents the Gamma function defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq180_HTML.gif for any positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq181_HTML.gif .

Substituting (26) and (25) into (24), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ27_HTML.gif
(27)
We can rewrite (27) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ28_HTML.gif
(28)
Assuming high signal-to-noise ratio (SNR), that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq183_HTML.gif , (28) can be evaluated as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ29_HTML.gif
(29)
The integral in (29) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq184_HTML.gif is given by [15]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ30_HTML.gif
(30)
Substituting (30) into (29), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ31_HTML.gif
(31)

Case 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq185_HTML.gif ).

We examine (23) and similarly define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq186_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq187_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq188_HTML.gif . Applying Chernoff bound, the PEP corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq189_HTML.gif is upper bounded by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq190_HTML.gif which is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ32_HTML.gif
(32)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ33_HTML.gif
(33)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ34_HTML.gif
(34)
The modified Bessel function of the first kind, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq191_HTML.gif in (33), is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ35_HTML.gif
(35)
Substituting (33) and (34) into (32), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ36_HTML.gif
(36)
Under the assumption of high SNR ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq192_HTML.gif ), (36) can be computed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ37_HTML.gif
(37)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq193_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq194_HTML.gif . After some mathematical manipulations [15], we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ38_HTML.gif
(38)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq195_HTML.gif is hypergeometric https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq196_HTML.gif -regularized function and is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ39_HTML.gif
(39)
Substituting (38) into (37), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ40_HTML.gif
(40)

Case 3 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq197_HTML.gif ).

Considering (22), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq198_HTML.gif can be calculated as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ41_HTML.gif
(41)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq199_HTML.gif , (41) can be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ42_HTML.gif
(42)
Let us define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ43_HTML.gif
(43)
By using some mathematical expansions [15], we can express https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq200_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ44_HTML.gif
(44)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq201_HTML.gif . Calculating the integral in (44), the value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq202_HTML.gif is evaluated as [15]
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ45_HTML.gif
(45)

where the incomplete Gamma function is defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq203_HTML.gif .

Substituting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq204_HTML.gif into (42), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ46_HTML.gif
(46)

Generalizing the three above cases with (31), (40) and (46), we can conclude that the diversity gain of our proposed DSTBC is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq205_HTML.gif by extracting the exponential terms of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq206_HTML.gif . Although the Rician fading parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq207_HTML.gif does not produce any diversity gain, it acts as a coding gain, and thus can improve the PEP.

We provide here one example of PEP calculation for the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq208_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq209_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq210_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq211_HTML.gif , the normalization factor in (9) can then be approximated as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq212_HTML.gif . From the result in (31), the PEP is upper bounded by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Equ47_HTML.gif
(47)

We observe that the diversity order of this case is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq213_HTML.gif + https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq214_HTML.gif . Additionally, PEP reduces to zero when the Rician parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq215_HTML.gif 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.

Figure 2 shows the BER performances of the proposed DSTBC for various combinations of channel lengths using frequency domain linear equalizer (FD-LE), such as MMSE receiver. The fading https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq217_HTML.gif are assumed to be Rayleigh fading. We assume that the value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq218_HTML.gif is fixed at 25 dB, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq219_HTML.gif dB, and plot the BER curves as a function of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq220_HTML.gif . Let us show the comparison of three cases.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Fig2_HTML.jpg
Figure 2

BER performance of DSTBC with FD-LE for various combinations of channel memory orders.

Case 1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq221_HTML.gif (black curve and square marker); the diversity order is calculated by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq222_HTML.gif .

Case 2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq223_HTML.gif (red curve and round marker); the diversity order is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq224_HTML.gif .

Case 3.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq225_HTML.gif (green curve and upper-triangular marker); the diversity order is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq226_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq227_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq228_HTML.gif (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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq229_HTML.gif .

The performance of DSTBC over wireless fading channels where the fading https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq230_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq231_HTML.gif are characterized by Rician and Rayleigh distributions, respectively, is drawn in Figure 3 with different values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq232_HTML.gif -factor and the assumption of channel memory order of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq233_HTML.gif . We remark that the better performance is achieved as the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq234_HTML.gif -factor of Rician fading increases. However, the slopes of the performance curve are almost the same at high SNR, that is, they achieve the same diversity gain. This is because the term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq235_HTML.gif only produces the coding gain to the PEP as we can see in the analysis. This confirms our conclusion about the achievable diversity gain from the PEP analysis mentioned in the previous section.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Fig3_HTML.jpg
Figure 3

BER performance of DSTBC with different values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq236_HTML.gif -factor over Rician fading channels.

The performances of two cases https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq237_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq238_HTML.gif for various combinations of channel memory orders are shown in Figure 4 to study the effects of LOS component and channel memory orders on the BER. For each value of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq239_HTML.gif , the BER curves corresponding to different channel memory orders are plotted to show the variations of slope due to different achievable diversity gains. We observe that the BER curves are shifted down if we increase https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq240_HTML.gif . That reflects the effect of LOS component on the coding gain through all range of SNR from low to high SNR. Meanwhile, the channel memory orders only have noticeable effects on the BER curves at high SNR caused by the change of diversity gain.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_Fig4_HTML.jpg
Figure 4

BER performance of DSTBC with different values of channel memory orders and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq241_HTML.gif -factor over Rician fading channels.

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq242_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F385872/MediaObjects/13638_2009_Article_1886_IEq243_HTML.gif -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

(1)
School of Electronics and Information, Kyung Hee University
(2)
School of Engineering and Computing, Glasgow Caledonian University

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Copyright

© Quoc-Tuan Vien 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.