J. Cent. South Univ. Technol. (2009) 16: 0280-0284

DOI: 10.1007/s11771-009-0048-9

Cooperative diversity based on rotation code

Xiong Xiong(熊 雄), Ge Jian-hua(葛建华), Li Jing(李 靖), Tang Yun-shuai(唐云帅)

 (State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an 710071, China)

                                                                                                

Abstract: In order to obtain coding gain along with diversity gain, rotation code was applied to cooperative diversity employing decoded-and-forward cooperative protocol. Different from the same two symbols transmitted in conventional repetition-coded scheme, two different symbols were transmitted separately in two successive timeslots in the proposed rotation-coded cooperative diversity. In this way, constellation spread in the available two-dimensional signal space rather than on a single-dimensional line in repetition-coded scheme, which was supposed to be responsible for the additional coding gain. Under the proposed cooperative transmission model, upper bounds for the symbol-error-rate (SER) of cooperative diversity based on repetition code and rotation code were derived respectively. Both analytical and simulated results show that cooperative diversity based on rotation code can obtain an improved coding gain (by about 2 dB) than repetition-coded scheme without the expense of power or bandwidth.

Key words: cooperative diversity; repetition code; rotation code; coding gain; symbol-error-rate

                                                                                                            

1 Introduction

Diversity is a powerful technique for combating the fading in wireless communications. Spatial diversity techniques based on multiple antennas are particularly attractive as they do not incur an expenditure of transmission time or bandwidth. However, the deployment of antenna array on mobile terminal is difficult due to the size and power limitation. In order to solve this problem, a new mode of gaining transmit diversity called cooperative diversity has been proposed and widely studied [1-7]. The basic idea is that single-antenna mobile users in the networks can share their antennas to generate a virtual multiple-antenna transmitter that allows them to achieve transmit diversity. Recently, many efforts have been focused on the design of cooperative protocols. Specifically, in Ref.[8], various cooperative protocols were proposed for wireless networks, in which one user serves as a relay to help the other user to forward information. The relay may simply amplify the received signal and forward it, which is termed as amplify-and-forward (AF), or first decode the received information and then forward the encoded symbols to the destination, which is referred to as decoded-and-forward (DF). For DF transmission, HUNTER et al [9-11] integrated user cooperation with channel coding, which resulted in coded cooperation (CC).

Unlike CC adopting forward error correction (FEC) code, cooperative system with DF scheme considered in Ref.[8] is uncoded. After detecting the symbol from the source, the relay directly forwards the same symbol, which can be seen as a repetition-coded scheme. Although this scheme achieves a diversity gain, it does not exploit the degrees of freedom available in the channel effectively because it simply repeats the same symbols over two different paths. By using more sophisticated codes, a coding gain can also be obtained beyond the diversity gain. In Ref.[12], one example of these possible codes called rotation code was introduced. It is natural to think of extending the rotation code to the DF cooperative scheme, which is just the theme of this paper. In the following, cooperative transmission model was presented firstly, and then an upper bound for the symbol-error-rate (SER) of cooperative diversity based on rotation code was derived. As a reference, the SER of conventional repetition-coded scheme was also analyzed. Through the theoretical and simulation comparison, it was found that cooperative diversity based on rotation code can obtain a better coding gain (by about 2 dB) than repetition-coded scheme without the expense of power or bandwidth.

2 System model

Consider a wireless network with one source, one relay and one destination. As shown in Fig.1, data are transmitted from the source terminal (S) to the destination terminal (D) with the assistance of the relay terminal (R). Relay employs DF cooperative scheme. Each terminal is equipped with a single antenna. All three channels experience flat Rayleigh, slow fading. Due to current limitation in radio implementation, half-duplex operation is assumed and time-division is employed to ensure this half-duplex.

Fig.1 Fading relay channel

Cooperation strategy consists of two phases. In Phase 1, the source broadcasts its information symbol x₁ to both the relay and destination. The received symbols y_r and y_d1 at the relay and destination are given by

y_r=h_srx₁+w_sr                                  (1)

y_d1=h_sdx₁+w_sd                                 (2)

where  w_sr and w_sd are additive noise terms at the relay and the destination respectively, which are modeled as zero-mean, mutually independent complex Gaussian random variables with variance N₀. h_sr and h_sd are the channel coefficients from the source to the relay and the destination respectively, and they are modeled as zero-mean complex Gaussian random variables with variances σ_sr² and σ_sd², respectively.

In Phase 2, if the relay is able to detect the transmitted symbol correctly, the relay forwards symbol x₂ to the destination, otherwise the source continues to transmit symbol x₂ to the destination. We refer to these two cases as cooperative case and non-cooperative case respectively. Symbol x₂ is transmitted according to symbol x₁, and there are various kinds of relationship between two symbols. In this paper, only two cases are considered. One case is that x₂ is the same as x₁, which is just the conventional repetition-coded cooperative diversity in Ref.[8]. The other is that x₁ and x₂ constitute

a codeword  from rotation code, which is termed

as rotation-coded cooperative diversity. The received symbol y_d2 at the destination in Phase 2 can be written as

Cooperative case: y_d2=h_rdx₂+w_d2                  (3)

Non-cooperative case: y_d2=h_sdx₂+w_d2              (4)

where  w_d2 is additive noise at the destination, which is modeled as a zero-mean complex Gaussian random variable with variance N₀ and independent of w_sr and w_sd. h_rd is the channel coefficient from the relay to the destination, which is modeled as a zero-mean complex Gaussian random variable with variance σ_rd². Due to the slow fading, channel coefficient h_sd in Eqn.(4) keeps the same as that in Eqn.(2).

In the cooperative transmission model above, the channel coefficients h_sr, h_sd and h_rd are assumed to be independent of each other and known to the appropriate receivers, but not known to, or not exploited by, the transmitters. At the destination, the received symbols y_d1 and y_d2 in two phases are jointly combined to detect the source information.

3 SER performance analysis

3.1 SER performance of repetition-coded cooperative diversity

In order to make a fair comparison with rotation-coded cooperative diversity, 4-PAM modulation (-3b, -b, b, 3b) is employed to ensure the same bit rate (2 bits over 2-symbol intervals). It is noted that the transmission above is only over the (real) I channel, but extension to both I/Q channels is immediate.       Four codewords (X_A, X_B, X_C and X_D) of repetition code onsisting of x₁ and x₂, i.e. , are shown in Fig.2. As

we see, x₁=x₂.

Fig.2 Four codewords of repetition code

With knowledge of the channel coefficients h_sr, h_sd and h_rd, the destination detects the transmitted symbol x₁ by jointly combining the received symbol y_d1 from the source (Eqn.(2)) and y_d2 from the relay (cooperative case: Eqn.(3)) or from the source (non-cooperative case: Eqn.(4)) by using maximum-ratio combining (MRC). Output signal-to-noise ratio (SNR) of cooperative   case and non-cooperative case can be gotten by the MRC

detector: and  respectively, where E_s=5b² is average transmitted energy per symbol. For 4-PAM, the conditional SER of cooperative and non-cooperative cases can be written as [13]

     (5)

              (6)

where Q(?) is the complementary cumulative distribution function of an N(0, 1) random variable.

Using Chernoff bound Q(x)≤(forx＞0) and averaging Eqns.(5) and (6) over the Rayleigh fading channel coefficients h_sd and h_rd, we can get average SER upper bounds of the cooperative and non-cooperative cases as follows:

P_e-co-rep≤(γ_sdγ_rd)^-1, P_e-non-rep≤γ_sd^-1          (7)

where   and  are the app-ropriate average received signal-to-noise ratios (SNRs).

The probability of non-cooperative case P_non-rep is the probability that the relay fails to detect x₁. This is equivalent to obtain the average SER of 4-PAM over Rayleigh fading channel which has been given in Ref.[14]:

                     (8)

where   is the average received SNR at the relay.

Combining cooperative case with non-cooperative case, the average SER upper bound of repetition-coded cooperative diversity is obtained:

P_e-rep=P_e-non-repP_non-rep+P_e-co-rep(1-P_non-rep)≤

(9)

As we can see, when the channel condition between the source and the relay (this channel is called inter-user channel) is perfect, i.e. γ_sr is large enough, only the second term in Eqn.(9) survive and full diversity order of two can be achieved. However, since the codewords of repetition code are packed on a single-dimensional line (Fig.2), it does not exploit the degrees of freedom available effectively, thereby repetition-coded scheme is inefficient in obtaining coding gain.

3.2 SER performance of rotation-coded cooperative diversity

Rotation code specifically designed to exploit time diversity was introduced in Ref.[12] and the error probability was derived. Similar to the deduction in Ref.[12], SER upper bound of the rotation-coded cooperative diversity can be derived. In the rotation-coded

cooperative diversity,  transmitted in the two

phases is a codeword from rotation code with 4 codewords (shown in Fig.3):

, ,

,

where  is a rotation matrix (θ∈(0, 2π)).

Fig.3 Four codewords of rotation code

With the channel coefficients h_sr, h_sd and h_rd, the

destination detects the transmitted codeword

through the received symbols y_d1 (Eqn.(2)) and y_d2 (Eqns.(3) or (4)). This is a detection problem in a complex vector space [12]. It is difficult to obtain an explicit expression for the exact average SER, so we try to get the union bound instead. Due to the symmetry of the rotation code, without loss of generality we can assume that X_A is transmitted. The union bound is given by:

P_e-rot≤P{X_A→X_B}+P{X_A→X_C}+P{X_A→X_D}      (10)

where  P{X_A→X_B} is the pairwise error probability of confusing X_A with X_B when there are only two hypotheses for transmitted signal. It is easy to calculate the pairwise error probability of binary signals detection in complex Gaussian noise, and pairwise error probabilities for cooperative and non-cooperative cases can be derived as follows:

Cooperative case:

 (11)

Non-cooperative case:

        (12)

 where  and E_s=a².

Substituting Eqns.(11) and (12) into Eqn.(10), we can derive the average SER upper bounds of cooperative and non-cooperative cases:

       (13)

 Now proceed to derive the probability of non-cooperative case P_non-rot. This probability is the SER of x₁ at the relay in Phase 1. As shown in Fig.3, x₁ is a symbol from 4-PAM whose amplitude levels are not equally spaced. Four levels of this PAM are -a(sinθ+cosθ), a(sinθ-cosθ), a(cosθ-sinθ) and a(sinθ+cosθ). Through the deduction, the SER of x₁ conditioned on inter-user channel fading coefficient h_sr is given as follows:

=

         (14)

Averaging Eqn.(14) over h_sr, we can obtain the probability of non-cooperative case P_non-rot:

P_non-rot=

(15)

where  and E_s=a².

With the combination of cooperative case and non-cooperative case, the average SER upper bound of rotation-coded cooperative diversity can be gotten:

P_e-rot= P_e-non-rotP_non-rot+P_e-co-rot(1-P_non-rot)≤

   (16)

When inter-user channel is sound noisy (i.e. γ_sr→0), Eqn.(16) is determined by the non-cooperative case, and

it becomes P_e-rot≤while the upper bound Eqn.(9) for repetition-coded cooperative diversity becomes  P_e-rep≤ As we can see, in bad inter-user channel case, neither rotation-coded nor repetition-coded cooperative diversity obtains diversity gain, but rotation-coded scheme has an improved coding gain over the repetition code scheme in terms of saving transmit power by a factor of 1.5 (about 1.8 dB).

When inter-user channel condition is perfect (i.e.  γ_sr→∞), Eqns.(16) and (9) depend on the cooperative case. Now Eqn.(16) can be written as P_e-rot≤ At the minimum bound P_e-rot≤ is reached. Likewise, the bound (Eqn.(9)) becomes P_e-rep≤ Comparing the two inequations, it is found that both cooperative diversity schemes achieve full diversity order of two, but rotation-coded scheme has an improved coding gain by  (about 2 dB).

Through two extreme cases above, we can conclude that rotation-coded cooperative diversity has an improved coding gain over the repetition-coded scheme in terms of saving transmit power by about 1.8-2.0 dB. This improvement comes from rotation code spreading its codewords in the two-dimensional space rather than packing them on a single-dimensional line as repetition code, thereby exploiting the available degrees of freedom more effectively. It is noted that a similar scheme that the relay does not simply repeat the detected symbol is found in Ref.[15], where constellation rearrangement is used at the relay. We find this constellation rearrangement is equivalent to the rotation code in the case that θ=arctan(1/2). Apparently, this is not the optimum case.

4 Simulation results

To make a fair comparison, it is assumed that the transmit powers in both cooperative diversity schemes keep the same, i.e., E_s=a²=5b² in the discussion above.

Fig.4 shows the SER performance difference between  the rotation-coded cooperative diversity (θ=  and repetition-coded cooperative diversity when inter-user channel average received SNR γ_sr is 0 dB, 15 dB and , respectively. As we can see, in each case, repetition-coded scheme and rotation-coded scheme have the same slope of the error probability curves, which means that they have the same diversity gain. But rotation-coded cooperative diversity has an improved coding gain over the repetition-coded scheme in terms of saving transmit power by about 2 dB, which fits the theoretical analysis above very well.

Fig.4 SER performance of repetition-coded and rotation-coded cooperative diversity using I channel

In Fig.5, the transmission over the I channel above is extended to both the I/Q channels. Hence, spectral efficiency is doubled, i.e. 4 bits rather than 2 bits information are transmitted over two-symbol intervals. It is also found that rotation-coded scheme still obtains a coding gain of about 2 dB over repetition-coded scheme.

Fig.5 SER performance of repetition-coded and rotation-coded cooperative diversity using I/Q channels

5 Conclusions

(1) To exploit the degrees of freedom available in the channel more effectively and obtain coding gain along with diversity gain, rotation code is extended to DF cooperative protocol instead of conventional repetition code.

(2) Both theoretical analysis and simulation results show rotation-coded cooperative diversity has an improved coding gain of about 2 dB over the repetition-coded scheme along with the same diversity gain. This improvement comes from rotation code spreading its codewords in the two-dimensional space rather than packing them on a single-dimensional line as repetition code. It is noticed that this coding gain does not incur any expenditure of transmission power or bandwidth. In addition, the proposed scheme is simple and brings no increase in complexity.

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Foundation item: Project(2006AA01Z270) supported by the National High Technology Research and Development Program of China; Project(U0635003) supported by the National Natural Science Foundation of Guangdong Province, China; Project(2007F07) supported by the National Science Foundation of Shaanxi Province, China

Received date: 2008-06-18; Accepted date: 2008-09-12

Corresponding author: GE Jian-hua, Professor, PhD; Tel: +86-29-88201009; E-mail: jhge@xidian.edu.cn

(Edited by YANG You-ping)