Search 닫기

Journal of information and communication convergence engineering 2017; 15(1): 21-27

Published online March 31, 2017

https://doi.org/10.6109/jicce.2017.15.1.21

© Korea Institute of Information and Communication Engineering

Outage Analysis and Power Allocation for Distributed Space-Time Coding-Based Cooperative Systems over Rayleigh Fading Channels

In-Ho Lee

University of British Columbia and Hankyong National University

Received: March 4, 2017; Accepted: March 17, 2017

In this research, we study the outage probability for distributed space-time coding-based cooperative (DSTC) systems with amplify-and-forward relaying over Rayleigh fading channels with a high temporal correlation where the direct link between the source and the destination is available. In particular, we derive the upper and lower bounds of the outage probability as well as their corresponding asymptotic expressions. In addition, using only the average channel powers for the source-to-relay and relay-to-destination links, we propose an efficient power allocation scheme between the source and the relay to minimize the asymptotic upper bound of the outage probability. Through a numerical investigation, we verify the analytical expressions as well as the effectiveness of the proposed efficient power allocation. The numerical results show that the lower and upper bounds tightly correspond to the exact outage probability, and the proposed efficient power allocation scheme provides an outage probability similar to that of the optimal power allocation scheme that minimizes the exact outage probability.

Keywords Amplify-and-forward relay, Cooperative systems, Distributed space-time coding, Outage analysis, Power allocation

Node cooperation techniques have attracted considerable attention from the perspective of achieving cooperative diversity. Distributed space-time coding [1] is a node cooperation technique, and distributed space-time coding-based cooperative (DSTC) systems have been well investigated in [2-6]. In [2], the bit error probability has been analyzed for DSTC systems using amplify-and-forward (AF) relaying, and the optimum power allocation for these systems has been presented assuming full instantaneous channel information at the source and the relay. In [3], the outage probability and the diversity–multiplexing tradeoff have been studied for DSTC systems using decode-and-forward relaying. In [4], adaptive buffer-aided DSTC schemes with AF relaying have been proposed, and their bit error probabilities have been investigated. In [5], assuming that instantaneous channel information is available at the relay, power allocation schemes have been proposed for DSTC systems with AF relaying in order to minimize the bit error probability and to maximize the data rate. In [6], using only the average channel power information, the authors have proposed power allocation for DSTC systems with AF relaying to minimize the outage probability, where the direct link between the source and the destination is ignored.

In this paper, we focus on DSTC systems with AF relaying, where the Alamouti code [7, 8] is used for distributed space-time coding. Analogous to previous works, in DSTC systems, two transmission phases are considered as follows: in the first phase, both the relay and the destination receive the original data signals from the source, whereas in the second phase, only the destination receives the space-time coded signals from both the source and the relay.

In practice, a continuous wireless channel between the transmitter and the receiver may have a temporal correlation. Thus, considering highly correlated channels over time, we assume that the channel for the source-to-destination (S-D) link is constant during two phases.

In this study, we analyze the outage probability for DSTC systems with AF relaying over Rayleigh fading channels with a high temporal correlation, where the direct line between the source and the destination is assumed to be available unlike in the system model considered in [6]. However, unfortunately, it is expression for the exact outage probability. Thus, we derive the upper and lower bounds of the outage probability as well as their corresponding asymptotic expressions. Further, in previous works, instantaneous channel information was required for power allocation. However, assuming that only the average channel power for the source-to-relay (S-R) and relay-to-destination (R-D) links is known at the source and the relay, we propose a scheme for efficient power allocation between the source and the relay to minimize the asymptotic upper bound of the outage probability, and verify the effectiveness of this scheme. We do so by comparing the outage performances and the proposed and the optimal power allocation scheme that minimizes the exact outage probability.

A. DSTC System Description

We consider a DSTC system consisting of three single-antenna nodes: a source, a relay, and a destination. As shown in Fig. 1, in the DSTC system, the source broadcasts signals to both the relay and the destination during the first time slot, and then, the relay and the source collaboratively transmit space-time coded signals to the destination over the second time slot. We assume perfect synchronization between the cooperative transmissions in the second time slot. After the destination receives the signals over the two time slots, it combines them using maximal ratio combining (MRC).

Fig. 1. Data transmissions in a distributed space-time coding-based cooperative system during (a) the first time slot and (b) the second time slot.

In this study, we focus on an AF relay, and thus, the space-time coded signals transmitted by the relay are generated from the noisy signals received from the source.

Further, the AF relay uses variable amplification for preventing the saturation of the relay amplifier and satisfying its transmit power constraint. We assume that the relay has the exact channel information for the S-R link, and the destination has the exact channel information for all the links, i.e., the S-D link, the S-R link, and the R-D link.

In the DSTC system, assuming that the channels are constant during a time slot, the complex channels for the S-D and S-R links in the first time slot are denoted by gSD(1) and gSR, respectively, and the complex channels for the S-D and R-D links in the second time slot are represented by gSD(2) and gRD, respectively. We assume that gSD(1), gRD, gSD(2), and gRD are complex Gaussian random variables with zero mean and ωSD, ωSR, ωSD, and ωRD variance, respectively. The random variables are independent of each other, but gSD(1) and gSD(2) can be dependent when the transmissions during the two successive time slots occur over the same frequency band. In particular, in low-mobility networks, the two consecutive channels for the S-D link are highly correlated. In order to focus on such a highly correlated case, in this study, we assume that gSD(1)=gSD(2), which is represented by gSD.

B. Signal-to-Noise Ratio for the DSTC System

Let rR(i,j) and rD(i,j) denote the received signals in the jth symbol duration in the ith time slot at the relay and the destination, respectively, and nR(i,j) and nD(i,j) denote the additive complex white Gaussian noise with zero mean and σ2 variance in the jth symbol duration in the ith time slot at the relay and the destination, respectively. Further, let PS(i) denote the transmit power at the source in the ith time slot, and PR represent the transmit power at the relay.

During the first time slot, the source sequentially transmits two complex symbols s1 and s2, and the relay and the destination receive the two symbols over two symbol durations as follows:

rR(1,i)=gSRPS(1)si+nR(1,i),

rD(1,i)=gSDPS(1)si+nD(1,i),

for i = 1, 2, where E[s1s1*]=E[s2s2*]=1 . Then, the relay compensates for the phase of received signals by using the normalized weight wR=gSR*/|gSR| for the DSTC transmission with the source, and amplifies them with the following variable amplification: α2=PR/(|gSR|2PS(1)+σ2). During the second time slot, the relay transmits αwRrR(1,2) and α(wRrR(1,1))* during two symbol durations, and at the same time, the source transmits s1 and s2*, respectively, where the Alamouti code is used for the DSTC transmission. Thus, the destination receives the DSTC signals as follows:

rD(2,1)=gSDPS(2)s1+gSDαwRrR(1,2)+nD(2,1),

rD(2,2)=gSDPS(2)s2*gRDα(wRrR(1,1))*+nD(2,2).

Then, at the destination, the received DSTC signals are decoupled as follows:

x1=gSD*PS(2)rD(2,1)α|gSR|gRDPS(1)rD(2,2)*,

x2=α|gSR|gRD*PS(1)rD(2,1)+gSDPS(2)rD(2,2)*.

Finally, the destination combines the decoupled signals and the preceding received signals by using MRC as follows:

(|gRD|2α2+1)1xi+gSD*PS(1)rD(1,i),

for i = 1, 2. The signal-to-noise ratio (SNR) of the MRC-combined signals is then obtained as follows:

γ=|gSD|2PS(2)+|gRD|2α2|gSR|2PS(1)|gRD|2α2σ2+σ2+|gSD|2PS(1)σ2.

Letting zSD=|gSD|2PS(2)/σ2, zSR=|gSR|2PS(1)/σ2, and zRD=|gRD|2PR/σ2, and inserting α2=PR/(|gSR|2PS(1)+σ2) into (8), we can rewrite the MRC output SNR as follows:

γ=zSDzSR+zSRzRD+zSDzSR+zRD+1+ηzSD,

where η=PS(1)/PS(2).

A. Lower and Upper Bounds on SNR

The MRC output SNR in (9) is upper bounded as follows:

γ=zSRzRDzSDzRDzSR+zRD+1+(1+η)zSD{zRD(zSRzSD)zSR+zRD+1+(1+η)zSDfor  zSRzSDzRD(zSRzSD)zSR+zRDzSD+(1+η)zSDfor  zSR>zSD{(1+η)zSDfor  zSRzSDmin{zRD+(1+η)zSD,zSR+ηzSD}for  zSR>zSD=γU.

Further, a lower bound on the MRC output SNR is given as follows:

γL=zSRzRDzSR+zRD+1+ηzSD.

B. Outage Probability Bounds

The outage probability is defined as the probability that the achievable data rate, (1/2)log2(1+γ), falls below a specified outage threshold, Rth . Here, note that the achievable data rate decreases by half since two symbols are transmitted for four symbol durations. Then, by assuming that λth = 22Rth−1, and using (10) and (11), we derive the lower and upper bounds on the outage probability for the DSTC system. The probability density functions (PDFs) of zSD , zSR , and zRD are given in [9] as follows:

pX(x)=1βXex/βX for  X=zSD,zSR,zRD,

where βzSD=ωSDPS(2)/σ2 , βzSR=ωSRPS(1)/σ2 , and βzRD=ωRDPR/σ2 . For brevity, let βSD = βzSD , βSR = βzSR , and βRD = βzRD .

Using (10) and the PDFs in (12), we obtain a lower bound on the outage probability for the DSTC system as follows:

Pr{γU<λth}=1 Pr{γU>λth}=1 Pr{min{zRD+(1+η)zSD,zSR+ηzSD}>λth,zSR>zSD} Pr{(1+η)zSD>λth,zSR<zSD}=1 Pr{zSD>λth1+η,zSR>zSD,zRD>0} Pr{zSD<λth1+η,zSR>λthηzSD,zRD>λth(1+ηzSD)} Pr{zSD>λth1+η,zSR<λth1+η}Pr{zSR>λth1+η,zSR<zSD}=1 1βSDλth1+ηez(1βSD+1βSR)dz1βSDeλth(1βSR+1βRD)×0λth1+ηez(1βSDηβSR1+ηβRD)dzeλth(1+η)βSD(1eλth(1+η)βSR) 1βSRλth1+ηez(1βSD+1βSR)dz=1 (βSRβRDβSRβRDηβSDβRD(1+η)βSDβSR)×(eλth(1βSR+1βRD)eλth1+η(1βSD+1βSR))eλth(1+η)βSD.

For high transmit SNRs (i.e., PS(1)/σ2, PS(2)/σ2, and PR/σ2 → ∞), the SNR lower bound in (11) can be approximated as follows:

γLmin{zSR,zRD}+ηzSD.

Let γmin = min{zSR, zRD} and ySD = ηzSD . Then, the cumulative distribution function of γmin is given as follows:

Fγmin=1Pr{zSR>x}Pr{zRD>x}=1ex(1βSR+1βRD).

Further, the PDF of ySD is given as follows:

pySD(x)=1ηβSDex/(ηβSD).

Using (14), Fγmin(x), and pySD(x), we can approximate an upper bound on the outage probability of the DSTC system for high transmit SNRs as follows:

Pr{γL<λth}Pr{γmin+ySD<λth} =0λthFγmin(λthz)pySD(z)dz =1eλthηβSD+βSRβRDβSRβRDηβSDβRDηβSDβSR       ×(eλthηβSDeλth(1βSR+1βRD)).

C. Asymptotic Upper and Lower Bounds and Diversity Order

Let PS(1)=PS(2)=ρ1Pt, and PR = ρ2Pt, where Pt denotes the total transmit power used for transmitting the two symbols in the DSTC system (i.e., Pt=PS(1)+PR+PS(2)), 2ρ1+ρ2 = 1, and ρ1, ρ2 > 0. Note that 0 < ρ1 < 0.5 and 0 < ρ2 < 1.

When Pt/σ2 → ∞ , by applying the approximation ea ≈ 1+a+a2/2 for a → 0 to (13) and (17), we can obtain the asymptotic lower and upper bounds on the outage probability for the high transmit SNR regime, respectively, as follows:

OL=(λth22ρ1ωSD)(Ptσ2)2[(1ρ1ωSR+2ρ2ωRD1ρ1ωSD)1×{(1ρ1ωSR+1ρ2ωRD)214(1ρ1ωSD+1ρ1ωSR)2}14ρ1ωSD],

OU=λth22ρ1ωSD(Ptσ2)2(1ρ1ωSR+1ρ2ωRD).

From (18) and (19), the diversity order for the DSTC system is obtained as follows:

limPt/σ2logOlog(Pt/σ2)=2,

where O represents the asymptotic outage probability bounds in (18) and (19), and implies that the diversity order for the DSTC system is 2.

In this section, assuming that only the average channel powers for the S-R and R-D links are known at the source and the relay, we propose an efficient power allocation scheme to improve the outage performance of the DSTC system. The efficient power allocation coefficients denoted by ρ1o and ρ2o for the source and the relay, respectively, are derived to minimize the upper bound of the outage probability in (19).

Inserting ρ2 = 1−2ρ1 into (19), we can rewrite the upper bound of the outage probability as follows:

OU=λth22ρ1ωSD(Ptσ2)2(1ρ1ωSR+1(12ρ1)ωRD).

Taking the derivative of OU in (21) with respect to ρ1 , we obtain the following:

dOUdρ1=λth22ρ1ωSD(Ptσ2)2[1ρ12ωSR1ρ1(12ρ1)ωRD1ρ12ωSR+2ωRD(ωRD2ρ1ωRD)2].

Using (22), we can derive the equation dOU / 1 = 0 as follows:

(4ωSR8ωRD)ρ12(ωSR8ωRD)ρ12ωRD=0.

From (23), ρ1o is finally obtained as follows:

ρ1o=ωSR8ωRD+(ωSR8ωRD)2+32(ωSR2ωRD)ωRD8(ωSR2ωRD),

where 0 < ρ1o < 0.5 since 2ρ1+ρ2 = 1 . Using (24), we also obtain ρ2o=12ρ1o .

To show that ρ1o in (24) achieves the minimum OU, we prove that d2OU/dρ12>0 as follows:

d2OUdρ12=λth22ρ12ωSD(Ptσ2)2[6ρ12ωSR+26ρ1ρ1(12ρ1)2ωRD            212ρ1(12ρ1)3ωRD]=λth22ρ12ωSD(Ptσ2)2[6ρ12ωSR+(24ρ12)2+(8312)ρ1ρ1(12ρ1)3ωRD]  >0.

In this section, we verify the analytical expressions for the lower and upper bounds on the outage probability in (13) and (17), respectively, as well as their corresponding asymptotic expressions in (18) and (19) by comparing with the simulation results of the exact outage probability. In addition, we verify the effectiveness of the proposed efficient power allocation scheme by comparing the simulation results of the exact outage probabilities for the optimal and the proposed power allocation schemes.

Figs. 2 and 3 show the upper and lower bounds on the outage probability of the DSTC system as well as their asymptotic results for ωSR = 2 , ωRD = 2 in Fig. 2 and ωSR = 5 , ωRD = 2 in Fig. 3, when ωSD = 1 , ρ1 = 0.3 , and Rth = 0.5 , 1 bps/Hz. The analytical results of the lower and upper bounds are obtained using (13) and (17), respectively, and the corresponding asymptotic results are obtained using (18) and (19), respectively. From the figures, we observe that the lower and upper bounds tightly correspond with the exact outage probability for the different average channel powers and outage thresholds, even in the low transmit SNR regime. Further, the figures demonstrate that the asymptotic results perfectly match the results of their corresponding lower and upper bounds for a high transmit SNR.

Fig. 2. Upper and lower bounds on the outage probability of the DSTC system and their asymptotic results when ωSD = 1 , ωSR = 2 , ωRD = 2 , and ρ1 = 0.3 .
Fig. 3. Upper and lower bounds on the outage probability of the DSTC system and their asymptotic results when ωSD = 1 , ωSR = 5 , ωRD = 2 , and ρ1 = 0.3 .

Fig. 4 shows the exact outage probability and the asymptotic upper bound on the outage probability for power allocation between the source and the relay when Pt / σ2 = 20 dB, ωSD = 1 , ωRD = 2 , ωSR = 2, 5, and Rth = 0.5, 1 bps/Hz. The figure illustrates that a value of ρ1 exists to minimize the outage probability. Further, the values of ρ1 to minimize the asymptotic upper bound and the exact outage probability are slightly different, but their outage probabilities are almost the same.

Fig. 4. Exact outage probability and the asymptotic upper bound on the outage probability for power allocation when ωSD = 1 , ωRD = 2 , and Pt / σ2= 20 dB.

Fig. 5 compares the exact outage probabilities for the optimal power allocation scheme and the proposed efficient power allocation scheme when ωSD = 1 , ωRD = 2 , ωSR = 2, 5, and Rth = 0.5, 1 bps/Hz. The optimal power allocation scheme minimizes the exact outage probability with respect to 0 < ρ1 < 0.5, for which results are obtained through simulations. The figure demonstrates that the outage probability results of the proposed efficient power allocation obtained using (24) are extremely similar to those of the optimal power allocation for different average channel powers and outage thresholds.

Fig. 5. Exact outage probabilities for the optimal and the proposed efficient power allocation schemes when ωSD = 1 and ωRD = 2 .

In this paper, we presented closed-form expressions for the lower and upper bounds on the outage probability for a DSTC system with AF relaying over Rayleigh fading channels with a high temporal correlation, which tightly correspond with the exact outage probability. Further, we provided the asymptotic expressions for the lower and upper bounds. In addition, a scheme for efficient power allocation using only the average channel powers for the source-to-relay and relay-to-destination links at the source and the relay is proposed to minimize the asymptotic upper bound of the outage probability. Numerical results show that the proposed efficient power allocation scheme achieves an outage performance similar to that of the optimal power allocation scheme.

This work was supported by ‘The Cross-Ministry Giga KOREA Project’ grant from the Ministry of Science, ICT and Future Planning, Korea.
  1. J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003.
    CrossRef
  2. P. A. Anghel and M. Kaveh, “On the performance of distributed space-time coding systems with one and two non-regenerative relays,” IEEE Transactions on Wireless Communications, vol. 5, no. 3, pp. 682-692, 2006.
    CrossRef
  3. Y. Zou, Y. D. Yao, and B. Zheng, “Opportunistic distributed space-time coding for decode-and-forward cooperation systems,” IEEE Transactions on Signal Processing, vol. 60, no. 4, pp. 1766-1781, 2012.
    CrossRef
  4. T. Peng and R. C. de Lamare, “Adaptive buffer-aided distributed space-time coding for cooperative wireless networks,” IEEE Transactions on Communications, vol. 64, no. 5, pp. 1888-1900, 2016.
    CrossRef
  5. T. Peng, R. C. de Lamare, and A. Schmeink, “Adaptive power allocation strategies for distributed space-time coding in cooperative MIMO networks,” IET Communications, vol. 8, no. 7, pp. 1141-1150, 2014.
    CrossRef
  6. J. Abouei, H. Bagheri, and A. K. Khandani, “An efficient adaptive distributed space-time coding scheme for cooperative relaying,” IEEE Transactions on Wireless Communications, vol. 8, no. 10, pp. 4957–4962, 2009.
    CrossRef
  7. G. Scutari and S. Barbarossa, “Distributed space-time coding for regenerative relay networks,” IEEE Transactions on Wireless Communications, vol. 4, no. 5, pp. 2387–2399, 2005.
    CrossRef
  8. S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal of Selected Areas in Communications, vol. 16, no. 8, pp. 1451-1458, 1998.
    CrossRef
  9. J. G. Proakis, Digital Communications, 4th ed. New York: McGraw- Hill, 2001.

In-Ho Lee

received his B.S., M.S., and Ph.D. in Electrical Engineering from Hanyang University, Ansan, Korea, in 2003, 2005, and 2008, respectively. He worked towards LTE-Advanced standardization at Samsung Electronics Co. from 2008 to 2010. He was a Post-Doctoral Fellow at the Department of Electrical Engineering, Hanyang University, Ansan, Korea, from April 2010 to March 2011. Since March 2011, he has been with the Department of Electrical, Electronic, and Control Engineering, Hankyong National University, Anseong, Korea. Further, since February 2017, he has been a visiting associate professor in the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, Canada. His present research interests include non-orthogonal multiple access, millimeter-wave wireless communications, cooperative communications, multi-hop relaying, transmission and reception of multiple-input and multiple-output communications, multicast communications, and multi-user channel state information feedback.


Article

Journal of information and communication convergence engineering 2017; 15(1): 21-27

Published online March 31, 2017 https://doi.org/10.6109/jicce.2017.15.1.21

Copyright © Korea Institute of Information and Communication Engineering.

Outage Analysis and Power Allocation for Distributed Space-Time Coding-Based Cooperative Systems over Rayleigh Fading Channels

In-Ho Lee

University of British Columbia and Hankyong National University

Received: March 4, 2017; Accepted: March 17, 2017

Abstract

In this research, we study the outage probability for distributed space-time coding-based cooperative (DSTC) systems with amplify-and-forward relaying over Rayleigh fading channels with a high temporal correlation where the direct link between the source and the destination is available. In particular, we derive the upper and lower bounds of the outage probability as well as their corresponding asymptotic expressions. In addition, using only the average channel powers for the source-to-relay and relay-to-destination links, we propose an efficient power allocation scheme between the source and the relay to minimize the asymptotic upper bound of the outage probability. Through a numerical investigation, we verify the analytical expressions as well as the effectiveness of the proposed efficient power allocation. The numerical results show that the lower and upper bounds tightly correspond to the exact outage probability, and the proposed efficient power allocation scheme provides an outage probability similar to that of the optimal power allocation scheme that minimizes the exact outage probability.

Keywords: Amplify-and-forward relay, Cooperative systems, Distributed space-time coding, Outage analysis, Power allocation

I. INTRODUCTION

Node cooperation techniques have attracted considerable attention from the perspective of achieving cooperative diversity. Distributed space-time coding [1] is a node cooperation technique, and distributed space-time coding-based cooperative (DSTC) systems have been well investigated in [2-6]. In [2], the bit error probability has been analyzed for DSTC systems using amplify-and-forward (AF) relaying, and the optimum power allocation for these systems has been presented assuming full instantaneous channel information at the source and the relay. In [3], the outage probability and the diversity–multiplexing tradeoff have been studied for DSTC systems using decode-and-forward relaying. In [4], adaptive buffer-aided DSTC schemes with AF relaying have been proposed, and their bit error probabilities have been investigated. In [5], assuming that instantaneous channel information is available at the relay, power allocation schemes have been proposed for DSTC systems with AF relaying in order to minimize the bit error probability and to maximize the data rate. In [6], using only the average channel power information, the authors have proposed power allocation for DSTC systems with AF relaying to minimize the outage probability, where the direct link between the source and the destination is ignored.

In this paper, we focus on DSTC systems with AF relaying, where the Alamouti code [7, 8] is used for distributed space-time coding. Analogous to previous works, in DSTC systems, two transmission phases are considered as follows: in the first phase, both the relay and the destination receive the original data signals from the source, whereas in the second phase, only the destination receives the space-time coded signals from both the source and the relay.

In practice, a continuous wireless channel between the transmitter and the receiver may have a temporal correlation. Thus, considering highly correlated channels over time, we assume that the channel for the source-to-destination (S-D) link is constant during two phases.

In this study, we analyze the outage probability for DSTC systems with AF relaying over Rayleigh fading channels with a high temporal correlation, where the direct line between the source and the destination is assumed to be available unlike in the system model considered in [6]. However, unfortunately, it is expression for the exact outage probability. Thus, we derive the upper and lower bounds of the outage probability as well as their corresponding asymptotic expressions. Further, in previous works, instantaneous channel information was required for power allocation. However, assuming that only the average channel power for the source-to-relay (S-R) and relay-to-destination (R-D) links is known at the source and the relay, we propose a scheme for efficient power allocation between the source and the relay to minimize the asymptotic upper bound of the outage probability, and verify the effectiveness of this scheme. We do so by comparing the outage performances and the proposed and the optimal power allocation scheme that minimizes the exact outage probability.

II. SYSTEM MODEL

A. DSTC System Description

We consider a DSTC system consisting of three single-antenna nodes: a source, a relay, and a destination. As shown in Fig. 1, in the DSTC system, the source broadcasts signals to both the relay and the destination during the first time slot, and then, the relay and the source collaboratively transmit space-time coded signals to the destination over the second time slot. We assume perfect synchronization between the cooperative transmissions in the second time slot. After the destination receives the signals over the two time slots, it combines them using maximal ratio combining (MRC).

Figure 1. Data transmissions in a distributed space-time coding-based cooperative system during (a) the first time slot and (b) the second time slot.

In this study, we focus on an AF relay, and thus, the space-time coded signals transmitted by the relay are generated from the noisy signals received from the source.

Further, the AF relay uses variable amplification for preventing the saturation of the relay amplifier and satisfying its transmit power constraint. We assume that the relay has the exact channel information for the S-R link, and the destination has the exact channel information for all the links, i.e., the S-D link, the S-R link, and the R-D link.

In the DSTC system, assuming that the channels are constant during a time slot, the complex channels for the S-D and S-R links in the first time slot are denoted by gSD(1) and gSR, respectively, and the complex channels for the S-D and R-D links in the second time slot are represented by gSD(2) and gRD, respectively. We assume that gSD(1), gRD, gSD(2), and gRD are complex Gaussian random variables with zero mean and ωSD, ωSR, ωSD, and ωRD variance, respectively. The random variables are independent of each other, but gSD(1) and gSD(2) can be dependent when the transmissions during the two successive time slots occur over the same frequency band. In particular, in low-mobility networks, the two consecutive channels for the S-D link are highly correlated. In order to focus on such a highly correlated case, in this study, we assume that gSD(1)=gSD(2), which is represented by gSD.

B. Signal-to-Noise Ratio for the DSTC System

Let rR(i,j) and rD(i,j) denote the received signals in the jth symbol duration in the ith time slot at the relay and the destination, respectively, and nR(i,j) and nD(i,j) denote the additive complex white Gaussian noise with zero mean and σ2 variance in the jth symbol duration in the ith time slot at the relay and the destination, respectively. Further, let PS(i) denote the transmit power at the source in the ith time slot, and PR represent the transmit power at the relay.

During the first time slot, the source sequentially transmits two complex symbols s1 and s2, and the relay and the destination receive the two symbols over two symbol durations as follows:

rR(1,i)=gSRPS(1)si+nR(1,i),

rD(1,i)=gSDPS(1)si+nD(1,i),

for i = 1, 2, where E[s1s1*]=E[s2s2*]=1 . Then, the relay compensates for the phase of received signals by using the normalized weight wR=gSR*/|gSR| for the DSTC transmission with the source, and amplifies them with the following variable amplification: α2=PR/(|gSR|2PS(1)+σ2). During the second time slot, the relay transmits αwRrR(1,2) and α(wRrR(1,1))* during two symbol durations, and at the same time, the source transmits s1 and s2*, respectively, where the Alamouti code is used for the DSTC transmission. Thus, the destination receives the DSTC signals as follows:

rD(2,1)=gSDPS(2)s1+gSDαwRrR(1,2)+nD(2,1),

rD(2,2)=gSDPS(2)s2*gRDα(wRrR(1,1))*+nD(2,2).

Then, at the destination, the received DSTC signals are decoupled as follows:

x1=gSD*PS(2)rD(2,1)α|gSR|gRDPS(1)rD(2,2)*,

x2=α|gSR|gRD*PS(1)rD(2,1)+gSDPS(2)rD(2,2)*.

Finally, the destination combines the decoupled signals and the preceding received signals by using MRC as follows:

(|gRD|2α2+1)1xi+gSD*PS(1)rD(1,i),

for i = 1, 2. The signal-to-noise ratio (SNR) of the MRC-combined signals is then obtained as follows:

γ=|gSD|2PS(2)+|gRD|2α2|gSR|2PS(1)|gRD|2α2σ2+σ2+|gSD|2PS(1)σ2.

Letting zSD=|gSD|2PS(2)/σ2, zSR=|gSR|2PS(1)/σ2, and zRD=|gRD|2PR/σ2, and inserting α2=PR/(|gSR|2PS(1)+σ2) into (8), we can rewrite the MRC output SNR as follows:

γ=zSDzSR+zSRzRD+zSDzSR+zRD+1+ηzSD,

where η=PS(1)/PS(2).

III. PERFORMANCE ANALYSIS

A. Lower and Upper Bounds on SNR

The MRC output SNR in (9) is upper bounded as follows:

γ=zSRzRDzSDzRDzSR+zRD+1+(1+η)zSD{zRD(zSRzSD)zSR+zRD+1+(1+η)zSDfor  zSRzSDzRD(zSRzSD)zSR+zRDzSD+(1+η)zSDfor  zSR>zSD{(1+η)zSDfor  zSRzSDmin{zRD+(1+η)zSD,zSR+ηzSD}for  zSR>zSD=γU.

Further, a lower bound on the MRC output SNR is given as follows:

γL=zSRzRDzSR+zRD+1+ηzSD.

B. Outage Probability Bounds

The outage probability is defined as the probability that the achievable data rate, (1/2)log2(1+γ), falls below a specified outage threshold, Rth . Here, note that the achievable data rate decreases by half since two symbols are transmitted for four symbol durations. Then, by assuming that λth = 22Rth−1, and using (10) and (11), we derive the lower and upper bounds on the outage probability for the DSTC system. The probability density functions (PDFs) of zSD , zSR , and zRD are given in [9] as follows:

pX(x)=1βXex/βX for  X=zSD,zSR,zRD,

where βzSD=ωSDPS(2)/σ2 , βzSR=ωSRPS(1)/σ2 , and βzRD=ωRDPR/σ2 . For brevity, let βSD = βzSD , βSR = βzSR , and βRD = βzRD .

Using (10) and the PDFs in (12), we obtain a lower bound on the outage probability for the DSTC system as follows:

Pr{γU<λth}=1 Pr{γU>λth}=1 Pr{min{zRD+(1+η)zSD,zSR+ηzSD}>λth,zSR>zSD} Pr{(1+η)zSD>λth,zSR<zSD}=1 Pr{zSD>λth1+η,zSR>zSD,zRD>0} Pr{zSD<λth1+η,zSR>λthηzSD,zRD>λth(1+ηzSD)} Pr{zSD>λth1+η,zSR<λth1+η}Pr{zSR>λth1+η,zSR<zSD}=1 1βSDλth1+ηez(1βSD+1βSR)dz1βSDeλth(1βSR+1βRD)×0λth1+ηez(1βSDηβSR1+ηβRD)dzeλth(1+η)βSD(1eλth(1+η)βSR) 1βSRλth1+ηez(1βSD+1βSR)dz=1 (βSRβRDβSRβRDηβSDβRD(1+η)βSDβSR)×(eλth(1βSR+1βRD)eλth1+η(1βSD+1βSR))eλth(1+η)βSD.

For high transmit SNRs (i.e., PS(1)/σ2, PS(2)/σ2, and PR/σ2 → ∞), the SNR lower bound in (11) can be approximated as follows:

γLmin{zSR,zRD}+ηzSD.

Let γmin = min{zSR, zRD} and ySD = ηzSD . Then, the cumulative distribution function of γmin is given as follows:

Fγmin=1Pr{zSR>x}Pr{zRD>x}=1ex(1βSR+1βRD).

Further, the PDF of ySD is given as follows:

pySD(x)=1ηβSDex/(ηβSD).

Using (14), Fγmin(x), and pySD(x), we can approximate an upper bound on the outage probability of the DSTC system for high transmit SNRs as follows:

Pr{γL<λth}Pr{γmin+ySD<λth} =0λthFγmin(λthz)pySD(z)dz =1eλthηβSD+βSRβRDβSRβRDηβSDβRDηβSDβSR       ×(eλthηβSDeλth(1βSR+1βRD)).

C. Asymptotic Upper and Lower Bounds and Diversity Order

Let PS(1)=PS(2)=ρ1Pt, and PR = ρ2Pt, where Pt denotes the total transmit power used for transmitting the two symbols in the DSTC system (i.e., Pt=PS(1)+PR+PS(2)), 2ρ1+ρ2 = 1, and ρ1, ρ2 > 0. Note that 0 < ρ1 < 0.5 and 0 < ρ2 < 1.

When Pt/σ2 → ∞ , by applying the approximation ea ≈ 1+a+a2/2 for a → 0 to (13) and (17), we can obtain the asymptotic lower and upper bounds on the outage probability for the high transmit SNR regime, respectively, as follows:

OL=(λth22ρ1ωSD)(Ptσ2)2[(1ρ1ωSR+2ρ2ωRD1ρ1ωSD)1×{(1ρ1ωSR+1ρ2ωRD)214(1ρ1ωSD+1ρ1ωSR)2}14ρ1ωSD],

OU=λth22ρ1ωSD(Ptσ2)2(1ρ1ωSR+1ρ2ωRD).

From (18) and (19), the diversity order for the DSTC system is obtained as follows:

limPt/σ2logOlog(Pt/σ2)=2,

where O represents the asymptotic outage probability bounds in (18) and (19), and implies that the diversity order for the DSTC system is 2.

IV. POWER ALLOCATION

In this section, assuming that only the average channel powers for the S-R and R-D links are known at the source and the relay, we propose an efficient power allocation scheme to improve the outage performance of the DSTC system. The efficient power allocation coefficients denoted by ρ1o and ρ2o for the source and the relay, respectively, are derived to minimize the upper bound of the outage probability in (19).

Inserting ρ2 = 1−2ρ1 into (19), we can rewrite the upper bound of the outage probability as follows:

OU=λth22ρ1ωSD(Ptσ2)2(1ρ1ωSR+1(12ρ1)ωRD).

Taking the derivative of OU in (21) with respect to ρ1 , we obtain the following:

dOUdρ1=λth22ρ1ωSD(Ptσ2)2[1ρ12ωSR1ρ1(12ρ1)ωRD1ρ12ωSR+2ωRD(ωRD2ρ1ωRD)2].

Using (22), we can derive the equation dOU / 1 = 0 as follows:

(4ωSR8ωRD)ρ12(ωSR8ωRD)ρ12ωRD=0.

From (23), ρ1o is finally obtained as follows:

ρ1o=ωSR8ωRD+(ωSR8ωRD)2+32(ωSR2ωRD)ωRD8(ωSR2ωRD),

where 0 < ρ1o < 0.5 since 2ρ1+ρ2 = 1 . Using (24), we also obtain ρ2o=12ρ1o .

To show that ρ1o in (24) achieves the minimum OU, we prove that d2OU/dρ12>0 as follows:

d2OUdρ12=λth22ρ12ωSD(Ptσ2)2[6ρ12ωSR+26ρ1ρ1(12ρ1)2ωRD            212ρ1(12ρ1)3ωRD]=λth22ρ12ωSD(Ptσ2)2[6ρ12ωSR+(24ρ12)2+(8312)ρ1ρ1(12ρ1)3ωRD]  >0.

V. NUMERICAL RESULTS

In this section, we verify the analytical expressions for the lower and upper bounds on the outage probability in (13) and (17), respectively, as well as their corresponding asymptotic expressions in (18) and (19) by comparing with the simulation results of the exact outage probability. In addition, we verify the effectiveness of the proposed efficient power allocation scheme by comparing the simulation results of the exact outage probabilities for the optimal and the proposed power allocation schemes.

Figs. 2 and 3 show the upper and lower bounds on the outage probability of the DSTC system as well as their asymptotic results for ωSR = 2 , ωRD = 2 in Fig. 2 and ωSR = 5 , ωRD = 2 in Fig. 3, when ωSD = 1 , ρ1 = 0.3 , and Rth = 0.5 , 1 bps/Hz. The analytical results of the lower and upper bounds are obtained using (13) and (17), respectively, and the corresponding asymptotic results are obtained using (18) and (19), respectively. From the figures, we observe that the lower and upper bounds tightly correspond with the exact outage probability for the different average channel powers and outage thresholds, even in the low transmit SNR regime. Further, the figures demonstrate that the asymptotic results perfectly match the results of their corresponding lower and upper bounds for a high transmit SNR.

Figure 2. Upper and lower bounds on the outage probability of the DSTC system and their asymptotic results when ωSD = 1 , ωSR = 2 , ωRD = 2 , and ρ1 = 0.3 .
Figure 3. Upper and lower bounds on the outage probability of the DSTC system and their asymptotic results when ωSD = 1 , ωSR = 5 , ωRD = 2 , and ρ1 = 0.3 .

Fig. 4 shows the exact outage probability and the asymptotic upper bound on the outage probability for power allocation between the source and the relay when Pt / σ2 = 20 dB, ωSD = 1 , ωRD = 2 , ωSR = 2, 5, and Rth = 0.5, 1 bps/Hz. The figure illustrates that a value of ρ1 exists to minimize the outage probability. Further, the values of ρ1 to minimize the asymptotic upper bound and the exact outage probability are slightly different, but their outage probabilities are almost the same.

Figure 4. Exact outage probability and the asymptotic upper bound on the outage probability for power allocation when ωSD = 1 , ωRD = 2 , and Pt / σ2= 20 dB.

Fig. 5 compares the exact outage probabilities for the optimal power allocation scheme and the proposed efficient power allocation scheme when ωSD = 1 , ωRD = 2 , ωSR = 2, 5, and Rth = 0.5, 1 bps/Hz. The optimal power allocation scheme minimizes the exact outage probability with respect to 0 < ρ1 < 0.5, for which results are obtained through simulations. The figure demonstrates that the outage probability results of the proposed efficient power allocation obtained using (24) are extremely similar to those of the optimal power allocation for different average channel powers and outage thresholds.

Figure 5. Exact outage probabilities for the optimal and the proposed efficient power allocation schemes when ωSD = 1 and ωRD = 2 .

VI. CONCLUSION

In this paper, we presented closed-form expressions for the lower and upper bounds on the outage probability for a DSTC system with AF relaying over Rayleigh fading channels with a high temporal correlation, which tightly correspond with the exact outage probability. Further, we provided the asymptotic expressions for the lower and upper bounds. In addition, a scheme for efficient power allocation using only the average channel powers for the source-to-relay and relay-to-destination links at the source and the relay is proposed to minimize the asymptotic upper bound of the outage probability. Numerical results show that the proposed efficient power allocation scheme achieves an outage performance similar to that of the optimal power allocation scheme.

Fig 1.

Figure 1.Data transmissions in a distributed space-time coding-based cooperative system during (a) the first time slot and (b) the second time slot.
Journal of Information and Communication Convergence Engineering 2017; 15: 21-27https://doi.org/10.6109/jicce.2017.15.1.21

Fig 2.

Figure 2.Upper and lower bounds on the outage probability of the DSTC system and their asymptotic results when ωSD = 1 , ωSR = 2 , ωRD = 2 , and ρ1 = 0.3 .
Journal of Information and Communication Convergence Engineering 2017; 15: 21-27https://doi.org/10.6109/jicce.2017.15.1.21

Fig 3.

Figure 3.Upper and lower bounds on the outage probability of the DSTC system and their asymptotic results when ωSD = 1 , ωSR = 5 , ωRD = 2 , and ρ1 = 0.3 .
Journal of Information and Communication Convergence Engineering 2017; 15: 21-27https://doi.org/10.6109/jicce.2017.15.1.21

Fig 4.

Figure 4.Exact outage probability and the asymptotic upper bound on the outage probability for power allocation when ωSD = 1 , ωRD = 2 , and Pt / σ2= 20 dB.
Journal of Information and Communication Convergence Engineering 2017; 15: 21-27https://doi.org/10.6109/jicce.2017.15.1.21

Fig 5.

Figure 5.Exact outage probabilities for the optimal and the proposed efficient power allocation schemes when ωSD = 1 and ωRD = 2 .
Journal of Information and Communication Convergence Engineering 2017; 15: 21-27https://doi.org/10.6109/jicce.2017.15.1.21

References

  1. J. N. Laneman and G. W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Transactions on Information Theory, vol. 49, no. 10, pp. 2415–2425, 2003.
    CrossRef
  2. P. A. Anghel and M. Kaveh, “On the performance of distributed space-time coding systems with one and two non-regenerative relays,” IEEE Transactions on Wireless Communications, vol. 5, no. 3, pp. 682-692, 2006.
    CrossRef
  3. Y. Zou, Y. D. Yao, and B. Zheng, “Opportunistic distributed space-time coding for decode-and-forward cooperation systems,” IEEE Transactions on Signal Processing, vol. 60, no. 4, pp. 1766-1781, 2012.
    CrossRef
  4. T. Peng and R. C. de Lamare, “Adaptive buffer-aided distributed space-time coding for cooperative wireless networks,” IEEE Transactions on Communications, vol. 64, no. 5, pp. 1888-1900, 2016.
    CrossRef
  5. T. Peng, R. C. de Lamare, and A. Schmeink, “Adaptive power allocation strategies for distributed space-time coding in cooperative MIMO networks,” IET Communications, vol. 8, no. 7, pp. 1141-1150, 2014.
    CrossRef
  6. J. Abouei, H. Bagheri, and A. K. Khandani, “An efficient adaptive distributed space-time coding scheme for cooperative relaying,” IEEE Transactions on Wireless Communications, vol. 8, no. 10, pp. 4957–4962, 2009.
    CrossRef
  7. G. Scutari and S. Barbarossa, “Distributed space-time coding for regenerative relay networks,” IEEE Transactions on Wireless Communications, vol. 4, no. 5, pp. 2387–2399, 2005.
    CrossRef
  8. S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Journal of Selected Areas in Communications, vol. 16, no. 8, pp. 1451-1458, 1998.
    CrossRef
  9. J. G. Proakis, Digital Communications, 4th ed. New York: McGraw- Hill, 2001.
JICCE
Dec 31, 2024 Vol.22 No.4, pp. 267~343

Stats or Metrics

Share this article on

  • line

Related articles in JICCE

Journal of Information and Communication Convergence Engineering Jouranl of information and
communication convergence engineering
(J. Inf. Commun. Converg. Eng.)

eISSN 2234-8883
pISSN 2234-8255